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aba47be702e7d14755e3125d54952b35aa1f5759357a05cb2d074179970fb626 | from .matexpr import MatrixExpr
from sympy.core.function import FunctionClass, Lambda
from sympy.core.symbol import Dummy
from sympy.core.sympify import _sympify, sympify
from sympy.matrices import Matrix
from sympy.functions.elementary.complexes import re, im
class FunctionMatrix(MatrixExpr):
"""Represents a matrix using a function (``Lambda``) which gives
outputs according to the coordinates of each matrix entries.
Parameters
==========
rows : nonnegative integer. Can be symbolic.
cols : nonnegative integer. Can be symbolic.
lamda : Function, Lambda or str
If it is a SymPy ``Function`` or ``Lambda`` instance,
it should be able to accept two arguments which represents the
matrix coordinates.
If it is a pure string containing Python ``lambda`` semantics,
it is interpreted by the SymPy parser and casted into a SymPy
``Lambda`` instance.
Examples
========
Creating a ``FunctionMatrix`` from ``Lambda``:
>>> from sympy import FunctionMatrix, symbols, Lambda, MatPow
>>> i, j, n, m = symbols('i,j,n,m')
>>> FunctionMatrix(n, m, Lambda((i, j), i + j))
FunctionMatrix(n, m, Lambda((i, j), i + j))
Creating a ``FunctionMatrix`` from a SymPy function:
>>> from sympy import KroneckerDelta
>>> X = FunctionMatrix(3, 3, KroneckerDelta)
>>> X.as_explicit()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
Creating a ``FunctionMatrix`` from a SymPy undefined function:
>>> from sympy import Function
>>> f = Function('f')
>>> X = FunctionMatrix(3, 3, f)
>>> X.as_explicit()
Matrix([
[f(0, 0), f(0, 1), f(0, 2)],
[f(1, 0), f(1, 1), f(1, 2)],
[f(2, 0), f(2, 1), f(2, 2)]])
Creating a ``FunctionMatrix`` from Python ``lambda``:
>>> FunctionMatrix(n, m, 'lambda i, j: i + j')
FunctionMatrix(n, m, Lambda((i, j), i + j))
Example of lazy evaluation of matrix product:
>>> Y = FunctionMatrix(1000, 1000, Lambda((i, j), i + j))
>>> isinstance(Y*Y, MatPow) # this is an expression object
True
>>> (Y**2)[10,10] # So this is evaluated lazily
342923500
Notes
=====
This class provides an alternative way to represent an extremely
dense matrix with entries in some form of a sequence, in a most
sparse way.
"""
def __new__(cls, rows, cols, lamda):
rows, cols = _sympify(rows), _sympify(cols)
cls._check_dim(rows)
cls._check_dim(cols)
lamda = sympify(lamda)
if not isinstance(lamda, (FunctionClass, Lambda)):
raise ValueError(
"{} should be compatible with SymPy function classes."
.format(lamda))
if 2 not in lamda.nargs:
raise ValueError(
'{} should be able to accept 2 arguments.'.format(lamda))
if not isinstance(lamda, Lambda):
i, j = Dummy('i'), Dummy('j')
lamda = Lambda((i, j), lamda(i, j))
return super().__new__(cls, rows, cols, lamda)
@property
def shape(self):
return self.args[0:2]
@property
def lamda(self):
return self.args[2]
def _entry(self, i, j, **kwargs):
return self.lamda(i, j)
def _eval_trace(self):
from sympy.matrices.expressions.trace import Trace
from sympy.concrete.summations import Sum
return Trace(self).rewrite(Sum).doit()
def _eval_as_real_imag(self):
return (re(Matrix(self)), im(Matrix(self)))
|
969bb89438563b3bcfc48c52d77e1c487cd61130caeb88f49e8bb38eef7d5b8c | from functools import reduce
import operator
from sympy.core import Basic, sympify
from sympy.core.add import add, Add, _could_extract_minus_sign
from sympy.core.sorting import default_sort_key
from sympy.functions import adjoint
from sympy.matrices.common import ShapeError
from sympy.matrices.matrices import MatrixBase
from sympy.matrices.expressions.transpose import transpose
from sympy.strategies import (rm_id, unpack, flatten, sort, condition,
exhaust, do_one, glom)
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.matrices.expressions.special import ZeroMatrix, GenericZeroMatrix
from sympy.utilities.iterables import sift
from sympy.utilities.exceptions import sympy_deprecation_warning
# XXX: MatAdd should perhaps not subclass directly from Add
class MatAdd(MatrixExpr, Add):
"""A Sum of Matrix Expressions
MatAdd inherits from and operates like SymPy Add
Examples
========
>>> from sympy import MatAdd, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> C = MatrixSymbol('C', 5, 5)
>>> MatAdd(A, B, C)
A + B + C
"""
is_MatAdd = True
identity = GenericZeroMatrix()
def __new__(cls, *args, evaluate=False, check=None, _sympify=True):
if not args:
return cls.identity
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericZeroMatrix().shape
args = list(filter(lambda i: cls.identity != i, args))
if _sympify:
args = list(map(sympify, args))
obj = Basic.__new__(cls, *args)
if check is not None:
sympy_deprecation_warning(
"Passing check to MatAdd is deprecated and the check argument will be removed in a future version.",
deprecated_since_version="1.11",
active_deprecations_target='remove-check-argument-from-matrix-operations')
if check in (True, None):
if not any(isinstance(i, MatrixExpr) for i in args):
return Add.fromiter(args)
validate(*args)
else:
sympy_deprecation_warning(
"Passing check=False to MatAdd is deprecated and the check argument will be removed in a future version.",
deprecated_since_version="1.11",
active_deprecations_target='remove-check-argument-from-matrix-operations')
if evaluate:
obj = cls._evaluate(obj)
return obj
@classmethod
def _evaluate(cls, expr):
return canonicalize(expr)
@property
def shape(self):
return self.args[0].shape
def could_extract_minus_sign(self):
return _could_extract_minus_sign(self)
def expand(self, **kwargs):
expanded = super(MatAdd, self).expand(**kwargs)
return self._evaluate(expanded)
def _entry(self, i, j, **kwargs):
return Add(*[arg._entry(i, j, **kwargs) for arg in self.args])
def _eval_transpose(self):
return MatAdd(*[transpose(arg) for arg in self.args]).doit()
def _eval_adjoint(self):
return MatAdd(*[adjoint(arg) for arg in self.args]).doit()
def _eval_trace(self):
from .trace import trace
return Add(*[trace(arg) for arg in self.args]).doit()
def doit(self, **hints):
deep = hints.get('deep', True)
if deep:
args = [arg.doit(**hints) for arg in self.args]
else:
args = self.args
return canonicalize(MatAdd(*args))
def _eval_derivative_matrix_lines(self, x):
add_lines = [arg._eval_derivative_matrix_lines(x) for arg in self.args]
return [j for i in add_lines for j in i]
add.register_handlerclass((Add, MatAdd), MatAdd)
def validate(*args):
if not all(arg.is_Matrix for arg in args):
raise TypeError("Mix of Matrix and Scalar symbols")
A = args[0]
for B in args[1:]:
if A.shape != B.shape:
raise ShapeError("Matrices %s and %s are not aligned"%(A, B))
factor_of = lambda arg: arg.as_coeff_mmul()[0]
matrix_of = lambda arg: unpack(arg.as_coeff_mmul()[1])
def combine(cnt, mat):
if cnt == 1:
return mat
else:
return cnt * mat
def merge_explicit(matadd):
""" Merge explicit MatrixBase arguments
Examples
========
>>> from sympy import MatrixSymbol, eye, Matrix, MatAdd, pprint
>>> from sympy.matrices.expressions.matadd import merge_explicit
>>> A = MatrixSymbol('A', 2, 2)
>>> B = eye(2)
>>> C = Matrix([[1, 2], [3, 4]])
>>> X = MatAdd(A, B, C)
>>> pprint(X)
[1 0] [1 2]
A + [ ] + [ ]
[0 1] [3 4]
>>> pprint(merge_explicit(X))
[2 2]
A + [ ]
[3 5]
"""
groups = sift(matadd.args, lambda arg: isinstance(arg, MatrixBase))
if len(groups[True]) > 1:
return MatAdd(*(groups[False] + [reduce(operator.add, groups[True])]))
else:
return matadd
rules = (rm_id(lambda x: x == 0 or isinstance(x, ZeroMatrix)),
unpack,
flatten,
glom(matrix_of, factor_of, combine),
merge_explicit,
sort(default_sort_key))
canonicalize = exhaust(condition(lambda x: isinstance(x, MatAdd),
do_one(*rules)))
|
11f970133e60d04d18375d837f30d379493e4320747cfe0163faa0cc21aa2253 | from sympy.core import I, symbols, Basic, Mul, S
from sympy.core.mul import mul
from sympy.functions import adjoint, transpose
from sympy.matrices import (Identity, Inverse, Matrix, MatrixSymbol, ZeroMatrix,
eye, ImmutableMatrix)
from sympy.matrices.expressions import Adjoint, Transpose, det, MatPow
from sympy.matrices.expressions.special import GenericIdentity
from sympy.matrices.expressions.matmul import (factor_in_front, remove_ids,
MatMul, combine_powers, any_zeros, unpack, only_squares)
from sympy.strategies import null_safe
from sympy.assumptions.ask import Q
from sympy.assumptions.refine import refine
from sympy.core.symbol import Symbol
from sympy.testing.pytest import XFAIL
n, m, l, k = symbols('n m l k', integer=True)
x = symbols('x')
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', n, n)
D = MatrixSymbol('D', n, n)
E = MatrixSymbol('E', m, n)
def test_evaluate():
assert MatMul(C, C, evaluate=True) == MatMul(C, C).doit()
def test_adjoint():
assert adjoint(A*B) == Adjoint(B)*Adjoint(A)
assert adjoint(2*A*B) == 2*Adjoint(B)*Adjoint(A)
assert adjoint(2*I*C) == -2*I*Adjoint(C)
M = Matrix(2, 2, [1, 2 + I, 3, 4])
MA = Matrix(2, 2, [1, 3, 2 - I, 4])
assert adjoint(M) == MA
assert adjoint(2*M) == 2*MA
assert adjoint(MatMul(2, M)) == MatMul(2, MA).doit()
def test_transpose():
assert transpose(A*B) == Transpose(B)*Transpose(A)
assert transpose(2*A*B) == 2*Transpose(B)*Transpose(A)
assert transpose(2*I*C) == 2*I*Transpose(C)
M = Matrix(2, 2, [1, 2 + I, 3, 4])
MT = Matrix(2, 2, [1, 3, 2 + I, 4])
assert transpose(M) == MT
assert transpose(2*M) == 2*MT
assert transpose(x*M) == x*MT
assert transpose(MatMul(2, M)) == MatMul(2, MT).doit()
def test_factor_in_front():
assert factor_in_front(MatMul(A, 2, B, evaluate=False)) ==\
MatMul(2, A, B, evaluate=False)
def test_remove_ids():
assert remove_ids(MatMul(A, Identity(m), B, evaluate=False)) == \
MatMul(A, B, evaluate=False)
assert null_safe(remove_ids)(MatMul(Identity(n), evaluate=False)) == \
MatMul(Identity(n), evaluate=False)
def test_combine_powers():
assert combine_powers(MatMul(D, Inverse(D), D, evaluate=False)) == \
MatMul(Identity(n), D, evaluate=False)
assert combine_powers(MatMul(B.T, Inverse(E*A), E, A, B, evaluate=False)) == \
MatMul(B.T, Identity(m), B, evaluate=False)
assert combine_powers(MatMul(A, E, Inverse(A*E), D, evaluate=False)) == \
MatMul(Identity(n), D, evaluate=False)
def test_any_zeros():
assert any_zeros(MatMul(A, ZeroMatrix(m, k), evaluate=False)) == \
ZeroMatrix(n, k)
def test_unpack():
assert unpack(MatMul(A, evaluate=False)) == A
x = MatMul(A, B)
assert unpack(x) == x
def test_only_squares():
assert only_squares(C) == [C]
assert only_squares(C, D) == [C, D]
assert only_squares(C, A, A.T, D) == [C, A*A.T, D]
def test_determinant():
assert det(2*C) == 2**n*det(C)
assert det(2*C*D) == 2**n*det(C)*det(D)
assert det(3*C*A*A.T*D) == 3**n*det(C)*det(A*A.T)*det(D)
def test_doit():
assert MatMul(C, 2, D).args == (C, 2, D)
assert MatMul(C, 2, D).doit().args == (2, C, D)
assert MatMul(C, Transpose(D*C)).args == (C, Transpose(D*C))
assert MatMul(C, Transpose(D*C)).doit(deep=True).args == (C, C.T, D.T)
def test_doit_drills_down():
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[2, 3], [4, 5]])
assert MatMul(X, MatPow(Y, 2)).doit() == X*Y**2
assert MatMul(C, Transpose(D*C)).doit().args == (C, C.T, D.T)
def test_doit_deep_false_still_canonical():
assert (MatMul(C, Transpose(D*C), 2).doit(deep=False).args ==
(2, C, Transpose(D*C)))
def test_matmul_scalar_Matrix_doit():
# Issue 9053
X = Matrix([[1, 2], [3, 4]])
assert MatMul(2, X).doit() == 2*X
def test_matmul_sympify():
assert isinstance(MatMul(eye(1), eye(1)).args[0], Basic)
def test_collapse_MatrixBase():
A = Matrix([[1, 1], [1, 1]])
B = Matrix([[1, 2], [3, 4]])
assert MatMul(A, B).doit() == ImmutableMatrix([[4, 6], [4, 6]])
def test_refine():
assert refine(C*C.T*D, Q.orthogonal(C)).doit() == D
kC = k*C
assert refine(kC*C.T, Q.orthogonal(C)).doit() == k*Identity(n)
assert refine(kC* kC.T, Q.orthogonal(C)).doit() == (k**2)*Identity(n)
def test_matmul_no_matrices():
assert MatMul(1) == 1
assert MatMul(n, m) == n*m
assert not isinstance(MatMul(n, m), MatMul)
def test_matmul_args_cnc():
assert MatMul(n, A, A.T).args_cnc() == [[n], [A, A.T]]
assert MatMul(A, A.T).args_cnc() == [[], [A, A.T]]
@XFAIL
def test_matmul_args_cnc_symbols():
# Not currently supported
a, b = symbols('a b', commutative=False)
assert MatMul(n, a, b, A, A.T).args_cnc() == [[n], [a, b, A, A.T]]
assert MatMul(n, a, A, b, A.T).args_cnc() == [[n], [a, A, b, A.T]]
def test_issue_12950():
M = Matrix([[Symbol("x")]]) * MatrixSymbol("A", 1, 1)
assert MatrixSymbol("A", 1, 1).as_explicit()[0]*Symbol('x') == M.as_explicit()[0]
def test_construction_with_Mul():
assert Mul(C, D) == MatMul(C, D)
assert Mul(D, C) == MatMul(D, C)
def test_construction_with_mul():
assert mul(C, D) == MatMul(C, D)
assert mul(D, C) == MatMul(D, C)
assert mul(C, D) != MatMul(D, C)
def test_generic_identity():
assert MatMul.identity == GenericIdentity()
assert MatMul.identity != S.One
def test_issue_23519():
N = Symbol("N", integer=True)
M1 = MatrixSymbol("M1", N, N)
M2 = MatrixSymbol("M2", N, N)
I = Identity(N)
z = (M2 + 2 * (M2 + I) * M1 + I)
assert z.coeff(M1) == 2*I + 2*M2
|
1072569b676788d99fd427ce0d048f9bf1c2095efb21ad056de36282c6095d96 | from sympy.matrices.expressions.trace import Trace
from sympy.testing.pytest import raises, slow
from sympy.matrices.expressions.blockmatrix import (
block_collapse, bc_matmul, bc_block_plus_ident, BlockDiagMatrix,
BlockMatrix, bc_dist, bc_matadd, bc_transpose, bc_inverse,
blockcut, reblock_2x2, deblock)
from sympy.matrices.expressions import (MatrixSymbol, Identity,
Inverse, trace, Transpose, det, ZeroMatrix, OneMatrix)
from sympy.matrices.common import NonInvertibleMatrixError
from sympy.matrices import (
Matrix, ImmutableMatrix, ImmutableSparseMatrix)
from sympy.core import Tuple, symbols, Expr, S
from sympy.functions import transpose, im, re
i, j, k, l, m, n, p = symbols('i:n, p', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
C = MatrixSymbol('C', n, n)
D = MatrixSymbol('D', n, n)
G = MatrixSymbol('G', n, n)
H = MatrixSymbol('H', n, n)
b1 = BlockMatrix([[G, H]])
b2 = BlockMatrix([[G], [H]])
def test_bc_matmul():
assert bc_matmul(H*b1*b2*G) == BlockMatrix([[(H*G*G + H*H*H)*G]])
def test_bc_matadd():
assert bc_matadd(BlockMatrix([[G, H]]) + BlockMatrix([[H, H]])) == \
BlockMatrix([[G+H, H+H]])
def test_bc_transpose():
assert bc_transpose(Transpose(BlockMatrix([[A, B], [C, D]]))) == \
BlockMatrix([[A.T, C.T], [B.T, D.T]])
def test_bc_dist_diag():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', m, m)
C = MatrixSymbol('C', l, l)
X = BlockDiagMatrix(A, B, C)
assert bc_dist(X+X).equals(BlockDiagMatrix(2*A, 2*B, 2*C))
def test_block_plus_ident():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
Z = MatrixSymbol('Z', n + m, n + m)
assert bc_block_plus_ident(X + Identity(m + n) + Z) == \
BlockDiagMatrix(Identity(n), Identity(m)) + X + Z
def test_BlockMatrix():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, k)
C = MatrixSymbol('C', l, m)
D = MatrixSymbol('D', l, k)
M = MatrixSymbol('M', m + k, p)
N = MatrixSymbol('N', l + n, k + m)
X = BlockMatrix(Matrix([[A, B], [C, D]]))
assert X.__class__(*X.args) == X
# block_collapse does nothing on normal inputs
E = MatrixSymbol('E', n, m)
assert block_collapse(A + 2*E) == A + 2*E
F = MatrixSymbol('F', m, m)
assert block_collapse(E.T*A*F) == E.T*A*F
assert X.shape == (l + n, k + m)
assert X.blockshape == (2, 2)
assert transpose(X) == BlockMatrix(Matrix([[A.T, C.T], [B.T, D.T]]))
assert transpose(X).shape == X.shape[::-1]
# Test that BlockMatrices and MatrixSymbols can still mix
assert (X*M).is_MatMul
assert X._blockmul(M).is_MatMul
assert (X*M).shape == (n + l, p)
assert (X + N).is_MatAdd
assert X._blockadd(N).is_MatAdd
assert (X + N).shape == X.shape
E = MatrixSymbol('E', m, 1)
F = MatrixSymbol('F', k, 1)
Y = BlockMatrix(Matrix([[E], [F]]))
assert (X*Y).shape == (l + n, 1)
assert block_collapse(X*Y).blocks[0, 0] == A*E + B*F
assert block_collapse(X*Y).blocks[1, 0] == C*E + D*F
# block_collapse passes down into container objects, transposes, and inverse
assert block_collapse(transpose(X*Y)) == transpose(block_collapse(X*Y))
assert block_collapse(Tuple(X*Y, 2*X)) == (
block_collapse(X*Y), block_collapse(2*X))
# Make sure that MatrixSymbols will enter 1x1 BlockMatrix if it simplifies
Ab = BlockMatrix([[A]])
Z = MatrixSymbol('Z', *A.shape)
assert block_collapse(Ab + Z) == A + Z
def test_block_collapse_explicit_matrices():
A = Matrix([[1, 2], [3, 4]])
assert block_collapse(BlockMatrix([[A]])) == A
A = ImmutableSparseMatrix([[1, 2], [3, 4]])
assert block_collapse(BlockMatrix([[A]])) == A
def test_issue_17624():
a = MatrixSymbol("a", 2, 2)
z = ZeroMatrix(2, 2)
b = BlockMatrix([[a, z], [z, z]])
assert block_collapse(b * b) == BlockMatrix([[a**2, z], [z, z]])
assert block_collapse(b * b * b) == BlockMatrix([[a**3, z], [z, z]])
def test_issue_18618():
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
assert A == Matrix(BlockDiagMatrix(A))
def test_BlockMatrix_trace():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
X = BlockMatrix([[A, B], [C, D]])
assert trace(X) == trace(A) + trace(D)
assert trace(BlockMatrix([ZeroMatrix(n, n)])) == 0
def test_BlockMatrix_Determinant():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
X = BlockMatrix([[A, B], [C, D]])
from sympy.assumptions.ask import Q
from sympy.assumptions.assume import assuming
with assuming(Q.invertible(A)):
assert det(X) == det(A) * det(X.schur('A'))
assert isinstance(det(X), Expr)
assert det(BlockMatrix([A])) == det(A)
assert det(BlockMatrix([ZeroMatrix(n, n)])) == 0
def test_squareBlockMatrix():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
Y = BlockMatrix([[A]])
assert X.is_square
Q = X + Identity(m + n)
assert (block_collapse(Q) ==
BlockMatrix([[A + Identity(n), B], [C, D + Identity(m)]]))
assert (X + MatrixSymbol('Q', n + m, n + m)).is_MatAdd
assert (X * MatrixSymbol('Q', n + m, n + m)).is_MatMul
assert block_collapse(Y.I) == A.I
assert isinstance(X.inverse(), Inverse)
assert not X.is_Identity
Z = BlockMatrix([[Identity(n), B], [C, D]])
assert not Z.is_Identity
def test_BlockMatrix_2x2_inverse_symbolic():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, k - m)
C = MatrixSymbol('C', k - n, m)
D = MatrixSymbol('D', k - n, k - m)
X = BlockMatrix([[A, B], [C, D]])
assert X.is_square and X.shape == (k, k)
assert isinstance(block_collapse(X.I), Inverse) # Can't invert when none of the blocks is square
# test code path where only A is invertible
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = ZeroMatrix(m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[A.I + A.I * B * X.schur('A').I * C * A.I, -A.I * B * X.schur('A').I],
[-X.schur('A').I * C * A.I, X.schur('A').I],
])
# test code path where only B is invertible
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, n)
C = ZeroMatrix(m, m)
D = MatrixSymbol('D', m, n)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[-X.schur('B').I * D * B.I, X.schur('B').I],
[B.I + B.I * A * X.schur('B').I * D * B.I, -B.I * A * X.schur('B').I],
])
# test code path where only C is invertible
A = MatrixSymbol('A', n, m)
B = ZeroMatrix(n, n)
C = MatrixSymbol('C', m, m)
D = MatrixSymbol('D', m, n)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[-C.I * D * X.schur('C').I, C.I + C.I * D * X.schur('C').I * A * C.I],
[X.schur('C').I, -X.schur('C').I * A * C.I],
])
# test code path where only D is invertible
A = ZeroMatrix(n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[X.schur('D').I, -X.schur('D').I * B * D.I],
[-D.I * C * X.schur('D').I, D.I + D.I * C * X.schur('D').I * B * D.I],
])
def test_BlockMatrix_2x2_inverse_numeric():
"""Test 2x2 block matrix inversion numerically for all 4 formulas"""
M = Matrix([[1, 2], [3, 4]])
# rank deficient matrices that have full rank when two of them combined
D1 = Matrix([[1, 2], [2, 4]])
D2 = Matrix([[1, 3], [3, 9]])
D3 = Matrix([[1, 4], [4, 16]])
assert D1.rank() == D2.rank() == D3.rank() == 1
assert (D1 + D2).rank() == (D2 + D3).rank() == (D3 + D1).rank() == 2
# Only A is invertible
K = BlockMatrix([[M, D1], [D2, D3]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
# Only B is invertible
K = BlockMatrix([[D1, M], [D2, D3]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
# Only C is invertible
K = BlockMatrix([[D1, D2], [M, D3]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
# Only D is invertible
K = BlockMatrix([[D1, D2], [D3, M]])
assert block_collapse(K.inv()).as_explicit() == K.as_explicit().inv()
@slow
def test_BlockMatrix_3x3_symbolic():
# Only test one of these, instead of all permutations, because it's slow
rowblocksizes = (n, m, k)
colblocksizes = (m, k, n)
K = BlockMatrix([
[MatrixSymbol('M%s%s' % (rows, cols), rows, cols) for cols in colblocksizes]
for rows in rowblocksizes
])
collapse = block_collapse(K.I)
assert isinstance(collapse, BlockMatrix)
def test_BlockDiagMatrix():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', m, m)
C = MatrixSymbol('C', l, l)
M = MatrixSymbol('M', n + m + l, n + m + l)
X = BlockDiagMatrix(A, B, C)
Y = BlockDiagMatrix(A, 2*B, 3*C)
assert X.blocks[1, 1] == B
assert X.shape == (n + m + l, n + m + l)
assert all(X.blocks[i, j].is_ZeroMatrix if i != j else X.blocks[i, j] in [A, B, C]
for i in range(3) for j in range(3))
assert X.__class__(*X.args) == X
assert X.get_diag_blocks() == (A, B, C)
assert isinstance(block_collapse(X.I * X), Identity)
assert bc_matmul(X*X) == BlockDiagMatrix(A*A, B*B, C*C)
assert block_collapse(X*X) == BlockDiagMatrix(A*A, B*B, C*C)
#XXX: should be == ??
assert block_collapse(X + X).equals(BlockDiagMatrix(2*A, 2*B, 2*C))
assert block_collapse(X*Y) == BlockDiagMatrix(A*A, 2*B*B, 3*C*C)
assert block_collapse(X + Y) == BlockDiagMatrix(2*A, 3*B, 4*C)
# Ensure that BlockDiagMatrices can still interact with normal MatrixExprs
assert (X*(2*M)).is_MatMul
assert (X + (2*M)).is_MatAdd
assert (X._blockmul(M)).is_MatMul
assert (X._blockadd(M)).is_MatAdd
def test_BlockDiagMatrix_nonsquare():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', k, l)
X = BlockDiagMatrix(A, B)
assert X.shape == (n + k, m + l)
assert X.shape == (n + k, m + l)
assert X.rowblocksizes == [n, k]
assert X.colblocksizes == [m, l]
C = MatrixSymbol('C', n, m)
D = MatrixSymbol('D', k, l)
Y = BlockDiagMatrix(C, D)
assert block_collapse(X + Y) == BlockDiagMatrix(A + C, B + D)
assert block_collapse(X * Y.T) == BlockDiagMatrix(A * C.T, B * D.T)
raises(NonInvertibleMatrixError, lambda: BlockDiagMatrix(A, C.T).inverse())
def test_BlockDiagMatrix_determinant():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', m, m)
assert det(BlockDiagMatrix()) == 1
assert det(BlockDiagMatrix(A)) == det(A)
assert det(BlockDiagMatrix(A, B)) == det(A) * det(B)
# non-square blocks
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', n, m)
assert det(BlockDiagMatrix(C, D)) == 0
def test_BlockDiagMatrix_trace():
assert trace(BlockDiagMatrix()) == 0
assert trace(BlockDiagMatrix(ZeroMatrix(n, n))) == 0
A = MatrixSymbol('A', n, n)
assert trace(BlockDiagMatrix(A)) == trace(A)
B = MatrixSymbol('B', m, m)
assert trace(BlockDiagMatrix(A, B)) == trace(A) + trace(B)
# non-square blocks
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', n, m)
assert isinstance(trace(BlockDiagMatrix(C, D)), Trace)
def test_BlockDiagMatrix_transpose():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', k, l)
assert transpose(BlockDiagMatrix()) == BlockDiagMatrix()
assert transpose(BlockDiagMatrix(A)) == BlockDiagMatrix(A.T)
assert transpose(BlockDiagMatrix(A, B)) == BlockDiagMatrix(A.T, B.T)
def test_issue_2460():
bdm1 = BlockDiagMatrix(Matrix([i]), Matrix([j]))
bdm2 = BlockDiagMatrix(Matrix([k]), Matrix([l]))
assert block_collapse(bdm1 + bdm2) == BlockDiagMatrix(Matrix([i + k]), Matrix([j + l]))
def test_blockcut():
A = MatrixSymbol('A', n, m)
B = blockcut(A, (n/2, n/2), (m/2, m/2))
assert B == BlockMatrix([[A[:n/2, :m/2], A[:n/2, m/2:]],
[A[n/2:, :m/2], A[n/2:, m/2:]]])
M = ImmutableMatrix(4, 4, range(16))
B = blockcut(M, (2, 2), (2, 2))
assert M == ImmutableMatrix(B)
B = blockcut(M, (1, 3), (2, 2))
assert ImmutableMatrix(B.blocks[0, 1]) == ImmutableMatrix([[2, 3]])
def test_reblock_2x2():
B = BlockMatrix([[MatrixSymbol('A_%d%d'%(i,j), 2, 2)
for j in range(3)]
for i in range(3)])
assert B.blocks.shape == (3, 3)
BB = reblock_2x2(B)
assert BB.blocks.shape == (2, 2)
assert B.shape == BB.shape
assert B.as_explicit() == BB.as_explicit()
def test_deblock():
B = BlockMatrix([[MatrixSymbol('A_%d%d'%(i,j), n, n)
for j in range(4)]
for i in range(4)])
assert deblock(reblock_2x2(B)) == B
def test_block_collapse_type():
bm1 = BlockDiagMatrix(ImmutableMatrix([1]), ImmutableMatrix([2]))
bm2 = BlockDiagMatrix(ImmutableMatrix([3]), ImmutableMatrix([4]))
assert bm1.T.__class__ == BlockDiagMatrix
assert block_collapse(bm1 - bm2).__class__ == BlockDiagMatrix
assert block_collapse(Inverse(bm1)).__class__ == BlockDiagMatrix
assert block_collapse(Transpose(bm1)).__class__ == BlockDiagMatrix
assert bc_transpose(Transpose(bm1)).__class__ == BlockDiagMatrix
assert bc_inverse(Inverse(bm1)).__class__ == BlockDiagMatrix
def test_invalid_block_matrix():
raises(ValueError, lambda: BlockMatrix([
[Identity(2), Identity(5)],
]))
raises(ValueError, lambda: BlockMatrix([
[Identity(n), Identity(m)],
]))
raises(ValueError, lambda: BlockMatrix([
[ZeroMatrix(n, n), ZeroMatrix(n, n)],
[ZeroMatrix(n, n - 1), ZeroMatrix(n, n + 1)],
]))
raises(ValueError, lambda: BlockMatrix([
[ZeroMatrix(n - 1, n), ZeroMatrix(n, n)],
[ZeroMatrix(n + 1, n), ZeroMatrix(n, n)],
]))
def test_block_lu_decomposition():
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
#LDU decomposition
L, D, U = X.LDUdecomposition()
assert block_collapse(L*D*U) == X
#UDL decomposition
U, D, L = X.UDLdecomposition()
assert block_collapse(U*D*L) == X
#LU decomposition
L, U = X.LUdecomposition()
assert block_collapse(L*U) == X
def test_issue_21866():
n = 10
I = Identity(n)
O = ZeroMatrix(n, n)
A = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ I, O, O, I ]])
Ainv = block_collapse(A.inv())
AinvT = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ -I, O, O, I ]])
assert Ainv == AinvT
def test_adjoint_and_special_matrices():
A = Identity(3)
B = OneMatrix(3, 2)
C = ZeroMatrix(2, 3)
D = Identity(2)
X = BlockMatrix([[A, B], [C, D]])
X2 = BlockMatrix([[A, S.ImaginaryUnit*B], [C, D]])
assert X.adjoint() == BlockMatrix([[A, ZeroMatrix(3, 2)], [OneMatrix(2, 3), D]])
assert re(X) == X
assert X2.adjoint() == BlockMatrix([[A, ZeroMatrix(3, 2)], [-S.ImaginaryUnit*OneMatrix(2, 3), D]])
assert im(X2) == BlockMatrix([[ZeroMatrix(3, 3), OneMatrix(3, 2)], [ZeroMatrix(2, 3), ZeroMatrix(2, 2)]])
|
7c6c27af3e773e2fc913a6b77d59971ddc3096f6eaba09d4785b9625367c5e19 | from sympy.core.add import Add
from sympy.core.expr import unchanged
from sympy.core.mul import Mul
from sympy.core.symbol import symbols
from sympy.core.relational import Eq
from sympy.concrete.summations import Sum
from sympy.functions.elementary.complexes import im, re
from sympy.functions.elementary.piecewise import Piecewise
from sympy.matrices.common import NonSquareMatrixError, ShapeError
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.special import (
ZeroMatrix, GenericZeroMatrix, Identity, GenericIdentity, OneMatrix)
from sympy.matrices.expressions.matmul import MatMul
from sympy.testing.pytest import raises
def test_zero_matrix_creation():
assert unchanged(ZeroMatrix, 2, 2)
assert unchanged(ZeroMatrix, 0, 0)
raises(ValueError, lambda: ZeroMatrix(-1, 2))
raises(ValueError, lambda: ZeroMatrix(2.0, 2))
raises(ValueError, lambda: ZeroMatrix(2j, 2))
raises(ValueError, lambda: ZeroMatrix(2, -1))
raises(ValueError, lambda: ZeroMatrix(2, 2.0))
raises(ValueError, lambda: ZeroMatrix(2, 2j))
n = symbols('n')
assert unchanged(ZeroMatrix, n, n)
n = symbols('n', integer=False)
raises(ValueError, lambda: ZeroMatrix(n, n))
n = symbols('n', negative=True)
raises(ValueError, lambda: ZeroMatrix(n, n))
def test_generic_zero_matrix():
z = GenericZeroMatrix()
n = symbols('n', integer=True)
A = MatrixSymbol("A", n, n)
assert z == z
assert z != A
assert A != z
assert z.is_ZeroMatrix
raises(TypeError, lambda: z.shape)
raises(TypeError, lambda: z.rows)
raises(TypeError, lambda: z.cols)
assert MatAdd() == z
assert MatAdd(z, A) == MatAdd(A)
# Make sure it is hashable
hash(z)
def test_identity_matrix_creation():
assert Identity(2)
assert Identity(0)
raises(ValueError, lambda: Identity(-1))
raises(ValueError, lambda: Identity(2.0))
raises(ValueError, lambda: Identity(2j))
n = symbols('n')
assert Identity(n)
n = symbols('n', integer=False)
raises(ValueError, lambda: Identity(n))
n = symbols('n', negative=True)
raises(ValueError, lambda: Identity(n))
def test_generic_identity():
I = GenericIdentity()
n = symbols('n', integer=True)
A = MatrixSymbol("A", n, n)
assert I == I
assert I != A
assert A != I
assert I.is_Identity
assert I**-1 == I
raises(TypeError, lambda: I.shape)
raises(TypeError, lambda: I.rows)
raises(TypeError, lambda: I.cols)
assert MatMul() == I
assert MatMul(I, A) == MatMul(A)
# Make sure it is hashable
hash(I)
def test_one_matrix_creation():
assert OneMatrix(2, 2)
assert OneMatrix(0, 0)
assert Eq(OneMatrix(1, 1), Identity(1))
raises(ValueError, lambda: OneMatrix(-1, 2))
raises(ValueError, lambda: OneMatrix(2.0, 2))
raises(ValueError, lambda: OneMatrix(2j, 2))
raises(ValueError, lambda: OneMatrix(2, -1))
raises(ValueError, lambda: OneMatrix(2, 2.0))
raises(ValueError, lambda: OneMatrix(2, 2j))
n = symbols('n')
assert OneMatrix(n, n)
n = symbols('n', integer=False)
raises(ValueError, lambda: OneMatrix(n, n))
n = symbols('n', negative=True)
raises(ValueError, lambda: OneMatrix(n, n))
def test_ZeroMatrix():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
Z = ZeroMatrix(n, m)
assert A + Z == A
assert A*Z.T == ZeroMatrix(n, n)
assert Z*A.T == ZeroMatrix(n, n)
assert A - A == ZeroMatrix(*A.shape)
assert Z
assert Z.transpose() == ZeroMatrix(m, n)
assert Z.conjugate() == Z
assert Z.adjoint() == ZeroMatrix(m, n)
assert re(Z) == Z
assert im(Z) == Z
assert ZeroMatrix(n, n)**0 == Identity(n)
with raises(NonSquareMatrixError):
Z**0
with raises(NonSquareMatrixError):
Z**1
with raises(NonSquareMatrixError):
Z**2
assert ZeroMatrix(3, 3).as_explicit() == ImmutableDenseMatrix.zeros(3, 3)
def test_ZeroMatrix_doit():
n = symbols('n', integer=True)
Znn = ZeroMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Znn.rows, Add)
assert Znn.doit() == ZeroMatrix(2*n, n)
assert isinstance(Znn.doit().rows, Mul)
def test_OneMatrix():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
a = MatrixSymbol('a', n, 1)
U = OneMatrix(n, m)
assert U.shape == (n, m)
assert isinstance(A + U, Add)
assert U.transpose() == OneMatrix(m, n)
assert U.conjugate() == U
assert U.adjoint() == OneMatrix(m, n)
assert re(U) == U
assert im(U) == ZeroMatrix(n, m)
assert OneMatrix(n, n) ** 0 == Identity(n)
with raises(NonSquareMatrixError):
U ** 0
with raises(NonSquareMatrixError):
U ** 1
with raises(NonSquareMatrixError):
U ** 2
with raises(ShapeError):
a + U
U = OneMatrix(n, n)
assert U[1, 2] == 1
U = OneMatrix(2, 3)
assert U.as_explicit() == ImmutableDenseMatrix.ones(2, 3)
def test_OneMatrix_doit():
n = symbols('n', integer=True)
Unn = OneMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Unn.rows, Add)
assert Unn.doit() == OneMatrix(2 * n, n)
assert isinstance(Unn.doit().rows, Mul)
def test_OneMatrix_mul():
n, m, k = symbols('n m k', integer=True)
w = MatrixSymbol('w', n, 1)
assert OneMatrix(n, m) * OneMatrix(m, k) == OneMatrix(n, k) * m
assert w * OneMatrix(1, 1) == w
assert OneMatrix(1, 1) * w.T == w.T
def test_Identity():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
i, j = symbols('i j')
In = Identity(n)
Im = Identity(m)
assert A*Im == A
assert In*A == A
assert In.transpose() == In
assert In.inverse() == In
assert In.conjugate() == In
assert In.adjoint() == In
assert re(In) == In
assert im(In) == ZeroMatrix(n, n)
assert In[i, j] != 0
assert Sum(In[i, j], (i, 0, n-1), (j, 0, n-1)).subs(n,3).doit() == 3
assert Sum(Sum(In[i, j], (i, 0, n-1)), (j, 0, n-1)).subs(n,3).doit() == 3
# If range exceeds the limit `(0, n-1)`, do not remove `Piecewise`:
expr = Sum(In[i, j], (i, 0, n-1))
assert expr.doit() == 1
expr = Sum(In[i, j], (i, 0, n-2))
assert expr.doit().dummy_eq(
Piecewise(
(1, (j >= 0) & (j <= n-2)),
(0, True)
)
)
expr = Sum(In[i, j], (i, 1, n-1))
assert expr.doit().dummy_eq(
Piecewise(
(1, (j >= 1) & (j <= n-1)),
(0, True)
)
)
assert Identity(3).as_explicit() == ImmutableDenseMatrix.eye(3)
def test_Identity_doit():
n = symbols('n', integer=True)
Inn = Identity(Add(n, n, evaluate=False))
assert isinstance(Inn.rows, Add)
assert Inn.doit() == Identity(2*n)
assert isinstance(Inn.doit().rows, Mul)
|
26afb8ce186a2fd9ac8192baac94d8f60f4d5dc26b7d9747b6fcb1fc6d67c6a0 | from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.special import (Identity, OneMatrix, ZeroMatrix)
from sympy.core import symbols
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.matrices import ShapeError, MatrixSymbol
from sympy.matrices.expressions import (HadamardProduct, hadamard_product, HadamardPower, hadamard_power)
n, m, k = symbols('n,m,k')
Z = MatrixSymbol('Z', n, n)
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, k)
def test_HadamardProduct():
assert HadamardProduct(A, B, A).shape == A.shape
raises(ShapeError, lambda: HadamardProduct(A, B.T))
raises(TypeError, lambda: HadamardProduct(A, n))
raises(TypeError, lambda: HadamardProduct(A, 1))
assert HadamardProduct(A, 2*B, -A)[1, 1] == \
-2 * A[1, 1] * B[1, 1] * A[1, 1]
mix = HadamardProduct(Z*A, B)*C
assert mix.shape == (n, k)
assert set(HadamardProduct(A, B, A).T.args) == {A.T, A.T, B.T}
def test_HadamardProduct_isnt_commutative():
assert HadamardProduct(A, B) != HadamardProduct(B, A)
def test_mixed_indexing():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
Z = MatrixSymbol('Z', 2, 2)
assert (X*HadamardProduct(Y, Z))[0, 0] == \
X[0, 0]*Y[0, 0]*Z[0, 0] + X[0, 1]*Y[1, 0]*Z[1, 0]
def test_canonicalize():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
with warns_deprecated_sympy():
expr = HadamardProduct(X, check=False)
assert isinstance(expr, HadamardProduct)
expr2 = expr.doit() # unpack is called
assert isinstance(expr2, MatrixSymbol)
Z = ZeroMatrix(2, 2)
U = OneMatrix(2, 2)
assert HadamardProduct(Z, X).doit() == Z
assert HadamardProduct(U, X, X, U).doit() == HadamardPower(X, 2)
assert HadamardProduct(X, U, Y).doit() == HadamardProduct(X, Y)
assert HadamardProduct(X, Z, U, Y).doit() == Z
def test_hadamard():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
B = MatrixSymbol('B', m, n)
C = MatrixSymbol('C', m, p)
X = MatrixSymbol('X', m, m)
I = Identity(m)
with raises(TypeError):
hadamard_product()
assert hadamard_product(A) == A
assert isinstance(hadamard_product(A, B), HadamardProduct)
assert hadamard_product(A, B).doit() == hadamard_product(A, B)
with raises(ShapeError):
hadamard_product(A, C)
hadamard_product(A, I)
assert hadamard_product(X, I) == X
assert isinstance(hadamard_product(X, I), MatrixSymbol)
a = MatrixSymbol("a", k, 1)
expr = MatAdd(ZeroMatrix(k, 1), OneMatrix(k, 1))
expr = HadamardProduct(expr, a)
assert expr.doit() == a
def test_hadamard_product_with_explicit_mat():
A = MatrixSymbol("A", 3, 3).as_explicit()
B = MatrixSymbol("B", 3, 3).as_explicit()
X = MatrixSymbol("X", 3, 3)
expr = hadamard_product(A, B)
ret = Matrix([i*j for i, j in zip(A, B)]).reshape(3, 3)
assert expr == ret
expr = hadamard_product(A, X, B)
assert expr == HadamardProduct(ret, X)
def test_hadamard_power():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
assert hadamard_power(A, 1) == A
assert isinstance(hadamard_power(A, 2), HadamardPower)
assert hadamard_power(A, n).T == hadamard_power(A.T, n)
assert hadamard_power(A, n)[0, 0] == A[0, 0]**n
assert hadamard_power(m, n) == m**n
raises(ValueError, lambda: hadamard_power(A, A))
def test_hadamard_power_explicit():
A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
a, b = symbols('a b')
assert HadamardPower(a, b) == a**b
assert HadamardPower(a, B).as_explicit() == \
Matrix([
[a**B[0, 0], a**B[0, 1]],
[a**B[1, 0], a**B[1, 1]]])
assert HadamardPower(A, b).as_explicit() == \
Matrix([
[A[0, 0]**b, A[0, 1]**b],
[A[1, 0]**b, A[1, 1]**b]])
assert HadamardPower(A, B).as_explicit() == \
Matrix([
[A[0, 0]**B[0, 0], A[0, 1]**B[0, 1]],
[A[1, 0]**B[1, 0], A[1, 1]**B[1, 1]]])
|
1d695e74e3ebdb12aadd7146af8e7c2dce5cd3081fd9ed0815ae23eb76df747d | import os
from tempfile import TemporaryDirectory
from sympy.concrete.summations import Sum
from sympy.core.numbers import (I, oo, pi)
from sympy.core.relational import Ne
from sympy.core.symbol import Symbol
from sympy.functions.elementary.exponential import (LambertW, exp, exp_polar, log)
from sympy.functions.elementary.miscellaneous import (real_root, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.functions.special.hyper import meijerg
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import And
from sympy.core.singleton import S
from sympy.core.sympify import sympify
from sympy.external import import_module
from sympy.plotting.plot import (
Plot, plot, plot_parametric, plot3d_parametric_line, plot3d,
plot3d_parametric_surface)
from sympy.plotting.plot import (
unset_show, plot_contour, PlotGrid, DefaultBackend, MatplotlibBackend,
TextBackend, BaseBackend)
from sympy.testing.pytest import skip, raises, warns, warns_deprecated_sympy
from sympy.utilities import lambdify as lambdify_
from sympy.utilities.exceptions import ignore_warnings
unset_show()
matplotlib = import_module(
'matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
class DummyBackendNotOk(BaseBackend):
""" Used to verify if users can create their own backends.
This backend is meant to raise NotImplementedError for methods `show`,
`save`, `close`.
"""
pass
class DummyBackendOk(BaseBackend):
""" Used to verify if users can create their own backends.
This backend is meant to pass all tests.
"""
def show(self):
pass
def save(self):
pass
def close(self):
pass
def test_plot_and_save_1():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
###
# Examples from the 'introduction' notebook
###
p = plot(x, legend=True, label='f1')
p = plot(x*sin(x), x*cos(x), label='f2')
p.extend(p)
p[0].line_color = lambda a: a
p[1].line_color = 'b'
p.title = 'Big title'
p.xlabel = 'the x axis'
p[1].label = 'straight line'
p.legend = True
p.aspect_ratio = (1, 1)
p.xlim = (-15, 20)
filename = 'test_basic_options_and_colors.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p.extend(plot(x + 1))
p.append(plot(x + 3, x**2)[1])
filename = 'test_plot_extend_append.png'
p.save(os.path.join(tmpdir, filename))
p[2] = plot(x**2, (x, -2, 3))
filename = 'test_plot_setitem.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(sin(x), (x, -2*pi, 4*pi))
filename = 'test_line_explicit.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(sin(x))
filename = 'test_line_default_range.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot((x**2, (x, -5, 5)), (x**3, (x, -3, 3)))
filename = 'test_line_multiple_range.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
raises(ValueError, lambda: plot(x, y))
#Piecewise plots
p = plot(Piecewise((1, x > 0), (0, True)), (x, -1, 1))
filename = 'test_plot_piecewise.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(Piecewise((x, x < 1), (x**2, True)), (x, -3, 3))
filename = 'test_plot_piecewise_2.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# test issue 7471
p1 = plot(x)
p2 = plot(3)
p1.extend(p2)
filename = 'test_horizontal_line.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# test issue 10925
f = Piecewise((-1, x < -1), (x, And(-1 <= x, x < 0)), \
(x**2, And(0 <= x, x < 1)), (x**3, x >= 1))
p = plot(f, (x, -3, 3))
filename = 'test_plot_piecewise_3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_2():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
#parametric 2d plots.
#Single plot with default range.
p = plot_parametric(sin(x), cos(x))
filename = 'test_parametric.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#Single plot with range.
p = plot_parametric(
sin(x), cos(x), (x, -5, 5), legend=True, label='parametric_plot')
filename = 'test_parametric_range.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#Multiple plots with same range.
p = plot_parametric((sin(x), cos(x)), (x, sin(x)))
filename = 'test_parametric_multiple.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#Multiple plots with different ranges.
p = plot_parametric(
(sin(x), cos(x), (x, -3, 3)), (x, sin(x), (x, -5, 5)))
filename = 'test_parametric_multiple_ranges.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#depth of recursion specified.
p = plot_parametric(x, sin(x), depth=13)
filename = 'test_recursion_depth.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#No adaptive sampling.
p = plot_parametric(cos(x), sin(x), adaptive=False, nb_of_points=500)
filename = 'test_adaptive.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
#3d parametric plots
p = plot3d_parametric_line(
sin(x), cos(x), x, legend=True, label='3d_parametric_plot')
filename = 'test_3d_line.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_line(
(sin(x), cos(x), x, (x, -5, 5)), (cos(x), sin(x), x, (x, -3, 3)))
filename = 'test_3d_line_multiple.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_line(sin(x), cos(x), x, nb_of_points=30)
filename = 'test_3d_line_points.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# 3d surface single plot.
p = plot3d(x * y)
filename = 'test_surface.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple 3D plots with same range.
p = plot3d(-x * y, x * y, (x, -5, 5))
filename = 'test_surface_multiple.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple 3D plots with different ranges.
p = plot3d(
(x * y, (x, -3, 3), (y, -3, 3)), (-x * y, (x, -3, 3), (y, -3, 3)))
filename = 'test_surface_multiple_ranges.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Single Parametric 3D plot
p = plot3d_parametric_surface(sin(x + y), cos(x - y), x - y)
filename = 'test_parametric_surface.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple Parametric 3D plots.
p = plot3d_parametric_surface(
(x*sin(z), x*cos(z), z, (x, -5, 5), (z, -5, 5)),
(sin(x + y), cos(x - y), x - y, (x, -5, 5), (y, -5, 5)))
filename = 'test_parametric_surface.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Single Contour plot.
p = plot_contour(sin(x)*sin(y), (x, -5, 5), (y, -5, 5))
filename = 'test_contour_plot.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple Contour plots with same range.
p = plot_contour(x**2 + y**2, x**3 + y**3, (x, -5, 5), (y, -5, 5))
filename = 'test_contour_plot.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# Multiple Contour plots with different range.
p = plot_contour(
(x**2 + y**2, (x, -5, 5), (y, -5, 5)),
(x**3 + y**3, (x, -3, 3), (y, -3, 3)))
filename = 'test_contour_plot.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_3():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
###
# Examples from the 'colors' notebook
###
p = plot(sin(x))
p[0].line_color = lambda a: a
filename = 'test_colors_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: b
filename = 'test_colors_line_arity2.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(x*sin(x), x*cos(x), (x, 0, 10))
p[0].line_color = lambda a: a
filename = 'test_colors_param_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: a
filename = 'test_colors_param_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: b
filename = 'test_colors_param_line_arity2b.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_line(sin(x) + 0.1*sin(x)*cos(7*x),
cos(x) + 0.1*cos(x)*cos(7*x),
0.1*sin(7*x),
(x, 0, 2*pi))
p[0].line_color = lambdify_(x, sin(4*x))
filename = 'test_colors_3d_line_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b: b
filename = 'test_colors_3d_line_arity2.png'
p.save(os.path.join(tmpdir, filename))
p[0].line_color = lambda a, b, c: c
filename = 'test_colors_3d_line_arity3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d(sin(x)*y, (x, 0, 6*pi), (y, -5, 5))
p[0].surface_color = lambda a: a
filename = 'test_colors_surface_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambda a, b: b
filename = 'test_colors_surface_arity2.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambda a, b, c: c
filename = 'test_colors_surface_arity3a.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambdify_((x, y, z), sqrt((x - 3*pi)**2 + y**2))
filename = 'test_colors_surface_arity3b.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot3d_parametric_surface(x * cos(4 * y), x * sin(4 * y), y,
(x, -1, 1), (y, -1, 1))
p[0].surface_color = lambda a: a
filename = 'test_colors_param_surf_arity1.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambda a, b: a*b
filename = 'test_colors_param_surf_arity2.png'
p.save(os.path.join(tmpdir, filename))
p[0].surface_color = lambdify_((x, y, z), sqrt(x**2 + y**2 + z**2))
filename = 'test_colors_param_surf_arity3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_4():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
###
# Examples from the 'advanced' notebook
###
# XXX: This raises the warning "The evaluation of the expression is
# problematic. We are trying a failback method that may still work. Please
# report this as a bug." It has to use the fallback because using evalf()
# is the only way to evaluate the integral. We should perhaps just remove
# that warning.
with TemporaryDirectory(prefix='sympy_') as tmpdir:
with warns(
UserWarning,
match="The evaluation of the expression is problematic",
test_stacklevel=False,
):
i = Integral(log((sin(x)**2 + 1)*sqrt(x**2 + 1)), (x, 0, y))
p = plot(i, (y, 1, 5))
filename = 'test_advanced_integral.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_5():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
s = Sum(1/x**y, (x, 1, oo))
p = plot(s, (y, 2, 10))
filename = 'test_advanced_inf_sum.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p = plot(Sum(1/x, (x, 1, y)), (y, 2, 10), show=False)
p[0].only_integers = True
p[0].steps = True
filename = 'test_advanced_fin_sum.png'
# XXX: This should be fixed in experimental_lambdify or by using
# ordinary lambdify so that it doesn't warn. The error results from
# passing an array of values as the integration limit.
#
# UserWarning: The evaluation of the expression is problematic. We are
# trying a failback method that may still work. Please report this as a
# bug.
with ignore_warnings(UserWarning):
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_plot_and_save_6():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
filename = 'test.png'
###
# Test expressions that can not be translated to np and generate complex
# results.
###
p = plot(sin(x) + I*cos(x))
p.save(os.path.join(tmpdir, filename))
with ignore_warnings(RuntimeWarning):
p = plot(sqrt(sqrt(-x)))
p.save(os.path.join(tmpdir, filename))
p = plot(LambertW(x))
p.save(os.path.join(tmpdir, filename))
p = plot(sqrt(LambertW(x)))
p.save(os.path.join(tmpdir, filename))
#Characteristic function of a StudentT distribution with nu=10
x1 = 5 * x**2 * exp_polar(-I*pi)/2
m1 = meijerg(((1 / 2,), ()), ((5, 0, 1 / 2), ()), x1)
x2 = 5*x**2 * exp_polar(I*pi)/2
m2 = meijerg(((1/2,), ()), ((5, 0, 1/2), ()), x2)
expr = (m1 + m2) / (48 * pi)
p = plot(expr, (x, 1e-6, 1e-2))
p.save(os.path.join(tmpdir, filename))
def test_plotgrid_and_save():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
y = Symbol('y')
with TemporaryDirectory(prefix='sympy_') as tmpdir:
p1 = plot(x)
p2 = plot_parametric((sin(x), cos(x)), (x, sin(x)), show=False)
p3 = plot_parametric(
cos(x), sin(x), adaptive=False, nb_of_points=500, show=False)
p4 = plot3d_parametric_line(sin(x), cos(x), x, show=False)
# symmetric grid
p = PlotGrid(2, 2, p1, p2, p3, p4)
filename = 'test_grid1.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
# grid size greater than the number of subplots
p = PlotGrid(3, 4, p1, p2, p3, p4)
filename = 'test_grid2.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
p5 = plot(cos(x),(x, -pi, pi), show=False)
p5[0].line_color = lambda a: a
p6 = plot(Piecewise((1, x > 0), (0, True)), (x, -1, 1), show=False)
p7 = plot_contour(
(x**2 + y**2, (x, -5, 5), (y, -5, 5)),
(x**3 + y**3, (x, -3, 3), (y, -3, 3)), show=False)
# unsymmetric grid (subplots in one line)
p = PlotGrid(1, 3, p5, p6, p7)
filename = 'test_grid3.png'
p.save(os.path.join(tmpdir, filename))
p._backend.close()
def test_append_issue_7140():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p1 = plot(x)
p2 = plot(x**2)
plot(x + 2)
# append a series
p2.append(p1[0])
assert len(p2._series) == 2
with raises(TypeError):
p1.append(p2)
with raises(TypeError):
p1.append(p2._series)
def test_issue_15265():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
eqn = sin(x)
p = plot(eqn, xlim=(-S.Pi, S.Pi), ylim=(-1, 1))
p._backend.close()
p = plot(eqn, xlim=(-1, 1), ylim=(-S.Pi, S.Pi))
p._backend.close()
p = plot(eqn, xlim=(-1, 1), ylim=(sympify('-3.14'), sympify('3.14')))
p._backend.close()
p = plot(eqn, xlim=(sympify('-3.14'), sympify('3.14')), ylim=(-1, 1))
p._backend.close()
raises(ValueError,
lambda: plot(eqn, xlim=(-S.ImaginaryUnit, 1), ylim=(-1, 1)))
raises(ValueError,
lambda: plot(eqn, xlim=(-1, 1), ylim=(-1, S.ImaginaryUnit)))
raises(ValueError,
lambda: plot(eqn, xlim=(S.NegativeInfinity, 1), ylim=(-1, 1)))
raises(ValueError,
lambda: plot(eqn, xlim=(-1, 1), ylim=(-1, S.Infinity)))
def test_empty_Plot():
if not matplotlib:
skip("Matplotlib not the default backend")
# No exception showing an empty plot
plot()
p = Plot()
p.show()
def test_issue_17405():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
f = x**0.3 - 10*x**3 + x**2
p = plot(f, (x, -10, 10), show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
# RuntimeWarning: invalid value encountered in double_scalars
with ignore_warnings(RuntimeWarning):
assert len(p[0].get_data()[0]) >= 30
def test_logplot_PR_16796():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(x, (x, .001, 100), xscale='log', show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
assert len(p[0].get_data()[0]) >= 30
assert p[0].end == 100.0
assert p[0].start == .001
def test_issue_16572():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(LambertW(x), show=False)
# Random number of segments, probably more than 50, but we want to see
# that there are segments generated, as opposed to when the bug was present
assert len(p[0].get_data()[0]) >= 30
def test_issue_11865():
if not matplotlib:
skip("Matplotlib not the default backend")
k = Symbol('k', integer=True)
f = Piecewise((-I*exp(I*pi*k)/k + I*exp(-I*pi*k)/k, Ne(k, 0)), (2*pi, True))
p = plot(f, show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
# and that there are no exceptions.
assert len(p[0].get_data()[0]) >= 30
def test_issue_11461():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(real_root((log(x/(x-2))), 3), show=False)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
# and that there are no exceptions.
assert len(p[0].get_data()[0]) >= 30
def test_issue_11764():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot_parametric(cos(x), sin(x), (x, 0, 2 * pi), aspect_ratio=(1,1), show=False)
assert p.aspect_ratio == (1, 1)
# Random number of segments, probably more than 100, but we want to see
# that there are segments generated, as opposed to when the bug was present
assert len(p[0].get_data()[0]) >= 30
def test_issue_13516():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
pm = plot(sin(x), backend="matplotlib", show=False)
assert pm.backend == MatplotlibBackend
assert len(pm[0].get_data()[0]) >= 30
pt = plot(sin(x), backend="text", show=False)
assert pt.backend == TextBackend
assert len(pt[0].get_data()[0]) >= 30
pd = plot(sin(x), backend="default", show=False)
assert pd.backend == DefaultBackend
assert len(pd[0].get_data()[0]) >= 30
p = plot(sin(x), show=False)
assert p.backend == DefaultBackend
assert len(p[0].get_data()[0]) >= 30
def test_plot_limits():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p = plot(x, x**2, (x, -10, 10))
backend = p._backend
xmin, xmax = backend.ax[0].get_xlim()
assert abs(xmin + 10) < 2
assert abs(xmax - 10) < 2
ymin, ymax = backend.ax[0].get_ylim()
assert abs(ymin + 10) < 10
assert abs(ymax - 100) < 10
def test_plot3d_parametric_line_limits():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
v1 = (2*cos(x), 2*sin(x), 2*x, (x, -5, 5))
v2 = (sin(x), cos(x), x, (x, -5, 5))
p = plot3d_parametric_line(v1, v2)
backend = p._backend
xmin, xmax = backend.ax[0].get_xlim()
assert abs(xmin + 2) < 1e-2
assert abs(xmax - 2) < 1e-2
ymin, ymax = backend.ax[0].get_ylim()
assert abs(ymin + 2) < 1e-2
assert abs(ymax - 2) < 1e-2
zmin, zmax = backend.ax[0].get_zlim()
assert abs(zmin + 10) < 1e-2
assert abs(zmax - 10) < 1e-2
p = plot3d_parametric_line(v2, v1)
backend = p._backend
xmin, xmax = backend.ax[0].get_xlim()
assert abs(xmin + 2) < 1e-2
assert abs(xmax - 2) < 1e-2
ymin, ymax = backend.ax[0].get_ylim()
assert abs(ymin + 2) < 1e-2
assert abs(ymax - 2) < 1e-2
zmin, zmax = backend.ax[0].get_zlim()
assert abs(zmin + 10) < 1e-2
assert abs(zmax - 10) < 1e-2
def test_plot_size():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
p1 = plot(sin(x), backend="matplotlib", size=(8, 4))
s1 = p1._backend.fig.get_size_inches()
assert (s1[0] == 8) and (s1[1] == 4)
p2 = plot(sin(x), backend="matplotlib", size=(5, 10))
s2 = p2._backend.fig.get_size_inches()
assert (s2[0] == 5) and (s2[1] == 10)
p3 = PlotGrid(2, 1, p1, p2, size=(6, 2))
s3 = p3._backend.fig.get_size_inches()
assert (s3[0] == 6) and (s3[1] == 2)
with raises(ValueError):
plot(sin(x), backend="matplotlib", size=(-1, 3))
def test_issue_20113():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
# verify the capability to use custom backends
with raises(TypeError):
plot(sin(x), backend=Plot, show=False)
p2 = plot(sin(x), backend=MatplotlibBackend, show=False)
assert p2.backend == MatplotlibBackend
assert len(p2[0].get_data()[0]) >= 30
p3 = plot(sin(x), backend=DummyBackendOk, show=False)
assert p3.backend == DummyBackendOk
assert len(p3[0].get_data()[0]) >= 30
# test for an improper coded backend
p4 = plot(sin(x), backend=DummyBackendNotOk, show=False)
assert p4.backend == DummyBackendNotOk
assert len(p4[0].get_data()[0]) >= 30
with raises(NotImplementedError):
p4.show()
with raises(NotImplementedError):
p4.save("test/path")
with raises(NotImplementedError):
p4._backend.close()
def test_custom_coloring():
x = Symbol('x')
y = Symbol('y')
plot(cos(x), line_color=lambda a: a)
plot(cos(x), line_color=1)
plot(cos(x), line_color="r")
plot_parametric(cos(x), sin(x), line_color=lambda a: a)
plot_parametric(cos(x), sin(x), line_color=1)
plot_parametric(cos(x), sin(x), line_color="r")
plot3d_parametric_line(cos(x), sin(x), x, line_color=lambda a: a)
plot3d_parametric_line(cos(x), sin(x), x, line_color=1)
plot3d_parametric_line(cos(x), sin(x), x, line_color="r")
plot3d_parametric_surface(cos(x + y), sin(x - y), x - y,
(x, -5, 5), (y, -5, 5),
surface_color=lambda a, b: a**2 + b**2)
plot3d_parametric_surface(cos(x + y), sin(x - y), x - y,
(x, -5, 5), (y, -5, 5),
surface_color=1)
plot3d_parametric_surface(cos(x + y), sin(x - y), x - y,
(x, -5, 5), (y, -5, 5),
surface_color="r")
plot3d(x*y, (x, -5, 5), (y, -5, 5),
surface_color=lambda a, b: a**2 + b**2)
plot3d(x*y, (x, -5, 5), (y, -5, 5), surface_color=1)
plot3d(x*y, (x, -5, 5), (y, -5, 5), surface_color="r")
def test_deprecated_get_segments():
if not matplotlib:
skip("Matplotlib not the default backend")
x = Symbol('x')
f = sin(x)
p = plot(f, (x, -10, 10), show=False)
with warns_deprecated_sympy():
p[0].get_segments()
|
56ead319e32351f04ba4b76832acc92bd81b71284d6fb7f53ab01c44dcabb5a8 | from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import sin
from sympy.plotting.textplot import textplot_str
from sympy.utilities.exceptions import ignore_warnings
def test_axes_alignment():
x = Symbol('x')
lines = [
' 1 | ..',
' | ... ',
' | .. ',
' | ... ',
' | ... ',
' | .. ',
' | ... ',
' | ... ',
' | .. ',
' | ... ',
' 0 |--------------------------...--------------------------',
' | ... ',
' | .. ',
' | ... ',
' | ... ',
' | .. ',
' | ... ',
' | ... ',
' | .. ',
' | ... ',
' -1 |_______________________________________________________',
' -1 0 1'
]
assert lines == list(textplot_str(x, -1, 1))
lines = [
' 1 | ..',
' | .... ',
' | ... ',
' | ... ',
' | .... ',
' | ... ',
' | ... ',
' | .... ',
' 0 |--------------------------...--------------------------',
' | .... ',
' | ... ',
' | ... ',
' | .... ',
' | ... ',
' | ... ',
' | .... ',
' -1 |_______________________________________________________',
' -1 0 1'
]
assert lines == list(textplot_str(x, -1, 1, H=17))
def test_singularity():
x = Symbol('x')
lines = [
' 54 | . ',
' | ',
' | ',
' | ',
' | ',' | ',
' | ',
' | ',
' | ',
' | ',
' 27.5 |--.----------------------------------------------------',
' | ',
' | ',
' | ',
' | . ',
' | \\ ',
' | \\ ',
' | .. ',
' | ... ',
' | ............. ',
' 1 |_______________________________________________________',
' 0 0.5 1'
]
assert lines == list(textplot_str(1/x, 0, 1))
lines = [
' 0 | ......',
' | ........ ',
' | ........ ',
' | ...... ',
' | ..... ',
' | .... ',
' | ... ',
' | .. ',
' | ... ',
' | / ',
' -2 |-------..----------------------------------------------',
' | / ',
' | / ',
' | / ',
' | . ',
' | ',
' | . ',
' | ',
' | ',
' | ',
' -4 |_______________________________________________________',
' 0 0.5 1'
]
# RuntimeWarning: divide by zero encountered in log
with ignore_warnings(RuntimeWarning):
assert lines == list(textplot_str(log(x), 0, 1))
def test_sinc():
x = Symbol('x')
lines = [
' 1 | . . ',
' | . . ',
' | ',
' | . . ',
' | ',
' | . . ',
' | ',
' | ',
' | . . ',
' | ',
' 0.4 |-------------------------------------------------------',
' | . . ',
' | ',
' | . . ',
' | ',
' | ..... ..... ',
' | .. \\ . . / .. ',
' | / \\ / \\ ',
' |/ \\ . . / \\',
' | \\ / \\ / ',
' -0.2 |_______________________________________________________',
' -10 0 10'
]
# RuntimeWarning: invalid value encountered in double_scalars
with ignore_warnings(RuntimeWarning):
assert lines == list(textplot_str(sin(x)/x, -10, 10))
def test_imaginary():
x = Symbol('x')
lines = [
' 1 | ..',
' | .. ',
' | ... ',
' | .. ',
' | .. ',
' | .. ',
' | .. ',
' | .. ',
' | .. ',
' | / ',
' 0.5 |----------------------------------/--------------------',
' | .. ',
' | / ',
' | . ',
' | ',
' | . ',
' | . ',
' | ',
' | ',
' | ',
' 0 |_______________________________________________________',
' -1 0 1'
]
# RuntimeWarning: invalid value encountered in sqrt
with ignore_warnings(RuntimeWarning):
assert list(textplot_str(sqrt(x), -1, 1)) == lines
lines = [
' 1 | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' 0 |-------------------------------------------------------',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' | ',
' -1 |_______________________________________________________',
' -1 0 1'
]
assert list(textplot_str(S.ImaginaryUnit, -1, 1)) == lines
|
eadcc0e81a06e8da7e54e7866e56f0145158e40027b3f42712124f85af01a539 | #!/usr/bin/env python
# -*- coding: utf-8 -*-
"""
A tool to generate AUTHORS. We started tracking authors before moving to git,
so we have to do some manual rearrangement of the git history authors in order
to get the order in AUTHORS. bin/mailmap_check.py should be run before
committing the results.
See here for instructions on using this script:
https://github.com/sympy/sympy/wiki/Development-workflow#update-mailmap
"""
from __future__ import unicode_literals
from __future__ import print_function
import sys
import os
if sys.version_info < (3, 8):
sys.exit("This script requires Python 3.8 or newer")
from pathlib import Path
from subprocess import run, PIPE
from collections import OrderedDict, defaultdict
from argparse import ArgumentParser
def sympy_dir():
return Path(__file__).resolve().parent.parent
# put sympy on the path
sys.path.insert(0, str(sympy_dir()))
import sympy
from sympy.utilities.misc import filldedent
from sympy.external.importtools import version_tuple
def main(*args):
parser = ArgumentParser(description='Update the .mailmap file')
parser.add_argument('--skip-last-commit', action='store_true',
help=filldedent("""
Do not check metadata from the most recent commit. This is used
when the script runs in CI to ignore the merge commit that is
implicitly created by github."""))
parser.add_argument('--update-authors', action='store_true',
help=filldedent("""
Also updates the AUTHORS file. DO NOT use this option as part of a
pull request. The AUTHORS file will be updated later at the time a
new version of SymPy is released."""))
args = parser.parse_args(args)
if not check_git_version():
return 1
# find who git knows ahout
try:
git_people = get_authors_from_git()
if args.skip_last_commit:
git_people_skip = get_authors_from_git(skip_last=True)
else:
git_people_skip = git_people
except AssertionError as msg:
print(red(msg))
return 1
lines_mailmap = read_lines(mailmap_path())
def key(line):
# return lower case first address on line or
# raise an error if not an entry
if '#' in line:
line = line.split('#')[0]
L, R = line.count("<"), line.count(">")
assert L == R and L in (1, 2)
return line.split(">", 1)[0].split("<")[1].lower()
who = OrderedDict()
for i, line in enumerate(lines_mailmap):
try:
who.setdefault(key(line), []).append(line)
except AssertionError:
who[i] = [line]
problems = False
missing = False
ambiguous = False
dups = defaultdict(list)
#
# Here we use the git people with the most recent commit skipped. This
# means we don't need to add .mailmap entries for the temporary merge
# commit created in CI on a PR.
#
for person in git_people_skip:
email = key(person)
dups[email].append(person)
if email not in who:
print(red("This author is not included in the .mailmap file:"))
print(person)
missing = True
elif not any(p.startswith(person) for p in who[email]):
print(red("Ambiguous names in .mailmap"))
print(red("This email address appears for multiple entries:"))
print('Person:', person)
print('Mailmap entries:')
for line in who[email]:
print(line)
ambiguous = True
if missing:
print(red(filldedent("""
The .mailmap file needs to be updated because there are commits with
unrecognised author/email metadata.
""")))
problems = True
if ambiguous:
print(red(filldedent("""
Lines should be added to .mailmap to indicate the correct name and
email aliases for all commits.
""")))
problems = True
for email, commitauthors in dups.items():
if len(commitauthors) > 2:
print(red(filldedent("""
The following commits are recorded with different metadata but the
same/ambiguous email address. The .mailmap file will need to be
updated.""")))
for author in commitauthors:
print(author)
problems = True
lines_mailmap_sorted = sort_lines_mailmap(lines_mailmap)
write_lines(mailmap_path(), lines_mailmap_sorted)
if lines_mailmap_sorted != lines_mailmap:
problems = True
print(red("The mailmap file was reordered"))
# Check if changes to AUTHORS file are also needed
#
# Here we don't skip the last commit. We need authors from the most recent
# commit if the AUTHORS file was updated.
lines_authors = make_authors_file_lines(git_people)
old_lines_authors = read_lines(authors_path())
for person in old_lines_authors[8:]:
if person not in git_people:
print(red("This author is in the AUTHORS file but not .mailmap:"))
print(person)
problems = True
if problems:
print(red(filldedent("""
For instructions on updating the .mailmap file see:
https://github.com/sympy/sympy/wiki/Development-workflow#add-your-name-and-email-address-to-the-mailmap-file""",
break_on_hyphens=False, break_long_words=False)))
else:
print(green("No changes needed in .mailmap"))
# Actually update the AUTHORS file (if --update-authors was passed)
authors_changed = update_authors_file(lines_authors, old_lines_authors, args.update_authors)
return int(problems) + int(authors_changed)
def update_authors_file(lines, old_lines, update_yesno):
if old_lines == lines:
print(green('No changes needed in AUTHORS.'))
return 0
# Actually write changes to the file?
if update_yesno:
write_lines(authors_path(), lines)
print(red("Changes were made in the authors file"))
# check for new additions
new_authors = []
for i in sorted(set(lines) - set(old_lines)):
try:
author_name(i)
new_authors.append(i)
except AssertionError:
continue
if new_authors:
if update_yesno:
print(yellow("The following authors were added to AUTHORS."))
else:
print(green(filldedent("""
The following authors will be added to the AUTHORS file at the
time of the next SymPy release.""")))
print()
for i in sorted(new_authors, key=lambda x: x.lower()):
print('\t%s' % i)
if new_authors and update_yesno:
return 1
else:
return 0
def check_git_version():
# check git version
minimal = '1.8.4.2'
git_ver = run(['git', '--version'], stdout=PIPE, encoding='utf-8').stdout[12:]
if version_tuple(git_ver) < version_tuple(minimal):
print(yellow("Please use a git version >= %s" % minimal))
return False
else:
return True
def authors_path():
return sympy_dir() / 'AUTHORS'
def mailmap_path():
return sympy_dir() / '.mailmap'
def red(text):
return "\033[31m%s\033[0m" % text
def yellow(text):
return "\033[33m%s\033[0m" % text
def green(text):
return "\033[32m%s\033[0m" % text
def author_name(line):
assert line.count("<") == line.count(">") == 1
assert line.endswith(">")
return line.split("<", 1)[0].strip()
def get_authors_from_git(skip_last=False):
git_command = ["git", "log", "--topo-order", "--reverse", "--format=%aN <%aE>"]
if skip_last:
# Skip the most recent commit. Used to ignore the merge commit created
# when this script runs in CI. We use HEAD^2 rather than HEAD^1 to
# select the parent commit that is part of the PR rather than the
# parent commit that was the previous tip of master.
git_command.append("HEAD^2")
git_people = run(git_command, stdout=PIPE, encoding='utf-8').stdout.strip().split("\n")
# remove duplicates, keeping the original order
git_people = list(OrderedDict.fromkeys(git_people))
# Do the few changes necessary in order to reproduce AUTHORS:
def move(l, i1, i2, who):
x = l.pop(i1)
# this will fail if the .mailmap is not right
assert who == author_name(x), \
'%s was not found at line %i' % (who, i1)
l.insert(i2, x)
move(git_people, 2, 0, 'Ondřej Čertík')
move(git_people, 42, 1, 'Fabian Pedregosa')
move(git_people, 22, 2, 'Jurjen N.E. Bos')
git_people.insert(4, "*Marc-Etienne M.Leveille <[email protected]>")
move(git_people, 10, 5, 'Brian Jorgensen')
git_people.insert(11, "*Ulrich Hecht <[email protected]>")
# this will fail if the .mailmap is not right
assert 'Kirill Smelkov' == author_name(git_people.pop(12)
), 'Kirill Smelkov was not found at line 12'
move(git_people, 12, 32, 'Sebastian Krämer')
move(git_people, 227, 35, 'Case Van Horsen')
git_people.insert(43, "*Dan <[email protected]>")
move(git_people, 57, 59, 'Aaron Meurer')
move(git_people, 58, 57, 'Andrew Docherty')
move(git_people, 67, 66, 'Chris Smith')
move(git_people, 79, 76, 'Kevin Goodsell')
git_people.insert(84, "*Chu-Ching Huang <[email protected]>")
move(git_people, 93, 92, 'James Pearson')
# this will fail if the .mailmap is not right
assert 'Sergey B Kirpichev' == author_name(git_people.pop(226)
), 'Sergey B Kirpichev was not found at line 226.'
index = git_people.index(
"azure-pipelines[bot] " +
"<azure-pipelines[bot]@users.noreply.github.com>")
git_people.pop(index)
index = git_people.index(
"whitesource-bolt-for-github[bot] " +
"<whitesource-bolt-for-github[bot]@users.noreply.github.com>")
git_people.pop(index)
return git_people
def make_authors_file_lines(git_people):
# define new lines for the file
header = filldedent("""
All people who contributed to SymPy by sending at least a patch or
more (in the order of the date of their first contribution), except
those who explicitly didn't want to be mentioned. People with a * next
to their names are not found in the metadata of the git history. This
file is generated automatically by running `./bin/authors_update.py`.
""").lstrip()
header_extra = "There are a total of %d authors." % len(git_people)
lines = header.splitlines()
lines.append('')
lines.append(header_extra)
lines.append('')
lines.extend(git_people)
return lines
def sort_lines_mailmap(lines):
for n, line in enumerate(lines):
if not line.startswith('#'):
header_end = n
break
header = lines[:header_end]
mailmap_lines = lines[header_end:]
return header + sorted(mailmap_lines)
def read_lines(path):
with open(path, 'r', encoding='utf-8') as fin:
return [line.strip() for line in fin.readlines()]
def write_lines(path, lines):
with open(path, 'w', encoding='utf-8') as fout:
fout.write('\n'.join(lines))
fout.write('\n')
if __name__ == "__main__":
import sys
sys.exit(main(*sys.argv[1:]))
|
5cef572aab0816cee4c9a838917981a9d3d4d0b84a9adc09d6b1b0b4994b928b | """
SymPy is a Python library for symbolic mathematics. It aims to become a
full-featured computer algebra system (CAS) while keeping the code as simple
as possible in order to be comprehensible and easily extensible. SymPy is
written entirely in Python. It depends on mpmath, and other external libraries
may be optionally for things like plotting support.
See the webpage for more information and documentation:
https://sympy.org
"""
import sys
if sys.version_info < (3, 8):
raise ImportError("Python version 3.8 or above is required for SymPy.")
del sys
try:
import mpmath
except ImportError:
raise ImportError("SymPy now depends on mpmath as an external library. "
"See https://docs.sympy.org/latest/install.html#mpmath for more information.")
del mpmath
from sympy.release import __version__
if 'dev' in __version__:
def enable_warnings():
import warnings
warnings.filterwarnings('default', '.*', DeprecationWarning, module='sympy.*')
del warnings
enable_warnings()
del enable_warnings
def __sympy_debug():
# helper function so we don't import os globally
import os
debug_str = os.getenv('SYMPY_DEBUG', 'False')
if debug_str in ('True', 'False'):
return eval(debug_str)
else:
raise RuntimeError("unrecognized value for SYMPY_DEBUG: %s" %
debug_str)
SYMPY_DEBUG = __sympy_debug() # type: bool
from .core import (sympify, SympifyError, cacheit, Basic, Atom,
preorder_traversal, S, Expr, AtomicExpr, UnevaluatedExpr, Symbol,
Wild, Dummy, symbols, var, Number, Float, Rational, Integer,
NumberSymbol, RealNumber, igcd, ilcm, seterr, E, I, nan, oo, pi, zoo,
AlgebraicNumber, comp, mod_inverse, Pow, integer_nthroot, integer_log,
Mul, prod, Add, Mod, Rel, Eq, Ne, Lt, Le, Gt, Ge, Equality,
GreaterThan, LessThan, Unequality, StrictGreaterThan, StrictLessThan,
vectorize, Lambda, WildFunction, Derivative, diff, FunctionClass,
Function, Subs, expand, PoleError, count_ops, expand_mul, expand_log,
expand_func, expand_trig, expand_complex, expand_multinomial, nfloat,
expand_power_base, expand_power_exp, arity, PrecisionExhausted, N,
evalf, Tuple, Dict, gcd_terms, factor_terms, factor_nc, evaluate,
Catalan, EulerGamma, GoldenRatio, TribonacciConstant, bottom_up, use,
postorder_traversal, default_sort_key, ordered)
from .logic import (to_cnf, to_dnf, to_nnf, And, Or, Not, Xor, Nand, Nor,
Implies, Equivalent, ITE, POSform, SOPform, simplify_logic, bool_map,
true, false, satisfiable)
from .assumptions import (AppliedPredicate, Predicate, AssumptionsContext,
assuming, Q, ask, register_handler, remove_handler, refine)
from .polys import (Poly, PurePoly, poly_from_expr, parallel_poly_from_expr,
degree, total_degree, degree_list, LC, LM, LT, pdiv, prem, pquo,
pexquo, div, rem, quo, exquo, half_gcdex, gcdex, invert,
subresultants, resultant, discriminant, cofactors, gcd_list, gcd,
lcm_list, lcm, terms_gcd, trunc, monic, content, primitive, compose,
decompose, sturm, gff_list, gff, sqf_norm, sqf_part, sqf_list, sqf,
factor_list, factor, intervals, refine_root, count_roots, real_roots,
nroots, ground_roots, nth_power_roots_poly, cancel, reduced, groebner,
is_zero_dimensional, GroebnerBasis, poly, symmetrize, horner,
interpolate, rational_interpolate, viete, together,
BasePolynomialError, ExactQuotientFailed, PolynomialDivisionFailed,
OperationNotSupported, HeuristicGCDFailed, HomomorphismFailed,
IsomorphismFailed, ExtraneousFactors, EvaluationFailed,
RefinementFailed, CoercionFailed, NotInvertible, NotReversible,
NotAlgebraic, DomainError, PolynomialError, UnificationFailed,
GeneratorsError, GeneratorsNeeded, ComputationFailed,
UnivariatePolynomialError, MultivariatePolynomialError,
PolificationFailed, OptionError, FlagError, minpoly,
minimal_polynomial, primitive_element, field_isomorphism,
to_number_field, isolate, round_two, prime_decomp, prime_valuation,
itermonomials, Monomial, lex, grlex,
grevlex, ilex, igrlex, igrevlex, CRootOf, rootof, RootOf,
ComplexRootOf, RootSum, roots, Domain, FiniteField, IntegerRing,
RationalField, RealField, ComplexField, PythonFiniteField,
GMPYFiniteField, PythonIntegerRing, GMPYIntegerRing, PythonRational,
GMPYRationalField, AlgebraicField, PolynomialRing, FractionField,
ExpressionDomain, FF_python, FF_gmpy, ZZ_python, ZZ_gmpy, QQ_python,
QQ_gmpy, GF, FF, ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX, EXRAW,
construct_domain, swinnerton_dyer_poly, cyclotomic_poly,
symmetric_poly, random_poly, interpolating_poly, jacobi_poly,
chebyshevt_poly, chebyshevu_poly, hermite_poly, legendre_poly,
laguerre_poly, apart, apart_list, assemble_partfrac_list, Options,
ring, xring, vring, sring, field, xfield, vfield, sfield)
from .series import (Order, O, limit, Limit, gruntz, series, approximants,
residue, EmptySequence, SeqPer, SeqFormula, sequence, SeqAdd, SeqMul,
fourier_series, fps, difference_delta, limit_seq)
from .functions import (factorial, factorial2, rf, ff, binomial,
RisingFactorial, FallingFactorial, subfactorial, carmichael,
fibonacci, lucas, motzkin, tribonacci, harmonic, bernoulli, bell, euler,
catalan, genocchi, partition, sqrt, root, Min, Max, Id, real_root, Rem,
cbrt, re, im, sign, Abs, conjugate, arg, polar_lift,
periodic_argument, unbranched_argument, principal_branch, transpose,
adjoint, polarify, unpolarify, sin, cos, tan, sec, csc, cot, sinc,
asin, acos, atan, asec, acsc, acot, atan2, exp_polar, exp, ln, log,
LambertW, sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh,
acoth, asech, acsch, floor, ceiling, frac, Piecewise, piecewise_fold,
piecewise_exclusive, erf, erfc, erfi, erf2, erfinv, erfcinv, erf2inv,
Ei, expint, E1, li, Li, Si, Ci, Shi, Chi, fresnels, fresnelc, gamma,
lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma,
multigamma, dirichlet_eta, zeta, lerchphi, polylog, stieltjes, Eijk,
LeviCivita, KroneckerDelta, SingularityFunction, DiracDelta, Heaviside,
bspline_basis, bspline_basis_set, interpolating_spline, besselj,
bessely, besseli, besselk, hankel1, hankel2, jn, yn, jn_zeros, hn1,
hn2, airyai, airybi, airyaiprime, airybiprime, marcumq, hyper,
meijerg, appellf1, legendre, assoc_legendre, hermite, chebyshevt,
chebyshevu, chebyshevu_root, chebyshevt_root, laguerre,
assoc_laguerre, gegenbauer, jacobi, jacobi_normalized, Ynm, Ynm_c,
Znm, elliptic_k, elliptic_f, elliptic_e, elliptic_pi, beta, mathieus,
mathieuc, mathieusprime, mathieucprime, riemann_xi, betainc, betainc_regularized)
from .ntheory import (nextprime, prevprime, prime, primepi, primerange,
randprime, Sieve, sieve, primorial, cycle_length, composite,
compositepi, isprime, divisors, proper_divisors, factorint,
multiplicity, perfect_power, pollard_pm1, pollard_rho, primefactors,
totient, trailing, divisor_count, proper_divisor_count, divisor_sigma,
factorrat, reduced_totient, primenu, primeomega,
mersenne_prime_exponent, is_perfect, is_mersenne_prime, is_abundant,
is_deficient, is_amicable, abundance, npartitions, is_primitive_root,
is_quad_residue, legendre_symbol, jacobi_symbol, n_order, sqrt_mod,
quadratic_residues, primitive_root, nthroot_mod, is_nthpow_residue,
sqrt_mod_iter, mobius, discrete_log, quadratic_congruence,
binomial_coefficients, binomial_coefficients_list,
multinomial_coefficients, continued_fraction_periodic,
continued_fraction_iterator, continued_fraction_reduce,
continued_fraction_convergents, continued_fraction, egyptian_fraction)
from .concrete import product, Product, summation, Sum
from .discrete import (fft, ifft, ntt, intt, fwht, ifwht, mobius_transform,
inverse_mobius_transform, convolution, covering_product,
intersecting_product)
from .simplify import (simplify, hypersimp, hypersimilar, logcombine,
separatevars, posify, besselsimp, kroneckersimp, signsimp,
nsimplify, FU, fu, sqrtdenest, cse, epath, EPath, hyperexpand,
collect, rcollect, radsimp, collect_const, fraction, numer, denom,
trigsimp, exptrigsimp, powsimp, powdenest, combsimp, gammasimp,
ratsimp, ratsimpmodprime)
from .sets import (Set, Interval, Union, EmptySet, FiniteSet, ProductSet,
Intersection, DisjointUnion, imageset, Complement, SymmetricDifference, ImageSet,
Range, ComplexRegion, Complexes, Reals, Contains, ConditionSet, Ordinal,
OmegaPower, ord0, PowerSet, Naturals, Naturals0, UniversalSet,
Integers, Rationals)
from .solvers import (solve, solve_linear_system, solve_linear_system_LU,
solve_undetermined_coeffs, nsolve, solve_linear, checksol, det_quick,
inv_quick, check_assumptions, failing_assumptions, diophantine,
rsolve, rsolve_poly, rsolve_ratio, rsolve_hyper, checkodesol,
classify_ode, dsolve, homogeneous_order, solve_poly_system,
solve_triangulated, pde_separate, pde_separate_add, pde_separate_mul,
pdsolve, classify_pde, checkpdesol, ode_order, reduce_inequalities,
reduce_abs_inequality, reduce_abs_inequalities, solve_poly_inequality,
solve_rational_inequalities, solve_univariate_inequality, decompogen,
solveset, linsolve, linear_eq_to_matrix, nonlinsolve, substitution)
from .matrices import (ShapeError, NonSquareMatrixError, GramSchmidt,
casoratian, diag, eye, hessian, jordan_cell, list2numpy, matrix2numpy,
matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2,
rot_axis3, symarray, wronskian, zeros, MutableDenseMatrix,
DeferredVector, MatrixBase, Matrix, MutableMatrix,
MutableSparseMatrix, banded, ImmutableDenseMatrix,
ImmutableSparseMatrix, ImmutableMatrix, SparseMatrix, MatrixSlice,
BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity, Inverse,
MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace, Transpose,
ZeroMatrix, OneMatrix, blockcut, block_collapse, matrix_symbols,
Adjoint, hadamard_product, HadamardProduct, HadamardPower,
Determinant, det, diagonalize_vector, DiagMatrix, DiagonalMatrix,
DiagonalOf, trace, DotProduct, kronecker_product, KroneckerProduct,
PermutationMatrix, MatrixPermute, Permanent, per)
from .geometry import (Point, Point2D, Point3D, Line, Ray, Segment, Line2D,
Segment2D, Ray2D, Line3D, Segment3D, Ray3D, Plane, Ellipse, Circle,
Polygon, RegularPolygon, Triangle, rad, deg, are_similar, centroid,
convex_hull, idiff, intersection, closest_points, farthest_points,
GeometryError, Curve, Parabola)
from .utilities import (flatten, group, take, subsets, variations,
numbered_symbols, cartes, capture, dict_merge, prefixes, postfixes,
sift, topological_sort, unflatten, has_dups, has_variety, reshape,
rotations, filldedent, lambdify, source,
threaded, xthreaded, public, memoize_property, timed)
from .integrals import (integrate, Integral, line_integrate, mellin_transform,
inverse_mellin_transform, MellinTransform, InverseMellinTransform,
laplace_transform, inverse_laplace_transform, LaplaceTransform,
InverseLaplaceTransform, fourier_transform, inverse_fourier_transform,
FourierTransform, InverseFourierTransform, sine_transform,
inverse_sine_transform, SineTransform, InverseSineTransform,
cosine_transform, inverse_cosine_transform, CosineTransform,
InverseCosineTransform, hankel_transform, inverse_hankel_transform,
HankelTransform, InverseHankelTransform, singularityintegrate)
from .tensor import (IndexedBase, Idx, Indexed, get_contraction_structure,
get_indices, shape, MutableDenseNDimArray, ImmutableDenseNDimArray,
MutableSparseNDimArray, ImmutableSparseNDimArray, NDimArray,
tensorproduct, tensorcontraction, tensordiagonal, derive_by_array,
permutedims, Array, DenseNDimArray, SparseNDimArray)
from .parsing import parse_expr
from .calculus import (euler_equations, singularities, is_increasing,
is_strictly_increasing, is_decreasing, is_strictly_decreasing,
is_monotonic, finite_diff_weights, apply_finite_diff,
differentiate_finite, periodicity, not_empty_in, AccumBounds,
is_convex, stationary_points, minimum, maximum)
from .algebras import Quaternion
from .printing import (pager_print, pretty, pretty_print, pprint,
pprint_use_unicode, pprint_try_use_unicode, latex, print_latex,
multiline_latex, mathml, print_mathml, python, print_python, pycode,
ccode, print_ccode, glsl_code, print_glsl, cxxcode, fcode,
print_fcode, rcode, print_rcode, jscode, print_jscode, julia_code,
mathematica_code, octave_code, rust_code, print_gtk, preview, srepr,
print_tree, StrPrinter, sstr, sstrrepr, TableForm, dotprint,
maple_code, print_maple_code)
from .testing import test, doctest
# This module causes conflicts with other modules:
# from .stats import *
# Adds about .04-.05 seconds of import time
# from combinatorics import *
# This module is slow to import:
#from physics import units
from .plotting import plot, textplot, plot_backends, plot_implicit, plot_parametric
from .interactive import init_session, init_printing, interactive_traversal
evalf._create_evalf_table()
__all__ = [
'__version__',
# sympy.core
'sympify', 'SympifyError', 'cacheit', 'Basic', 'Atom',
'preorder_traversal', 'S', 'Expr', 'AtomicExpr', 'UnevaluatedExpr',
'Symbol', 'Wild', 'Dummy', 'symbols', 'var', 'Number', 'Float',
'Rational', 'Integer', 'NumberSymbol', 'RealNumber', 'igcd', 'ilcm',
'seterr', 'E', 'I', 'nan', 'oo', 'pi', 'zoo', 'AlgebraicNumber', 'comp',
'mod_inverse', 'Pow', 'integer_nthroot', 'integer_log', 'Mul', 'prod',
'Add', 'Mod', 'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge', 'Equality',
'GreaterThan', 'LessThan', 'Unequality', 'StrictGreaterThan',
'StrictLessThan', 'vectorize', 'Lambda', 'WildFunction', 'Derivative',
'diff', 'FunctionClass', 'Function', 'Subs', 'expand', 'PoleError',
'count_ops', 'expand_mul', 'expand_log', 'expand_func', 'expand_trig',
'expand_complex', 'expand_multinomial', 'nfloat', 'expand_power_base',
'expand_power_exp', 'arity', 'PrecisionExhausted', 'N', 'evalf', 'Tuple',
'Dict', 'gcd_terms', 'factor_terms', 'factor_nc', 'evaluate', 'Catalan',
'EulerGamma', 'GoldenRatio', 'TribonacciConstant', 'bottom_up', 'use',
'postorder_traversal', 'default_sort_key', 'ordered',
# sympy.logic
'to_cnf', 'to_dnf', 'to_nnf', 'And', 'Or', 'Not', 'Xor', 'Nand', 'Nor',
'Implies', 'Equivalent', 'ITE', 'POSform', 'SOPform', 'simplify_logic',
'bool_map', 'true', 'false', 'satisfiable',
# sympy.assumptions
'AppliedPredicate', 'Predicate', 'AssumptionsContext', 'assuming', 'Q',
'ask', 'register_handler', 'remove_handler', 'refine',
# sympy.polys
'Poly', 'PurePoly', 'poly_from_expr', 'parallel_poly_from_expr', 'degree',
'total_degree', 'degree_list', 'LC', 'LM', 'LT', 'pdiv', 'prem', 'pquo',
'pexquo', 'div', 'rem', 'quo', 'exquo', 'half_gcdex', 'gcdex', 'invert',
'subresultants', 'resultant', 'discriminant', 'cofactors', 'gcd_list',
'gcd', 'lcm_list', 'lcm', 'terms_gcd', 'trunc', 'monic', 'content',
'primitive', 'compose', 'decompose', 'sturm', 'gff_list', 'gff',
'sqf_norm', 'sqf_part', 'sqf_list', 'sqf', 'factor_list', 'factor',
'intervals', 'refine_root', 'count_roots', 'real_roots', 'nroots',
'ground_roots', 'nth_power_roots_poly', 'cancel', 'reduced', 'groebner',
'is_zero_dimensional', 'GroebnerBasis', 'poly', 'symmetrize', 'horner',
'interpolate', 'rational_interpolate', 'viete', 'together',
'BasePolynomialError', 'ExactQuotientFailed', 'PolynomialDivisionFailed',
'OperationNotSupported', 'HeuristicGCDFailed', 'HomomorphismFailed',
'IsomorphismFailed', 'ExtraneousFactors', 'EvaluationFailed',
'RefinementFailed', 'CoercionFailed', 'NotInvertible', 'NotReversible',
'NotAlgebraic', 'DomainError', 'PolynomialError', 'UnificationFailed',
'GeneratorsError', 'GeneratorsNeeded', 'ComputationFailed',
'UnivariatePolynomialError', 'MultivariatePolynomialError',
'PolificationFailed', 'OptionError', 'FlagError', 'minpoly',
'minimal_polynomial', 'primitive_element', 'field_isomorphism',
'to_number_field', 'isolate', 'round_two', 'prime_decomp',
'prime_valuation', 'itermonomials', 'Monomial', 'lex', 'grlex',
'grevlex', 'ilex', 'igrlex', 'igrevlex', 'CRootOf', 'rootof', 'RootOf',
'ComplexRootOf', 'RootSum', 'roots', 'Domain', 'FiniteField',
'IntegerRing', 'RationalField', 'RealField', 'ComplexField',
'PythonFiniteField', 'GMPYFiniteField', 'PythonIntegerRing',
'GMPYIntegerRing', 'PythonRational', 'GMPYRationalField',
'AlgebraicField', 'PolynomialRing', 'FractionField', 'ExpressionDomain',
'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy', 'QQ_python', 'QQ_gmpy',
'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR', 'CC', 'EX', 'EXRAW',
'construct_domain', 'swinnerton_dyer_poly', 'cyclotomic_poly',
'symmetric_poly', 'random_poly', 'interpolating_poly', 'jacobi_poly',
'chebyshevt_poly', 'chebyshevu_poly', 'hermite_poly', 'legendre_poly',
'laguerre_poly', 'apart', 'apart_list', 'assemble_partfrac_list',
'Options', 'ring', 'xring', 'vring', 'sring', 'field', 'xfield', 'vfield',
'sfield',
# sympy.series
'Order', 'O', 'limit', 'Limit', 'gruntz', 'series', 'approximants',
'residue', 'EmptySequence', 'SeqPer', 'SeqFormula', 'sequence', 'SeqAdd',
'SeqMul', 'fourier_series', 'fps', 'difference_delta', 'limit_seq',
# sympy.functions
'factorial', 'factorial2', 'rf', 'ff', 'binomial', 'RisingFactorial',
'FallingFactorial', 'subfactorial', 'carmichael', 'fibonacci', 'lucas',
'motzkin', 'tribonacci', 'harmonic', 'bernoulli', 'bell', 'euler', 'catalan',
'genocchi', 'partition', 'sqrt', 'root', 'Min', 'Max', 'Id', 'real_root', 'Rem',
'cbrt', 're', 'im', 'sign', 'Abs', 'conjugate', 'arg', 'polar_lift',
'periodic_argument', 'unbranched_argument', 'principal_branch',
'transpose', 'adjoint', 'polarify', 'unpolarify', 'sin', 'cos', 'tan',
'sec', 'csc', 'cot', 'sinc', 'asin', 'acos', 'atan', 'asec', 'acsc',
'acot', 'atan2', 'exp_polar', 'exp', 'ln', 'log', 'LambertW', 'sinh',
'cosh', 'tanh', 'coth', 'sech', 'csch', 'asinh', 'acosh', 'atanh',
'acoth', 'asech', 'acsch', 'floor', 'ceiling', 'frac', 'Piecewise',
'piecewise_fold', 'piecewise_exclusive', 'erf', 'erfc', 'erfi', 'erf2',
'erfinv', 'erfcinv', 'erf2inv', 'Ei', 'expint', 'E1', 'li', 'Li', 'Si',
'Ci', 'Shi', 'Chi', 'fresnels', 'fresnelc', 'gamma', 'lowergamma',
'uppergamma', 'polygamma', 'loggamma', 'digamma', 'trigamma', 'multigamma',
'dirichlet_eta', 'zeta', 'lerchphi', 'polylog', 'stieltjes', 'Eijk', 'LeviCivita',
'KroneckerDelta', 'SingularityFunction', 'DiracDelta', 'Heaviside',
'bspline_basis', 'bspline_basis_set', 'interpolating_spline', 'besselj',
'bessely', 'besseli', 'besselk', 'hankel1', 'hankel2', 'jn', 'yn',
'jn_zeros', 'hn1', 'hn2', 'airyai', 'airybi', 'airyaiprime',
'airybiprime', 'marcumq', 'hyper', 'meijerg', 'appellf1', 'legendre',
'assoc_legendre', 'hermite', 'chebyshevt', 'chebyshevu',
'chebyshevu_root', 'chebyshevt_root', 'laguerre', 'assoc_laguerre',
'gegenbauer', 'jacobi', 'jacobi_normalized', 'Ynm', 'Ynm_c', 'Znm',
'elliptic_k', 'elliptic_f', 'elliptic_e', 'elliptic_pi', 'beta',
'mathieus', 'mathieuc', 'mathieusprime', 'mathieucprime', 'riemann_xi','betainc',
'betainc_regularized',
# sympy.ntheory
'nextprime', 'prevprime', 'prime', 'primepi', 'primerange', 'randprime',
'Sieve', 'sieve', 'primorial', 'cycle_length', 'composite', 'compositepi',
'isprime', 'divisors', 'proper_divisors', 'factorint', 'multiplicity',
'perfect_power', 'pollard_pm1', 'pollard_rho', 'primefactors', 'totient',
'trailing', 'divisor_count', 'proper_divisor_count', 'divisor_sigma',
'factorrat', 'reduced_totient', 'primenu', 'primeomega',
'mersenne_prime_exponent', 'is_perfect', 'is_mersenne_prime',
'is_abundant', 'is_deficient', 'is_amicable', 'abundance', 'npartitions',
'is_primitive_root', 'is_quad_residue', 'legendre_symbol',
'jacobi_symbol', 'n_order', 'sqrt_mod', 'quadratic_residues',
'primitive_root', 'nthroot_mod', 'is_nthpow_residue', 'sqrt_mod_iter',
'mobius', 'discrete_log', 'quadratic_congruence', 'binomial_coefficients',
'binomial_coefficients_list', 'multinomial_coefficients',
'continued_fraction_periodic', 'continued_fraction_iterator',
'continued_fraction_reduce', 'continued_fraction_convergents',
'continued_fraction', 'egyptian_fraction',
# sympy.concrete
'product', 'Product', 'summation', 'Sum',
# sympy.discrete
'fft', 'ifft', 'ntt', 'intt', 'fwht', 'ifwht', 'mobius_transform',
'inverse_mobius_transform', 'convolution', 'covering_product',
'intersecting_product',
# sympy.simplify
'simplify', 'hypersimp', 'hypersimilar', 'logcombine', 'separatevars',
'posify', 'besselsimp', 'kroneckersimp', 'signsimp',
'nsimplify', 'FU', 'fu', 'sqrtdenest', 'cse', 'epath', 'EPath',
'hyperexpand', 'collect', 'rcollect', 'radsimp', 'collect_const',
'fraction', 'numer', 'denom', 'trigsimp', 'exptrigsimp', 'powsimp',
'powdenest', 'combsimp', 'gammasimp', 'ratsimp', 'ratsimpmodprime',
# sympy.sets
'Set', 'Interval', 'Union', 'EmptySet', 'FiniteSet', 'ProductSet',
'Intersection', 'imageset', 'DisjointUnion', 'Complement', 'SymmetricDifference',
'ImageSet', 'Range', 'ComplexRegion', 'Reals', 'Contains', 'ConditionSet',
'Ordinal', 'OmegaPower', 'ord0', 'PowerSet', 'Naturals',
'Naturals0', 'UniversalSet', 'Integers', 'Rationals', 'Complexes',
# sympy.solvers
'solve', 'solve_linear_system', 'solve_linear_system_LU',
'solve_undetermined_coeffs', 'nsolve', 'solve_linear', 'checksol',
'det_quick', 'inv_quick', 'check_assumptions', 'failing_assumptions',
'diophantine', 'rsolve', 'rsolve_poly', 'rsolve_ratio', 'rsolve_hyper',
'checkodesol', 'classify_ode', 'dsolve', 'homogeneous_order',
'solve_poly_system', 'solve_triangulated', 'pde_separate',
'pde_separate_add', 'pde_separate_mul', 'pdsolve', 'classify_pde',
'checkpdesol', 'ode_order', 'reduce_inequalities',
'reduce_abs_inequality', 'reduce_abs_inequalities',
'solve_poly_inequality', 'solve_rational_inequalities',
'solve_univariate_inequality', 'decompogen', 'solveset', 'linsolve',
'linear_eq_to_matrix', 'nonlinsolve', 'substitution',
# sympy.matrices
'ShapeError', 'NonSquareMatrixError', 'GramSchmidt', 'casoratian', 'diag',
'eye', 'hessian', 'jordan_cell', 'list2numpy', 'matrix2numpy',
'matrix_multiply_elementwise', 'ones', 'randMatrix', 'rot_axis1',
'rot_axis2', 'rot_axis3', 'symarray', 'wronskian', 'zeros',
'MutableDenseMatrix', 'DeferredVector', 'MatrixBase', 'Matrix',
'MutableMatrix', 'MutableSparseMatrix', 'banded', 'ImmutableDenseMatrix',
'ImmutableSparseMatrix', 'ImmutableMatrix', 'SparseMatrix', 'MatrixSlice',
'BlockDiagMatrix', 'BlockMatrix', 'FunctionMatrix', 'Identity', 'Inverse',
'MatAdd', 'MatMul', 'MatPow', 'MatrixExpr', 'MatrixSymbol', 'Trace',
'Transpose', 'ZeroMatrix', 'OneMatrix', 'blockcut', 'block_collapse',
'matrix_symbols', 'Adjoint', 'hadamard_product', 'HadamardProduct',
'HadamardPower', 'Determinant', 'det', 'diagonalize_vector', 'DiagMatrix',
'DiagonalMatrix', 'DiagonalOf', 'trace', 'DotProduct',
'kronecker_product', 'KroneckerProduct', 'PermutationMatrix',
'MatrixPermute', 'Permanent', 'per',
# sympy.geometry
'Point', 'Point2D', 'Point3D', 'Line', 'Ray', 'Segment', 'Line2D',
'Segment2D', 'Ray2D', 'Line3D', 'Segment3D', 'Ray3D', 'Plane', 'Ellipse',
'Circle', 'Polygon', 'RegularPolygon', 'Triangle', 'rad', 'deg',
'are_similar', 'centroid', 'convex_hull', 'idiff', 'intersection',
'closest_points', 'farthest_points', 'GeometryError', 'Curve', 'Parabola',
# sympy.utilities
'flatten', 'group', 'take', 'subsets', 'variations', 'numbered_symbols',
'cartes', 'capture', 'dict_merge', 'prefixes', 'postfixes', 'sift',
'topological_sort', 'unflatten', 'has_dups', 'has_variety', 'reshape',
'rotations', 'filldedent', 'lambdify', 'source', 'threaded', 'xthreaded',
'public', 'memoize_property', 'timed',
# sympy.integrals
'integrate', 'Integral', 'line_integrate', 'mellin_transform',
'inverse_mellin_transform', 'MellinTransform', 'InverseMellinTransform',
'laplace_transform', 'inverse_laplace_transform', 'LaplaceTransform',
'InverseLaplaceTransform', 'fourier_transform',
'inverse_fourier_transform', 'FourierTransform',
'InverseFourierTransform', 'sine_transform', 'inverse_sine_transform',
'SineTransform', 'InverseSineTransform', 'cosine_transform',
'inverse_cosine_transform', 'CosineTransform', 'InverseCosineTransform',
'hankel_transform', 'inverse_hankel_transform', 'HankelTransform',
'InverseHankelTransform', 'singularityintegrate',
# sympy.tensor
'IndexedBase', 'Idx', 'Indexed', 'get_contraction_structure',
'get_indices', 'shape', 'MutableDenseNDimArray', 'ImmutableDenseNDimArray',
'MutableSparseNDimArray', 'ImmutableSparseNDimArray', 'NDimArray',
'tensorproduct', 'tensorcontraction', 'tensordiagonal', 'derive_by_array',
'permutedims', 'Array', 'DenseNDimArray', 'SparseNDimArray',
# sympy.parsing
'parse_expr',
# sympy.calculus
'euler_equations', 'singularities', 'is_increasing',
'is_strictly_increasing', 'is_decreasing', 'is_strictly_decreasing',
'is_monotonic', 'finite_diff_weights', 'apply_finite_diff',
'differentiate_finite', 'periodicity', 'not_empty_in',
'AccumBounds', 'is_convex', 'stationary_points', 'minimum', 'maximum',
# sympy.algebras
'Quaternion',
# sympy.printing
'pager_print', 'pretty', 'pretty_print', 'pprint', 'pprint_use_unicode',
'pprint_try_use_unicode', 'latex', 'print_latex', 'multiline_latex',
'mathml', 'print_mathml', 'python', 'print_python', 'pycode', 'ccode',
'print_ccode', 'glsl_code', 'print_glsl', 'cxxcode', 'fcode',
'print_fcode', 'rcode', 'print_rcode', 'jscode', 'print_jscode',
'julia_code', 'mathematica_code', 'octave_code', 'rust_code', 'print_gtk',
'preview', 'srepr', 'print_tree', 'StrPrinter', 'sstr', 'sstrrepr',
'TableForm', 'dotprint', 'maple_code', 'print_maple_code',
# sympy.plotting
'plot', 'textplot', 'plot_backends', 'plot_implicit', 'plot_parametric',
# sympy.interactive
'init_session', 'init_printing', 'interactive_traversal',
# sympy.testing
'test', 'doctest',
]
#===========================================================================#
# #
# XXX: The names below were importable before SymPy 1.6 using #
# #
# from sympy import * #
# #
# This happened implicitly because there was no __all__ defined in this #
# __init__.py file. Not every package is imported. The list matches what #
# would have been imported before. It is possible that these packages will #
# not be imported by a star-import from sympy in future. #
# #
#===========================================================================#
__all__.extend((
'algebras',
'assumptions',
'calculus',
'concrete',
'discrete',
'external',
'functions',
'geometry',
'interactive',
'multipledispatch',
'ntheory',
'parsing',
'plotting',
'polys',
'printing',
'release',
'strategies',
'tensor',
'utilities',
))
|
585dc1a4220ce390b35ba7a10e2b6bd9829711425da7280cf65aaad45de35b57 | """
Contains
========
FlorySchulz
Geometric
Hermite
Logarithmic
NegativeBinomial
Poisson
Skellam
YuleSimon
Zeta
"""
from sympy.concrete.summations import Sum
from sympy.core.basic import Basic
from sympy.core.function import Lambda
from sympy.core.numbers import I
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import (binomial, factorial)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.bessel import besseli
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.hyper import hyper
from sympy.functions.special.zeta_functions import (polylog, zeta)
from sympy.stats.drv import SingleDiscreteDistribution, SingleDiscretePSpace
from sympy.stats.rv import _value_check, is_random
__all__ = ['FlorySchulz',
'Geometric',
'Hermite',
'Logarithmic',
'NegativeBinomial',
'Poisson',
'Skellam',
'YuleSimon',
'Zeta'
]
def rv(symbol, cls, *args, **kwargs):
args = list(map(sympify, args))
dist = cls(*args)
if kwargs.pop('check', True):
dist.check(*args)
pspace = SingleDiscretePSpace(symbol, dist)
if any(is_random(arg) for arg in args):
from sympy.stats.compound_rv import CompoundPSpace, CompoundDistribution
pspace = CompoundPSpace(symbol, CompoundDistribution(dist))
return pspace.value
class DiscreteDistributionHandmade(SingleDiscreteDistribution):
_argnames = ('pdf',)
def __new__(cls, pdf, set=S.Integers):
return Basic.__new__(cls, pdf, set)
@property
def set(self):
return self.args[1]
@staticmethod
def check(pdf, set):
x = Dummy('x')
val = Sum(pdf(x), (x, set._inf, set._sup)).doit()
_value_check(Eq(val, 1) != S.false, "The pdf is incorrect on the given set.")
def DiscreteRV(symbol, density, set=S.Integers, **kwargs):
"""
Create a Discrete Random Variable given the following:
Parameters
==========
symbol : Symbol
Represents name of the random variable.
density : Expression containing symbol
Represents probability density function.
set : set
Represents the region where the pdf is valid, by default is real line.
check : bool
If True, it will check whether the given density
integrates to 1 over the given set. If False, it
will not perform this check. Default is False.
Examples
========
>>> from sympy.stats import DiscreteRV, P, E
>>> from sympy import Rational, Symbol
>>> x = Symbol('x')
>>> n = 10
>>> density = Rational(1, 10)
>>> X = DiscreteRV(x, density, set=set(range(n)))
>>> E(X)
9/2
>>> P(X>3)
3/5
Returns
=======
RandomSymbol
"""
set = sympify(set)
pdf = Piecewise((density, set.as_relational(symbol)), (0, True))
pdf = Lambda(symbol, pdf)
# have a default of False while `rv` should have a default of True
kwargs['check'] = kwargs.pop('check', False)
return rv(symbol.name, DiscreteDistributionHandmade, pdf, set, **kwargs)
#-------------------------------------------------------------------------------
# Flory-Schulz distribution ------------------------------------------------------------
class FlorySchulzDistribution(SingleDiscreteDistribution):
_argnames = ('a',)
set = S.Naturals
@staticmethod
def check(a):
_value_check((0 < a, a < 1), "a must be between 0 and 1")
def pdf(self, k):
a = self.a
return (a**2 * k * (1 - a)**(k - 1))
def _characteristic_function(self, t):
a = self.a
return a**2*exp(I*t)/((1 + (a - 1)*exp(I*t))**2)
def _moment_generating_function(self, t):
a = self.a
return a**2*exp(t)/((1 + (a - 1)*exp(t))**2)
def FlorySchulz(name, a):
r"""
Create a discrete random variable with a FlorySchulz distribution.
The density of the FlorySchulz distribution is given by
.. math::
f(k) := (a^2) k (1 - a)^{k-1}
Parameters
==========
a : A real number between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import density, E, variance, FlorySchulz
>>> from sympy import Symbol, S
>>> a = S.One / 5
>>> z = Symbol("z")
>>> X = FlorySchulz("x", a)
>>> density(X)(z)
(5/4)**(1 - z)*z/25
>>> E(X)
9
>>> variance(X)
40
References
==========
https://en.wikipedia.org/wiki/Flory%E2%80%93Schulz_distribution
"""
return rv(name, FlorySchulzDistribution, a)
#-------------------------------------------------------------------------------
# Geometric distribution ------------------------------------------------------------
class GeometricDistribution(SingleDiscreteDistribution):
_argnames = ('p',)
set = S.Naturals
@staticmethod
def check(p):
_value_check((0 < p, p <= 1), "p must be between 0 and 1")
def pdf(self, k):
return (1 - self.p)**(k - 1) * self.p
def _characteristic_function(self, t):
p = self.p
return p * exp(I*t) / (1 - (1 - p)*exp(I*t))
def _moment_generating_function(self, t):
p = self.p
return p * exp(t) / (1 - (1 - p) * exp(t))
def Geometric(name, p):
r"""
Create a discrete random variable with a Geometric distribution.
Explanation
===========
The density of the Geometric distribution is given by
.. math::
f(k) := p (1 - p)^{k - 1}
Parameters
==========
p : A probability between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Geometric, density, E, variance
>>> from sympy import Symbol, S
>>> p = S.One / 5
>>> z = Symbol("z")
>>> X = Geometric("x", p)
>>> density(X)(z)
(5/4)**(1 - z)/5
>>> E(X)
5
>>> variance(X)
20
References
==========
.. [1] https://en.wikipedia.org/wiki/Geometric_distribution
.. [2] http://mathworld.wolfram.com/GeometricDistribution.html
"""
return rv(name, GeometricDistribution, p)
#-------------------------------------------------------------------------------
# Hermite distribution ---------------------------------------------------------
class HermiteDistribution(SingleDiscreteDistribution):
_argnames = ('a1', 'a2')
set = S.Naturals0
@staticmethod
def check(a1, a2):
_value_check(a1.is_nonnegative, 'Parameter a1 must be >= 0.')
_value_check(a2.is_nonnegative, 'Parameter a2 must be >= 0.')
def pdf(self, k):
a1, a2 = self.a1, self.a2
term1 = exp(-(a1 + a2))
j = Dummy("j", integer=True)
num = a1**(k - 2*j) * a2**j
den = factorial(k - 2*j) * factorial(j)
return term1 * Sum(num/den, (j, 0, k//2)).doit()
def _moment_generating_function(self, t):
a1, a2 = self.a1, self.a2
term1 = a1 * (exp(t) - 1)
term2 = a2 * (exp(2*t) - 1)
return exp(term1 + term2)
def _characteristic_function(self, t):
a1, a2 = self.a1, self.a2
term1 = a1 * (exp(I*t) - 1)
term2 = a2 * (exp(2*I*t) - 1)
return exp(term1 + term2)
def Hermite(name, a1, a2):
r"""
Create a discrete random variable with a Hermite distribution.
Explanation
===========
The density of the Hermite distribution is given by
.. math::
f(x):= e^{-a_1 -a_2}\sum_{j=0}^{\left \lfloor x/2 \right \rfloor}
\frac{a_{1}^{x-2j}a_{2}^{j}}{(x-2j)!j!}
Parameters
==========
a1 : A Positive number greater than equal to 0.
a2 : A Positive number greater than equal to 0.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Hermite, density, E, variance
>>> from sympy import Symbol
>>> a1 = Symbol("a1", positive=True)
>>> a2 = Symbol("a2", positive=True)
>>> x = Symbol("x")
>>> H = Hermite("H", a1=5, a2=4)
>>> density(H)(2)
33*exp(-9)/2
>>> E(H)
13
>>> variance(H)
21
References
==========
.. [1] https://en.wikipedia.org/wiki/Hermite_distribution
"""
return rv(name, HermiteDistribution, a1, a2)
#-------------------------------------------------------------------------------
# Logarithmic distribution ------------------------------------------------------------
class LogarithmicDistribution(SingleDiscreteDistribution):
_argnames = ('p',)
set = S.Naturals
@staticmethod
def check(p):
_value_check((p > 0, p < 1), "p should be between 0 and 1")
def pdf(self, k):
p = self.p
return (-1) * p**k / (k * log(1 - p))
def _characteristic_function(self, t):
p = self.p
return log(1 - p * exp(I*t)) / log(1 - p)
def _moment_generating_function(self, t):
p = self.p
return log(1 - p * exp(t)) / log(1 - p)
def Logarithmic(name, p):
r"""
Create a discrete random variable with a Logarithmic distribution.
Explanation
===========
The density of the Logarithmic distribution is given by
.. math::
f(k) := \frac{-p^k}{k \ln{(1 - p)}}
Parameters
==========
p : A value between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Logarithmic, density, E, variance
>>> from sympy import Symbol, S
>>> p = S.One / 5
>>> z = Symbol("z")
>>> X = Logarithmic("x", p)
>>> density(X)(z)
-1/(5**z*z*log(4/5))
>>> E(X)
-1/(-4*log(5) + 8*log(2))
>>> variance(X)
-1/((-4*log(5) + 8*log(2))*(-2*log(5) + 4*log(2))) + 1/(-64*log(2)*log(5) + 64*log(2)**2 + 16*log(5)**2) - 10/(-32*log(5) + 64*log(2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Logarithmic_distribution
.. [2] http://mathworld.wolfram.com/LogarithmicDistribution.html
"""
return rv(name, LogarithmicDistribution, p)
#-------------------------------------------------------------------------------
# Negative binomial distribution ------------------------------------------------------------
class NegativeBinomialDistribution(SingleDiscreteDistribution):
_argnames = ('r', 'p')
set = S.Naturals0
@staticmethod
def check(r, p):
_value_check(r > 0, 'r should be positive')
_value_check((p > 0, p < 1), 'p should be between 0 and 1')
def pdf(self, k):
r = self.r
p = self.p
return binomial(k + r - 1, k) * (1 - p)**r * p**k
def _characteristic_function(self, t):
r = self.r
p = self.p
return ((1 - p) / (1 - p * exp(I*t)))**r
def _moment_generating_function(self, t):
r = self.r
p = self.p
return ((1 - p) / (1 - p * exp(t)))**r
def NegativeBinomial(name, r, p):
r"""
Create a discrete random variable with a Negative Binomial distribution.
Explanation
===========
The density of the Negative Binomial distribution is given by
.. math::
f(k) := \binom{k + r - 1}{k} (1 - p)^r p^k
Parameters
==========
r : A positive value
p : A value between 0 and 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import NegativeBinomial, density, E, variance
>>> from sympy import Symbol, S
>>> r = 5
>>> p = S.One / 5
>>> z = Symbol("z")
>>> X = NegativeBinomial("x", r, p)
>>> density(X)(z)
1024*binomial(z + 4, z)/(3125*5**z)
>>> E(X)
5/4
>>> variance(X)
25/16
References
==========
.. [1] https://en.wikipedia.org/wiki/Negative_binomial_distribution
.. [2] http://mathworld.wolfram.com/NegativeBinomialDistribution.html
"""
return rv(name, NegativeBinomialDistribution, r, p)
#-------------------------------------------------------------------------------
# Poisson distribution ------------------------------------------------------------
class PoissonDistribution(SingleDiscreteDistribution):
_argnames = ('lamda',)
set = S.Naturals0
@staticmethod
def check(lamda):
_value_check(lamda > 0, "Lambda must be positive")
def pdf(self, k):
return self.lamda**k / factorial(k) * exp(-self.lamda)
def _characteristic_function(self, t):
return exp(self.lamda * (exp(I*t) - 1))
def _moment_generating_function(self, t):
return exp(self.lamda * (exp(t) - 1))
def Poisson(name, lamda):
r"""
Create a discrete random variable with a Poisson distribution.
Explanation
===========
The density of the Poisson distribution is given by
.. math::
f(k) := \frac{\lambda^{k} e^{- \lambda}}{k!}
Parameters
==========
lamda : Positive number, a rate
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Poisson, density, E, variance
>>> from sympy import Symbol, simplify
>>> rate = Symbol("lambda", positive=True)
>>> z = Symbol("z")
>>> X = Poisson("x", rate)
>>> density(X)(z)
lambda**z*exp(-lambda)/factorial(z)
>>> E(X)
lambda
>>> simplify(variance(X))
lambda
References
==========
.. [1] https://en.wikipedia.org/wiki/Poisson_distribution
.. [2] http://mathworld.wolfram.com/PoissonDistribution.html
"""
return rv(name, PoissonDistribution, lamda)
# -----------------------------------------------------------------------------
# Skellam distribution --------------------------------------------------------
class SkellamDistribution(SingleDiscreteDistribution):
_argnames = ('mu1', 'mu2')
set = S.Integers
@staticmethod
def check(mu1, mu2):
_value_check(mu1 >= 0, 'Parameter mu1 must be >= 0')
_value_check(mu2 >= 0, 'Parameter mu2 must be >= 0')
def pdf(self, k):
(mu1, mu2) = (self.mu1, self.mu2)
term1 = exp(-(mu1 + mu2)) * (mu1 / mu2) ** (k / 2)
term2 = besseli(k, 2 * sqrt(mu1 * mu2))
return term1 * term2
def _cdf(self, x):
raise NotImplementedError(
"Skellam doesn't have closed form for the CDF.")
def _characteristic_function(self, t):
(mu1, mu2) = (self.mu1, self.mu2)
return exp(-(mu1 + mu2) + mu1 * exp(I * t) + mu2 * exp(-I * t))
def _moment_generating_function(self, t):
(mu1, mu2) = (self.mu1, self.mu2)
return exp(-(mu1 + mu2) + mu1 * exp(t) + mu2 * exp(-t))
def Skellam(name, mu1, mu2):
r"""
Create a discrete random variable with a Skellam distribution.
Explanation
===========
The Skellam is the distribution of the difference N1 - N2
of two statistically independent random variables N1 and N2
each Poisson-distributed with respective expected values mu1 and mu2.
The density of the Skellam distribution is given by
.. math::
f(k) := e^{-(\mu_1+\mu_2)}(\frac{\mu_1}{\mu_2})^{k/2}I_k(2\sqrt{\mu_1\mu_2})
Parameters
==========
mu1 : A non-negative value
mu2 : A non-negative value
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Skellam, density, E, variance
>>> from sympy import Symbol, pprint
>>> z = Symbol("z", integer=True)
>>> mu1 = Symbol("mu1", positive=True)
>>> mu2 = Symbol("mu2", positive=True)
>>> X = Skellam("x", mu1, mu2)
>>> pprint(density(X)(z), use_unicode=False)
z
-
2
/mu1\ -mu1 - mu2 / _____ _____\
|---| *e *besseli\z, 2*\/ mu1 *\/ mu2 /
\mu2/
>>> E(X)
mu1 - mu2
>>> variance(X).expand()
mu1 + mu2
References
==========
.. [1] https://en.wikipedia.org/wiki/Skellam_distribution
"""
return rv(name, SkellamDistribution, mu1, mu2)
#-------------------------------------------------------------------------------
# Yule-Simon distribution ------------------------------------------------------------
class YuleSimonDistribution(SingleDiscreteDistribution):
_argnames = ('rho',)
set = S.Naturals
@staticmethod
def check(rho):
_value_check(rho > 0, 'rho should be positive')
def pdf(self, k):
rho = self.rho
return rho * beta(k, rho + 1)
def _cdf(self, x):
return Piecewise((1 - floor(x) * beta(floor(x), self.rho + 1), x >= 1), (0, True))
def _characteristic_function(self, t):
rho = self.rho
return rho * hyper((1, 1), (rho + 2,), exp(I*t)) * exp(I*t) / (rho + 1)
def _moment_generating_function(self, t):
rho = self.rho
return rho * hyper((1, 1), (rho + 2,), exp(t)) * exp(t) / (rho + 1)
def YuleSimon(name, rho):
r"""
Create a discrete random variable with a Yule-Simon distribution.
Explanation
===========
The density of the Yule-Simon distribution is given by
.. math::
f(k) := \rho B(k, \rho + 1)
Parameters
==========
rho : A positive value
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import YuleSimon, density, E, variance
>>> from sympy import Symbol, simplify
>>> p = 5
>>> z = Symbol("z")
>>> X = YuleSimon("x", p)
>>> density(X)(z)
5*beta(z, 6)
>>> simplify(E(X))
5/4
>>> simplify(variance(X))
25/48
References
==========
.. [1] https://en.wikipedia.org/wiki/Yule%E2%80%93Simon_distribution
"""
return rv(name, YuleSimonDistribution, rho)
#-------------------------------------------------------------------------------
# Zeta distribution ------------------------------------------------------------
class ZetaDistribution(SingleDiscreteDistribution):
_argnames = ('s',)
set = S.Naturals
@staticmethod
def check(s):
_value_check(s > 1, 's should be greater than 1')
def pdf(self, k):
s = self.s
return 1 / (k**s * zeta(s))
def _characteristic_function(self, t):
return polylog(self.s, exp(I*t)) / zeta(self.s)
def _moment_generating_function(self, t):
return polylog(self.s, exp(t)) / zeta(self.s)
def Zeta(name, s):
r"""
Create a discrete random variable with a Zeta distribution.
Explanation
===========
The density of the Zeta distribution is given by
.. math::
f(k) := \frac{1}{k^s \zeta{(s)}}
Parameters
==========
s : A value greater than 1
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Zeta, density, E, variance
>>> from sympy import Symbol
>>> s = 5
>>> z = Symbol("z")
>>> X = Zeta("x", s)
>>> density(X)(z)
1/(z**5*zeta(5))
>>> E(X)
pi**4/(90*zeta(5))
>>> variance(X)
-pi**8/(8100*zeta(5)**2) + zeta(3)/zeta(5)
References
==========
.. [1] https://en.wikipedia.org/wiki/Zeta_distribution
"""
return rv(name, ZetaDistribution, s)
|
e32d0d119c7960eafc4728253271b45ae3aecd5ba52a151c3a1eefc1696fd04b | from sympy.core import S, sympify, Expr, Dummy, Add, Mul
from sympy.core.cache import cacheit
from sympy.core.containers import Tuple
from sympy.core.function import Function, PoleError, expand_power_base, expand_log
from sympy.core.sorting import default_sort_key
from sympy.functions.elementary.exponential import exp, log
from sympy.sets.sets import Complement
from sympy.utilities.iterables import uniq, is_sequence
class Order(Expr):
r""" Represents the limiting behavior of some function.
Explanation
===========
The order of a function characterizes the function based on the limiting
behavior of the function as it goes to some limit. Only taking the limit
point to be a number is currently supported. This is expressed in
big O notation [1]_.
The formal definition for the order of a function `g(x)` about a point `a`
is such that `g(x) = O(f(x))` as `x \rightarrow a` if and only if for any
`\delta > 0` there exists a `M > 0` such that `|g(x)| \leq M|f(x)|` for
`|x-a| < \delta`. This is equivalent to `\lim_{x \rightarrow a}
\sup |g(x)/f(x)| < \infty`.
Let's illustrate it on the following example by taking the expansion of
`\sin(x)` about 0:
.. math ::
\sin(x) = x - x^3/3! + O(x^5)
where in this case `O(x^5) = x^5/5! - x^7/7! + \cdots`. By the definition
of `O`, for any `\delta > 0` there is an `M` such that:
.. math ::
|x^5/5! - x^7/7! + ....| <= M|x^5| \text{ for } |x| < \delta
or by the alternate definition:
.. math ::
\lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| < \infty
which surely is true, because
.. math ::
\lim_{x \rightarrow 0} | (x^5/5! - x^7/7! + ....) / x^5| = 1/5!
As it is usually used, the order of a function can be intuitively thought
of representing all terms of powers greater than the one specified. For
example, `O(x^3)` corresponds to any terms proportional to `x^3,
x^4,\ldots` and any higher power. For a polynomial, this leaves terms
proportional to `x^2`, `x` and constants.
Examples
========
>>> from sympy import O, oo, cos, pi
>>> from sympy.abc import x, y
>>> O(x + x**2)
O(x)
>>> O(x + x**2, (x, 0))
O(x)
>>> O(x + x**2, (x, oo))
O(x**2, (x, oo))
>>> O(1 + x*y)
O(1, x, y)
>>> O(1 + x*y, (x, 0), (y, 0))
O(1, x, y)
>>> O(1 + x*y, (x, oo), (y, oo))
O(x*y, (x, oo), (y, oo))
>>> O(1) in O(1, x)
True
>>> O(1, x) in O(1)
False
>>> O(x) in O(1, x)
True
>>> O(x**2) in O(x)
True
>>> O(x)*x
O(x**2)
>>> O(x) - O(x)
O(x)
>>> O(cos(x))
O(1)
>>> O(cos(x), (x, pi/2))
O(x - pi/2, (x, pi/2))
References
==========
.. [1] `Big O notation <https://en.wikipedia.org/wiki/Big_O_notation>`_
Notes
=====
In ``O(f(x), x)`` the expression ``f(x)`` is assumed to have a leading
term. ``O(f(x), x)`` is automatically transformed to
``O(f(x).as_leading_term(x),x)``.
``O(expr*f(x), x)`` is ``O(f(x), x)``
``O(expr, x)`` is ``O(1)``
``O(0, x)`` is 0.
Multivariate O is also supported:
``O(f(x, y), x, y)`` is transformed to
``O(f(x, y).as_leading_term(x,y).as_leading_term(y), x, y)``
In the multivariate case, it is assumed the limits w.r.t. the various
symbols commute.
If no symbols are passed then all symbols in the expression are used
and the limit point is assumed to be zero.
"""
is_Order = True
__slots__ = ()
@cacheit
def __new__(cls, expr, *args, **kwargs):
expr = sympify(expr)
if not args:
if expr.is_Order:
variables = expr.variables
point = expr.point
else:
variables = list(expr.free_symbols)
point = [S.Zero]*len(variables)
else:
args = list(args if is_sequence(args) else [args])
variables, point = [], []
if is_sequence(args[0]):
for a in args:
v, p = list(map(sympify, a))
variables.append(v)
point.append(p)
else:
variables = list(map(sympify, args))
point = [S.Zero]*len(variables)
if not all(v.is_symbol for v in variables):
raise TypeError('Variables are not symbols, got %s' % variables)
if len(list(uniq(variables))) != len(variables):
raise ValueError('Variables are supposed to be unique symbols, got %s' % variables)
if expr.is_Order:
expr_vp = dict(expr.args[1:])
new_vp = dict(expr_vp)
vp = dict(zip(variables, point))
for v, p in vp.items():
if v in new_vp.keys():
if p != new_vp[v]:
raise NotImplementedError(
"Mixing Order at different points is not supported.")
else:
new_vp[v] = p
if set(expr_vp.keys()) == set(new_vp.keys()):
return expr
else:
variables = list(new_vp.keys())
point = [new_vp[v] for v in variables]
if expr is S.NaN:
return S.NaN
if any(x in p.free_symbols for x in variables for p in point):
raise ValueError('Got %s as a point.' % point)
if variables:
if any(p != point[0] for p in point):
raise NotImplementedError(
"Multivariable orders at different points are not supported.")
if point[0] in (S.Infinity, S.Infinity*S.ImaginaryUnit):
s = {k: 1/Dummy() for k in variables}
rs = {1/v: 1/k for k, v in s.items()}
ps = [S.Zero for p in point]
elif point[0] in (S.NegativeInfinity, S.NegativeInfinity*S.ImaginaryUnit):
s = {k: -1/Dummy() for k in variables}
rs = {-1/v: -1/k for k, v in s.items()}
ps = [S.Zero for p in point]
elif point[0] is not S.Zero:
s = {k: Dummy() + point[0] for k in variables}
rs = {(v - point[0]).together(): k - point[0] for k, v in s.items()}
ps = [S.Zero for p in point]
else:
s = ()
rs = ()
ps = list(point)
expr = expr.subs(s)
if expr.is_Add:
expr = expr.factor()
if s:
args = tuple([r[0] for r in rs.items()])
else:
args = tuple(variables)
if len(variables) > 1:
# XXX: better way? We need this expand() to
# workaround e.g: expr = x*(x + y).
# (x*(x + y)).as_leading_term(x, y) currently returns
# x*y (wrong order term!). That's why we want to deal with
# expand()'ed expr (handled in "if expr.is_Add" branch below).
expr = expr.expand()
old_expr = None
while old_expr != expr:
old_expr = expr
if expr.is_Add:
lst = expr.extract_leading_order(args)
expr = Add(*[f.expr for (e, f) in lst])
elif expr:
try:
expr = expr.as_leading_term(*args)
except PoleError:
if isinstance(expr, Function) or\
all(isinstance(arg, Function) for arg in expr.args):
# It is not possible to simplify an expression
# containing only functions (which raise error on
# call to leading term) further
pass
else:
orders = []
pts = tuple(zip(args, ps))
for arg in expr.args:
try:
lt = arg.as_leading_term(*args)
except PoleError:
lt = arg
if lt not in args:
order = Order(lt)
else:
order = Order(lt, *pts)
orders.append(order)
if expr.is_Add:
new_expr = Order(Add(*orders), *pts)
if new_expr.is_Add:
new_expr = Order(Add(*[a.expr for a in new_expr.args]), *pts)
expr = new_expr.expr
elif expr.is_Mul:
expr = Mul(*[a.expr for a in orders])
elif expr.is_Pow:
e = expr.exp
b = expr.base
expr = exp(e * log(b))
# It would probably be better to handle this somewhere
# else. This is needed for a testcase in which there is a
# symbol with the assumptions zero=True.
if expr.is_zero:
expr = S.Zero
else:
expr = expr.as_independent(*args, as_Add=False)[1]
expr = expand_power_base(expr)
expr = expand_log(expr)
if len(args) == 1:
# The definition of O(f(x)) symbol explicitly stated that
# the argument of f(x) is irrelevant. That's why we can
# combine some power exponents (only "on top" of the
# expression tree for f(x)), e.g.:
# x**p * (-x)**q -> x**(p+q) for real p, q.
x = args[0]
margs = list(Mul.make_args(
expr.as_independent(x, as_Add=False)[1]))
for i, t in enumerate(margs):
if t.is_Pow:
b, q = t.args
if b in (x, -x) and q.is_real and not q.has(x):
margs[i] = x**q
elif b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
elif b.is_Mul and b.args[0] is S.NegativeOne:
b = -b
if b.is_Pow and not b.exp.has(x):
b, r = b.args
if b in (x, -x) and r.is_real:
margs[i] = x**(r*q)
expr = Mul(*margs)
expr = expr.subs(rs)
if expr.is_Order:
expr = expr.expr
if not expr.has(*variables) and not expr.is_zero:
expr = S.One
# create Order instance:
vp = dict(zip(variables, point))
variables.sort(key=default_sort_key)
point = [vp[v] for v in variables]
args = (expr,) + Tuple(*zip(variables, point))
obj = Expr.__new__(cls, *args)
return obj
def _eval_nseries(self, x, n, logx, cdir=0):
return self
@property
def expr(self):
return self.args[0]
@property
def variables(self):
if self.args[1:]:
return tuple(x[0] for x in self.args[1:])
else:
return ()
@property
def point(self):
if self.args[1:]:
return tuple(x[1] for x in self.args[1:])
else:
return ()
@property
def free_symbols(self):
return self.expr.free_symbols | set(self.variables)
def _eval_power(b, e):
if e.is_Number and e.is_nonnegative:
return b.func(b.expr ** e, *b.args[1:])
if e == O(1):
return b
return
def as_expr_variables(self, order_symbols):
if order_symbols is None:
order_symbols = self.args[1:]
else:
if (not all(o[1] == order_symbols[0][1] for o in order_symbols) and
not all(p == self.point[0] for p in self.point)): # pragma: no cover
raise NotImplementedError('Order at points other than 0 '
'or oo not supported, got %s as a point.' % self.point)
if order_symbols and order_symbols[0][1] != self.point[0]:
raise NotImplementedError(
"Multiplying Order at different points is not supported.")
order_symbols = dict(order_symbols)
for s, p in dict(self.args[1:]).items():
if s not in order_symbols.keys():
order_symbols[s] = p
order_symbols = sorted(order_symbols.items(), key=lambda x: default_sort_key(x[0]))
return self.expr, tuple(order_symbols)
def removeO(self):
return S.Zero
def getO(self):
return self
@cacheit
def contains(self, expr):
r"""
Return True if expr belongs to Order(self.expr, \*self.variables).
Return False if self belongs to expr.
Return None if the inclusion relation cannot be determined
(e.g. when self and expr have different symbols).
"""
expr = sympify(expr)
if expr.is_zero:
return True
if expr is S.NaN:
return False
point = self.point[0] if self.point else S.Zero
if expr.is_Order:
if (any(p != point for p in expr.point) or
any(p != point for p in self.point)):
return None
if expr.expr == self.expr:
# O(1) + O(1), O(1) + O(1, x), etc.
return all(x in self.args[1:] for x in expr.args[1:])
if expr.expr.is_Add:
return all(self.contains(x) for x in expr.expr.args)
if self.expr.is_Add and point.is_zero:
return any(self.func(x, *self.args[1:]).contains(expr)
for x in self.expr.args)
if self.variables and expr.variables:
common_symbols = tuple(
[s for s in self.variables if s in expr.variables])
elif self.variables:
common_symbols = self.variables
else:
common_symbols = expr.variables
if not common_symbols:
return None
if (self.expr.is_Pow and len(self.variables) == 1
and self.variables == expr.variables):
symbol = self.variables[0]
other = expr.expr.as_independent(symbol, as_Add=False)[1]
if (other.is_Pow and other.base == symbol and
self.expr.base == symbol):
if point.is_zero:
rv = (self.expr.exp - other.exp).is_nonpositive
if point.is_infinite:
rv = (self.expr.exp - other.exp).is_nonnegative
if rv is not None:
return rv
from sympy.simplify.powsimp import powsimp
r = None
ratio = self.expr/expr.expr
ratio = powsimp(ratio, deep=True, combine='exp')
for s in common_symbols:
from sympy.series.limits import Limit
l = Limit(ratio, s, point).doit(heuristics=False)
if not isinstance(l, Limit):
l = l != 0
else:
l = None
if r is None:
r = l
else:
if r != l:
return
return r
if self.expr.is_Pow and len(self.variables) == 1:
symbol = self.variables[0]
other = expr.as_independent(symbol, as_Add=False)[1]
if (other.is_Pow and other.base == symbol and
self.expr.base == symbol):
if point.is_zero:
rv = (self.expr.exp - other.exp).is_nonpositive
if point.is_infinite:
rv = (self.expr.exp - other.exp).is_nonnegative
if rv is not None:
return rv
obj = self.func(expr, *self.args[1:])
return self.contains(obj)
def __contains__(self, other):
result = self.contains(other)
if result is None:
raise TypeError('contains did not evaluate to a bool')
return result
def _eval_subs(self, old, new):
if old in self.variables:
newexpr = self.expr.subs(old, new)
i = self.variables.index(old)
newvars = list(self.variables)
newpt = list(self.point)
if new.is_symbol:
newvars[i] = new
else:
syms = new.free_symbols
if len(syms) == 1 or old in syms:
if old in syms:
var = self.variables[i]
else:
var = syms.pop()
# First, try to substitute self.point in the "new"
# expr to see if this is a fixed point.
# E.g. O(y).subs(y, sin(x))
point = new.subs(var, self.point[i])
if point != self.point[i]:
from sympy.solvers.solveset import solveset
d = Dummy()
sol = solveset(old - new.subs(var, d), d)
if isinstance(sol, Complement):
e1 = sol.args[0]
e2 = sol.args[1]
sol = set(e1) - set(e2)
res = [dict(zip((d, ), sol))]
point = d.subs(res[0]).limit(old, self.point[i])
newvars[i] = var
newpt[i] = point
elif old not in syms:
del newvars[i], newpt[i]
if not syms and new == self.point[i]:
newvars.extend(syms)
newpt.extend([S.Zero]*len(syms))
else:
return
return Order(newexpr, *zip(newvars, newpt))
def _eval_conjugate(self):
expr = self.expr._eval_conjugate()
if expr is not None:
return self.func(expr, *self.args[1:])
def _eval_derivative(self, x):
return self.func(self.expr.diff(x), *self.args[1:]) or self
def _eval_transpose(self):
expr = self.expr._eval_transpose()
if expr is not None:
return self.func(expr, *self.args[1:])
def __neg__(self):
return self
O = Order
|
6a7db5fe136ecf2ad4f787e4dc7c1ccbd82a41d5c4bc2bc65ef6f12c86b123e5 | from itertools import combinations_with_replacement
from sympy.core import symbols, Add, Dummy
from sympy.core.numbers import Rational
from sympy.polys import cancel, ComputationFailed, parallel_poly_from_expr, reduced, Poly
from sympy.polys.monomials import Monomial, monomial_div
from sympy.polys.polyerrors import DomainError, PolificationFailed
from sympy.utilities.misc import debug
def ratsimp(expr):
"""
Put an expression over a common denominator, cancel and reduce.
Examples
========
>>> from sympy import ratsimp
>>> from sympy.abc import x, y
>>> ratsimp(1/x + 1/y)
(x + y)/(x*y)
"""
f, g = cancel(expr).as_numer_denom()
try:
Q, r = reduced(f, [g], field=True, expand=False)
except ComputationFailed:
return f/g
return Add(*Q) + cancel(r/g)
def ratsimpmodprime(expr, G, *gens, quick=True, polynomial=False, **args):
"""
Simplifies a rational expression ``expr`` modulo the prime ideal
generated by ``G``. ``G`` should be a Groebner basis of the
ideal.
Examples
========
>>> from sympy.simplify.ratsimp import ratsimpmodprime
>>> from sympy.abc import x, y
>>> eq = (x + y**5 + y)/(x - y)
>>> ratsimpmodprime(eq, [x*y**5 - x - y], x, y, order='lex')
(-x**2 - x*y - x - y)/(-x**2 + x*y)
If ``polynomial`` is ``False``, the algorithm computes a rational
simplification which minimizes the sum of the total degrees of
the numerator and the denominator.
If ``polynomial`` is ``True``, this function just brings numerator and
denominator into a canonical form. This is much faster, but has
potentially worse results.
References
==========
.. [1] M. Monagan, R. Pearce, Rational Simplification Modulo a Polynomial
Ideal, http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.163.6984
(specifically, the second algorithm)
"""
from sympy.solvers.solvers import solve
debug('ratsimpmodprime', expr)
# usual preparation of polynomials:
num, denom = cancel(expr).as_numer_denom()
try:
polys, opt = parallel_poly_from_expr([num, denom] + G, *gens, **args)
except PolificationFailed:
return expr
domain = opt.domain
if domain.has_assoc_Field:
opt.domain = domain.get_field()
else:
raise DomainError(
"Cannot compute rational simplification over %s" % domain)
# compute only once
leading_monomials = [g.LM(opt.order) for g in polys[2:]]
tested = set()
def staircase(n):
"""
Compute all monomials with degree less than ``n`` that are
not divisible by any element of ``leading_monomials``.
"""
if n == 0:
return [1]
S = []
for mi in combinations_with_replacement(range(len(opt.gens)), n):
m = [0]*len(opt.gens)
for i in mi:
m[i] += 1
if all(monomial_div(m, lmg) is None for lmg in
leading_monomials):
S.append(m)
return [Monomial(s).as_expr(*opt.gens) for s in S] + staircase(n - 1)
def _ratsimpmodprime(a, b, allsol, N=0, D=0):
r"""
Computes a rational simplification of ``a/b`` which minimizes
the sum of the total degrees of the numerator and the denominator.
Explanation
===========
The algorithm proceeds by looking at ``a * d - b * c`` modulo
the ideal generated by ``G`` for some ``c`` and ``d`` with degree
less than ``a`` and ``b`` respectively.
The coefficients of ``c`` and ``d`` are indeterminates and thus
the coefficients of the normalform of ``a * d - b * c`` are
linear polynomials in these indeterminates.
If these linear polynomials, considered as system of
equations, have a nontrivial solution, then `\frac{a}{b}
\equiv \frac{c}{d}` modulo the ideal generated by ``G``. So,
by construction, the degree of ``c`` and ``d`` is less than
the degree of ``a`` and ``b``, so a simpler representation
has been found.
After a simpler representation has been found, the algorithm
tries to reduce the degree of the numerator and denominator
and returns the result afterwards.
As an extension, if quick=False, we look at all possible degrees such
that the total degree is less than *or equal to* the best current
solution. We retain a list of all solutions of minimal degree, and try
to find the best one at the end.
"""
c, d = a, b
steps = 0
maxdeg = a.total_degree() + b.total_degree()
if quick:
bound = maxdeg - 1
else:
bound = maxdeg
while N + D <= bound:
if (N, D) in tested:
break
tested.add((N, D))
M1 = staircase(N)
M2 = staircase(D)
debug('%s / %s: %s, %s' % (N, D, M1, M2))
Cs = symbols("c:%d" % len(M1), cls=Dummy)
Ds = symbols("d:%d" % len(M2), cls=Dummy)
ng = Cs + Ds
c_hat = Poly(
sum([Cs[i] * M1[i] for i in range(len(M1))]), opt.gens + ng)
d_hat = Poly(
sum([Ds[i] * M2[i] for i in range(len(M2))]), opt.gens + ng)
r = reduced(a * d_hat - b * c_hat, G, opt.gens + ng,
order=opt.order, polys=True)[1]
S = Poly(r, gens=opt.gens).coeffs()
sol = solve(S, Cs + Ds, particular=True, quick=True)
if sol and not all(s == 0 for s in sol.values()):
c = c_hat.subs(sol)
d = d_hat.subs(sol)
# The "free" variables occurring before as parameters
# might still be in the substituted c, d, so set them
# to the value chosen before:
c = c.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds))))))
d = d.subs(dict(list(zip(Cs + Ds, [1] * (len(Cs) + len(Ds))))))
c = Poly(c, opt.gens)
d = Poly(d, opt.gens)
if d == 0:
raise ValueError('Ideal not prime?')
allsol.append((c_hat, d_hat, S, Cs + Ds))
if N + D != maxdeg:
allsol = [allsol[-1]]
break
steps += 1
N += 1
D += 1
if steps > 0:
c, d, allsol = _ratsimpmodprime(c, d, allsol, N, D - steps)
c, d, allsol = _ratsimpmodprime(c, d, allsol, N - steps, D)
return c, d, allsol
# preprocessing. this improves performance a bit when deg(num)
# and deg(denom) are large:
num = reduced(num, G, opt.gens, order=opt.order)[1]
denom = reduced(denom, G, opt.gens, order=opt.order)[1]
if polynomial:
return (num/denom).cancel()
c, d, allsol = _ratsimpmodprime(
Poly(num, opt.gens, domain=opt.domain), Poly(denom, opt.gens, domain=opt.domain), [])
if not quick and allsol:
debug('Looking for best minimal solution. Got: %s' % len(allsol))
newsol = []
for c_hat, d_hat, S, ng in allsol:
sol = solve(S, ng, particular=True, quick=False)
# all values of sol should be numbers; if not, solve is broken
newsol.append((c_hat.subs(sol), d_hat.subs(sol)))
c, d = min(newsol, key=lambda x: len(x[0].terms()) + len(x[1].terms()))
if not domain.is_Field:
cn, c = c.clear_denoms(convert=True)
dn, d = d.clear_denoms(convert=True)
r = Rational(cn, dn)
else:
r = Rational(1)
return (c*r.q)/(d*r.p)
|
07cb4f43aca7dae0813007b0cb54864650a6fbcf12d2fd4d1c787e75915c8424 | """A functions module, includes all the standard functions.
Combinatorial - factorial, fibonacci, harmonic, bernoulli...
Elementary - hyperbolic, trigonometric, exponential, floor and ceiling, sqrt...
Special - gamma, zeta,spherical harmonics...
"""
from sympy.functions.combinatorial.factorials import (factorial, factorial2,
rf, ff, binomial, RisingFactorial, FallingFactorial, subfactorial)
from sympy.functions.combinatorial.numbers import (carmichael, fibonacci, lucas, tribonacci,
harmonic, bernoulli, bell, euler, catalan, genocchi, partition, motzkin)
from sympy.functions.elementary.miscellaneous import (sqrt, root, Min, Max,
Id, real_root, cbrt, Rem)
from sympy.functions.elementary.complexes import (re, im, sign, Abs,
conjugate, arg, polar_lift, periodic_argument, unbranched_argument,
principal_branch, transpose, adjoint, polarify, unpolarify)
from sympy.functions.elementary.trigonometric import (sin, cos, tan,
sec, csc, cot, sinc, asin, acos, atan, asec, acsc, acot, atan2)
from sympy.functions.elementary.exponential import (exp_polar, exp, log,
LambertW)
from sympy.functions.elementary.hyperbolic import (sinh, cosh, tanh, coth,
sech, csch, asinh, acosh, atanh, acoth, asech, acsch)
from sympy.functions.elementary.integers import floor, ceiling, frac
from sympy.functions.elementary.piecewise import (Piecewise, piecewise_fold,
piecewise_exclusive)
from sympy.functions.special.error_functions import (erf, erfc, erfi, erf2,
erfinv, erfcinv, erf2inv, Ei, expint, E1, li, Li, Si, Ci, Shi, Chi,
fresnels, fresnelc)
from sympy.functions.special.gamma_functions import (gamma, lowergamma,
uppergamma, polygamma, loggamma, digamma, trigamma, multigamma)
from sympy.functions.special.zeta_functions import (dirichlet_eta, zeta,
lerchphi, polylog, stieltjes, riemann_xi)
from sympy.functions.special.tensor_functions import (Eijk, LeviCivita,
KroneckerDelta)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.delta_functions import DiracDelta, Heaviside
from sympy.functions.special.bsplines import bspline_basis, bspline_basis_set, interpolating_spline
from sympy.functions.special.bessel import (besselj, bessely, besseli, besselk,
hankel1, hankel2, jn, yn, jn_zeros, hn1, hn2, airyai, airybi, airyaiprime, airybiprime, marcumq)
from sympy.functions.special.hyper import hyper, meijerg, appellf1
from sympy.functions.special.polynomials import (legendre, assoc_legendre,
hermite, chebyshevt, chebyshevu, chebyshevu_root, chebyshevt_root,
laguerre, assoc_laguerre, gegenbauer, jacobi, jacobi_normalized)
from sympy.functions.special.spherical_harmonics import Ynm, Ynm_c, Znm
from sympy.functions.special.elliptic_integrals import (elliptic_k,
elliptic_f, elliptic_e, elliptic_pi)
from sympy.functions.special.beta_functions import beta, betainc, betainc_regularized
from sympy.functions.special.mathieu_functions import (mathieus, mathieuc,
mathieusprime, mathieucprime)
ln = log
__all__ = [
'factorial', 'factorial2', 'rf', 'ff', 'binomial', 'RisingFactorial',
'FallingFactorial', 'subfactorial',
'carmichael', 'fibonacci', 'lucas', 'motzkin', 'tribonacci', 'harmonic',
'bernoulli', 'bell', 'euler', 'catalan', 'genocchi', 'partition',
'sqrt', 'root', 'Min', 'Max', 'Id', 'real_root', 'cbrt', 'Rem',
're', 'im', 'sign', 'Abs', 'conjugate', 'arg', 'polar_lift',
'periodic_argument', 'unbranched_argument', 'principal_branch',
'transpose', 'adjoint', 'polarify', 'unpolarify',
'sin', 'cos', 'tan', 'sec', 'csc', 'cot', 'sinc', 'asin', 'acos', 'atan',
'asec', 'acsc', 'acot', 'atan2',
'exp_polar', 'exp', 'ln', 'log', 'LambertW',
'sinh', 'cosh', 'tanh', 'coth', 'sech', 'csch', 'asinh', 'acosh', 'atanh',
'acoth', 'asech', 'acsch',
'floor', 'ceiling', 'frac',
'Piecewise', 'piecewise_fold', 'piecewise_exclusive',
'erf', 'erfc', 'erfi', 'erf2', 'erfinv', 'erfcinv', 'erf2inv', 'Ei',
'expint', 'E1', 'li', 'Li', 'Si', 'Ci', 'Shi', 'Chi', 'fresnels',
'fresnelc',
'gamma', 'lowergamma', 'uppergamma', 'polygamma', 'loggamma', 'digamma',
'trigamma', 'multigamma',
'dirichlet_eta', 'zeta', 'lerchphi', 'polylog', 'stieltjes', 'riemann_xi',
'Eijk', 'LeviCivita', 'KroneckerDelta',
'SingularityFunction',
'DiracDelta', 'Heaviside',
'bspline_basis', 'bspline_basis_set', 'interpolating_spline',
'besselj', 'bessely', 'besseli', 'besselk', 'hankel1', 'hankel2', 'jn',
'yn', 'jn_zeros', 'hn1', 'hn2', 'airyai', 'airybi', 'airyaiprime',
'airybiprime', 'marcumq',
'hyper', 'meijerg', 'appellf1',
'legendre', 'assoc_legendre', 'hermite', 'chebyshevt', 'chebyshevu',
'chebyshevu_root', 'chebyshevt_root', 'laguerre', 'assoc_laguerre',
'gegenbauer', 'jacobi', 'jacobi_normalized',
'Ynm', 'Ynm_c', 'Znm',
'elliptic_k', 'elliptic_f', 'elliptic_e', 'elliptic_pi',
'beta', 'betainc', 'betainc_regularized',
'mathieus', 'mathieuc', 'mathieusprime', 'mathieucprime',
]
|
09495eb28c58af789c411bf386f20b1882889229f4df685979b31cfd7bc7a34f | r"""
This module is intended for solving recurrences or, in other words,
difference equations. Currently supported are linear, inhomogeneous
equations with polynomial or rational coefficients.
The solutions are obtained among polynomials, rational functions,
hypergeometric terms, or combinations of hypergeometric term which
are pairwise dissimilar.
``rsolve_X`` functions were meant as a low level interface
for ``rsolve`` which would use Mathematica's syntax.
Given a recurrence relation:
.. math:: a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) +
... + a_{0}(n) y(n) = f(n)
where `k > 0` and `a_{i}(n)` are polynomials in `n`. To use
``rsolve_X`` we need to put all coefficients in to a list ``L`` of
`k+1` elements the following way:
``L = [a_{0}(n), ..., a_{k-1}(n), a_{k}(n)]``
where ``L[i]``, for `i=0, \ldots, k`, maps to
`a_{i}(n) y(n+i)` (`y(n+i)` is implicit).
For example if we would like to compute `m`-th Bernoulli polynomial
up to a constant (example was taken from rsolve_poly docstring),
then we would use `b(n+1) - b(n) = m n^{m-1}` recurrence, which
has solution `b(n) = B_m + C`.
Then ``L = [-1, 1]`` and `f(n) = m n^(m-1)` and finally for `m=4`:
>>> from sympy import Symbol, bernoulli, rsolve_poly
>>> n = Symbol('n', integer=True)
>>> rsolve_poly([-1, 1], 4*n**3, n)
C0 + n**4 - 2*n**3 + n**2
>>> bernoulli(4, n)
n**4 - 2*n**3 + n**2 - 1/30
For the sake of completeness, `f(n)` can be:
[1] a polynomial -> rsolve_poly
[2] a rational function -> rsolve_ratio
[3] a hypergeometric function -> rsolve_hyper
"""
from collections import defaultdict
from sympy.concrete import product
from sympy.core.singleton import S
from sympy.core.numbers import Rational, I
from sympy.core.symbol import Symbol, Wild, Dummy
from sympy.core.relational import Equality
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.sorting import default_sort_key
from sympy.core.sympify import sympify
from sympy.simplify import simplify, hypersimp, hypersimilar # type: ignore
from sympy.solvers import solve, solve_undetermined_coeffs
from sympy.polys import Poly, quo, gcd, lcm, roots, resultant
from sympy.functions import binomial, factorial, FallingFactorial, RisingFactorial
from sympy.matrices import Matrix, casoratian
from sympy.utilities.iterables import numbered_symbols
def rsolve_poly(coeffs, f, n, shift=0, **hints):
r"""
Given linear recurrence operator `\operatorname{L}` of order
`k` with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f`, where `f` is a polynomial, we seek for
all polynomial solutions over field `K` of characteristic zero.
The algorithm performs two basic steps:
(1) Compute degree `N` of the general polynomial solution.
(2) Find all polynomials of degree `N` or less
of `\operatorname{L} y = f`.
There are two methods for computing the polynomial solutions.
If the degree bound is relatively small, i.e. it's smaller than
or equal to the order of the recurrence, then naive method of
undetermined coefficients is being used. This gives a system
of algebraic equations with `N+1` unknowns.
In the other case, the algorithm performs transformation of the
initial equation to an equivalent one for which the system of
algebraic equations has only `r` indeterminates. This method is
quite sophisticated (in comparison with the naive one) and was
invented together by Abramov, Bronstein and Petkovsek.
It is possible to generalize the algorithm implemented here to
the case of linear q-difference and differential equations.
Lets say that we would like to compute `m`-th Bernoulli polynomial
up to a constant. For this we can use `b(n+1) - b(n) = m n^{m-1}`
recurrence, which has solution `b(n) = B_m + C`. For example:
>>> from sympy import Symbol, rsolve_poly
>>> n = Symbol('n', integer=True)
>>> rsolve_poly([-1, 1], 4*n**3, n)
C0 + n**4 - 2*n**3 + n**2
References
==========
.. [1] S. A. Abramov, M. Bronstein and M. Petkovsek, On polynomial
solutions of linear operator equations, in: T. Levelt, ed.,
Proc. ISSAC '95, ACM Press, New York, 1995, 290-296.
.. [2] M. Petkovsek, Hypergeometric solutions of linear recurrences
with polynomial coefficients, J. Symbolic Computation,
14 (1992), 243-264.
.. [3] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.
"""
f = sympify(f)
if not f.is_polynomial(n):
return None
homogeneous = f.is_zero
r = len(coeffs) - 1
coeffs = [Poly(coeff, n) for coeff in coeffs]
polys = [Poly(0, n)]*(r + 1)
terms = [(S.Zero, S.NegativeInfinity)]*(r + 1)
for i in range(r + 1):
for j in range(i, r + 1):
polys[i] += coeffs[j]*(binomial(j, i).as_poly(n))
if not polys[i].is_zero:
(exp,), coeff = polys[i].LT()
terms[i] = (coeff, exp)
d = b = terms[0][1]
for i in range(1, r + 1):
if terms[i][1] > d:
d = terms[i][1]
if terms[i][1] - i > b:
b = terms[i][1] - i
d, b = int(d), int(b)
x = Dummy('x')
degree_poly = S.Zero
for i in range(r + 1):
if terms[i][1] - i == b:
degree_poly += terms[i][0]*FallingFactorial(x, i)
nni_roots = list(roots(degree_poly, x, filter='Z',
predicate=lambda r: r >= 0).keys())
if nni_roots:
N = [max(nni_roots)]
else:
N = []
if homogeneous:
N += [-b - 1]
else:
N += [f.as_poly(n).degree() - b, -b - 1]
N = int(max(N))
if N < 0:
if homogeneous:
if hints.get('symbols', False):
return (S.Zero, [])
else:
return S.Zero
else:
return None
if N <= r:
C = []
y = E = S.Zero
for i in range(N + 1):
C.append(Symbol('C' + str(i + shift)))
y += C[i] * n**i
for i in range(r + 1):
E += coeffs[i].as_expr()*y.subs(n, n + i)
solutions = solve_undetermined_coeffs(E - f, C, n)
if solutions is not None:
_C = C
C = [c for c in C if (c not in solutions)]
result = y.subs(solutions)
else:
return None # TBD
else:
A = r
U = N + A + b + 1
nni_roots = list(roots(polys[r], filter='Z',
predicate=lambda r: r >= 0).keys())
if nni_roots != []:
a = max(nni_roots) + 1
else:
a = S.Zero
def _zero_vector(k):
return [S.Zero] * k
def _one_vector(k):
return [S.One] * k
def _delta(p, k):
B = S.One
D = p.subs(n, a + k)
for i in range(1, k + 1):
B *= Rational(i - k - 1, i)
D += B * p.subs(n, a + k - i)
return D
alpha = {}
for i in range(-A, d + 1):
I = _one_vector(d + 1)
for k in range(1, d + 1):
I[k] = I[k - 1] * (x + i - k + 1)/k
alpha[i] = S.Zero
for j in range(A + 1):
for k in range(d + 1):
B = binomial(k, i + j)
D = _delta(polys[j].as_expr(), k)
alpha[i] += I[k]*B*D
V = Matrix(U, A, lambda i, j: int(i == j))
if homogeneous:
for i in range(A, U):
v = _zero_vector(A)
for k in range(1, A + b + 1):
if i - k < 0:
break
B = alpha[k - A].subs(x, i - k)
for j in range(A):
v[j] += B * V[i - k, j]
denom = alpha[-A].subs(x, i)
for j in range(A):
V[i, j] = -v[j] / denom
else:
G = _zero_vector(U)
for i in range(A, U):
v = _zero_vector(A)
g = S.Zero
for k in range(1, A + b + 1):
if i - k < 0:
break
B = alpha[k - A].subs(x, i - k)
for j in range(A):
v[j] += B * V[i - k, j]
g += B * G[i - k]
denom = alpha[-A].subs(x, i)
for j in range(A):
V[i, j] = -v[j] / denom
G[i] = (_delta(f, i - A) - g) / denom
P, Q = _one_vector(U), _zero_vector(A)
for i in range(1, U):
P[i] = (P[i - 1] * (n - a - i + 1)/i).expand()
for i in range(A):
Q[i] = Add(*[(v*p).expand() for v, p in zip(V[:, i], P)])
if not homogeneous:
h = Add(*[(g*p).expand() for g, p in zip(G, P)])
C = [Symbol('C' + str(i + shift)) for i in range(A)]
g = lambda i: Add(*[c*_delta(q, i) for c, q in zip(C, Q)])
if homogeneous:
E = [g(i) for i in range(N + 1, U)]
else:
E = [g(i) + _delta(h, i) for i in range(N + 1, U)]
if E != []:
solutions = solve(E, *C)
if not solutions:
if homogeneous:
if hints.get('symbols', False):
return (S.Zero, [])
else:
return S.Zero
else:
return None
else:
solutions = {}
if homogeneous:
result = S.Zero
else:
result = h
_C = C[:]
for c, q in list(zip(C, Q)):
if c in solutions:
s = solutions[c]*q
C.remove(c)
else:
s = c*q
result += s.expand()
if C != _C:
# renumber so they are contiguous
result = result.xreplace(dict(zip(C, _C)))
C = _C[:len(C)]
if hints.get('symbols', False):
return (result, C)
else:
return result
def rsolve_ratio(coeffs, f, n, **hints):
r"""
Given linear recurrence operator `\operatorname{L}` of order `k`
with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f`, where `f` is a polynomial, we seek
for all rational solutions over field `K` of characteristic zero.
This procedure accepts only polynomials, however if you are
interested in solving recurrence with rational coefficients
then use ``rsolve`` which will pre-process the given equation
and run this procedure with polynomial arguments.
The algorithm performs two basic steps:
(1) Compute polynomial `v(n)` which can be used as universal
denominator of any rational solution of equation
`\operatorname{L} y = f`.
(2) Construct new linear difference equation by substitution
`y(n) = u(n)/v(n)` and solve it for `u(n)` finding all its
polynomial solutions. Return ``None`` if none were found.
The algorithm implemented here is a revised version of the original
Abramov's algorithm, developed in 1989. The new approach is much
simpler to implement and has better overall efficiency. This
method can be easily adapted to the q-difference equations case.
Besides finding rational solutions alone, this functions is
an important part of Hyper algorithm where it is used to find
a particular solution for the inhomogeneous part of a recurrence.
Examples
========
>>> from sympy.abc import x
>>> from sympy.solvers.recurr import rsolve_ratio
>>> rsolve_ratio([-2*x**3 + x**2 + 2*x - 1, 2*x**3 + x**2 - 6*x,
... - 2*x**3 - 11*x**2 - 18*x - 9, 2*x**3 + 13*x**2 + 22*x + 8], 0, x)
C0*(2*x - 3)/(2*(x**2 - 1))
References
==========
.. [1] S. A. Abramov, Rational solutions of linear difference
and q-difference equations with polynomial coefficients,
in: T. Levelt, ed., Proc. ISSAC '95, ACM Press, New York,
1995, 285-289
See Also
========
rsolve_hyper
"""
f = sympify(f)
if not f.is_polynomial(n):
return None
coeffs = list(map(sympify, coeffs))
r = len(coeffs) - 1
A, B = coeffs[r], coeffs[0]
A = A.subs(n, n - r).expand()
h = Dummy('h')
res = resultant(A, B.subs(n, n + h), n)
if not res.is_polynomial(h):
p, q = res.as_numer_denom()
res = quo(p, q, h)
nni_roots = list(roots(res, h, filter='Z',
predicate=lambda r: r >= 0).keys())
if not nni_roots:
return rsolve_poly(coeffs, f, n, **hints)
else:
C, numers = S.One, [S.Zero]*(r + 1)
for i in range(int(max(nni_roots)), -1, -1):
d = gcd(A, B.subs(n, n + i), n)
A = quo(A, d, n)
B = quo(B, d.subs(n, n - i), n)
C *= Mul(*[d.subs(n, n - j) for j in range(i + 1)])
denoms = [C.subs(n, n + i) for i in range(r + 1)]
for i in range(r + 1):
g = gcd(coeffs[i], denoms[i], n)
numers[i] = quo(coeffs[i], g, n)
denoms[i] = quo(denoms[i], g, n)
for i in range(r + 1):
numers[i] *= Mul(*(denoms[:i] + denoms[i + 1:]))
result = rsolve_poly(numers, f * Mul(*denoms), n, **hints)
if result is not None:
if hints.get('symbols', False):
return (simplify(result[0] / C), result[1])
else:
return simplify(result / C)
else:
return None
def rsolve_hyper(coeffs, f, n, **hints):
r"""
Given linear recurrence operator `\operatorname{L}` of order `k`
with polynomial coefficients and inhomogeneous equation
`\operatorname{L} y = f` we seek for all hypergeometric solutions
over field `K` of characteristic zero.
The inhomogeneous part can be either hypergeometric or a sum
of a fixed number of pairwise dissimilar hypergeometric terms.
The algorithm performs three basic steps:
(1) Group together similar hypergeometric terms in the
inhomogeneous part of `\operatorname{L} y = f`, and find
particular solution using Abramov's algorithm.
(2) Compute generating set of `\operatorname{L}` and find basis
in it, so that all solutions are linearly independent.
(3) Form final solution with the number of arbitrary
constants equal to dimension of basis of `\operatorname{L}`.
Term `a(n)` is hypergeometric if it is annihilated by first order
linear difference equations with polynomial coefficients or, in
simpler words, if consecutive term ratio is a rational function.
The output of this procedure is a linear combination of fixed
number of hypergeometric terms. However the underlying method
can generate larger class of solutions - D'Alembertian terms.
Note also that this method not only computes the kernel of the
inhomogeneous equation, but also reduces in to a basis so that
solutions generated by this procedure are linearly independent
Examples
========
>>> from sympy.solvers import rsolve_hyper
>>> from sympy.abc import x
>>> rsolve_hyper([-1, -1, 1], 0, x)
C0*(1/2 - sqrt(5)/2)**x + C1*(1/2 + sqrt(5)/2)**x
>>> rsolve_hyper([-1, 1], 1 + x, x)
C0 + x*(x + 1)/2
References
==========
.. [1] M. Petkovsek, Hypergeometric solutions of linear recurrences
with polynomial coefficients, J. Symbolic Computation,
14 (1992), 243-264.
.. [2] M. Petkovsek, H. S. Wilf, D. Zeilberger, A = B, 1996.
"""
coeffs = list(map(sympify, coeffs))
f = sympify(f)
r, kernel, symbols = len(coeffs) - 1, [], set()
if not f.is_zero:
if f.is_Add:
similar = {}
for g in f.expand().args:
if not g.is_hypergeometric(n):
return None
for h in similar.keys():
if hypersimilar(g, h, n):
similar[h] += g
break
else:
similar[g] = S.Zero
inhomogeneous = [g + h for g, h in similar.items()]
elif f.is_hypergeometric(n):
inhomogeneous = [f]
else:
return None
for i, g in enumerate(inhomogeneous):
coeff, polys = S.One, coeffs[:]
denoms = [S.One]*(r + 1)
s = hypersimp(g, n)
for j in range(1, r + 1):
coeff *= s.subs(n, n + j - 1)
p, q = coeff.as_numer_denom()
polys[j] *= p
denoms[j] = q
for j in range(r + 1):
polys[j] *= Mul(*(denoms[:j] + denoms[j + 1:]))
# FIXME: The call to rsolve_ratio below should suffice (rsolve_poly
# call can be removed) but the XFAIL test_rsolve_ratio_missed must
# be fixed first.
R = rsolve_ratio(polys, Mul(*denoms), n, symbols=True)
if R is not None:
R, syms = R
if syms:
R = R.subs(zip(syms, [0]*len(syms)))
else:
R = rsolve_poly(polys, Mul(*denoms), n)
if R:
inhomogeneous[i] *= R
else:
return None
result = Add(*inhomogeneous)
result = simplify(result)
else:
result = S.Zero
Z = Dummy('Z')
p, q = coeffs[0], coeffs[r].subs(n, n - r + 1)
p_factors = [z for z in roots(p, n).keys()]
q_factors = [z for z in roots(q, n).keys()]
factors = [(S.One, S.One)]
for p in p_factors:
for q in q_factors:
if p.is_integer and q.is_integer and p <= q:
continue
else:
factors += [(n - p, n - q)]
p = [(n - p, S.One) for p in p_factors]
q = [(S.One, n - q) for q in q_factors]
factors = p + factors + q
for A, B in factors:
polys, degrees = [], []
D = A*B.subs(n, n + r - 1)
for i in range(r + 1):
a = Mul(*[A.subs(n, n + j) for j in range(i)])
b = Mul(*[B.subs(n, n + j) for j in range(i, r)])
poly = quo(coeffs[i]*a*b, D, n)
polys.append(poly.as_poly(n))
if not poly.is_zero:
degrees.append(polys[i].degree())
if degrees:
d, poly = max(degrees), S.Zero
else:
return None
for i in range(r + 1):
coeff = polys[i].nth(d)
if coeff is not S.Zero:
poly += coeff * Z**i
for z in roots(poly, Z).keys():
if z.is_zero:
continue
recurr_coeffs = [polys[i].as_expr()*z**i for i in range(r + 1)]
if d == 0 and 0 != Add(*[recurr_coeffs[j]*j for j in range(1, r + 1)]):
# faster inline check (than calling rsolve_poly) for a
# constant solution to a constant coefficient recurrence.
sol = [Symbol("C" + str(len(symbols)))]
else:
sol, syms = rsolve_poly(recurr_coeffs, 0, n, len(symbols), symbols=True)
sol = sol.collect(syms)
sol = [sol.coeff(s) for s in syms]
for C in sol:
ratio = z * A * C.subs(n, n + 1) / B / C
ratio = simplify(ratio)
# If there is a nonnegative root in the denominator of the ratio,
# this indicates that the term y(n_root) is zero, and one should
# start the product with the term y(n_root + 1).
n0 = 0
for n_root in roots(ratio.as_numer_denom()[1], n).keys():
if n_root.has(I):
return None
elif (n0 < (n_root + 1)) == True:
n0 = n_root + 1
K = product(ratio, (n, n0, n - 1))
if K.has(factorial, FallingFactorial, RisingFactorial):
K = simplify(K)
if casoratian(kernel + [K], n, zero=False) != 0:
kernel.append(K)
kernel.sort(key=default_sort_key)
sk = list(zip(numbered_symbols('C'), kernel))
for C, ker in sk:
result += C * ker
if hints.get('symbols', False):
# XXX: This returns the symbols in a non-deterministic order
symbols |= {s for s, k in sk}
return (result, list(symbols))
else:
return result
def rsolve(f, y, init=None):
r"""
Solve univariate recurrence with rational coefficients.
Given `k`-th order linear recurrence `\operatorname{L} y = f`,
or equivalently:
.. math:: a_{k}(n) y(n+k) + a_{k-1}(n) y(n+k-1) +
\cdots + a_{0}(n) y(n) = f(n)
where `a_{i}(n)`, for `i=0, \ldots, k`, are polynomials or rational
functions in `n`, and `f` is a hypergeometric function or a sum
of a fixed number of pairwise dissimilar hypergeometric terms in
`n`, finds all solutions or returns ``None``, if none were found.
Initial conditions can be given as a dictionary in two forms:
(1) ``{ n_0 : v_0, n_1 : v_1, ..., n_m : v_m}``
(2) ``{y(n_0) : v_0, y(n_1) : v_1, ..., y(n_m) : v_m}``
or as a list ``L`` of values:
``L = [v_0, v_1, ..., v_m]``
where ``L[i] = v_i``, for `i=0, \ldots, m`, maps to `y(n_i)`.
Examples
========
Lets consider the following recurrence:
.. math:: (n - 1) y(n + 2) - (n^2 + 3 n - 2) y(n + 1) +
2 n (n + 1) y(n) = 0
>>> from sympy import Function, rsolve
>>> from sympy.abc import n
>>> y = Function('y')
>>> f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
>>> rsolve(f, y(n))
2**n*C0 + C1*factorial(n)
>>> rsolve(f, y(n), {y(0):0, y(1):3})
3*2**n - 3*factorial(n)
See Also
========
rsolve_poly, rsolve_ratio, rsolve_hyper
"""
if isinstance(f, Equality):
f = f.lhs - f.rhs
n = y.args[0]
k = Wild('k', exclude=(n,))
# Preprocess user input to allow things like
# y(n) + a*(y(n + 1) + y(n - 1))/2
f = f.expand().collect(y.func(Wild('m', integer=True)))
h_part = defaultdict(list)
i_part = []
for g in Add.make_args(f):
coeff, dep = g.as_coeff_mul(y.func)
if not dep:
i_part.append(coeff)
continue
for h in dep:
if h.is_Function and h.func == y.func:
result = h.args[0].match(n + k)
if result is not None:
h_part[int(result[k])].append(coeff)
continue
raise ValueError(
"'%s(%s + k)' expected, got '%s'" % (y.func, n, h))
for k in h_part:
h_part[k] = Add(*h_part[k])
h_part.default_factory = lambda: 0
i_part = Add(*i_part)
for k, coeff in h_part.items():
h_part[k] = simplify(coeff)
common = S.One
if not i_part.is_zero and not i_part.is_hypergeometric(n) and \
not (i_part.is_Add and all(map(lambda x: x.is_hypergeometric(n), i_part.expand().args))):
raise ValueError("The independent term should be a sum of hypergeometric functions, got '%s'" % i_part)
for coeff in h_part.values():
if coeff.is_rational_function(n):
if not coeff.is_polynomial(n):
common = lcm(common, coeff.as_numer_denom()[1], n)
else:
raise ValueError(
"Polynomial or rational function expected, got '%s'" % coeff)
i_numer, i_denom = i_part.as_numer_denom()
if i_denom.is_polynomial(n):
common = lcm(common, i_denom, n)
if common is not S.One:
for k, coeff in h_part.items():
numer, denom = coeff.as_numer_denom()
h_part[k] = numer*quo(common, denom, n)
i_part = i_numer*quo(common, i_denom, n)
K_min = min(h_part.keys())
if K_min < 0:
K = abs(K_min)
H_part = defaultdict(lambda: S.Zero)
i_part = i_part.subs(n, n + K).expand()
common = common.subs(n, n + K).expand()
for k, coeff in h_part.items():
H_part[k + K] = coeff.subs(n, n + K).expand()
else:
H_part = h_part
K_max = max(H_part.keys())
coeffs = [H_part[i] for i in range(K_max + 1)]
result = rsolve_hyper(coeffs, -i_part, n, symbols=True)
if result is None:
return None
solution, symbols = result
if init in ({}, []):
init = None
if symbols and init is not None:
if isinstance(init, list):
init = {i: init[i] for i in range(len(init))}
equations = []
for k, v in init.items():
try:
i = int(k)
except TypeError:
if k.is_Function and k.func == y.func:
i = int(k.args[0])
else:
raise ValueError("Integer or term expected, got '%s'" % k)
eq = solution.subs(n, i) - v
if eq.has(S.NaN):
eq = solution.limit(n, i) - v
equations.append(eq)
result = solve(equations, *symbols)
if not result:
return None
else:
solution = solution.subs(result)
return solution
|
ce82e5385b00f3efa500deec69e0b3c798b6757218717a582bc0ae87addebe28 | """
This module contains functions to:
- solve a single equation for a single variable, in any domain either real or complex.
- solve a single transcendental equation for a single variable in any domain either real or complex.
(currently supports solving in real domain only)
- solve a system of linear equations with N variables and M equations.
- solve a system of Non Linear Equations with N variables and M equations
"""
from sympy.core.sympify import sympify
from sympy.core import (S, Pow, Dummy, pi, Expr, Wild, Mul, Equality,
Add, Basic)
from sympy.core.containers import Tuple
from sympy.core.function import (Lambda, expand_complex, AppliedUndef,
expand_log, _mexpand, expand_trig, nfloat)
from sympy.core.mod import Mod
from sympy.core.numbers import igcd, I, Number, Rational, oo, ilcm
from sympy.core.power import integer_log
from sympy.core.relational import Eq, Ne, Relational
from sympy.core.sorting import default_sort_key, ordered
from sympy.core.symbol import Symbol, _uniquely_named_symbol
from sympy.core.sympify import _sympify
from sympy.polys.matrices.linsolve import _linear_eq_to_dict
from sympy.polys.polyroots import UnsolvableFactorError
from sympy.simplify.simplify import simplify, fraction, trigsimp, nsimplify
from sympy.simplify import powdenest, logcombine
from sympy.functions import (log, tan, cot, sin, cos, sec, csc, exp,
acos, asin, acsc, asec,
piecewise_fold, Piecewise)
from sympy.functions.elementary.complexes import Abs, arg, re, im
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
from sympy.functions.elementary.miscellaneous import real_root
from sympy.functions.elementary.trigonometric import TrigonometricFunction
from sympy.logic.boolalg import And, BooleanTrue
from sympy.sets import (FiniteSet, imageset, Interval, Intersection,
Union, ConditionSet, ImageSet, Complement, Contains)
from sympy.sets.sets import Set, ProductSet
from sympy.matrices import zeros, Matrix, MatrixBase
from sympy.ntheory import totient
from sympy.ntheory.factor_ import divisors
from sympy.ntheory.residue_ntheory import discrete_log, nthroot_mod
from sympy.polys import (roots, Poly, degree, together, PolynomialError,
RootOf, factor, lcm, gcd)
from sympy.polys.polyerrors import CoercionFailed
from sympy.polys.polytools import invert, groebner, poly
from sympy.polys.solvers import (sympy_eqs_to_ring, solve_lin_sys,
PolyNonlinearError)
from sympy.polys.matrices.linsolve import _linsolve
from sympy.solvers.solvers import (checksol, denoms, unrad,
_simple_dens, recast_to_symbols)
from sympy.solvers.polysys import solve_poly_system
from sympy.utilities import filldedent
from sympy.utilities.iterables import (numbered_symbols, has_dups,
is_sequence, iterable)
from sympy.calculus.util import periodicity, continuous_domain, function_range
from types import GeneratorType
class NonlinearError(ValueError):
"""Raised when unexpectedly encountering nonlinear equations"""
pass
_rc = Dummy("R", real=True), Dummy("C", complex=True)
def _masked(f, *atoms):
"""Return ``f``, with all objects given by ``atoms`` replaced with
Dummy symbols, ``d``, and the list of replacements, ``(d, e)``,
where ``e`` is an object of type given by ``atoms`` in which
any other instances of atoms have been recursively replaced with
Dummy symbols, too. The tuples are ordered so that if they are
applied in sequence, the origin ``f`` will be restored.
Examples
========
>>> from sympy import cos
>>> from sympy.abc import x
>>> from sympy.solvers.solveset import _masked
>>> f = cos(cos(x) + 1)
>>> f, reps = _masked(cos(1 + cos(x)), cos)
>>> f
_a1
>>> reps
[(_a1, cos(_a0 + 1)), (_a0, cos(x))]
>>> for d, e in reps:
... f = f.xreplace({d: e})
>>> f
cos(cos(x) + 1)
"""
sym = numbered_symbols('a', cls=Dummy, real=True)
mask = []
for a in ordered(f.atoms(*atoms)):
for i in mask:
a = a.replace(*i)
mask.append((a, next(sym)))
for i, (o, n) in enumerate(mask):
f = f.replace(o, n)
mask[i] = (n, o)
mask = list(reversed(mask))
return f, mask
def _invert(f_x, y, x, domain=S.Complexes):
r"""
Reduce the complex valued equation $f(x) = y$ to a set of equations
$$\left\{g(x) = h_1(y),\ g(x) = h_2(y),\ \dots,\ g(x) = h_n(y) \right\}$$
where $g(x)$ is a simpler function than $f(x)$. The return value is a tuple
$(g(x), \mathrm{set}_h)$, where $g(x)$ is a function of $x$ and $\mathrm{set}_h$ is
the set of function $\left\{h_1(y), h_2(y), \dots, h_n(y)\right\}$.
Here, $y$ is not necessarily a symbol.
$\mathrm{set}_h$ contains the functions, along with the information
about the domain in which they are valid, through set
operations. For instance, if :math:`y = |x| - n` is inverted
in the real domain, then $\mathrm{set}_h$ is not simply
$\{-n, n\}$ as the nature of `n` is unknown; rather, it is:
$$ \left(\left[0, \infty\right) \cap \left\{n\right\}\right) \cup
\left(\left(-\infty, 0\right] \cap \left\{- n\right\}\right)$$
By default, the complex domain is used which means that inverting even
seemingly simple functions like $\exp(x)$ will give very different
results from those obtained in the real domain.
(In the case of $\exp(x)$, the inversion via $\log$ is multi-valued
in the complex domain, having infinitely many branches.)
If you are working with real values only (or you are not sure which
function to use) you should probably set the domain to
``S.Reals`` (or use ``invert_real`` which does that automatically).
Examples
========
>>> from sympy.solvers.solveset import invert_complex, invert_real
>>> from sympy.abc import x, y
>>> from sympy import exp
When does exp(x) == y?
>>> invert_complex(exp(x), y, x)
(x, ImageSet(Lambda(_n, I*(2*_n*pi + arg(y)) + log(Abs(y))), Integers))
>>> invert_real(exp(x), y, x)
(x, Intersection({log(y)}, Reals))
When does exp(x) == 1?
>>> invert_complex(exp(x), 1, x)
(x, ImageSet(Lambda(_n, 2*_n*I*pi), Integers))
>>> invert_real(exp(x), 1, x)
(x, {0})
See Also
========
invert_real, invert_complex
"""
x = sympify(x)
if not x.is_Symbol:
raise ValueError("x must be a symbol")
f_x = sympify(f_x)
if x not in f_x.free_symbols:
raise ValueError("Inverse of constant function doesn't exist")
y = sympify(y)
if x in y.free_symbols:
raise ValueError("y should be independent of x ")
if domain.is_subset(S.Reals):
x1, s = _invert_real(f_x, FiniteSet(y), x)
else:
x1, s = _invert_complex(f_x, FiniteSet(y), x)
if not isinstance(s, FiniteSet) or x1 != x:
return x1, s
# Avoid adding gratuitous intersections with S.Complexes. Actual
# conditions should be handled by the respective inverters.
if domain is S.Complexes:
return x1, s
else:
return x1, s.intersection(domain)
invert_complex = _invert
def invert_real(f_x, y, x):
"""
Inverts a real-valued function. Same as :func:`invert_complex`, but sets
the domain to ``S.Reals`` before inverting.
"""
return _invert(f_x, y, x, S.Reals)
def _invert_real(f, g_ys, symbol):
"""Helper function for _invert."""
if f == symbol or g_ys is S.EmptySet:
return (f, g_ys)
n = Dummy('n', real=True)
if isinstance(f, exp) or (f.is_Pow and f.base == S.Exp1):
return _invert_real(f.exp,
imageset(Lambda(n, log(n)), g_ys),
symbol)
if hasattr(f, 'inverse') and f.inverse() is not None and not isinstance(f, (
TrigonometricFunction,
HyperbolicFunction,
)):
if len(f.args) > 1:
raise ValueError("Only functions with one argument are supported.")
return _invert_real(f.args[0],
imageset(Lambda(n, f.inverse()(n)), g_ys),
symbol)
if isinstance(f, Abs):
return _invert_abs(f.args[0], g_ys, symbol)
if f.is_Add:
# f = g + h
g, h = f.as_independent(symbol)
if g is not S.Zero:
return _invert_real(h, imageset(Lambda(n, n - g), g_ys), symbol)
if f.is_Mul:
# f = g*h
g, h = f.as_independent(symbol)
if g is not S.One:
return _invert_real(h, imageset(Lambda(n, n/g), g_ys), symbol)
if f.is_Pow:
base, expo = f.args
base_has_sym = base.has(symbol)
expo_has_sym = expo.has(symbol)
if not expo_has_sym:
if expo.is_rational:
num, den = expo.as_numer_denom()
if den % 2 == 0 and num % 2 == 1 and den.is_zero is False:
# Here we have f(x)**(num/den) = y
# where den is nonzero and even and y is an element
# of the set g_ys.
# den is even, so we are only interested in the cases
# where both f(x) and y are positive.
# Restricting y to be positive (using the set g_ys_pos)
# means that y**(den/num) is always positive.
# Therefore it isn't necessary to also constrain f(x)
# to be positive because we are only going to
# find solutions of f(x) = y**(d/n)
# where the rhs is already required to be positive.
root = Lambda(n, real_root(n, expo))
g_ys_pos = g_ys & Interval(0, oo)
res = imageset(root, g_ys_pos)
_inv, _set = _invert_real(base, res, symbol)
return (_inv, _set)
if den % 2 == 1:
root = Lambda(n, real_root(n, expo))
res = imageset(root, g_ys)
if num % 2 == 0:
neg_res = imageset(Lambda(n, -n), res)
return _invert_real(base, res + neg_res, symbol)
if num % 2 == 1:
return _invert_real(base, res, symbol)
elif expo.is_irrational:
root = Lambda(n, real_root(n, expo))
g_ys_pos = g_ys & Interval(0, oo)
res = imageset(root, g_ys_pos)
return _invert_real(base, res, symbol)
else:
# indeterminate exponent, e.g. Float or parity of
# num, den of rational could not be determined
pass # use default return
if not base_has_sym:
rhs = g_ys.args[0]
if base.is_positive:
return _invert_real(expo,
imageset(Lambda(n, log(n, base, evaluate=False)), g_ys), symbol)
elif base.is_negative:
s, b = integer_log(rhs, base)
if b:
return _invert_real(expo, FiniteSet(s), symbol)
else:
return (expo, S.EmptySet)
elif base.is_zero:
one = Eq(rhs, 1)
if one == S.true:
# special case: 0**x - 1
return _invert_real(expo, FiniteSet(0), symbol)
elif one == S.false:
return (expo, S.EmptySet)
if isinstance(f, TrigonometricFunction):
if isinstance(g_ys, FiniteSet):
def inv(trig):
if isinstance(trig, (sin, csc)):
F = asin if isinstance(trig, sin) else acsc
return (lambda a: n*pi + S.NegativeOne**n*F(a),)
if isinstance(trig, (cos, sec)):
F = acos if isinstance(trig, cos) else asec
return (
lambda a: 2*n*pi + F(a),
lambda a: 2*n*pi - F(a),)
if isinstance(trig, (tan, cot)):
return (lambda a: n*pi + trig.inverse()(a),)
n = Dummy('n', integer=True)
invs = S.EmptySet
for L in inv(f):
invs += Union(*[imageset(Lambda(n, L(g)), S.Integers) for g in g_ys])
return _invert_real(f.args[0], invs, symbol)
return (f, g_ys)
def _invert_complex(f, g_ys, symbol):
"""Helper function for _invert."""
if f == symbol or g_ys is S.EmptySet:
return (f, g_ys)
n = Dummy('n')
if f.is_Add:
# f = g + h
g, h = f.as_independent(symbol)
if g is not S.Zero:
return _invert_complex(h, imageset(Lambda(n, n - g), g_ys), symbol)
if f.is_Mul:
# f = g*h
g, h = f.as_independent(symbol)
if g is not S.One:
if g in {S.NegativeInfinity, S.ComplexInfinity, S.Infinity}:
return (h, S.EmptySet)
return _invert_complex(h, imageset(Lambda(n, n/g), g_ys), symbol)
if f.is_Pow:
base, expo = f.args
# special case: g**r = 0
# Could be improved like `_invert_real` to handle more general cases.
if expo.is_Rational and g_ys == FiniteSet(0):
if expo.is_positive:
return _invert_complex(base, g_ys, symbol)
if hasattr(f, 'inverse') and f.inverse() is not None and \
not isinstance(f, TrigonometricFunction) and \
not isinstance(f, HyperbolicFunction) and \
not isinstance(f, exp):
if len(f.args) > 1:
raise ValueError("Only functions with one argument are supported.")
return _invert_complex(f.args[0],
imageset(Lambda(n, f.inverse()(n)), g_ys), symbol)
if isinstance(f, exp) or (f.is_Pow and f.base == S.Exp1):
if isinstance(g_ys, ImageSet):
# can solve upto `(d*exp(exp(...(exp(a*x + b))...) + c)` format.
# Further can be improved to `(d*exp(exp(...(exp(a*x**n + b*x**(n-1) + ... + f))...) + c)`.
g_ys_expr = g_ys.lamda.expr
g_ys_vars = g_ys.lamda.variables
k = Dummy('k{}'.format(len(g_ys_vars)))
g_ys_vars_1 = (k,) + g_ys_vars
exp_invs = Union(*[imageset(Lambda((g_ys_vars_1,), (I*(2*k*pi + arg(g_ys_expr))
+ log(Abs(g_ys_expr)))), S.Integers**(len(g_ys_vars_1)))])
return _invert_complex(f.exp, exp_invs, symbol)
elif isinstance(g_ys, FiniteSet):
exp_invs = Union(*[imageset(Lambda(n, I*(2*n*pi + arg(g_y)) +
log(Abs(g_y))), S.Integers)
for g_y in g_ys if g_y != 0])
return _invert_complex(f.exp, exp_invs, symbol)
return (f, g_ys)
def _invert_abs(f, g_ys, symbol):
"""Helper function for inverting absolute value functions.
Returns the complete result of inverting an absolute value
function along with the conditions which must also be satisfied.
If it is certain that all these conditions are met, a :class:`~.FiniteSet`
of all possible solutions is returned. If any condition cannot be
satisfied, an :class:`~.EmptySet` is returned. Otherwise, a
:class:`~.ConditionSet` of the solutions, with all the required conditions
specified, is returned.
"""
if not g_ys.is_FiniteSet:
# this could be used for FiniteSet, but the
# results are more compact if they aren't, e.g.
# ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n}) vs
# Union(Intersection(Interval(0, oo), {n}), Intersection(Interval(-oo, 0), {-n}))
# for the solution of abs(x) - n
pos = Intersection(g_ys, Interval(0, S.Infinity))
parg = _invert_real(f, pos, symbol)
narg = _invert_real(-f, pos, symbol)
if parg[0] != narg[0]:
raise NotImplementedError
return parg[0], Union(narg[1], parg[1])
# check conditions: all these must be true. If any are unknown
# then return them as conditions which must be satisfied
unknown = []
for a in g_ys.args:
ok = a.is_nonnegative if a.is_Number else a.is_positive
if ok is None:
unknown.append(a)
elif not ok:
return symbol, S.EmptySet
if unknown:
conditions = And(*[Contains(i, Interval(0, oo))
for i in unknown])
else:
conditions = True
n = Dummy('n', real=True)
# this is slightly different than above: instead of solving
# +/-f on positive values, here we solve for f on +/- g_ys
g_x, values = _invert_real(f, Union(
imageset(Lambda(n, n), g_ys),
imageset(Lambda(n, -n), g_ys)), symbol)
return g_x, ConditionSet(g_x, conditions, values)
def domain_check(f, symbol, p):
"""Returns False if point p is infinite or any subexpression of f
is infinite or becomes so after replacing symbol with p. If none of
these conditions is met then True will be returned.
Examples
========
>>> from sympy import Mul, oo
>>> from sympy.abc import x
>>> from sympy.solvers.solveset import domain_check
>>> g = 1/(1 + (1/(x + 1))**2)
>>> domain_check(g, x, -1)
False
>>> domain_check(x**2, x, 0)
True
>>> domain_check(1/x, x, oo)
False
* The function relies on the assumption that the original form
of the equation has not been changed by automatic simplification.
>>> domain_check(x/x, x, 0) # x/x is automatically simplified to 1
True
* To deal with automatic evaluations use evaluate=False:
>>> domain_check(Mul(x, 1/x, evaluate=False), x, 0)
False
"""
f, p = sympify(f), sympify(p)
if p.is_infinite:
return False
return _domain_check(f, symbol, p)
def _domain_check(f, symbol, p):
# helper for domain check
if f.is_Atom and f.is_finite:
return True
elif f.subs(symbol, p).is_infinite:
return False
elif isinstance(f, Piecewise):
# Check the cases of the Piecewise in turn. There might be invalid
# expressions in later cases that don't apply e.g.
# solveset(Piecewise((0, Eq(x, 0)), (1/x, True)), x)
for expr, cond in f.args:
condsubs = cond.subs(symbol, p)
if condsubs is S.false:
continue
elif condsubs is S.true:
return _domain_check(expr, symbol, p)
else:
# We don't know which case of the Piecewise holds. On this
# basis we cannot decide whether any solution is in or out of
# the domain. Ideally this function would allow returning a
# symbolic condition for the validity of the solution that
# could be handled in the calling code. In the mean time we'll
# give this particular solution the benefit of the doubt and
# let it pass.
return True
else:
# TODO : We should not blindly recurse through all args of arbitrary expressions like this
return all(_domain_check(g, symbol, p)
for g in f.args)
def _is_finite_with_finite_vars(f, domain=S.Complexes):
"""
Return True if the given expression is finite. For symbols that
do not assign a value for `complex` and/or `real`, the domain will
be used to assign a value; symbols that do not assign a value
for `finite` will be made finite. All other assumptions are
left unmodified.
"""
def assumptions(s):
A = s.assumptions0
A.setdefault('finite', A.get('finite', True))
if domain.is_subset(S.Reals):
# if this gets set it will make complex=True, too
A.setdefault('real', True)
else:
# don't change 'real' because being complex implies
# nothing about being real
A.setdefault('complex', True)
return A
reps = {s: Dummy(**assumptions(s)) for s in f.free_symbols}
return f.xreplace(reps).is_finite
def _is_function_class_equation(func_class, f, symbol):
""" Tests whether the equation is an equation of the given function class.
The given equation belongs to the given function class if it is
comprised of functions of the function class which are multiplied by
or added to expressions independent of the symbol. In addition, the
arguments of all such functions must be linear in the symbol as well.
Examples
========
>>> from sympy.solvers.solveset import _is_function_class_equation
>>> from sympy import tan, sin, tanh, sinh, exp
>>> from sympy.abc import x
>>> from sympy.functions.elementary.trigonometric import TrigonometricFunction
>>> from sympy.functions.elementary.hyperbolic import HyperbolicFunction
>>> _is_function_class_equation(TrigonometricFunction, exp(x) + tan(x), x)
False
>>> _is_function_class_equation(TrigonometricFunction, tan(x) + sin(x), x)
True
>>> _is_function_class_equation(TrigonometricFunction, tan(x**2), x)
False
>>> _is_function_class_equation(TrigonometricFunction, tan(x + 2), x)
True
>>> _is_function_class_equation(HyperbolicFunction, tanh(x) + sinh(x), x)
True
"""
if f.is_Mul or f.is_Add:
return all(_is_function_class_equation(func_class, arg, symbol)
for arg in f.args)
if f.is_Pow:
if not f.exp.has(symbol):
return _is_function_class_equation(func_class, f.base, symbol)
else:
return False
if not f.has(symbol):
return True
if isinstance(f, func_class):
try:
g = Poly(f.args[0], symbol)
return g.degree() <= 1
except PolynomialError:
return False
else:
return False
def _solve_as_rational(f, symbol, domain):
""" solve rational functions"""
f = together(_mexpand(f, recursive=True), deep=True)
g, h = fraction(f)
if not h.has(symbol):
try:
return _solve_as_poly(g, symbol, domain)
except NotImplementedError:
# The polynomial formed from g could end up having
# coefficients in a ring over which finding roots
# isn't implemented yet, e.g. ZZ[a] for some symbol a
return ConditionSet(symbol, Eq(f, 0), domain)
except CoercionFailed:
# contained oo, zoo or nan
return S.EmptySet
else:
valid_solns = _solveset(g, symbol, domain)
invalid_solns = _solveset(h, symbol, domain)
return valid_solns - invalid_solns
class _SolveTrig1Error(Exception):
"""Raised when _solve_trig1 heuristics do not apply"""
def _solve_trig(f, symbol, domain):
"""Function to call other helpers to solve trigonometric equations """
sol = None
try:
sol = _solve_trig1(f, symbol, domain)
except _SolveTrig1Error:
try:
sol = _solve_trig2(f, symbol, domain)
except ValueError:
raise NotImplementedError(filldedent('''
Solution to this kind of trigonometric equations
is yet to be implemented'''))
return sol
def _solve_trig1(f, symbol, domain):
"""Primary solver for trigonometric and hyperbolic equations
Returns either the solution set as a ConditionSet (auto-evaluated to a
union of ImageSets if no variables besides 'symbol' are involved) or
raises _SolveTrig1Error if f == 0 cannot be solved.
Notes
=====
Algorithm:
1. Do a change of variable x -> mu*x in arguments to trigonometric and
hyperbolic functions, in order to reduce them to small integers. (This
step is crucial to keep the degrees of the polynomials of step 4 low.)
2. Rewrite trigonometric/hyperbolic functions as exponentials.
3. Proceed to a 2nd change of variable, replacing exp(I*x) or exp(x) by y.
4. Solve the resulting rational equation.
5. Use invert_complex or invert_real to return to the original variable.
6. If the coefficients of 'symbol' were symbolic in nature, add the
necessary consistency conditions in a ConditionSet.
"""
# Prepare change of variable
x = Dummy('x')
if _is_function_class_equation(HyperbolicFunction, f, symbol):
cov = exp(x)
inverter = invert_real if domain.is_subset(S.Reals) else invert_complex
else:
cov = exp(I*x)
inverter = invert_complex
f = trigsimp(f)
f_original = f
trig_functions = f.atoms(TrigonometricFunction, HyperbolicFunction)
trig_arguments = [e.args[0] for e in trig_functions]
# trigsimp may have reduced the equation to an expression
# that is independent of 'symbol' (e.g. cos**2+sin**2)
if not any(a.has(symbol) for a in trig_arguments):
return solveset(f_original, symbol, domain)
denominators = []
numerators = []
for ar in trig_arguments:
try:
poly_ar = Poly(ar, symbol)
except PolynomialError:
raise _SolveTrig1Error("trig argument is not a polynomial")
if poly_ar.degree() > 1: # degree >1 still bad
raise _SolveTrig1Error("degree of variable must not exceed one")
if poly_ar.degree() == 0: # degree 0, don't care
continue
c = poly_ar.all_coeffs()[0] # got the coefficient of 'symbol'
numerators.append(fraction(c)[0])
denominators.append(fraction(c)[1])
mu = lcm(denominators)/gcd(numerators)
f = f.subs(symbol, mu*x)
f = f.rewrite(exp)
f = together(f)
g, h = fraction(f)
y = Dummy('y')
g, h = g.expand(), h.expand()
g, h = g.subs(cov, y), h.subs(cov, y)
if g.has(x) or h.has(x):
raise _SolveTrig1Error("change of variable not possible")
solns = solveset_complex(g, y) - solveset_complex(h, y)
if isinstance(solns, ConditionSet):
raise _SolveTrig1Error("polynomial has ConditionSet solution")
if isinstance(solns, FiniteSet):
if any(isinstance(s, RootOf) for s in solns):
raise _SolveTrig1Error("polynomial results in RootOf object")
# revert the change of variable
cov = cov.subs(x, symbol/mu)
result = Union(*[inverter(cov, s, symbol)[1] for s in solns])
# In case of symbolic coefficients, the solution set is only valid
# if numerator and denominator of mu are non-zero.
if mu.has(Symbol):
syms = (mu).atoms(Symbol)
munum, muden = fraction(mu)
condnum = munum.as_independent(*syms, as_Add=False)[1]
condden = muden.as_independent(*syms, as_Add=False)[1]
cond = And(Ne(condnum, 0), Ne(condden, 0))
else:
cond = True
# Actual conditions are returned as part of the ConditionSet. Adding an
# intersection with C would only complicate some solution sets due to
# current limitations of intersection code. (e.g. #19154)
if domain is S.Complexes:
# This is a slight abuse of ConditionSet. Ideally this should
# be some kind of "PiecewiseSet". (See #19507 discussion)
return ConditionSet(symbol, cond, result)
else:
return ConditionSet(symbol, cond, Intersection(result, domain))
elif solns is S.EmptySet:
return S.EmptySet
else:
raise _SolveTrig1Error("polynomial solutions must form FiniteSet")
def _solve_trig2(f, symbol, domain):
"""Secondary helper to solve trigonometric equations,
called when first helper fails """
f = trigsimp(f)
f_original = f
trig_functions = f.atoms(sin, cos, tan, sec, cot, csc)
trig_arguments = [e.args[0] for e in trig_functions]
denominators = []
numerators = []
# todo: This solver can be extended to hyperbolics if the
# analogous change of variable to tanh (instead of tan)
# is used.
if not trig_functions:
return ConditionSet(symbol, Eq(f_original, 0), domain)
# todo: The pre-processing below (extraction of numerators, denominators,
# gcd, lcm, mu, etc.) should be updated to the enhanced version in
# _solve_trig1. (See #19507)
for ar in trig_arguments:
try:
poly_ar = Poly(ar, symbol)
except PolynomialError:
raise ValueError("give up, we cannot solve if this is not a polynomial in x")
if poly_ar.degree() > 1: # degree >1 still bad
raise ValueError("degree of variable inside polynomial should not exceed one")
if poly_ar.degree() == 0: # degree 0, don't care
continue
c = poly_ar.all_coeffs()[0] # got the coefficient of 'symbol'
try:
numerators.append(Rational(c).p)
denominators.append(Rational(c).q)
except TypeError:
return ConditionSet(symbol, Eq(f_original, 0), domain)
x = Dummy('x')
# ilcm() and igcd() require more than one argument
if len(numerators) > 1:
mu = Rational(2)*ilcm(*denominators)/igcd(*numerators)
else:
assert len(numerators) == 1
mu = Rational(2)*denominators[0]/numerators[0]
f = f.subs(symbol, mu*x)
f = f.rewrite(tan)
f = expand_trig(f)
f = together(f)
g, h = fraction(f)
y = Dummy('y')
g, h = g.expand(), h.expand()
g, h = g.subs(tan(x), y), h.subs(tan(x), y)
if g.has(x) or h.has(x):
return ConditionSet(symbol, Eq(f_original, 0), domain)
solns = solveset(g, y, S.Reals) - solveset(h, y, S.Reals)
if isinstance(solns, FiniteSet):
result = Union(*[invert_real(tan(symbol/mu), s, symbol)[1]
for s in solns])
dsol = invert_real(tan(symbol/mu), oo, symbol)[1]
if degree(h) > degree(g): # If degree(denom)>degree(num) then there
result = Union(result, dsol) # would be another sol at Lim(denom-->oo)
return Intersection(result, domain)
elif solns is S.EmptySet:
return S.EmptySet
else:
return ConditionSet(symbol, Eq(f_original, 0), S.Reals)
def _solve_as_poly(f, symbol, domain=S.Complexes):
"""
Solve the equation using polynomial techniques if it already is a
polynomial equation or, with a change of variables, can be made so.
"""
result = None
if f.is_polynomial(symbol):
solns = roots(f, symbol, cubics=True, quartics=True,
quintics=True, domain='EX')
num_roots = sum(solns.values())
if degree(f, symbol) <= num_roots:
result = FiniteSet(*solns.keys())
else:
poly = Poly(f, symbol)
solns = poly.all_roots()
if poly.degree() <= len(solns):
result = FiniteSet(*solns)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
poly = Poly(f)
if poly is None:
result = ConditionSet(symbol, Eq(f, 0), domain)
gens = [g for g in poly.gens if g.has(symbol)]
if len(gens) == 1:
poly = Poly(poly, gens[0])
gen = poly.gen
deg = poly.degree()
poly = Poly(poly.as_expr(), poly.gen, composite=True)
poly_solns = FiniteSet(*roots(poly, cubics=True, quartics=True,
quintics=True).keys())
if len(poly_solns) < deg:
result = ConditionSet(symbol, Eq(f, 0), domain)
if gen != symbol:
y = Dummy('y')
inverter = invert_real if domain.is_subset(S.Reals) else invert_complex
lhs, rhs_s = inverter(gen, y, symbol)
if lhs == symbol:
result = Union(*[rhs_s.subs(y, s) for s in poly_solns])
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
else:
result = ConditionSet(symbol, Eq(f, 0), domain)
if result is not None:
if isinstance(result, FiniteSet):
# this is to simplify solutions like -sqrt(-I) to sqrt(2)/2
# - sqrt(2)*I/2. We are not expanding for solution with symbols
# or undefined functions because that makes the solution more complicated.
# For example, expand_complex(a) returns re(a) + I*im(a)
if all(s.atoms(Symbol, AppliedUndef) == set() and not isinstance(s, RootOf)
for s in result):
s = Dummy('s')
result = imageset(Lambda(s, expand_complex(s)), result)
if isinstance(result, FiniteSet) and domain != S.Complexes:
# Avoid adding gratuitous intersections with S.Complexes. Actual
# conditions should be handled elsewhere.
result = result.intersection(domain)
return result
else:
return ConditionSet(symbol, Eq(f, 0), domain)
def _solve_radical(f, unradf, symbol, solveset_solver):
""" Helper function to solve equations with radicals """
res = unradf
eq, cov = res if res else (f, [])
if not cov:
result = solveset_solver(eq, symbol) - \
Union(*[solveset_solver(g, symbol) for g in denoms(f, symbol)])
else:
y, yeq = cov
if not solveset_solver(y - I, y):
yreal = Dummy('yreal', real=True)
yeq = yeq.xreplace({y: yreal})
eq = eq.xreplace({y: yreal})
y = yreal
g_y_s = solveset_solver(yeq, symbol)
f_y_sols = solveset_solver(eq, y)
result = Union(*[imageset(Lambda(y, g_y), f_y_sols)
for g_y in g_y_s])
if not isinstance(result, FiniteSet):
solution_set = result
else:
f_set = [] # solutions for FiniteSet
c_set = [] # solutions for ConditionSet
for s in result:
if checksol(f, symbol, s):
f_set.append(s)
else:
c_set.append(s)
solution_set = FiniteSet(*f_set) + ConditionSet(symbol, Eq(f, 0), FiniteSet(*c_set))
return solution_set
def _solve_abs(f, symbol, domain):
""" Helper function to solve equation involving absolute value function """
if not domain.is_subset(S.Reals):
raise ValueError(filldedent('''
Absolute values cannot be inverted in the
complex domain.'''))
p, q, r = Wild('p'), Wild('q'), Wild('r')
pattern_match = f.match(p*Abs(q) + r) or {}
f_p, f_q, f_r = [pattern_match.get(i, S.Zero) for i in (p, q, r)]
if not (f_p.is_zero or f_q.is_zero):
domain = continuous_domain(f_q, symbol, domain)
from .inequalities import solve_univariate_inequality
q_pos_cond = solve_univariate_inequality(f_q >= 0, symbol,
relational=False, domain=domain, continuous=True)
q_neg_cond = q_pos_cond.complement(domain)
sols_q_pos = solveset_real(f_p*f_q + f_r,
symbol).intersect(q_pos_cond)
sols_q_neg = solveset_real(f_p*(-f_q) + f_r,
symbol).intersect(q_neg_cond)
return Union(sols_q_pos, sols_q_neg)
else:
return ConditionSet(symbol, Eq(f, 0), domain)
def solve_decomposition(f, symbol, domain):
"""
Function to solve equations via the principle of "Decomposition
and Rewriting".
Examples
========
>>> from sympy import exp, sin, Symbol, pprint, S
>>> from sympy.solvers.solveset import solve_decomposition as sd
>>> x = Symbol('x')
>>> f1 = exp(2*x) - 3*exp(x) + 2
>>> sd(f1, x, S.Reals)
{0, log(2)}
>>> f2 = sin(x)**2 + 2*sin(x) + 1
>>> pprint(sd(f2, x, S.Reals), use_unicode=False)
3*pi
{2*n*pi + ---- | n in Integers}
2
>>> f3 = sin(x + 2)
>>> pprint(sd(f3, x, S.Reals), use_unicode=False)
{2*n*pi - 2 | n in Integers} U {2*n*pi - 2 + pi | n in Integers}
"""
from sympy.solvers.decompogen import decompogen
# decompose the given function
g_s = decompogen(f, symbol)
# `y_s` represents the set of values for which the function `g` is to be
# solved.
# `solutions` represent the solutions of the equations `g = y_s` or
# `g = 0` depending on the type of `y_s`.
# As we are interested in solving the equation: f = 0
y_s = FiniteSet(0)
for g in g_s:
frange = function_range(g, symbol, domain)
y_s = Intersection(frange, y_s)
result = S.EmptySet
if isinstance(y_s, FiniteSet):
for y in y_s:
solutions = solveset(Eq(g, y), symbol, domain)
if not isinstance(solutions, ConditionSet):
result += solutions
else:
if isinstance(y_s, ImageSet):
iter_iset = (y_s,)
elif isinstance(y_s, Union):
iter_iset = y_s.args
elif y_s is S.EmptySet:
# y_s is not in the range of g in g_s, so no solution exists
#in the given domain
return S.EmptySet
for iset in iter_iset:
new_solutions = solveset(Eq(iset.lamda.expr, g), symbol, domain)
dummy_var = tuple(iset.lamda.expr.free_symbols)[0]
(base_set,) = iset.base_sets
if isinstance(new_solutions, FiniteSet):
new_exprs = new_solutions
elif isinstance(new_solutions, Intersection):
if isinstance(new_solutions.args[1], FiniteSet):
new_exprs = new_solutions.args[1]
for new_expr in new_exprs:
result += ImageSet(Lambda(dummy_var, new_expr), base_set)
if result is S.EmptySet:
return ConditionSet(symbol, Eq(f, 0), domain)
y_s = result
return y_s
def _solveset(f, symbol, domain, _check=False):
"""Helper for solveset to return a result from an expression
that has already been sympify'ed and is known to contain the
given symbol."""
# _check controls whether the answer is checked or not
from sympy.simplify.simplify import signsimp
if isinstance(f, BooleanTrue):
return domain
orig_f = f
if f.is_Mul:
coeff, f = f.as_independent(symbol, as_Add=False)
if coeff in {S.ComplexInfinity, S.NegativeInfinity, S.Infinity}:
f = together(orig_f)
elif f.is_Add:
a, h = f.as_independent(symbol)
m, h = h.as_independent(symbol, as_Add=False)
if m not in {S.ComplexInfinity, S.Zero, S.Infinity,
S.NegativeInfinity}:
f = a/m + h # XXX condition `m != 0` should be added to soln
# assign the solvers to use
solver = lambda f, x, domain=domain: _solveset(f, x, domain)
inverter = lambda f, rhs, symbol: _invert(f, rhs, symbol, domain)
result = S.EmptySet
if f.expand().is_zero:
return domain
elif not f.has(symbol):
return S.EmptySet
elif f.is_Mul and all(_is_finite_with_finite_vars(m, domain)
for m in f.args):
# if f(x) and g(x) are both finite we can say that the solution of
# f(x)*g(x) == 0 is same as Union(f(x) == 0, g(x) == 0) is not true in
# general. g(x) can grow to infinitely large for the values where
# f(x) == 0. To be sure that we are not silently allowing any
# wrong solutions we are using this technique only if both f and g are
# finite for a finite input.
result = Union(*[solver(m, symbol) for m in f.args])
elif _is_function_class_equation(TrigonometricFunction, f, symbol) or \
_is_function_class_equation(HyperbolicFunction, f, symbol):
result = _solve_trig(f, symbol, domain)
elif isinstance(f, arg):
a = f.args[0]
result = Intersection(_solveset(re(a) > 0, symbol, domain),
_solveset(im(a), symbol, domain))
elif f.is_Piecewise:
expr_set_pairs = f.as_expr_set_pairs(domain)
for (expr, in_set) in expr_set_pairs:
if in_set.is_Relational:
in_set = in_set.as_set()
solns = solver(expr, symbol, in_set)
result += solns
elif isinstance(f, Eq):
result = solver(Add(f.lhs, - f.rhs, evaluate=False), symbol, domain)
elif f.is_Relational:
from .inequalities import solve_univariate_inequality
try:
result = solve_univariate_inequality(
f, symbol, domain=domain, relational=False)
except NotImplementedError:
result = ConditionSet(symbol, f, domain)
return result
elif _is_modular(f, symbol):
result = _solve_modular(f, symbol, domain)
else:
lhs, rhs_s = inverter(f, 0, symbol)
if lhs == symbol:
# do some very minimal simplification since
# repeated inversion may have left the result
# in a state that other solvers (e.g. poly)
# would have simplified; this is done here
# rather than in the inverter since here it
# is only done once whereas there it would
# be repeated for each step of the inversion
if isinstance(rhs_s, FiniteSet):
rhs_s = FiniteSet(*[Mul(*
signsimp(i).as_content_primitive())
for i in rhs_s])
result = rhs_s
elif isinstance(rhs_s, FiniteSet):
for equation in [lhs - rhs for rhs in rhs_s]:
if equation == f:
u = unrad(f, symbol)
if u:
result += _solve_radical(equation, u,
symbol,
solver)
elif equation.has(Abs):
result += _solve_abs(f, symbol, domain)
else:
result_rational = _solve_as_rational(equation, symbol, domain)
if not isinstance(result_rational, ConditionSet):
result += result_rational
else:
# may be a transcendental type equation
t_result = _transolve(equation, symbol, domain)
if isinstance(t_result, ConditionSet):
# might need factoring; this is expensive so we
# have delayed until now. To avoid recursion
# errors look for a non-trivial factoring into
# a product of symbol dependent terms; I think
# that something that factors as a Pow would
# have already been recognized by now.
factored = equation.factor()
if factored.is_Mul and equation != factored:
_, dep = factored.as_independent(symbol)
if not dep.is_Add:
# non-trivial factoring of equation
# but use form with constants
# in case they need special handling
t_results = []
for fac in Mul.make_args(factored):
if fac.has(symbol):
t_results.append(solver(fac, symbol))
t_result = Union(*t_results)
result += t_result
else:
result += solver(equation, symbol)
elif rhs_s is not S.EmptySet:
result = ConditionSet(symbol, Eq(f, 0), domain)
if isinstance(result, ConditionSet):
if isinstance(f, Expr):
num, den = f.as_numer_denom()
if den.has(symbol):
_result = _solveset(num, symbol, domain)
if not isinstance(_result, ConditionSet):
singularities = _solveset(den, symbol, domain)
result = _result - singularities
if _check:
if isinstance(result, ConditionSet):
# it wasn't solved or has enumerated all conditions
# -- leave it alone
return result
# whittle away all but the symbol-containing core
# to use this for testing
if isinstance(orig_f, Expr):
fx = orig_f.as_independent(symbol, as_Add=True)[1]
fx = fx.as_independent(symbol, as_Add=False)[1]
else:
fx = orig_f
if isinstance(result, FiniteSet):
# check the result for invalid solutions
result = FiniteSet(*[s for s in result
if isinstance(s, RootOf)
or domain_check(fx, symbol, s)])
return result
def _is_modular(f, symbol):
"""
Helper function to check below mentioned types of modular equations.
``A - Mod(B, C) = 0``
A -> This can or cannot be a function of symbol.
B -> This is surely a function of symbol.
C -> It is an integer.
Parameters
==========
f : Expr
The equation to be checked.
symbol : Symbol
The concerned variable for which the equation is to be checked.
Examples
========
>>> from sympy import symbols, exp, Mod
>>> from sympy.solvers.solveset import _is_modular as check
>>> x, y = symbols('x y')
>>> check(Mod(x, 3) - 1, x)
True
>>> check(Mod(x, 3) - 1, y)
False
>>> check(Mod(x, 3)**2 - 5, x)
False
>>> check(Mod(x, 3)**2 - y, x)
False
>>> check(exp(Mod(x, 3)) - 1, x)
False
>>> check(Mod(3, y) - 1, y)
False
"""
if not f.has(Mod):
return False
# extract modterms from f.
modterms = list(f.atoms(Mod))
return (len(modterms) == 1 and # only one Mod should be present
modterms[0].args[0].has(symbol) and # B-> function of symbol
modterms[0].args[1].is_integer and # C-> to be an integer.
any(isinstance(term, Mod)
for term in list(_term_factors(f))) # free from other funcs
)
def _invert_modular(modterm, rhs, n, symbol):
"""
Helper function to invert modular equation.
``Mod(a, m) - rhs = 0``
Generally it is inverted as (a, ImageSet(Lambda(n, m*n + rhs), S.Integers)).
More simplified form will be returned if possible.
If it is not invertible then (modterm, rhs) is returned.
The following cases arise while inverting equation ``Mod(a, m) - rhs = 0``:
1. If a is symbol then m*n + rhs is the required solution.
2. If a is an instance of ``Add`` then we try to find two symbol independent
parts of a and the symbol independent part gets tranferred to the other
side and again the ``_invert_modular`` is called on the symbol
dependent part.
3. If a is an instance of ``Mul`` then same as we done in ``Add`` we separate
out the symbol dependent and symbol independent parts and transfer the
symbol independent part to the rhs with the help of invert and again the
``_invert_modular`` is called on the symbol dependent part.
4. If a is an instance of ``Pow`` then two cases arise as following:
- If a is of type (symbol_indep)**(symbol_dep) then the remainder is
evaluated with the help of discrete_log function and then the least
period is being found out with the help of totient function.
period*n + remainder is the required solution in this case.
For reference: (https://en.wikipedia.org/wiki/Euler's_theorem)
- If a is of type (symbol_dep)**(symbol_indep) then we try to find all
primitive solutions list with the help of nthroot_mod function.
m*n + rem is the general solution where rem belongs to solutions list
from nthroot_mod function.
Parameters
==========
modterm, rhs : Expr
The modular equation to be inverted, ``modterm - rhs = 0``
symbol : Symbol
The variable in the equation to be inverted.
n : Dummy
Dummy variable for output g_n.
Returns
=======
A tuple (f_x, g_n) is being returned where f_x is modular independent function
of symbol and g_n being set of values f_x can have.
Examples
========
>>> from sympy import symbols, exp, Mod, Dummy, S
>>> from sympy.solvers.solveset import _invert_modular as invert_modular
>>> x, y = symbols('x y')
>>> n = Dummy('n')
>>> invert_modular(Mod(exp(x), 7), S(5), n, x)
(Mod(exp(x), 7), 5)
>>> invert_modular(Mod(x, 7), S(5), n, x)
(x, ImageSet(Lambda(_n, 7*_n + 5), Integers))
>>> invert_modular(Mod(3*x + 8, 7), S(5), n, x)
(x, ImageSet(Lambda(_n, 7*_n + 6), Integers))
>>> invert_modular(Mod(x**4, 7), S(5), n, x)
(x, EmptySet)
>>> invert_modular(Mod(2**(x**2 + x + 1), 7), S(2), n, x)
(x**2 + x + 1, ImageSet(Lambda(_n, 3*_n + 1), Naturals0))
"""
a, m = modterm.args
if rhs.is_real is False or any(term.is_real is False
for term in list(_term_factors(a))):
# Check for complex arguments
return modterm, rhs
if abs(rhs) >= abs(m):
# if rhs has value greater than value of m.
return symbol, S.EmptySet
if a == symbol:
return symbol, ImageSet(Lambda(n, m*n + rhs), S.Integers)
if a.is_Add:
# g + h = a
g, h = a.as_independent(symbol)
if g is not S.Zero:
x_indep_term = rhs - Mod(g, m)
return _invert_modular(Mod(h, m), Mod(x_indep_term, m), n, symbol)
if a.is_Mul:
# g*h = a
g, h = a.as_independent(symbol)
if g is not S.One:
x_indep_term = rhs*invert(g, m)
return _invert_modular(Mod(h, m), Mod(x_indep_term, m), n, symbol)
if a.is_Pow:
# base**expo = a
base, expo = a.args
if expo.has(symbol) and not base.has(symbol):
# remainder -> solution independent of n of equation.
# m, rhs are made coprime by dividing igcd(m, rhs)
try:
remainder = discrete_log(m / igcd(m, rhs), rhs, a.base)
except ValueError: # log does not exist
return modterm, rhs
# period -> coefficient of n in the solution and also referred as
# the least period of expo in which it is repeats itself.
# (a**(totient(m)) - 1) divides m. Here is link of theorem:
# (https://en.wikipedia.org/wiki/Euler's_theorem)
period = totient(m)
for p in divisors(period):
# there might a lesser period exist than totient(m).
if pow(a.base, p, m / igcd(m, a.base)) == 1:
period = p
break
# recursion is not applied here since _invert_modular is currently
# not smart enough to handle infinite rhs as here expo has infinite
# rhs = ImageSet(Lambda(n, period*n + remainder), S.Naturals0).
return expo, ImageSet(Lambda(n, period*n + remainder), S.Naturals0)
elif base.has(symbol) and not expo.has(symbol):
try:
remainder_list = nthroot_mod(rhs, expo, m, all_roots=True)
if remainder_list == []:
return symbol, S.EmptySet
except (ValueError, NotImplementedError):
return modterm, rhs
g_n = S.EmptySet
for rem in remainder_list:
g_n += ImageSet(Lambda(n, m*n + rem), S.Integers)
return base, g_n
return modterm, rhs
def _solve_modular(f, symbol, domain):
r"""
Helper function for solving modular equations of type ``A - Mod(B, C) = 0``,
where A can or cannot be a function of symbol, B is surely a function of
symbol and C is an integer.
Currently ``_solve_modular`` is only able to solve cases
where A is not a function of symbol.
Parameters
==========
f : Expr
The modular equation to be solved, ``f = 0``
symbol : Symbol
The variable in the equation to be solved.
domain : Set
A set over which the equation is solved. It has to be a subset of
Integers.
Returns
=======
A set of integer solutions satisfying the given modular equation.
A ``ConditionSet`` if the equation is unsolvable.
Examples
========
>>> from sympy.solvers.solveset import _solve_modular as solve_modulo
>>> from sympy import S, Symbol, sin, Intersection, Interval, Mod
>>> x = Symbol('x')
>>> solve_modulo(Mod(5*x - 8, 7) - 3, x, S.Integers)
ImageSet(Lambda(_n, 7*_n + 5), Integers)
>>> solve_modulo(Mod(5*x - 8, 7) - 3, x, S.Reals) # domain should be subset of integers.
ConditionSet(x, Eq(Mod(5*x + 6, 7) - 3, 0), Reals)
>>> solve_modulo(-7 + Mod(x, 5), x, S.Integers)
EmptySet
>>> solve_modulo(Mod(12**x, 21) - 18, x, S.Integers)
ImageSet(Lambda(_n, 6*_n + 2), Naturals0)
>>> solve_modulo(Mod(sin(x), 7) - 3, x, S.Integers) # not solvable
ConditionSet(x, Eq(Mod(sin(x), 7) - 3, 0), Integers)
>>> solve_modulo(3 - Mod(x, 5), x, Intersection(S.Integers, Interval(0, 100)))
Intersection(ImageSet(Lambda(_n, 5*_n + 3), Integers), Range(0, 101, 1))
"""
# extract modterm and g_y from f
unsolved_result = ConditionSet(symbol, Eq(f, 0), domain)
modterm = list(f.atoms(Mod))[0]
rhs = -S.One*(f.subs(modterm, S.Zero))
if f.as_coefficients_dict()[modterm].is_negative:
# checks if coefficient of modterm is negative in main equation.
rhs *= -S.One
if not domain.is_subset(S.Integers):
return unsolved_result
if rhs.has(symbol):
# TODO Case: A-> function of symbol, can be extended here
# in future.
return unsolved_result
n = Dummy('n', integer=True)
f_x, g_n = _invert_modular(modterm, rhs, n, symbol)
if f_x == modterm and g_n == rhs:
return unsolved_result
if f_x == symbol:
if domain is not S.Integers:
return domain.intersect(g_n)
return g_n
if isinstance(g_n, ImageSet):
lamda_expr = g_n.lamda.expr
lamda_vars = g_n.lamda.variables
base_sets = g_n.base_sets
sol_set = _solveset(f_x - lamda_expr, symbol, S.Integers)
if isinstance(sol_set, FiniteSet):
tmp_sol = S.EmptySet
for sol in sol_set:
tmp_sol += ImageSet(Lambda(lamda_vars, sol), *base_sets)
sol_set = tmp_sol
else:
sol_set = ImageSet(Lambda(lamda_vars, sol_set), *base_sets)
return domain.intersect(sol_set)
return unsolved_result
def _term_factors(f):
"""
Iterator to get the factors of all terms present
in the given equation.
Parameters
==========
f : Expr
Equation that needs to be addressed
Returns
=======
Factors of all terms present in the equation.
Examples
========
>>> from sympy import symbols
>>> from sympy.solvers.solveset import _term_factors
>>> x = symbols('x')
>>> list(_term_factors(-2 - x**2 + x*(x + 1)))
[-2, -1, x**2, x, x + 1]
"""
for add_arg in Add.make_args(f):
yield from Mul.make_args(add_arg)
def _solve_exponential(lhs, rhs, symbol, domain):
r"""
Helper function for solving (supported) exponential equations.
Exponential equations are the sum of (currently) at most
two terms with one or both of them having a power with a
symbol-dependent exponent.
For example
.. math:: 5^{2x + 3} - 5^{3x - 1}
.. math:: 4^{5 - 9x} - e^{2 - x}
Parameters
==========
lhs, rhs : Expr
The exponential equation to be solved, `lhs = rhs`
symbol : Symbol
The variable in which the equation is solved
domain : Set
A set over which the equation is solved.
Returns
=======
A set of solutions satisfying the given equation.
A ``ConditionSet`` if the equation is unsolvable or
if the assumptions are not properly defined, in that case
a different style of ``ConditionSet`` is returned having the
solution(s) of the equation with the desired assumptions.
Examples
========
>>> from sympy.solvers.solveset import _solve_exponential as solve_expo
>>> from sympy import symbols, S
>>> x = symbols('x', real=True)
>>> a, b = symbols('a b')
>>> solve_expo(2**x + 3**x - 5**x, 0, x, S.Reals) # not solvable
ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), Reals)
>>> solve_expo(a**x - b**x, 0, x, S.Reals) # solvable but incorrect assumptions
ConditionSet(x, (a > 0) & (b > 0), {0})
>>> solve_expo(3**(2*x) - 2**(x + 3), 0, x, S.Reals)
{-3*log(2)/(-2*log(3) + log(2))}
>>> solve_expo(2**x - 4**x, 0, x, S.Reals)
{0}
* Proof of correctness of the method
The logarithm function is the inverse of the exponential function.
The defining relation between exponentiation and logarithm is:
.. math:: {\log_b x} = y \enspace if \enspace b^y = x
Therefore if we are given an equation with exponent terms, we can
convert every term to its corresponding logarithmic form. This is
achieved by taking logarithms and expanding the equation using
logarithmic identities so that it can easily be handled by ``solveset``.
For example:
.. math:: 3^{2x} = 2^{x + 3}
Taking log both sides will reduce the equation to
.. math:: (2x)\log(3) = (x + 3)\log(2)
This form can be easily handed by ``solveset``.
"""
unsolved_result = ConditionSet(symbol, Eq(lhs - rhs, 0), domain)
newlhs = powdenest(lhs)
if lhs != newlhs:
# it may also be advantageous to factor the new expr
neweq = factor(newlhs - rhs)
if neweq != (lhs - rhs):
return _solveset(neweq, symbol, domain) # try again with _solveset
if not (isinstance(lhs, Add) and len(lhs.args) == 2):
# solving for the sum of more than two powers is possible
# but not yet implemented
return unsolved_result
if rhs != 0:
return unsolved_result
a, b = list(ordered(lhs.args))
a_term = a.as_independent(symbol)[1]
b_term = b.as_independent(symbol)[1]
a_base, a_exp = a_term.as_base_exp()
b_base, b_exp = b_term.as_base_exp()
if domain.is_subset(S.Reals):
conditions = And(
a_base > 0,
b_base > 0,
Eq(im(a_exp), 0),
Eq(im(b_exp), 0))
else:
conditions = And(
Ne(a_base, 0),
Ne(b_base, 0))
L, R = map(lambda i: expand_log(log(i), force=True), (a, -b))
solutions = _solveset(L - R, symbol, domain)
return ConditionSet(symbol, conditions, solutions)
def _is_exponential(f, symbol):
r"""
Return ``True`` if one or more terms contain ``symbol`` only in
exponents, else ``False``.
Parameters
==========
f : Expr
The equation to be checked
symbol : Symbol
The variable in which the equation is checked
Examples
========
>>> from sympy import symbols, cos, exp
>>> from sympy.solvers.solveset import _is_exponential as check
>>> x, y = symbols('x y')
>>> check(y, y)
False
>>> check(x**y - 1, y)
True
>>> check(x**y*2**y - 1, y)
True
>>> check(exp(x + 3) + 3**x, x)
True
>>> check(cos(2**x), x)
False
* Philosophy behind the helper
The function extracts each term of the equation and checks if it is
of exponential form w.r.t ``symbol``.
"""
rv = False
for expr_arg in _term_factors(f):
if symbol not in expr_arg.free_symbols:
continue
if (isinstance(expr_arg, Pow) and
symbol not in expr_arg.base.free_symbols or
isinstance(expr_arg, exp)):
rv = True # symbol in exponent
else:
return False # dependent on symbol in non-exponential way
return rv
def _solve_logarithm(lhs, rhs, symbol, domain):
r"""
Helper to solve logarithmic equations which are reducible
to a single instance of `\log`.
Logarithmic equations are (currently) the equations that contains
`\log` terms which can be reduced to a single `\log` term or
a constant using various logarithmic identities.
For example:
.. math:: \log(x) + \log(x - 4)
can be reduced to:
.. math:: \log(x(x - 4))
Parameters
==========
lhs, rhs : Expr
The logarithmic equation to be solved, `lhs = rhs`
symbol : Symbol
The variable in which the equation is solved
domain : Set
A set over which the equation is solved.
Returns
=======
A set of solutions satisfying the given equation.
A ``ConditionSet`` if the equation is unsolvable.
Examples
========
>>> from sympy import symbols, log, S
>>> from sympy.solvers.solveset import _solve_logarithm as solve_log
>>> x = symbols('x')
>>> f = log(x - 3) + log(x + 3)
>>> solve_log(f, 0, x, S.Reals)
{-sqrt(10), sqrt(10)}
* Proof of correctness
A logarithm is another way to write exponent and is defined by
.. math:: {\log_b x} = y \enspace if \enspace b^y = x
When one side of the equation contains a single logarithm, the
equation can be solved by rewriting the equation as an equivalent
exponential equation as defined above. But if one side contains
more than one logarithm, we need to use the properties of logarithm
to condense it into a single logarithm.
Take for example
.. math:: \log(2x) - 15 = 0
contains single logarithm, therefore we can directly rewrite it to
exponential form as
.. math:: x = \frac{e^{15}}{2}
But if the equation has more than one logarithm as
.. math:: \log(x - 3) + \log(x + 3) = 0
we use logarithmic identities to convert it into a reduced form
Using,
.. math:: \log(a) + \log(b) = \log(ab)
the equation becomes,
.. math:: \log((x - 3)(x + 3))
This equation contains one logarithm and can be solved by rewriting
to exponents.
"""
new_lhs = logcombine(lhs, force=True)
new_f = new_lhs - rhs
return _solveset(new_f, symbol, domain)
def _is_logarithmic(f, symbol):
r"""
Return ``True`` if the equation is in the form
`a\log(f(x)) + b\log(g(x)) + ... + c` else ``False``.
Parameters
==========
f : Expr
The equation to be checked
symbol : Symbol
The variable in which the equation is checked
Returns
=======
``True`` if the equation is logarithmic otherwise ``False``.
Examples
========
>>> from sympy import symbols, tan, log
>>> from sympy.solvers.solveset import _is_logarithmic as check
>>> x, y = symbols('x y')
>>> check(log(x + 2) - log(x + 3), x)
True
>>> check(tan(log(2*x)), x)
False
>>> check(x*log(x), x)
False
>>> check(x + log(x), x)
False
>>> check(y + log(x), x)
True
* Philosophy behind the helper
The function extracts each term and checks whether it is
logarithmic w.r.t ``symbol``.
"""
rv = False
for term in Add.make_args(f):
saw_log = False
for term_arg in Mul.make_args(term):
if symbol not in term_arg.free_symbols:
continue
if isinstance(term_arg, log):
if saw_log:
return False # more than one log in term
saw_log = True
else:
return False # dependent on symbol in non-log way
if saw_log:
rv = True
return rv
def _is_lambert(f, symbol):
r"""
If this returns ``False`` then the Lambert solver (``_solve_lambert``) will not be called.
Explanation
===========
Quick check for cases that the Lambert solver might be able to handle.
1. Equations containing more than two operands and `symbol`s involving any of
`Pow`, `exp`, `HyperbolicFunction`,`TrigonometricFunction`, `log` terms.
2. In `Pow`, `exp` the exponent should have `symbol` whereas for
`HyperbolicFunction`,`TrigonometricFunction`, `log` should contain `symbol`.
3. For `HyperbolicFunction`,`TrigonometricFunction` the number of trigonometric functions in
equation should be less than number of symbols. (since `A*cos(x) + B*sin(x) - c`
is not the Lambert type).
Some forms of lambert equations are:
1. X**X = C
2. X*(B*log(X) + D)**A = C
3. A*log(B*X + A) + d*X = C
4. (B*X + A)*exp(d*X + g) = C
5. g*exp(B*X + h) - B*X = C
6. A*D**(E*X + g) - B*X = C
7. A*cos(X) + B*sin(X) - D*X = C
8. A*cosh(X) + B*sinh(X) - D*X = C
Where X is any variable,
A, B, C, D, E are any constants,
g, h are linear functions or log terms.
Parameters
==========
f : Expr
The equation to be checked
symbol : Symbol
The variable in which the equation is checked
Returns
=======
If this returns ``False`` then the Lambert solver (``_solve_lambert``) will not be called.
Examples
========
>>> from sympy.solvers.solveset import _is_lambert
>>> from sympy import symbols, cosh, sinh, log
>>> x = symbols('x')
>>> _is_lambert(3*log(x) - x*log(3), x)
True
>>> _is_lambert(log(log(x - 3)) + log(x-3), x)
True
>>> _is_lambert(cosh(x) - sinh(x), x)
False
>>> _is_lambert((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x)
True
See Also
========
_solve_lambert
"""
term_factors = list(_term_factors(f.expand()))
# total number of symbols in equation
no_of_symbols = len([arg for arg in term_factors if arg.has(symbol)])
# total number of trigonometric terms in equation
no_of_trig = len([arg for arg in term_factors \
if arg.has(HyperbolicFunction, TrigonometricFunction)])
if f.is_Add and no_of_symbols >= 2:
# `log`, `HyperbolicFunction`, `TrigonometricFunction` should have symbols
# and no_of_trig < no_of_symbols
lambert_funcs = (log, HyperbolicFunction, TrigonometricFunction)
if any(isinstance(arg, lambert_funcs)\
for arg in term_factors if arg.has(symbol)):
if no_of_trig < no_of_symbols:
return True
# here, `Pow`, `exp` exponent should have symbols
elif any(isinstance(arg, (Pow, exp)) \
for arg in term_factors if (arg.as_base_exp()[1]).has(symbol)):
return True
return False
def _transolve(f, symbol, domain):
r"""
Function to solve transcendental equations. It is a helper to
``solveset`` and should be used internally. ``_transolve``
currently supports the following class of equations:
- Exponential equations
- Logarithmic equations
Parameters
==========
f : Any transcendental equation that needs to be solved.
This needs to be an expression, which is assumed
to be equal to ``0``.
symbol : The variable for which the equation is solved.
This needs to be of class ``Symbol``.
domain : A set over which the equation is solved.
This needs to be of class ``Set``.
Returns
=======
Set
A set of values for ``symbol`` for which ``f`` is equal to
zero. An ``EmptySet`` is returned if ``f`` does not have solutions
in respective domain. A ``ConditionSet`` is returned as unsolved
object if algorithms to evaluate complete solution are not
yet implemented.
How to use ``_transolve``
=========================
``_transolve`` should not be used as an independent function, because
it assumes that the equation (``f``) and the ``symbol`` comes from
``solveset`` and might have undergone a few modification(s).
To use ``_transolve`` as an independent function the equation (``f``)
and the ``symbol`` should be passed as they would have been by
``solveset``.
Examples
========
>>> from sympy.solvers.solveset import _transolve as transolve
>>> from sympy.solvers.solvers import _tsolve as tsolve
>>> from sympy import symbols, S, pprint
>>> x = symbols('x', real=True) # assumption added
>>> transolve(5**(x - 3) - 3**(2*x + 1), x, S.Reals)
{-(log(3) + 3*log(5))/(-log(5) + 2*log(3))}
How ``_transolve`` works
========================
``_transolve`` uses two types of helper functions to solve equations
of a particular class:
Identifying helpers: To determine whether a given equation
belongs to a certain class of equation or not. Returns either
``True`` or ``False``.
Solving helpers: Once an equation is identified, a corresponding
helper either solves the equation or returns a form of the equation
that ``solveset`` might better be able to handle.
* Philosophy behind the module
The purpose of ``_transolve`` is to take equations which are not
already polynomial in their generator(s) and to either recast them
as such through a valid transformation or to solve them outright.
A pair of helper functions for each class of supported
transcendental functions are employed for this purpose. One
identifies the transcendental form of an equation and the other
either solves it or recasts it into a tractable form that can be
solved by ``solveset``.
For example, an equation in the form `ab^{f(x)} - cd^{g(x)} = 0`
can be transformed to
`\log(a) + f(x)\log(b) - \log(c) - g(x)\log(d) = 0`
(under certain assumptions) and this can be solved with ``solveset``
if `f(x)` and `g(x)` are in polynomial form.
How ``_transolve`` is better than ``_tsolve``
=============================================
1) Better output
``_transolve`` provides expressions in a more simplified form.
Consider a simple exponential equation
>>> f = 3**(2*x) - 2**(x + 3)
>>> pprint(transolve(f, x, S.Reals), use_unicode=False)
-3*log(2)
{------------------}
-2*log(3) + log(2)
>>> pprint(tsolve(f, x), use_unicode=False)
/ 3 \
| --------|
| log(2/9)|
[-log\2 /]
2) Extensible
The API of ``_transolve`` is designed such that it is easily
extensible, i.e. the code that solves a given class of
equations is encapsulated in a helper and not mixed in with
the code of ``_transolve`` itself.
3) Modular
``_transolve`` is designed to be modular i.e, for every class of
equation a separate helper for identification and solving is
implemented. This makes it easy to change or modify any of the
method implemented directly in the helpers without interfering
with the actual structure of the API.
4) Faster Computation
Solving equation via ``_transolve`` is much faster as compared to
``_tsolve``. In ``solve``, attempts are made computing every possibility
to get the solutions. This series of attempts makes solving a bit
slow. In ``_transolve``, computation begins only after a particular
type of equation is identified.
How to add new class of equations
=================================
Adding a new class of equation solver is a three-step procedure:
- Identify the type of the equations
Determine the type of the class of equations to which they belong:
it could be of ``Add``, ``Pow``, etc. types. Separate internal functions
are used for each type. Write identification and solving helpers
and use them from within the routine for the given type of equation
(after adding it, if necessary). Something like:
.. code-block:: python
def add_type(lhs, rhs, x):
....
if _is_exponential(lhs, x):
new_eq = _solve_exponential(lhs, rhs, x)
....
rhs, lhs = eq.as_independent(x)
if lhs.is_Add:
result = add_type(lhs, rhs, x)
- Define the identification helper.
- Define the solving helper.
Apart from this, a few other things needs to be taken care while
adding an equation solver:
- Naming conventions:
Name of the identification helper should be as
``_is_class`` where class will be the name or abbreviation
of the class of equation. The solving helper will be named as
``_solve_class``.
For example: for exponential equations it becomes
``_is_exponential`` and ``_solve_expo``.
- The identifying helpers should take two input parameters,
the equation to be checked and the variable for which a solution
is being sought, while solving helpers would require an additional
domain parameter.
- Be sure to consider corner cases.
- Add tests for each helper.
- Add a docstring to your helper that describes the method
implemented.
The documentation of the helpers should identify:
- the purpose of the helper,
- the method used to identify and solve the equation,
- a proof of correctness
- the return values of the helpers
"""
def add_type(lhs, rhs, symbol, domain):
"""
Helper for ``_transolve`` to handle equations of
``Add`` type, i.e. equations taking the form as
``a*f(x) + b*g(x) + .... = c``.
For example: 4**x + 8**x = 0
"""
result = ConditionSet(symbol, Eq(lhs - rhs, 0), domain)
# check if it is exponential type equation
if _is_exponential(lhs, symbol):
result = _solve_exponential(lhs, rhs, symbol, domain)
# check if it is logarithmic type equation
elif _is_logarithmic(lhs, symbol):
result = _solve_logarithm(lhs, rhs, symbol, domain)
return result
result = ConditionSet(symbol, Eq(f, 0), domain)
# invert_complex handles the call to the desired inverter based
# on the domain specified.
lhs, rhs_s = invert_complex(f, 0, symbol, domain)
if isinstance(rhs_s, FiniteSet):
assert (len(rhs_s.args)) == 1
rhs = rhs_s.args[0]
if lhs.is_Add:
result = add_type(lhs, rhs, symbol, domain)
else:
result = rhs_s
return result
def solveset(f, symbol=None, domain=S.Complexes):
r"""Solves a given inequality or equation with set as output
Parameters
==========
f : Expr or a relational.
The target equation or inequality
symbol : Symbol
The variable for which the equation is solved
domain : Set
The domain over which the equation is solved
Returns
=======
Set
A set of values for `symbol` for which `f` is True or is equal to
zero. An :class:`~.EmptySet` is returned if `f` is False or nonzero.
A :class:`~.ConditionSet` is returned as unsolved object if algorithms
to evaluate complete solution are not yet implemented.
``solveset`` claims to be complete in the solution set that it returns.
Raises
======
NotImplementedError
The algorithms to solve inequalities in complex domain are
not yet implemented.
ValueError
The input is not valid.
RuntimeError
It is a bug, please report to the github issue tracker.
Notes
=====
Python interprets 0 and 1 as False and True, respectively, but
in this function they refer to solutions of an expression. So 0 and 1
return the domain and EmptySet, respectively, while True and False
return the opposite (as they are assumed to be solutions of relational
expressions).
See Also
========
solveset_real: solver for real domain
solveset_complex: solver for complex domain
Examples
========
>>> from sympy import exp, sin, Symbol, pprint, S, Eq
>>> from sympy.solvers.solveset import solveset, solveset_real
* The default domain is complex. Not specifying a domain will lead
to the solving of the equation in the complex domain (and this
is not affected by the assumptions on the symbol):
>>> x = Symbol('x')
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers}
>>> x = Symbol('x', real=True)
>>> pprint(solveset(exp(x) - 1, x), use_unicode=False)
{2*n*I*pi | n in Integers}
* If you want to use ``solveset`` to solve the equation in the
real domain, provide a real domain. (Using ``solveset_real``
does this automatically.)
>>> R = S.Reals
>>> x = Symbol('x')
>>> solveset(exp(x) - 1, x, R)
{0}
>>> solveset_real(exp(x) - 1, x)
{0}
The solution is unaffected by assumptions on the symbol:
>>> p = Symbol('p', positive=True)
>>> pprint(solveset(p**2 - 4))
{-2, 2}
When a :class:`~.ConditionSet` is returned, symbols with assumptions that
would alter the set are replaced with more generic symbols:
>>> i = Symbol('i', imaginary=True)
>>> solveset(Eq(i**2 + i*sin(i), 1), i, domain=S.Reals)
ConditionSet(_R, Eq(_R**2 + _R*sin(_R) - 1, 0), Reals)
* Inequalities can be solved over the real domain only. Use of a complex
domain leads to a NotImplementedError.
>>> solveset(exp(x) > 1, x, R)
Interval.open(0, oo)
"""
f = sympify(f)
symbol = sympify(symbol)
if f is S.true:
return domain
if f is S.false:
return S.EmptySet
if not isinstance(f, (Expr, Relational, Number)):
raise ValueError("%s is not a valid SymPy expression" % f)
if not isinstance(symbol, (Expr, Relational)) and symbol is not None:
raise ValueError("%s is not a valid SymPy symbol" % (symbol,))
if not isinstance(domain, Set):
raise ValueError("%s is not a valid domain" %(domain))
free_symbols = f.free_symbols
if f.has(Piecewise):
f = piecewise_fold(f)
if symbol is None and not free_symbols:
b = Eq(f, 0)
if b is S.true:
return domain
elif b is S.false:
return S.EmptySet
else:
raise NotImplementedError(filldedent('''
relationship between value and 0 is unknown: %s''' % b))
if symbol is None:
if len(free_symbols) == 1:
symbol = free_symbols.pop()
elif free_symbols:
raise ValueError(filldedent('''
The independent variable must be specified for a
multivariate equation.'''))
elif not isinstance(symbol, Symbol):
f, s, swap = recast_to_symbols([f], [symbol])
# the xreplace will be needed if a ConditionSet is returned
return solveset(f[0], s[0], domain).xreplace(swap)
# solveset should ignore assumptions on symbols
if symbol not in _rc:
x = _rc[0] if domain.is_subset(S.Reals) else _rc[1]
rv = solveset(f.xreplace({symbol: x}), x, domain)
# try to use the original symbol if possible
try:
_rv = rv.xreplace({x: symbol})
except TypeError:
_rv = rv
if rv.dummy_eq(_rv):
rv = _rv
return rv
# Abs has its own handling method which avoids the
# rewriting property that the first piece of abs(x)
# is for x >= 0 and the 2nd piece for x < 0 -- solutions
# can look better if the 2nd condition is x <= 0. Since
# the solution is a set, duplication of results is not
# an issue, e.g. {y, -y} when y is 0 will be {0}
f, mask = _masked(f, Abs)
f = f.rewrite(Piecewise) # everything that's not an Abs
for d, e in mask:
# everything *in* an Abs
e = e.func(e.args[0].rewrite(Piecewise))
f = f.xreplace({d: e})
f = piecewise_fold(f)
return _solveset(f, symbol, domain, _check=True)
def solveset_real(f, symbol):
return solveset(f, symbol, S.Reals)
def solveset_complex(f, symbol):
return solveset(f, symbol, S.Complexes)
def _solveset_multi(eqs, syms, domains):
'''Basic implementation of a multivariate solveset.
For internal use (not ready for public consumption)'''
rep = {}
for sym, dom in zip(syms, domains):
if dom is S.Reals:
rep[sym] = Symbol(sym.name, real=True)
eqs = [eq.subs(rep) for eq in eqs]
syms = [sym.subs(rep) for sym in syms]
syms = tuple(syms)
if len(eqs) == 0:
return ProductSet(*domains)
if len(syms) == 1:
sym = syms[0]
domain = domains[0]
solsets = [solveset(eq, sym, domain) for eq in eqs]
solset = Intersection(*solsets)
return ImageSet(Lambda((sym,), (sym,)), solset).doit()
eqs = sorted(eqs, key=lambda eq: len(eq.free_symbols & set(syms)))
for n, eq in enumerate(eqs):
sols = []
all_handled = True
for sym in syms:
if sym not in eq.free_symbols:
continue
sol = solveset(eq, sym, domains[syms.index(sym)])
if isinstance(sol, FiniteSet):
i = syms.index(sym)
symsp = syms[:i] + syms[i+1:]
domainsp = domains[:i] + domains[i+1:]
eqsp = eqs[:n] + eqs[n+1:]
for s in sol:
eqsp_sub = [eq.subs(sym, s) for eq in eqsp]
sol_others = _solveset_multi(eqsp_sub, symsp, domainsp)
fun = Lambda((symsp,), symsp[:i] + (s,) + symsp[i:])
sols.append(ImageSet(fun, sol_others).doit())
else:
all_handled = False
if all_handled:
return Union(*sols)
def solvify(f, symbol, domain):
"""Solves an equation using solveset and returns the solution in accordance
with the `solve` output API.
Returns
=======
We classify the output based on the type of solution returned by `solveset`.
Solution | Output
----------------------------------------
FiniteSet | list
ImageSet, | list (if `f` is periodic)
Union |
Union | list (with FiniteSet)
EmptySet | empty list
Others | None
Raises
======
NotImplementedError
A ConditionSet is the input.
Examples
========
>>> from sympy.solvers.solveset import solvify
>>> from sympy.abc import x
>>> from sympy import S, tan, sin, exp
>>> solvify(x**2 - 9, x, S.Reals)
[-3, 3]
>>> solvify(sin(x) - 1, x, S.Reals)
[pi/2]
>>> solvify(tan(x), x, S.Reals)
[0]
>>> solvify(exp(x) - 1, x, S.Complexes)
>>> solvify(exp(x) - 1, x, S.Reals)
[0]
"""
solution_set = solveset(f, symbol, domain)
result = None
if solution_set is S.EmptySet:
result = []
elif isinstance(solution_set, ConditionSet):
raise NotImplementedError('solveset is unable to solve this equation.')
elif isinstance(solution_set, FiniteSet):
result = list(solution_set)
else:
period = periodicity(f, symbol)
if period is not None:
solutions = S.EmptySet
iter_solutions = ()
if isinstance(solution_set, ImageSet):
iter_solutions = (solution_set,)
elif isinstance(solution_set, Union):
if all(isinstance(i, ImageSet) for i in solution_set.args):
iter_solutions = solution_set.args
for solution in iter_solutions:
solutions += solution.intersect(Interval(0, period, False, True))
if isinstance(solutions, FiniteSet):
result = list(solutions)
else:
solution = solution_set.intersect(domain)
if isinstance(solution, Union):
# concerned about only FiniteSet with Union but not about ImageSet
# if required could be extend
if any(isinstance(i, FiniteSet) for i in solution.args):
result = [sol for soln in solution.args \
for sol in soln.args if isinstance(soln,FiniteSet)]
else:
return None
elif isinstance(solution, FiniteSet):
result += solution
return result
###############################################################################
################################ LINSOLVE #####################################
###############################################################################
def linear_coeffs(eq, *syms, dict=False):
"""Return a list whose elements are the coefficients of the
corresponding symbols in the sum of terms in ``eq``.
The additive constant is returned as the last element of the
list.
Raises
======
NonlinearError
The equation contains a nonlinear term
ValueError
duplicate or unordered symbols are passed
Parameters
==========
dict - (default False) when True, return coefficients as a
dictionary with coefficients keyed to syms that were present;
key 1 gives the constant term
Examples
========
>>> from sympy.solvers.solveset import linear_coeffs
>>> from sympy.abc import x, y, z
>>> linear_coeffs(3*x + 2*y - 1, x, y)
[3, 2, -1]
It is not necessary to expand the expression:
>>> linear_coeffs(x + y*(z*(x*3 + 2) + 3), x)
[3*y*z + 1, y*(2*z + 3)]
When nonlinear is detected, an error will be raised:
* even if they would cancel after expansion (so the
situation does not pass silently past the caller's
attention)
>>> eq = 1/x*(x - 1) + 1/x
>>> linear_coeffs(eq.expand(), x)
[0, 1]
>>> linear_coeffs(eq, x)
Traceback (most recent call last):
...
NonlinearError:
nonlinear in given generators
* when there are cross terms
>>> linear_coeffs(x*(y + 1), x, y)
Traceback (most recent call last):
...
NonlinearError:
symbol-dependent cross-terms encountered
* when there are terms that contain an expression
dependent on the symbols that is not linear
>>> linear_coeffs(x**2, x)
Traceback (most recent call last):
...
NonlinearError:
nonlinear in given generators
"""
eq = _sympify(eq)
if len(syms) == 1 and iterable(syms[0]) and not isinstance(syms[0], Basic):
raise ValueError('expecting unpacked symbols, *syms')
symset = set(syms)
if len(symset) != len(syms):
raise ValueError('duplicate symbols given')
try:
d, c = _linear_eq_to_dict([eq], symset)
d = d[0]
c = c[0]
except PolyNonlinearError as err:
raise NonlinearError(str(err))
if dict:
if c:
d[S.One] = c
return d
rv = [S.Zero]*(len(syms) + 1)
rv[-1] = c
for i, k in enumerate(syms):
if k not in d:
continue
rv[i] = d[k]
return rv
def linear_eq_to_matrix(equations, *symbols):
r"""
Converts a given System of Equations into Matrix form.
Here `equations` must be a linear system of equations in
`symbols`. Element ``M[i, j]`` corresponds to the coefficient
of the jth symbol in the ith equation.
The Matrix form corresponds to the augmented matrix form.
For example:
.. math:: 4x + 2y + 3z = 1
.. math:: 3x + y + z = -6
.. math:: 2x + 4y + 9z = 2
This system will return $A$ and $b$ as:
$$ A = \left[\begin{array}{ccc}
4 & 2 & 3 \\
3 & 1 & 1 \\
2 & 4 & 9
\end{array}\right] \ \ b = \left[\begin{array}{c}
1 \\ -6 \\ 2
\end{array}\right] $$
The only simplification performed is to convert
``Eq(a, b)`` $\Rightarrow a - b$.
Raises
======
NonlinearError
The equations contain a nonlinear term.
ValueError
The symbols are not given or are not unique.
Examples
========
>>> from sympy import linear_eq_to_matrix, symbols
>>> c, x, y, z = symbols('c, x, y, z')
The coefficients (numerical or symbolic) of the symbols will
be returned as matrices:
>>> eqns = [c*x + z - 1 - c, y + z, x - y]
>>> A, b = linear_eq_to_matrix(eqns, [x, y, z])
>>> A
Matrix([
[c, 0, 1],
[0, 1, 1],
[1, -1, 0]])
>>> b
Matrix([
[c + 1],
[ 0],
[ 0]])
This routine does not simplify expressions and will raise an error
if nonlinearity is encountered:
>>> eqns = [
... (x**2 - 3*x)/(x - 3) - 3,
... y**2 - 3*y - y*(y - 4) + x - 4]
>>> linear_eq_to_matrix(eqns, [x, y])
Traceback (most recent call last):
...
NonlinearError:
symbol-dependent term can be ignored using `strict=False`
Simplifying these equations will discard the removable singularity
in the first and reveal the linear structure of the second:
>>> [e.simplify() for e in eqns]
[x - 3, x + y - 4]
Any such simplification needed to eliminate nonlinear terms must
be done *before* calling this routine.
"""
if not symbols:
raise ValueError(filldedent('''
Symbols must be given, for which coefficients
are to be found.
'''))
if hasattr(symbols[0], '__iter__'):
symbols = symbols[0]
if has_dups(symbols):
raise ValueError('Symbols must be unique')
equations = sympify(equations)
if isinstance(equations, MatrixBase):
equations = list(equations)
elif isinstance(equations, (Expr, Eq)):
equations = [equations]
elif not is_sequence(equations):
raise ValueError(filldedent('''
Equation(s) must be given as a sequence, Expr,
Eq or Matrix.
'''))
# construct the dictionaries
try:
eq, c = _linear_eq_to_dict(equations, symbols)
except PolyNonlinearError as err:
raise NonlinearError(str(err))
# prepare output matrices
n, m = shape = len(eq), len(symbols)
ix = dict(zip(symbols, range(m)))
A = zeros(*shape)
for row, d in enumerate(eq):
for k in d:
col = ix[k]
A[row, col] = d[k]
b = Matrix(n, 1, [-i for i in c])
return A, b
def linsolve(system, *symbols):
r"""
Solve system of $N$ linear equations with $M$ variables; both
underdetermined and overdetermined systems are supported.
The possible number of solutions is zero, one or infinite.
Zero solutions throws a ValueError, whereas infinite
solutions are represented parametrically in terms of the given
symbols. For unique solution a :class:`~.FiniteSet` of ordered tuples
is returned.
All standard input formats are supported:
For the given set of equations, the respective input types
are given below:
.. math:: 3x + 2y - z = 1
.. math:: 2x - 2y + 4z = -2
.. math:: 2x - y + 2z = 0
* Augmented matrix form, ``system`` given below:
$$ \text{system} = \left[{array}{cccc}
3 & 2 & -1 & 1\\
2 & -2 & 4 & -2\\
2 & -1 & 2 & 0
\end{array}\right] $$
::
system = Matrix([[3, 2, -1, 1], [2, -2, 4, -2], [2, -1, 2, 0]])
* List of equations form
::
system = [3x + 2y - z - 1, 2x - 2y + 4z + 2, 2x - y + 2z]
* Input $A$ and $b$ in matrix form (from $Ax = b$) are given as:
$$ A = \left[\begin{array}{ccc}
3 & 2 & -1 \\
2 & -2 & 4 \\
2 & -1 & 2
\end{array}\right] \ \ b = \left[\begin{array}{c}
1 \\ -2 \\ 0
\end{array}\right] $$
::
A = Matrix([[3, 2, -1], [2, -2, 4], [2, -1, 2]])
b = Matrix([[1], [-2], [0]])
system = (A, b)
Symbols can always be passed but are actually only needed
when 1) a system of equations is being passed and 2) the
system is passed as an underdetermined matrix and one wants
to control the name of the free variables in the result.
An error is raised if no symbols are used for case 1, but if
no symbols are provided for case 2, internally generated symbols
will be provided. When providing symbols for case 2, there should
be at least as many symbols are there are columns in matrix A.
The algorithm used here is Gauss-Jordan elimination, which
results, after elimination, in a row echelon form matrix.
Returns
=======
A FiniteSet containing an ordered tuple of values for the
unknowns for which the `system` has a solution. (Wrapping
the tuple in FiniteSet is used to maintain a consistent
output format throughout solveset.)
Returns EmptySet, if the linear system is inconsistent.
Raises
======
ValueError
The input is not valid.
The symbols are not given.
Examples
========
>>> from sympy import Matrix, linsolve, symbols
>>> x, y, z = symbols("x, y, z")
>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
>>> b = Matrix([3, 6, 9])
>>> A
Matrix([
[1, 2, 3],
[4, 5, 6],
[7, 8, 10]])
>>> b
Matrix([
[3],
[6],
[9]])
>>> linsolve((A, b), [x, y, z])
{(-1, 2, 0)}
* Parametric Solution: In case the system is underdetermined, the
function will return a parametric solution in terms of the given
symbols. Those that are free will be returned unchanged. e.g. in
the system below, `z` is returned as the solution for variable z;
it can take on any value.
>>> A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
>>> b = Matrix([3, 6, 9])
>>> linsolve((A, b), x, y, z)
{(z - 1, 2 - 2*z, z)}
If no symbols are given, internally generated symbols will be used.
The ``tau0`` in the third position indicates (as before) that the third
variable -- whatever it is named -- can take on any value:
>>> linsolve((A, b))
{(tau0 - 1, 2 - 2*tau0, tau0)}
* List of equations as input
>>> Eqns = [3*x + 2*y - z - 1, 2*x - 2*y + 4*z + 2, - x + y/2 - z]
>>> linsolve(Eqns, x, y, z)
{(1, -2, -2)}
* Augmented matrix as input
>>> aug = Matrix([[2, 1, 3, 1], [2, 6, 8, 3], [6, 8, 18, 5]])
>>> aug
Matrix([
[2, 1, 3, 1],
[2, 6, 8, 3],
[6, 8, 18, 5]])
>>> linsolve(aug, x, y, z)
{(3/10, 2/5, 0)}
* Solve for symbolic coefficients
>>> a, b, c, d, e, f = symbols('a, b, c, d, e, f')
>>> eqns = [a*x + b*y - c, d*x + e*y - f]
>>> linsolve(eqns, x, y)
{((-b*f + c*e)/(a*e - b*d), (a*f - c*d)/(a*e - b*d))}
* A degenerate system returns solution as set of given
symbols.
>>> system = Matrix(([0, 0, 0], [0, 0, 0], [0, 0, 0]))
>>> linsolve(system, x, y)
{(x, y)}
* For an empty system linsolve returns empty set
>>> linsolve([], x)
EmptySet
* An error is raised if any nonlinearity is detected, even
if it could be removed with expansion
>>> linsolve([x*(1/x - 1)], x)
Traceback (most recent call last):
...
NonlinearError: nonlinear term: 1/x
>>> linsolve([x*(y + 1)], x, y)
Traceback (most recent call last):
...
NonlinearError: nonlinear cross-term: x*(y + 1)
>>> linsolve([x**2 - 1], x)
Traceback (most recent call last):
...
NonlinearError: nonlinear term: x**2
"""
if not system:
return S.EmptySet
# If second argument is an iterable
if symbols and hasattr(symbols[0], '__iter__'):
symbols = symbols[0]
sym_gen = isinstance(symbols, GeneratorType)
dup_msg = 'duplicate symbols given'
b = None # if we don't get b the input was bad
# unpack system
if hasattr(system, '__iter__'):
# 1). (A, b)
if len(system) == 2 and isinstance(system[0], MatrixBase):
A, b = system
# 2). (eq1, eq2, ...)
if not isinstance(system[0], MatrixBase):
if sym_gen or not symbols:
raise ValueError(filldedent('''
When passing a system of equations, the explicit
symbols for which a solution is being sought must
be given as a sequence, too.
'''))
if len(set(symbols)) != len(symbols):
raise ValueError(dup_msg)
#
# Pass to the sparse solver implemented in polys. It is important
# that we do not attempt to convert the equations to a matrix
# because that would be very inefficient for large sparse systems
# of equations.
#
eqs = system
eqs = [sympify(eq) for eq in eqs]
try:
sol = _linsolve(eqs, symbols)
except PolyNonlinearError as exc:
# e.g. cos(x) contains an element of the set of generators
raise NonlinearError(str(exc))
if sol is None:
return S.EmptySet
sol = FiniteSet(Tuple(*(sol.get(sym, sym) for sym in symbols)))
return sol
elif isinstance(system, MatrixBase) and not (
symbols and not isinstance(symbols, GeneratorType) and
isinstance(symbols[0], MatrixBase)):
# 3). A augmented with b
A, b = system[:, :-1], system[:, -1:]
if b is None:
raise ValueError("Invalid arguments")
if sym_gen:
symbols = [next(symbols) for i in range(A.cols)]
symset = set(symbols)
if any(symset & (A.free_symbols | b.free_symbols)):
raise ValueError(filldedent('''
At least one of the symbols provided
already appears in the system to be solved.
One way to avoid this is to use Dummy symbols in
the generator, e.g. numbered_symbols('%s', cls=Dummy)
''' % symbols[0].name.rstrip('1234567890')))
elif len(symset) != len(symbols):
raise ValueError(dup_msg)
if not symbols:
symbols = [Dummy() for _ in range(A.cols)]
name = _uniquely_named_symbol('tau', (A, b),
compare=lambda i: str(i).rstrip('1234567890')).name
gen = numbered_symbols(name)
else:
gen = None
# This is just a wrapper for solve_lin_sys
eqs = []
rows = A.tolist()
for rowi, bi in zip(rows, b):
terms = [elem * sym for elem, sym in zip(rowi, symbols) if elem]
terms.append(-bi)
eqs.append(Add(*terms))
eqs, ring = sympy_eqs_to_ring(eqs, symbols)
sol = solve_lin_sys(eqs, ring, _raw=False)
if sol is None:
return S.EmptySet
#sol = {sym:val for sym, val in sol.items() if sym != val}
sol = FiniteSet(Tuple(*(sol.get(sym, sym) for sym in symbols)))
if gen is not None:
solsym = sol.free_symbols
rep = {sym: next(gen) for sym in symbols if sym in solsym}
sol = sol.subs(rep)
return sol
##############################################################################
# ------------------------------nonlinsolve ---------------------------------#
##############################################################################
def _return_conditionset(eqs, symbols):
# return conditionset
eqs = (Eq(lhs, 0) for lhs in eqs)
condition_set = ConditionSet(
Tuple(*symbols), And(*eqs), S.Complexes**len(symbols))
return condition_set
def substitution(system, symbols, result=[{}], known_symbols=[],
exclude=[], all_symbols=None):
r"""
Solves the `system` using substitution method. It is used in
:func:`~.nonlinsolve`. This will be called from :func:`~.nonlinsolve` when any
equation(s) is non polynomial equation.
Parameters
==========
system : list of equations
The target system of equations
symbols : list of symbols to be solved.
The variable(s) for which the system is solved
known_symbols : list of solved symbols
Values are known for these variable(s)
result : An empty list or list of dict
If No symbol values is known then empty list otherwise
symbol as keys and corresponding value in dict.
exclude : Set of expression.
Mostly denominator expression(s) of the equations of the system.
Final solution should not satisfy these expressions.
all_symbols : known_symbols + symbols(unsolved).
Returns
=======
A FiniteSet of ordered tuple of values of `all_symbols` for which the
`system` has solution. Order of values in the tuple is same as symbols
present in the parameter `all_symbols`. If parameter `all_symbols` is None
then same as symbols present in the parameter `symbols`.
Please note that general FiniteSet is unordered, the solution returned
here is not simply a FiniteSet of solutions, rather it is a FiniteSet of
ordered tuple, i.e. the first & only argument to FiniteSet is a tuple of
solutions, which is ordered, & hence the returned solution is ordered.
Also note that solution could also have been returned as an ordered tuple,
FiniteSet is just a wrapper `{}` around the tuple. It has no other
significance except for the fact it is just used to maintain a consistent
output format throughout the solveset.
Raises
======
ValueError
The input is not valid.
The symbols are not given.
AttributeError
The input symbols are not :class:`~.Symbol` type.
Examples
========
>>> from sympy import symbols, substitution
>>> x, y = symbols('x, y', real=True)
>>> substitution([x + y], [x], [{y: 1}], [y], set([]), [x, y])
{(-1, 1)}
* When you want a soln not satisfying $x + 1 = 0$
>>> substitution([x + y], [x], [{y: 1}], [y], set([x + 1]), [y, x])
EmptySet
>>> substitution([x + y], [x], [{y: 1}], [y], set([x - 1]), [y, x])
{(1, -1)}
>>> substitution([x + y - 1, y - x**2 + 5], [x, y])
{(-3, 4), (2, -1)}
* Returns both real and complex solution
>>> x, y, z = symbols('x, y, z')
>>> from sympy import exp, sin
>>> substitution([exp(x) - sin(y), y**2 - 4], [x, y])
{(ImageSet(Lambda(_n, I*(2*_n*pi + pi) + log(sin(2))), Integers), -2),
(ImageSet(Lambda(_n, 2*_n*I*pi + log(sin(2))), Integers), 2)}
>>> eqs = [z**2 + exp(2*x) - sin(y), -3 + exp(-y)]
>>> substitution(eqs, [y, z])
{(-log(3), -sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), sqrt(-exp(2*x) - sin(log(3)))),
(ImageSet(Lambda(_n, 2*_n*I*pi - log(3)), Integers),
ImageSet(Lambda(_n, -sqrt(-exp(2*x) + sin(2*_n*I*pi - log(3)))), Integers)),
(ImageSet(Lambda(_n, 2*_n*I*pi - log(3)), Integers),
ImageSet(Lambda(_n, sqrt(-exp(2*x) + sin(2*_n*I*pi - log(3)))), Integers))}
"""
if not system:
return S.EmptySet
if not symbols:
msg = ('Symbols must be given, for which solution of the '
'system is to be found.')
raise ValueError(filldedent(msg))
if not is_sequence(symbols):
msg = ('symbols should be given as a sequence, e.g. a list.'
'Not type %s: %s')
raise TypeError(filldedent(msg % (type(symbols), symbols)))
if not getattr(symbols[0], 'is_Symbol', False):
msg = ('Iterable of symbols must be given as '
'second argument, not type %s: %s')
raise ValueError(filldedent(msg % (type(symbols[0]), symbols[0])))
# By default `all_symbols` will be same as `symbols`
if all_symbols is None:
all_symbols = symbols
old_result = result
# storing complements and intersection for particular symbol
complements = {}
intersections = {}
# when total_solveset_call equals total_conditionset
# it means that solveset failed to solve all eqs.
total_conditionset = -1
total_solveset_call = -1
def _unsolved_syms(eq, sort=False):
"""Returns the unsolved symbol present
in the equation `eq`.
"""
free = eq.free_symbols
unsolved = (free - set(known_symbols)) & set(all_symbols)
if sort:
unsolved = list(unsolved)
unsolved.sort(key=default_sort_key)
return unsolved
# end of _unsolved_syms()
# sort such that equation with the fewest potential symbols is first.
# means eq with less number of variable first in the list.
eqs_in_better_order = list(
ordered(system, lambda _: len(_unsolved_syms(_))))
def add_intersection_complement(result, intersection_dict, complement_dict):
# If solveset has returned some intersection/complement
# for any symbol, it will be added in the final solution.
final_result = []
for res in result:
res_copy = res
for key_res, value_res in res.items():
intersect_set, complement_set = None, None
for key_sym, value_sym in intersection_dict.items():
if key_sym == key_res:
intersect_set = value_sym
for key_sym, value_sym in complement_dict.items():
if key_sym == key_res:
complement_set = value_sym
if intersect_set or complement_set:
new_value = FiniteSet(value_res)
if intersect_set and intersect_set != S.Complexes:
new_value = Intersection(new_value, intersect_set)
if complement_set:
new_value = Complement(new_value, complement_set)
if new_value is S.EmptySet:
res_copy = None
break
elif new_value.is_FiniteSet and len(new_value) == 1:
res_copy[key_res] = set(new_value).pop()
else:
res_copy[key_res] = new_value
if res_copy is not None:
final_result.append(res_copy)
return final_result
# end of def add_intersection_complement()
def _extract_main_soln(sym, sol, soln_imageset):
"""Separate the Complements, Intersections, ImageSet lambda expr and
its base_set. This function returns the unmasks sol from different classes
of sets and also returns the appended ImageSet elements in a
soln_imageset (dict: where key as unmasked element and value as ImageSet).
"""
# if there is union, then need to check
# Complement, Intersection, Imageset.
# Order should not be changed.
if isinstance(sol, ConditionSet):
# extracts any solution in ConditionSet
sol = sol.base_set
if isinstance(sol, Complement):
# extract solution and complement
complements[sym] = sol.args[1]
sol = sol.args[0]
# complement will be added at the end
# using `add_intersection_complement` method
# if there is union of Imageset or other in soln.
# no testcase is written for this if block
if isinstance(sol, Union):
sol_args = sol.args
sol = S.EmptySet
# We need in sequence so append finteset elements
# and then imageset or other.
for sol_arg2 in sol_args:
if isinstance(sol_arg2, FiniteSet):
sol += sol_arg2
else:
# ImageSet, Intersection, complement then
# append them directly
sol += FiniteSet(sol_arg2)
if isinstance(sol, Intersection):
# Interval/Set will be at 0th index always
if sol.args[0] not in (S.Reals, S.Complexes):
# Sometimes solveset returns soln with intersection
# S.Reals or S.Complexes. We don't consider that
# intersection.
intersections[sym] = sol.args[0]
sol = sol.args[1]
# after intersection and complement Imageset should
# be checked.
if isinstance(sol, ImageSet):
soln_imagest = sol
expr2 = sol.lamda.expr
sol = FiniteSet(expr2)
soln_imageset[expr2] = soln_imagest
if not isinstance(sol, FiniteSet):
sol = FiniteSet(sol)
return sol, soln_imageset
# end of def _extract_main_soln()
# helper function for _append_new_soln
def _check_exclude(rnew, imgset_yes):
rnew_ = rnew
if imgset_yes:
# replace all dummy variables (Imageset lambda variables)
# with zero before `checksol`. Considering fundamental soln
# for `checksol`.
rnew_copy = rnew.copy()
dummy_n = imgset_yes[0]
for key_res, value_res in rnew_copy.items():
rnew_copy[key_res] = value_res.subs(dummy_n, 0)
rnew_ = rnew_copy
# satisfy_exclude == true if it satisfies the expr of `exclude` list.
try:
# something like : `Mod(-log(3), 2*I*pi)` can't be
# simplified right now, so `checksol` returns `TypeError`.
# when this issue is fixed this try block should be
# removed. Mod(-log(3), 2*I*pi) == -log(3)
satisfy_exclude = any(
checksol(d, rnew_) for d in exclude)
except TypeError:
satisfy_exclude = None
return satisfy_exclude
# end of def _check_exclude()
# helper function for _append_new_soln
def _restore_imgset(rnew, original_imageset, newresult):
restore_sym = set(rnew.keys()) & \
set(original_imageset.keys())
for key_sym in restore_sym:
img = original_imageset[key_sym]
rnew[key_sym] = img
if rnew not in newresult:
newresult.append(rnew)
# end of def _restore_imgset()
def _append_eq(eq, result, res, delete_soln, n=None):
u = Dummy('u')
if n:
eq = eq.subs(n, 0)
satisfy = eq if eq in (True, False) else checksol(u, u, eq, minimal=True)
if satisfy is False:
delete_soln = True
res = {}
else:
result.append(res)
return result, res, delete_soln
def _append_new_soln(rnew, sym, sol, imgset_yes, soln_imageset,
original_imageset, newresult, eq=None):
"""If `rnew` (A dict <symbol: soln>) contains valid soln
append it to `newresult` list.
`imgset_yes` is (base, dummy_var) if there was imageset in previously
calculated result(otherwise empty tuple). `original_imageset` is dict
of imageset expr and imageset from this result.
`soln_imageset` dict of imageset expr and imageset of new soln.
"""
satisfy_exclude = _check_exclude(rnew, imgset_yes)
delete_soln = False
# soln should not satisfy expr present in `exclude` list.
if not satisfy_exclude:
local_n = None
# if it is imageset
if imgset_yes:
local_n = imgset_yes[0]
base = imgset_yes[1]
if sym and sol:
# when `sym` and `sol` is `None` means no new
# soln. In that case we will append rnew directly after
# substituting original imagesets in rnew values if present
# (second last line of this function using _restore_imgset)
dummy_list = list(sol.atoms(Dummy))
# use one dummy `n` which is in
# previous imageset
local_n_list = [
local_n for i in range(
0, len(dummy_list))]
dummy_zip = zip(dummy_list, local_n_list)
lam = Lambda(local_n, sol.subs(dummy_zip))
rnew[sym] = ImageSet(lam, base)
if eq is not None:
newresult, rnew, delete_soln = _append_eq(
eq, newresult, rnew, delete_soln, local_n)
elif eq is not None:
newresult, rnew, delete_soln = _append_eq(
eq, newresult, rnew, delete_soln)
elif sol in soln_imageset.keys():
rnew[sym] = soln_imageset[sol]
# restore original imageset
_restore_imgset(rnew, original_imageset, newresult)
else:
newresult.append(rnew)
elif satisfy_exclude:
delete_soln = True
rnew = {}
_restore_imgset(rnew, original_imageset, newresult)
return newresult, delete_soln
# end of def _append_new_soln()
def _new_order_result(result, eq):
# separate first, second priority. `res` that makes `eq` value equals
# to zero, should be used first then other result(second priority).
# If it is not done then we may miss some soln.
first_priority = []
second_priority = []
for res in result:
if not any(isinstance(val, ImageSet) for val in res.values()):
if eq.subs(res) == 0:
first_priority.append(res)
else:
second_priority.append(res)
if first_priority or second_priority:
return first_priority + second_priority
return result
def _solve_using_known_values(result, solver):
"""Solves the system using already known solution
(result contains the dict <symbol: value>).
solver is :func:`~.solveset_complex` or :func:`~.solveset_real`.
"""
# stores imageset <expr: imageset(Lambda(n, expr), base)>.
soln_imageset = {}
total_solvest_call = 0
total_conditionst = 0
# sort such that equation with the fewest potential symbols is first.
# means eq with less variable first
for index, eq in enumerate(eqs_in_better_order):
newresult = []
original_imageset = {}
# if imageset expr is used to solve other symbol
imgset_yes = False
result = _new_order_result(result, eq)
for res in result:
got_symbol = set() # symbols solved in one iteration
# find the imageset and use its expr.
for key_res, value_res in res.items():
if isinstance(value_res, ImageSet):
res[key_res] = value_res.lamda.expr
original_imageset[key_res] = value_res
dummy_n = value_res.lamda.expr.atoms(Dummy).pop()
(base,) = value_res.base_sets
imgset_yes = (dummy_n, base)
# update eq with everything that is known so far
eq2 = eq.subs(res).expand()
unsolved_syms = _unsolved_syms(eq2, sort=True)
if not unsolved_syms:
if res:
newresult, delete_res = _append_new_soln(
res, None, None, imgset_yes, soln_imageset,
original_imageset, newresult, eq2)
if delete_res:
# `delete_res` is true, means substituting `res` in
# eq2 doesn't return `zero` or deleting the `res`
# (a soln) since it staisfies expr of `exclude`
# list.
result.remove(res)
continue # skip as it's independent of desired symbols
depen1, depen2 = (eq2.rewrite(Add)).as_independent(*unsolved_syms)
if (depen1.has(Abs) or depen2.has(Abs)) and solver == solveset_complex:
# Absolute values cannot be inverted in the
# complex domain
continue
soln_imageset = {}
for sym in unsolved_syms:
not_solvable = False
try:
soln = solver(eq2, sym)
total_solvest_call += 1
soln_new = S.EmptySet
if isinstance(soln, Complement):
# separate solution and complement
complements[sym] = soln.args[1]
soln = soln.args[0]
# complement will be added at the end
if isinstance(soln, Intersection):
# Interval will be at 0th index always
if soln.args[0] != Interval(-oo, oo):
# sometimes solveset returns soln
# with intersection S.Reals, to confirm that
# soln is in domain=S.Reals
intersections[sym] = soln.args[0]
soln_new += soln.args[1]
soln = soln_new if soln_new else soln
if index > 0 and solver == solveset_real:
# one symbol's real soln, another symbol may have
# corresponding complex soln.
if not isinstance(soln, (ImageSet, ConditionSet)):
soln += solveset_complex(eq2, sym) # might give ValueError with Abs
except (NotImplementedError, ValueError):
# If solveset is not able to solve equation `eq2`. Next
# time we may get soln using next equation `eq2`
continue
if isinstance(soln, ConditionSet):
if soln.base_set in (S.Reals, S.Complexes):
soln = S.EmptySet
# don't do `continue` we may get soln
# in terms of other symbol(s)
not_solvable = True
total_conditionst += 1
else:
soln = soln.base_set
if soln is not S.EmptySet:
soln, soln_imageset = _extract_main_soln(
sym, soln, soln_imageset)
for sol in soln:
# sol is not a `Union` since we checked it
# before this loop
sol, soln_imageset = _extract_main_soln(
sym, sol, soln_imageset)
sol = set(sol).pop()
free = sol.free_symbols
if got_symbol and any(
ss in free for ss in got_symbol
):
# sol depends on previously solved symbols
# then continue
continue
rnew = res.copy()
# put each solution in res and append the new result
# in the new result list (solution for symbol `s`)
# along with old results.
for k, v in res.items():
if isinstance(v, Expr) and isinstance(sol, Expr):
# if any unsolved symbol is present
# Then subs known value
rnew[k] = v.subs(sym, sol)
# and add this new solution
if sol in soln_imageset.keys():
# replace all lambda variables with 0.
imgst = soln_imageset[sol]
rnew[sym] = imgst.lamda(
*[0 for i in range(0, len(
imgst.lamda.variables))])
else:
rnew[sym] = sol
newresult, delete_res = _append_new_soln(
rnew, sym, sol, imgset_yes, soln_imageset,
original_imageset, newresult)
if delete_res:
# deleting the `res` (a soln) since it staisfies
# eq of `exclude` list
result.remove(res)
# solution got for sym
if not not_solvable:
got_symbol.add(sym)
# next time use this new soln
if newresult:
result = newresult
return result, total_solvest_call, total_conditionst
# end def _solve_using_know_values()
new_result_real, solve_call1, cnd_call1 = _solve_using_known_values(
old_result, solveset_real)
new_result_complex, solve_call2, cnd_call2 = _solve_using_known_values(
old_result, solveset_complex)
# If total_solveset_call is equal to total_conditionset
# then solveset failed to solve all of the equations.
# In this case we return a ConditionSet here.
total_conditionset += (cnd_call1 + cnd_call2)
total_solveset_call += (solve_call1 + solve_call2)
if total_conditionset == total_solveset_call and total_solveset_call != -1:
return _return_conditionset(eqs_in_better_order, all_symbols)
# don't keep duplicate solutions
filtered_complex = []
for i in list(new_result_complex):
for j in list(new_result_real):
if i.keys() != j.keys():
continue
if all(a.dummy_eq(b) for a, b in zip(i.values(), j.values()) \
if not (isinstance(a, int) and isinstance(b, int))):
break
else:
filtered_complex.append(i)
# overall result
result = new_result_real + filtered_complex
result_all_variables = []
result_infinite = []
for res in result:
if not res:
# means {None : None}
continue
# If length < len(all_symbols) means infinite soln.
# Some or all the soln is dependent on 1 symbol.
# eg. {x: y+2} then final soln {x: y+2, y: y}
if len(res) < len(all_symbols):
solved_symbols = res.keys()
unsolved = list(filter(
lambda x: x not in solved_symbols, all_symbols))
for unsolved_sym in unsolved:
res[unsolved_sym] = unsolved_sym
result_infinite.append(res)
if res not in result_all_variables:
result_all_variables.append(res)
if result_infinite:
# we have general soln
# eg : [{x: -1, y : 1}, {x : -y, y: y}] then
# return [{x : -y, y : y}]
result_all_variables = result_infinite
if intersections or complements:
result_all_variables = add_intersection_complement(
result_all_variables, intersections, complements)
# convert to ordered tuple
result = S.EmptySet
for r in result_all_variables:
temp = [r[symb] for symb in all_symbols]
result += FiniteSet(tuple(temp))
return result
# end of def substitution()
def _solveset_work(system, symbols):
soln = solveset(system[0], symbols[0])
if isinstance(soln, FiniteSet):
_soln = FiniteSet(*[tuple((s,)) for s in soln])
return _soln
else:
return FiniteSet(tuple(FiniteSet(soln)))
def _handle_positive_dimensional(polys, symbols, denominators):
from sympy.polys.polytools import groebner
# substitution method where new system is groebner basis of the system
_symbols = list(symbols)
_symbols.sort(key=default_sort_key)
basis = groebner(polys, _symbols, polys=True)
new_system = []
for poly_eq in basis:
new_system.append(poly_eq.as_expr())
result = [{}]
result = substitution(
new_system, symbols, result, [],
denominators)
return result
# end of def _handle_positive_dimensional()
def _handle_zero_dimensional(polys, symbols, system):
# solve 0 dimensional poly system using `solve_poly_system`
result = solve_poly_system(polys, *symbols)
# May be some extra soln is added because
# we used `unrad` in `_separate_poly_nonpoly`, so
# need to check and remove if it is not a soln.
result_update = S.EmptySet
for res in result:
dict_sym_value = dict(list(zip(symbols, res)))
if all(checksol(eq, dict_sym_value) for eq in system):
result_update += FiniteSet(res)
return result_update
# end of def _handle_zero_dimensional()
def _separate_poly_nonpoly(system, symbols):
polys = []
polys_expr = []
nonpolys = []
# unrad_changed stores a list of expressions containing
# radicals that were processed using unrad
# this is useful if solutions need to be checked later.
unrad_changed = []
denominators = set()
poly = None
for eq in system:
# Store denom expressions that contain symbols
denominators.update(_simple_dens(eq, symbols))
# Convert equality to expression
if isinstance(eq, Equality):
eq = eq.rewrite(Add)
# try to remove sqrt and rational power
without_radicals = unrad(simplify(eq), *symbols)
if without_radicals:
unrad_changed.append(eq)
eq_unrad, cov = without_radicals
if not cov:
eq = eq_unrad
if isinstance(eq, Expr):
eq = eq.as_numer_denom()[0]
poly = eq.as_poly(*symbols, extension=True)
elif simplify(eq).is_number:
continue
if poly is not None:
polys.append(poly)
polys_expr.append(poly.as_expr())
else:
nonpolys.append(eq)
return polys, polys_expr, nonpolys, denominators, unrad_changed
# end of def _separate_poly_nonpoly()
def _handle_poly(polys, symbols):
# _handle_poly(polys, symbols) -> (poly_sol, poly_eqs)
#
# We will return possible solution information to nonlinsolve as well as a
# new system of polynomial equations to be solved if we cannot solve
# everything directly here. The new system of polynomial equations will be
# a lex-order Groebner basis for the original system. The lex basis
# hopefully separate some of the variables and equations and give something
# easier for substitution to work with.
# The format for representing solution sets in nonlinsolve and substitution
# is a list of dicts. These are the special cases:
no_information = [{}] # No equations solved yet
no_solutions = [] # The system is inconsistent and has no solutions.
# If there is no need to attempt further solution of these equations then
# we return no equations:
no_equations = []
inexact = any(not p.domain.is_Exact for p in polys)
if inexact:
# The use of Groebner over RR is likely to result incorrectly in an
# inconsistent Groebner basis. So, convert any float coefficients to
# Rational before computing the Groebner basis.
polys = [poly(nsimplify(p, rational=True)) for p in polys]
# Compute a Groebner basis in grevlex order wrt the ordering given. We will
# try to convert this to lex order later. Usually it seems to be more
# efficient to compute a lex order basis by computing a grevlex basis and
# converting to lex with fglm.
basis = groebner(polys, symbols, order='grevlex', polys=False)
#
# No solutions (inconsistent equations)?
#
if 1 in basis:
# No solutions:
poly_sol = no_solutions
poly_eqs = no_equations
#
# Finite number of solutions (zero-dimensional case)
#
elif basis.is_zero_dimensional:
# Convert Groebner basis to lex ordering
basis = basis.fglm('lex')
# Convert polynomial coefficients back to float before calling
# solve_poly_system
if inexact:
basis = [nfloat(p) for p in basis]
# Solve the zero-dimensional case using solve_poly_system if possible.
# If some polynomials have factors that cannot be solved in radicals
# then this will fail. Using solve_poly_system(..., strict=True)
# ensures that we either get a complete solution set in radicals or
# UnsolvableFactorError will be raised.
try:
result = solve_poly_system(basis, *symbols, strict=True)
except UnsolvableFactorError:
# Failure... not fully solvable in radicals. Return the lex-order
# basis for substitution to handle.
poly_sol = no_information
poly_eqs = list(basis)
else:
# Success! We have a finite solution set and solve_poly_system has
# succeeded in finding all solutions. Return the solutions and also
# an empty list of remaining equations to be solved.
poly_sol = [dict(zip(symbols, res)) for res in result]
poly_eqs = no_equations
#
# Infinite families of solutions (positive-dimensional case)
#
else:
# In this case the grevlex basis cannot be converted to lex using the
# fglm method and also solve_poly_system cannot solve the equations. We
# would like to return a lex basis but since we can't use fglm we
# compute the lex basis directly here. The time required to recompute
# the basis is generally significantly less than the time required by
# substitution to solve the new system.
poly_sol = no_information
poly_eqs = list(groebner(polys, symbols, order='lex', polys=False))
if inexact:
poly_eqs = [nfloat(p) for p in poly_eqs]
return poly_sol, poly_eqs
def nonlinsolve(system, *symbols):
r"""
Solve system of $N$ nonlinear equations with $M$ variables, which means both
under and overdetermined systems are supported. Positive dimensional
system is also supported (A system with infinitely many solutions is said
to be positive-dimensional). In a positive dimensional system the solution will
be dependent on at least one symbol. Returns both real solution
and complex solution (if they exist).
Parameters
==========
system : list of equations
The target system of equations
symbols : list of Symbols
symbols should be given as a sequence eg. list
Returns
=======
A :class:`~.FiniteSet` of ordered tuple of values of `symbols` for which the `system`
has solution. Order of values in the tuple is same as symbols present in
the parameter `symbols`.
Please note that general :class:`~.FiniteSet` is unordered, the solution
returned here is not simply a :class:`~.FiniteSet` of solutions, rather it
is a :class:`~.FiniteSet` of ordered tuple, i.e. the first and only
argument to :class:`~.FiniteSet` is a tuple of solutions, which is
ordered, and, hence ,the returned solution is ordered.
Also note that solution could also have been returned as an ordered tuple,
FiniteSet is just a wrapper ``{}`` around the tuple. It has no other
significance except for the fact it is just used to maintain a consistent
output format throughout the solveset.
For the given set of equations, the respective input types
are given below:
.. math:: xy - 1 = 0
.. math:: 4x^2 + y^2 - 5 = 0
::
system = [x*y - 1, 4*x**2 + y**2 - 5]
symbols = [x, y]
Raises
======
ValueError
The input is not valid.
The symbols are not given.
AttributeError
The input symbols are not `Symbol` type.
Examples
========
>>> from sympy import symbols, nonlinsolve
>>> x, y, z = symbols('x, y, z', real=True)
>>> nonlinsolve([x*y - 1, 4*x**2 + y**2 - 5], [x, y])
{(-1, -1), (-1/2, -2), (1/2, 2), (1, 1)}
1. Positive dimensional system and complements:
>>> from sympy import pprint
>>> from sympy.polys.polytools import is_zero_dimensional
>>> a, b, c, d = symbols('a, b, c, d', extended_real=True)
>>> eq1 = a + b + c + d
>>> eq2 = a*b + b*c + c*d + d*a
>>> eq3 = a*b*c + b*c*d + c*d*a + d*a*b
>>> eq4 = a*b*c*d - 1
>>> system = [eq1, eq2, eq3, eq4]
>>> is_zero_dimensional(system)
False
>>> pprint(nonlinsolve(system, [a, b, c, d]), use_unicode=False)
-1 1 1 -1
{(---, -d, -, {d} \ {0}), (-, -d, ---, {d} \ {0})}
d d d d
>>> nonlinsolve([(x+y)**2 - 4, x + y - 2], [x, y])
{(2 - y, y)}
2. If some of the equations are non-polynomial then `nonlinsolve`
will call the ``substitution`` function and return real and complex solutions,
if present.
>>> from sympy import exp, sin
>>> nonlinsolve([exp(x) - sin(y), y**2 - 4], [x, y])
{(ImageSet(Lambda(_n, I*(2*_n*pi + pi) + log(sin(2))), Integers), -2),
(ImageSet(Lambda(_n, 2*_n*I*pi + log(sin(2))), Integers), 2)}
3. If system is non-linear polynomial and zero-dimensional then it
returns both solution (real and complex solutions, if present) using
:func:`~.solve_poly_system`:
>>> from sympy import sqrt
>>> nonlinsolve([x**2 - 2*y**2 -2, x*y - 2], [x, y])
{(-2, -1), (2, 1), (-sqrt(2)*I, sqrt(2)*I), (sqrt(2)*I, -sqrt(2)*I)}
4. ``nonlinsolve`` can solve some linear (zero or positive dimensional)
system (because it uses the :func:`sympy.polys.polytools.groebner` function to get the
groebner basis and then uses the ``substitution`` function basis as the
new `system`). But it is not recommended to solve linear system using
``nonlinsolve``, because :func:`~.linsolve` is better for general linear systems.
>>> nonlinsolve([x + 2*y -z - 3, x - y - 4*z + 9, y + z - 4], [x, y, z])
{(3*z - 5, 4 - z, z)}
5. System having polynomial equations and only real solution is
solved using :func:`~.solve_poly_system`:
>>> e1 = sqrt(x**2 + y**2) - 10
>>> e2 = sqrt(y**2 + (-x + 10)**2) - 3
>>> nonlinsolve((e1, e2), (x, y))
{(191/20, -3*sqrt(391)/20), (191/20, 3*sqrt(391)/20)}
>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [x, y])
{(1, 2), (1 - sqrt(5), 2 + sqrt(5)), (1 + sqrt(5), 2 - sqrt(5))}
>>> nonlinsolve([x**2 + 2/y - 2, x + y - 3], [y, x])
{(2, 1), (2 - sqrt(5), 1 + sqrt(5)), (2 + sqrt(5), 1 - sqrt(5))}
6. It is better to use symbols instead of trigonometric functions or
:class:`~.Function`. For example, replace $\sin(x)$ with a symbol, replace
$f(x)$ with a symbol and so on. Get a solution from ``nonlinsolve`` and then
use :func:`~.solveset` to get the value of $x$.
How nonlinsolve is better than old solver ``_solve_system`` :
=============================================================
1. A positive dimensional system solver: nonlinsolve can return
solution for positive dimensional system. It finds the
Groebner Basis of the positive dimensional system(calling it as
basis) then we can start solving equation(having least number of
variable first in the basis) using solveset and substituting that
solved solutions into other equation(of basis) to get solution in
terms of minimum variables. Here the important thing is how we
are substituting the known values and in which equations.
2. Real and complex solutions: nonlinsolve returns both real
and complex solution. If all the equations in the system are polynomial
then using :func:`~.solve_poly_system` both real and complex solution is returned.
If all the equations in the system are not polynomial equation then goes to
``substitution`` method with this polynomial and non polynomial equation(s),
to solve for unsolved variables. Here to solve for particular variable
solveset_real and solveset_complex is used. For both real and complex
solution ``_solve_using_known_values`` is used inside ``substitution``
(``substitution`` will be called when any non-polynomial equation is present).
If a solution is valid its general solution is added to the final result.
3. :class:`~.Complement` and :class:`~.Intersection` will be added:
nonlinsolve maintains dict for complements and intersections. If solveset
find complements or/and intersections with any interval or set during the
execution of ``substitution`` function, then complement or/and
intersection for that variable is added before returning final solution.
"""
if not system:
return S.EmptySet
if not symbols:
msg = ('Symbols must be given, for which solution of the '
'system is to be found.')
raise ValueError(filldedent(msg))
if hasattr(symbols[0], '__iter__'):
symbols = symbols[0]
if not is_sequence(symbols) or not symbols:
msg = ('Symbols must be given, for which solution of the '
'system is to be found.')
raise IndexError(filldedent(msg))
symbols = list(map(_sympify, symbols))
system, symbols, swap = recast_to_symbols(system, symbols)
if swap:
soln = nonlinsolve(system, symbols)
return FiniteSet(*[tuple(i.xreplace(swap) for i in s) for s in soln])
if len(system) == 1 and len(symbols) == 1:
return _solveset_work(system, symbols)
# main code of def nonlinsolve() starts from here
polys, polys_expr, nonpolys, denominators, unrad_changed = \
_separate_poly_nonpoly(system, symbols)
poly_eqs = []
poly_sol = [{}]
if polys:
poly_sol, poly_eqs = _handle_poly(polys, symbols)
if poly_sol and poly_sol[0]:
poly_syms = set().union(*(eq.free_symbols for eq in polys))
unrad_syms = set().union(*(eq.free_symbols for eq in unrad_changed))
if unrad_syms == poly_syms and unrad_changed:
# if all the symbols have been solved by _handle_poly
# and unrad has been used then check solutions
poly_sol = [sol for sol in poly_sol if checksol(unrad_changed, sol)]
# Collect together the unsolved polynomials with the non-polynomial
# equations.
remaining = poly_eqs + nonpolys
# to_tuple converts a solution dictionary to a tuple containing the
# value for each symbol
to_tuple = lambda sol: tuple(sol[s] for s in symbols)
if not remaining:
# If there is nothing left to solve then return the solution from
# solve_poly_system directly.
return FiniteSet(*map(to_tuple, poly_sol))
else:
# Here we handle:
#
# 1. The Groebner basis if solve_poly_system failed.
# 2. The Groebner basis in the positive-dimensional case.
# 3. Any non-polynomial equations
#
# If solve_poly_system did succeed then we pass those solutions in as
# preliminary results.
subs_res = substitution(remaining, symbols, result=poly_sol, exclude=denominators)
if not isinstance(subs_res, FiniteSet):
return subs_res
# check solutions produced by substitution. Currently, checking is done for
# only those solutions which have non-Set variable values.
if unrad_changed:
result = [dict(zip(symbols, sol)) for sol in subs_res.args]
correct_sols = [sol for sol in result if any(isinstance(v, Set) for v in sol)
or checksol(unrad_changed, sol) != False]
return FiniteSet(*map(to_tuple, correct_sols))
else:
return subs_res
|
69be7ab0efeafeab2985f6b42cb201099e7bfc8ebd728ba7c6f2d51d4d04cb11 | """Solvers of systems of polynomial equations. """
import itertools
from sympy.core import S
from sympy.core.sorting import default_sort_key
from sympy.polys import Poly, groebner, roots
from sympy.polys.polytools import parallel_poly_from_expr
from sympy.polys.polyerrors import (ComputationFailed,
PolificationFailed, CoercionFailed)
from sympy.simplify import rcollect
from sympy.utilities import postfixes
from sympy.utilities.misc import filldedent
class SolveFailed(Exception):
"""Raised when solver's conditions were not met. """
def solve_poly_system(seq, *gens, strict=False, **args):
"""
returns a list of solutions for the system of polynomial equations
or else None.
Parameters
==========
seq: a list/tuple/set
Listing all the equations that are needed to be solved
gens: generators
generators of the equations in seq for which we want the
solutions
strict: a boolean (default is False)
if strict is True, NotImplementedError will be raised if
the solution is known to be incomplete (which can occur if
not all solutions are expressible in radicals)
args: Keyword arguments
Special options for solving the equations.
Returns
=======
List[Tuple]
a list of tuples with elements being solutions for the
symbols in the order they were passed as gens
None
None is returned when the computed basis contains only the ground.
Examples
========
>>> from sympy import solve_poly_system
>>> from sympy.abc import x, y
>>> solve_poly_system([x*y - 2*y, 2*y**2 - x**2], x, y)
[(0, 0), (2, -sqrt(2)), (2, sqrt(2))]
>>> solve_poly_system([x**5 - x + y**3, y**2 - 1], x, y, strict=True)
Traceback (most recent call last):
...
UnsolvableFactorError
"""
try:
polys, opt = parallel_poly_from_expr(seq, *gens, **args)
except PolificationFailed as exc:
raise ComputationFailed('solve_poly_system', len(seq), exc)
if len(polys) == len(opt.gens) == 2:
f, g = polys
if all(i <= 2 for i in f.degree_list() + g.degree_list()):
try:
return solve_biquadratic(f, g, opt)
except SolveFailed:
pass
return solve_generic(polys, opt, strict=strict)
def solve_biquadratic(f, g, opt):
"""Solve a system of two bivariate quadratic polynomial equations.
Parameters
==========
f: a single Expr or Poly
First equation
g: a single Expr or Poly
Second Equation
opt: an Options object
For specifying keyword arguments and generators
Returns
=======
List[Tuple]
a list of tuples with elements being solutions for the
symbols in the order they were passed as gens
None
None is returned when the computed basis contains only the ground.
Examples
========
>>> from sympy import Options, Poly
>>> from sympy.abc import x, y
>>> from sympy.solvers.polysys import solve_biquadratic
>>> NewOption = Options((x, y), {'domain': 'ZZ'})
>>> a = Poly(y**2 - 4 + x, y, x, domain='ZZ')
>>> b = Poly(y*2 + 3*x - 7, y, x, domain='ZZ')
>>> solve_biquadratic(a, b, NewOption)
[(1/3, 3), (41/27, 11/9)]
>>> a = Poly(y + x**2 - 3, y, x, domain='ZZ')
>>> b = Poly(-y + x - 4, y, x, domain='ZZ')
>>> solve_biquadratic(a, b, NewOption)
[(7/2 - sqrt(29)/2, -sqrt(29)/2 - 1/2), (sqrt(29)/2 + 7/2, -1/2 + \
sqrt(29)/2)]
"""
G = groebner([f, g])
if len(G) == 1 and G[0].is_ground:
return None
if len(G) != 2:
raise SolveFailed
x, y = opt.gens
p, q = G
if not p.gcd(q).is_ground:
# not 0-dimensional
raise SolveFailed
p = Poly(p, x, expand=False)
p_roots = [rcollect(expr, y) for expr in roots(p).keys()]
q = q.ltrim(-1)
q_roots = list(roots(q).keys())
solutions = [(p_root.subs(y, q_root), q_root) for q_root, p_root in
itertools.product(q_roots, p_roots)]
return sorted(solutions, key=default_sort_key)
def solve_generic(polys, opt, strict=False):
"""
Solve a generic system of polynomial equations.
Returns all possible solutions over C[x_1, x_2, ..., x_m] of a
set F = { f_1, f_2, ..., f_n } of polynomial equations, using
Groebner basis approach. For now only zero-dimensional systems
are supported, which means F can have at most a finite number
of solutions. If the basis contains only the ground, None is
returned.
The algorithm works by the fact that, supposing G is the basis
of F with respect to an elimination order (here lexicographic
order is used), G and F generate the same ideal, they have the
same set of solutions. By the elimination property, if G is a
reduced, zero-dimensional Groebner basis, then there exists an
univariate polynomial in G (in its last variable). This can be
solved by computing its roots. Substituting all computed roots
for the last (eliminated) variable in other elements of G, new
polynomial system is generated. Applying the above procedure
recursively, a finite number of solutions can be found.
The ability of finding all solutions by this procedure depends
on the root finding algorithms. If no solutions were found, it
means only that roots() failed, but the system is solvable. To
overcome this difficulty use numerical algorithms instead.
Parameters
==========
polys: a list/tuple/set
Listing all the polynomial equations that are needed to be solved
opt: an Options object
For specifying keyword arguments and generators
strict: a boolean
If strict is True, NotImplementedError will be raised if the solution
is known to be incomplete
Returns
=======
List[Tuple]
a list of tuples with elements being solutions for the
symbols in the order they were passed as gens
None
None is returned when the computed basis contains only the ground.
References
==========
.. [Buchberger01] B. Buchberger, Groebner Bases: A Short
Introduction for Systems Theorists, In: R. Moreno-Diaz,
B. Buchberger, J.L. Freire, Proceedings of EUROCAST'01,
February, 2001
.. [Cox97] D. Cox, J. Little, D. O'Shea, Ideals, Varieties
and Algorithms, Springer, Second Edition, 1997, pp. 112
Raises
========
NotImplementedError
If the system is not zero-dimensional (does not have a finite
number of solutions)
UnsolvableFactorError
If ``strict`` is True and not all solution components are
expressible in radicals
Examples
========
>>> from sympy import Poly, Options
>>> from sympy.solvers.polysys import solve_generic
>>> from sympy.abc import x, y
>>> NewOption = Options((x, y), {'domain': 'ZZ'})
>>> a = Poly(x - y + 5, x, y, domain='ZZ')
>>> b = Poly(x + y - 3, x, y, domain='ZZ')
>>> solve_generic([a, b], NewOption)
[(-1, 4)]
>>> a = Poly(x - 2*y + 5, x, y, domain='ZZ')
>>> b = Poly(2*x - y - 3, x, y, domain='ZZ')
>>> solve_generic([a, b], NewOption)
[(11/3, 13/3)]
>>> a = Poly(x**2 + y, x, y, domain='ZZ')
>>> b = Poly(x + y*4, x, y, domain='ZZ')
>>> solve_generic([a, b], NewOption)
[(0, 0), (1/4, -1/16)]
>>> a = Poly(x**5 - x + y**3, x, y, domain='ZZ')
>>> b = Poly(y**2 - 1, x, y, domain='ZZ')
>>> solve_generic([a, b], NewOption, strict=True)
Traceback (most recent call last):
...
UnsolvableFactorError
"""
def _is_univariate(f):
"""Returns True if 'f' is univariate in its last variable. """
for monom in f.monoms():
if any(monom[:-1]):
return False
return True
def _subs_root(f, gen, zero):
"""Replace generator with a root so that the result is nice. """
p = f.as_expr({gen: zero})
if f.degree(gen) >= 2:
p = p.expand(deep=False)
return p
def _solve_reduced_system(system, gens, entry=False):
"""Recursively solves reduced polynomial systems. """
if len(system) == len(gens) == 1:
# the below line will produce UnsolvableFactorError if
# strict=True and the solution from `roots` is incomplete
zeros = list(roots(system[0], gens[-1], strict=strict).keys())
return [(zero,) for zero in zeros]
basis = groebner(system, gens, polys=True)
if len(basis) == 1 and basis[0].is_ground:
if not entry:
return []
else:
return None
univariate = list(filter(_is_univariate, basis))
if len(basis) < len(gens):
raise NotImplementedError(filldedent('''
only zero-dimensional systems supported
(finite number of solutions)
'''))
if len(univariate) == 1:
f = univariate.pop()
else:
raise NotImplementedError(filldedent('''
only zero-dimensional systems supported
(finite number of solutions)
'''))
gens = f.gens
gen = gens[-1]
# the below line will produce UnsolvableFactorError if
# strict=True and the solution from `roots` is incomplete
zeros = list(roots(f.ltrim(gen), strict=strict).keys())
if not zeros:
return []
if len(basis) == 1:
return [(zero,) for zero in zeros]
solutions = []
for zero in zeros:
new_system = []
new_gens = gens[:-1]
for b in basis[:-1]:
eq = _subs_root(b, gen, zero)
if eq is not S.Zero:
new_system.append(eq)
for solution in _solve_reduced_system(new_system, new_gens):
solutions.append(solution + (zero,))
if solutions and len(solutions[0]) != len(gens):
raise NotImplementedError(filldedent('''
only zero-dimensional systems supported
(finite number of solutions)
'''))
return solutions
try:
result = _solve_reduced_system(polys, opt.gens, entry=True)
except CoercionFailed:
raise NotImplementedError
if result is not None:
return sorted(result, key=default_sort_key)
def solve_triangulated(polys, *gens, **args):
"""
Solve a polynomial system using Gianni-Kalkbrenner algorithm.
The algorithm proceeds by computing one Groebner basis in the ground
domain and then by iteratively computing polynomial factorizations in
appropriately constructed algebraic extensions of the ground domain.
Parameters
==========
polys: a list/tuple/set
Listing all the equations that are needed to be solved
gens: generators
generators of the equations in polys for which we want the
solutions
args: Keyword arguments
Special options for solving the equations
Returns
=======
List[Tuple]
A List of tuples. Solutions for symbols that satisfy the
equations listed in polys
Examples
========
>>> from sympy import solve_triangulated
>>> from sympy.abc import x, y, z
>>> F = [x**2 + y + z - 1, x + y**2 + z - 1, x + y + z**2 - 1]
>>> solve_triangulated(F, x, y, z)
[(0, 0, 1), (0, 1, 0), (1, 0, 0)]
References
==========
1. Patrizia Gianni, Teo Mora, Algebraic Solution of System of
Polynomial Equations using Groebner Bases, AAECC-5 on Applied Algebra,
Algebraic Algorithms and Error-Correcting Codes, LNCS 356 247--257, 1989
"""
G = groebner(polys, gens, polys=True)
G = list(reversed(G))
domain = args.get('domain')
if domain is not None:
for i, g in enumerate(G):
G[i] = g.set_domain(domain)
f, G = G[0].ltrim(-1), G[1:]
dom = f.get_domain()
zeros = f.ground_roots()
solutions = set()
for zero in zeros:
solutions.add(((zero,), dom))
var_seq = reversed(gens[:-1])
vars_seq = postfixes(gens[1:])
for var, vars in zip(var_seq, vars_seq):
_solutions = set()
for values, dom in solutions:
H, mapping = [], list(zip(vars, values))
for g in G:
_vars = (var,) + vars
if g.has_only_gens(*_vars) and g.degree(var) != 0:
h = g.ltrim(var).eval(dict(mapping))
if g.degree(var) == h.degree():
H.append(h)
p = min(H, key=lambda h: h.degree())
zeros = p.ground_roots()
for zero in zeros:
if not zero.is_Rational:
dom_zero = dom.algebraic_field(zero)
else:
dom_zero = dom
_solutions.add(((zero,) + values, dom_zero))
solutions = _solutions
solutions = list(solutions)
for i, (solution, _) in enumerate(solutions):
solutions[i] = solution
return sorted(solutions, key=default_sort_key)
|
b2527dcf9ccb295fa86b26289b85294b84ff16c46204486460df74259a1d902a | """
This module contain solvers for all kinds of equations:
- algebraic or transcendental, use solve()
- recurrence, use rsolve()
- differential, use dsolve()
- nonlinear (numerically), use nsolve()
(you will need a good starting point)
"""
from sympy.core import (S, Add, Symbol, Dummy, Expr, Mul)
from sympy.core.assumptions import check_assumptions
from sympy.core.exprtools import factor_terms
from sympy.core.function import (expand_mul, expand_log, Derivative,
AppliedUndef, UndefinedFunction, nfloat,
Function, expand_power_exp, _mexpand, expand,
expand_func)
from sympy.core.logic import fuzzy_not
from sympy.core.numbers import ilcm, Float, Rational, _illegal
from sympy.core.power import integer_log, Pow
from sympy.core.relational import Eq, Ne
from sympy.core.sorting import ordered, default_sort_key
from sympy.core.sympify import sympify, _sympify
from sympy.core.traversal import preorder_traversal
from sympy.logic.boolalg import And, BooleanAtom
from sympy.functions import (log, exp, LambertW, cos, sin, tan, acos, asin, atan,
Abs, re, im, arg, sqrt, atan2)
from sympy.functions.combinatorial.factorials import binomial
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
from sympy.functions.elementary.piecewise import piecewise_fold, Piecewise
from sympy.functions.elementary.trigonometric import TrigonometricFunction
from sympy.integrals.integrals import Integral
from sympy.ntheory.factor_ import divisors
from sympy.simplify import (simplify, collect, powsimp, posify, # type: ignore
powdenest, nsimplify, denom, logcombine, sqrtdenest, fraction,
separatevars)
from sympy.simplify.sqrtdenest import sqrt_depth
from sympy.simplify.fu import TR1, TR2i
from sympy.matrices.common import NonInvertibleMatrixError
from sympy.matrices import Matrix, zeros
from sympy.polys import roots, cancel, factor, Poly
from sympy.polys.polyerrors import GeneratorsNeeded, PolynomialError
from sympy.polys.solvers import sympy_eqs_to_ring, solve_lin_sys
from sympy.utilities.lambdify import lambdify
from sympy.utilities.misc import filldedent, debug
from sympy.utilities.iterables import (connected_components,
generate_bell, uniq, iterable, is_sequence, subsets, flatten)
from sympy.utilities.decorator import conserve_mpmath_dps
from mpmath import findroot
from sympy.solvers.polysys import solve_poly_system
from types import GeneratorType
from collections import defaultdict
from itertools import combinations, product
import warnings
def recast_to_symbols(eqs, symbols):
"""
Return (e, s, d) where e and s are versions of *eqs* and
*symbols* in which any non-Symbol objects in *symbols* have
been replaced with generic Dummy symbols and d is a dictionary
that can be used to restore the original expressions.
Examples
========
>>> from sympy.solvers.solvers import recast_to_symbols
>>> from sympy import symbols, Function
>>> x, y = symbols('x y')
>>> fx = Function('f')(x)
>>> eqs, syms = [fx + 1, x, y], [fx, y]
>>> e, s, d = recast_to_symbols(eqs, syms); (e, s, d)
([_X0 + 1, x, y], [_X0, y], {_X0: f(x)})
The original equations and symbols can be restored using d:
>>> assert [i.xreplace(d) for i in eqs] == eqs
>>> assert [d.get(i, i) for i in s] == syms
"""
if not iterable(eqs) and iterable(symbols):
raise ValueError('Both eqs and symbols must be iterable')
orig = list(symbols)
symbols = list(ordered(symbols))
swap_sym = {}
i = 0
for j, s in enumerate(symbols):
if not isinstance(s, Symbol) and s not in swap_sym:
swap_sym[s] = Dummy('X%d' % i)
i += 1
new_f = []
for i in eqs:
isubs = getattr(i, 'subs', None)
if isubs is not None:
new_f.append(isubs(swap_sym))
else:
new_f.append(i)
restore = {v: k for k, v in swap_sym.items()}
return new_f, [swap_sym.get(i, i) for i in orig], restore
def _ispow(e):
"""Return True if e is a Pow or is exp."""
return isinstance(e, Expr) and (e.is_Pow or isinstance(e, exp))
def _simple_dens(f, symbols):
# when checking if a denominator is zero, we can just check the
# base of powers with nonzero exponents since if the base is zero
# the power will be zero, too. To keep it simple and fast, we
# limit simplification to exponents that are Numbers
dens = set()
for d in denoms(f, symbols):
if d.is_Pow and d.exp.is_Number:
if d.exp.is_zero:
continue # foo**0 is never 0
d = d.base
dens.add(d)
return dens
def denoms(eq, *symbols):
"""
Return (recursively) set of all denominators that appear in *eq*
that contain any symbol in *symbols*; if *symbols* are not
provided then all denominators will be returned.
Examples
========
>>> from sympy.solvers.solvers import denoms
>>> from sympy.abc import x, y, z
>>> denoms(x/y)
{y}
>>> denoms(x/(y*z))
{y, z}
>>> denoms(3/x + y/z)
{x, z}
>>> denoms(x/2 + y/z)
{2, z}
If *symbols* are provided then only denominators containing
those symbols will be returned:
>>> denoms(1/x + 1/y + 1/z, y, z)
{y, z}
"""
pot = preorder_traversal(eq)
dens = set()
for p in pot:
# Here p might be Tuple or Relational
# Expr subtrees (e.g. lhs and rhs) will be traversed after by pot
if not isinstance(p, Expr):
continue
den = denom(p)
if den is S.One:
continue
for d in Mul.make_args(den):
dens.add(d)
if not symbols:
return dens
elif len(symbols) == 1:
if iterable(symbols[0]):
symbols = symbols[0]
return {d for d in dens if any(s in d.free_symbols for s in symbols)}
def checksol(f, symbol, sol=None, **flags):
"""
Checks whether sol is a solution of equation f == 0.
Explanation
===========
Input can be either a single symbol and corresponding value
or a dictionary of symbols and values. When given as a dictionary
and flag ``simplify=True``, the values in the dictionary will be
simplified. *f* can be a single equation or an iterable of equations.
A solution must satisfy all equations in *f* to be considered valid;
if a solution does not satisfy any equation, False is returned; if one or
more checks are inconclusive (and none are False) then None is returned.
Examples
========
>>> from sympy import checksol, symbols
>>> x, y = symbols('x,y')
>>> checksol(x**4 - 1, x, 1)
True
>>> checksol(x**4 - 1, x, 0)
False
>>> checksol(x**2 + y**2 - 5**2, {x: 3, y: 4})
True
To check if an expression is zero using ``checksol()``, pass it
as *f* and send an empty dictionary for *symbol*:
>>> checksol(x**2 + x - x*(x + 1), {})
True
None is returned if ``checksol()`` could not conclude.
flags:
'numerical=True (default)'
do a fast numerical check if ``f`` has only one symbol.
'minimal=True (default is False)'
a very fast, minimal testing.
'warn=True (default is False)'
show a warning if checksol() could not conclude.
'simplify=True (default)'
simplify solution before substituting into function and
simplify the function before trying specific simplifications
'force=True (default is False)'
make positive all symbols without assumptions regarding sign.
"""
from sympy.physics.units import Unit
minimal = flags.get('minimal', False)
if sol is not None:
sol = {symbol: sol}
elif isinstance(symbol, dict):
sol = symbol
else:
msg = 'Expecting (sym, val) or ({sym: val}, None) but got (%s, %s)'
raise ValueError(msg % (symbol, sol))
if iterable(f):
if not f:
raise ValueError('no functions to check')
rv = True
for fi in f:
check = checksol(fi, sol, **flags)
if check:
continue
if check is False:
return False
rv = None # don't return, wait to see if there's a False
return rv
f = _sympify(f)
if f.is_number:
return f.is_zero
if isinstance(f, Poly):
f = f.as_expr()
elif isinstance(f, (Eq, Ne)):
if f.rhs in (S.true, S.false):
f = f.reversed
B, E = f.args
if isinstance(B, BooleanAtom):
f = f.subs(sol)
if not f.is_Boolean:
return
else:
f = f.rewrite(Add, evaluate=False)
if isinstance(f, BooleanAtom):
return bool(f)
elif not f.is_Relational and not f:
return True
illegal = set(_illegal)
if any(sympify(v).atoms() & illegal for k, v in sol.items()):
return False
was = f
attempt = -1
numerical = flags.get('numerical', True)
while 1:
attempt += 1
if attempt == 0:
val = f.subs(sol)
if isinstance(val, Mul):
val = val.as_independent(Unit)[0]
if val.atoms() & illegal:
return False
elif attempt == 1:
if not val.is_number:
if not val.is_constant(*list(sol.keys()), simplify=not minimal):
return False
# there are free symbols -- simple expansion might work
_, val = val.as_content_primitive()
val = _mexpand(val.as_numer_denom()[0], recursive=True)
elif attempt == 2:
if minimal:
return
if flags.get('simplify', True):
for k in sol:
sol[k] = simplify(sol[k])
# start over without the failed expanded form, possibly
# with a simplified solution
val = simplify(f.subs(sol))
if flags.get('force', True):
val, reps = posify(val)
# expansion may work now, so try again and check
exval = _mexpand(val, recursive=True)
if exval.is_number:
# we can decide now
val = exval
else:
# if there are no radicals and no functions then this can't be
# zero anymore -- can it?
pot = preorder_traversal(expand_mul(val))
seen = set()
saw_pow_func = False
for p in pot:
if p in seen:
continue
seen.add(p)
if p.is_Pow and not p.exp.is_Integer:
saw_pow_func = True
elif p.is_Function:
saw_pow_func = True
elif isinstance(p, UndefinedFunction):
saw_pow_func = True
if saw_pow_func:
break
if saw_pow_func is False:
return False
if flags.get('force', True):
# don't do a zero check with the positive assumptions in place
val = val.subs(reps)
nz = fuzzy_not(val.is_zero)
if nz is not None:
# issue 5673: nz may be True even when False
# so these are just hacks to keep a false positive
# from being returned
# HACK 1: LambertW (issue 5673)
if val.is_number and val.has(LambertW):
# don't eval this to verify solution since if we got here,
# numerical must be False
return None
# add other HACKs here if necessary, otherwise we assume
# the nz value is correct
return not nz
break
if numerical and val.is_number:
return (abs(val.n(18).n(12, chop=True)) < 1e-9) is S.true
if val == was:
continue
elif val.is_Rational:
return val == 0
was = val
if flags.get('warn', False):
warnings.warn("\n\tWarning: could not verify solution %s." % sol)
# returns None if it can't conclude
# TODO: improve solution testing
def solve(f, *symbols, **flags):
r"""
Algebraically solves equations and systems of equations.
Explanation
===========
Currently supported:
- polynomial
- transcendental
- piecewise combinations of the above
- systems of linear and polynomial equations
- systems containing relational expressions
- systems implied by undetermined coefficients
Examples
========
The default output varies according to the input and might
be a list (possibly empty), a dictionary, a list of
dictionaries or tuples, or an expression involving relationals.
For specifics regarding different forms of output that may appear, see :ref:`solve_output`.
Let it suffice here to say that to obtain a uniform output from
`solve` use ``dict=True`` or ``set=True`` (see below).
>>> from sympy import solve, Poly, Eq, Matrix, Symbol
>>> from sympy.abc import x, y, z, a, b
The expressions that are passed can be Expr, Equality, or Poly
classes (or lists of the same); a Matrix is considered to be a
list of all the elements of the matrix:
>>> solve(x - 3, x)
[3]
>>> solve(Eq(x, 3), x)
[3]
>>> solve(Poly(x - 3), x)
[3]
>>> solve(Matrix([[x, x + y]]), x, y) == solve([x, x + y], x, y)
True
If no symbols are indicated to be of interest and the equation is
univariate, a list of values is returned; otherwise, the keys in
a dictionary will indicate which (of all the variables used in
the expression(s)) variables and solutions were found:
>>> solve(x**2 - 4)
[-2, 2]
>>> solve((x - a)*(y - b))
[{a: x}, {b: y}]
>>> solve([x - 3, y - 1])
{x: 3, y: 1}
>>> solve([x - 3, y**2 - 1])
[{x: 3, y: -1}, {x: 3, y: 1}]
If you pass symbols for which solutions are sought, the output will vary
depending on the number of symbols you passed, whether you are passing
a list of expressions or not, and whether a linear system was solved.
Uniform output is attained by using ``dict=True`` or ``set=True``.
>>> #### *** feel free to skip to the stars below *** ####
>>> from sympy import TableForm
>>> h = [None, ';|;'.join(['e', 's', 'solve(e, s)', 'solve(e, s, dict=True)',
... 'solve(e, s, set=True)']).split(';')]
>>> t = []
>>> for e, s in [
... (x - y, y),
... (x - y, [x, y]),
... (x**2 - y, [x, y]),
... ([x - 3, y -1], [x, y]),
... ]:
... how = [{}, dict(dict=True), dict(set=True)]
... res = [solve(e, s, **f) for f in how]
... t.append([e, '|', s, '|'] + [res[0], '|', res[1], '|', res[2]])
...
>>> # ******************************************************* #
>>> TableForm(t, headings=h, alignments="<")
e | s | solve(e, s) | solve(e, s, dict=True) | solve(e, s, set=True)
---------------------------------------------------------------------------------------
x - y | y | [x] | [{y: x}] | ([y], {(x,)})
x - y | [x, y] | [(y, y)] | [{x: y}] | ([x, y], {(y, y)})
x**2 - y | [x, y] | [(x, x**2)] | [{y: x**2}] | ([x, y], {(x, x**2)})
[x - 3, y - 1] | [x, y] | {x: 3, y: 1} | [{x: 3, y: 1}] | ([x, y], {(3, 1)})
* If any equation does not depend on the symbol(s) given, it will be
eliminated from the equation set and an answer may be given
implicitly in terms of variables that were not of interest:
>>> solve([x - y, y - 3], x)
{x: y}
When you pass all but one of the free symbols, an attempt
is made to find a single solution based on the method of
undetermined coefficients. If it succeeds, a dictionary of values
is returned. If you want an algebraic solutions for one
or more of the symbols, pass the expression to be solved in a list:
>>> e = a*x + b - 2*x - 3
>>> solve(e, [a, b])
{a: 2, b: 3}
>>> solve([e], [a, b])
{a: -b/x + (2*x + 3)/x}
When there is no solution for any given symbol which will make all
expressions zero, the empty list is returned (or an empty set in
the tuple when ``set=True``):
>>> from sympy import sqrt
>>> solve(3, x)
[]
>>> solve(x - 3, y)
[]
>>> solve(sqrt(x) + 1, x, set=True)
([x], set())
When an object other than a Symbol is given as a symbol, it is
isolated algebraically and an implicit solution may be obtained.
This is mostly provided as a convenience to save you from replacing
the object with a Symbol and solving for that Symbol. It will only
work if the specified object can be replaced with a Symbol using the
subs method:
>>> from sympy import exp, Function
>>> f = Function('f')
>>> solve(f(x) - x, f(x))
[x]
>>> solve(f(x).diff(x) - f(x) - x, f(x).diff(x))
[x + f(x)]
>>> solve(f(x).diff(x) - f(x) - x, f(x))
[-x + Derivative(f(x), x)]
>>> solve(x + exp(x)**2, exp(x), set=True)
([exp(x)], {(-sqrt(-x),), (sqrt(-x),)})
>>> from sympy import Indexed, IndexedBase, Tuple
>>> A = IndexedBase('A')
>>> eqs = Tuple(A[1] + A[2] - 3, A[1] - A[2] + 1)
>>> solve(eqs, eqs.atoms(Indexed))
{A[1]: 1, A[2]: 2}
* To solve for a function within a derivative, use :func:`~.dsolve`.
To solve for a symbol implicitly, use implicit=True:
>>> solve(x + exp(x), x)
[-LambertW(1)]
>>> solve(x + exp(x), x, implicit=True)
[-exp(x)]
It is possible to solve for anything in an expression that can be
replaced with a symbol using :obj:`~sympy.core.basic.Basic.subs`:
>>> solve(x + 2 + sqrt(3), x + 2)
[-sqrt(3)]
>>> solve((x + 2 + sqrt(3), x + 4 + y), y, x + 2)
{y: -2 + sqrt(3), x + 2: -sqrt(3)}
* Nothing heroic is done in this implicit solving so you may end up
with a symbol still in the solution:
>>> eqs = (x*y + 3*y + sqrt(3), x + 4 + y)
>>> solve(eqs, y, x + 2)
{y: -sqrt(3)/(x + 3), x + 2: -2*x/(x + 3) - 6/(x + 3) + sqrt(3)/(x + 3)}
>>> solve(eqs, y*x, x)
{x: -y - 4, x*y: -3*y - sqrt(3)}
* If you attempt to solve for a number, remember that the number
you have obtained does not necessarily mean that the value is
equivalent to the expression obtained:
>>> solve(sqrt(2) - 1, 1)
[sqrt(2)]
>>> solve(x - y + 1, 1) # /!\ -1 is targeted, too
[x/(y - 1)]
>>> [_.subs(z, -1) for _ in solve((x - y + 1).subs(-1, z), 1)]
[-x + y]
**Additional Examples**
``solve()`` with check=True (default) will run through the symbol tags to
eliminate unwanted solutions. If no assumptions are included, all possible
solutions will be returned:
>>> x = Symbol("x")
>>> solve(x**2 - 1)
[-1, 1]
By setting the ``positive`` flag, only one solution will be returned:
>>> pos = Symbol("pos", positive=True)
>>> solve(pos**2 - 1)
[1]
When the solutions are checked, those that make any denominator zero
are automatically excluded. If you do not want to exclude such solutions,
then use the check=False option:
>>> from sympy import sin, limit
>>> solve(sin(x)/x) # 0 is excluded
[pi]
If ``check=False``, then a solution to the numerator being zero is found
but the value of $x = 0$ is a spurious solution since $\sin(x)/x$ has the well
known limit (without discontinuity) of 1 at $x = 0$:
>>> solve(sin(x)/x, check=False)
[0, pi]
In the following case, however, the limit exists and is equal to the
value of $x = 0$ that is excluded when check=True:
>>> eq = x**2*(1/x - z**2/x)
>>> solve(eq, x)
[]
>>> solve(eq, x, check=False)
[0]
>>> limit(eq, x, 0, '-')
0
>>> limit(eq, x, 0, '+')
0
**Solving Relationships**
When one or more expressions passed to ``solve`` is a relational,
a relational result is returned (and the ``dict`` and ``set`` flags
are ignored):
>>> solve(x < 3)
(-oo < x) & (x < 3)
>>> solve([x < 3, x**2 > 4], x)
((-oo < x) & (x < -2)) | ((2 < x) & (x < 3))
>>> solve([x + y - 3, x > 3], x)
(3 < x) & (x < oo) & Eq(x, 3 - y)
Although checking of assumptions on symbols in relationals
is not done, setting assumptions will affect how certain
relationals might automatically simplify:
>>> solve(x**2 > 4)
((-oo < x) & (x < -2)) | ((2 < x) & (x < oo))
>>> r = Symbol('r', real=True)
>>> solve(r**2 > 4)
(2 < r) | (r < -2)
There is currently no algorithm in SymPy that allows you to use
relationships to resolve more than one variable. So the following
does not determine that ``q < 0`` (and trying to solve for ``r``
and ``q`` will raise an error):
>>> from sympy import symbols
>>> r, q = symbols('r, q', real=True)
>>> solve([r + q - 3, r > 3], r)
(3 < r) & Eq(r, 3 - q)
You can directly call the routine that ``solve`` calls
when it encounters a relational: :func:`~.reduce_inequalities`.
It treats Expr like Equality.
>>> from sympy import reduce_inequalities
>>> reduce_inequalities([x**2 - 4])
Eq(x, -2) | Eq(x, 2)
If each relationship contains only one symbol of interest,
the expressions can be processed for multiple symbols:
>>> reduce_inequalities([0 <= x - 1, y < 3], [x, y])
(-oo < y) & (1 <= x) & (x < oo) & (y < 3)
But an error is raised if any relationship has more than one
symbol of interest:
>>> reduce_inequalities([0 <= x*y - 1, y < 3], [x, y])
Traceback (most recent call last):
...
NotImplementedError:
inequality has more than one symbol of interest.
**Disabling High-Order Explicit Solutions**
When solving polynomial expressions, you might not want explicit solutions
(which can be quite long). If the expression is univariate, ``CRootOf``
instances will be returned instead:
>>> solve(x**3 - x + 1)
[-1/((-1/2 - sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)) -
(-1/2 - sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)/3,
-(-1/2 + sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)/3 -
1/((-1/2 + sqrt(3)*I/2)*(3*sqrt(69)/2 + 27/2)**(1/3)),
-(3*sqrt(69)/2 + 27/2)**(1/3)/3 -
1/(3*sqrt(69)/2 + 27/2)**(1/3)]
>>> solve(x**3 - x + 1, cubics=False)
[CRootOf(x**3 - x + 1, 0),
CRootOf(x**3 - x + 1, 1),
CRootOf(x**3 - x + 1, 2)]
If the expression is multivariate, no solution might be returned:
>>> solve(x**3 - x + a, x, cubics=False)
[]
Sometimes solutions will be obtained even when a flag is False because the
expression could be factored. In the following example, the equation can
be factored as the product of a linear and a quadratic factor so explicit
solutions (which did not require solving a cubic expression) are obtained:
>>> eq = x**3 + 3*x**2 + x - 1
>>> solve(eq, cubics=False)
[-1, -1 + sqrt(2), -sqrt(2) - 1]
**Solving Equations Involving Radicals**
Because of SymPy's use of the principle root, some solutions
to radical equations will be missed unless check=False:
>>> from sympy import root
>>> eq = root(x**3 - 3*x**2, 3) + 1 - x
>>> solve(eq)
[]
>>> solve(eq, check=False)
[1/3]
In the above example, there is only a single solution to the
equation. Other expressions will yield spurious roots which
must be checked manually; roots which give a negative argument
to odd-powered radicals will also need special checking:
>>> from sympy import real_root, S
>>> eq = root(x, 3) - root(x, 5) + S(1)/7
>>> solve(eq) # this gives 2 solutions but misses a 3rd
[CRootOf(7*x**5 - 7*x**3 + 1, 1)**15,
CRootOf(7*x**5 - 7*x**3 + 1, 2)**15]
>>> sol = solve(eq, check=False)
>>> [abs(eq.subs(x,i).n(2)) for i in sol]
[0.48, 0.e-110, 0.e-110, 0.052, 0.052]
The first solution is negative so ``real_root`` must be used to see that it
satisfies the expression:
>>> abs(real_root(eq.subs(x, sol[0])).n(2))
0.e-110
If the roots of the equation are not real then more care will be
necessary to find the roots, especially for higher order equations.
Consider the following expression:
>>> expr = root(x, 3) - root(x, 5)
We will construct a known value for this expression at x = 3 by selecting
the 1-th root for each radical:
>>> expr1 = root(x, 3, 1) - root(x, 5, 1)
>>> v = expr1.subs(x, -3)
The ``solve`` function is unable to find any exact roots to this equation:
>>> eq = Eq(expr, v); eq1 = Eq(expr1, v)
>>> solve(eq, check=False), solve(eq1, check=False)
([], [])
The function ``unrad``, however, can be used to get a form of the equation
for which numerical roots can be found:
>>> from sympy.solvers.solvers import unrad
>>> from sympy import nroots
>>> e, (p, cov) = unrad(eq)
>>> pvals = nroots(e)
>>> inversion = solve(cov, x)[0]
>>> xvals = [inversion.subs(p, i) for i in pvals]
Although ``eq`` or ``eq1`` could have been used to find ``xvals``, the
solution can only be verified with ``expr1``:
>>> z = expr - v
>>> [xi.n(chop=1e-9) for xi in xvals if abs(z.subs(x, xi).n()) < 1e-9]
[]
>>> z1 = expr1 - v
>>> [xi.n(chop=1e-9) for xi in xvals if abs(z1.subs(x, xi).n()) < 1e-9]
[-3.0]
Parameters
==========
f :
- a single Expr or Poly that must be zero
- an Equality
- a Relational expression
- a Boolean
- iterable of one or more of the above
symbols : (object(s) to solve for) specified as
- none given (other non-numeric objects will be used)
- single symbol
- denested list of symbols
(e.g., ``solve(f, x, y)``)
- ordered iterable of symbols
(e.g., ``solve(f, [x, y])``)
flags :
dict=True (default is False)
Return list (perhaps empty) of solution mappings.
set=True (default is False)
Return list of symbols and set of tuple(s) of solution(s).
exclude=[] (default)
Do not try to solve for any of the free symbols in exclude;
if expressions are given, the free symbols in them will
be extracted automatically.
check=True (default)
If False, do not do any testing of solutions. This can be
useful if you want to include solutions that make any
denominator zero.
numerical=True (default)
Do a fast numerical check if *f* has only one symbol.
minimal=True (default is False)
A very fast, minimal testing.
warn=True (default is False)
Show a warning if ``checksol()`` could not conclude.
simplify=True (default)
Simplify all but polynomials of order 3 or greater before
returning them and (if check is not False) use the
general simplify function on the solutions and the
expression obtained when they are substituted into the
function which should be zero.
force=True (default is False)
Make positive all symbols without assumptions regarding sign.
rational=True (default)
Recast Floats as Rational; if this option is not used, the
system containing Floats may fail to solve because of issues
with polys. If rational=None, Floats will be recast as
rationals but the answer will be recast as Floats. If the
flag is False then nothing will be done to the Floats.
manual=True (default is False)
Do not use the polys/matrix method to solve a system of
equations, solve them one at a time as you might "manually."
implicit=True (default is False)
Allows ``solve`` to return a solution for a pattern in terms of
other functions that contain that pattern; this is only
needed if the pattern is inside of some invertible function
like cos, exp, ect.
particular=True (default is False)
Instructs ``solve`` to try to find a particular solution to
a linear system with as many zeros as possible; this is very
expensive.
quick=True (default is False; ``particular`` must be True)
Selects a fast heuristic to find a solution with many zeros
whereas a value of False uses the very slow method guaranteed
to find the largest number of zeros possible.
cubics=True (default)
Return explicit solutions when cubic expressions are encountered.
When False, quartics and quintics are disabled, too.
quartics=True (default)
Return explicit solutions when quartic expressions are encountered.
When False, quintics are disabled, too.
quintics=True (default)
Return explicit solutions (if possible) when quintic expressions
are encountered.
See Also
========
rsolve: For solving recurrence relationships
dsolve: For solving differential equations
"""
from .inequalities import reduce_inequalities
# checking/recording flags
###########################################################################
# set solver types explicitly; as soon as one is False
# all the rest will be False
hints = ('cubics', 'quartics', 'quintics')
default = True
for k in hints:
default = flags.setdefault(k, bool(flags.get(k, default)))
# allow solution to contain symbol if True:
implicit = flags.get('implicit', False)
# record desire to see warnings
warn = flags.get('warn', False)
# this flag will be needed for quick exits below, so record
# now -- but don't record `dict` yet since it might change
as_set = flags.get('set', False)
# keeping track of how f was passed
bare_f = not iterable(f)
# check flag usage for particular/quick which should only be used
# with systems of equations
if flags.get('quick', None) is not None:
if not flags.get('particular', None):
raise ValueError('when using `quick`, `particular` should be True')
if flags.get('particular', False) and bare_f:
raise ValueError(filldedent("""
The 'particular/quick' flag is usually used with systems of
equations. Either pass your equation in a list or
consider using a solver like `diophantine` if you are
looking for a solution in integers."""))
# sympify everything, creating list of expressions and list of symbols
###########################################################################
def _sympified_list(w):
return list(map(sympify, w if iterable(w) else [w]))
f, symbols = (_sympified_list(w) for w in [f, symbols])
# preprocess symbol(s)
###########################################################################
ordered_symbols = None # were the symbols in a well defined order?
if not symbols:
# get symbols from equations
symbols = set().union(*[fi.free_symbols for fi in f])
if len(symbols) < len(f):
for fi in f:
pot = preorder_traversal(fi)
for p in pot:
if isinstance(p, AppliedUndef):
if not as_set:
flags['dict'] = True # better show symbols
symbols.add(p)
pot.skip() # don't go any deeper
ordered_symbols = False
symbols = list(ordered(symbols)) # to make it canonical
else:
if len(symbols) == 1 and iterable(symbols[0]):
symbols = symbols[0]
ordered_symbols = symbols and is_sequence(symbols,
include=GeneratorType)
_symbols = list(uniq(symbols))
if len(_symbols) != len(symbols):
ordered_symbols = False
symbols = list(ordered(symbols))
else:
symbols = _symbols
# check for duplicates
if len(symbols) != len(set(symbols)):
raise ValueError('duplicate symbols given')
# remove those not of interest
exclude = flags.pop('exclude', set())
if exclude:
if isinstance(exclude, Expr):
exclude = [exclude]
exclude = set().union(*[e.free_symbols for e in sympify(exclude)])
symbols = [s for s in symbols if s not in exclude]
# preprocess equation(s)
###########################################################################
# automatically ignore True values
if isinstance(f, list):
f = [s for s in f if s is not S.true]
# handle canonicalization of equation types
for i, fi in enumerate(f):
if isinstance(fi, (Eq, Ne)):
if 'ImmutableDenseMatrix' in [type(a).__name__ for a in fi.args]:
fi = fi.lhs - fi.rhs
else:
L, R = fi.args
if isinstance(R, BooleanAtom):
L, R = R, L
if isinstance(L, BooleanAtom):
if isinstance(fi, Ne):
L = ~L
if R.is_Relational:
fi = ~R if L is S.false else R
elif R.is_Symbol:
return L
elif R.is_Boolean and (~R).is_Symbol:
return ~L
else:
raise NotImplementedError(filldedent('''
Unanticipated argument of Eq when other arg
is True or False.
'''))
else:
fi = fi.rewrite(Add, evaluate=False)
f[i] = fi
# *** dispatch and handle as a system of relationals
# **************************************************
if fi.is_Relational:
if len(symbols) != 1:
raise ValueError("can only solve for one symbol at a time")
if warn and symbols[0].assumptions0:
warnings.warn(filldedent("""
\tWarning: assumptions about variable '%s' are
not handled currently.""" % symbols[0]))
return reduce_inequalities(f, symbols=symbols)
# convert Poly to expression
if isinstance(fi, Poly):
f[i] = fi.as_expr()
# rewrite hyperbolics in terms of exp if they have symbols of
# interest
f[i] = f[i].replace(lambda w: isinstance(w, HyperbolicFunction) and \
w.has_free(*symbols), lambda w: w.rewrite(exp))
# if we have a Matrix, we need to iterate over its elements again
if f[i].is_Matrix:
bare_f = False
f.extend(list(f[i]))
f[i] = S.Zero
# if we can split it into real and imaginary parts then do so
freei = f[i].free_symbols
if freei and all(s.is_extended_real or s.is_imaginary for s in freei):
fr, fi = f[i].as_real_imag()
# accept as long as new re, im, arg or atan2 are not introduced
had = f[i].atoms(re, im, arg, atan2)
if fr and fi and fr != fi and not any(
i.atoms(re, im, arg, atan2) - had for i in (fr, fi)):
if bare_f:
bare_f = False
f[i: i + 1] = [fr, fi]
# real/imag handling -----------------------------
if any(isinstance(fi, (bool, BooleanAtom)) for fi in f):
if as_set:
return [], set()
return []
for i, fi in enumerate(f):
# Abs
while True:
was = fi
fi = fi.replace(Abs, lambda arg:
separatevars(Abs(arg)).rewrite(Piecewise) if arg.has(*symbols)
else Abs(arg))
if was == fi:
break
for e in fi.find(Abs):
if e.has(*symbols):
raise NotImplementedError('solving %s when the argument '
'is not real or imaginary.' % e)
# arg
fi = fi.replace(arg, lambda a: arg(a).rewrite(atan2).rewrite(atan))
# save changes
f[i] = fi
# see if re(s) or im(s) appear
freim = [fi for fi in f if fi.has(re, im)]
if freim:
irf = []
for s in symbols:
if s.is_real or s.is_imaginary:
continue # neither re(x) nor im(x) will appear
# if re(s) or im(s) appear, the auxiliary equation must be present
if any(fi.has(re(s), im(s)) for fi in freim):
irf.append((s, re(s) + S.ImaginaryUnit*im(s)))
if irf:
for s, rhs in irf:
f = [fi.xreplace({s: rhs}) for fi in f] + [s - rhs]
symbols.extend([re(s), im(s)])
if bare_f:
bare_f = False
flags['dict'] = True
# end of real/imag handling -----------------------------
# we can solve for non-symbol entities by replacing them with Dummy symbols
f, symbols, swap_sym = recast_to_symbols(f, symbols)
# this set of symbols (perhaps recast) is needed below
symset = set(symbols)
# get rid of equations that have no symbols of interest; we don't
# try to solve them because the user didn't ask and they might be
# hard to solve; this means that solutions may be given in terms
# of the eliminated equations e.g. solve((x-y, y-3), x) -> {x: y}
newf = []
for fi in f:
# let the solver handle equations that..
# - have no symbols but are expressions
# - have symbols of interest
# - have no symbols of interest but are constant
# but when an expression is not constant and has no symbols of
# interest, it can't change what we obtain for a solution from
# the remaining equations so we don't include it; and if it's
# zero it can be removed and if it's not zero, there is no
# solution for the equation set as a whole
#
# The reason for doing this filtering is to allow an answer
# to be obtained to queries like solve((x - y, y), x); without
# this mod the return value is []
ok = False
if fi.free_symbols & symset:
ok = True
else:
if fi.is_number:
if fi.is_Number:
if fi.is_zero:
continue
return []
ok = True
else:
if fi.is_constant():
ok = True
if ok:
newf.append(fi)
if not newf:
if as_set:
return symbols, set()
return []
f = newf
del newf
# mask off any Object that we aren't going to invert: Derivative,
# Integral, etc... so that solving for anything that they contain will
# give an implicit solution
seen = set()
non_inverts = set()
for fi in f:
pot = preorder_traversal(fi)
for p in pot:
if not isinstance(p, Expr) or isinstance(p, Piecewise):
pass
elif (isinstance(p, bool) or
not p.args or
p in symset or
p.is_Add or p.is_Mul or
p.is_Pow and not implicit or
p.is_Function and not implicit) and p.func not in (re, im):
continue
elif p not in seen:
seen.add(p)
if p.free_symbols & symset:
non_inverts.add(p)
else:
continue
pot.skip()
del seen
non_inverts = dict(list(zip(non_inverts, [Dummy() for _ in non_inverts])))
f = [fi.subs(non_inverts) for fi in f]
# Both xreplace and subs are needed below: xreplace to force substitution
# inside Derivative, subs to handle non-straightforward substitutions
non_inverts = [(v, k.xreplace(swap_sym).subs(swap_sym)) for k, v in non_inverts.items()]
# rationalize Floats
floats = False
if flags.get('rational', True) is not False:
for i, fi in enumerate(f):
if fi.has(Float):
floats = True
f[i] = nsimplify(fi, rational=True)
# capture any denominators before rewriting since
# they may disappear after the rewrite, e.g. issue 14779
flags['_denominators'] = _simple_dens(f[0], symbols)
# Any embedded piecewise functions need to be brought out to the
# top level so that the appropriate strategy gets selected.
# However, this is necessary only if one of the piecewise
# functions depends on one of the symbols we are solving for.
def _has_piecewise(e):
if e.is_Piecewise:
return e.has(*symbols)
return any(_has_piecewise(a) for a in e.args)
for i, fi in enumerate(f):
if _has_piecewise(fi):
f[i] = piecewise_fold(fi)
#
# try to get a solution
###########################################################################
if bare_f:
solution = None
if len(symbols) != 1:
solution = _solve_undetermined(f[0], symbols, flags)
if not solution:
solution = _solve(f[0], *symbols, **flags)
else:
linear, solution = _solve_system(f, symbols, **flags)
assert type(solution) is list
assert not solution or type(solution[0]) is dict, solution
#
# postprocessing
###########################################################################
# capture as_dict flag now (as_set already captured)
as_dict = flags.get('dict', False)
# define how solution will get unpacked
tuple_format = lambda s: [tuple([i.get(x, x) for x in symbols]) for i in s]
if as_dict or as_set:
unpack = None
elif bare_f:
if len(symbols) == 1:
unpack = lambda s: [i[symbols[0]] for i in s]
elif len(solution) == 1 and len(solution[0]) == len(symbols):
# undetermined linear coeffs solution
unpack = lambda s: s[0]
elif ordered_symbols:
unpack = tuple_format
else:
unpack = lambda s: s
else:
if solution:
if linear and len(solution) == 1:
# if you want the tuple solution for the linear
# case, use `set=True`
unpack = lambda s: s[0]
elif ordered_symbols:
unpack = tuple_format
else:
unpack = lambda s: s
else:
unpack = None
# Restore masked-off objects
if non_inverts and type(solution) is list:
solution = [{k: v.subs(non_inverts) for k, v in s.items()}
for s in solution]
# Restore original "symbols" if a dictionary is returned.
# This is not necessary for
# - the single univariate equation case
# since the symbol will have been removed from the solution;
# - the nonlinear poly_system since that only supports zero-dimensional
# systems and those results come back as a list
#
# ** unless there were Derivatives with the symbols, but those were handled
# above.
if swap_sym:
symbols = [swap_sym.get(k, k) for k in symbols]
for i, sol in enumerate(solution):
solution[i] = {swap_sym.get(k, k): v.subs(swap_sym)
for k, v in sol.items()}
# Get assumptions about symbols, to filter solutions.
# Note that if assumptions about a solution can't be verified, it is still
# returned.
check = flags.get('check', True)
# restore floats
if floats and solution and flags.get('rational', None) is None:
solution = nfloat(solution, exponent=False)
if check and solution: # assumption checking
warn = flags.get('warn', False)
got_None = [] # solutions for which one or more symbols gave None
no_False = [] # solutions for which no symbols gave False
for sol in solution:
a_None = False
for symb, val in sol.items():
test = check_assumptions(val, **symb.assumptions0)
if test:
continue
if test is False:
break
a_None = True
else:
no_False.append(sol)
if a_None:
got_None.append(sol)
solution = no_False
if warn and got_None:
warnings.warn(filldedent("""
\tWarning: assumptions concerning following solution(s)
cannot be checked:""" + '\n\t' +
', '.join(str(s) for s in got_None)))
#
# done
###########################################################################
if not solution:
if as_set:
return symbols, set()
return []
# make orderings canonical for list of dictionaries
if not as_set: # for set, no point in ordering
solution = [{k: s[k] for k in ordered(s)} for s in solution]
solution.sort(key=default_sort_key)
if not (as_set or as_dict):
return unpack(solution)
if as_dict:
return solution
# set output: (symbols, {t1, t2, ...}) from list of dictionaries;
# include all symbols for those that like a verbose solution
# and to resolve any differences in dictionary keys.
#
# The set results can easily be used to make a verbose dict as
# k, v = solve(eqs, syms, set=True)
# sol = [dict(zip(k,i)) for i in v]
#
if ordered_symbols:
k = symbols # keep preferred order
else:
# just unify the symbols for which solutions were found
k = list(ordered(set(flatten(tuple(i.keys()) for i in solution))))
return k, {tuple([s.get(ki, ki) for ki in k]) for s in solution}
def _solve_undetermined(g, symbols, flags):
# helper to find independent variable for undetermined coefficients
# call and to return a list with a single dictionary else None
# a*x + 2*x + b - c
symset = set(symbols) # {a, b}
free = g.free_symbols # {a, b, c, x}
ex = free - symset # {c, x}
if len(ex) != 1:
ind, dep = g.as_independent(*symbols) # (2*x - c, a*x + b)
ex = ind.free_symbols & dep.free_symbols # {x, c} & {a, x, b} -> {x}
if len(ex) != 1: # e.g. (a + b)*x + b - c -> {c}, {a, b, x}
ex = dep.free_symbols - symset # {x} = {a, b, x}-{a, b}
if len(ex) == 1:
ex = ex.pop()
sol = solve_undetermined_coeffs(g, symbols, ex, **flags)
if type(sol) is dict:
return [sol]
def _solve(f, *symbols, **flags):
"""Return a checked solution for *f* in terms of one or more of the
symbols in the form of a list of dictionaries.
If no method is implemented to solve the equation, a NotImplementedError
will be raised. In the case that conversion of an expression to a Poly
gives None a ValueError will be raised.
"""
not_impl_msg = "No algorithms are implemented to solve equation %s"
if len(symbols) != 1:
# look for solutions for desired symbols that are independent
# of symbols already solved for, e.g. if we solve for x = y
# then no symbol having x in its solution will be returned.
# First solve for linear symbols (since that is easier and limits
# solution size) and then proceed with symbols appearing
# in a non-linear fashion. Ideally, if one is solving a single
# expression for several symbols, they would have to be
# appear in factors of an expression, but we do not here
# attempt factorization. XXX perhaps handling a Mul
# should come first in this routine whether there is
# one or several symbols.
nonlin_s = []
got_s = set()
rhs_s = set()
result = []
for s in symbols:
xi, v = solve_linear(f, symbols=[s])
if xi == s:
# no need to check but we should simplify if desired
if flags.get('simplify', True):
v = simplify(v)
vfree = v.free_symbols
if vfree & got_s:
# was linear, but has redundant relationship
# e.g. x - y = 0 has y == x is redundant for x == y
# so ignore
continue
rhs_s |= vfree
got_s.add(xi)
result.append({xi: v})
elif xi: # there might be a non-linear solution if xi is not 0
nonlin_s.append(s)
if not nonlin_s:
return result
for s in nonlin_s:
try:
soln = _solve(f, s, **flags)
for sol in soln:
if sol[s].free_symbols & got_s:
# depends on previously solved symbols: ignore
continue
got_s.add(s)
result.append(sol)
except NotImplementedError:
continue
if got_s:
return result
else:
raise NotImplementedError(not_impl_msg % f)
# solve f for a single variable
symbol = symbols[0]
# expand binomials only if it has the unknown symbol
f = f.replace(lambda e: isinstance(e, binomial) and e.has(symbol),
lambda e: expand_func(e))
# checking will be done unless it is turned off before making a
# recursive call; the variables `checkdens` and `check` are
# captured here (for reference below) in case flag value changes
flags['check'] = checkdens = check = flags.pop('check', True)
# build up solutions if f is a Mul
if f.is_Mul:
result = set()
for m in f.args:
if m in {S.NegativeInfinity, S.ComplexInfinity, S.Infinity}:
result = set()
break
soln = _vsolve(m, symbol, **flags)
result.update(set(soln))
result = [{symbol: v} for v in result]
if check:
# all solutions have been checked but now we must
# check that the solutions do not set denominators
# in any factor to zero
dens = flags.get('_denominators', _simple_dens(f, symbols))
result = [s for s in result if
not any(checksol(den, s, **flags) for den in
dens)]
# set flags for quick exit at end; solutions for each
# factor were already checked and simplified
check = False
flags['simplify'] = False
elif f.is_Piecewise:
result = set()
for i, (expr, cond) in enumerate(f.args):
if expr.is_zero:
raise NotImplementedError(
'solve cannot represent interval solutions')
candidates = _vsolve(expr, symbol, **flags)
# the explicit condition for this expr is the current cond
# and none of the previous conditions
args = [~c for _, c in f.args[:i]] + [cond]
cond = And(*args)
for candidate in candidates:
if candidate in result:
# an unconditional value was already there
continue
try:
v = cond.subs(symbol, candidate)
_eval_simplify = getattr(v, '_eval_simplify', None)
if _eval_simplify is not None:
# unconditionally take the simpification of v
v = _eval_simplify(ratio=2, measure=lambda x: 1)
except TypeError:
# incompatible type with condition(s)
continue
if v == False:
continue
if v == True:
result.add(candidate)
else:
result.add(Piecewise(
(candidate, v),
(S.NaN, True)))
# solutions already checked and simplified
# ****************************************
return [{symbol: r} for r in result]
else:
# first see if it really depends on symbol and whether there
# is only a linear solution
f_num, sol = solve_linear(f, symbols=symbols)
if f_num.is_zero or sol is S.NaN:
return []
elif f_num.is_Symbol:
# no need to check but simplify if desired
if flags.get('simplify', True):
sol = simplify(sol)
return [{f_num: sol}]
poly = None
# check for a single Add generator
if not f_num.is_Add:
add_args = [i for i in f_num.atoms(Add)
if symbol in i.free_symbols]
if len(add_args) == 1:
gen = add_args[0]
spart = gen.as_independent(symbol)[1].as_base_exp()[0]
if spart == symbol:
try:
poly = Poly(f_num, spart)
except PolynomialError:
pass
result = False # no solution was obtained
msg = '' # there is no failure message
# Poly is generally robust enough to convert anything to
# a polynomial and tell us the different generators that it
# contains, so we will inspect the generators identified by
# polys to figure out what to do.
# try to identify a single generator that will allow us to solve this
# as a polynomial, followed (perhaps) by a change of variables if the
# generator is not a symbol
try:
if poly is None:
poly = Poly(f_num)
if poly is None:
raise ValueError('could not convert %s to Poly' % f_num)
except GeneratorsNeeded:
simplified_f = simplify(f_num)
if simplified_f != f_num:
return _solve(simplified_f, symbol, **flags)
raise ValueError('expression appears to be a constant')
gens = [g for g in poly.gens if g.has(symbol)]
def _as_base_q(x):
"""Return (b**e, q) for x = b**(p*e/q) where p/q is the leading
Rational of the exponent of x, e.g. exp(-2*x/3) -> (exp(x), 3)
"""
b, e = x.as_base_exp()
if e.is_Rational:
return b, e.q
if not e.is_Mul:
return x, 1
c, ee = e.as_coeff_Mul()
if c.is_Rational and c is not S.One: # c could be a Float
return b**ee, c.q
return x, 1
if len(gens) > 1:
# If there is more than one generator, it could be that the
# generators have the same base but different powers, e.g.
# >>> Poly(exp(x) + 1/exp(x))
# Poly(exp(-x) + exp(x), exp(-x), exp(x), domain='ZZ')
#
# If unrad was not disabled then there should be no rational
# exponents appearing as in
# >>> Poly(sqrt(x) + sqrt(sqrt(x)))
# Poly(sqrt(x) + x**(1/4), sqrt(x), x**(1/4), domain='ZZ')
bases, qs = list(zip(*[_as_base_q(g) for g in gens]))
bases = set(bases)
if len(bases) > 1 or not all(q == 1 for q in qs):
funcs = {b for b in bases if b.is_Function}
trig = {_ for _ in funcs if
isinstance(_, TrigonometricFunction)}
other = funcs - trig
if not other and len(funcs.intersection(trig)) > 1:
newf = None
if f_num.is_Add and len(f_num.args) == 2:
# check for sin(x)**p = cos(x)**p
_args = f_num.args
t = a, b = [i.atoms(Function).intersection(
trig) for i in _args]
if all(len(i) == 1 for i in t):
a, b = [i.pop() for i in t]
if isinstance(a, cos):
a, b = b, a
_args = _args[::-1]
if isinstance(a, sin) and isinstance(b, cos
) and a.args[0] == b.args[0]:
# sin(x) + cos(x) = 0 -> tan(x) + 1 = 0
newf, _d = (TR2i(_args[0]/_args[1]) + 1
).as_numer_denom()
if not _d.is_Number:
newf = None
if newf is None:
newf = TR1(f_num).rewrite(tan)
if newf != f_num:
# don't check the rewritten form --check
# solutions in the un-rewritten form below
flags['check'] = False
result = _solve(newf, symbol, **flags)
flags['check'] = check
# just a simple case - see if replacement of single function
# clears all symbol-dependent functions, e.g.
# log(x) - log(log(x) - 1) - 3 can be solved even though it has
# two generators.
if result is False and funcs:
funcs = list(ordered(funcs)) # put shallowest function first
f1 = funcs[0]
t = Dummy('t')
# perform the substitution
ftry = f_num.subs(f1, t)
# if no Functions left, we can proceed with usual solve
if not ftry.has(symbol):
cv_sols = _solve(ftry, t, **flags)
cv_inv = list(ordered(_vsolve(t - f1, symbol, **flags)))[0]
result = [{symbol: cv_inv.subs(sol)} for sol in cv_sols]
if result is False:
msg = 'multiple generators %s' % gens
else:
# e.g. case where gens are exp(x), exp(-x)
u = bases.pop()
t = Dummy('t')
inv = _vsolve(u - t, symbol, **flags)
if isinstance(u, (Pow, exp)):
# this will be resolved by factor in _tsolve but we might
# as well try a simple expansion here to get things in
# order so something like the following will work now without
# having to factor:
#
# >>> eq = (exp(I*(-x-2))+exp(I*(x+2)))
# >>> eq.subs(exp(x),y) # fails
# exp(I*(-x - 2)) + exp(I*(x + 2))
# >>> eq.expand().subs(exp(x),y) # works
# y**I*exp(2*I) + y**(-I)*exp(-2*I)
def _expand(p):
b, e = p.as_base_exp()
e = expand_mul(e)
return expand_power_exp(b**e)
ftry = f_num.replace(
lambda w: w.is_Pow or isinstance(w, exp),
_expand).subs(u, t)
if not ftry.has(symbol):
soln = _solve(ftry, t, **flags)
result = [{symbol: i.subs(s)} for i in inv for s in soln]
elif len(gens) == 1:
# There is only one generator that we are interested in, but
# there may have been more than one generator identified by
# polys (e.g. for symbols other than the one we are interested
# in) so recast the poly in terms of our generator of interest.
# Also use composite=True with f_num since Poly won't update
# poly as documented in issue 8810.
poly = Poly(f_num, gens[0], composite=True)
# if we aren't on the tsolve-pass, use roots
if not flags.pop('tsolve', False):
soln = None
deg = poly.degree()
flags['tsolve'] = True
hints = ('cubics', 'quartics', 'quintics')
solvers = {h: flags.get(h) for h in hints}
soln = roots(poly, **solvers)
if sum(soln.values()) < deg:
# e.g. roots(32*x**5 + 400*x**4 + 2032*x**3 +
# 5000*x**2 + 6250*x + 3189) -> {}
# so all_roots is used and RootOf instances are
# returned *unless* the system is multivariate
# or high-order EX domain.
try:
soln = poly.all_roots()
except NotImplementedError:
if not flags.get('incomplete', True):
raise NotImplementedError(
filldedent('''
Neither high-order multivariate polynomials
nor sorting of EX-domain polynomials is supported.
If you want to see any results, pass keyword incomplete=True to
solve; to see numerical values of roots
for univariate expressions, use nroots.
'''))
else:
pass
else:
soln = list(soln.keys())
if soln is not None:
u = poly.gen
if u != symbol:
try:
t = Dummy('t')
inv = _vsolve(u - t, symbol, **flags)
soln = {i.subs(t, s) for i in inv for s in soln}
except NotImplementedError:
# perhaps _tsolve can handle f_num
soln = None
else:
check = False # only dens need to be checked
if soln is not None:
if len(soln) > 2:
# if the flag wasn't set then unset it since high-order
# results are quite long. Perhaps one could base this
# decision on a certain critical length of the
# roots. In addition, wester test M2 has an expression
# whose roots can be shown to be real with the
# unsimplified form of the solution whereas only one of
# the simplified forms appears to be real.
flags['simplify'] = flags.get('simplify', False)
if soln is not None:
result = [{symbol: v} for v in soln]
# fallback if above fails
# -----------------------
if result is False:
# try unrad
if flags.pop('_unrad', True):
try:
u = unrad(f_num, symbol)
except (ValueError, NotImplementedError):
u = False
if u:
eq, cov = u
if cov:
isym, ieq = cov
inv = _vsolve(ieq, symbol, **flags)[0]
rv = {inv.subs(xi) for xi in _solve(eq, isym, **flags)}
else:
try:
rv = set(_vsolve(eq, symbol, **flags))
except NotImplementedError:
rv = None
if rv is not None:
result = [{symbol: v} for v in rv]
# if the flag wasn't set then unset it since unrad results
# can be quite long or of very high order
flags['simplify'] = flags.get('simplify', False)
else:
pass # for coverage
# try _tsolve
if result is False:
flags.pop('tsolve', None) # allow tsolve to be used on next pass
try:
soln = _tsolve(f_num, symbol, **flags)
if soln is not None:
result = [{symbol: v} for v in soln]
except PolynomialError:
pass
# ----------- end of fallback ----------------------------
if result is False:
raise NotImplementedError('\n'.join([msg, not_impl_msg % f]))
if flags.get('simplify', True):
result = [{k: d[k].simplify() for k in d} for d in result]
# we just simplified the solution so we now set the flag to
# False so the simplification doesn't happen again in checksol()
flags['simplify'] = False
if checkdens:
# reject any result that makes any denom. affirmatively 0;
# if in doubt, keep it
dens = _simple_dens(f, symbols)
result = [r for r in result if
not any(checksol(d, r, **flags)
for d in dens)]
if check:
# keep only results if the check is not False
result = [r for r in result if
checksol(f_num, r, **flags) is not False]
return result
def _solve_system(exprs, symbols, **flags):
"""return ``(linear, solution)`` where ``linear`` is True
if the system was linear, else False; ``solution``
is a list of dictionaries giving solutions for the symbols
"""
if not exprs:
return False, []
if flags.pop('_split', True):
# Split the system into connected components
V = exprs
symsset = set(symbols)
exprsyms = {e: e.free_symbols & symsset for e in exprs}
E = []
sym_indices = {sym: i for i, sym in enumerate(symbols)}
for n, e1 in enumerate(exprs):
for e2 in exprs[:n]:
# Equations are connected if they share a symbol
if exprsyms[e1] & exprsyms[e2]:
E.append((e1, e2))
G = V, E
subexprs = connected_components(G)
if len(subexprs) > 1:
subsols = []
linear = True
for subexpr in subexprs:
subsyms = set()
for e in subexpr:
subsyms |= exprsyms[e]
subsyms = list(sorted(subsyms, key = lambda x: sym_indices[x]))
flags['_split'] = False # skip split step
_linear, subsol = _solve_system(subexpr, subsyms, **flags)
if linear:
linear = linear and _linear
if not isinstance(subsol, list):
subsol = [subsol]
subsols.append(subsol)
# Full solution is cartesion product of subsystems
sols = []
for soldicts in product(*subsols):
sols.append(dict(item for sd in soldicts
for item in sd.items()))
return linear, sols
polys = []
dens = set()
failed = []
result = []
solved_syms = []
linear = True
manual = flags.get('manual', False)
checkdens = check = flags.get('check', True)
for j, g in enumerate(exprs):
dens.update(_simple_dens(g, symbols))
i, d = _invert(g, *symbols)
if d in symbols:
if linear:
linear = solve_linear(g, 0, [d])[0] == d
g = d - i
g = g.as_numer_denom()[0]
if manual:
failed.append(g)
continue
poly = g.as_poly(*symbols, extension=True)
if poly is not None:
polys.append(poly)
else:
failed.append(g)
if polys:
if all(p.is_linear for p in polys):
n, m = len(polys), len(symbols)
matrix = zeros(n, m + 1)
for i, poly in enumerate(polys):
for monom, coeff in poly.terms():
try:
j = monom.index(1)
matrix[i, j] = coeff
except ValueError:
matrix[i, m] = -coeff
# returns a dictionary ({symbols: values}) or None
if flags.pop('particular', False):
result = minsolve_linear_system(matrix, *symbols, **flags)
else:
result = solve_linear_system(matrix, *symbols, **flags)
result = [result] if result else []
if failed:
if result:
solved_syms = list(result[0].keys()) # there is only one result dict
else:
solved_syms = []
# linear doesn't change
else:
linear = False
if len(symbols) > len(polys):
free = set().union(*[p.free_symbols for p in polys])
free = list(ordered(free.intersection(symbols)))
got_s = set()
result = []
for syms in subsets(free, len(polys)):
try:
# returns [], None or list of tuples
res = solve_poly_system(polys, *syms)
if res:
for r in set(res):
skip = False
for r1 in r:
if got_s and any(ss in r1.free_symbols
for ss in got_s):
# sol depends on previously
# solved symbols: discard it
skip = True
if not skip:
got_s.update(syms)
result.append(dict(list(zip(syms, r))))
except NotImplementedError:
pass
if got_s:
solved_syms = list(got_s)
else:
raise NotImplementedError('no valid subset found')
else:
try:
result = solve_poly_system(polys, *symbols)
if result:
solved_syms = symbols
result = [dict(list(zip(solved_syms, r))) for r in set(result)]
except NotImplementedError:
failed.extend([g.as_expr() for g in polys])
solved_syms = []
# convert None or [] to [{}]
result = result or [{}]
if failed:
linear = False
# For each failed equation, see if we can solve for one of the
# remaining symbols from that equation. If so, we update the
# solution set and continue with the next failed equation,
# repeating until we are done or we get an equation that can't
# be solved.
def _ok_syms(e, sort=False):
rv = e.free_symbols & legal
# Solve first for symbols that have lower degree in the equation.
# Ideally we want to solve firstly for symbols that appear linearly
# with rational coefficients e.g. if e = x*y + z then we should
# solve for z first.
def key(sym):
ep = e.as_poly(sym)
if ep is None:
complexity = (S.Infinity, S.Infinity, S.Infinity)
else:
coeff_syms = ep.LC().free_symbols
complexity = (ep.degree(), len(coeff_syms & rv), len(coeff_syms))
return complexity + (default_sort_key(sym),)
if sort:
rv = sorted(rv, key=key)
return rv
legal = set(symbols) # what we are interested in
# sort so equation with the fewest potential symbols is first
u = Dummy() # used in solution checking
for eq in ordered(failed, lambda _: len(_ok_syms(_))):
newresult = []
bad_results = []
hit = False
for r in result:
got_s = set()
# update eq with everything that is known so far
eq2 = eq.subs(r)
# if check is True then we see if it satisfies this
# equation, otherwise we just accept it
if check and r:
b = checksol(u, u, eq2, minimal=True)
if b is not None:
# this solution is sufficient to know whether
# it is valid or not so we either accept or
# reject it, then continue
if b:
newresult.append(r)
else:
bad_results.append(r)
continue
# search for a symbol amongst those available that
# can be solved for
ok_syms = _ok_syms(eq2, sort=True)
if not ok_syms:
if r:
newresult.append(r)
break # skip as it's independent of desired symbols
for s in ok_syms:
try:
soln = _vsolve(eq2, s, **flags)
except NotImplementedError:
continue
# put each solution in r and append the now-expanded
# result in the new result list; use copy since the
# solution for s is being added in-place
for sol in soln:
if got_s and any(ss in sol.free_symbols for ss in got_s):
# sol depends on previously solved symbols: discard it
continue
rnew = r.copy()
for k, v in r.items():
rnew[k] = v.subs(s, sol)
# and add this new solution
rnew[s] = sol
# check that it is independent of previous solutions
iset = set(rnew.items())
for i in newresult:
if len(i) < len(iset) and not set(i.items()) - iset:
# this is a superset of a known solution that
# is smaller
break
else:
# keep it
newresult.append(rnew)
hit = True
got_s.add(s)
if not hit:
raise NotImplementedError('could not solve %s' % eq2)
else:
result = newresult
for b in bad_results:
if b in result:
result.remove(b)
if not result:
return False, []
# rely on linear/polynomial system solvers to simplify
# XXX the following tests show that the expressions
# returned are not the same as they would be if simplify
# were applied to this:
# sympy/solvers/ode/tests/test_systems/test__classify_linear_system
# sympy/solvers/tests/test_solvers/test_issue_4886
# so the docs should be updated to reflect that or else
# the following should be `bool(failed) or not linear`
default_simplify = bool(failed)
if flags.get('simplify', default_simplify):
for r in result:
for k in r:
r[k] = simplify(r[k])
flags['simplify'] = False # don't need to do so in checksol now
if checkdens:
result = [r for r in result
if not any(checksol(d, r, **flags) for d in dens)]
if check and not linear:
result = [r for r in result
if not any(checksol(e, r, **flags) is False for e in exprs)]
result = [r for r in result if r]
return linear, result
def solve_linear(lhs, rhs=0, symbols=[], exclude=[]):
r"""
Return a tuple derived from ``f = lhs - rhs`` that is one of
the following: ``(0, 1)``, ``(0, 0)``, ``(symbol, solution)``, ``(n, d)``.
Explanation
===========
``(0, 1)`` meaning that ``f`` is independent of the symbols in *symbols*
that are not in *exclude*.
``(0, 0)`` meaning that there is no solution to the equation amongst the
symbols given. If the first element of the tuple is not zero, then the
function is guaranteed to be dependent on a symbol in *symbols*.
``(symbol, solution)`` where symbol appears linearly in the numerator of
``f``, is in *symbols* (if given), and is not in *exclude* (if given). No
simplification is done to ``f`` other than a ``mul=True`` expansion, so the
solution will correspond strictly to a unique solution.
``(n, d)`` where ``n`` and ``d`` are the numerator and denominator of ``f``
when the numerator was not linear in any symbol of interest; ``n`` will
never be a symbol unless a solution for that symbol was found (in which case
the second element is the solution, not the denominator).
Examples
========
>>> from sympy import cancel, Pow
``f`` is independent of the symbols in *symbols* that are not in
*exclude*:
>>> from sympy import cos, sin, solve_linear
>>> from sympy.abc import x, y, z
>>> eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0
>>> solve_linear(eq)
(0, 1)
>>> eq = cos(x)**2 + sin(x)**2 # = 1
>>> solve_linear(eq)
(0, 1)
>>> solve_linear(x, exclude=[x])
(0, 1)
The variable ``x`` appears as a linear variable in each of the
following:
>>> solve_linear(x + y**2)
(x, -y**2)
>>> solve_linear(1/x - y**2)
(x, y**(-2))
When not linear in ``x`` or ``y`` then the numerator and denominator are
returned:
>>> solve_linear(x**2/y**2 - 3)
(x**2 - 3*y**2, y**2)
If the numerator of the expression is a symbol, then ``(0, 0)`` is
returned if the solution for that symbol would have set any
denominator to 0:
>>> eq = 1/(1/x - 2)
>>> eq.as_numer_denom()
(x, 1 - 2*x)
>>> solve_linear(eq)
(0, 0)
But automatic rewriting may cause a symbol in the denominator to
appear in the numerator so a solution will be returned:
>>> (1/x)**-1
x
>>> solve_linear((1/x)**-1)
(x, 0)
Use an unevaluated expression to avoid this:
>>> solve_linear(Pow(1/x, -1, evaluate=False))
(0, 0)
If ``x`` is allowed to cancel in the following expression, then it
appears to be linear in ``x``, but this sort of cancellation is not
done by ``solve_linear`` so the solution will always satisfy the
original expression without causing a division by zero error.
>>> eq = x**2*(1/x - z**2/x)
>>> solve_linear(cancel(eq))
(x, 0)
>>> solve_linear(eq)
(x**2*(1 - z**2), x)
A list of symbols for which a solution is desired may be given:
>>> solve_linear(x + y + z, symbols=[y])
(y, -x - z)
A list of symbols to ignore may also be given:
>>> solve_linear(x + y + z, exclude=[x])
(y, -x - z)
(A solution for ``y`` is obtained because it is the first variable
from the canonically sorted list of symbols that had a linear
solution.)
"""
if isinstance(lhs, Eq):
if rhs:
raise ValueError(filldedent('''
If lhs is an Equality, rhs must be 0 but was %s''' % rhs))
rhs = lhs.rhs
lhs = lhs.lhs
dens = None
eq = lhs - rhs
n, d = eq.as_numer_denom()
if not n:
return S.Zero, S.One
free = n.free_symbols
if not symbols:
symbols = free
else:
bad = [s for s in symbols if not s.is_Symbol]
if bad:
if len(bad) == 1:
bad = bad[0]
if len(symbols) == 1:
eg = 'solve(%s, %s)' % (eq, symbols[0])
else:
eg = 'solve(%s, *%s)' % (eq, list(symbols))
raise ValueError(filldedent('''
solve_linear only handles symbols, not %s. To isolate
non-symbols use solve, e.g. >>> %s <<<.
''' % (bad, eg)))
symbols = free.intersection(symbols)
symbols = symbols.difference(exclude)
if not symbols:
return S.Zero, S.One
# derivatives are easy to do but tricky to analyze to see if they
# are going to disallow a linear solution, so for simplicity we
# just evaluate the ones that have the symbols of interest
derivs = defaultdict(list)
for der in n.atoms(Derivative):
csym = der.free_symbols & symbols
for c in csym:
derivs[c].append(der)
all_zero = True
for xi in sorted(symbols, key=default_sort_key): # canonical order
# if there are derivatives in this var, calculate them now
if isinstance(derivs[xi], list):
derivs[xi] = {der: der.doit() for der in derivs[xi]}
newn = n.subs(derivs[xi])
dnewn_dxi = newn.diff(xi)
# dnewn_dxi can be nonzero if it survives differentation by any
# of its free symbols
free = dnewn_dxi.free_symbols
if dnewn_dxi and (not free or any(dnewn_dxi.diff(s) for s in free) or free == symbols):
all_zero = False
if dnewn_dxi is S.NaN:
break
if xi not in dnewn_dxi.free_symbols:
vi = -1/dnewn_dxi*(newn.subs(xi, 0))
if dens is None:
dens = _simple_dens(eq, symbols)
if not any(checksol(di, {xi: vi}, minimal=True) is True
for di in dens):
# simplify any trivial integral
irep = [(i, i.doit()) for i in vi.atoms(Integral) if
i.function.is_number]
# do a slight bit of simplification
vi = expand_mul(vi.subs(irep))
return xi, vi
if all_zero:
return S.Zero, S.One
if n.is_Symbol: # no solution for this symbol was found
return S.Zero, S.Zero
return n, d
def minsolve_linear_system(system, *symbols, **flags):
r"""
Find a particular solution to a linear system.
Explanation
===========
In particular, try to find a solution with the minimal possible number
of non-zero variables using a naive algorithm with exponential complexity.
If ``quick=True``, a heuristic is used.
"""
quick = flags.get('quick', False)
# Check if there are any non-zero solutions at all
s0 = solve_linear_system(system, *symbols, **flags)
if not s0 or all(v == 0 for v in s0.values()):
return s0
if quick:
# We just solve the system and try to heuristically find a nice
# solution.
s = solve_linear_system(system, *symbols)
def update(determined, solution):
delete = []
for k, v in solution.items():
solution[k] = v.subs(determined)
if not solution[k].free_symbols:
delete.append(k)
determined[k] = solution[k]
for k in delete:
del solution[k]
determined = {}
update(determined, s)
while s:
# NOTE sort by default_sort_key to get deterministic result
k = max((k for k in s.values()),
key=lambda x: (len(x.free_symbols), default_sort_key(x)))
kfree = k.free_symbols
x = next(reversed(list(ordered(kfree))))
if len(kfree) != 1:
determined[x] = S.Zero
else:
val = _vsolve(k, x, check=False)[0]
if not val and not any(v.subs(x, val) for v in s.values()):
determined[x] = S.One
else:
determined[x] = val
update(determined, s)
return determined
else:
# We try to select n variables which we want to be non-zero.
# All others will be assumed zero. We try to solve the modified system.
# If there is a non-trivial solution, just set the free variables to
# one. If we do this for increasing n, trying all combinations of
# variables, we will find an optimal solution.
# We speed up slightly by starting at one less than the number of
# variables the quick method manages.
N = len(symbols)
bestsol = minsolve_linear_system(system, *symbols, quick=True)
n0 = len([x for x in bestsol.values() if x != 0])
for n in range(n0 - 1, 1, -1):
debug('minsolve: %s' % n)
thissol = None
for nonzeros in combinations(list(range(N)), n):
subm = Matrix([system.col(i).T for i in nonzeros] + [system.col(-1).T]).T
s = solve_linear_system(subm, *[symbols[i] for i in nonzeros])
if s and not all(v == 0 for v in s.values()):
subs = [(symbols[v], S.One) for v in nonzeros]
for k, v in s.items():
s[k] = v.subs(subs)
for sym in symbols:
if sym not in s:
if symbols.index(sym) in nonzeros:
s[sym] = S.One
else:
s[sym] = S.Zero
thissol = s
break
if thissol is None:
break
bestsol = thissol
return bestsol
def solve_linear_system(system, *symbols, **flags):
r"""
Solve system of $N$ linear equations with $M$ variables, which means
both under- and overdetermined systems are supported.
Explanation
===========
The possible number of solutions is zero, one, or infinite. Respectively,
this procedure will return None or a dictionary with solutions. In the
case of underdetermined systems, all arbitrary parameters are skipped.
This may cause a situation in which an empty dictionary is returned.
In that case, all symbols can be assigned arbitrary values.
Input to this function is a $N\times M + 1$ matrix, which means it has
to be in augmented form. If you prefer to enter $N$ equations and $M$
unknowns then use ``solve(Neqs, *Msymbols)`` instead. Note: a local
copy of the matrix is made by this routine so the matrix that is
passed will not be modified.
The algorithm used here is fraction-free Gaussian elimination,
which results, after elimination, in an upper-triangular matrix.
Then solutions are found using back-substitution. This approach
is more efficient and compact than the Gauss-Jordan method.
Examples
========
>>> from sympy import Matrix, solve_linear_system
>>> from sympy.abc import x, y
Solve the following system::
x + 4 y == 2
-2 x + y == 14
>>> system = Matrix(( (1, 4, 2), (-2, 1, 14)))
>>> solve_linear_system(system, x, y)
{x: -6, y: 2}
A degenerate system returns an empty dictionary:
>>> system = Matrix(( (0,0,0), (0,0,0) ))
>>> solve_linear_system(system, x, y)
{}
"""
assert system.shape[1] == len(symbols) + 1
# This is just a wrapper for solve_lin_sys
eqs = list(system * Matrix(symbols + (-1,)))
eqs, ring = sympy_eqs_to_ring(eqs, symbols)
sol = solve_lin_sys(eqs, ring, _raw=False)
if sol is not None:
sol = {sym:val for sym, val in sol.items() if sym != val}
return sol
def solve_undetermined_coeffs(equ, coeffs, sym, **flags):
r"""
Solve equation of a type $p(x; a_1, \ldots, a_k) = q(x)$ where both
$p$ and $q$ are univariate polynomials that depend on $k$ parameters.
Explanation
===========
The result of this function is a dictionary with symbolic values of those
parameters with respect to coefficients in $q$, [] if there is no
solution, else None if the system was not recognized.
This function accepts both equations class instances and ordinary
SymPy expressions. Specification of parameters and variables is
obligatory for efficiency and simplicity reasons.
Examples
========
>>> from sympy import Eq, solve_undetermined_coeffs
>>> from sympy.abc import a, b, c, x
>>> solve_undetermined_coeffs(Eq(2*a*x + a+b, x), [a, b], x)
{a: 1/2, b: -1/2}
>>> solve_undetermined_coeffs(Eq(a*c*x + a+b, x), [a, b], x)
{a: 1/c, b: -1/c}
>>> solve_undetermined_coeffs(0, [a], x)
[]
>>> solve_undetermined_coeffs(a**2*x + 2*x + b + 3, [a, b], x) is None
True
"""
if isinstance(equ, Eq):
# got equation, so move all the
# terms to the left hand side
equ = equ.lhs - equ.rhs
equ = cancel(equ).as_numer_denom()[0]
system = list(collect(equ.expand(), sym, evaluate=False).values())
if not any(equ.has(sym) for equ in system):
# consecutive powers in the input expressions have
# been successfully collected, so solve remaining
# system using Gaussian elimination algorithm
_flags = dict(flags, set=None, dict=True)
sol = solve(system, *coeffs, **_flags)
if len(sol) == 1:
return sol[0]
elif not sol:
return []
def solve_linear_system_LU(matrix, syms):
"""
Solves the augmented matrix system using ``LUsolve`` and returns a
dictionary in which solutions are keyed to the symbols of *syms* as ordered.
Explanation
===========
The matrix must be invertible.
Examples
========
>>> from sympy import Matrix, solve_linear_system_LU
>>> from sympy.abc import x, y, z
>>> solve_linear_system_LU(Matrix([
... [1, 2, 0, 1],
... [3, 2, 2, 1],
... [2, 0, 0, 1]]), [x, y, z])
{x: 1/2, y: 1/4, z: -1/2}
See Also
========
LUsolve
"""
if matrix.rows != matrix.cols - 1:
raise ValueError("Rows should be equal to columns - 1")
A = matrix[:matrix.rows, :matrix.rows]
b = matrix[:, matrix.cols - 1:]
soln = A.LUsolve(b)
solutions = {}
for i in range(soln.rows):
solutions[syms[i]] = soln[i, 0]
return solutions
def det_perm(M):
"""
Return the determinant of *M* by using permutations to select factors.
Explanation
===========
For sizes larger than 8 the number of permutations becomes prohibitively
large, or if there are no symbols in the matrix, it is better to use the
standard determinant routines (e.g., ``M.det()``.)
See Also
========
det_minor
det_quick
"""
args = []
s = True
n = M.rows
list_ = M.flat()
for perm in generate_bell(n):
fac = []
idx = 0
for j in perm:
fac.append(list_[idx + j])
idx += n
term = Mul(*fac) # disaster with unevaluated Mul -- takes forever for n=7
args.append(term if s else -term)
s = not s
return Add(*args)
def det_minor(M):
"""
Return the ``det(M)`` computed from minors without
introducing new nesting in products.
See Also
========
det_perm
det_quick
"""
n = M.rows
if n == 2:
return M[0, 0]*M[1, 1] - M[1, 0]*M[0, 1]
else:
return sum([(1, -1)[i % 2]*Add(*[M[0, i]*d for d in
Add.make_args(det_minor(M.minor_submatrix(0, i)))])
if M[0, i] else S.Zero for i in range(n)])
def det_quick(M, method=None):
"""
Return ``det(M)`` assuming that either
there are lots of zeros or the size of the matrix
is small. If this assumption is not met, then the normal
Matrix.det function will be used with method = ``method``.
See Also
========
det_minor
det_perm
"""
if any(i.has(Symbol) for i in M):
if M.rows < 8 and all(i.has(Symbol) for i in M):
return det_perm(M)
return det_minor(M)
else:
return M.det(method=method) if method else M.det()
def inv_quick(M):
"""Return the inverse of ``M``, assuming that either
there are lots of zeros or the size of the matrix
is small.
"""
if not all(i.is_Number for i in M):
if not any(i.is_Number for i in M):
det = lambda _: det_perm(_)
else:
det = lambda _: det_minor(_)
else:
return M.inv()
n = M.rows
d = det(M)
if d == S.Zero:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
ret = zeros(n)
s1 = -1
for i in range(n):
s = s1 = -s1
for j in range(n):
di = det(M.minor_submatrix(i, j))
ret[j, i] = s*di/d
s = -s
return ret
# these are functions that have multiple inverse values per period
multi_inverses = {
sin: lambda x: (asin(x), S.Pi - asin(x)),
cos: lambda x: (acos(x), 2*S.Pi - acos(x)),
}
def _vsolve(e, s, **flags):
"""return list of scalar values for the solution of e for symbol s"""
return [i[s] for i in _solve(e, s, **flags)]
def _tsolve(eq, sym, **flags):
"""
Helper for ``_solve`` that solves a transcendental equation with respect
to the given symbol. Various equations containing powers and logarithms,
can be solved.
There is currently no guarantee that all solutions will be returned or
that a real solution will be favored over a complex one.
Either a list of potential solutions will be returned or None will be
returned (in the case that no method was known to get a solution
for the equation). All other errors (like the inability to cast an
expression as a Poly) are unhandled.
Examples
========
>>> from sympy import log
>>> from sympy.solvers.solvers import _tsolve as tsolve
>>> from sympy.abc import x
>>> set(tsolve(3**(2*x + 5) - 4, x))
{(-5*log(3)/2 + log(2) + I*pi)/log(3), -5/2 + log(2)/log(3)}
>>> tsolve(log(x) + 2*x, x)
[LambertW(2)/2]
"""
if 'tsolve_saw' not in flags:
flags['tsolve_saw'] = []
if eq in flags['tsolve_saw']:
return None
else:
flags['tsolve_saw'].append(eq)
rhs, lhs = _invert(eq, sym)
if lhs == sym:
return [rhs]
try:
if lhs.is_Add:
# it's time to try factoring; powdenest is used
# to try get powers in standard form for better factoring
f = factor(powdenest(lhs - rhs))
if f.is_Mul:
return _vsolve(f, sym, **flags)
if rhs:
f = logcombine(lhs, force=flags.get('force', True))
if f.count(log) != lhs.count(log):
if isinstance(f, log):
return _vsolve(f.args[0] - exp(rhs), sym, **flags)
return _tsolve(f - rhs, sym, **flags)
elif lhs.is_Pow:
if lhs.exp.is_Integer:
if lhs - rhs != eq:
return _vsolve(lhs - rhs, sym, **flags)
if sym not in lhs.exp.free_symbols:
return _vsolve(lhs.base - rhs**(1/lhs.exp), sym, **flags)
# _tsolve calls this with Dummy before passing the actual number in.
if any(t.is_Dummy for t in rhs.free_symbols):
raise NotImplementedError # _tsolve will call here again...
# a ** g(x) == 0
if not rhs:
# f(x)**g(x) only has solutions where f(x) == 0 and g(x) != 0 at
# the same place
sol_base = _vsolve(lhs.base, sym, **flags)
return [s for s in sol_base if lhs.exp.subs(sym, s) != 0] # XXX use checksol here?
# a ** g(x) == b
if not lhs.base.has(sym):
if lhs.base == 0:
return _vsolve(lhs.exp, sym, **flags) if rhs != 0 else []
# Gets most solutions...
if lhs.base == rhs.as_base_exp()[0]:
# handles case when bases are equal
sol = _vsolve(lhs.exp - rhs.as_base_exp()[1], sym, **flags)
else:
# handles cases when bases are not equal and exp
# may or may not be equal
f = exp(log(lhs.base)*lhs.exp) - exp(log(rhs))
sol = _vsolve(f, sym, **flags)
# Check for duplicate solutions
def equal(expr1, expr2):
_ = Dummy()
eq = checksol(expr1 - _, _, expr2)
if eq is None:
if nsimplify(expr1) != nsimplify(expr2):
return False
# they might be coincidentally the same
# so check more rigorously
eq = expr1.equals(expr2) # XXX expensive but necessary?
return eq
# Guess a rational exponent
e_rat = nsimplify(log(abs(rhs))/log(abs(lhs.base)))
e_rat = simplify(posify(e_rat)[0])
n, d = fraction(e_rat)
if expand(lhs.base**n - rhs**d) == 0:
sol = [s for s in sol if not equal(lhs.exp.subs(sym, s), e_rat)]
sol.extend(_vsolve(lhs.exp - e_rat, sym, **flags))
return list(set(sol))
# f(x) ** g(x) == c
else:
sol = []
logform = lhs.exp*log(lhs.base) - log(rhs)
if logform != lhs - rhs:
try:
sol.extend(_vsolve(logform, sym, **flags))
except NotImplementedError:
pass
# Collect possible solutions and check with substitution later.
check = []
if rhs == 1:
# f(x) ** g(x) = 1 -- g(x)=0 or f(x)=+-1
check.extend(_vsolve(lhs.exp, sym, **flags))
check.extend(_vsolve(lhs.base - 1, sym, **flags))
check.extend(_vsolve(lhs.base + 1, sym, **flags))
elif rhs.is_Rational:
for d in (i for i in divisors(abs(rhs.p)) if i != 1):
e, t = integer_log(rhs.p, d)
if not t:
continue # rhs.p != d**b
for s in divisors(abs(rhs.q)):
if s**e== rhs.q:
r = Rational(d, s)
check.extend(_vsolve(lhs.base - r, sym, **flags))
check.extend(_vsolve(lhs.base + r, sym, **flags))
check.extend(_vsolve(lhs.exp - e, sym, **flags))
elif rhs.is_irrational:
b_l, e_l = lhs.base.as_base_exp()
n, d = (e_l*lhs.exp).as_numer_denom()
b, e = sqrtdenest(rhs).as_base_exp()
check = [sqrtdenest(i) for i in (_vsolve(lhs.base - b, sym, **flags))]
check.extend([sqrtdenest(i) for i in (_vsolve(lhs.exp - e, sym, **flags))])
if e_l*d != 1:
check.extend(_vsolve(b_l**n - rhs**(e_l*d), sym, **flags))
for s in check:
ok = checksol(eq, sym, s)
if ok is None:
ok = eq.subs(sym, s).equals(0)
if ok:
sol.append(s)
return list(set(sol))
elif lhs.is_Function and len(lhs.args) == 1:
if lhs.func in multi_inverses:
# sin(x) = 1/3 -> x - asin(1/3) & x - (pi - asin(1/3))
soln = []
for i in multi_inverses[type(lhs)](rhs):
soln.extend(_vsolve(lhs.args[0] - i, sym, **flags))
return list(set(soln))
elif lhs.func == LambertW:
return _vsolve(lhs.args[0] - rhs*exp(rhs), sym, **flags)
rewrite = lhs.rewrite(exp)
if rewrite != lhs:
return _vsolve(rewrite - rhs, sym, **flags)
except NotImplementedError:
pass
# maybe it is a lambert pattern
if flags.pop('bivariate', True):
# lambert forms may need some help being recognized, e.g. changing
# 2**(3*x) + x**3*log(2)**3 + 3*x**2*log(2)**2 + 3*x*log(2) + 1
# to 2**(3*x) + (x*log(2) + 1)**3
# make generator in log have exponent of 1
logs = eq.atoms(log)
spow = min(
{i.exp for j in logs for i in j.atoms(Pow)
if i.base == sym} or {1})
if spow != 1:
p = sym**spow
u = Dummy('bivariate-cov')
ueq = eq.subs(p, u)
if not ueq.has_free(sym):
sol = _vsolve(ueq, u, **flags)
inv = _vsolve(p - u, sym)
return [i.subs(u, s) for i in inv for s in sol]
g = _filtered_gens(eq.as_poly(), sym)
up_or_log = set()
for gi in g:
if isinstance(gi, (exp, log)) or (gi.is_Pow and gi.base == S.Exp1):
up_or_log.add(gi)
elif gi.is_Pow:
gisimp = powdenest(expand_power_exp(gi))
if gisimp.is_Pow and sym in gisimp.exp.free_symbols:
up_or_log.add(gi)
eq_down = expand_log(expand_power_exp(eq)).subs(
dict(list(zip(up_or_log, [0]*len(up_or_log)))))
eq = expand_power_exp(factor(eq_down, deep=True) + (eq - eq_down))
rhs, lhs = _invert(eq, sym)
if lhs.has(sym):
try:
poly = lhs.as_poly()
g = _filtered_gens(poly, sym)
_eq = lhs - rhs
sols = _solve_lambert(_eq, sym, g)
# use a simplified form if it satisfies eq
# and has fewer operations
for n, s in enumerate(sols):
ns = nsimplify(s)
if ns != s and ns.count_ops() <= s.count_ops():
ok = checksol(_eq, sym, ns)
if ok is None:
ok = _eq.subs(sym, ns).equals(0)
if ok:
sols[n] = ns
return sols
except NotImplementedError:
# maybe it's a convoluted function
if len(g) == 2:
try:
gpu = bivariate_type(lhs - rhs, *g)
if gpu is None:
raise NotImplementedError
g, p, u = gpu
flags['bivariate'] = False
inversion = _tsolve(g - u, sym, **flags)
if inversion:
sol = _vsolve(p, u, **flags)
return list({i.subs(u, s)
for i in inversion for s in sol})
except NotImplementedError:
pass
else:
pass
if flags.pop('force', True):
flags['force'] = False
pos, reps = posify(lhs - rhs)
if rhs == S.ComplexInfinity:
return []
for u, s in reps.items():
if s == sym:
break
else:
u = sym
if pos.has(u):
try:
soln = _vsolve(pos, u, **flags)
return [s.subs(reps) for s in soln]
except NotImplementedError:
pass
else:
pass # here for coverage
return # here for coverage
# TODO: option for calculating J numerically
@conserve_mpmath_dps
def nsolve(*args, dict=False, **kwargs):
r"""
Solve a nonlinear equation system numerically: ``nsolve(f, [args,] x0,
modules=['mpmath'], **kwargs)``.
Explanation
===========
``f`` is a vector function of symbolic expressions representing the system.
*args* are the variables. If there is only one variable, this argument can
be omitted. ``x0`` is a starting vector close to a solution.
Use the modules keyword to specify which modules should be used to
evaluate the function and the Jacobian matrix. Make sure to use a module
that supports matrices. For more information on the syntax, please see the
docstring of ``lambdify``.
If the keyword arguments contain ``dict=True`` (default is False) ``nsolve``
will return a list (perhaps empty) of solution mappings. This might be
especially useful if you want to use ``nsolve`` as a fallback to solve since
using the dict argument for both methods produces return values of
consistent type structure. Please note: to keep this consistent with
``solve``, the solution will be returned in a list even though ``nsolve``
(currently at least) only finds one solution at a time.
Overdetermined systems are supported.
Examples
========
>>> from sympy import Symbol, nsolve
>>> import mpmath
>>> mpmath.mp.dps = 15
>>> x1 = Symbol('x1')
>>> x2 = Symbol('x2')
>>> f1 = 3 * x1**2 - 2 * x2**2 - 1
>>> f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8
>>> print(nsolve((f1, f2), (x1, x2), (-1, 1)))
Matrix([[-1.19287309935246], [1.27844411169911]])
For one-dimensional functions the syntax is simplified:
>>> from sympy import sin, nsolve
>>> from sympy.abc import x
>>> nsolve(sin(x), x, 2)
3.14159265358979
>>> nsolve(sin(x), 2)
3.14159265358979
To solve with higher precision than the default, use the prec argument:
>>> from sympy import cos
>>> nsolve(cos(x) - x, 1)
0.739085133215161
>>> nsolve(cos(x) - x, 1, prec=50)
0.73908513321516064165531208767387340401341175890076
>>> cos(_)
0.73908513321516064165531208767387340401341175890076
To solve for complex roots of real functions, a nonreal initial point
must be specified:
>>> from sympy import I
>>> nsolve(x**2 + 2, I)
1.4142135623731*I
``mpmath.findroot`` is used and you can find their more extensive
documentation, especially concerning keyword parameters and
available solvers. Note, however, that functions which are very
steep near the root, the verification of the solution may fail. In
this case you should use the flag ``verify=False`` and
independently verify the solution.
>>> from sympy import cos, cosh
>>> f = cos(x)*cosh(x) - 1
>>> nsolve(f, 3.14*100)
Traceback (most recent call last):
...
ValueError: Could not find root within given tolerance. (1.39267e+230 > 2.1684e-19)
>>> ans = nsolve(f, 3.14*100, verify=False); ans
312.588469032184
>>> f.subs(x, ans).n(2)
2.1e+121
>>> (f/f.diff(x)).subs(x, ans).n(2)
7.4e-15
One might safely skip the verification if bounds of the root are known
and a bisection method is used:
>>> bounds = lambda i: (3.14*i, 3.14*(i + 1))
>>> nsolve(f, bounds(100), solver='bisect', verify=False)
315.730061685774
Alternatively, a function may be better behaved when the
denominator is ignored. Since this is not always the case, however,
the decision of what function to use is left to the discretion of
the user.
>>> eq = x**2/(1 - x)/(1 - 2*x)**2 - 100
>>> nsolve(eq, 0.46)
Traceback (most recent call last):
...
ValueError: Could not find root within given tolerance. (10000 > 2.1684e-19)
Try another starting point or tweak arguments.
>>> nsolve(eq.as_numer_denom()[0], 0.46)
0.46792545969349058
"""
# there are several other SymPy functions that use method= so
# guard against that here
if 'method' in kwargs:
raise ValueError(filldedent('''
Keyword "method" should not be used in this context. When using
some mpmath solvers directly, the keyword "method" is
used, but when using nsolve (and findroot) the keyword to use is
"solver".'''))
if 'prec' in kwargs:
import mpmath
mpmath.mp.dps = kwargs.pop('prec')
# keyword argument to return result as a dictionary
as_dict = dict
from builtins import dict # to unhide the builtin
# interpret arguments
if len(args) == 3:
f = args[0]
fargs = args[1]
x0 = args[2]
if iterable(fargs) and iterable(x0):
if len(x0) != len(fargs):
raise TypeError('nsolve expected exactly %i guess vectors, got %i'
% (len(fargs), len(x0)))
elif len(args) == 2:
f = args[0]
fargs = None
x0 = args[1]
if iterable(f):
raise TypeError('nsolve expected 3 arguments, got 2')
elif len(args) < 2:
raise TypeError('nsolve expected at least 2 arguments, got %i'
% len(args))
else:
raise TypeError('nsolve expected at most 3 arguments, got %i'
% len(args))
modules = kwargs.get('modules', ['mpmath'])
if iterable(f):
f = list(f)
for i, fi in enumerate(f):
if isinstance(fi, Eq):
f[i] = fi.lhs - fi.rhs
f = Matrix(f).T
if iterable(x0):
x0 = list(x0)
if not isinstance(f, Matrix):
# assume it's a SymPy expression
if isinstance(f, Eq):
f = f.lhs - f.rhs
syms = f.free_symbols
if fargs is None:
fargs = syms.copy().pop()
if not (len(syms) == 1 and (fargs in syms or fargs[0] in syms)):
raise ValueError(filldedent('''
expected a one-dimensional and numerical function'''))
# the function is much better behaved if there is no denominator
# but sending the numerator is left to the user since sometimes
# the function is better behaved when the denominator is present
# e.g., issue 11768
f = lambdify(fargs, f, modules)
x = sympify(findroot(f, x0, **kwargs))
if as_dict:
return [{fargs: x}]
return x
if len(fargs) > f.cols:
raise NotImplementedError(filldedent('''
need at least as many equations as variables'''))
verbose = kwargs.get('verbose', False)
if verbose:
print('f(x):')
print(f)
# derive Jacobian
J = f.jacobian(fargs)
if verbose:
print('J(x):')
print(J)
# create functions
f = lambdify(fargs, f.T, modules)
J = lambdify(fargs, J, modules)
# solve the system numerically
x = findroot(f, x0, J=J, **kwargs)
if as_dict:
return [dict(zip(fargs, [sympify(xi) for xi in x]))]
return Matrix(x)
def _invert(eq, *symbols, **kwargs):
"""
Return tuple (i, d) where ``i`` is independent of *symbols* and ``d``
contains symbols.
Explanation
===========
``i`` and ``d`` are obtained after recursively using algebraic inversion
until an uninvertible ``d`` remains. If there are no free symbols then
``d`` will be zero. Some (but not necessarily all) solutions to the
expression ``i - d`` will be related to the solutions of the original
expression.
Examples
========
>>> from sympy.solvers.solvers import _invert as invert
>>> from sympy import sqrt, cos
>>> from sympy.abc import x, y
>>> invert(x - 3)
(3, x)
>>> invert(3)
(3, 0)
>>> invert(2*cos(x) - 1)
(1/2, cos(x))
>>> invert(sqrt(x) - 3)
(3, sqrt(x))
>>> invert(sqrt(x) + y, x)
(-y, sqrt(x))
>>> invert(sqrt(x) + y, y)
(-sqrt(x), y)
>>> invert(sqrt(x) + y, x, y)
(0, sqrt(x) + y)
If there is more than one symbol in a power's base and the exponent
is not an Integer, then the principal root will be used for the
inversion:
>>> invert(sqrt(x + y) - 2)
(4, x + y)
>>> invert(sqrt(x + y) - 2)
(4, x + y)
If the exponent is an Integer, setting ``integer_power`` to True
will force the principal root to be selected:
>>> invert(x**2 - 4, integer_power=True)
(2, x)
"""
eq = sympify(eq)
if eq.args:
# make sure we are working with flat eq
eq = eq.func(*eq.args)
free = eq.free_symbols
if not symbols:
symbols = free
if not free & set(symbols):
return eq, S.Zero
dointpow = bool(kwargs.get('integer_power', False))
lhs = eq
rhs = S.Zero
while True:
was = lhs
while True:
indep, dep = lhs.as_independent(*symbols)
# dep + indep == rhs
if lhs.is_Add:
# this indicates we have done it all
if indep.is_zero:
break
lhs = dep
rhs -= indep
# dep * indep == rhs
else:
# this indicates we have done it all
if indep is S.One:
break
lhs = dep
rhs /= indep
# collect like-terms in symbols
if lhs.is_Add:
terms = {}
for a in lhs.args:
i, d = a.as_independent(*symbols)
terms.setdefault(d, []).append(i)
if any(len(v) > 1 for v in terms.values()):
args = []
for d, i in terms.items():
if len(i) > 1:
args.append(Add(*i)*d)
else:
args.append(i[0]*d)
lhs = Add(*args)
# if it's a two-term Add with rhs = 0 and two powers we can get the
# dependent terms together, e.g. 3*f(x) + 2*g(x) -> f(x)/g(x) = -2/3
if lhs.is_Add and not rhs and len(lhs.args) == 2 and \
not lhs.is_polynomial(*symbols):
a, b = ordered(lhs.args)
ai, ad = a.as_independent(*symbols)
bi, bd = b.as_independent(*symbols)
if any(_ispow(i) for i in (ad, bd)):
a_base, a_exp = ad.as_base_exp()
b_base, b_exp = bd.as_base_exp()
if a_base == b_base:
# a = -b
lhs = powsimp(powdenest(ad/bd))
rhs = -bi/ai
else:
rat = ad/bd
_lhs = powsimp(ad/bd)
if _lhs != rat:
lhs = _lhs
rhs = -bi/ai
elif ai == -bi:
if isinstance(ad, Function) and ad.func == bd.func:
if len(ad.args) == len(bd.args) == 1:
lhs = ad.args[0] - bd.args[0]
elif len(ad.args) == len(bd.args):
# should be able to solve
# f(x, y) - f(2 - x, 0) == 0 -> x == 1
raise NotImplementedError(
'equal function with more than 1 argument')
else:
raise ValueError(
'function with different numbers of args')
elif lhs.is_Mul and any(_ispow(a) for a in lhs.args):
lhs = powsimp(powdenest(lhs))
if lhs.is_Function:
if hasattr(lhs, 'inverse') and lhs.inverse() is not None and len(lhs.args) == 1:
# -1
# f(x) = g -> x = f (g)
#
# /!\ inverse should not be defined if there are multiple values
# for the function -- these are handled in _tsolve
#
rhs = lhs.inverse()(rhs)
lhs = lhs.args[0]
elif isinstance(lhs, atan2):
y, x = lhs.args
lhs = 2*atan(y/(sqrt(x**2 + y**2) + x))
elif lhs.func == rhs.func:
if len(lhs.args) == len(rhs.args) == 1:
lhs = lhs.args[0]
rhs = rhs.args[0]
elif len(lhs.args) == len(rhs.args):
# should be able to solve
# f(x, y) == f(2, 3) -> x == 2
# f(x, x + y) == f(2, 3) -> x == 2
raise NotImplementedError(
'equal function with more than 1 argument')
else:
raise ValueError(
'function with different numbers of args')
if rhs and lhs.is_Pow and lhs.exp.is_Integer and lhs.exp < 0:
lhs = 1/lhs
rhs = 1/rhs
# base**a = b -> base = b**(1/a) if
# a is an Integer and dointpow=True (this gives real branch of root)
# a is not an Integer and the equation is multivariate and the
# base has more than 1 symbol in it
# The rationale for this is that right now the multi-system solvers
# doesn't try to resolve generators to see, for example, if the whole
# system is written in terms of sqrt(x + y) so it will just fail, so we
# do that step here.
if lhs.is_Pow and (
lhs.exp.is_Integer and dointpow or not lhs.exp.is_Integer and
len(symbols) > 1 and len(lhs.base.free_symbols & set(symbols)) > 1):
rhs = rhs**(1/lhs.exp)
lhs = lhs.base
if lhs == was:
break
return rhs, lhs
def unrad(eq, *syms, **flags):
"""
Remove radicals with symbolic arguments and return (eq, cov),
None, or raise an error.
Explanation
===========
None is returned if there are no radicals to remove.
NotImplementedError is raised if there are radicals and they cannot be
removed or if the relationship between the original symbols and the
change of variable needed to rewrite the system as a polynomial cannot
be solved.
Otherwise the tuple, ``(eq, cov)``, is returned where:
*eq*, ``cov``
*eq* is an equation without radicals (in the symbol(s) of
interest) whose solutions are a superset of the solutions to the
original expression. *eq* might be rewritten in terms of a new
variable; the relationship to the original variables is given by
``cov`` which is a list containing ``v`` and ``v**p - b`` where
``p`` is the power needed to clear the radical and ``b`` is the
radical now expressed as a polynomial in the symbols of interest.
For example, for sqrt(2 - x) the tuple would be
``(c, c**2 - 2 + x)``. The solutions of *eq* will contain
solutions to the original equation (if there are any).
*syms*
An iterable of symbols which, if provided, will limit the focus of
radical removal: only radicals with one or more of the symbols of
interest will be cleared. All free symbols are used if *syms* is not
set.
*flags* are used internally for communication during recursive calls.
Two options are also recognized:
``take``, when defined, is interpreted as a single-argument function
that returns True if a given Pow should be handled.
Radicals can be removed from an expression if:
* All bases of the radicals are the same; a change of variables is
done in this case.
* If all radicals appear in one term of the expression.
* There are only four terms with sqrt() factors or there are less than
four terms having sqrt() factors.
* There are only two terms with radicals.
Examples
========
>>> from sympy.solvers.solvers import unrad
>>> from sympy.abc import x
>>> from sympy import sqrt, Rational, root
>>> unrad(sqrt(x)*x**Rational(1, 3) + 2)
(x**5 - 64, [])
>>> unrad(sqrt(x) + root(x + 1, 3))
(-x**3 + x**2 + 2*x + 1, [])
>>> eq = sqrt(x) + root(x, 3) - 2
>>> unrad(eq)
(_p**3 + _p**2 - 2, [_p, _p**6 - x])
"""
uflags = dict(check=False, simplify=False)
def _cov(p, e):
if cov:
# XXX - uncovered
oldp, olde = cov
if Poly(e, p).degree(p) in (1, 2):
cov[:] = [p, olde.subs(oldp, _vsolve(e, p, **uflags)[0])]
else:
raise NotImplementedError
else:
cov[:] = [p, e]
def _canonical(eq, cov):
if cov:
# change symbol to vanilla so no solutions are eliminated
p, e = cov
rep = {p: Dummy(p.name)}
eq = eq.xreplace(rep)
cov = [p.xreplace(rep), e.xreplace(rep)]
# remove constants and powers of factors since these don't change
# the location of the root; XXX should factor or factor_terms be used?
eq = factor_terms(_mexpand(eq.as_numer_denom()[0], recursive=True), clear=True)
if eq.is_Mul:
args = []
for f in eq.args:
if f.is_number:
continue
if f.is_Pow:
args.append(f.base)
else:
args.append(f)
eq = Mul(*args) # leave as Mul for more efficient solving
# make the sign canonical
margs = list(Mul.make_args(eq))
changed = False
for i, m in enumerate(margs):
if m.could_extract_minus_sign():
margs[i] = -m
changed = True
if changed:
eq = Mul(*margs, evaluate=False)
return eq, cov
def _Q(pow):
# return leading Rational of denominator of Pow's exponent
c = pow.as_base_exp()[1].as_coeff_Mul()[0]
if not c.is_Rational:
return S.One
return c.q
# define the _take method that will determine whether a term is of interest
def _take(d):
# return True if coefficient of any factor's exponent's den is not 1
for pow in Mul.make_args(d):
if not pow.is_Pow:
continue
if _Q(pow) == 1:
continue
if pow.free_symbols & syms:
return True
return False
_take = flags.setdefault('_take', _take)
if isinstance(eq, Eq):
eq = eq.lhs - eq.rhs # XXX legacy Eq as Eqn support
elif not isinstance(eq, Expr):
return
cov, nwas, rpt = [flags.setdefault(k, v) for k, v in
sorted(dict(cov=[], n=None, rpt=0).items())]
# preconditioning
eq = powdenest(factor_terms(eq, radical=True, clear=True))
eq = eq.as_numer_denom()[0]
eq = _mexpand(eq, recursive=True)
if eq.is_number:
return
# see if there are radicals in symbols of interest
syms = set(syms) or eq.free_symbols # _take uses this
poly = eq.as_poly()
gens = [g for g in poly.gens if _take(g)]
if not gens:
return
# recast poly in terms of eigen-gens
poly = eq.as_poly(*gens)
# not a polynomial e.g. 1 + sqrt(x)*exp(sqrt(x)) with gen sqrt(x)
if poly is None:
return
# - an exponent has a symbol of interest (don't handle)
if any(g.exp.has(*syms) for g in gens):
return
def _rads_bases_lcm(poly):
# if all the bases are the same or all the radicals are in one
# term, `lcm` will be the lcm of the denominators of the
# exponents of the radicals
lcm = 1
rads = set()
bases = set()
for g in poly.gens:
q = _Q(g)
if q != 1:
rads.add(g)
lcm = ilcm(lcm, q)
bases.add(g.base)
return rads, bases, lcm
rads, bases, lcm = _rads_bases_lcm(poly)
covsym = Dummy('p', nonnegative=True)
# only keep in syms symbols that actually appear in radicals;
# and update gens
newsyms = set()
for r in rads:
newsyms.update(syms & r.free_symbols)
if newsyms != syms:
syms = newsyms
# get terms together that have common generators
drad = dict(list(zip(rads, list(range(len(rads))))))
rterms = {(): []}
args = Add.make_args(poly.as_expr())
for t in args:
if _take(t):
common = set(t.as_poly().gens).intersection(rads)
key = tuple(sorted([drad[i] for i in common]))
else:
key = ()
rterms.setdefault(key, []).append(t)
others = Add(*rterms.pop(()))
rterms = [Add(*rterms[k]) for k in rterms.keys()]
# the output will depend on the order terms are processed, so
# make it canonical quickly
rterms = list(reversed(list(ordered(rterms))))
ok = False # we don't have a solution yet
depth = sqrt_depth(eq)
if len(rterms) == 1 and not (rterms[0].is_Add and lcm > 2):
eq = rterms[0]**lcm - ((-others)**lcm)
ok = True
else:
if len(rterms) == 1 and rterms[0].is_Add:
rterms = list(rterms[0].args)
if len(bases) == 1:
b = bases.pop()
if len(syms) > 1:
x = b.free_symbols
else:
x = syms
x = list(ordered(x))[0]
try:
inv = _vsolve(covsym**lcm - b, x, **uflags)
if not inv:
raise NotImplementedError
eq = poly.as_expr().subs(b, covsym**lcm).subs(x, inv[0])
_cov(covsym, covsym**lcm - b)
return _canonical(eq, cov)
except NotImplementedError:
pass
if len(rterms) == 2:
if not others:
eq = rterms[0]**lcm - (-rterms[1])**lcm
ok = True
elif not log(lcm, 2).is_Integer:
# the lcm-is-power-of-two case is handled below
r0, r1 = rterms
if flags.get('_reverse', False):
r1, r0 = r0, r1
i0 = _rads0, _bases0, lcm0 = _rads_bases_lcm(r0.as_poly())
i1 = _rads1, _bases1, lcm1 = _rads_bases_lcm(r1.as_poly())
for reverse in range(2):
if reverse:
i0, i1 = i1, i0
r0, r1 = r1, r0
_rads1, _, lcm1 = i1
_rads1 = Mul(*_rads1)
t1 = _rads1**lcm1
c = covsym**lcm1 - t1
for x in syms:
try:
sol = _vsolve(c, x, **uflags)
if not sol:
raise NotImplementedError
neweq = r0.subs(x, sol[0]) + covsym*r1/_rads1 + \
others
tmp = unrad(neweq, covsym)
if tmp:
eq, newcov = tmp
if newcov:
newp, newc = newcov
_cov(newp, c.subs(covsym,
_vsolve(newc, covsym, **uflags)[0]))
else:
_cov(covsym, c)
else:
eq = neweq
_cov(covsym, c)
ok = True
break
except NotImplementedError:
if reverse:
raise NotImplementedError(
'no successful change of variable found')
else:
pass
if ok:
break
elif len(rterms) == 3:
# two cube roots and another with order less than 5
# (so an analytical solution can be found) or a base
# that matches one of the cube root bases
info = [_rads_bases_lcm(i.as_poly()) for i in rterms]
RAD = 0
BASES = 1
LCM = 2
if info[0][LCM] != 3:
info.append(info.pop(0))
rterms.append(rterms.pop(0))
elif info[1][LCM] != 3:
info.append(info.pop(1))
rterms.append(rterms.pop(1))
if info[0][LCM] == info[1][LCM] == 3:
if info[1][BASES] != info[2][BASES]:
info[0], info[1] = info[1], info[0]
rterms[0], rterms[1] = rterms[1], rterms[0]
if info[1][BASES] == info[2][BASES]:
eq = rterms[0]**3 + (rterms[1] + rterms[2] + others)**3
ok = True
elif info[2][LCM] < 5:
# a*root(A, 3) + b*root(B, 3) + others = c
a, b, c, d, A, B = [Dummy(i) for i in 'abcdAB']
# zz represents the unraded expression into which the
# specifics for this case are substituted
zz = (c - d)*(A**3*a**9 + 3*A**2*B*a**6*b**3 -
3*A**2*a**6*c**3 + 9*A**2*a**6*c**2*d - 9*A**2*a**6*c*d**2 +
3*A**2*a**6*d**3 + 3*A*B**2*a**3*b**6 + 21*A*B*a**3*b**3*c**3 -
63*A*B*a**3*b**3*c**2*d + 63*A*B*a**3*b**3*c*d**2 -
21*A*B*a**3*b**3*d**3 + 3*A*a**3*c**6 - 18*A*a**3*c**5*d +
45*A*a**3*c**4*d**2 - 60*A*a**3*c**3*d**3 + 45*A*a**3*c**2*d**4 -
18*A*a**3*c*d**5 + 3*A*a**3*d**6 + B**3*b**9 - 3*B**2*b**6*c**3 +
9*B**2*b**6*c**2*d - 9*B**2*b**6*c*d**2 + 3*B**2*b**6*d**3 +
3*B*b**3*c**6 - 18*B*b**3*c**5*d + 45*B*b**3*c**4*d**2 -
60*B*b**3*c**3*d**3 + 45*B*b**3*c**2*d**4 - 18*B*b**3*c*d**5 +
3*B*b**3*d**6 - c**9 + 9*c**8*d - 36*c**7*d**2 + 84*c**6*d**3 -
126*c**5*d**4 + 126*c**4*d**5 - 84*c**3*d**6 + 36*c**2*d**7 -
9*c*d**8 + d**9)
def _t(i):
b = Mul(*info[i][RAD])
return cancel(rterms[i]/b), Mul(*info[i][BASES])
aa, AA = _t(0)
bb, BB = _t(1)
cc = -rterms[2]
dd = others
eq = zz.xreplace(dict(zip(
(a, A, b, B, c, d),
(aa, AA, bb, BB, cc, dd))))
ok = True
# handle power-of-2 cases
if not ok:
if log(lcm, 2).is_Integer and (not others and
len(rterms) == 4 or len(rterms) < 4):
def _norm2(a, b):
return a**2 + b**2 + 2*a*b
if len(rterms) == 4:
# (r0+r1)**2 - (r2+r3)**2
r0, r1, r2, r3 = rterms
eq = _norm2(r0, r1) - _norm2(r2, r3)
ok = True
elif len(rterms) == 3:
# (r1+r2)**2 - (r0+others)**2
r0, r1, r2 = rterms
eq = _norm2(r1, r2) - _norm2(r0, others)
ok = True
elif len(rterms) == 2:
# r0**2 - (r1+others)**2
r0, r1 = rterms
eq = r0**2 - _norm2(r1, others)
ok = True
new_depth = sqrt_depth(eq) if ok else depth
rpt += 1 # XXX how many repeats with others unchanging is enough?
if not ok or (
nwas is not None and len(rterms) == nwas and
new_depth is not None and new_depth == depth and
rpt > 3):
raise NotImplementedError('Cannot remove all radicals')
flags.update(dict(cov=cov, n=len(rterms), rpt=rpt))
neq = unrad(eq, *syms, **flags)
if neq:
eq, cov = neq
eq, cov = _canonical(eq, cov)
return eq, cov
# delayed imports
from sympy.solvers.bivariate import (
bivariate_type, _solve_lambert, _filtered_gens)
|
92b708db57a6efd30d539e1e6098f1a24fcefc3c3822dca6dc9bb9bcc6d416f0 | """
This module provides convenient functions to transform SymPy expressions to
lambda functions which can be used to calculate numerical values very fast.
"""
from typing import Any, Dict as tDict
import builtins
import inspect
import keyword
import textwrap
import linecache
# Required despite static analysis claiming it is not used
from sympy.external import import_module # noqa:F401
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.iterables import (is_sequence, iterable,
NotIterable, flatten)
from sympy.utilities.misc import filldedent
__doctest_requires__ = {('lambdify',): ['numpy', 'tensorflow']}
# Default namespaces, letting us define translations that can't be defined
# by simple variable maps, like I => 1j
MATH_DEFAULT = {} # type: tDict[str, Any]
MPMATH_DEFAULT = {} # type: tDict[str, Any]
NUMPY_DEFAULT = {"I": 1j} # type: tDict[str, Any]
SCIPY_DEFAULT = {"I": 1j} # type: tDict[str, Any]
CUPY_DEFAULT = {"I": 1j} # type: tDict[str, Any]
JAX_DEFAULT = {"I": 1j} # type: tDict[str, Any]
TENSORFLOW_DEFAULT = {} # type: tDict[str, Any]
SYMPY_DEFAULT = {} # type: tDict[str, Any]
NUMEXPR_DEFAULT = {} # type: tDict[str, Any]
# These are the namespaces the lambda functions will use.
# These are separate from the names above because they are modified
# throughout this file, whereas the defaults should remain unmodified.
MATH = MATH_DEFAULT.copy()
MPMATH = MPMATH_DEFAULT.copy()
NUMPY = NUMPY_DEFAULT.copy()
SCIPY = SCIPY_DEFAULT.copy()
CUPY = CUPY_DEFAULT.copy()
JAX = JAX_DEFAULT.copy()
TENSORFLOW = TENSORFLOW_DEFAULT.copy()
SYMPY = SYMPY_DEFAULT.copy()
NUMEXPR = NUMEXPR_DEFAULT.copy()
# Mappings between SymPy and other modules function names.
MATH_TRANSLATIONS = {
"ceiling": "ceil",
"E": "e",
"ln": "log",
}
# NOTE: This dictionary is reused in Function._eval_evalf to allow subclasses
# of Function to automatically evalf.
MPMATH_TRANSLATIONS = {
"Abs": "fabs",
"elliptic_k": "ellipk",
"elliptic_f": "ellipf",
"elliptic_e": "ellipe",
"elliptic_pi": "ellippi",
"ceiling": "ceil",
"chebyshevt": "chebyt",
"chebyshevu": "chebyu",
"E": "e",
"I": "j",
"ln": "log",
#"lowergamma":"lower_gamma",
"oo": "inf",
#"uppergamma":"upper_gamma",
"LambertW": "lambertw",
"MutableDenseMatrix": "matrix",
"ImmutableDenseMatrix": "matrix",
"conjugate": "conj",
"dirichlet_eta": "altzeta",
"Ei": "ei",
"Shi": "shi",
"Chi": "chi",
"Si": "si",
"Ci": "ci",
"RisingFactorial": "rf",
"FallingFactorial": "ff",
"betainc_regularized": "betainc",
}
NUMPY_TRANSLATIONS = {
"Heaviside": "heaviside",
} # type: tDict[str, str]
SCIPY_TRANSLATIONS = {} # type: tDict[str, str]
CUPY_TRANSLATIONS = {} # type: tDict[str, str]
JAX_TRANSLATIONS = {} # type: tDict[str, str]
TENSORFLOW_TRANSLATIONS = {} # type: tDict[str, str]
NUMEXPR_TRANSLATIONS = {} # type: tDict[str, str]
# Available modules:
MODULES = {
"math": (MATH, MATH_DEFAULT, MATH_TRANSLATIONS, ("from math import *",)),
"mpmath": (MPMATH, MPMATH_DEFAULT, MPMATH_TRANSLATIONS, ("from mpmath import *",)),
"numpy": (NUMPY, NUMPY_DEFAULT, NUMPY_TRANSLATIONS, ("import numpy; from numpy import *; from numpy.linalg import *",)),
"scipy": (SCIPY, SCIPY_DEFAULT, SCIPY_TRANSLATIONS, ("import scipy; import numpy; from scipy.special import *",)),
"cupy": (CUPY, CUPY_DEFAULT, CUPY_TRANSLATIONS, ("import cupy",)),
"jax": (JAX, JAX_DEFAULT, JAX_TRANSLATIONS, ("import jax",)),
"tensorflow": (TENSORFLOW, TENSORFLOW_DEFAULT, TENSORFLOW_TRANSLATIONS, ("import tensorflow",)),
"sympy": (SYMPY, SYMPY_DEFAULT, {}, (
"from sympy.functions import *",
"from sympy.matrices import *",
"from sympy import Integral, pi, oo, nan, zoo, E, I",)),
"numexpr" : (NUMEXPR, NUMEXPR_DEFAULT, NUMEXPR_TRANSLATIONS,
("import_module('numexpr')", )),
}
def _import(module, reload=False):
"""
Creates a global translation dictionary for module.
The argument module has to be one of the following strings: "math",
"mpmath", "numpy", "sympy", "tensorflow", "jax".
These dictionaries map names of Python functions to their equivalent in
other modules.
"""
try:
namespace, namespace_default, translations, import_commands = MODULES[
module]
except KeyError:
raise NameError(
"'%s' module cannot be used for lambdification" % module)
# Clear namespace or exit
if namespace != namespace_default:
# The namespace was already generated, don't do it again if not forced.
if reload:
namespace.clear()
namespace.update(namespace_default)
else:
return
for import_command in import_commands:
if import_command.startswith('import_module'):
module = eval(import_command)
if module is not None:
namespace.update(module.__dict__)
continue
else:
try:
exec(import_command, {}, namespace)
continue
except ImportError:
pass
raise ImportError(
"Cannot import '%s' with '%s' command" % (module, import_command))
# Add translated names to namespace
for sympyname, translation in translations.items():
namespace[sympyname] = namespace[translation]
# For computing the modulus of a SymPy expression we use the builtin abs
# function, instead of the previously used fabs function for all
# translation modules. This is because the fabs function in the math
# module does not accept complex valued arguments. (see issue 9474). The
# only exception, where we don't use the builtin abs function is the
# mpmath translation module, because mpmath.fabs returns mpf objects in
# contrast to abs().
if 'Abs' not in namespace:
namespace['Abs'] = abs
# Used for dynamically generated filenames that are inserted into the
# linecache.
_lambdify_generated_counter = 1
@doctest_depends_on(modules=('numpy', 'scipy', 'tensorflow',), python_version=(3,))
def lambdify(args, expr, modules=None, printer=None, use_imps=True,
dummify=False, cse=False):
"""Convert a SymPy expression into a function that allows for fast
numeric evaluation.
.. warning::
This function uses ``exec``, and thus should not be used on
unsanitized input.
.. deprecated:: 1.7
Passing a set for the *args* parameter is deprecated as sets are
unordered. Use an ordered iterable such as a list or tuple.
Explanation
===========
For example, to convert the SymPy expression ``sin(x) + cos(x)`` to an
equivalent NumPy function that numerically evaluates it:
>>> from sympy import sin, cos, symbols, lambdify
>>> import numpy as np
>>> x = symbols('x')
>>> expr = sin(x) + cos(x)
>>> expr
sin(x) + cos(x)
>>> f = lambdify(x, expr, 'numpy')
>>> a = np.array([1, 2])
>>> f(a)
[1.38177329 0.49315059]
The primary purpose of this function is to provide a bridge from SymPy
expressions to numerical libraries such as NumPy, SciPy, NumExpr, mpmath,
and tensorflow. In general, SymPy functions do not work with objects from
other libraries, such as NumPy arrays, and functions from numeric
libraries like NumPy or mpmath do not work on SymPy expressions.
``lambdify`` bridges the two by converting a SymPy expression to an
equivalent numeric function.
The basic workflow with ``lambdify`` is to first create a SymPy expression
representing whatever mathematical function you wish to evaluate. This
should be done using only SymPy functions and expressions. Then, use
``lambdify`` to convert this to an equivalent function for numerical
evaluation. For instance, above we created ``expr`` using the SymPy symbol
``x`` and SymPy functions ``sin`` and ``cos``, then converted it to an
equivalent NumPy function ``f``, and called it on a NumPy array ``a``.
Parameters
==========
args : List[Symbol]
A variable or a list of variables whose nesting represents the
nesting of the arguments that will be passed to the function.
Variables can be symbols, undefined functions, or matrix symbols.
>>> from sympy import Eq
>>> from sympy.abc import x, y, z
The list of variables should match the structure of how the
arguments will be passed to the function. Simply enclose the
parameters as they will be passed in a list.
To call a function like ``f(x)`` then ``[x]``
should be the first argument to ``lambdify``; for this
case a single ``x`` can also be used:
>>> f = lambdify(x, x + 1)
>>> f(1)
2
>>> f = lambdify([x], x + 1)
>>> f(1)
2
To call a function like ``f(x, y)`` then ``[x, y]`` will
be the first argument of the ``lambdify``:
>>> f = lambdify([x, y], x + y)
>>> f(1, 1)
2
To call a function with a single 3-element tuple like
``f((x, y, z))`` then ``[(x, y, z)]`` will be the first
argument of the ``lambdify``:
>>> f = lambdify([(x, y, z)], Eq(z**2, x**2 + y**2))
>>> f((3, 4, 5))
True
If two args will be passed and the first is a scalar but
the second is a tuple with two arguments then the items
in the list should match that structure:
>>> f = lambdify([x, (y, z)], x + y + z)
>>> f(1, (2, 3))
6
expr : Expr
An expression, list of expressions, or matrix to be evaluated.
Lists may be nested.
If the expression is a list, the output will also be a list.
>>> f = lambdify(x, [x, [x + 1, x + 2]])
>>> f(1)
[1, [2, 3]]
If it is a matrix, an array will be returned (for the NumPy module).
>>> from sympy import Matrix
>>> f = lambdify(x, Matrix([x, x + 1]))
>>> f(1)
[[1]
[2]]
Note that the argument order here (variables then expression) is used
to emulate the Python ``lambda`` keyword. ``lambdify(x, expr)`` works
(roughly) like ``lambda x: expr``
(see :ref:`lambdify-how-it-works` below).
modules : str, optional
Specifies the numeric library to use.
If not specified, *modules* defaults to:
- ``["scipy", "numpy"]`` if SciPy is installed
- ``["numpy"]`` if only NumPy is installed
- ``["math", "mpmath", "sympy"]`` if neither is installed.
That is, SymPy functions are replaced as far as possible by
either ``scipy`` or ``numpy`` functions if available, and Python's
standard library ``math``, or ``mpmath`` functions otherwise.
*modules* can be one of the following types:
- The strings ``"math"``, ``"mpmath"``, ``"numpy"``, ``"numexpr"``,
``"scipy"``, ``"sympy"``, or ``"tensorflow"`` or ``"jax"``. This uses the
corresponding printer and namespace mapping for that module.
- A module (e.g., ``math``). This uses the global namespace of the
module. If the module is one of the above known modules, it will
also use the corresponding printer and namespace mapping
(i.e., ``modules=numpy`` is equivalent to ``modules="numpy"``).
- A dictionary that maps names of SymPy functions to arbitrary
functions
(e.g., ``{'sin': custom_sin}``).
- A list that contains a mix of the arguments above, with higher
priority given to entries appearing first
(e.g., to use the NumPy module but override the ``sin`` function
with a custom version, you can use
``[{'sin': custom_sin}, 'numpy']``).
dummify : bool, optional
Whether or not the variables in the provided expression that are not
valid Python identifiers are substituted with dummy symbols.
This allows for undefined functions like ``Function('f')(t)`` to be
supplied as arguments. By default, the variables are only dummified
if they are not valid Python identifiers.
Set ``dummify=True`` to replace all arguments with dummy symbols
(if ``args`` is not a string) - for example, to ensure that the
arguments do not redefine any built-in names.
cse : bool, or callable, optional
Large expressions can be computed more efficiently when
common subexpressions are identified and precomputed before
being used multiple time. Finding the subexpressions will make
creation of the 'lambdify' function slower, however.
When ``True``, ``sympy.simplify.cse`` is used, otherwise (the default)
the user may pass a function matching the ``cse`` signature.
Examples
========
>>> from sympy.utilities.lambdify import implemented_function
>>> from sympy import sqrt, sin, Matrix
>>> from sympy import Function
>>> from sympy.abc import w, x, y, z
>>> f = lambdify(x, x**2)
>>> f(2)
4
>>> f = lambdify((x, y, z), [z, y, x])
>>> f(1,2,3)
[3, 2, 1]
>>> f = lambdify(x, sqrt(x))
>>> f(4)
2.0
>>> f = lambdify((x, y), sin(x*y)**2)
>>> f(0, 5)
0.0
>>> row = lambdify((x, y), Matrix((x, x + y)).T, modules='sympy')
>>> row(1, 2)
Matrix([[1, 3]])
``lambdify`` can be used to translate SymPy expressions into mpmath
functions. This may be preferable to using ``evalf`` (which uses mpmath on
the backend) in some cases.
>>> f = lambdify(x, sin(x), 'mpmath')
>>> f(1)
0.8414709848078965
Tuple arguments are handled and the lambdified function should
be called with the same type of arguments as were used to create
the function:
>>> f = lambdify((x, (y, z)), x + y)
>>> f(1, (2, 4))
3
The ``flatten`` function can be used to always work with flattened
arguments:
>>> from sympy.utilities.iterables import flatten
>>> args = w, (x, (y, z))
>>> vals = 1, (2, (3, 4))
>>> f = lambdify(flatten(args), w + x + y + z)
>>> f(*flatten(vals))
10
Functions present in ``expr`` can also carry their own numerical
implementations, in a callable attached to the ``_imp_`` attribute. This
can be used with undefined functions using the ``implemented_function``
factory:
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> func = lambdify(x, f(x))
>>> func(4)
5
``lambdify`` always prefers ``_imp_`` implementations to implementations
in other namespaces, unless the ``use_imps`` input parameter is False.
Usage with Tensorflow:
>>> import tensorflow as tf
>>> from sympy import Max, sin, lambdify
>>> from sympy.abc import x
>>> f = Max(x, sin(x))
>>> func = lambdify(x, f, 'tensorflow')
After tensorflow v2, eager execution is enabled by default.
If you want to get the compatible result across tensorflow v1 and v2
as same as this tutorial, run this line.
>>> tf.compat.v1.enable_eager_execution()
If you have eager execution enabled, you can get the result out
immediately as you can use numpy.
If you pass tensorflow objects, you may get an ``EagerTensor``
object instead of value.
>>> result = func(tf.constant(1.0))
>>> print(result)
tf.Tensor(1.0, shape=(), dtype=float32)
>>> print(result.__class__)
<class 'tensorflow.python.framework.ops.EagerTensor'>
You can use ``.numpy()`` to get the numpy value of the tensor.
>>> result.numpy()
1.0
>>> var = tf.Variable(2.0)
>>> result = func(var) # also works for tf.Variable and tf.Placeholder
>>> result.numpy()
2.0
And it works with any shape array.
>>> tensor = tf.constant([[1.0, 2.0], [3.0, 4.0]])
>>> result = func(tensor)
>>> result.numpy()
[[1. 2.]
[3. 4.]]
Notes
=====
- For functions involving large array calculations, numexpr can provide a
significant speedup over numpy. Please note that the available functions
for numexpr are more limited than numpy but can be expanded with
``implemented_function`` and user defined subclasses of Function. If
specified, numexpr may be the only option in modules. The official list
of numexpr functions can be found at:
https://numexpr.readthedocs.io/en/latest/user_guide.html#supported-functions
- In the above examples, the generated functions can accept scalar
values or numpy arrays as arguments. However, in some cases
the generated function relies on the input being a numpy array:
>>> import numpy
>>> from sympy import Piecewise
>>> from sympy.testing.pytest import ignore_warnings
>>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "numpy")
>>> with ignore_warnings(RuntimeWarning):
... f(numpy.array([-1, 0, 1, 2]))
[-1. 0. 1. 0.5]
>>> f(0)
Traceback (most recent call last):
...
ZeroDivisionError: division by zero
In such cases, the input should be wrapped in a numpy array:
>>> with ignore_warnings(RuntimeWarning):
... float(f(numpy.array([0])))
0.0
Or if numpy functionality is not required another module can be used:
>>> f = lambdify(x, Piecewise((x, x <= 1), (1/x, x > 1)), "math")
>>> f(0)
0
.. _lambdify-how-it-works:
How it works
============
When using this function, it helps a great deal to have an idea of what it
is doing. At its core, lambdify is nothing more than a namespace
translation, on top of a special printer that makes some corner cases work
properly.
To understand lambdify, first we must properly understand how Python
namespaces work. Say we had two files. One called ``sin_cos_sympy.py``,
with
.. code:: python
# sin_cos_sympy.py
from sympy.functions.elementary.trigonometric import (cos, sin)
def sin_cos(x):
return sin(x) + cos(x)
and one called ``sin_cos_numpy.py`` with
.. code:: python
# sin_cos_numpy.py
from numpy import sin, cos
def sin_cos(x):
return sin(x) + cos(x)
The two files define an identical function ``sin_cos``. However, in the
first file, ``sin`` and ``cos`` are defined as the SymPy ``sin`` and
``cos``. In the second, they are defined as the NumPy versions.
If we were to import the first file and use the ``sin_cos`` function, we
would get something like
>>> from sin_cos_sympy import sin_cos # doctest: +SKIP
>>> sin_cos(1) # doctest: +SKIP
cos(1) + sin(1)
On the other hand, if we imported ``sin_cos`` from the second file, we
would get
>>> from sin_cos_numpy import sin_cos # doctest: +SKIP
>>> sin_cos(1) # doctest: +SKIP
1.38177329068
In the first case we got a symbolic output, because it used the symbolic
``sin`` and ``cos`` functions from SymPy. In the second, we got a numeric
result, because ``sin_cos`` used the numeric ``sin`` and ``cos`` functions
from NumPy. But notice that the versions of ``sin`` and ``cos`` that were
used was not inherent to the ``sin_cos`` function definition. Both
``sin_cos`` definitions are exactly the same. Rather, it was based on the
names defined at the module where the ``sin_cos`` function was defined.
The key point here is that when function in Python references a name that
is not defined in the function, that name is looked up in the "global"
namespace of the module where that function is defined.
Now, in Python, we can emulate this behavior without actually writing a
file to disk using the ``exec`` function. ``exec`` takes a string
containing a block of Python code, and a dictionary that should contain
the global variables of the module. It then executes the code "in" that
dictionary, as if it were the module globals. The following is equivalent
to the ``sin_cos`` defined in ``sin_cos_sympy.py``:
>>> import sympy
>>> module_dictionary = {'sin': sympy.sin, 'cos': sympy.cos}
>>> exec('''
... def sin_cos(x):
... return sin(x) + cos(x)
... ''', module_dictionary)
>>> sin_cos = module_dictionary['sin_cos']
>>> sin_cos(1)
cos(1) + sin(1)
and similarly with ``sin_cos_numpy``:
>>> import numpy
>>> module_dictionary = {'sin': numpy.sin, 'cos': numpy.cos}
>>> exec('''
... def sin_cos(x):
... return sin(x) + cos(x)
... ''', module_dictionary)
>>> sin_cos = module_dictionary['sin_cos']
>>> sin_cos(1)
1.38177329068
So now we can get an idea of how ``lambdify`` works. The name "lambdify"
comes from the fact that we can think of something like ``lambdify(x,
sin(x) + cos(x), 'numpy')`` as ``lambda x: sin(x) + cos(x)``, where
``sin`` and ``cos`` come from the ``numpy`` namespace. This is also why
the symbols argument is first in ``lambdify``, as opposed to most SymPy
functions where it comes after the expression: to better mimic the
``lambda`` keyword.
``lambdify`` takes the input expression (like ``sin(x) + cos(x)``) and
1. Converts it to a string
2. Creates a module globals dictionary based on the modules that are
passed in (by default, it uses the NumPy module)
3. Creates the string ``"def func({vars}): return {expr}"``, where ``{vars}`` is the
list of variables separated by commas, and ``{expr}`` is the string
created in step 1., then ``exec``s that string with the module globals
namespace and returns ``func``.
In fact, functions returned by ``lambdify`` support inspection. So you can
see exactly how they are defined by using ``inspect.getsource``, or ``??`` if you
are using IPython or the Jupyter notebook.
>>> f = lambdify(x, sin(x) + cos(x))
>>> import inspect
>>> print(inspect.getsource(f))
def _lambdifygenerated(x):
return sin(x) + cos(x)
This shows us the source code of the function, but not the namespace it
was defined in. We can inspect that by looking at the ``__globals__``
attribute of ``f``:
>>> f.__globals__['sin']
<ufunc 'sin'>
>>> f.__globals__['cos']
<ufunc 'cos'>
>>> f.__globals__['sin'] is numpy.sin
True
This shows us that ``sin`` and ``cos`` in the namespace of ``f`` will be
``numpy.sin`` and ``numpy.cos``.
Note that there are some convenience layers in each of these steps, but at
the core, this is how ``lambdify`` works. Step 1 is done using the
``LambdaPrinter`` printers defined in the printing module (see
:mod:`sympy.printing.lambdarepr`). This allows different SymPy expressions
to define how they should be converted to a string for different modules.
You can change which printer ``lambdify`` uses by passing a custom printer
in to the ``printer`` argument.
Step 2 is augmented by certain translations. There are default
translations for each module, but you can provide your own by passing a
list to the ``modules`` argument. For instance,
>>> def mysin(x):
... print('taking the sin of', x)
... return numpy.sin(x)
...
>>> f = lambdify(x, sin(x), [{'sin': mysin}, 'numpy'])
>>> f(1)
taking the sin of 1
0.8414709848078965
The globals dictionary is generated from the list by merging the
dictionary ``{'sin': mysin}`` and the module dictionary for NumPy. The
merging is done so that earlier items take precedence, which is why
``mysin`` is used above instead of ``numpy.sin``.
If you want to modify the way ``lambdify`` works for a given function, it
is usually easiest to do so by modifying the globals dictionary as such.
In more complicated cases, it may be necessary to create and pass in a
custom printer.
Finally, step 3 is augmented with certain convenience operations, such as
the addition of a docstring.
Understanding how ``lambdify`` works can make it easier to avoid certain
gotchas when using it. For instance, a common mistake is to create a
lambdified function for one module (say, NumPy), and pass it objects from
another (say, a SymPy expression).
For instance, say we create
>>> from sympy.abc import x
>>> f = lambdify(x, x + 1, 'numpy')
Now if we pass in a NumPy array, we get that array plus 1
>>> import numpy
>>> a = numpy.array([1, 2])
>>> f(a)
[2 3]
But what happens if you make the mistake of passing in a SymPy expression
instead of a NumPy array:
>>> f(x + 1)
x + 2
This worked, but it was only by accident. Now take a different lambdified
function:
>>> from sympy import sin
>>> g = lambdify(x, x + sin(x), 'numpy')
This works as expected on NumPy arrays:
>>> g(a)
[1.84147098 2.90929743]
But if we try to pass in a SymPy expression, it fails
>>> try:
... g(x + 1)
... # NumPy release after 1.17 raises TypeError instead of
... # AttributeError
... except (AttributeError, TypeError):
... raise AttributeError() # doctest: +IGNORE_EXCEPTION_DETAIL
Traceback (most recent call last):
...
AttributeError:
Now, let's look at what happened. The reason this fails is that ``g``
calls ``numpy.sin`` on the input expression, and ``numpy.sin`` does not
know how to operate on a SymPy object. **As a general rule, NumPy
functions do not know how to operate on SymPy expressions, and SymPy
functions do not know how to operate on NumPy arrays. This is why lambdify
exists: to provide a bridge between SymPy and NumPy.**
However, why is it that ``f`` did work? That's because ``f`` does not call
any functions, it only adds 1. So the resulting function that is created,
``def _lambdifygenerated(x): return x + 1`` does not depend on the globals
namespace it is defined in. Thus it works, but only by accident. A future
version of ``lambdify`` may remove this behavior.
Be aware that certain implementation details described here may change in
future versions of SymPy. The API of passing in custom modules and
printers will not change, but the details of how a lambda function is
created may change. However, the basic idea will remain the same, and
understanding it will be helpful to understanding the behavior of
lambdify.
**In general: you should create lambdified functions for one module (say,
NumPy), and only pass it input types that are compatible with that module
(say, NumPy arrays).** Remember that by default, if the ``module``
argument is not provided, ``lambdify`` creates functions using the NumPy
and SciPy namespaces.
"""
from sympy.core.symbol import Symbol
from sympy.core.expr import Expr
# If the user hasn't specified any modules, use what is available.
if modules is None:
try:
_import("scipy")
except ImportError:
try:
_import("numpy")
except ImportError:
# Use either numpy (if available) or python.math where possible.
# XXX: This leads to different behaviour on different systems and
# might be the reason for irreproducible errors.
modules = ["math", "mpmath", "sympy"]
else:
modules = ["numpy"]
else:
modules = ["numpy", "scipy"]
# Get the needed namespaces.
namespaces = []
# First find any function implementations
if use_imps:
namespaces.append(_imp_namespace(expr))
# Check for dict before iterating
if isinstance(modules, (dict, str)) or not hasattr(modules, '__iter__'):
namespaces.append(modules)
else:
# consistency check
if _module_present('numexpr', modules) and len(modules) > 1:
raise TypeError("numexpr must be the only item in 'modules'")
namespaces += list(modules)
# fill namespace with first having highest priority
namespace = {} # type: tDict[str, Any]
for m in namespaces[::-1]:
buf = _get_namespace(m)
namespace.update(buf)
if hasattr(expr, "atoms"):
#Try if you can extract symbols from the expression.
#Move on if expr.atoms in not implemented.
syms = expr.atoms(Symbol)
for term in syms:
namespace.update({str(term): term})
if printer is None:
if _module_present('mpmath', namespaces):
from sympy.printing.pycode import MpmathPrinter as Printer # type: ignore
elif _module_present('scipy', namespaces):
from sympy.printing.numpy import SciPyPrinter as Printer # type: ignore
elif _module_present('numpy', namespaces):
from sympy.printing.numpy import NumPyPrinter as Printer # type: ignore
elif _module_present('cupy', namespaces):
from sympy.printing.numpy import CuPyPrinter as Printer # type: ignore
elif _module_present('jax', namespaces):
from sympy.printing.numpy import JaxPrinter as Printer # type: ignore
elif _module_present('numexpr', namespaces):
from sympy.printing.lambdarepr import NumExprPrinter as Printer # type: ignore
elif _module_present('tensorflow', namespaces):
from sympy.printing.tensorflow import TensorflowPrinter as Printer # type: ignore
elif _module_present('sympy', namespaces):
from sympy.printing.pycode import SymPyPrinter as Printer # type: ignore
else:
from sympy.printing.pycode import PythonCodePrinter as Printer # type: ignore
user_functions = {}
for m in namespaces[::-1]:
if isinstance(m, dict):
for k in m:
user_functions[k] = k
printer = Printer({'fully_qualified_modules': False, 'inline': True,
'allow_unknown_functions': True,
'user_functions': user_functions})
if isinstance(args, set):
sympy_deprecation_warning(
"""
Passing the function arguments to lambdify() as a set is deprecated. This
leads to unpredictable results since sets are unordered. Instead, use a list
or tuple for the function arguments.
""",
deprecated_since_version="1.6.3",
active_deprecations_target="deprecated-lambdify-arguments-set",
)
# Get the names of the args, for creating a docstring
iterable_args = (args,) if isinstance(args, Expr) else args
names = []
# Grab the callers frame, for getting the names by inspection (if needed)
callers_local_vars = inspect.currentframe().f_back.f_locals.items() # type: ignore
for n, var in enumerate(iterable_args):
if hasattr(var, 'name'):
names.append(var.name)
else:
# It's an iterable. Try to get name by inspection of calling frame.
name_list = [var_name for var_name, var_val in callers_local_vars
if var_val is var]
if len(name_list) == 1:
names.append(name_list[0])
else:
# Cannot infer name with certainty. arg_# will have to do.
names.append('arg_' + str(n))
# Create the function definition code and execute it
funcname = '_lambdifygenerated'
if _module_present('tensorflow', namespaces):
funcprinter = _TensorflowEvaluatorPrinter(printer, dummify) # type: _EvaluatorPrinter
else:
funcprinter = _EvaluatorPrinter(printer, dummify)
if cse == True:
from sympy.simplify.cse_main import cse as _cse
cses, _expr = _cse(expr, list=False)
elif callable(cse):
cses, _expr = cse(expr)
else:
cses, _expr = (), expr
funcstr = funcprinter.doprint(funcname, iterable_args, _expr, cses=cses)
# Collect the module imports from the code printers.
imp_mod_lines = []
for mod, keys in (getattr(printer, 'module_imports', None) or {}).items():
for k in keys:
if k not in namespace:
ln = "from %s import %s" % (mod, k)
try:
exec(ln, {}, namespace)
except ImportError:
# Tensorflow 2.0 has issues with importing a specific
# function from its submodule.
# https://github.com/tensorflow/tensorflow/issues/33022
ln = "%s = %s.%s" % (k, mod, k)
exec(ln, {}, namespace)
imp_mod_lines.append(ln)
# Provide lambda expression with builtins, and compatible implementation of range
namespace.update({'builtins':builtins, 'range':range})
funclocals = {} # type: tDict[str, Any]
global _lambdify_generated_counter
filename = '<lambdifygenerated-%s>' % _lambdify_generated_counter
_lambdify_generated_counter += 1
c = compile(funcstr, filename, 'exec')
exec(c, namespace, funclocals)
# mtime has to be None or else linecache.checkcache will remove it
linecache.cache[filename] = (len(funcstr), None, funcstr.splitlines(True), filename) # type: ignore
func = funclocals[funcname]
# Apply the docstring
sig = "func({})".format(", ".join(str(i) for i in names))
sig = textwrap.fill(sig, subsequent_indent=' '*8)
expr_str = str(expr)
if len(expr_str) > 78:
expr_str = textwrap.wrap(expr_str, 75)[0] + '...'
func.__doc__ = (
"Created with lambdify. Signature:\n\n"
"{sig}\n\n"
"Expression:\n\n"
"{expr}\n\n"
"Source code:\n\n"
"{src}\n\n"
"Imported modules:\n\n"
"{imp_mods}"
).format(sig=sig, expr=expr_str, src=funcstr, imp_mods='\n'.join(imp_mod_lines))
return func
def _module_present(modname, modlist):
if modname in modlist:
return True
for m in modlist:
if hasattr(m, '__name__') and m.__name__ == modname:
return True
return False
def _get_namespace(m):
"""
This is used by _lambdify to parse its arguments.
"""
if isinstance(m, str):
_import(m)
return MODULES[m][0]
elif isinstance(m, dict):
return m
elif hasattr(m, "__dict__"):
return m.__dict__
else:
raise TypeError("Argument must be either a string, dict or module but it is: %s" % m)
def _recursive_to_string(doprint, arg):
"""Functions in lambdify accept both SymPy types and non-SymPy types such as python
lists and tuples. This method ensures that we only call the doprint method of the
printer with SymPy types (so that the printer safely can use SymPy-methods)."""
from sympy.matrices.common import MatrixOperations
from sympy.core.basic import Basic
if isinstance(arg, (Basic, MatrixOperations)):
return doprint(arg)
elif iterable(arg):
if isinstance(arg, list):
left, right = "[", "]"
elif isinstance(arg, tuple):
left, right = "(", ",)"
else:
raise NotImplementedError("unhandled type: %s, %s" % (type(arg), arg))
return left +', '.join(_recursive_to_string(doprint, e) for e in arg) + right
elif isinstance(arg, str):
return arg
else:
return doprint(arg)
def lambdastr(args, expr, printer=None, dummify=None):
"""
Returns a string that can be evaluated to a lambda function.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.utilities.lambdify import lambdastr
>>> lambdastr(x, x**2)
'lambda x: (x**2)'
>>> lambdastr((x,y,z), [z,y,x])
'lambda x,y,z: ([z, y, x])'
Although tuples may not appear as arguments to lambda in Python 3,
lambdastr will create a lambda function that will unpack the original
arguments so that nested arguments can be handled:
>>> lambdastr((x, (y, z)), x + y)
'lambda _0,_1: (lambda x,y,z: (x + y))(_0,_1[0],_1[1])'
"""
# Transforming everything to strings.
from sympy.matrices import DeferredVector
from sympy.core.basic import Basic
from sympy.core.function import (Derivative, Function)
from sympy.core.symbol import (Dummy, Symbol)
from sympy.core.sympify import sympify
if printer is not None:
if inspect.isfunction(printer):
lambdarepr = printer
else:
if inspect.isclass(printer):
lambdarepr = lambda expr: printer().doprint(expr)
else:
lambdarepr = lambda expr: printer.doprint(expr)
else:
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import lambdarepr
def sub_args(args, dummies_dict):
if isinstance(args, str):
return args
elif isinstance(args, DeferredVector):
return str(args)
elif iterable(args):
dummies = flatten([sub_args(a, dummies_dict) for a in args])
return ",".join(str(a) for a in dummies)
else:
# replace these with Dummy symbols
if isinstance(args, (Function, Symbol, Derivative)):
dummies = Dummy()
dummies_dict.update({args : dummies})
return str(dummies)
else:
return str(args)
def sub_expr(expr, dummies_dict):
expr = sympify(expr)
# dict/tuple are sympified to Basic
if isinstance(expr, Basic):
expr = expr.xreplace(dummies_dict)
# list is not sympified to Basic
elif isinstance(expr, list):
expr = [sub_expr(a, dummies_dict) for a in expr]
return expr
# Transform args
def isiter(l):
return iterable(l, exclude=(str, DeferredVector, NotIterable))
def flat_indexes(iterable):
n = 0
for el in iterable:
if isiter(el):
for ndeep in flat_indexes(el):
yield (n,) + ndeep
else:
yield (n,)
n += 1
if dummify is None:
dummify = any(isinstance(a, Basic) and
a.atoms(Function, Derivative) for a in (
args if isiter(args) else [args]))
if isiter(args) and any(isiter(i) for i in args):
dum_args = [str(Dummy(str(i))) for i in range(len(args))]
indexed_args = ','.join([
dum_args[ind[0]] + ''.join(["[%s]" % k for k in ind[1:]])
for ind in flat_indexes(args)])
lstr = lambdastr(flatten(args), expr, printer=printer, dummify=dummify)
return 'lambda %s: (%s)(%s)' % (','.join(dum_args), lstr, indexed_args)
dummies_dict = {}
if dummify:
args = sub_args(args, dummies_dict)
else:
if isinstance(args, str):
pass
elif iterable(args, exclude=DeferredVector):
args = ",".join(str(a) for a in args)
# Transform expr
if dummify:
if isinstance(expr, str):
pass
else:
expr = sub_expr(expr, dummies_dict)
expr = _recursive_to_string(lambdarepr, expr)
return "lambda %s: (%s)" % (args, expr)
class _EvaluatorPrinter:
def __init__(self, printer=None, dummify=False):
self._dummify = dummify
#XXX: This has to be done here because of circular imports
from sympy.printing.lambdarepr import LambdaPrinter
if printer is None:
printer = LambdaPrinter()
if inspect.isfunction(printer):
self._exprrepr = printer
else:
if inspect.isclass(printer):
printer = printer()
self._exprrepr = printer.doprint
#if hasattr(printer, '_print_Symbol'):
# symbolrepr = printer._print_Symbol
#if hasattr(printer, '_print_Dummy'):
# dummyrepr = printer._print_Dummy
# Used to print the generated function arguments in a standard way
self._argrepr = LambdaPrinter().doprint
def doprint(self, funcname, args, expr, *, cses=()):
"""
Returns the function definition code as a string.
"""
from sympy.core.symbol import Dummy
funcbody = []
if not iterable(args):
args = [args]
if cses:
subvars, subexprs = zip(*cses)
try:
exprs = expr + list(subexprs)
except TypeError:
try:
exprs = expr + tuple(subexprs)
except TypeError:
expr = [expr]
exprs = expr + list(subexprs)
argstrs, exprs = self._preprocess(args, exprs)
expr, subexprs = exprs[:len(expr)], exprs[len(expr):]
cses = zip(subvars, subexprs)
else:
argstrs, expr = self._preprocess(args, expr)
# Generate argument unpacking and final argument list
funcargs = []
unpackings = []
for argstr in argstrs:
if iterable(argstr):
funcargs.append(self._argrepr(Dummy()))
unpackings.extend(self._print_unpacking(argstr, funcargs[-1]))
else:
funcargs.append(argstr)
funcsig = 'def {}({}):'.format(funcname, ', '.join(funcargs))
# Wrap input arguments before unpacking
funcbody.extend(self._print_funcargwrapping(funcargs))
funcbody.extend(unpackings)
for s, e in cses:
if e is None:
funcbody.append('del {}'.format(s))
else:
funcbody.append('{} = {}'.format(s, self._exprrepr(e)))
str_expr = _recursive_to_string(self._exprrepr, expr)
if '\n' in str_expr:
str_expr = '({})'.format(str_expr)
funcbody.append('return {}'.format(str_expr))
funclines = [funcsig]
funclines.extend([' ' + line for line in funcbody])
return '\n'.join(funclines) + '\n'
@classmethod
def _is_safe_ident(cls, ident):
return isinstance(ident, str) and ident.isidentifier() \
and not keyword.iskeyword(ident)
def _preprocess(self, args, expr):
"""Preprocess args, expr to replace arguments that do not map
to valid Python identifiers.
Returns string form of args, and updated expr.
"""
from sympy.core.basic import Basic
from sympy.core.sorting import ordered
from sympy.core.function import (Derivative, Function)
from sympy.core.symbol import Dummy, uniquely_named_symbol
from sympy.matrices import DeferredVector
from sympy.core.expr import Expr
# Args of type Dummy can cause name collisions with args
# of type Symbol. Force dummify of everything in this
# situation.
dummify = self._dummify or any(
isinstance(arg, Dummy) for arg in flatten(args))
argstrs = [None]*len(args)
for arg, i in reversed(list(ordered(zip(args, range(len(args)))))):
if iterable(arg):
s, expr = self._preprocess(arg, expr)
elif isinstance(arg, DeferredVector):
s = str(arg)
elif isinstance(arg, Basic) and arg.is_symbol:
s = self._argrepr(arg)
if dummify or not self._is_safe_ident(s):
dummy = Dummy()
if isinstance(expr, Expr):
dummy = uniquely_named_symbol(
dummy.name, expr, modify=lambda s: '_' + s)
s = self._argrepr(dummy)
expr = self._subexpr(expr, {arg: dummy})
elif dummify or isinstance(arg, (Function, Derivative)):
dummy = Dummy()
s = self._argrepr(dummy)
expr = self._subexpr(expr, {arg: dummy})
else:
s = str(arg)
argstrs[i] = s
return argstrs, expr
def _subexpr(self, expr, dummies_dict):
from sympy.matrices import DeferredVector
from sympy.core.sympify import sympify
expr = sympify(expr)
xreplace = getattr(expr, 'xreplace', None)
if xreplace is not None:
expr = xreplace(dummies_dict)
else:
if isinstance(expr, DeferredVector):
pass
elif isinstance(expr, dict):
k = [self._subexpr(sympify(a), dummies_dict) for a in expr.keys()]
v = [self._subexpr(sympify(a), dummies_dict) for a in expr.values()]
expr = dict(zip(k, v))
elif isinstance(expr, tuple):
expr = tuple(self._subexpr(sympify(a), dummies_dict) for a in expr)
elif isinstance(expr, list):
expr = [self._subexpr(sympify(a), dummies_dict) for a in expr]
return expr
def _print_funcargwrapping(self, args):
"""Generate argument wrapping code.
args is the argument list of the generated function (strings).
Return value is a list of lines of code that will be inserted at
the beginning of the function definition.
"""
return []
def _print_unpacking(self, unpackto, arg):
"""Generate argument unpacking code.
arg is the function argument to be unpacked (a string), and
unpackto is a list or nested lists of the variable names (strings) to
unpack to.
"""
def unpack_lhs(lvalues):
return '[{}]'.format(', '.join(
unpack_lhs(val) if iterable(val) else val for val in lvalues))
return ['{} = {}'.format(unpack_lhs(unpackto), arg)]
class _TensorflowEvaluatorPrinter(_EvaluatorPrinter):
def _print_unpacking(self, lvalues, rvalue):
"""Generate argument unpacking code.
This method is used when the input value is not interable,
but can be indexed (see issue #14655).
"""
def flat_indexes(elems):
n = 0
for el in elems:
if iterable(el):
for ndeep in flat_indexes(el):
yield (n,) + ndeep
else:
yield (n,)
n += 1
indexed = ', '.join('{}[{}]'.format(rvalue, ']['.join(map(str, ind)))
for ind in flat_indexes(lvalues))
return ['[{}] = [{}]'.format(', '.join(flatten(lvalues)), indexed)]
def _imp_namespace(expr, namespace=None):
""" Return namespace dict with function implementations
We need to search for functions in anything that can be thrown at
us - that is - anything that could be passed as ``expr``. Examples
include SymPy expressions, as well as tuples, lists and dicts that may
contain SymPy expressions.
Parameters
----------
expr : object
Something passed to lambdify, that will generate valid code from
``str(expr)``.
namespace : None or mapping
Namespace to fill. None results in new empty dict
Returns
-------
namespace : dict
dict with keys of implemented function names within ``expr`` and
corresponding values being the numerical implementation of
function
Examples
========
>>> from sympy.abc import x
>>> from sympy.utilities.lambdify import implemented_function, _imp_namespace
>>> from sympy import Function
>>> f = implemented_function(Function('f'), lambda x: x+1)
>>> g = implemented_function(Function('g'), lambda x: x*10)
>>> namespace = _imp_namespace(f(g(x)))
>>> sorted(namespace.keys())
['f', 'g']
"""
# Delayed import to avoid circular imports
from sympy.core.function import FunctionClass
if namespace is None:
namespace = {}
# tuples, lists, dicts are valid expressions
if is_sequence(expr):
for arg in expr:
_imp_namespace(arg, namespace)
return namespace
elif isinstance(expr, dict):
for key, val in expr.items():
# functions can be in dictionary keys
_imp_namespace(key, namespace)
_imp_namespace(val, namespace)
return namespace
# SymPy expressions may be Functions themselves
func = getattr(expr, 'func', None)
if isinstance(func, FunctionClass):
imp = getattr(func, '_imp_', None)
if imp is not None:
name = expr.func.__name__
if name in namespace and namespace[name] != imp:
raise ValueError('We found more than one '
'implementation with name '
'"%s"' % name)
namespace[name] = imp
# and / or they may take Functions as arguments
if hasattr(expr, 'args'):
for arg in expr.args:
_imp_namespace(arg, namespace)
return namespace
def implemented_function(symfunc, implementation):
""" Add numerical ``implementation`` to function ``symfunc``.
``symfunc`` can be an ``UndefinedFunction`` instance, or a name string.
In the latter case we create an ``UndefinedFunction`` instance with that
name.
Be aware that this is a quick workaround, not a general method to create
special symbolic functions. If you want to create a symbolic function to be
used by all the machinery of SymPy you should subclass the ``Function``
class.
Parameters
----------
symfunc : ``str`` or ``UndefinedFunction`` instance
If ``str``, then create new ``UndefinedFunction`` with this as
name. If ``symfunc`` is an Undefined function, create a new function
with the same name and the implemented function attached.
implementation : callable
numerical implementation to be called by ``evalf()`` or ``lambdify``
Returns
-------
afunc : sympy.FunctionClass instance
function with attached implementation
Examples
========
>>> from sympy.abc import x
>>> from sympy.utilities.lambdify import implemented_function
>>> from sympy import lambdify
>>> f = implemented_function('f', lambda x: x+1)
>>> lam_f = lambdify(x, f(x))
>>> lam_f(4)
5
"""
# Delayed import to avoid circular imports
from sympy.core.function import UndefinedFunction
# if name, create function to hold implementation
kwargs = {}
if isinstance(symfunc, UndefinedFunction):
kwargs = symfunc._kwargs
symfunc = symfunc.__name__
if isinstance(symfunc, str):
# Keyword arguments to UndefinedFunction are added as attributes to
# the created class.
symfunc = UndefinedFunction(
symfunc, _imp_=staticmethod(implementation), **kwargs)
elif not isinstance(symfunc, UndefinedFunction):
raise ValueError(filldedent('''
symfunc should be either a string or
an UndefinedFunction instance.'''))
return symfunc
|
28e3b189aded9c1c9d8f34d30fc959791764ae43e9be6bf8e1de233f51469277 | from sympy.core import S
from sympy.core.function import Lambda
from sympy.core.power import Pow
from .pycode import PythonCodePrinter, _known_functions_math, _print_known_const, _print_known_func, _unpack_integral_limits, ArrayPrinter
from .codeprinter import CodePrinter
_not_in_numpy = 'erf erfc factorial gamma loggamma'.split()
_in_numpy = [(k, v) for k, v in _known_functions_math.items() if k not in _not_in_numpy]
_known_functions_numpy = dict(_in_numpy, **{
'acos': 'arccos',
'acosh': 'arccosh',
'asin': 'arcsin',
'asinh': 'arcsinh',
'atan': 'arctan',
'atan2': 'arctan2',
'atanh': 'arctanh',
'exp2': 'exp2',
'sign': 'sign',
'logaddexp': 'logaddexp',
'logaddexp2': 'logaddexp2',
})
_known_constants_numpy = {
'Exp1': 'e',
'Pi': 'pi',
'EulerGamma': 'euler_gamma',
'NaN': 'nan',
'Infinity': 'PINF',
'NegativeInfinity': 'NINF'
}
_numpy_known_functions = {k: 'numpy.' + v for k, v in _known_functions_numpy.items()}
_numpy_known_constants = {k: 'numpy.' + v for k, v in _known_constants_numpy.items()}
class NumPyPrinter(ArrayPrinter, PythonCodePrinter):
"""
Numpy printer which handles vectorized piecewise functions,
logical operators, etc.
"""
_module = 'numpy'
_kf = _numpy_known_functions
_kc = _numpy_known_constants
def __init__(self, settings=None):
"""
`settings` is passed to CodePrinter.__init__()
`module` specifies the array module to use, currently 'NumPy', 'CuPy'
or 'JAX'.
"""
self.language = "Python with {}".format(self._module)
self.printmethod = "_{}code".format(self._module)
self._kf = {**PythonCodePrinter._kf, **self._kf}
super().__init__(settings=settings)
def _print_seq(self, seq):
"General sequence printer: converts to tuple"
# Print tuples here instead of lists because numba supports
# tuples in nopython mode.
delimiter=', '
return '({},)'.format(delimiter.join(self._print(item) for item in seq))
def _print_MatMul(self, expr):
"Matrix multiplication printer"
if expr.as_coeff_matrices()[0] is not S.One:
expr_list = expr.as_coeff_matrices()[1]+[(expr.as_coeff_matrices()[0])]
return '({})'.format(').dot('.join(self._print(i) for i in expr_list))
return '({})'.format(').dot('.join(self._print(i) for i in expr.args))
def _print_MatPow(self, expr):
"Matrix power printer"
return '{}({}, {})'.format(self._module_format(self._module + '.linalg.matrix_power'),
self._print(expr.args[0]), self._print(expr.args[1]))
def _print_Inverse(self, expr):
"Matrix inverse printer"
return '{}({})'.format(self._module_format(self._module + '.linalg.inv'),
self._print(expr.args[0]))
def _print_DotProduct(self, expr):
# DotProduct allows any shape order, but numpy.dot does matrix
# multiplication, so we have to make sure it gets 1 x n by n x 1.
arg1, arg2 = expr.args
if arg1.shape[0] != 1:
arg1 = arg1.T
if arg2.shape[1] != 1:
arg2 = arg2.T
return "%s(%s, %s)" % (self._module_format(self._module + '.dot'),
self._print(arg1),
self._print(arg2))
def _print_MatrixSolve(self, expr):
return "%s(%s, %s)" % (self._module_format(self._module + '.linalg.solve'),
self._print(expr.matrix),
self._print(expr.vector))
def _print_ZeroMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.zeros'),
self._print(expr.shape))
def _print_OneMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.ones'),
self._print(expr.shape))
def _print_FunctionMatrix(self, expr):
from sympy.abc import i, j
lamda = expr.lamda
if not isinstance(lamda, Lambda):
lamda = Lambda((i, j), lamda(i, j))
return '{}(lambda {}: {}, {})'.format(self._module_format(self._module + '.fromfunction'),
', '.join(self._print(arg) for arg in lamda.args[0]),
self._print(lamda.args[1]), self._print(expr.shape))
def _print_HadamardProduct(self, expr):
func = self._module_format(self._module + '.multiply')
return ''.join('{}({}, '.format(func, self._print(arg)) \
for arg in expr.args[:-1]) + "{}{}".format(self._print(expr.args[-1]),
')' * (len(expr.args) - 1))
def _print_KroneckerProduct(self, expr):
func = self._module_format(self._module + '.kron')
return ''.join('{}({}, '.format(func, self._print(arg)) \
for arg in expr.args[:-1]) + "{}{}".format(self._print(expr.args[-1]),
')' * (len(expr.args) - 1))
def _print_Adjoint(self, expr):
return '{}({}({}))'.format(
self._module_format(self._module + '.conjugate'),
self._module_format(self._module + '.transpose'),
self._print(expr.args[0]))
def _print_DiagonalOf(self, expr):
vect = '{}({})'.format(
self._module_format(self._module + '.diag'),
self._print(expr.arg))
return '{}({}, (-1, 1))'.format(
self._module_format(self._module + '.reshape'), vect)
def _print_DiagMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.diagflat'),
self._print(expr.args[0]))
def _print_DiagonalMatrix(self, expr):
return '{}({}, {}({}, {}))'.format(self._module_format(self._module + '.multiply'),
self._print(expr.arg), self._module_format(self._module + '.eye'),
self._print(expr.shape[0]), self._print(expr.shape[1]))
def _print_Piecewise(self, expr):
"Piecewise function printer"
from sympy.logic.boolalg import ITE, simplify_logic
def print_cond(cond):
""" Problem having an ITE in the cond. """
if cond.has(ITE):
return self._print(simplify_logic(cond))
else:
return self._print(cond)
exprs = '[{}]'.format(','.join(self._print(arg.expr) for arg in expr.args))
conds = '[{}]'.format(','.join(print_cond(arg.cond) for arg in expr.args))
# If [default_value, True] is a (expr, cond) sequence in a Piecewise object
# it will behave the same as passing the 'default' kwarg to select()
# *as long as* it is the last element in expr.args.
# If this is not the case, it may be triggered prematurely.
return '{}({}, {}, default={})'.format(
self._module_format(self._module + '.select'), conds, exprs,
self._print(S.NaN))
def _print_Relational(self, expr):
"Relational printer for Equality and Unequality"
op = {
'==' :'equal',
'!=' :'not_equal',
'<' :'less',
'<=' :'less_equal',
'>' :'greater',
'>=' :'greater_equal',
}
if expr.rel_op in op:
lhs = self._print(expr.lhs)
rhs = self._print(expr.rhs)
return '{op}({lhs}, {rhs})'.format(op=self._module_format(self._module + '.'+op[expr.rel_op]),
lhs=lhs, rhs=rhs)
return super()._print_Relational(expr)
def _print_And(self, expr):
"Logical And printer"
# We have to override LambdaPrinter because it uses Python 'and' keyword.
# If LambdaPrinter didn't define it, we could use StrPrinter's
# version of the function and add 'logical_and' to NUMPY_TRANSLATIONS.
return '{}.reduce(({}))'.format(self._module_format(self._module + '.logical_and'), ','.join(self._print(i) for i in expr.args))
def _print_Or(self, expr):
"Logical Or printer"
# We have to override LambdaPrinter because it uses Python 'or' keyword.
# If LambdaPrinter didn't define it, we could use StrPrinter's
# version of the function and add 'logical_or' to NUMPY_TRANSLATIONS.
return '{}.reduce(({}))'.format(self._module_format(self._module + '.logical_or'), ','.join(self._print(i) for i in expr.args))
def _print_Not(self, expr):
"Logical Not printer"
# We have to override LambdaPrinter because it uses Python 'not' keyword.
# If LambdaPrinter didn't define it, we would still have to define our
# own because StrPrinter doesn't define it.
return '{}({})'.format(self._module_format(self._module + '.logical_not'), ','.join(self._print(i) for i in expr.args))
def _print_Pow(self, expr, rational=False):
# XXX Workaround for negative integer power error
if expr.exp.is_integer and expr.exp.is_negative:
expr = Pow(expr.base, expr.exp.evalf(), evaluate=False)
return self._hprint_Pow(expr, rational=rational, sqrt=self._module + '.sqrt')
def _print_Min(self, expr):
return '{}(({}), axis=0)'.format(self._module_format(self._module + '.amin'), ','.join(self._print(i) for i in expr.args))
def _print_Max(self, expr):
return '{}(({}), axis=0)'.format(self._module_format(self._module + '.amax'), ','.join(self._print(i) for i in expr.args))
def _print_arg(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.angle'), self._print(expr.args[0]))
def _print_im(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.imag'), self._print(expr.args[0]))
def _print_Mod(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.mod'), ', '.join(
map(lambda arg: self._print(arg), expr.args)))
def _print_re(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.real'), self._print(expr.args[0]))
def _print_sinc(self, expr):
return "%s(%s)" % (self._module_format(self._module + '.sinc'), self._print(expr.args[0]/S.Pi))
def _print_MatrixBase(self, expr):
func = self.known_functions.get(expr.__class__.__name__, None)
if func is None:
func = self._module_format(self._module + '.array')
return "%s(%s)" % (func, self._print(expr.tolist()))
def _print_Identity(self, expr):
shape = expr.shape
if all(dim.is_Integer for dim in shape):
return "%s(%s)" % (self._module_format(self._module + '.eye'), self._print(expr.shape[0]))
else:
raise NotImplementedError("Symbolic matrix dimensions are not yet supported for identity matrices")
def _print_BlockMatrix(self, expr):
return '{}({})'.format(self._module_format(self._module + '.block'),
self._print(expr.args[0].tolist()))
def _print_NDimArray(self, expr):
if len(expr.shape) == 1:
return self._module + '.array(' + self._print(expr.args[0]) + ')'
if len(expr.shape) == 2:
return self._print(expr.tomatrix())
# Should be possible to extend to more dimensions
return CodePrinter._print_not_supported(self, expr)
_add = "add"
_einsum = "einsum"
_transpose = "transpose"
_ones = "ones"
_zeros = "zeros"
_print_lowergamma = CodePrinter._print_not_supported
_print_uppergamma = CodePrinter._print_not_supported
_print_fresnelc = CodePrinter._print_not_supported
_print_fresnels = CodePrinter._print_not_supported
for func in _numpy_known_functions:
setattr(NumPyPrinter, f'_print_{func}', _print_known_func)
for const in _numpy_known_constants:
setattr(NumPyPrinter, f'_print_{const}', _print_known_const)
_known_functions_scipy_special = {
'Ei': 'expi',
'erf': 'erf',
'erfc': 'erfc',
'besselj': 'jv',
'bessely': 'yv',
'besseli': 'iv',
'besselk': 'kv',
'cosm1': 'cosm1',
'factorial': 'factorial',
'gamma': 'gamma',
'loggamma': 'gammaln',
'digamma': 'psi',
'RisingFactorial': 'poch',
'jacobi': 'eval_jacobi',
'gegenbauer': 'eval_gegenbauer',
'chebyshevt': 'eval_chebyt',
'chebyshevu': 'eval_chebyu',
'legendre': 'eval_legendre',
'hermite': 'eval_hermite',
'laguerre': 'eval_laguerre',
'assoc_laguerre': 'eval_genlaguerre',
'beta': 'beta',
'LambertW' : 'lambertw',
}
_known_constants_scipy_constants = {
'GoldenRatio': 'golden_ratio',
'Pi': 'pi',
}
_scipy_known_functions = {k : "scipy.special." + v for k, v in _known_functions_scipy_special.items()}
_scipy_known_constants = {k : "scipy.constants." + v for k, v in _known_constants_scipy_constants.items()}
class SciPyPrinter(NumPyPrinter):
_kf = {**NumPyPrinter._kf, **_scipy_known_functions}
_kc = {**NumPyPrinter._kc, **_scipy_known_constants}
def __init__(self, settings=None):
super().__init__(settings=settings)
self.language = "Python with SciPy and NumPy"
def _print_SparseRepMatrix(self, expr):
i, j, data = [], [], []
for (r, c), v in expr.todok().items():
i.append(r)
j.append(c)
data.append(v)
return "{name}(({data}, ({i}, {j})), shape={shape})".format(
name=self._module_format('scipy.sparse.coo_matrix'),
data=data, i=i, j=j, shape=expr.shape
)
_print_ImmutableSparseMatrix = _print_SparseRepMatrix
# SciPy's lpmv has a different order of arguments from assoc_legendre
def _print_assoc_legendre(self, expr):
return "{0}({2}, {1}, {3})".format(
self._module_format('scipy.special.lpmv'),
self._print(expr.args[0]),
self._print(expr.args[1]),
self._print(expr.args[2]))
def _print_lowergamma(self, expr):
return "{0}({2})*{1}({2}, {3})".format(
self._module_format('scipy.special.gamma'),
self._module_format('scipy.special.gammainc'),
self._print(expr.args[0]),
self._print(expr.args[1]))
def _print_uppergamma(self, expr):
return "{0}({2})*{1}({2}, {3})".format(
self._module_format('scipy.special.gamma'),
self._module_format('scipy.special.gammaincc'),
self._print(expr.args[0]),
self._print(expr.args[1]))
def _print_betainc(self, expr):
betainc = self._module_format('scipy.special.betainc')
beta = self._module_format('scipy.special.beta')
args = [self._print(arg) for arg in expr.args]
return f"({betainc}({args[0]}, {args[1]}, {args[3]}) - {betainc}({args[0]}, {args[1]}, {args[2]})) \
* {beta}({args[0]}, {args[1]})"
def _print_betainc_regularized(self, expr):
return "{0}({1}, {2}, {4}) - {0}({1}, {2}, {3})".format(
self._module_format('scipy.special.betainc'),
self._print(expr.args[0]),
self._print(expr.args[1]),
self._print(expr.args[2]),
self._print(expr.args[3]))
def _print_fresnels(self, expr):
return "{}({})[0]".format(
self._module_format("scipy.special.fresnel"),
self._print(expr.args[0]))
def _print_fresnelc(self, expr):
return "{}({})[1]".format(
self._module_format("scipy.special.fresnel"),
self._print(expr.args[0]))
def _print_airyai(self, expr):
return "{}({})[0]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_airyaiprime(self, expr):
return "{}({})[1]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_airybi(self, expr):
return "{}({})[2]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_airybiprime(self, expr):
return "{}({})[3]".format(
self._module_format("scipy.special.airy"),
self._print(expr.args[0]))
def _print_Integral(self, e):
integration_vars, limits = _unpack_integral_limits(e)
if len(limits) == 1:
# nicer (but not necessary) to prefer quad over nquad for 1D case
module_str = self._module_format("scipy.integrate.quad")
limit_str = "%s, %s" % tuple(map(self._print, limits[0]))
else:
module_str = self._module_format("scipy.integrate.nquad")
limit_str = "({})".format(", ".join(
"(%s, %s)" % tuple(map(self._print, l)) for l in limits))
return "{}(lambda {}: {}, {})[0]".format(
module_str,
", ".join(map(self._print, integration_vars)),
self._print(e.args[0]),
limit_str)
for func in _scipy_known_functions:
setattr(SciPyPrinter, f'_print_{func}', _print_known_func)
for const in _scipy_known_constants:
setattr(SciPyPrinter, f'_print_{const}', _print_known_const)
_cupy_known_functions = {k : "cupy." + v for k, v in _known_functions_numpy.items()}
_cupy_known_constants = {k : "cupy." + v for k, v in _known_constants_numpy.items()}
class CuPyPrinter(NumPyPrinter):
"""
CuPy printer which handles vectorized piecewise functions,
logical operators, etc.
"""
_module = 'cupy'
_kf = _cupy_known_functions
_kc = _cupy_known_constants
def __init__(self, settings=None):
super().__init__(settings=settings)
for func in _cupy_known_functions:
setattr(CuPyPrinter, f'_print_{func}', _print_known_func)
for const in _cupy_known_constants:
setattr(CuPyPrinter, f'_print_{const}', _print_known_const)
_jax_known_functions = {k: 'jax.numpy.' + v for k, v in _known_functions_numpy.items()}
_jax_known_constants = {k: 'jax.numpy.' + v for k, v in _known_constants_numpy.items()}
class JaxPrinter(NumPyPrinter):
"""
JAX printer which handles vectorized piecewise functions,
logical operators, etc.
"""
_module = "jax.numpy"
_kf = _jax_known_functions
_kc = _jax_known_constants
def __init__(self, settings=None):
super().__init__(settings=settings)
# These need specific override to allow for the lack of "jax.numpy.reduce"
def _print_And(self, expr):
"Logical And printer"
return "{}({}.asarray([{}]), axis=0)".format(
self._module_format(self._module + ".all"),
self._module_format(self._module),
",".join(self._print(i) for i in expr.args),
)
def _print_Or(self, expr):
"Logical Or printer"
return "{}({}.asarray([{}]), axis=0)".format(
self._module_format(self._module + ".any"),
self._module_format(self._module),
",".join(self._print(i) for i in expr.args),
)
for func in _jax_known_functions:
setattr(JaxPrinter, f'_print_{func}', _print_known_func)
for const in _jax_known_constants:
setattr(JaxPrinter, f'_print_{const}', _print_known_const)
|
013da8d845ca4ad34ff3255860f9288cf763b665c86f14b78f811aad7b33fc9d | from typing import Any, Dict as tDict, Set as tSet, Tuple as tTuple
from functools import wraps
from sympy.core import Add, Expr, Mul, Pow, S, sympify, Float
from sympy.core.basic import Basic
from sympy.core.expr import UnevaluatedExpr
from sympy.core.function import Lambda
from sympy.core.mul import _keep_coeff
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Symbol
from sympy.functions.elementary.complexes import re
from sympy.printing.str import StrPrinter
from sympy.printing.precedence import precedence, PRECEDENCE
class requires:
""" Decorator for registering requirements on print methods. """
def __init__(self, **kwargs):
self._req = kwargs
def __call__(self, method):
def _method_wrapper(self_, *args, **kwargs):
for k, v in self._req.items():
getattr(self_, k).update(v)
return method(self_, *args, **kwargs)
return wraps(method)(_method_wrapper)
class AssignmentError(Exception):
"""
Raised if an assignment variable for a loop is missing.
"""
pass
def _convert_python_lists(arg):
if isinstance(arg, list):
from sympy.codegen.abstract_nodes import List
return List(*(_convert_python_lists(e) for e in arg))
elif isinstance(arg, tuple):
return tuple(_convert_python_lists(e) for e in arg)
else:
return arg
class CodePrinter(StrPrinter):
"""
The base class for code-printing subclasses.
"""
_operators = {
'and': '&&',
'or': '||',
'not': '!',
}
_default_settings = {
'order': None,
'full_prec': 'auto',
'error_on_reserved': False,
'reserved_word_suffix': '_',
'human': True,
'inline': False,
'allow_unknown_functions': False,
} # type: tDict[str, Any]
# Functions which are "simple" to rewrite to other functions that
# may be supported
# function_to_rewrite : (function_to_rewrite_to, iterable_with_other_functions_required)
_rewriteable_functions = {
'cot': ('tan', []),
'csc': ('sin', []),
'sec': ('cos', []),
'acot': ('atan', []),
'acsc': ('asin', []),
'asec': ('acos', []),
'coth': ('exp', []),
'csch': ('exp', []),
'sech': ('exp', []),
'acoth': ('log', []),
'acsch': ('log', []),
'asech': ('log', []),
'catalan': ('gamma', []),
'fibonacci': ('sqrt', []),
'lucas': ('sqrt', []),
'beta': ('gamma', []),
'sinc': ('sin', ['Piecewise']),
'Mod': ('floor', []),
'factorial': ('gamma', []),
'factorial2': ('gamma', ['Piecewise']),
'subfactorial': ('uppergamma', []),
'RisingFactorial': ('gamma', ['Piecewise']),
'FallingFactorial': ('gamma', ['Piecewise']),
'binomial': ('gamma', []),
'frac': ('floor', []),
'Max': ('Piecewise', []),
'Min': ('Piecewise', []),
'Heaviside': ('Piecewise', []),
'erf2': ('erf', []),
'erfc': ('erf', []),
'Li': ('li', []),
'Ei': ('li', []),
'dirichlet_eta': ('zeta', []),
'riemann_xi': ('zeta', ['gamma']),
}
def __init__(self, settings=None):
super().__init__(settings=settings)
if not hasattr(self, 'reserved_words'):
self.reserved_words = set()
def _handle_UnevaluatedExpr(self, expr):
return expr.replace(re, lambda arg: arg if isinstance(
arg, UnevaluatedExpr) and arg.args[0].is_real else re(arg))
def doprint(self, expr, assign_to=None):
"""
Print the expression as code.
Parameters
----------
expr : Expression
The expression to be printed.
assign_to : Symbol, string, MatrixSymbol, list of strings or Symbols (optional)
If provided, the printed code will set the expression to a variable or multiple variables
with the name or names given in ``assign_to``.
"""
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.codegen.ast import CodeBlock, Assignment
def _handle_assign_to(expr, assign_to):
if assign_to is None:
return sympify(expr)
if isinstance(assign_to, (list, tuple)):
if len(expr) != len(assign_to):
raise ValueError('Failed to assign an expression of length {} to {} variables'.format(len(expr), len(assign_to)))
return CodeBlock(*[_handle_assign_to(lhs, rhs) for lhs, rhs in zip(expr, assign_to)])
if isinstance(assign_to, str):
if expr.is_Matrix:
assign_to = MatrixSymbol(assign_to, *expr.shape)
else:
assign_to = Symbol(assign_to)
elif not isinstance(assign_to, Basic):
raise TypeError("{} cannot assign to object of type {}".format(
type(self).__name__, type(assign_to)))
return Assignment(assign_to, expr)
expr = _convert_python_lists(expr)
expr = _handle_assign_to(expr, assign_to)
# Remove re(...) nodes due to UnevaluatedExpr.is_real always is None:
expr = self._handle_UnevaluatedExpr(expr)
# keep a set of expressions that are not strictly translatable to Code
# and number constants that must be declared and initialized
self._not_supported = set()
self._number_symbols = set() # type: tSet[tTuple[Expr, Float]]
lines = self._print(expr).splitlines()
# format the output
if self._settings["human"]:
frontlines = []
if self._not_supported:
frontlines.append(self._get_comment(
"Not supported in {}:".format(self.language)))
for expr in sorted(self._not_supported, key=str):
frontlines.append(self._get_comment(type(expr).__name__))
for name, value in sorted(self._number_symbols, key=str):
frontlines.append(self._declare_number_const(name, value))
lines = frontlines + lines
lines = self._format_code(lines)
result = "\n".join(lines)
else:
lines = self._format_code(lines)
num_syms = {(k, self._print(v)) for k, v in self._number_symbols}
result = (num_syms, self._not_supported, "\n".join(lines))
self._not_supported = set()
self._number_symbols = set()
return result
def _doprint_loops(self, expr, assign_to=None):
# Here we print an expression that contains Indexed objects, they
# correspond to arrays in the generated code. The low-level implementation
# involves looping over array elements and possibly storing results in temporary
# variables or accumulate it in the assign_to object.
if self._settings.get('contract', True):
from sympy.tensor import get_contraction_structure
# Setup loops over non-dummy indices -- all terms need these
indices = self._get_expression_indices(expr, assign_to)
# Setup loops over dummy indices -- each term needs separate treatment
dummies = get_contraction_structure(expr)
else:
indices = []
dummies = {None: (expr,)}
openloop, closeloop = self._get_loop_opening_ending(indices)
# terms with no summations first
if None in dummies:
text = StrPrinter.doprint(self, Add(*dummies[None]))
else:
# If all terms have summations we must initialize array to Zero
text = StrPrinter.doprint(self, 0)
# skip redundant assignments (where lhs == rhs)
lhs_printed = self._print(assign_to)
lines = []
if text != lhs_printed:
lines.extend(openloop)
if assign_to is not None:
text = self._get_statement("%s = %s" % (lhs_printed, text))
lines.append(text)
lines.extend(closeloop)
# then terms with summations
for d in dummies:
if isinstance(d, tuple):
indices = self._sort_optimized(d, expr)
openloop_d, closeloop_d = self._get_loop_opening_ending(
indices)
for term in dummies[d]:
if term in dummies and not ([list(f.keys()) for f in dummies[term]]
== [[None] for f in dummies[term]]):
# If one factor in the term has it's own internal
# contractions, those must be computed first.
# (temporary variables?)
raise NotImplementedError(
"FIXME: no support for contractions in factor yet")
else:
# We need the lhs expression as an accumulator for
# the loops, i.e
#
# for (int d=0; d < dim; d++){
# lhs[] = lhs[] + term[][d]
# } ^.................. the accumulator
#
# We check if the expression already contains the
# lhs, and raise an exception if it does, as that
# syntax is currently undefined. FIXME: What would be
# a good interpretation?
if assign_to is None:
raise AssignmentError(
"need assignment variable for loops")
if term.has(assign_to):
raise ValueError("FIXME: lhs present in rhs,\
this is undefined in CodePrinter")
lines.extend(openloop)
lines.extend(openloop_d)
text = "%s = %s" % (lhs_printed, StrPrinter.doprint(
self, assign_to + term))
lines.append(self._get_statement(text))
lines.extend(closeloop_d)
lines.extend(closeloop)
return "\n".join(lines)
def _get_expression_indices(self, expr, assign_to):
from sympy.tensor import get_indices
rinds, junk = get_indices(expr)
linds, junk = get_indices(assign_to)
# support broadcast of scalar
if linds and not rinds:
rinds = linds
if rinds != linds:
raise ValueError("lhs indices must match non-dummy"
" rhs indices in %s" % expr)
return self._sort_optimized(rinds, assign_to)
def _sort_optimized(self, indices, expr):
from sympy.tensor.indexed import Indexed
if not indices:
return []
# determine optimized loop order by giving a score to each index
# the index with the highest score are put in the innermost loop.
score_table = {}
for i in indices:
score_table[i] = 0
arrays = expr.atoms(Indexed)
for arr in arrays:
for p, ind in enumerate(arr.indices):
try:
score_table[ind] += self._rate_index_position(p)
except KeyError:
pass
return sorted(indices, key=lambda x: score_table[x])
def _rate_index_position(self, p):
"""function to calculate score based on position among indices
This method is used to sort loops in an optimized order, see
CodePrinter._sort_optimized()
"""
raise NotImplementedError("This function must be implemented by "
"subclass of CodePrinter.")
def _get_statement(self, codestring):
"""Formats a codestring with the proper line ending."""
raise NotImplementedError("This function must be implemented by "
"subclass of CodePrinter.")
def _get_comment(self, text):
"""Formats a text string as a comment."""
raise NotImplementedError("This function must be implemented by "
"subclass of CodePrinter.")
def _declare_number_const(self, name, value):
"""Declare a numeric constant at the top of a function"""
raise NotImplementedError("This function must be implemented by "
"subclass of CodePrinter.")
def _format_code(self, lines):
"""Take in a list of lines of code, and format them accordingly.
This may include indenting, wrapping long lines, etc..."""
raise NotImplementedError("This function must be implemented by "
"subclass of CodePrinter.")
def _get_loop_opening_ending(self, indices):
"""Returns a tuple (open_lines, close_lines) containing lists
of codelines"""
raise NotImplementedError("This function must be implemented by "
"subclass of CodePrinter.")
def _print_Dummy(self, expr):
if expr.name.startswith('Dummy_'):
return '_' + expr.name
else:
return '%s_%d' % (expr.name, expr.dummy_index)
def _print_CodeBlock(self, expr):
return '\n'.join([self._print(i) for i in expr.args])
def _print_String(self, string):
return str(string)
def _print_QuotedString(self, arg):
return '"%s"' % arg.text
def _print_Comment(self, string):
return self._get_comment(str(string))
def _print_Assignment(self, expr):
from sympy.codegen.ast import Assignment
from sympy.functions.elementary.piecewise import Piecewise
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.tensor.indexed import IndexedBase
lhs = expr.lhs
rhs = expr.rhs
# We special case assignments that take multiple lines
if isinstance(expr.rhs, Piecewise):
# Here we modify Piecewise so each expression is now
# an Assignment, and then continue on the print.
expressions = []
conditions = []
for (e, c) in rhs.args:
expressions.append(Assignment(lhs, e))
conditions.append(c)
temp = Piecewise(*zip(expressions, conditions))
return self._print(temp)
elif isinstance(lhs, MatrixSymbol):
# Here we form an Assignment for each element in the array,
# printing each one.
lines = []
for (i, j) in self._traverse_matrix_indices(lhs):
temp = Assignment(lhs[i, j], rhs[i, j])
code0 = self._print(temp)
lines.append(code0)
return "\n".join(lines)
elif self._settings.get("contract", False) and (lhs.has(IndexedBase) or
rhs.has(IndexedBase)):
# Here we check if there is looping to be done, and if so
# print the required loops.
return self._doprint_loops(rhs, lhs)
else:
lhs_code = self._print(lhs)
rhs_code = self._print(rhs)
return self._get_statement("%s = %s" % (lhs_code, rhs_code))
def _print_AugmentedAssignment(self, expr):
lhs_code = self._print(expr.lhs)
rhs_code = self._print(expr.rhs)
return self._get_statement("{} {} {}".format(
*map(lambda arg: self._print(arg),
[lhs_code, expr.op, rhs_code])))
def _print_FunctionCall(self, expr):
return '%s(%s)' % (
expr.name,
', '.join(map(lambda arg: self._print(arg),
expr.function_args)))
def _print_Variable(self, expr):
return self._print(expr.symbol)
def _print_Symbol(self, expr):
name = super()._print_Symbol(expr)
if name in self.reserved_words:
if self._settings['error_on_reserved']:
msg = ('This expression includes the symbol "{}" which is a '
'reserved keyword in this language.')
raise ValueError(msg.format(name))
return name + self._settings['reserved_word_suffix']
else:
return name
def _can_print(self, name):
""" Check if function ``name`` is either a known function or has its own
printing method. Used to check if rewriting is possible."""
return name in self.known_functions or getattr(self, '_print_{}'.format(name), False)
def _print_Function(self, expr):
if expr.func.__name__ in self.known_functions:
cond_func = self.known_functions[expr.func.__name__]
if isinstance(cond_func, str):
return "%s(%s)" % (cond_func, self.stringify(expr.args, ", "))
else:
for cond, func in cond_func:
if cond(*expr.args):
break
if func is not None:
try:
return func(*[self.parenthesize(item, 0) for item in expr.args])
except TypeError:
return "%s(%s)" % (func, self.stringify(expr.args, ", "))
elif hasattr(expr, '_imp_') and isinstance(expr._imp_, Lambda):
# inlined function
return self._print(expr._imp_(*expr.args))
elif expr.func.__name__ in self._rewriteable_functions:
# Simple rewrite to supported function possible
target_f, required_fs = self._rewriteable_functions[expr.func.__name__]
if self._can_print(target_f) and all(self._can_print(f) for f in required_fs):
return self._print(expr.rewrite(target_f))
if expr.is_Function and self._settings.get('allow_unknown_functions', False):
return '%s(%s)' % (self._print(expr.func), ', '.join(map(self._print, expr.args)))
else:
return self._print_not_supported(expr)
_print_Expr = _print_Function
# Don't inherit the str-printer method for Heaviside to the code printers
_print_Heaviside = None
def _print_NumberSymbol(self, expr):
if self._settings.get("inline", False):
return self._print(Float(expr.evalf(self._settings["precision"])))
else:
# A Number symbol that is not implemented here or with _printmethod
# is registered and evaluated
self._number_symbols.add((expr,
Float(expr.evalf(self._settings["precision"]))))
return str(expr)
def _print_Catalan(self, expr):
return self._print_NumberSymbol(expr)
def _print_EulerGamma(self, expr):
return self._print_NumberSymbol(expr)
def _print_GoldenRatio(self, expr):
return self._print_NumberSymbol(expr)
def _print_TribonacciConstant(self, expr):
return self._print_NumberSymbol(expr)
def _print_Exp1(self, expr):
return self._print_NumberSymbol(expr)
def _print_Pi(self, expr):
return self._print_NumberSymbol(expr)
def _print_And(self, expr):
PREC = precedence(expr)
return (" %s " % self._operators['and']).join(self.parenthesize(a, PREC)
for a in sorted(expr.args, key=default_sort_key))
def _print_Or(self, expr):
PREC = precedence(expr)
return (" %s " % self._operators['or']).join(self.parenthesize(a, PREC)
for a in sorted(expr.args, key=default_sort_key))
def _print_Xor(self, expr):
if self._operators.get('xor') is None:
return self._print(expr.to_nnf())
PREC = precedence(expr)
return (" %s " % self._operators['xor']).join(self.parenthesize(a, PREC)
for a in expr.args)
def _print_Equivalent(self, expr):
if self._operators.get('equivalent') is None:
return self._print(expr.to_nnf())
PREC = precedence(expr)
return (" %s " % self._operators['equivalent']).join(self.parenthesize(a, PREC)
for a in expr.args)
def _print_Not(self, expr):
PREC = precedence(expr)
return self._operators['not'] + self.parenthesize(expr.args[0], PREC)
def _print_BooleanFunction(self, expr):
return self._print(expr.to_nnf())
def _print_Mul(self, expr):
prec = precedence(expr)
c, e = expr.as_coeff_Mul()
if c < 0:
expr = _keep_coeff(-c, e)
sign = "-"
else:
sign = ""
a = [] # items in the numerator
b = [] # items that are in the denominator (if any)
pow_paren = [] # Will collect all pow with more than one base element and exp = -1
if self.order not in ('old', 'none'):
args = expr.as_ordered_factors()
else:
# use make_args in case expr was something like -x -> x
args = Mul.make_args(expr)
# Gather args for numerator/denominator
for item in args:
if item.is_commutative and item.is_Pow and item.exp.is_Rational and item.exp.is_negative:
if item.exp != -1:
b.append(Pow(item.base, -item.exp, evaluate=False))
else:
if len(item.args[0].args) != 1 and isinstance(item.base, Mul): # To avoid situations like #14160
pow_paren.append(item)
b.append(Pow(item.base, -item.exp))
else:
a.append(item)
a = a or [S.One]
if len(a) == 1 and sign == "-":
# Unary minus does not have a SymPy class, and hence there's no
# precedence weight associated with it, Python's unary minus has
# an operator precedence between multiplication and exponentiation,
# so we use this to compute a weight.
a_str = [self.parenthesize(a[0], 0.5*(PRECEDENCE["Pow"]+PRECEDENCE["Mul"]))]
else:
a_str = [self.parenthesize(x, prec) for x in a]
b_str = [self.parenthesize(x, prec) for x in b]
# To parenthesize Pow with exp = -1 and having more than one Symbol
for item in pow_paren:
if item.base in b:
b_str[b.index(item.base)] = "(%s)" % b_str[b.index(item.base)]
if not b:
return sign + '*'.join(a_str)
elif len(b) == 1:
return sign + '*'.join(a_str) + "/" + b_str[0]
else:
return sign + '*'.join(a_str) + "/(%s)" % '*'.join(b_str)
def _print_not_supported(self, expr):
try:
self._not_supported.add(expr)
except TypeError:
# not hashable
pass
return self.emptyPrinter(expr)
# The following can not be simply translated into C or Fortran
_print_Basic = _print_not_supported
_print_ComplexInfinity = _print_not_supported
_print_Derivative = _print_not_supported
_print_ExprCondPair = _print_not_supported
_print_GeometryEntity = _print_not_supported
_print_Infinity = _print_not_supported
_print_Integral = _print_not_supported
_print_Interval = _print_not_supported
_print_AccumulationBounds = _print_not_supported
_print_Limit = _print_not_supported
_print_MatrixBase = _print_not_supported
_print_DeferredVector = _print_not_supported
_print_NaN = _print_not_supported
_print_NegativeInfinity = _print_not_supported
_print_Order = _print_not_supported
_print_RootOf = _print_not_supported
_print_RootsOf = _print_not_supported
_print_RootSum = _print_not_supported
_print_Uniform = _print_not_supported
_print_Unit = _print_not_supported
_print_Wild = _print_not_supported
_print_WildFunction = _print_not_supported
_print_Relational = _print_not_supported
# Code printer functions. These are included in this file so that they can be
# imported in the top-level __init__.py without importing the sympy.codegen
# module.
def ccode(expr, assign_to=None, standard='c99', **settings):
"""Converts an expr to a string of c code
Parameters
==========
expr : Expr
A SymPy expression to be converted.
assign_to : optional
When given, the argument is used as the name of the variable to which
the expression is assigned. Can be a string, ``Symbol``,
``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of
line-wrapping, or for expressions that generate multi-line statements.
standard : str, optional
String specifying the standard. If your compiler supports a more modern
standard you may set this to 'c99' to allow the printer to use more math
functions. [default='c89'].
precision : integer, optional
The precision for numbers such as pi [default=17].
user_functions : dict, optional
A dictionary where the keys are string representations of either
``FunctionClass`` or ``UndefinedFunction`` instances and the values
are their desired C string representations. Alternatively, the
dictionary value can be a list of tuples i.e. [(argument_test,
cfunction_string)] or [(argument_test, cfunction_formater)]. See below
for examples.
dereference : iterable, optional
An iterable of symbols that should be dereferenced in the printed code
expression. These would be values passed by address to the function.
For example, if ``dereference=[a]``, the resulting code would print
``(*a)`` instead of ``a``.
human : bool, optional
If True, the result is a single string that may contain some constant
declarations for the number symbols. If False, the same information is
returned in a tuple of (symbols_to_declare, not_supported_functions,
code_text). [default=True].
contract: bool, optional
If True, ``Indexed`` instances are assumed to obey tensor contraction
rules and the corresponding nested loops over indices are generated.
Setting contract=False will not generate loops, instead the user is
responsible to provide values for the indices in the code.
[default=True].
Examples
========
>>> from sympy import ccode, symbols, Rational, sin, ceiling, Abs, Function
>>> x, tau = symbols("x, tau")
>>> expr = (2*tau)**Rational(7, 2)
>>> ccode(expr)
'8*M_SQRT2*pow(tau, 7.0/2.0)'
>>> ccode(expr, math_macros={})
'8*sqrt(2)*pow(tau, 7.0/2.0)'
>>> ccode(sin(x), assign_to="s")
's = sin(x);'
>>> from sympy.codegen.ast import real, float80
>>> ccode(expr, type_aliases={real: float80})
'8*M_SQRT2l*powl(tau, 7.0L/2.0L)'
Simple custom printing can be defined for certain types by passing a
dictionary of {"type" : "function"} to the ``user_functions`` kwarg.
Alternatively, the dictionary value can be a list of tuples i.e.
[(argument_test, cfunction_string)].
>>> custom_functions = {
... "ceiling": "CEIL",
... "Abs": [(lambda x: not x.is_integer, "fabs"),
... (lambda x: x.is_integer, "ABS")],
... "func": "f"
... }
>>> func = Function('func')
>>> ccode(func(Abs(x) + ceiling(x)), standard='C89', user_functions=custom_functions)
'f(fabs(x) + CEIL(x))'
or if the C-function takes a subset of the original arguments:
>>> ccode(2**x + 3**x, standard='C99', user_functions={'Pow': [
... (lambda b, e: b == 2, lambda b, e: 'exp2(%s)' % e),
... (lambda b, e: b != 2, 'pow')]})
'exp2(x) + pow(3, x)'
``Piecewise`` expressions are converted into conditionals. If an
``assign_to`` variable is provided an if statement is created, otherwise
the ternary operator is used. Note that if the ``Piecewise`` lacks a
default term, represented by ``(expr, True)`` then an error will be thrown.
This is to prevent generating an expression that may not evaluate to
anything.
>>> from sympy import Piecewise
>>> expr = Piecewise((x + 1, x > 0), (x, True))
>>> print(ccode(expr, tau, standard='C89'))
if (x > 0) {
tau = x + 1;
}
else {
tau = x;
}
Support for loops is provided through ``Indexed`` types. With
``contract=True`` these expressions will be turned into loops, whereas
``contract=False`` will just print the assignment expression that should be
looped over:
>>> from sympy import Eq, IndexedBase, Idx
>>> len_y = 5
>>> y = IndexedBase('y', shape=(len_y,))
>>> t = IndexedBase('t', shape=(len_y,))
>>> Dy = IndexedBase('Dy', shape=(len_y-1,))
>>> i = Idx('i', len_y-1)
>>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i]))
>>> ccode(e.rhs, assign_to=e.lhs, contract=False, standard='C89')
'Dy[i] = (y[i + 1] - y[i])/(t[i + 1] - t[i]);'
Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions
must be provided to ``assign_to``. Note that any expression that can be
generated normally can also exist inside a Matrix:
>>> from sympy import Matrix, MatrixSymbol
>>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)])
>>> A = MatrixSymbol('A', 3, 1)
>>> print(ccode(mat, A, standard='C89'))
A[0] = pow(x, 2);
if (x > 0) {
A[1] = x + 1;
}
else {
A[1] = x;
}
A[2] = sin(x);
"""
from sympy.printing.c import c_code_printers
return c_code_printers[standard.lower()](settings).doprint(expr, assign_to)
def print_ccode(expr, **settings):
"""Prints C representation of the given expression."""
print(ccode(expr, **settings))
def fcode(expr, assign_to=None, **settings):
"""Converts an expr to a string of fortran code
Parameters
==========
expr : Expr
A SymPy expression to be converted.
assign_to : optional
When given, the argument is used as the name of the variable to which
the expression is assigned. Can be a string, ``Symbol``,
``MatrixSymbol``, or ``Indexed`` type. This is helpful in case of
line-wrapping, or for expressions that generate multi-line statements.
precision : integer, optional
DEPRECATED. Use type_mappings instead. The precision for numbers such
as pi [default=17].
user_functions : dict, optional
A dictionary where keys are ``FunctionClass`` instances and values are
their string representations. Alternatively, the dictionary value can
be a list of tuples i.e. [(argument_test, cfunction_string)]. See below
for examples.
human : bool, optional
If True, the result is a single string that may contain some constant
declarations for the number symbols. If False, the same information is
returned in a tuple of (symbols_to_declare, not_supported_functions,
code_text). [default=True].
contract: bool, optional
If True, ``Indexed`` instances are assumed to obey tensor contraction
rules and the corresponding nested loops over indices are generated.
Setting contract=False will not generate loops, instead the user is
responsible to provide values for the indices in the code.
[default=True].
source_format : optional
The source format can be either 'fixed' or 'free'. [default='fixed']
standard : integer, optional
The Fortran standard to be followed. This is specified as an integer.
Acceptable standards are 66, 77, 90, 95, 2003, and 2008. Default is 77.
Note that currently the only distinction internally is between
standards before 95, and those 95 and after. This may change later as
more features are added.
name_mangling : bool, optional
If True, then the variables that would become identical in
case-insensitive Fortran are mangled by appending different number
of ``_`` at the end. If False, SymPy Will not interfere with naming of
variables. [default=True]
Examples
========
>>> from sympy import fcode, symbols, Rational, sin, ceiling, floor
>>> x, tau = symbols("x, tau")
>>> fcode((2*tau)**Rational(7, 2))
' 8*sqrt(2.0d0)*tau**(7.0d0/2.0d0)'
>>> fcode(sin(x), assign_to="s")
' s = sin(x)'
Custom printing can be defined for certain types by passing a dictionary of
"type" : "function" to the ``user_functions`` kwarg. Alternatively, the
dictionary value can be a list of tuples i.e. [(argument_test,
cfunction_string)].
>>> custom_functions = {
... "ceiling": "CEIL",
... "floor": [(lambda x: not x.is_integer, "FLOOR1"),
... (lambda x: x.is_integer, "FLOOR2")]
... }
>>> fcode(floor(x) + ceiling(x), user_functions=custom_functions)
' CEIL(x) + FLOOR1(x)'
``Piecewise`` expressions are converted into conditionals. If an
``assign_to`` variable is provided an if statement is created, otherwise
the ternary operator is used. Note that if the ``Piecewise`` lacks a
default term, represented by ``(expr, True)`` then an error will be thrown.
This is to prevent generating an expression that may not evaluate to
anything.
>>> from sympy import Piecewise
>>> expr = Piecewise((x + 1, x > 0), (x, True))
>>> print(fcode(expr, tau))
if (x > 0) then
tau = x + 1
else
tau = x
end if
Support for loops is provided through ``Indexed`` types. With
``contract=True`` these expressions will be turned into loops, whereas
``contract=False`` will just print the assignment expression that should be
looped over:
>>> from sympy import Eq, IndexedBase, Idx
>>> len_y = 5
>>> y = IndexedBase('y', shape=(len_y,))
>>> t = IndexedBase('t', shape=(len_y,))
>>> Dy = IndexedBase('Dy', shape=(len_y-1,))
>>> i = Idx('i', len_y-1)
>>> e=Eq(Dy[i], (y[i+1]-y[i])/(t[i+1]-t[i]))
>>> fcode(e.rhs, assign_to=e.lhs, contract=False)
' Dy(i) = (y(i + 1) - y(i))/(t(i + 1) - t(i))'
Matrices are also supported, but a ``MatrixSymbol`` of the same dimensions
must be provided to ``assign_to``. Note that any expression that can be
generated normally can also exist inside a Matrix:
>>> from sympy import Matrix, MatrixSymbol
>>> mat = Matrix([x**2, Piecewise((x + 1, x > 0), (x, True)), sin(x)])
>>> A = MatrixSymbol('A', 3, 1)
>>> print(fcode(mat, A))
A(1, 1) = x**2
if (x > 0) then
A(2, 1) = x + 1
else
A(2, 1) = x
end if
A(3, 1) = sin(x)
"""
from sympy.printing.fortran import FCodePrinter
return FCodePrinter(settings).doprint(expr, assign_to)
def print_fcode(expr, **settings):
"""Prints the Fortran representation of the given expression.
See fcode for the meaning of the optional arguments.
"""
print(fcode(expr, **settings))
def cxxcode(expr, assign_to=None, standard='c++11', **settings):
""" C++ equivalent of :func:`~.ccode`. """
from sympy.printing.cxx import cxx_code_printers
return cxx_code_printers[standard.lower()](settings).doprint(expr, assign_to)
|
2fefa74cc87c42633bd2810841ebb4ef03b11c47fdef0bfb3fe1f045a2e843e8 | """Base class for all the objects in SymPy"""
from __future__ import annotations
from collections import defaultdict
from collections.abc import Mapping
from itertools import chain, zip_longest
from .assumptions import ManagedProperties
from .cache import cacheit
from .core import BasicMeta
from .sympify import _sympify, sympify, SympifyError, _external_converter
from .sorting import ordered
from .kind import Kind, UndefinedKind
from ._print_helpers import Printable
from sympy.utilities.decorator import deprecated
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.iterables import iterable, numbered_symbols
from sympy.utilities.misc import filldedent, func_name
from inspect import getmro
def as_Basic(expr):
"""Return expr as a Basic instance using strict sympify
or raise a TypeError; this is just a wrapper to _sympify,
raising a TypeError instead of a SympifyError."""
try:
return _sympify(expr)
except SympifyError:
raise TypeError(
'Argument must be a Basic object, not `%s`' % func_name(
expr))
class Basic(Printable, metaclass=ManagedProperties):
"""
Base class for all SymPy objects.
Notes and conventions
=====================
1) Always use ``.args``, when accessing parameters of some instance:
>>> from sympy import cot
>>> from sympy.abc import x, y
>>> cot(x).args
(x,)
>>> cot(x).args[0]
x
>>> (x*y).args
(x, y)
>>> (x*y).args[1]
y
2) Never use internal methods or variables (the ones prefixed with ``_``):
>>> cot(x)._args # do not use this, use cot(x).args instead
(x,)
3) By "SymPy object" we mean something that can be returned by
``sympify``. But not all objects one encounters using SymPy are
subclasses of Basic. For example, mutable objects are not:
>>> from sympy import Basic, Matrix, sympify
>>> A = Matrix([[1, 2], [3, 4]]).as_mutable()
>>> isinstance(A, Basic)
False
>>> B = sympify(A)
>>> isinstance(B, Basic)
True
"""
__slots__ = ('_mhash', # hash value
'_args', # arguments
'_assumptions'
)
_args: tuple[Basic, ...]
_mhash: int | None
# To be overridden with True in the appropriate subclasses
is_number = False
is_Atom = False
is_Symbol = False
is_symbol = False
is_Indexed = False
is_Dummy = False
is_Wild = False
is_Function = False
is_Add = False
is_Mul = False
is_Pow = False
is_Number = False
is_Float = False
is_Rational = False
is_Integer = False
is_NumberSymbol = False
is_Order = False
is_Derivative = False
is_Piecewise = False
is_Poly = False
is_AlgebraicNumber = False
is_Relational = False
is_Equality = False
is_Boolean = False
is_Not = False
is_Matrix = False
is_Vector = False
is_Point = False
is_MatAdd = False
is_MatMul = False
is_real: bool | None
is_zero: bool | None
is_negative: bool | None
is_commutative: bool | None
kind: Kind = UndefinedKind
def __new__(cls, *args):
obj = object.__new__(cls)
obj._assumptions = cls.default_assumptions
obj._mhash = None # will be set by __hash__ method.
obj._args = args # all items in args must be Basic objects
return obj
def copy(self):
return self.func(*self.args)
def __getnewargs__(self):
return self.args
def __getstate__(self):
return None
def __setstate__(self, state):
for name, value in state.items():
setattr(self, name, value)
def __reduce_ex__(self, protocol):
if protocol < 2:
msg = "Only pickle protocol 2 or higher is supported by SymPy"
raise NotImplementedError(msg)
return super().__reduce_ex__(protocol)
def __hash__(self) -> int:
# hash cannot be cached using cache_it because infinite recurrence
# occurs as hash is needed for setting cache dictionary keys
h = self._mhash
if h is None:
h = hash((type(self).__name__,) + self._hashable_content())
self._mhash = h
return h
def _hashable_content(self):
"""Return a tuple of information about self that can be used to
compute the hash. If a class defines additional attributes,
like ``name`` in Symbol, then this method should be updated
accordingly to return such relevant attributes.
Defining more than _hashable_content is necessary if __eq__ has
been defined by a class. See note about this in Basic.__eq__."""
return self._args
@property
def assumptions0(self):
"""
Return object `type` assumptions.
For example:
Symbol('x', real=True)
Symbol('x', integer=True)
are different objects. In other words, besides Python type (Symbol in
this case), the initial assumptions are also forming their typeinfo.
Examples
========
>>> from sympy import Symbol
>>> from sympy.abc import x
>>> x.assumptions0
{'commutative': True}
>>> x = Symbol("x", positive=True)
>>> x.assumptions0
{'commutative': True, 'complex': True, 'extended_negative': False,
'extended_nonnegative': True, 'extended_nonpositive': False,
'extended_nonzero': True, 'extended_positive': True, 'extended_real':
True, 'finite': True, 'hermitian': True, 'imaginary': False,
'infinite': False, 'negative': False, 'nonnegative': True,
'nonpositive': False, 'nonzero': True, 'positive': True, 'real':
True, 'zero': False}
"""
return {}
def compare(self, other):
"""
Return -1, 0, 1 if the object is smaller, equal, or greater than other.
Not in the mathematical sense. If the object is of a different type
from the "other" then their classes are ordered according to
the sorted_classes list.
Examples
========
>>> from sympy.abc import x, y
>>> x.compare(y)
-1
>>> x.compare(x)
0
>>> y.compare(x)
1
"""
# all redefinitions of __cmp__ method should start with the
# following lines:
if self is other:
return 0
n1 = self.__class__
n2 = other.__class__
c = (n1 > n2) - (n1 < n2)
if c:
return c
#
st = self._hashable_content()
ot = other._hashable_content()
c = (len(st) > len(ot)) - (len(st) < len(ot))
if c:
return c
for l, r in zip(st, ot):
l = Basic(*l) if isinstance(l, frozenset) else l
r = Basic(*r) if isinstance(r, frozenset) else r
if isinstance(l, Basic):
c = l.compare(r)
else:
c = (l > r) - (l < r)
if c:
return c
return 0
@staticmethod
def _compare_pretty(a, b):
from sympy.series.order import Order
if isinstance(a, Order) and not isinstance(b, Order):
return 1
if not isinstance(a, Order) and isinstance(b, Order):
return -1
if a.is_Rational and b.is_Rational:
l = a.p * b.q
r = b.p * a.q
return (l > r) - (l < r)
else:
from .symbol import Wild
p1, p2, p3 = Wild("p1"), Wild("p2"), Wild("p3")
r_a = a.match(p1 * p2**p3)
if r_a and p3 in r_a:
a3 = r_a[p3]
r_b = b.match(p1 * p2**p3)
if r_b and p3 in r_b:
b3 = r_b[p3]
c = Basic.compare(a3, b3)
if c != 0:
return c
return Basic.compare(a, b)
@classmethod
def fromiter(cls, args, **assumptions):
"""
Create a new object from an iterable.
This is a convenience function that allows one to create objects from
any iterable, without having to convert to a list or tuple first.
Examples
========
>>> from sympy import Tuple
>>> Tuple.fromiter(i for i in range(5))
(0, 1, 2, 3, 4)
"""
return cls(*tuple(args), **assumptions)
@classmethod
def class_key(cls):
"""Nice order of classes. """
return 5, 0, cls.__name__
@cacheit
def sort_key(self, order=None):
"""
Return a sort key.
Examples
========
>>> from sympy import S, I
>>> sorted([S(1)/2, I, -I], key=lambda x: x.sort_key())
[1/2, -I, I]
>>> S("[x, 1/x, 1/x**2, x**2, x**(1/2), x**(1/4), x**(3/2)]")
[x, 1/x, x**(-2), x**2, sqrt(x), x**(1/4), x**(3/2)]
>>> sorted(_, key=lambda x: x.sort_key())
[x**(-2), 1/x, x**(1/4), sqrt(x), x, x**(3/2), x**2]
"""
# XXX: remove this when issue 5169 is fixed
def inner_key(arg):
if isinstance(arg, Basic):
return arg.sort_key(order)
else:
return arg
args = self._sorted_args
args = len(args), tuple([inner_key(arg) for arg in args])
return self.class_key(), args, S.One.sort_key(), S.One
def _do_eq_sympify(self, other):
"""Returns a boolean indicating whether a == b when either a
or b is not a Basic. This is only done for types that were either
added to `converter` by a 3rd party or when the object has `_sympy_`
defined. This essentially reuses the code in `_sympify` that is
specific for this use case. Non-user defined types that are meant
to work with SymPy should be handled directly in the __eq__ methods
of the `Basic` classes it could equate to and not be converted. Note
that after conversion, `==` is used again since it is not
neccesarily clear whether `self` or `other`'s __eq__ method needs
to be used."""
for superclass in type(other).__mro__:
conv = _external_converter.get(superclass)
if conv is not None:
return self == conv(other)
if hasattr(other, '_sympy_'):
return self == other._sympy_()
return NotImplemented
def __eq__(self, other):
"""Return a boolean indicating whether a == b on the basis of
their symbolic trees.
This is the same as a.compare(b) == 0 but faster.
Notes
=====
If a class that overrides __eq__() needs to retain the
implementation of __hash__() from a parent class, the
interpreter must be told this explicitly by setting
__hash__ : Callable[[object], int] = <ParentClass>.__hash__.
Otherwise the inheritance of __hash__() will be blocked,
just as if __hash__ had been explicitly set to None.
References
==========
from http://docs.python.org/dev/reference/datamodel.html#object.__hash__
"""
if self is other:
return True
if not isinstance(other, Basic):
return self._do_eq_sympify(other)
# check for pure number expr
if not (self.is_Number and other.is_Number) and (
type(self) != type(other)):
return False
a, b = self._hashable_content(), other._hashable_content()
if a != b:
return False
# check number *in* an expression
for a, b in zip(a, b):
if not isinstance(a, Basic):
continue
if a.is_Number and type(a) != type(b):
return False
return True
def __ne__(self, other):
"""``a != b`` -> Compare two symbolic trees and see whether they are different
this is the same as:
``a.compare(b) != 0``
but faster
"""
return not self == other
def dummy_eq(self, other, symbol=None):
"""
Compare two expressions and handle dummy symbols.
Examples
========
>>> from sympy import Dummy
>>> from sympy.abc import x, y
>>> u = Dummy('u')
>>> (u**2 + 1).dummy_eq(x**2 + 1)
True
>>> (u**2 + 1) == (x**2 + 1)
False
>>> (u**2 + y).dummy_eq(x**2 + y, x)
True
>>> (u**2 + y).dummy_eq(x**2 + y, y)
False
"""
s = self.as_dummy()
o = _sympify(other)
o = o.as_dummy()
dummy_symbols = [i for i in s.free_symbols if i.is_Dummy]
if len(dummy_symbols) == 1:
dummy = dummy_symbols.pop()
else:
return s == o
if symbol is None:
symbols = o.free_symbols
if len(symbols) == 1:
symbol = symbols.pop()
else:
return s == o
tmp = dummy.__class__()
return s.xreplace({dummy: tmp}) == o.xreplace({symbol: tmp})
def atoms(self, *types):
"""Returns the atoms that form the current object.
By default, only objects that are truly atomic and cannot
be divided into smaller pieces are returned: symbols, numbers,
and number symbols like I and pi. It is possible to request
atoms of any type, however, as demonstrated below.
Examples
========
>>> from sympy import I, pi, sin
>>> from sympy.abc import x, y
>>> (1 + x + 2*sin(y + I*pi)).atoms()
{1, 2, I, pi, x, y}
If one or more types are given, the results will contain only
those types of atoms.
>>> from sympy import Number, NumberSymbol, Symbol
>>> (1 + x + 2*sin(y + I*pi)).atoms(Symbol)
{x, y}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number)
{1, 2}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol)
{1, 2, pi}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Number, NumberSymbol, I)
{1, 2, I, pi}
Note that I (imaginary unit) and zoo (complex infinity) are special
types of number symbols and are not part of the NumberSymbol class.
The type can be given implicitly, too:
>>> (1 + x + 2*sin(y + I*pi)).atoms(x) # x is a Symbol
{x, y}
Be careful to check your assumptions when using the implicit option
since ``S(1).is_Integer = True`` but ``type(S(1))`` is ``One``, a special type
of SymPy atom, while ``type(S(2))`` is type ``Integer`` and will find all
integers in an expression:
>>> from sympy import S
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(1))
{1}
>>> (1 + x + 2*sin(y + I*pi)).atoms(S(2))
{1, 2}
Finally, arguments to atoms() can select more than atomic atoms: any
SymPy type (loaded in core/__init__.py) can be listed as an argument
and those types of "atoms" as found in scanning the arguments of the
expression recursively:
>>> from sympy import Function, Mul
>>> from sympy.core.function import AppliedUndef
>>> f = Function('f')
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(Function)
{f(x), sin(y + I*pi)}
>>> (1 + f(x) + 2*sin(y + I*pi)).atoms(AppliedUndef)
{f(x)}
>>> (1 + x + 2*sin(y + I*pi)).atoms(Mul)
{I*pi, 2*sin(y + I*pi)}
"""
if types:
types = tuple(
[t if isinstance(t, type) else type(t) for t in types])
nodes = _preorder_traversal(self)
if types:
result = {node for node in nodes if isinstance(node, types)}
else:
result = {node for node in nodes if not node.args}
return result
@property
def free_symbols(self) -> set[Basic]:
"""Return from the atoms of self those which are free symbols.
Not all free symbols are ``Symbol``. Eg: IndexedBase('I')[0].free_symbols
For most expressions, all symbols are free symbols. For some classes
this is not true. e.g. Integrals use Symbols for the dummy variables
which are bound variables, so Integral has a method to return all
symbols except those. Derivative keeps track of symbols with respect
to which it will perform a derivative; those are
bound variables, too, so it has its own free_symbols method.
Any other method that uses bound variables should implement a
free_symbols method."""
empty: set[Basic] = set()
return empty.union(*(a.free_symbols for a in self.args))
@property
def expr_free_symbols(self):
sympy_deprecation_warning("""
The expr_free_symbols property is deprecated. Use free_symbols to get
the free symbols of an expression.
""",
deprecated_since_version="1.9",
active_deprecations_target="deprecated-expr-free-symbols")
return set()
def as_dummy(self):
"""Return the expression with any objects having structurally
bound symbols replaced with unique, canonical symbols within
the object in which they appear and having only the default
assumption for commutativity being True. When applied to a
symbol a new symbol having only the same commutativity will be
returned.
Examples
========
>>> from sympy import Integral, Symbol
>>> from sympy.abc import x
>>> r = Symbol('r', real=True)
>>> Integral(r, (r, x)).as_dummy()
Integral(_0, (_0, x))
>>> _.variables[0].is_real is None
True
>>> r.as_dummy()
_r
Notes
=====
Any object that has structurally bound variables should have
a property, `bound_symbols` that returns those symbols
appearing in the object.
"""
from .symbol import Dummy, Symbol
def can(x):
# mask free that shadow bound
free = x.free_symbols
bound = set(x.bound_symbols)
d = {i: Dummy() for i in bound & free}
x = x.subs(d)
# replace bound with canonical names
x = x.xreplace(x.canonical_variables)
# return after undoing masking
return x.xreplace({v: k for k, v in d.items()})
if not self.has(Symbol):
return self
return self.replace(
lambda x: hasattr(x, 'bound_symbols'),
can,
simultaneous=False)
@property
def canonical_variables(self):
"""Return a dictionary mapping any variable defined in
``self.bound_symbols`` to Symbols that do not clash
with any free symbols in the expression.
Examples
========
>>> from sympy import Lambda
>>> from sympy.abc import x
>>> Lambda(x, 2*x).canonical_variables
{x: _0}
"""
if not hasattr(self, 'bound_symbols'):
return {}
dums = numbered_symbols('_')
reps = {}
# watch out for free symbol that are not in bound symbols;
# those that are in bound symbols are about to get changed
bound = self.bound_symbols
names = {i.name for i in self.free_symbols - set(bound)}
for b in bound:
d = next(dums)
if b.is_Symbol:
while d.name in names:
d = next(dums)
reps[b] = d
return reps
def rcall(self, *args):
"""Apply on the argument recursively through the expression tree.
This method is used to simulate a common abuse of notation for
operators. For instance, in SymPy the following will not work:
``(x+Lambda(y, 2*y))(z) == x+2*z``,
however, you can use:
>>> from sympy import Lambda
>>> from sympy.abc import x, y, z
>>> (x + Lambda(y, 2*y)).rcall(z)
x + 2*z
"""
return Basic._recursive_call(self, args)
@staticmethod
def _recursive_call(expr_to_call, on_args):
"""Helper for rcall method."""
from .symbol import Symbol
def the_call_method_is_overridden(expr):
for cls in getmro(type(expr)):
if '__call__' in cls.__dict__:
return cls != Basic
if callable(expr_to_call) and the_call_method_is_overridden(expr_to_call):
if isinstance(expr_to_call, Symbol): # XXX When you call a Symbol it is
return expr_to_call # transformed into an UndefFunction
else:
return expr_to_call(*on_args)
elif expr_to_call.args:
args = [Basic._recursive_call(
sub, on_args) for sub in expr_to_call.args]
return type(expr_to_call)(*args)
else:
return expr_to_call
def is_hypergeometric(self, k):
from sympy.simplify.simplify import hypersimp
from sympy.functions.elementary.piecewise import Piecewise
if self.has(Piecewise):
return None
return hypersimp(self, k) is not None
@property
def is_comparable(self):
"""Return True if self can be computed to a real number
(or already is a real number) with precision, else False.
Examples
========
>>> from sympy import exp_polar, pi, I
>>> (I*exp_polar(I*pi/2)).is_comparable
True
>>> (I*exp_polar(I*pi*2)).is_comparable
False
A False result does not mean that `self` cannot be rewritten
into a form that would be comparable. For example, the
difference computed below is zero but without simplification
it does not evaluate to a zero with precision:
>>> e = 2**pi*(1 + 2**pi)
>>> dif = e - e.expand()
>>> dif.is_comparable
False
>>> dif.n(2)._prec
1
"""
is_extended_real = self.is_extended_real
if is_extended_real is False:
return False
if not self.is_number:
return False
# don't re-eval numbers that are already evaluated since
# this will create spurious precision
n, i = [p.evalf(2) if not p.is_Number else p
for p in self.as_real_imag()]
if not (i.is_Number and n.is_Number):
return False
if i:
# if _prec = 1 we can't decide and if not,
# the answer is False because numbers with
# imaginary parts can't be compared
# so return False
return False
else:
return n._prec != 1
@property
def func(self):
"""
The top-level function in an expression.
The following should hold for all objects::
>> x == x.func(*x.args)
Examples
========
>>> from sympy.abc import x
>>> a = 2*x
>>> a.func
<class 'sympy.core.mul.Mul'>
>>> a.args
(2, x)
>>> a.func(*a.args)
2*x
>>> a == a.func(*a.args)
True
"""
return self.__class__
@property
def args(self) -> tuple[Basic, ...]:
"""Returns a tuple of arguments of 'self'.
Examples
========
>>> from sympy import cot
>>> from sympy.abc import x, y
>>> cot(x).args
(x,)
>>> cot(x).args[0]
x
>>> (x*y).args
(x, y)
>>> (x*y).args[1]
y
Notes
=====
Never use self._args, always use self.args.
Only use _args in __new__ when creating a new function.
Do not override .args() from Basic (so that it is easy to
change the interface in the future if needed).
"""
return self._args
@property
def _sorted_args(self):
"""
The same as ``args``. Derived classes which do not fix an
order on their arguments should override this method to
produce the sorted representation.
"""
return self.args
def as_content_primitive(self, radical=False, clear=True):
"""A stub to allow Basic args (like Tuple) to be skipped when computing
the content and primitive components of an expression.
See Also
========
sympy.core.expr.Expr.as_content_primitive
"""
return S.One, self
def subs(self, *args, **kwargs):
"""
Substitutes old for new in an expression after sympifying args.
`args` is either:
- two arguments, e.g. foo.subs(old, new)
- one iterable argument, e.g. foo.subs(iterable). The iterable may be
o an iterable container with (old, new) pairs. In this case the
replacements are processed in the order given with successive
patterns possibly affecting replacements already made.
o a dict or set whose key/value items correspond to old/new pairs.
In this case the old/new pairs will be sorted by op count and in
case of a tie, by number of args and the default_sort_key. The
resulting sorted list is then processed as an iterable container
(see previous).
If the keyword ``simultaneous`` is True, the subexpressions will not be
evaluated until all the substitutions have been made.
Examples
========
>>> from sympy import pi, exp, limit, oo
>>> from sympy.abc import x, y
>>> (1 + x*y).subs(x, pi)
pi*y + 1
>>> (1 + x*y).subs({x:pi, y:2})
1 + 2*pi
>>> (1 + x*y).subs([(x, pi), (y, 2)])
1 + 2*pi
>>> reps = [(y, x**2), (x, 2)]
>>> (x + y).subs(reps)
6
>>> (x + y).subs(reversed(reps))
x**2 + 2
>>> (x**2 + x**4).subs(x**2, y)
y**2 + y
To replace only the x**2 but not the x**4, use xreplace:
>>> (x**2 + x**4).xreplace({x**2: y})
x**4 + y
To delay evaluation until all substitutions have been made,
set the keyword ``simultaneous`` to True:
>>> (x/y).subs([(x, 0), (y, 0)])
0
>>> (x/y).subs([(x, 0), (y, 0)], simultaneous=True)
nan
This has the added feature of not allowing subsequent substitutions
to affect those already made:
>>> ((x + y)/y).subs({x + y: y, y: x + y})
1
>>> ((x + y)/y).subs({x + y: y, y: x + y}, simultaneous=True)
y/(x + y)
In order to obtain a canonical result, unordered iterables are
sorted by count_op length, number of arguments and by the
default_sort_key to break any ties. All other iterables are left
unsorted.
>>> from sympy import sqrt, sin, cos
>>> from sympy.abc import a, b, c, d, e
>>> A = (sqrt(sin(2*x)), a)
>>> B = (sin(2*x), b)
>>> C = (cos(2*x), c)
>>> D = (x, d)
>>> E = (exp(x), e)
>>> expr = sqrt(sin(2*x))*sin(exp(x)*x)*cos(2*x) + sin(2*x)
>>> expr.subs(dict([A, B, C, D, E]))
a*c*sin(d*e) + b
The resulting expression represents a literal replacement of the
old arguments with the new arguments. This may not reflect the
limiting behavior of the expression:
>>> (x**3 - 3*x).subs({x: oo})
nan
>>> limit(x**3 - 3*x, x, oo)
oo
If the substitution will be followed by numerical
evaluation, it is better to pass the substitution to
evalf as
>>> (1/x).evalf(subs={x: 3.0}, n=21)
0.333333333333333333333
rather than
>>> (1/x).subs({x: 3.0}).evalf(21)
0.333333333333333314830
as the former will ensure that the desired level of precision is
obtained.
See Also
========
replace: replacement capable of doing wildcard-like matching,
parsing of match, and conditional replacements
xreplace: exact node replacement in expr tree; also capable of
using matching rules
sympy.core.evalf.EvalfMixin.evalf: calculates the given formula to a desired level of precision
"""
from .containers import Dict
from .symbol import Dummy, Symbol
from .numbers import _illegal
unordered = False
if len(args) == 1:
sequence = args[0]
if isinstance(sequence, set):
unordered = True
elif isinstance(sequence, (Dict, Mapping)):
unordered = True
sequence = sequence.items()
elif not iterable(sequence):
raise ValueError(filldedent("""
When a single argument is passed to subs
it should be a dictionary of old: new pairs or an iterable
of (old, new) tuples."""))
elif len(args) == 2:
sequence = [args]
else:
raise ValueError("subs accepts either 1 or 2 arguments")
def sympify_old(old):
if isinstance(old, str):
# Use Symbol rather than parse_expr for old
return Symbol(old)
elif isinstance(old, type):
# Allow a type e.g. Function('f') or sin
return sympify(old, strict=False)
else:
return sympify(old, strict=True)
def sympify_new(new):
if isinstance(new, (str, type)):
# Allow a type or parse a string input
return sympify(new, strict=False)
else:
return sympify(new, strict=True)
sequence = [(sympify_old(s1), sympify_new(s2)) for s1, s2 in sequence]
# skip if there is no change
sequence = [(s1, s2) for s1, s2 in sequence if not _aresame(s1, s2)]
simultaneous = kwargs.pop('simultaneous', False)
if unordered:
from .sorting import _nodes, default_sort_key
sequence = dict(sequence)
# order so more complex items are first and items
# of identical complexity are ordered so
# f(x) < f(y) < x < y
# \___ 2 __/ \_1_/ <- number of nodes
#
# For more complex ordering use an unordered sequence.
k = list(ordered(sequence, default=False, keys=(
lambda x: -_nodes(x),
default_sort_key,
)))
sequence = [(k, sequence[k]) for k in k]
# do infinities first
if not simultaneous:
redo = [i for i, seq in enumerate(sequence) if seq[1] in _illegal]
for i in reversed(redo):
sequence.insert(0, sequence.pop(i))
if simultaneous: # XXX should this be the default for dict subs?
reps = {}
rv = self
kwargs['hack2'] = True
m = Dummy('subs_m')
for old, new in sequence:
com = new.is_commutative
if com is None:
com = True
d = Dummy('subs_d', commutative=com)
# using d*m so Subs will be used on dummy variables
# in things like Derivative(f(x, y), x) in which x
# is both free and bound
rv = rv._subs(old, d*m, **kwargs)
if not isinstance(rv, Basic):
break
reps[d] = new
reps[m] = S.One # get rid of m
return rv.xreplace(reps)
else:
rv = self
for old, new in sequence:
rv = rv._subs(old, new, **kwargs)
if not isinstance(rv, Basic):
break
return rv
@cacheit
def _subs(self, old, new, **hints):
"""Substitutes an expression old -> new.
If self is not equal to old then _eval_subs is called.
If _eval_subs does not want to make any special replacement
then a None is received which indicates that the fallback
should be applied wherein a search for replacements is made
amongst the arguments of self.
>>> from sympy import Add
>>> from sympy.abc import x, y, z
Examples
========
Add's _eval_subs knows how to target x + y in the following
so it makes the change:
>>> (x + y + z).subs(x + y, 1)
z + 1
Add's _eval_subs does not need to know how to find x + y in
the following:
>>> Add._eval_subs(z*(x + y) + 3, x + y, 1) is None
True
The returned None will cause the fallback routine to traverse the args and
pass the z*(x + y) arg to Mul where the change will take place and the
substitution will succeed:
>>> (z*(x + y) + 3).subs(x + y, 1)
z + 3
** Developers Notes **
An _eval_subs routine for a class should be written if:
1) any arguments are not instances of Basic (e.g. bool, tuple);
2) some arguments should not be targeted (as in integration
variables);
3) if there is something other than a literal replacement
that should be attempted (as in Piecewise where the condition
may be updated without doing a replacement).
If it is overridden, here are some special cases that might arise:
1) If it turns out that no special change was made and all
the original sub-arguments should be checked for
replacements then None should be returned.
2) If it is necessary to do substitutions on a portion of
the expression then _subs should be called. _subs will
handle the case of any sub-expression being equal to old
(which usually would not be the case) while its fallback
will handle the recursion into the sub-arguments. For
example, after Add's _eval_subs removes some matching terms
it must process the remaining terms so it calls _subs
on each of the un-matched terms and then adds them
onto the terms previously obtained.
3) If the initial expression should remain unchanged then
the original expression should be returned. (Whenever an
expression is returned, modified or not, no further
substitution of old -> new is attempted.) Sum's _eval_subs
routine uses this strategy when a substitution is attempted
on any of its summation variables.
"""
def fallback(self, old, new):
"""
Try to replace old with new in any of self's arguments.
"""
hit = False
args = list(self.args)
for i, arg in enumerate(args):
if not hasattr(arg, '_eval_subs'):
continue
arg = arg._subs(old, new, **hints)
if not _aresame(arg, args[i]):
hit = True
args[i] = arg
if hit:
rv = self.func(*args)
hack2 = hints.get('hack2', False)
if hack2 and self.is_Mul and not rv.is_Mul: # 2-arg hack
coeff = S.One
nonnumber = []
for i in args:
if i.is_Number:
coeff *= i
else:
nonnumber.append(i)
nonnumber = self.func(*nonnumber)
if coeff is S.One:
return nonnumber
else:
return self.func(coeff, nonnumber, evaluate=False)
return rv
return self
if _aresame(self, old):
return new
rv = self._eval_subs(old, new)
if rv is None:
rv = fallback(self, old, new)
return rv
def _eval_subs(self, old, new):
"""Override this stub if you want to do anything more than
attempt a replacement of old with new in the arguments of self.
See also
========
_subs
"""
return None
def xreplace(self, rule):
"""
Replace occurrences of objects within the expression.
Parameters
==========
rule : dict-like
Expresses a replacement rule
Returns
=======
xreplace : the result of the replacement
Examples
========
>>> from sympy import symbols, pi, exp
>>> x, y, z = symbols('x y z')
>>> (1 + x*y).xreplace({x: pi})
pi*y + 1
>>> (1 + x*y).xreplace({x: pi, y: 2})
1 + 2*pi
Replacements occur only if an entire node in the expression tree is
matched:
>>> (x*y + z).xreplace({x*y: pi})
z + pi
>>> (x*y*z).xreplace({x*y: pi})
x*y*z
>>> (2*x).xreplace({2*x: y, x: z})
y
>>> (2*2*x).xreplace({2*x: y, x: z})
4*z
>>> (x + y + 2).xreplace({x + y: 2})
x + y + 2
>>> (x + 2 + exp(x + 2)).xreplace({x + 2: y})
x + exp(y) + 2
xreplace does not differentiate between free and bound symbols. In the
following, subs(x, y) would not change x since it is a bound symbol,
but xreplace does:
>>> from sympy import Integral
>>> Integral(x, (x, 1, 2*x)).xreplace({x: y})
Integral(y, (y, 1, 2*y))
Trying to replace x with an expression raises an error:
>>> Integral(x, (x, 1, 2*x)).xreplace({x: 2*y}) # doctest: +SKIP
ValueError: Invalid limits given: ((2*y, 1, 4*y),)
See Also
========
replace: replacement capable of doing wildcard-like matching,
parsing of match, and conditional replacements
subs: substitution of subexpressions as defined by the objects
themselves.
"""
value, _ = self._xreplace(rule)
return value
def _xreplace(self, rule):
"""
Helper for xreplace. Tracks whether a replacement actually occurred.
"""
if self in rule:
return rule[self], True
elif rule:
args = []
changed = False
for a in self.args:
_xreplace = getattr(a, '_xreplace', None)
if _xreplace is not None:
a_xr = _xreplace(rule)
args.append(a_xr[0])
changed |= a_xr[1]
else:
args.append(a)
args = tuple(args)
if changed:
return self.func(*args), True
return self, False
@cacheit
def has(self, *patterns):
"""
Test whether any subexpression matches any of the patterns.
Examples
========
>>> from sympy import sin
>>> from sympy.abc import x, y, z
>>> (x**2 + sin(x*y)).has(z)
False
>>> (x**2 + sin(x*y)).has(x, y, z)
True
>>> x.has(x)
True
Note ``has`` is a structural algorithm with no knowledge of
mathematics. Consider the following half-open interval:
>>> from sympy import Interval
>>> i = Interval.Lopen(0, 5); i
Interval.Lopen(0, 5)
>>> i.args
(0, 5, True, False)
>>> i.has(4) # there is no "4" in the arguments
False
>>> i.has(0) # there *is* a "0" in the arguments
True
Instead, use ``contains`` to determine whether a number is in the
interval or not:
>>> i.contains(4)
True
>>> i.contains(0)
False
Note that ``expr.has(*patterns)`` is exactly equivalent to
``any(expr.has(p) for p in patterns)``. In particular, ``False`` is
returned when the list of patterns is empty.
>>> x.has()
False
"""
return self._has(iterargs, *patterns)
def has_xfree(self, s: set[Basic]):
"""return True if self has any of the patterns in s as a
free argument, else False. This is like `Basic.has_free`
but this will only report exact argument matches.
Examples
========
>>> from sympy import Function
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> f(x).has_xfree({f})
False
>>> f(x).has_xfree({f(x)})
True
>>> f(x + 1).has_xfree({x})
True
>>> f(x + 1).has_xfree({x + 1})
True
>>> f(x + y + 1).has_xfree({x + 1})
False
"""
# protect O(1) containment check by requiring:
if type(s) is not set:
raise TypeError('expecting set argument')
return any(a in s for a in iterfreeargs(self))
@cacheit
def has_free(self, *patterns):
"""return True if self has object(s) ``x`` as a free expression
else False.
Examples
========
>>> from sympy import Integral, Function
>>> from sympy.abc import x, y
>>> f = Function('f')
>>> g = Function('g')
>>> expr = Integral(f(x), (f(x), 1, g(y)))
>>> expr.free_symbols
{y}
>>> expr.has_free(g(y))
True
>>> expr.has_free(*(x, f(x)))
False
This works for subexpressions and types, too:
>>> expr.has_free(g)
True
>>> (x + y + 1).has_free(y + 1)
True
"""
if not patterns:
return False
p0 = patterns[0]
if len(patterns) == 1 and iterable(p0) and not isinstance(p0, Basic):
# Basic can contain iterables (though not non-Basic, ideally)
# but don't encourage mixed passing patterns
raise TypeError(filldedent('''
Expecting 1 or more Basic args, not a single
non-Basic iterable. Don't forget to unpack
iterables: `eq.has_free(*patterns)`'''))
# try quick test first
s = set(patterns)
rv = self.has_xfree(s)
if rv:
return rv
# now try matching through slower _has
return self._has(iterfreeargs, *patterns)
def _has(self, iterargs, *patterns):
# separate out types and unhashable objects
type_set = set() # only types
p_set = set() # hashable non-types
for p in patterns:
if isinstance(p, BasicMeta):
type_set.add(p)
continue
if not isinstance(p, Basic):
try:
p = _sympify(p)
except SympifyError:
continue # Basic won't have this in it
p_set.add(p) # fails if object defines __eq__ but
# doesn't define __hash__
types = tuple(type_set) #
for i in iterargs(self): #
if i in p_set: # <--- here, too
return True
if isinstance(i, types):
return True
# use matcher if defined, e.g. operations defines
# matcher that checks for exact subset containment,
# (x + y + 1).has(x + 1) -> True
for i in p_set - type_set: # types don't have matchers
if not hasattr(i, '_has_matcher'):
continue
match = i._has_matcher()
if any(match(arg) for arg in iterargs(self)):
return True
# no success
return False
def replace(self, query, value, map=False, simultaneous=True, exact=None):
"""
Replace matching subexpressions of ``self`` with ``value``.
If ``map = True`` then also return the mapping {old: new} where ``old``
was a sub-expression found with query and ``new`` is the replacement
value for it. If the expression itself does not match the query, then
the returned value will be ``self.xreplace(map)`` otherwise it should
be ``self.subs(ordered(map.items()))``.
Traverses an expression tree and performs replacement of matching
subexpressions from the bottom to the top of the tree. The default
approach is to do the replacement in a simultaneous fashion so
changes made are targeted only once. If this is not desired or causes
problems, ``simultaneous`` can be set to False.
In addition, if an expression containing more than one Wild symbol
is being used to match subexpressions and the ``exact`` flag is None
it will be set to True so the match will only succeed if all non-zero
values are received for each Wild that appears in the match pattern.
Setting this to False accepts a match of 0; while setting it True
accepts all matches that have a 0 in them. See example below for
cautions.
The list of possible combinations of queries and replacement values
is listed below:
Examples
========
Initial setup
>>> from sympy import log, sin, cos, tan, Wild, Mul, Add
>>> from sympy.abc import x, y
>>> f = log(sin(x)) + tan(sin(x**2))
1.1. type -> type
obj.replace(type, newtype)
When object of type ``type`` is found, replace it with the
result of passing its argument(s) to ``newtype``.
>>> f.replace(sin, cos)
log(cos(x)) + tan(cos(x**2))
>>> sin(x).replace(sin, cos, map=True)
(cos(x), {sin(x): cos(x)})
>>> (x*y).replace(Mul, Add)
x + y
1.2. type -> func
obj.replace(type, func)
When object of type ``type`` is found, apply ``func`` to its
argument(s). ``func`` must be written to handle the number
of arguments of ``type``.
>>> f.replace(sin, lambda arg: sin(2*arg))
log(sin(2*x)) + tan(sin(2*x**2))
>>> (x*y).replace(Mul, lambda *args: sin(2*Mul(*args)))
sin(2*x*y)
2.1. pattern -> expr
obj.replace(pattern(wild), expr(wild))
Replace subexpressions matching ``pattern`` with the expression
written in terms of the Wild symbols in ``pattern``.
>>> a, b = map(Wild, 'ab')
>>> f.replace(sin(a), tan(a))
log(tan(x)) + tan(tan(x**2))
>>> f.replace(sin(a), tan(a/2))
log(tan(x/2)) + tan(tan(x**2/2))
>>> f.replace(sin(a), a)
log(x) + tan(x**2)
>>> (x*y).replace(a*x, a)
y
Matching is exact by default when more than one Wild symbol
is used: matching fails unless the match gives non-zero
values for all Wild symbols:
>>> (2*x + y).replace(a*x + b, b - a)
y - 2
>>> (2*x).replace(a*x + b, b - a)
2*x
When set to False, the results may be non-intuitive:
>>> (2*x).replace(a*x + b, b - a, exact=False)
2/x
2.2. pattern -> func
obj.replace(pattern(wild), lambda wild: expr(wild))
All behavior is the same as in 2.1 but now a function in terms of
pattern variables is used rather than an expression:
>>> f.replace(sin(a), lambda a: sin(2*a))
log(sin(2*x)) + tan(sin(2*x**2))
3.1. func -> func
obj.replace(filter, func)
Replace subexpression ``e`` with ``func(e)`` if ``filter(e)``
is True.
>>> g = 2*sin(x**3)
>>> g.replace(lambda expr: expr.is_Number, lambda expr: expr**2)
4*sin(x**9)
The expression itself is also targeted by the query but is done in
such a fashion that changes are not made twice.
>>> e = x*(x*y + 1)
>>> e.replace(lambda x: x.is_Mul, lambda x: 2*x)
2*x*(2*x*y + 1)
When matching a single symbol, `exact` will default to True, but
this may or may not be the behavior that is desired:
Here, we want `exact=False`:
>>> from sympy import Function
>>> f = Function('f')
>>> e = f(1) + f(0)
>>> q = f(a), lambda a: f(a + 1)
>>> e.replace(*q, exact=False)
f(1) + f(2)
>>> e.replace(*q, exact=True)
f(0) + f(2)
But here, the nature of matching makes selecting
the right setting tricky:
>>> e = x**(1 + y)
>>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=False)
x
>>> (x**(1 + y)).replace(x**(1 + a), lambda a: x**-a, exact=True)
x**(-x - y + 1)
>>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=False)
x
>>> (x**y).replace(x**(1 + a), lambda a: x**-a, exact=True)
x**(1 - y)
It is probably better to use a different form of the query
that describes the target expression more precisely:
>>> (1 + x**(1 + y)).replace(
... lambda x: x.is_Pow and x.exp.is_Add and x.exp.args[0] == 1,
... lambda x: x.base**(1 - (x.exp - 1)))
...
x**(1 - y) + 1
See Also
========
subs: substitution of subexpressions as defined by the objects
themselves.
xreplace: exact node replacement in expr tree; also capable of
using matching rules
"""
try:
query = _sympify(query)
except SympifyError:
pass
try:
value = _sympify(value)
except SympifyError:
pass
if isinstance(query, type):
_query = lambda expr: isinstance(expr, query)
if isinstance(value, type):
_value = lambda expr, result: value(*expr.args)
elif callable(value):
_value = lambda expr, result: value(*expr.args)
else:
raise TypeError(
"given a type, replace() expects another "
"type or a callable")
elif isinstance(query, Basic):
_query = lambda expr: expr.match(query)
if exact is None:
from .symbol import Wild
exact = (len(query.atoms(Wild)) > 1)
if isinstance(value, Basic):
if exact:
_value = lambda expr, result: (value.subs(result)
if all(result.values()) else expr)
else:
_value = lambda expr, result: value.subs(result)
elif callable(value):
# match dictionary keys get the trailing underscore stripped
# from them and are then passed as keywords to the callable;
# if ``exact`` is True, only accept match if there are no null
# values amongst those matched.
if exact:
_value = lambda expr, result: (value(**
{str(k)[:-1]: v for k, v in result.items()})
if all(val for val in result.values()) else expr)
else:
_value = lambda expr, result: value(**
{str(k)[:-1]: v for k, v in result.items()})
else:
raise TypeError(
"given an expression, replace() expects "
"another expression or a callable")
elif callable(query):
_query = query
if callable(value):
_value = lambda expr, result: value(expr)
else:
raise TypeError(
"given a callable, replace() expects "
"another callable")
else:
raise TypeError(
"first argument to replace() must be a "
"type, an expression or a callable")
def walk(rv, F):
"""Apply ``F`` to args and then to result.
"""
args = getattr(rv, 'args', None)
if args is not None:
if args:
newargs = tuple([walk(a, F) for a in args])
if args != newargs:
rv = rv.func(*newargs)
if simultaneous:
# if rv is something that was already
# matched (that was changed) then skip
# applying F again
for i, e in enumerate(args):
if rv == e and e != newargs[i]:
return rv
rv = F(rv)
return rv
mapping = {} # changes that took place
def rec_replace(expr):
result = _query(expr)
if result or result == {}:
v = _value(expr, result)
if v is not None and v != expr:
if map:
mapping[expr] = v
expr = v
return expr
rv = walk(self, rec_replace)
return (rv, mapping) if map else rv
def find(self, query, group=False):
"""Find all subexpressions matching a query. """
query = _make_find_query(query)
results = list(filter(query, _preorder_traversal(self)))
if not group:
return set(results)
else:
groups = {}
for result in results:
if result in groups:
groups[result] += 1
else:
groups[result] = 1
return groups
def count(self, query):
"""Count the number of matching subexpressions. """
query = _make_find_query(query)
return sum(bool(query(sub)) for sub in _preorder_traversal(self))
def matches(self, expr, repl_dict=None, old=False):
"""
Helper method for match() that looks for a match between Wild symbols
in self and expressions in expr.
Examples
========
>>> from sympy import symbols, Wild, Basic
>>> a, b, c = symbols('a b c')
>>> x = Wild('x')
>>> Basic(a + x, x).matches(Basic(a + b, c)) is None
True
>>> Basic(a + x, x).matches(Basic(a + b + c, b + c))
{x_: b + c}
"""
expr = sympify(expr)
if not isinstance(expr, self.__class__):
return None
if repl_dict is None:
repl_dict = {}
else:
repl_dict = repl_dict.copy()
if self == expr:
return repl_dict
if len(self.args) != len(expr.args):
return None
d = repl_dict # already a copy
for arg, other_arg in zip(self.args, expr.args):
if arg == other_arg:
continue
if arg.is_Relational:
try:
d = arg.xreplace(d).matches(other_arg, d, old=old)
except TypeError: # Should be InvalidComparisonError when introduced
d = None
else:
d = arg.xreplace(d).matches(other_arg, d, old=old)
if d is None:
return None
return d
def match(self, pattern, old=False):
"""
Pattern matching.
Wild symbols match all.
Return ``None`` when expression (self) does not match
with pattern. Otherwise return a dictionary such that::
pattern.xreplace(self.match(pattern)) == self
Examples
========
>>> from sympy import Wild, Sum
>>> from sympy.abc import x, y
>>> p = Wild("p")
>>> q = Wild("q")
>>> r = Wild("r")
>>> e = (x+y)**(x+y)
>>> e.match(p**p)
{p_: x + y}
>>> e.match(p**q)
{p_: x + y, q_: x + y}
>>> e = (2*x)**2
>>> e.match(p*q**r)
{p_: 4, q_: x, r_: 2}
>>> (p*q**r).xreplace(e.match(p*q**r))
4*x**2
Structurally bound symbols are ignored during matching:
>>> Sum(x, (x, 1, 2)).match(Sum(y, (y, 1, p)))
{p_: 2}
But they can be identified if desired:
>>> Sum(x, (x, 1, 2)).match(Sum(q, (q, 1, p)))
{p_: 2, q_: x}
The ``old`` flag will give the old-style pattern matching where
expressions and patterns are essentially solved to give the
match. Both of the following give None unless ``old=True``:
>>> (x - 2).match(p - x, old=True)
{p_: 2*x - 2}
>>> (2/x).match(p*x, old=True)
{p_: 2/x**2}
"""
pattern = sympify(pattern)
# match non-bound symbols
canonical = lambda x: x if x.is_Symbol else x.as_dummy()
m = canonical(pattern).matches(canonical(self), old=old)
if m is None:
return m
from .symbol import Wild
from .function import WildFunction
wild = pattern.atoms(Wild, WildFunction)
# sanity check
if set(m) - wild:
raise ValueError(filldedent('''
Some `matches` routine did not use a copy of repl_dict
and injected unexpected symbols. Report this as an
error at https://github.com/sympy/sympy/issues'''))
# now see if bound symbols were requested
bwild = wild - set(m)
if not bwild:
return m
# replace free-Wild symbols in pattern with match result
# so they will match but not be in the next match
wpat = pattern.xreplace(m)
# identify remaining bound wild
w = wpat.matches(self, old=old)
# add them to m
if w:
m.update(w)
# done
return m
def count_ops(self, visual=None):
"""wrapper for count_ops that returns the operation count."""
from .function import count_ops
return count_ops(self, visual)
def doit(self, **hints):
"""Evaluate objects that are not evaluated by default like limits,
integrals, sums and products. All objects of this kind will be
evaluated recursively, unless some species were excluded via 'hints'
or unless the 'deep' hint was set to 'False'.
>>> from sympy import Integral
>>> from sympy.abc import x
>>> 2*Integral(x, x)
2*Integral(x, x)
>>> (2*Integral(x, x)).doit()
x**2
>>> (2*Integral(x, x)).doit(deep=False)
2*Integral(x, x)
"""
if hints.get('deep', True):
terms = [term.doit(**hints) if isinstance(term, Basic) else term
for term in self.args]
return self.func(*terms)
else:
return self
def simplify(self, **kwargs):
"""See the simplify function in sympy.simplify"""
from sympy.simplify.simplify import simplify
return simplify(self, **kwargs)
def refine(self, assumption=True):
"""See the refine function in sympy.assumptions"""
from sympy.assumptions.refine import refine
return refine(self, assumption)
def _eval_derivative_n_times(self, s, n):
# This is the default evaluator for derivatives (as called by `diff`
# and `Derivative`), it will attempt a loop to derive the expression
# `n` times by calling the corresponding `_eval_derivative` method,
# while leaving the derivative unevaluated if `n` is symbolic. This
# method should be overridden if the object has a closed form for its
# symbolic n-th derivative.
from .numbers import Integer
if isinstance(n, (int, Integer)):
obj = self
for i in range(n):
obj2 = obj._eval_derivative(s)
if obj == obj2 or obj2 is None:
break
obj = obj2
return obj2
else:
return None
def rewrite(self, *args, deep=True, **hints):
"""
Rewrite *self* using a defined rule.
Rewriting transforms an expression to another, which is mathematically
equivalent but structurally different. For example you can rewrite
trigonometric functions as complex exponentials or combinatorial
functions as gamma function.
This method takes a *pattern* and a *rule* as positional arguments.
*pattern* is optional parameter which defines the types of expressions
that will be transformed. If it is not passed, all possible expressions
will be rewritten. *rule* defines how the expression will be rewritten.
Parameters
==========
args : *rule*, or *pattern* and *rule*.
- *pattern* is a type or an iterable of types.
- *rule* can be any object.
deep : bool, optional.
If ``True``, subexpressions are recursively transformed. Default is
``True``.
Examples
========
If *pattern* is unspecified, all possible expressions are transformed.
>>> from sympy import cos, sin, exp, I
>>> from sympy.abc import x
>>> expr = cos(x) + I*sin(x)
>>> expr.rewrite(exp)
exp(I*x)
Pattern can be a type or an iterable of types.
>>> expr.rewrite(sin, exp)
exp(I*x)/2 + cos(x) - exp(-I*x)/2
>>> expr.rewrite([cos,], exp)
exp(I*x)/2 + I*sin(x) + exp(-I*x)/2
>>> expr.rewrite([cos, sin], exp)
exp(I*x)
Rewriting behavior can be implemented by defining ``_eval_rewrite()``
method.
>>> from sympy import Expr, sqrt, pi
>>> class MySin(Expr):
... def _eval_rewrite(self, rule, args, **hints):
... x, = args
... if rule == cos:
... return cos(pi/2 - x, evaluate=False)
... if rule == sqrt:
... return sqrt(1 - cos(x)**2)
>>> MySin(MySin(x)).rewrite(cos)
cos(-cos(-x + pi/2) + pi/2)
>>> MySin(x).rewrite(sqrt)
sqrt(1 - cos(x)**2)
Defining ``_eval_rewrite_as_[...]()`` method is supported for backwards
compatibility reason. This may be removed in the future and using it is
discouraged.
>>> class MySin(Expr):
... def _eval_rewrite_as_cos(self, *args, **hints):
... x, = args
... return cos(pi/2 - x, evaluate=False)
>>> MySin(x).rewrite(cos)
cos(-x + pi/2)
"""
if not args:
return self
hints.update(deep=deep)
pattern = args[:-1]
rule = args[-1]
# support old design by _eval_rewrite_as_[...] method
if isinstance(rule, str):
method = "_eval_rewrite_as_%s" % rule
elif hasattr(rule, "__name__"):
# rule is class or function
clsname = rule.__name__
method = "_eval_rewrite_as_%s" % clsname
else:
# rule is instance
clsname = rule.__class__.__name__
method = "_eval_rewrite_as_%s" % clsname
if pattern:
if iterable(pattern[0]):
pattern = pattern[0]
pattern = tuple(p for p in pattern if self.has(p))
if not pattern:
return self
# hereafter, empty pattern is interpreted as all pattern.
return self._rewrite(pattern, rule, method, **hints)
def _rewrite(self, pattern, rule, method, **hints):
deep = hints.pop('deep', True)
if deep:
args = [a._rewrite(pattern, rule, method, **hints)
for a in self.args]
else:
args = self.args
if not pattern or any(isinstance(self, p) for p in pattern):
meth = getattr(self, method, None)
if meth is not None:
rewritten = meth(*args, **hints)
else:
rewritten = self._eval_rewrite(rule, args, **hints)
if rewritten is not None:
return rewritten
if not args:
return self
return self.func(*args)
def _eval_rewrite(self, rule, args, **hints):
return None
_constructor_postprocessor_mapping = {} # type: ignore
@classmethod
def _exec_constructor_postprocessors(cls, obj):
# WARNING: This API is experimental.
# This is an experimental API that introduces constructor
# postprosessors for SymPy Core elements. If an argument of a SymPy
# expression has a `_constructor_postprocessor_mapping` attribute, it will
# be interpreted as a dictionary containing lists of postprocessing
# functions for matching expression node names.
clsname = obj.__class__.__name__
postprocessors = defaultdict(list)
for i in obj.args:
try:
postprocessor_mappings = (
Basic._constructor_postprocessor_mapping[cls].items()
for cls in type(i).mro()
if cls in Basic._constructor_postprocessor_mapping
)
for k, v in chain.from_iterable(postprocessor_mappings):
postprocessors[k].extend([j for j in v if j not in postprocessors[k]])
except TypeError:
pass
for f in postprocessors.get(clsname, []):
obj = f(obj)
return obj
def _sage_(self):
"""
Convert *self* to a symbolic expression of SageMath.
This version of the method is merely a placeholder.
"""
old_method = self._sage_
from sage.interfaces.sympy import sympy_init
sympy_init() # may monkey-patch _sage_ method into self's class or superclasses
if old_method == self._sage_:
raise NotImplementedError('conversion to SageMath is not implemented')
else:
# call the freshly monkey-patched method
return self._sage_()
def could_extract_minus_sign(self):
return False # see Expr.could_extract_minus_sign
class Atom(Basic):
"""
A parent class for atomic things. An atom is an expression with no subexpressions.
Examples
========
Symbol, Number, Rational, Integer, ...
But not: Add, Mul, Pow, ...
"""
is_Atom = True
__slots__ = ()
def matches(self, expr, repl_dict=None, old=False):
if self == expr:
if repl_dict is None:
return {}
return repl_dict.copy()
def xreplace(self, rule, hack2=False):
return rule.get(self, self)
def doit(self, **hints):
return self
@classmethod
def class_key(cls):
return 2, 0, cls.__name__
@cacheit
def sort_key(self, order=None):
return self.class_key(), (1, (str(self),)), S.One.sort_key(), S.One
def _eval_simplify(self, **kwargs):
return self
@property
def _sorted_args(self):
# this is here as a safeguard against accidentally using _sorted_args
# on Atoms -- they cannot be rebuilt as atom.func(*atom._sorted_args)
# since there are no args. So the calling routine should be checking
# to see that this property is not called for Atoms.
raise AttributeError('Atoms have no args. It might be necessary'
' to make a check for Atoms in the calling code.')
def _aresame(a, b):
"""Return True if a and b are structurally the same, else False.
Examples
========
In SymPy (as in Python) two numbers compare the same if they
have the same underlying base-2 representation even though
they may not be the same type:
>>> from sympy import S
>>> 2.0 == S(2)
True
>>> 0.5 == S.Half
True
This routine was written to provide a query for such cases that
would give false when the types do not match:
>>> from sympy.core.basic import _aresame
>>> _aresame(S(2.0), S(2))
False
"""
from .numbers import Number
from .function import AppliedUndef, UndefinedFunction as UndefFunc
if isinstance(a, Number) and isinstance(b, Number):
return a == b and a.__class__ == b.__class__
for i, j in zip_longest(_preorder_traversal(a), _preorder_traversal(b)):
if i != j or type(i) != type(j):
if ((isinstance(i, UndefFunc) and isinstance(j, UndefFunc)) or
(isinstance(i, AppliedUndef) and isinstance(j, AppliedUndef))):
if i.class_key() != j.class_key():
return False
else:
return False
return True
def _ne(a, b):
# use this as a second test after `a != b` if you want to make
# sure that things are truly equal, e.g.
# a, b = 0.5, S.Half
# a !=b or _ne(a, b) -> True
from .numbers import Number
# 0.5 == S.Half
if isinstance(a, Number) and isinstance(b, Number):
return a.__class__ != b.__class__
def _atomic(e, recursive=False):
"""Return atom-like quantities as far as substitution is
concerned: Derivatives, Functions and Symbols. Do not
return any 'atoms' that are inside such quantities unless
they also appear outside, too, unless `recursive` is True.
Examples
========
>>> from sympy import Derivative, Function, cos
>>> from sympy.abc import x, y
>>> from sympy.core.basic import _atomic
>>> f = Function('f')
>>> _atomic(x + y)
{x, y}
>>> _atomic(x + f(y))
{x, f(y)}
>>> _atomic(Derivative(f(x), x) + cos(x) + y)
{y, cos(x), Derivative(f(x), x)}
"""
pot = _preorder_traversal(e)
seen = set()
if isinstance(e, Basic):
free = getattr(e, "free_symbols", None)
if free is None:
return {e}
else:
return set()
from .symbol import Symbol
from .function import Derivative, Function
atoms = set()
for p in pot:
if p in seen:
pot.skip()
continue
seen.add(p)
if isinstance(p, Symbol) and p in free:
atoms.add(p)
elif isinstance(p, (Derivative, Function)):
if not recursive:
pot.skip()
atoms.add(p)
return atoms
def _make_find_query(query):
"""Convert the argument of Basic.find() into a callable"""
try:
query = _sympify(query)
except SympifyError:
pass
if isinstance(query, type):
return lambda expr: isinstance(expr, query)
elif isinstance(query, Basic):
return lambda expr: expr.match(query) is not None
return query
# Delayed to avoid cyclic import
from .singleton import S
from .traversal import (preorder_traversal as _preorder_traversal,
iterargs, iterfreeargs)
preorder_traversal = deprecated(
"""
Using preorder_traversal from the sympy.core.basic submodule is
deprecated.
Instead, use preorder_traversal from the top-level sympy namespace, like
sympy.preorder_traversal
""",
deprecated_since_version="1.10",
active_deprecations_target="deprecated-traversal-functions-moved",
)(_preorder_traversal)
|
aeb4dfa1e1c3553616a9fe016657dc1b1384b2a5b60bb5520087087cf0cb47e5 | from __future__ import annotations
from typing import Callable
from math import log as _log, sqrt as _sqrt
from itertools import product
from .sympify import _sympify
from .cache import cacheit
from .singleton import S
from .expr import Expr
from .evalf import PrecisionExhausted
from .function import (expand_complex, expand_multinomial,
expand_mul, _mexpand, PoleError)
from .logic import fuzzy_bool, fuzzy_not, fuzzy_and, fuzzy_or
from .parameters import global_parameters
from .relational import is_gt, is_lt
from .kind import NumberKind, UndefinedKind
from sympy.external.gmpy import HAS_GMPY, gmpy
from sympy.utilities.iterables import sift
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.misc import as_int
from sympy.multipledispatch import Dispatcher
from mpmath.libmp import sqrtrem as mpmath_sqrtrem
def isqrt(n):
"""Return the largest integer less than or equal to sqrt(n)."""
if n < 0:
raise ValueError("n must be nonnegative")
n = int(n)
# Fast path: with IEEE 754 binary64 floats and a correctly-rounded
# math.sqrt, int(math.sqrt(n)) works for any integer n satisfying 0 <= n <
# 4503599761588224 = 2**52 + 2**27. But Python doesn't guarantee either
# IEEE 754 format floats *or* correct rounding of math.sqrt, so check the
# answer and fall back to the slow method if necessary.
if n < 4503599761588224:
s = int(_sqrt(n))
if 0 <= n - s*s <= 2*s:
return s
return integer_nthroot(n, 2)[0]
def integer_nthroot(y, n):
"""
Return a tuple containing x = floor(y**(1/n))
and a boolean indicating whether the result is exact (that is,
whether x**n == y).
Examples
========
>>> from sympy import integer_nthroot
>>> integer_nthroot(16, 2)
(4, True)
>>> integer_nthroot(26, 2)
(5, False)
To simply determine if a number is a perfect square, the is_square
function should be used:
>>> from sympy.ntheory.primetest import is_square
>>> is_square(26)
False
See Also
========
sympy.ntheory.primetest.is_square
integer_log
"""
y, n = as_int(y), as_int(n)
if y < 0:
raise ValueError("y must be nonnegative")
if n < 1:
raise ValueError("n must be positive")
if HAS_GMPY and n < 2**63:
# Currently it works only for n < 2**63, else it produces TypeError
# sympy issue: https://github.com/sympy/sympy/issues/18374
# gmpy2 issue: https://github.com/aleaxit/gmpy/issues/257
if HAS_GMPY >= 2:
x, t = gmpy.iroot(y, n)
else:
x, t = gmpy.root(y, n)
return as_int(x), bool(t)
return _integer_nthroot_python(y, n)
def _integer_nthroot_python(y, n):
if y in (0, 1):
return y, True
if n == 1:
return y, True
if n == 2:
x, rem = mpmath_sqrtrem(y)
return int(x), not rem
if n >= y.bit_length():
return 1, False
# Get initial estimate for Newton's method. Care must be taken to
# avoid overflow
try:
guess = int(y**(1./n) + 0.5)
except OverflowError:
exp = _log(y, 2)/n
if exp > 53:
shift = int(exp - 53)
guess = int(2.0**(exp - shift) + 1) << shift
else:
guess = int(2.0**exp)
if guess > 2**50:
# Newton iteration
xprev, x = -1, guess
while 1:
t = x**(n - 1)
xprev, x = x, ((n - 1)*x + y//t)//n
if abs(x - xprev) < 2:
break
else:
x = guess
# Compensate
t = x**n
while t < y:
x += 1
t = x**n
while t > y:
x -= 1
t = x**n
return int(x), t == y # int converts long to int if possible
def integer_log(y, x):
r"""
Returns ``(e, bool)`` where e is the largest nonnegative integer
such that :math:`|y| \geq |x^e|` and ``bool`` is True if $y = x^e$.
Examples
========
>>> from sympy import integer_log
>>> integer_log(125, 5)
(3, True)
>>> integer_log(17, 9)
(1, False)
>>> integer_log(4, -2)
(2, True)
>>> integer_log(-125,-5)
(3, True)
See Also
========
integer_nthroot
sympy.ntheory.primetest.is_square
sympy.ntheory.factor_.multiplicity
sympy.ntheory.factor_.perfect_power
"""
if x == 1:
raise ValueError('x cannot take value as 1')
if y == 0:
raise ValueError('y cannot take value as 0')
if x in (-2, 2):
x = int(x)
y = as_int(y)
e = y.bit_length() - 1
return e, x**e == y
if x < 0:
n, b = integer_log(y if y > 0 else -y, -x)
return n, b and bool(n % 2 if y < 0 else not n % 2)
x = as_int(x)
y = as_int(y)
r = e = 0
while y >= x:
d = x
m = 1
while y >= d:
y, rem = divmod(y, d)
r = r or rem
e += m
if y > d:
d *= d
m *= 2
return e, r == 0 and y == 1
class Pow(Expr):
"""
Defines the expression x**y as "x raised to a power y"
.. deprecated:: 1.7
Using arguments that aren't subclasses of :class:`~.Expr` in core
operators (:class:`~.Mul`, :class:`~.Add`, and :class:`~.Pow`) is
deprecated. See :ref:`non-expr-args-deprecated` for details.
Singleton definitions involving (0, 1, -1, oo, -oo, I, -I):
+--------------+---------+-----------------------------------------------+
| expr | value | reason |
+==============+=========+===============================================+
| z**0 | 1 | Although arguments over 0**0 exist, see [2]. |
+--------------+---------+-----------------------------------------------+
| z**1 | z | |
+--------------+---------+-----------------------------------------------+
| (-oo)**(-1) | 0 | |
+--------------+---------+-----------------------------------------------+
| (-1)**-1 | -1 | |
+--------------+---------+-----------------------------------------------+
| S.Zero**-1 | zoo | This is not strictly true, as 0**-1 may be |
| | | undefined, but is convenient in some contexts |
| | | where the base is assumed to be positive. |
+--------------+---------+-----------------------------------------------+
| 1**-1 | 1 | |
+--------------+---------+-----------------------------------------------+
| oo**-1 | 0 | |
+--------------+---------+-----------------------------------------------+
| 0**oo | 0 | Because for all complex numbers z near |
| | | 0, z**oo -> 0. |
+--------------+---------+-----------------------------------------------+
| 0**-oo | zoo | This is not strictly true, as 0**oo may be |
| | | oscillating between positive and negative |
| | | values or rotating in the complex plane. |
| | | It is convenient, however, when the base |
| | | is positive. |
+--------------+---------+-----------------------------------------------+
| 1**oo | nan | Because there are various cases where |
| 1**-oo | | lim(x(t),t)=1, lim(y(t),t)=oo (or -oo), |
| | | but lim( x(t)**y(t), t) != 1. See [3]. |
+--------------+---------+-----------------------------------------------+
| b**zoo | nan | Because b**z has no limit as z -> zoo |
+--------------+---------+-----------------------------------------------+
| (-1)**oo | nan | Because of oscillations in the limit. |
| (-1)**(-oo) | | |
+--------------+---------+-----------------------------------------------+
| oo**oo | oo | |
+--------------+---------+-----------------------------------------------+
| oo**-oo | 0 | |
+--------------+---------+-----------------------------------------------+
| (-oo)**oo | nan | |
| (-oo)**-oo | | |
+--------------+---------+-----------------------------------------------+
| oo**I | nan | oo**e could probably be best thought of as |
| (-oo)**I | | the limit of x**e for real x as x tends to |
| | | oo. If e is I, then the limit does not exist |
| | | and nan is used to indicate that. |
+--------------+---------+-----------------------------------------------+
| oo**(1+I) | zoo | If the real part of e is positive, then the |
| (-oo)**(1+I) | | limit of abs(x**e) is oo. So the limit value |
| | | is zoo. |
+--------------+---------+-----------------------------------------------+
| oo**(-1+I) | 0 | If the real part of e is negative, then the |
| -oo**(-1+I) | | limit is 0. |
+--------------+---------+-----------------------------------------------+
Because symbolic computations are more flexible than floating point
calculations and we prefer to never return an incorrect answer,
we choose not to conform to all IEEE 754 conventions. This helps
us avoid extra test-case code in the calculation of limits.
See Also
========
sympy.core.numbers.Infinity
sympy.core.numbers.NegativeInfinity
sympy.core.numbers.NaN
References
==========
.. [1] https://en.wikipedia.org/wiki/Exponentiation
.. [2] https://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero
.. [3] https://en.wikipedia.org/wiki/Indeterminate_forms
"""
is_Pow = True
__slots__ = ('is_commutative',)
args: tuple[Expr, Expr]
_args: tuple[Expr, Expr]
@cacheit
def __new__(cls, b, e, evaluate=None):
if evaluate is None:
evaluate = global_parameters.evaluate
b = _sympify(b)
e = _sympify(e)
# XXX: This can be removed when non-Expr args are disallowed rather
# than deprecated.
from .relational import Relational
if isinstance(b, Relational) or isinstance(e, Relational):
raise TypeError('Relational cannot be used in Pow')
# XXX: This should raise TypeError once deprecation period is over:
for arg in [b, e]:
if not isinstance(arg, Expr):
sympy_deprecation_warning(
f"""
Using non-Expr arguments in Pow is deprecated (in this case, one of the
arguments is of type {type(arg).__name__!r}).
If you really did intend to construct a power with this base, use the **
operator instead.""",
deprecated_since_version="1.7",
active_deprecations_target="non-expr-args-deprecated",
stacklevel=4,
)
if evaluate:
if e is S.ComplexInfinity:
return S.NaN
if e is S.Infinity:
if is_gt(b, S.One):
return S.Infinity
if is_gt(b, S.NegativeOne) and is_lt(b, S.One):
return S.Zero
if is_lt(b, S.NegativeOne):
if b.is_finite:
return S.ComplexInfinity
if b.is_finite is False:
return S.NaN
if e is S.Zero:
return S.One
elif e is S.One:
return b
elif e == -1 and not b:
return S.ComplexInfinity
elif e.__class__.__name__ == "AccumulationBounds":
if b == S.Exp1:
from sympy.calculus.accumulationbounds import AccumBounds
return AccumBounds(Pow(b, e.min), Pow(b, e.max))
# autosimplification if base is a number and exp odd/even
# if base is Number then the base will end up positive; we
# do not do this with arbitrary expressions since symbolic
# cancellation might occur as in (x - 1)/(1 - x) -> -1. If
# we returned Piecewise((-1, Ne(x, 1))) for such cases then
# we could do this...but we don't
elif (e.is_Symbol and e.is_integer or e.is_Integer
) and (b.is_number and b.is_Mul or b.is_Number
) and b.could_extract_minus_sign():
if e.is_even:
b = -b
elif e.is_odd:
return -Pow(-b, e)
if S.NaN in (b, e): # XXX S.NaN**x -> S.NaN under assumption that x != 0
return S.NaN
elif b is S.One:
if abs(e).is_infinite:
return S.NaN
return S.One
else:
# recognize base as E
from sympy.functions.elementary.exponential import exp_polar
if not e.is_Atom and b is not S.Exp1 and not isinstance(b, exp_polar):
from .exprtools import factor_terms
from sympy.functions.elementary.exponential import log
from sympy.simplify.radsimp import fraction
c, ex = factor_terms(e, sign=False).as_coeff_Mul()
num, den = fraction(ex)
if isinstance(den, log) and den.args[0] == b:
return S.Exp1**(c*num)
elif den.is_Add:
from sympy.functions.elementary.complexes import sign, im
s = sign(im(b))
if s.is_Number and s and den == \
log(-factor_terms(b, sign=False)) + s*S.ImaginaryUnit*S.Pi:
return S.Exp1**(c*num)
obj = b._eval_power(e)
if obj is not None:
return obj
obj = Expr.__new__(cls, b, e)
obj = cls._exec_constructor_postprocessors(obj)
if not isinstance(obj, Pow):
return obj
obj.is_commutative = (b.is_commutative and e.is_commutative)
return obj
def inverse(self, argindex=1):
if self.base == S.Exp1:
from sympy.functions.elementary.exponential import log
return log
return None
@property
def base(self) -> Expr:
return self._args[0]
@property
def exp(self) -> Expr:
return self._args[1]
@property
def kind(self):
if self.exp.kind is NumberKind:
return self.base.kind
else:
return UndefinedKind
@classmethod
def class_key(cls):
return 3, 2, cls.__name__
def _eval_refine(self, assumptions):
from sympy.assumptions.ask import ask, Q
b, e = self.as_base_exp()
if ask(Q.integer(e), assumptions) and b.could_extract_minus_sign():
if ask(Q.even(e), assumptions):
return Pow(-b, e)
elif ask(Q.odd(e), assumptions):
return -Pow(-b, e)
def _eval_power(self, other):
b, e = self.as_base_exp()
if b is S.NaN:
return (b**e)**other # let __new__ handle it
s = None
if other.is_integer:
s = 1
elif b.is_polar: # e.g. exp_polar, besselj, var('p', polar=True)...
s = 1
elif e.is_extended_real is not None:
from sympy.functions.elementary.complexes import arg, im, re, sign
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.integers import floor
# helper functions ===========================
def _half(e):
"""Return True if the exponent has a literal 2 as the
denominator, else None."""
if getattr(e, 'q', None) == 2:
return True
n, d = e.as_numer_denom()
if n.is_integer and d == 2:
return True
def _n2(e):
"""Return ``e`` evaluated to a Number with 2 significant
digits, else None."""
try:
rv = e.evalf(2, strict=True)
if rv.is_Number:
return rv
except PrecisionExhausted:
pass
# ===================================================
if e.is_extended_real:
# we need _half(other) with constant floor or
# floor(S.Half - e*arg(b)/2/pi) == 0
# handle -1 as special case
if e == -1:
# floor arg. is 1/2 + arg(b)/2/pi
if _half(other):
if b.is_negative is True:
return S.NegativeOne**other*Pow(-b, e*other)
elif b.is_negative is False: # XXX ok if im(b) != 0?
return Pow(b, -other)
elif e.is_even:
if b.is_extended_real:
b = abs(b)
if b.is_imaginary:
b = abs(im(b))*S.ImaginaryUnit
if (abs(e) < 1) == True or e == 1:
s = 1 # floor = 0
elif b.is_extended_nonnegative:
s = 1 # floor = 0
elif re(b).is_extended_nonnegative and (abs(e) < 2) == True:
s = 1 # floor = 0
elif _half(other):
s = exp(2*S.Pi*S.ImaginaryUnit*other*floor(
S.Half - e*arg(b)/(2*S.Pi)))
if s.is_extended_real and _n2(sign(s) - s) == 0:
s = sign(s)
else:
s = None
else:
# e.is_extended_real is False requires:
# _half(other) with constant floor or
# floor(S.Half - im(e*log(b))/2/pi) == 0
try:
s = exp(2*S.ImaginaryUnit*S.Pi*other*
floor(S.Half - im(e*log(b))/2/S.Pi))
# be careful to test that s is -1 or 1 b/c sign(I) == I:
# so check that s is real
if s.is_extended_real and _n2(sign(s) - s) == 0:
s = sign(s)
else:
s = None
except PrecisionExhausted:
s = None
if s is not None:
return s*Pow(b, e*other)
def _eval_Mod(self, q):
r"""A dispatched function to compute `b^e \bmod q`, dispatched
by ``Mod``.
Notes
=====
Algorithms:
1. For unevaluated integer power, use built-in ``pow`` function
with 3 arguments, if powers are not too large wrt base.
2. For very large powers, use totient reduction if $e \ge \log(m)$.
Bound on m, is for safe factorization memory wise i.e. $m^{1/4}$.
For pollard-rho to be faster than built-in pow $\log(e) > m^{1/4}$
check is added.
3. For any unevaluated power found in `b` or `e`, the step 2
will be recursed down to the base and the exponent
such that the $b \bmod q$ becomes the new base and
$\phi(q) + e \bmod \phi(q)$ becomes the new exponent, and then
the computation for the reduced expression can be done.
"""
base, exp = self.base, self.exp
if exp.is_integer and exp.is_positive:
if q.is_integer and base % q == 0:
return S.Zero
from sympy.ntheory.factor_ import totient
if base.is_Integer and exp.is_Integer and q.is_Integer:
b, e, m = int(base), int(exp), int(q)
mb = m.bit_length()
if mb <= 80 and e >= mb and e.bit_length()**4 >= m:
phi = int(totient(m))
return Integer(pow(b, phi + e%phi, m))
return Integer(pow(b, e, m))
from .mod import Mod
if isinstance(base, Pow) and base.is_integer and base.is_number:
base = Mod(base, q)
return Mod(Pow(base, exp, evaluate=False), q)
if isinstance(exp, Pow) and exp.is_integer and exp.is_number:
bit_length = int(q).bit_length()
# XXX Mod-Pow actually attempts to do a hanging evaluation
# if this dispatched function returns None.
# May need some fixes in the dispatcher itself.
if bit_length <= 80:
phi = totient(q)
exp = phi + Mod(exp, phi)
return Mod(Pow(base, exp, evaluate=False), q)
def _eval_is_even(self):
if self.exp.is_integer and self.exp.is_positive:
return self.base.is_even
def _eval_is_negative(self):
ext_neg = Pow._eval_is_extended_negative(self)
if ext_neg is True:
return self.is_finite
return ext_neg
def _eval_is_extended_positive(self):
if self.base == self.exp:
if self.base.is_extended_nonnegative:
return True
elif self.base.is_positive:
if self.exp.is_real:
return True
elif self.base.is_extended_negative:
if self.exp.is_even:
return True
if self.exp.is_odd:
return False
elif self.base.is_zero:
if self.exp.is_extended_real:
return self.exp.is_zero
elif self.base.is_extended_nonpositive:
if self.exp.is_odd:
return False
elif self.base.is_imaginary:
if self.exp.is_integer:
m = self.exp % 4
if m.is_zero:
return True
if m.is_integer and m.is_zero is False:
return False
if self.exp.is_imaginary:
from sympy.functions.elementary.exponential import log
return log(self.base).is_imaginary
def _eval_is_extended_negative(self):
if self.exp is S.Half:
if self.base.is_complex or self.base.is_extended_real:
return False
if self.base.is_extended_negative:
if self.exp.is_odd and self.base.is_finite:
return True
if self.exp.is_even:
return False
elif self.base.is_extended_positive:
if self.exp.is_extended_real:
return False
elif self.base.is_zero:
if self.exp.is_extended_real:
return False
elif self.base.is_extended_nonnegative:
if self.exp.is_extended_nonnegative:
return False
elif self.base.is_extended_nonpositive:
if self.exp.is_even:
return False
elif self.base.is_extended_real:
if self.exp.is_even:
return False
def _eval_is_zero(self):
if self.base.is_zero:
if self.exp.is_extended_positive:
return True
elif self.exp.is_extended_nonpositive:
return False
elif self.base == S.Exp1:
return self.exp is S.NegativeInfinity
elif self.base.is_zero is False:
if self.base.is_finite and self.exp.is_finite:
return False
elif self.exp.is_negative:
return self.base.is_infinite
elif self.exp.is_nonnegative:
return False
elif self.exp.is_infinite and self.exp.is_extended_real:
if (1 - abs(self.base)).is_extended_positive:
return self.exp.is_extended_positive
elif (1 - abs(self.base)).is_extended_negative:
return self.exp.is_extended_negative
elif self.base.is_finite and self.exp.is_negative:
# when self.base.is_zero is None
return False
def _eval_is_integer(self):
b, e = self.args
if b.is_rational:
if b.is_integer is False and e.is_positive:
return False # rat**nonneg
if b.is_integer and e.is_integer:
if b is S.NegativeOne:
return True
if e.is_nonnegative or e.is_positive:
return True
if b.is_integer and e.is_negative and (e.is_finite or e.is_integer):
if fuzzy_not((b - 1).is_zero) and fuzzy_not((b + 1).is_zero):
return False
if b.is_Number and e.is_Number:
check = self.func(*self.args)
return check.is_Integer
if e.is_negative and b.is_positive and (b - 1).is_positive:
return False
if e.is_negative and b.is_negative and (b + 1).is_negative:
return False
def _eval_is_extended_real(self):
if self.base is S.Exp1:
if self.exp.is_extended_real:
return True
elif self.exp.is_imaginary:
return (2*S.ImaginaryUnit*self.exp/S.Pi).is_even
from sympy.functions.elementary.exponential import log, exp
real_b = self.base.is_extended_real
if real_b is None:
if self.base.func == exp and self.base.exp.is_imaginary:
return self.exp.is_imaginary
if self.base.func == Pow and self.base.base is S.Exp1 and self.base.exp.is_imaginary:
return self.exp.is_imaginary
return
real_e = self.exp.is_extended_real
if real_e is None:
return
if real_b and real_e:
if self.base.is_extended_positive:
return True
elif self.base.is_extended_nonnegative and self.exp.is_extended_nonnegative:
return True
elif self.exp.is_integer and self.base.is_extended_nonzero:
return True
elif self.exp.is_integer and self.exp.is_nonnegative:
return True
elif self.base.is_extended_negative:
if self.exp.is_Rational:
return False
if real_e and self.exp.is_extended_negative and self.base.is_zero is False:
return Pow(self.base, -self.exp).is_extended_real
im_b = self.base.is_imaginary
im_e = self.exp.is_imaginary
if im_b:
if self.exp.is_integer:
if self.exp.is_even:
return True
elif self.exp.is_odd:
return False
elif im_e and log(self.base).is_imaginary:
return True
elif self.exp.is_Add:
c, a = self.exp.as_coeff_Add()
if c and c.is_Integer:
return Mul(
self.base**c, self.base**a, evaluate=False).is_extended_real
elif self.base in (-S.ImaginaryUnit, S.ImaginaryUnit):
if (self.exp/2).is_integer is False:
return False
if real_b and im_e:
if self.base is S.NegativeOne:
return True
c = self.exp.coeff(S.ImaginaryUnit)
if c:
if self.base.is_rational and c.is_rational:
if self.base.is_nonzero and (self.base - 1).is_nonzero and c.is_nonzero:
return False
ok = (c*log(self.base)/S.Pi).is_integer
if ok is not None:
return ok
if real_b is False and real_e: # we already know it's not imag
from sympy.functions.elementary.complexes import arg
i = arg(self.base)*self.exp/S.Pi
if i.is_complex: # finite
return i.is_integer
def _eval_is_complex(self):
if self.base == S.Exp1:
return fuzzy_or([self.exp.is_complex, self.exp.is_extended_negative])
if all(a.is_complex for a in self.args) and self._eval_is_finite():
return True
def _eval_is_imaginary(self):
if self.base.is_commutative is False:
return False
if self.base.is_imaginary:
if self.exp.is_integer:
odd = self.exp.is_odd
if odd is not None:
return odd
return
if self.base == S.Exp1:
f = 2 * self.exp / (S.Pi*S.ImaginaryUnit)
# exp(pi*integer) = 1 or -1, so not imaginary
if f.is_even:
return False
# exp(pi*integer + pi/2) = I or -I, so it is imaginary
if f.is_odd:
return True
return None
if self.exp.is_imaginary:
from sympy.functions.elementary.exponential import log
imlog = log(self.base).is_imaginary
if imlog is not None:
return False # I**i -> real; (2*I)**i -> complex ==> not imaginary
if self.base.is_extended_real and self.exp.is_extended_real:
if self.base.is_positive:
return False
else:
rat = self.exp.is_rational
if not rat:
return rat
if self.exp.is_integer:
return False
else:
half = (2*self.exp).is_integer
if half:
return self.base.is_negative
return half
if self.base.is_extended_real is False: # we already know it's not imag
from sympy.functions.elementary.complexes import arg
i = arg(self.base)*self.exp/S.Pi
isodd = (2*i).is_odd
if isodd is not None:
return isodd
def _eval_is_odd(self):
if self.exp.is_integer:
if self.exp.is_positive:
return self.base.is_odd
elif self.exp.is_nonnegative and self.base.is_odd:
return True
elif self.base is S.NegativeOne:
return True
def _eval_is_finite(self):
if self.exp.is_negative:
if self.base.is_zero:
return False
if self.base.is_infinite or self.base.is_nonzero:
return True
c1 = self.base.is_finite
if c1 is None:
return
c2 = self.exp.is_finite
if c2 is None:
return
if c1 and c2:
if self.exp.is_nonnegative or fuzzy_not(self.base.is_zero):
return True
def _eval_is_prime(self):
'''
An integer raised to the n(>=2)-th power cannot be a prime.
'''
if self.base.is_integer and self.exp.is_integer and (self.exp - 1).is_positive:
return False
def _eval_is_composite(self):
"""
A power is composite if both base and exponent are greater than 1
"""
if (self.base.is_integer and self.exp.is_integer and
((self.base - 1).is_positive and (self.exp - 1).is_positive or
(self.base + 1).is_negative and self.exp.is_positive and self.exp.is_even)):
return True
def _eval_is_polar(self):
return self.base.is_polar
def _eval_subs(self, old, new):
from sympy.calculus.accumulationbounds import AccumBounds
if isinstance(self.exp, AccumBounds):
b = self.base.subs(old, new)
e = self.exp.subs(old, new)
if isinstance(e, AccumBounds):
return e.__rpow__(b)
return self.func(b, e)
from sympy.functions.elementary.exponential import exp, log
def _check(ct1, ct2, old):
"""Return (bool, pow, remainder_pow) where, if bool is True, then the
exponent of Pow `old` will combine with `pow` so the substitution
is valid, otherwise bool will be False.
For noncommutative objects, `pow` will be an integer, and a factor
`Pow(old.base, remainder_pow)` needs to be included. If there is
no such factor, None is returned. For commutative objects,
remainder_pow is always None.
cti are the coefficient and terms of an exponent of self or old
In this _eval_subs routine a change like (b**(2*x)).subs(b**x, y)
will give y**2 since (b**x)**2 == b**(2*x); if that equality does
not hold then the substitution should not occur so `bool` will be
False.
"""
coeff1, terms1 = ct1
coeff2, terms2 = ct2
if terms1 == terms2:
if old.is_commutative:
# Allow fractional powers for commutative objects
pow = coeff1/coeff2
try:
as_int(pow, strict=False)
combines = True
except ValueError:
b, e = old.as_base_exp()
# These conditions ensure that (b**e)**f == b**(e*f) for any f
combines = b.is_positive and e.is_real or b.is_nonnegative and e.is_nonnegative
return combines, pow, None
else:
# With noncommutative symbols, substitute only integer powers
if not isinstance(terms1, tuple):
terms1 = (terms1,)
if not all(term.is_integer for term in terms1):
return False, None, None
try:
# Round pow toward zero
pow, remainder = divmod(as_int(coeff1), as_int(coeff2))
if pow < 0 and remainder != 0:
pow += 1
remainder -= as_int(coeff2)
if remainder == 0:
remainder_pow = None
else:
remainder_pow = Mul(remainder, *terms1)
return True, pow, remainder_pow
except ValueError:
# Can't substitute
pass
return False, None, None
if old == self.base or (old == exp and self.base == S.Exp1):
if new.is_Function and isinstance(new, Callable):
return new(self.exp._subs(old, new))
else:
return new**self.exp._subs(old, new)
# issue 10829: (4**x - 3*y + 2).subs(2**x, y) -> y**2 - 3*y + 2
if isinstance(old, self.func) and self.exp == old.exp:
l = log(self.base, old.base)
if l.is_Number:
return Pow(new, l)
if isinstance(old, self.func) and self.base == old.base:
if self.exp.is_Add is False:
ct1 = self.exp.as_independent(Symbol, as_Add=False)
ct2 = old.exp.as_independent(Symbol, as_Add=False)
ok, pow, remainder_pow = _check(ct1, ct2, old)
if ok:
# issue 5180: (x**(6*y)).subs(x**(3*y),z)->z**2
result = self.func(new, pow)
if remainder_pow is not None:
result = Mul(result, Pow(old.base, remainder_pow))
return result
else: # b**(6*x + a).subs(b**(3*x), y) -> y**2 * b**a
# exp(exp(x) + exp(x**2)).subs(exp(exp(x)), w) -> w * exp(exp(x**2))
oarg = old.exp
new_l = []
o_al = []
ct2 = oarg.as_coeff_mul()
for a in self.exp.args:
newa = a._subs(old, new)
ct1 = newa.as_coeff_mul()
ok, pow, remainder_pow = _check(ct1, ct2, old)
if ok:
new_l.append(new**pow)
if remainder_pow is not None:
o_al.append(remainder_pow)
continue
elif not old.is_commutative and not newa.is_integer:
# If any term in the exponent is non-integer,
# we do not do any substitutions in the noncommutative case
return
o_al.append(newa)
if new_l:
expo = Add(*o_al)
new_l.append(Pow(self.base, expo, evaluate=False) if expo != 1 else self.base)
return Mul(*new_l)
if (isinstance(old, exp) or (old.is_Pow and old.base is S.Exp1)) and self.exp.is_extended_real and self.base.is_positive:
ct1 = old.exp.as_independent(Symbol, as_Add=False)
ct2 = (self.exp*log(self.base)).as_independent(
Symbol, as_Add=False)
ok, pow, remainder_pow = _check(ct1, ct2, old)
if ok:
result = self.func(new, pow) # (2**x).subs(exp(x*log(2)), z) -> z
if remainder_pow is not None:
result = Mul(result, Pow(old.base, remainder_pow))
return result
def as_base_exp(self):
"""Return base and exp of self.
Explanation
===========
If base a Rational less than 1, then return 1/Rational, -exp.
If this extra processing is not needed, the base and exp
properties will give the raw arguments.
Examples
========
>>> from sympy import Pow, S
>>> p = Pow(S.Half, 2, evaluate=False)
>>> p.as_base_exp()
(2, -2)
>>> p.args
(1/2, 2)
>>> p.base, p.exp
(1/2, 2)
"""
b, e = self.args
if b.is_Rational and b.p < b.q and b.p > 0:
return 1/b, -e
return b, e
def _eval_adjoint(self):
from sympy.functions.elementary.complexes import adjoint
i, p = self.exp.is_integer, self.base.is_positive
if i:
return adjoint(self.base)**self.exp
if p:
return self.base**adjoint(self.exp)
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return adjoint(expanded)
def _eval_conjugate(self):
from sympy.functions.elementary.complexes import conjugate as c
i, p = self.exp.is_integer, self.base.is_positive
if i:
return c(self.base)**self.exp
if p:
return self.base**c(self.exp)
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return c(expanded)
if self.is_extended_real:
return self
def _eval_transpose(self):
from sympy.functions.elementary.complexes import transpose
if self.base == S.Exp1:
return self.func(S.Exp1, self.exp.transpose())
i, p = self.exp.is_integer, (self.base.is_complex or self.base.is_infinite)
if p:
return self.base**self.exp
if i:
return transpose(self.base)**self.exp
if i is False and p is False:
expanded = expand_complex(self)
if expanded != self:
return transpose(expanded)
def _eval_expand_power_exp(self, **hints):
"""a**(n + m) -> a**n*a**m"""
b = self.base
e = self.exp
if b == S.Exp1:
from sympy.concrete.summations import Sum
if isinstance(e, Sum) and e.is_commutative:
from sympy.concrete.products import Product
return Product(self.func(b, e.function), *e.limits)
if e.is_Add and e.is_commutative:
return Mul(*[self.func(b, x) for x in e.args])
return self.func(b, e)
def _eval_expand_power_base(self, **hints):
"""(a*b)**n -> a**n * b**n"""
force = hints.get('force', False)
b = self.base
e = self.exp
if not b.is_Mul:
return self
cargs, nc = b.args_cnc(split_1=False)
# expand each term - this is top-level-only
# expansion but we have to watch out for things
# that don't have an _eval_expand method
if nc:
nc = [i._eval_expand_power_base(**hints)
if hasattr(i, '_eval_expand_power_base') else i
for i in nc]
if e.is_Integer:
if e.is_positive:
rv = Mul(*nc*e)
else:
rv = Mul(*[i**-1 for i in nc[::-1]]*-e)
if cargs:
rv *= Mul(*cargs)**e
return rv
if not cargs:
return self.func(Mul(*nc), e, evaluate=False)
nc = [Mul(*nc)]
# sift the commutative bases
other, maybe_real = sift(cargs, lambda x: x.is_extended_real is False,
binary=True)
def pred(x):
if x is S.ImaginaryUnit:
return S.ImaginaryUnit
polar = x.is_polar
if polar:
return True
if polar is None:
return fuzzy_bool(x.is_extended_nonnegative)
sifted = sift(maybe_real, pred)
nonneg = sifted[True]
other += sifted[None]
neg = sifted[False]
imag = sifted[S.ImaginaryUnit]
if imag:
I = S.ImaginaryUnit
i = len(imag) % 4
if i == 0:
pass
elif i == 1:
other.append(I)
elif i == 2:
if neg:
nonn = -neg.pop()
if nonn is not S.One:
nonneg.append(nonn)
else:
neg.append(S.NegativeOne)
else:
if neg:
nonn = -neg.pop()
if nonn is not S.One:
nonneg.append(nonn)
else:
neg.append(S.NegativeOne)
other.append(I)
del imag
# bring out the bases that can be separated from the base
if force or e.is_integer:
# treat all commutatives the same and put nc in other
cargs = nonneg + neg + other
other = nc
else:
# this is just like what is happening automatically, except
# that now we are doing it for an arbitrary exponent for which
# no automatic expansion is done
assert not e.is_Integer
# handle negatives by making them all positive and putting
# the residual -1 in other
if len(neg) > 1:
o = S.One
if not other and neg[0].is_Number:
o *= neg.pop(0)
if len(neg) % 2:
o = -o
for n in neg:
nonneg.append(-n)
if o is not S.One:
other.append(o)
elif neg and other:
if neg[0].is_Number and neg[0] is not S.NegativeOne:
other.append(S.NegativeOne)
nonneg.append(-neg[0])
else:
other.extend(neg)
else:
other.extend(neg)
del neg
cargs = nonneg
other += nc
rv = S.One
if cargs:
if e.is_Rational:
npow, cargs = sift(cargs, lambda x: x.is_Pow and
x.exp.is_Rational and x.base.is_number,
binary=True)
rv = Mul(*[self.func(b.func(*b.args), e) for b in npow])
rv *= Mul(*[self.func(b, e, evaluate=False) for b in cargs])
if other:
rv *= self.func(Mul(*other), e, evaluate=False)
return rv
def _eval_expand_multinomial(self, **hints):
"""(a + b + ..)**n -> a**n + n*a**(n-1)*b + .., n is nonzero integer"""
base, exp = self.args
result = self
if exp.is_Rational and exp.p > 0 and base.is_Add:
if not exp.is_Integer:
n = Integer(exp.p // exp.q)
if not n:
return result
else:
radical, result = self.func(base, exp - n), []
expanded_base_n = self.func(base, n)
if expanded_base_n.is_Pow:
expanded_base_n = \
expanded_base_n._eval_expand_multinomial()
for term in Add.make_args(expanded_base_n):
result.append(term*radical)
return Add(*result)
n = int(exp)
if base.is_commutative:
order_terms, other_terms = [], []
for b in base.args:
if b.is_Order:
order_terms.append(b)
else:
other_terms.append(b)
if order_terms:
# (f(x) + O(x^n))^m -> f(x)^m + m*f(x)^{m-1} *O(x^n)
f = Add(*other_terms)
o = Add(*order_terms)
if n == 2:
return expand_multinomial(f**n, deep=False) + n*f*o
else:
g = expand_multinomial(f**(n - 1), deep=False)
return expand_mul(f*g, deep=False) + n*g*o
if base.is_number:
# Efficiently expand expressions of the form (a + b*I)**n
# where 'a' and 'b' are real numbers and 'n' is integer.
a, b = base.as_real_imag()
if a.is_Rational and b.is_Rational:
if not a.is_Integer:
if not b.is_Integer:
k = self.func(a.q * b.q, n)
a, b = a.p*b.q, a.q*b.p
else:
k = self.func(a.q, n)
a, b = a.p, a.q*b
elif not b.is_Integer:
k = self.func(b.q, n)
a, b = a*b.q, b.p
else:
k = 1
a, b, c, d = int(a), int(b), 1, 0
while n:
if n & 1:
c, d = a*c - b*d, b*c + a*d
n -= 1
a, b = a*a - b*b, 2*a*b
n //= 2
I = S.ImaginaryUnit
if k == 1:
return c + I*d
else:
return Integer(c)/k + I*d/k
p = other_terms
# (x + y)**3 -> x**3 + 3*x**2*y + 3*x*y**2 + y**3
# in this particular example:
# p = [x,y]; n = 3
# so now it's easy to get the correct result -- we get the
# coefficients first:
from sympy.ntheory.multinomial import multinomial_coefficients
from sympy.polys.polyutils import basic_from_dict
expansion_dict = multinomial_coefficients(len(p), n)
# in our example: {(3, 0): 1, (1, 2): 3, (0, 3): 1, (2, 1): 3}
# and now construct the expression.
return basic_from_dict(expansion_dict, *p)
else:
if n == 2:
return Add(*[f*g for f in base.args for g in base.args])
else:
multi = (base**(n - 1))._eval_expand_multinomial()
if multi.is_Add:
return Add(*[f*g for f in base.args
for g in multi.args])
else:
# XXX can this ever happen if base was an Add?
return Add(*[f*multi for f in base.args])
elif (exp.is_Rational and exp.p < 0 and base.is_Add and
abs(exp.p) > exp.q):
return 1 / self.func(base, -exp)._eval_expand_multinomial()
elif exp.is_Add and base.is_Number:
# a + b a b
# n --> n n, where n, a, b are Numbers
coeff, tail = S.One, S.Zero
for term in exp.args:
if term.is_Number:
coeff *= self.func(base, term)
else:
tail += term
return coeff * self.func(base, tail)
else:
return result
def as_real_imag(self, deep=True, **hints):
if self.exp.is_Integer:
from sympy.polys.polytools import poly
exp = self.exp
re_e, im_e = self.base.as_real_imag(deep=deep)
if not im_e:
return self, S.Zero
a, b = symbols('a b', cls=Dummy)
if exp >= 0:
if re_e.is_Number and im_e.is_Number:
# We can be more efficient in this case
expr = expand_multinomial(self.base**exp)
if expr != self:
return expr.as_real_imag()
expr = poly(
(a + b)**exp) # a = re, b = im; expr = (a + b*I)**exp
else:
mag = re_e**2 + im_e**2
re_e, im_e = re_e/mag, -im_e/mag
if re_e.is_Number and im_e.is_Number:
# We can be more efficient in this case
expr = expand_multinomial((re_e + im_e*S.ImaginaryUnit)**-exp)
if expr != self:
return expr.as_real_imag()
expr = poly((a + b)**-exp)
# Terms with even b powers will be real
r = [i for i in expr.terms() if not i[0][1] % 2]
re_part = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
# Terms with odd b powers will be imaginary
r = [i for i in expr.terms() if i[0][1] % 4 == 1]
im_part1 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
r = [i for i in expr.terms() if i[0][1] % 4 == 3]
im_part3 = Add(*[cc*a**aa*b**bb for (aa, bb), cc in r])
return (re_part.subs({a: re_e, b: S.ImaginaryUnit*im_e}),
im_part1.subs({a: re_e, b: im_e}) + im_part3.subs({a: re_e, b: -im_e}))
from sympy.functions.elementary.trigonometric import atan2, cos, sin
if self.exp.is_Rational:
re_e, im_e = self.base.as_real_imag(deep=deep)
if im_e.is_zero and self.exp is S.Half:
if re_e.is_extended_nonnegative:
return self, S.Zero
if re_e.is_extended_nonpositive:
return S.Zero, (-self.base)**self.exp
# XXX: This is not totally correct since for x**(p/q) with
# x being imaginary there are actually q roots, but
# only a single one is returned from here.
r = self.func(self.func(re_e, 2) + self.func(im_e, 2), S.Half)
t = atan2(im_e, re_e)
rp, tp = self.func(r, self.exp), t*self.exp
return rp*cos(tp), rp*sin(tp)
elif self.base is S.Exp1:
from sympy.functions.elementary.exponential import exp
re_e, im_e = self.exp.as_real_imag()
if deep:
re_e = re_e.expand(deep, **hints)
im_e = im_e.expand(deep, **hints)
c, s = cos(im_e), sin(im_e)
return exp(re_e)*c, exp(re_e)*s
else:
from sympy.functions.elementary.complexes import im, re
if deep:
hints['complex'] = False
expanded = self.expand(deep, **hints)
if hints.get('ignore') == expanded:
return None
else:
return (re(expanded), im(expanded))
else:
return re(self), im(self)
def _eval_derivative(self, s):
from sympy.functions.elementary.exponential import log
dbase = self.base.diff(s)
dexp = self.exp.diff(s)
return self * (dexp * log(self.base) + dbase * self.exp/self.base)
def _eval_evalf(self, prec):
base, exp = self.as_base_exp()
if base == S.Exp1:
# Use mpmath function associated to class "exp":
from sympy.functions.elementary.exponential import exp as exp_function
return exp_function(self.exp, evaluate=False)._eval_evalf(prec)
base = base._evalf(prec)
if not exp.is_Integer:
exp = exp._evalf(prec)
if exp.is_negative and base.is_number and base.is_extended_real is False:
base = base.conjugate() / (base * base.conjugate())._evalf(prec)
exp = -exp
return self.func(base, exp).expand()
return self.func(base, exp)
def _eval_is_polynomial(self, syms):
if self.exp.has(*syms):
return False
if self.base.has(*syms):
return bool(self.base._eval_is_polynomial(syms) and
self.exp.is_Integer and (self.exp >= 0))
else:
return True
def _eval_is_rational(self):
# The evaluation of self.func below can be very expensive in the case
# of integer**integer if the exponent is large. We should try to exit
# before that if possible:
if (self.exp.is_integer and self.base.is_rational
and fuzzy_not(fuzzy_and([self.exp.is_negative, self.base.is_zero]))):
return True
p = self.func(*self.as_base_exp()) # in case it's unevaluated
if not p.is_Pow:
return p.is_rational
b, e = p.as_base_exp()
if e.is_Rational and b.is_Rational:
# we didn't check that e is not an Integer
# because Rational**Integer autosimplifies
return False
if e.is_integer:
if b.is_rational:
if fuzzy_not(b.is_zero) or e.is_nonnegative:
return True
if b == e: # always rational, even for 0**0
return True
elif b.is_irrational:
return e.is_zero
if b is S.Exp1:
if e.is_rational and e.is_nonzero:
return False
def _eval_is_algebraic(self):
def _is_one(expr):
try:
return (expr - 1).is_zero
except ValueError:
# when the operation is not allowed
return False
if self.base.is_zero or _is_one(self.base):
return True
elif self.base is S.Exp1:
s = self.func(*self.args)
if s.func == self.func:
if self.exp.is_nonzero:
if self.exp.is_algebraic:
return False
elif (self.exp/S.Pi).is_rational:
return False
elif (self.exp/(S.ImaginaryUnit*S.Pi)).is_rational:
return True
else:
return s.is_algebraic
elif self.exp.is_rational:
if self.base.is_algebraic is False:
return self.exp.is_zero
if self.base.is_zero is False:
if self.exp.is_nonzero:
return self.base.is_algebraic
elif self.base.is_algebraic:
return True
if self.exp.is_positive:
return self.base.is_algebraic
elif self.base.is_algebraic and self.exp.is_algebraic:
if ((fuzzy_not(self.base.is_zero)
and fuzzy_not(_is_one(self.base)))
or self.base.is_integer is False
or self.base.is_irrational):
return self.exp.is_rational
def _eval_is_rational_function(self, syms):
if self.exp.has(*syms):
return False
if self.base.has(*syms):
return self.base._eval_is_rational_function(syms) and \
self.exp.is_Integer
else:
return True
def _eval_is_meromorphic(self, x, a):
# f**g is meromorphic if g is an integer and f is meromorphic.
# E**(log(f)*g) is meromorphic if log(f)*g is meromorphic
# and finite.
base_merom = self.base._eval_is_meromorphic(x, a)
exp_integer = self.exp.is_Integer
if exp_integer:
return base_merom
exp_merom = self.exp._eval_is_meromorphic(x, a)
if base_merom is False:
# f**g = E**(log(f)*g) may be meromorphic if the
# singularities of log(f) and g cancel each other,
# for example, if g = 1/log(f). Hence,
return False if exp_merom else None
elif base_merom is None:
return None
b = self.base.subs(x, a)
# b is extended complex as base is meromorphic.
# log(base) is finite and meromorphic when b != 0, zoo.
b_zero = b.is_zero
if b_zero:
log_defined = False
else:
log_defined = fuzzy_and((b.is_finite, fuzzy_not(b_zero)))
if log_defined is False: # zero or pole of base
return exp_integer # False or None
elif log_defined is None:
return None
if not exp_merom:
return exp_merom # False or None
return self.exp.subs(x, a).is_finite
def _eval_is_algebraic_expr(self, syms):
if self.exp.has(*syms):
return False
if self.base.has(*syms):
return self.base._eval_is_algebraic_expr(syms) and \
self.exp.is_Rational
else:
return True
def _eval_rewrite_as_exp(self, base, expo, **kwargs):
from sympy.functions.elementary.exponential import exp, log
if base.is_zero or base.has(exp) or expo.has(exp):
return base**expo
if base.has(Symbol):
# delay evaluation if expo is non symbolic
# (as exp(x*log(5)) automatically reduces to x**5)
if global_parameters.exp_is_pow:
return Pow(S.Exp1, log(base)*expo, evaluate=expo.has(Symbol))
else:
return exp(log(base)*expo, evaluate=expo.has(Symbol))
else:
from sympy.functions.elementary.complexes import arg, Abs
return exp((log(Abs(base)) + S.ImaginaryUnit*arg(base))*expo)
def as_numer_denom(self):
if not self.is_commutative:
return self, S.One
base, exp = self.as_base_exp()
n, d = base.as_numer_denom()
# this should be the same as ExpBase.as_numer_denom wrt
# exponent handling
neg_exp = exp.is_negative
if exp.is_Mul and not neg_exp and not exp.is_positive:
neg_exp = exp.could_extract_minus_sign()
int_exp = exp.is_integer
# the denominator cannot be separated from the numerator if
# its sign is unknown unless the exponent is an integer, e.g.
# sqrt(a/b) != sqrt(a)/sqrt(b) when a=1 and b=-1. But if the
# denominator is negative the numerator and denominator can
# be negated and the denominator (now positive) separated.
if not (d.is_extended_real or int_exp):
n = base
d = S.One
dnonpos = d.is_nonpositive
if dnonpos:
n, d = -n, -d
elif dnonpos is None and not int_exp:
n = base
d = S.One
if neg_exp:
n, d = d, n
exp = -exp
if exp.is_infinite:
if n is S.One and d is not S.One:
return n, self.func(d, exp)
if n is not S.One and d is S.One:
return self.func(n, exp), d
return self.func(n, exp), self.func(d, exp)
def matches(self, expr, repl_dict=None, old=False):
expr = _sympify(expr)
if repl_dict is None:
repl_dict = {}
# special case, pattern = 1 and expr.exp can match to 0
if expr is S.One:
d = self.exp.matches(S.Zero, repl_dict)
if d is not None:
return d
# make sure the expression to be matched is an Expr
if not isinstance(expr, Expr):
return None
b, e = expr.as_base_exp()
# special case number
sb, se = self.as_base_exp()
if sb.is_Symbol and se.is_Integer and expr:
if e.is_rational:
return sb.matches(b**(e/se), repl_dict)
return sb.matches(expr**(1/se), repl_dict)
d = repl_dict.copy()
d = self.base.matches(b, d)
if d is None:
return None
d = self.exp.xreplace(d).matches(e, d)
if d is None:
return Expr.matches(self, expr, repl_dict)
return d
def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE! This function is an important part of the gruntz algorithm
# for computing limits. It has to return a generalized power
# series with coefficients in C(log, log(x)). In more detail:
# It has to return an expression
# c_0*x**e_0 + c_1*x**e_1 + ... (finitely many terms)
# where e_i are numbers (not necessarily integers) and c_i are
# expressions involving only numbers, the log function, and log(x).
# The series expansion of b**e is computed as follows:
# 1) We express b as f*(1 + g) where f is the leading term of b.
# g has order O(x**d) where d is strictly positive.
# 2) Then b**e = (f**e)*((1 + g)**e).
# (1 + g)**e is computed using binomial series.
from sympy.functions.elementary.exponential import exp, log
from sympy.series.limits import limit
from sympy.series.order import Order
if self.base is S.Exp1:
e_series = self.exp.nseries(x, n=n, logx=logx)
if e_series.is_Order:
return 1 + e_series
e0 = limit(e_series.removeO(), x, 0)
if e0 is S.NegativeInfinity:
return Order(x**n, x)
if e0 is S.Infinity:
return self
t = e_series - e0
exp_series = term = exp(e0)
# series of exp(e0 + t) in t
for i in range(1, n):
term *= t/i
term = term.nseries(x, n=n, logx=logx)
exp_series += term
exp_series += Order(t**n, x)
from sympy.simplify.powsimp import powsimp
return powsimp(exp_series, deep=True, combine='exp')
from sympy.simplify.powsimp import powdenest
from .numbers import _illegal
self = powdenest(self, force=True).trigsimp()
b, e = self.as_base_exp()
if e.has(*_illegal):
raise PoleError()
if e.has(x):
return exp(e*log(b))._eval_nseries(x, n=n, logx=logx, cdir=cdir)
if logx is not None and b.has(log):
from .symbol import Wild
c, ex = symbols('c, ex', cls=Wild, exclude=[x])
b = b.replace(log(c*x**ex), log(c) + ex*logx)
self = b**e
b = b.removeO()
try:
from sympy.functions.special.gamma_functions import polygamma
if b.has(polygamma, S.EulerGamma) and logx is not None:
raise ValueError()
_, m = b.leadterm(x)
except (ValueError, NotImplementedError, PoleError):
b = b._eval_nseries(x, n=max(2, n), logx=logx, cdir=cdir).removeO()
if b.has(S.NaN, S.ComplexInfinity):
raise NotImplementedError()
_, m = b.leadterm(x)
if e.has(log):
from sympy.simplify.simplify import logcombine
e = logcombine(e).cancel()
if not (m.is_zero or e.is_number and e.is_real):
res = exp(e*log(b))._eval_nseries(x, n=n, logx=logx, cdir=cdir)
if res is exp(e*log(b)):
return self
return res
f = b.as_leading_term(x, logx=logx)
g = (b/f - S.One).cancel(expand=False)
if not m.is_number:
raise NotImplementedError()
maxpow = n - m*e
if maxpow.is_negative:
return Order(x**(m*e), x)
if g.is_zero:
r = f**e
if r != self:
r += Order(x**n, x)
return r
def coeff_exp(term, x):
coeff, exp = S.One, S.Zero
for factor in Mul.make_args(term):
if factor.has(x):
base, exp = factor.as_base_exp()
if base != x:
try:
return term.leadterm(x)
except ValueError:
return term, S.Zero
else:
coeff *= factor
return coeff, exp
def mul(d1, d2):
res = {}
for e1, e2 in product(d1, d2):
ex = e1 + e2
if ex < maxpow:
res[ex] = res.get(ex, S.Zero) + d1[e1]*d2[e2]
return res
try:
c, d = g.leadterm(x, logx=logx)
except (ValueError, NotImplementedError):
if limit(g/x**maxpow, x, 0) == 0:
# g has higher order zero
return f**e + e*f**e*g # first term of binomial series
else:
raise NotImplementedError()
if c.is_Float and d == S.Zero:
# Convert floats like 0.5 to exact SymPy numbers like S.Half, to
# prevent rounding errors which can induce wrong values of d leading
# to a NotImplementedError being returned from the block below.
from sympy.simplify.simplify import nsimplify
_, d = nsimplify(g).leadterm(x, logx=logx)
if not d.is_positive:
g = g.simplify()
if g.is_zero:
return f**e
_, d = g.leadterm(x, logx=logx)
if not d.is_positive:
g = ((b - f)/f).expand()
_, d = g.leadterm(x, logx=logx)
if not d.is_positive:
raise NotImplementedError()
from sympy.functions.elementary.integers import ceiling
gpoly = g._eval_nseries(x, n=ceiling(maxpow), logx=logx, cdir=cdir).removeO()
gterms = {}
for term in Add.make_args(gpoly):
co1, e1 = coeff_exp(term, x)
gterms[e1] = gterms.get(e1, S.Zero) + co1
k = S.One
terms = {S.Zero: S.One}
tk = gterms
from sympy.functions.combinatorial.factorials import factorial, ff
while (k*d - maxpow).is_negative:
coeff = ff(e, k)/factorial(k)
for ex in tk:
terms[ex] = terms.get(ex, S.Zero) + coeff*tk[ex]
tk = mul(tk, gterms)
k += S.One
from sympy.functions.elementary.complexes import im
if not e.is_integer and m.is_zero and f.is_negative:
ndir = (b - f).dir(x, cdir)
if im(ndir).is_negative:
inco, inex = coeff_exp(f**e*(-1)**(-2*e), x)
elif im(ndir).is_zero:
inco, inex = coeff_exp(exp(e*log(b)).as_leading_term(x, logx=logx, cdir=cdir), x)
else:
inco, inex = coeff_exp(f**e, x)
else:
inco, inex = coeff_exp(f**e, x)
res = S.Zero
for e1 in terms:
ex = e1 + inex
res += terms[e1]*inco*x**(ex)
if not (e.is_integer and e.is_positive and (e*d - n).is_nonpositive and
res == _mexpand(self)):
res += Order(x**n, x)
return res
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.functions.elementary.exponential import exp, log
e = self.exp
b = self.base
if self.base is S.Exp1:
arg = e.as_leading_term(x, logx=logx)
arg0 = arg.subs(x, 0)
if arg0 is S.NaN:
arg0 = arg.limit(x, 0)
if arg0.is_infinite is False:
return S.Exp1**arg0
raise PoleError("Cannot expand %s around 0" % (self))
elif e.has(x):
lt = exp(e * log(b))
return lt.as_leading_term(x, logx=logx, cdir=cdir)
else:
from sympy.functions.elementary.complexes import im
try:
f = b.as_leading_term(x, logx=logx, cdir=cdir)
except PoleError:
return self
if not e.is_integer and f.is_negative and not f.has(x):
ndir = (b - f).dir(x, cdir)
if im(ndir).is_negative:
# Normally, f**e would evaluate to exp(e*log(f)) but on branch cuts
# an other value is expected through the following computation
# exp(e*(log(f) - 2*pi*I)) == f**e*exp(-2*e*pi*I) == f**e*(-1)**(-2*e).
return self.func(f, e) * (-1)**(-2*e)
elif im(ndir).is_zero:
log_leadterm = log(b)._eval_as_leading_term(x, logx=logx, cdir=cdir)
if log_leadterm.is_infinite is False:
return exp(e*log_leadterm)
return self.func(f, e)
@cacheit
def _taylor_term(self, n, x, *previous_terms): # of (1 + x)**e
from sympy.functions.combinatorial.factorials import binomial
return binomial(self.exp, n) * self.func(x, n)
def taylor_term(self, n, x, *previous_terms):
if self.base is not S.Exp1:
return super().taylor_term(n, x, *previous_terms)
if n < 0:
return S.Zero
if n == 0:
return S.One
from .sympify import sympify
x = sympify(x)
if previous_terms:
p = previous_terms[-1]
if p is not None:
return p * x / n
from sympy.functions.combinatorial.factorials import factorial
return x**n/factorial(n)
def _eval_rewrite_as_sin(self, base, exp):
if self.base is S.Exp1:
from sympy.functions.elementary.trigonometric import sin
return sin(S.ImaginaryUnit*self.exp + S.Pi/2) - S.ImaginaryUnit*sin(S.ImaginaryUnit*self.exp)
def _eval_rewrite_as_cos(self, base, exp):
if self.base is S.Exp1:
from sympy.functions.elementary.trigonometric import cos
return cos(S.ImaginaryUnit*self.exp) + S.ImaginaryUnit*cos(S.ImaginaryUnit*self.exp + S.Pi/2)
def _eval_rewrite_as_tanh(self, base, exp):
if self.base is S.Exp1:
from sympy.functions.elementary.hyperbolic import tanh
return (1 + tanh(self.exp/2))/(1 - tanh(self.exp/2))
def _eval_rewrite_as_sqrt(self, base, exp, **kwargs):
from sympy.functions.elementary.trigonometric import sin, cos
if base is not S.Exp1:
return None
if exp.is_Mul:
coeff = exp.coeff(S.Pi * S.ImaginaryUnit)
if coeff and coeff.is_number:
cosine, sine = cos(S.Pi*coeff), sin(S.Pi*coeff)
if not isinstance(cosine, cos) and not isinstance (sine, sin):
return cosine + S.ImaginaryUnit*sine
def as_content_primitive(self, radical=False, clear=True):
"""Return the tuple (R, self/R) where R is the positive Rational
extracted from self.
Examples
========
>>> from sympy import sqrt
>>> sqrt(4 + 4*sqrt(2)).as_content_primitive()
(2, sqrt(1 + sqrt(2)))
>>> sqrt(3 + 3*sqrt(2)).as_content_primitive()
(1, sqrt(3)*sqrt(1 + sqrt(2)))
>>> from sympy import expand_power_base, powsimp, Mul
>>> from sympy.abc import x, y
>>> ((2*x + 2)**2).as_content_primitive()
(4, (x + 1)**2)
>>> (4**((1 + y)/2)).as_content_primitive()
(2, 4**(y/2))
>>> (3**((1 + y)/2)).as_content_primitive()
(1, 3**((y + 1)/2))
>>> (3**((5 + y)/2)).as_content_primitive()
(9, 3**((y + 1)/2))
>>> eq = 3**(2 + 2*x)
>>> powsimp(eq) == eq
True
>>> eq.as_content_primitive()
(9, 3**(2*x))
>>> powsimp(Mul(*_))
3**(2*x + 2)
>>> eq = (2 + 2*x)**y
>>> s = expand_power_base(eq); s.is_Mul, s
(False, (2*x + 2)**y)
>>> eq.as_content_primitive()
(1, (2*(x + 1))**y)
>>> s = expand_power_base(_[1]); s.is_Mul, s
(True, 2**y*(x + 1)**y)
See docstring of Expr.as_content_primitive for more examples.
"""
b, e = self.as_base_exp()
b = _keep_coeff(*b.as_content_primitive(radical=radical, clear=clear))
ce, pe = e.as_content_primitive(radical=radical, clear=clear)
if b.is_Rational:
#e
#= ce*pe
#= ce*(h + t)
#= ce*h + ce*t
#=> self
#= b**(ce*h)*b**(ce*t)
#= b**(cehp/cehq)*b**(ce*t)
#= b**(iceh + r/cehq)*b**(ce*t)
#= b**(iceh)*b**(r/cehq)*b**(ce*t)
#= b**(iceh)*b**(ce*t + r/cehq)
h, t = pe.as_coeff_Add()
if h.is_Rational and b != S.Zero:
ceh = ce*h
c = self.func(b, ceh)
r = S.Zero
if not c.is_Rational:
iceh, r = divmod(ceh.p, ceh.q)
c = self.func(b, iceh)
return c, self.func(b, _keep_coeff(ce, t + r/ce/ceh.q))
e = _keep_coeff(ce, pe)
# b**e = (h*t)**e = h**e*t**e = c*m*t**e
if e.is_Rational and b.is_Mul:
h, t = b.as_content_primitive(radical=radical, clear=clear) # h is positive
c, m = self.func(h, e).as_coeff_Mul() # so c is positive
m, me = m.as_base_exp()
if m is S.One or me == e: # probably always true
# return the following, not return c, m*Pow(t, e)
# which would change Pow into Mul; we let SymPy
# decide what to do by using the unevaluated Mul, e.g
# should it stay as sqrt(2 + 2*sqrt(5)) or become
# sqrt(2)*sqrt(1 + sqrt(5))
return c, self.func(_keep_coeff(m, t), e)
return S.One, self.func(b, e)
def is_constant(self, *wrt, **flags):
expr = self
if flags.get('simplify', True):
expr = expr.simplify()
b, e = expr.as_base_exp()
bz = b.equals(0)
if bz: # recalculate with assumptions in case it's unevaluated
new = b**e
if new != expr:
return new.is_constant()
econ = e.is_constant(*wrt)
bcon = b.is_constant(*wrt)
if bcon:
if econ:
return True
bz = b.equals(0)
if bz is False:
return False
elif bcon is None:
return None
return e.equals(0)
def _eval_difference_delta(self, n, step):
b, e = self.args
if e.has(n) and not b.has(n):
new_e = e.subs(n, n + step)
return (b**(new_e - e) - 1) * self
power = Dispatcher('power')
power.add((object, object), Pow)
from .add import Add
from .numbers import Integer
from .mul import Mul, _keep_coeff
from .symbol import Symbol, Dummy, symbols
|
de6854f9ed41585920f59d548138f6312612aa45ddceadd2923e974cd042b412 | """Tools for manipulating of large commutative expressions. """
from .add import Add
from .mul import Mul, _keep_coeff
from .power import Pow
from .basic import Basic
from .expr import Expr
from .sympify import sympify
from .numbers import Rational, Integer, Number, I
from .singleton import S
from .sorting import default_sort_key, ordered
from .symbol import Dummy
from .traversal import preorder_traversal
from .coreerrors import NonCommutativeExpression
from .containers import Tuple, Dict
from sympy.external.gmpy import SYMPY_INTS
from sympy.utilities.iterables import (common_prefix, common_suffix,
variations, iterable, is_sequence)
from collections import defaultdict
from typing import Tuple as tTuple
_eps = Dummy(positive=True)
def _isnumber(i):
return isinstance(i, (SYMPY_INTS, float)) or i.is_Number
def _monotonic_sign(self):
"""Return the value closest to 0 that ``self`` may have if all symbols
are signed and the result is uniformly the same sign for all values of symbols.
If a symbol is only signed but not known to be an
integer or the result is 0 then a symbol representative of the sign of self
will be returned. Otherwise, None is returned if a) the sign could be positive
or negative or b) self is not in one of the following forms:
- L(x, y, ...) + A: a function linear in all symbols x, y, ... with an
additive constant; if A is zero then the function can be a monomial whose
sign is monotonic over the range of the variables, e.g. (x + 1)**3 if x is
nonnegative.
- A/L(x, y, ...) + B: the inverse of a function linear in all symbols x, y, ...
that does not have a sign change from positive to negative for any set
of values for the variables.
- M(x, y, ...) + A: a monomial M whose factors are all signed and a constant, A.
- A/M(x, y, ...) + B: the inverse of a monomial and constants A and B.
- P(x): a univariate polynomial
Examples
========
>>> from sympy.core.exprtools import _monotonic_sign as F
>>> from sympy import Dummy
>>> nn = Dummy(integer=True, nonnegative=True)
>>> p = Dummy(integer=True, positive=True)
>>> p2 = Dummy(integer=True, positive=True)
>>> F(nn + 1)
1
>>> F(p - 1)
_nneg
>>> F(nn*p + 1)
1
>>> F(p2*p + 1)
2
>>> F(nn - 1) # could be negative, zero or positive
"""
if not self.is_extended_real:
return
if (-self).is_Symbol:
rv = _monotonic_sign(-self)
return rv if rv is None else -rv
if not self.is_Add and self.as_numer_denom()[1].is_number:
s = self
if s.is_prime:
if s.is_odd:
return Integer(3)
else:
return Integer(2)
elif s.is_composite:
if s.is_odd:
return Integer(9)
else:
return Integer(4)
elif s.is_positive:
if s.is_even:
if s.is_prime is False:
return Integer(4)
else:
return Integer(2)
elif s.is_integer:
return S.One
else:
return _eps
elif s.is_extended_negative:
if s.is_even:
return Integer(-2)
elif s.is_integer:
return S.NegativeOne
else:
return -_eps
if s.is_zero or s.is_extended_nonpositive or s.is_extended_nonnegative:
return S.Zero
return None
# univariate polynomial
free = self.free_symbols
if len(free) == 1:
if self.is_polynomial():
from sympy.polys.polytools import real_roots
from sympy.polys.polyroots import roots
from sympy.polys.polyerrors import PolynomialError
x = free.pop()
x0 = _monotonic_sign(x)
if x0 in (_eps, -_eps):
x0 = S.Zero
if x0 is not None:
d = self.diff(x)
if d.is_number:
currentroots = []
else:
try:
currentroots = real_roots(d)
except (PolynomialError, NotImplementedError):
currentroots = [r for r in roots(d, x) if r.is_extended_real]
y = self.subs(x, x0)
if x.is_nonnegative and all(
(r - x0).is_nonpositive for r in currentroots):
if y.is_nonnegative and d.is_positive:
if y:
return y if y.is_positive else Dummy('pos', positive=True)
else:
return Dummy('nneg', nonnegative=True)
if y.is_nonpositive and d.is_negative:
if y:
return y if y.is_negative else Dummy('neg', negative=True)
else:
return Dummy('npos', nonpositive=True)
elif x.is_nonpositive and all(
(r - x0).is_nonnegative for r in currentroots):
if y.is_nonnegative and d.is_negative:
if y:
return Dummy('pos', positive=True)
else:
return Dummy('nneg', nonnegative=True)
if y.is_nonpositive and d.is_positive:
if y:
return Dummy('neg', negative=True)
else:
return Dummy('npos', nonpositive=True)
else:
n, d = self.as_numer_denom()
den = None
if n.is_number:
den = _monotonic_sign(d)
elif not d.is_number:
if _monotonic_sign(n) is not None:
den = _monotonic_sign(d)
if den is not None and (den.is_positive or den.is_negative):
v = n*den
if v.is_positive:
return Dummy('pos', positive=True)
elif v.is_nonnegative:
return Dummy('nneg', nonnegative=True)
elif v.is_negative:
return Dummy('neg', negative=True)
elif v.is_nonpositive:
return Dummy('npos', nonpositive=True)
return None
# multivariate
c, a = self.as_coeff_Add()
v = None
if not a.is_polynomial():
# F/A or A/F where A is a number and F is a signed, rational monomial
n, d = a.as_numer_denom()
if not (n.is_number or d.is_number):
return
if (
a.is_Mul or a.is_Pow) and \
a.is_rational and \
all(p.exp.is_Integer for p in a.atoms(Pow) if p.is_Pow) and \
(a.is_positive or a.is_negative):
v = S.One
for ai in Mul.make_args(a):
if ai.is_number:
v *= ai
continue
reps = {}
for x in ai.free_symbols:
reps[x] = _monotonic_sign(x)
if reps[x] is None:
return
v *= ai.subs(reps)
elif c:
# signed linear expression
if not any(p for p in a.atoms(Pow) if not p.is_number) and (a.is_nonpositive or a.is_nonnegative):
free = list(a.free_symbols)
p = {}
for i in free:
v = _monotonic_sign(i)
if v is None:
return
p[i] = v or (_eps if i.is_nonnegative else -_eps)
v = a.xreplace(p)
if v is not None:
rv = v + c
if v.is_nonnegative and rv.is_positive:
return rv.subs(_eps, 0)
if v.is_nonpositive and rv.is_negative:
return rv.subs(_eps, 0)
def decompose_power(expr: Expr) -> tTuple[Expr, int]:
"""
Decompose power into symbolic base and integer exponent.
Examples
========
>>> from sympy.core.exprtools import decompose_power
>>> from sympy.abc import x, y
>>> from sympy import exp
>>> decompose_power(x)
(x, 1)
>>> decompose_power(x**2)
(x, 2)
>>> decompose_power(exp(2*y/3))
(exp(y/3), 2)
"""
base, exp = expr.as_base_exp()
if exp.is_Number:
if exp.is_Rational:
if not exp.is_Integer:
base = Pow(base, Rational(1, exp.q)) # type: ignore
e = exp.p # type: ignore
else:
base, e = expr, 1
else:
exp, tail = exp.as_coeff_Mul(rational=True)
if exp is S.NegativeOne:
base, e = Pow(base, tail), -1
elif exp is not S.One:
# todo: after dropping python 3.7 support, use overload and Literal
# in as_coeff_Mul to make exp Rational, and remove these 2 ignores
tail = _keep_coeff(Rational(1, exp.q), tail) # type: ignore
base, e = Pow(base, tail), exp.p # type: ignore
else:
base, e = expr, 1
return base, e
def decompose_power_rat(expr: Expr) -> tTuple[Expr, Rational]:
"""
Decompose power into symbolic base and rational exponent;
if the exponent is not a Rational, then separate only the
integer coefficient.
Examples
========
>>> from sympy.core.exprtools import decompose_power_rat
>>> from sympy.abc import x
>>> from sympy import sqrt, exp
>>> decompose_power_rat(sqrt(x))
(x, 1/2)
>>> decompose_power_rat(exp(-3*x/2))
(exp(x/2), -3)
"""
_ = base, exp = expr.as_base_exp()
return _ if exp.is_Rational else decompose_power(expr)
class Factors:
"""Efficient representation of ``f_1*f_2*...*f_n``."""
__slots__ = ('factors', 'gens')
def __init__(self, factors=None): # Factors
"""Initialize Factors from dict or expr.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x
>>> from sympy import I
>>> e = 2*x**3
>>> Factors(e)
Factors({2: 1, x: 3})
>>> Factors(e.as_powers_dict())
Factors({2: 1, x: 3})
>>> f = _
>>> f.factors # underlying dictionary
{2: 1, x: 3}
>>> f.gens # base of each factor
frozenset({2, x})
>>> Factors(0)
Factors({0: 1})
>>> Factors(I)
Factors({I: 1})
Notes
=====
Although a dictionary can be passed, only minimal checking is
performed: powers of -1 and I are made canonical.
"""
if isinstance(factors, (SYMPY_INTS, float)):
factors = S(factors)
if isinstance(factors, Factors):
factors = factors.factors.copy()
elif factors in (None, S.One):
factors = {}
elif factors is S.Zero or factors == 0:
factors = {S.Zero: S.One}
elif isinstance(factors, Number):
n = factors
factors = {}
if n < 0:
factors[S.NegativeOne] = S.One
n = -n
if n is not S.One:
if n.is_Float or n.is_Integer or n is S.Infinity:
factors[n] = S.One
elif n.is_Rational:
# since we're processing Numbers, the denominator is
# stored with a negative exponent; all other factors
# are left .
if n.p != 1:
factors[Integer(n.p)] = S.One
factors[Integer(n.q)] = S.NegativeOne
else:
raise ValueError('Expected Float|Rational|Integer, not %s' % n)
elif isinstance(factors, Basic) and not factors.args:
factors = {factors: S.One}
elif isinstance(factors, Expr):
c, nc = factors.args_cnc()
i = c.count(I)
for _ in range(i):
c.remove(I)
factors = dict(Mul._from_args(c).as_powers_dict())
# Handle all rational Coefficients
for f in list(factors.keys()):
if isinstance(f, Rational) and not isinstance(f, Integer):
p, q = Integer(f.p), Integer(f.q)
factors[p] = (factors[p] if p in factors else S.Zero) + factors[f]
factors[q] = (factors[q] if q in factors else S.Zero) - factors[f]
factors.pop(f)
if i:
factors[I] = factors.get(I, S.Zero) + i
if nc:
factors[Mul(*nc, evaluate=False)] = S.One
else:
factors = factors.copy() # /!\ should be dict-like
# tidy up -/+1 and I exponents if Rational
handle = [k for k in factors if k is I or k in (-1, 1)]
if handle:
i1 = S.One
for k in handle:
if not _isnumber(factors[k]):
continue
i1 *= k**factors.pop(k)
if i1 is not S.One:
for a in i1.args if i1.is_Mul else [i1]: # at worst, -1.0*I*(-1)**e
if a is S.NegativeOne:
factors[a] = S.One
elif a is I:
factors[I] = S.One
elif a.is_Pow:
factors[a.base] = factors.get(a.base, S.Zero) + a.exp
elif a == 1:
factors[a] = S.One
elif a == -1:
factors[-a] = S.One
factors[S.NegativeOne] = S.One
else:
raise ValueError('unexpected factor in i1: %s' % a)
self.factors = factors
keys = getattr(factors, 'keys', None)
if keys is None:
raise TypeError('expecting Expr or dictionary')
self.gens = frozenset(keys())
def __hash__(self): # Factors
keys = tuple(ordered(self.factors.keys()))
values = [self.factors[k] for k in keys]
return hash((keys, values))
def __repr__(self): # Factors
return "Factors({%s})" % ', '.join(
['%s: %s' % (k, v) for k, v in ordered(self.factors.items())])
@property
def is_zero(self): # Factors
"""
>>> from sympy.core.exprtools import Factors
>>> Factors(0).is_zero
True
"""
f = self.factors
return len(f) == 1 and S.Zero in f
@property
def is_one(self): # Factors
"""
>>> from sympy.core.exprtools import Factors
>>> Factors(1).is_one
True
"""
return not self.factors
def as_expr(self): # Factors
"""Return the underlying expression.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y
>>> Factors((x*y**2).as_powers_dict()).as_expr()
x*y**2
"""
args = []
for factor, exp in self.factors.items():
if exp != 1:
if isinstance(exp, Integer):
b, e = factor.as_base_exp()
e = _keep_coeff(exp, e)
args.append(b**e)
else:
args.append(factor**exp)
else:
args.append(factor)
return Mul(*args)
def mul(self, other): # Factors
"""Return Factors of ``self * other``.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.mul(b)
Factors({x: 2, y: 3, z: -1})
>>> a*b
Factors({x: 2, y: 3, z: -1})
"""
if not isinstance(other, Factors):
other = Factors(other)
if any(f.is_zero for f in (self, other)):
return Factors(S.Zero)
factors = dict(self.factors)
for factor, exp in other.factors.items():
if factor in factors:
exp = factors[factor] + exp
if not exp:
del factors[factor]
continue
factors[factor] = exp
return Factors(factors)
def normal(self, other):
"""Return ``self`` and ``other`` with ``gcd`` removed from each.
The only differences between this and method ``div`` is that this
is 1) optimized for the case when there are few factors in common and
2) this does not raise an error if ``other`` is zero.
See Also
========
div
"""
if not isinstance(other, Factors):
other = Factors(other)
if other.is_zero:
return (Factors(), Factors(S.Zero))
if self.is_zero:
return (Factors(S.Zero), Factors())
self_factors = dict(self.factors)
other_factors = dict(other.factors)
for factor, self_exp in self.factors.items():
try:
other_exp = other.factors[factor]
except KeyError:
continue
exp = self_exp - other_exp
if not exp:
del self_factors[factor]
del other_factors[factor]
elif _isnumber(exp):
if exp > 0:
self_factors[factor] = exp
del other_factors[factor]
else:
del self_factors[factor]
other_factors[factor] = -exp
else:
r = self_exp.extract_additively(other_exp)
if r is not None:
if r:
self_factors[factor] = r
del other_factors[factor]
else: # should be handled already
del self_factors[factor]
del other_factors[factor]
else:
sc, sa = self_exp.as_coeff_Add()
if sc:
oc, oa = other_exp.as_coeff_Add()
diff = sc - oc
if diff > 0:
self_factors[factor] -= oc
other_exp = oa
elif diff < 0:
self_factors[factor] -= sc
other_factors[factor] -= sc
other_exp = oa - diff
else:
self_factors[factor] = sa
other_exp = oa
if other_exp:
other_factors[factor] = other_exp
else:
del other_factors[factor]
return Factors(self_factors), Factors(other_factors)
def div(self, other): # Factors
"""Return ``self`` and ``other`` with ``gcd`` removed from each.
This is optimized for the case when there are many factors in common.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> from sympy import S
>>> a = Factors((x*y**2).as_powers_dict())
>>> a.div(a)
(Factors({}), Factors({}))
>>> a.div(x*z)
(Factors({y: 2}), Factors({z: 1}))
The ``/`` operator only gives ``quo``:
>>> a/x
Factors({y: 2})
Factors treats its factors as though they are all in the numerator, so
if you violate this assumption the results will be correct but will
not strictly correspond to the numerator and denominator of the ratio:
>>> a.div(x/z)
(Factors({y: 2}), Factors({z: -1}))
Factors is also naive about bases: it does not attempt any denesting
of Rational-base terms, for example the following does not become
2**(2*x)/2.
>>> Factors(2**(2*x + 2)).div(S(8))
(Factors({2: 2*x + 2}), Factors({8: 1}))
factor_terms can clean up such Rational-bases powers:
>>> from sympy import factor_terms
>>> n, d = Factors(2**(2*x + 2)).div(S(8))
>>> n.as_expr()/d.as_expr()
2**(2*x + 2)/8
>>> factor_terms(_)
2**(2*x)/2
"""
quo, rem = dict(self.factors), {}
if not isinstance(other, Factors):
other = Factors(other)
if other.is_zero:
raise ZeroDivisionError
if self.is_zero:
return (Factors(S.Zero), Factors())
for factor, exp in other.factors.items():
if factor in quo:
d = quo[factor] - exp
if _isnumber(d):
if d <= 0:
del quo[factor]
if d >= 0:
if d:
quo[factor] = d
continue
exp = -d
else:
r = quo[factor].extract_additively(exp)
if r is not None:
if r:
quo[factor] = r
else: # should be handled already
del quo[factor]
else:
other_exp = exp
sc, sa = quo[factor].as_coeff_Add()
if sc:
oc, oa = other_exp.as_coeff_Add()
diff = sc - oc
if diff > 0:
quo[factor] -= oc
other_exp = oa
elif diff < 0:
quo[factor] -= sc
other_exp = oa - diff
else:
quo[factor] = sa
other_exp = oa
if other_exp:
rem[factor] = other_exp
else:
assert factor not in rem
continue
rem[factor] = exp
return Factors(quo), Factors(rem)
def quo(self, other): # Factors
"""Return numerator Factor of ``self / other``.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.quo(b) # same as a/b
Factors({y: 1})
"""
return self.div(other)[0]
def rem(self, other): # Factors
"""Return denominator Factors of ``self / other``.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.rem(b)
Factors({z: -1})
>>> a.rem(a)
Factors({})
"""
return self.div(other)[1]
def pow(self, other): # Factors
"""Return self raised to a non-negative integer power.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y
>>> a = Factors((x*y**2).as_powers_dict())
>>> a**2
Factors({x: 2, y: 4})
"""
if isinstance(other, Factors):
other = other.as_expr()
if other.is_Integer:
other = int(other)
if isinstance(other, SYMPY_INTS) and other >= 0:
factors = {}
if other:
for factor, exp in self.factors.items():
factors[factor] = exp*other
return Factors(factors)
else:
raise ValueError("expected non-negative integer, got %s" % other)
def gcd(self, other): # Factors
"""Return Factors of ``gcd(self, other)``. The keys are
the intersection of factors with the minimum exponent for
each factor.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.gcd(b)
Factors({x: 1, y: 1})
"""
if not isinstance(other, Factors):
other = Factors(other)
if other.is_zero:
return Factors(self.factors)
factors = {}
for factor, exp in self.factors.items():
factor, exp = sympify(factor), sympify(exp)
if factor in other.factors:
lt = (exp - other.factors[factor]).is_negative
if lt == True:
factors[factor] = exp
elif lt == False:
factors[factor] = other.factors[factor]
return Factors(factors)
def lcm(self, other): # Factors
"""Return Factors of ``lcm(self, other)`` which are
the union of factors with the maximum exponent for
each factor.
Examples
========
>>> from sympy.core.exprtools import Factors
>>> from sympy.abc import x, y, z
>>> a = Factors((x*y**2).as_powers_dict())
>>> b = Factors((x*y/z).as_powers_dict())
>>> a.lcm(b)
Factors({x: 1, y: 2, z: -1})
"""
if not isinstance(other, Factors):
other = Factors(other)
if any(f.is_zero for f in (self, other)):
return Factors(S.Zero)
factors = dict(self.factors)
for factor, exp in other.factors.items():
if factor in factors:
exp = max(exp, factors[factor])
factors[factor] = exp
return Factors(factors)
def __mul__(self, other): # Factors
return self.mul(other)
def __divmod__(self, other): # Factors
return self.div(other)
def __truediv__(self, other): # Factors
return self.quo(other)
def __mod__(self, other): # Factors
return self.rem(other)
def __pow__(self, other): # Factors
return self.pow(other)
def __eq__(self, other): # Factors
if not isinstance(other, Factors):
other = Factors(other)
return self.factors == other.factors
def __ne__(self, other): # Factors
return not self == other
class Term:
"""Efficient representation of ``coeff*(numer/denom)``. """
__slots__ = ('coeff', 'numer', 'denom')
def __init__(self, term, numer=None, denom=None): # Term
if numer is None and denom is None:
if not term.is_commutative:
raise NonCommutativeExpression(
'commutative expression expected')
coeff, factors = term.as_coeff_mul()
numer, denom = defaultdict(int), defaultdict(int)
for factor in factors:
base, exp = decompose_power(factor)
if base.is_Add:
cont, base = base.primitive()
coeff *= cont**exp
if exp > 0:
numer[base] += exp
else:
denom[base] += -exp
numer = Factors(numer)
denom = Factors(denom)
else:
coeff = term
if numer is None:
numer = Factors()
if denom is None:
denom = Factors()
self.coeff = coeff
self.numer = numer
self.denom = denom
def __hash__(self): # Term
return hash((self.coeff, self.numer, self.denom))
def __repr__(self): # Term
return "Term(%s, %s, %s)" % (self.coeff, self.numer, self.denom)
def as_expr(self): # Term
return self.coeff*(self.numer.as_expr()/self.denom.as_expr())
def mul(self, other): # Term
coeff = self.coeff*other.coeff
numer = self.numer.mul(other.numer)
denom = self.denom.mul(other.denom)
numer, denom = numer.normal(denom)
return Term(coeff, numer, denom)
def inv(self): # Term
return Term(1/self.coeff, self.denom, self.numer)
def quo(self, other): # Term
return self.mul(other.inv())
def pow(self, other): # Term
if other < 0:
return self.inv().pow(-other)
else:
return Term(self.coeff ** other,
self.numer.pow(other),
self.denom.pow(other))
def gcd(self, other): # Term
return Term(self.coeff.gcd(other.coeff),
self.numer.gcd(other.numer),
self.denom.gcd(other.denom))
def lcm(self, other): # Term
return Term(self.coeff.lcm(other.coeff),
self.numer.lcm(other.numer),
self.denom.lcm(other.denom))
def __mul__(self, other): # Term
if isinstance(other, Term):
return self.mul(other)
else:
return NotImplemented
def __truediv__(self, other): # Term
if isinstance(other, Term):
return self.quo(other)
else:
return NotImplemented
def __pow__(self, other): # Term
if isinstance(other, SYMPY_INTS):
return self.pow(other)
else:
return NotImplemented
def __eq__(self, other): # Term
return (self.coeff == other.coeff and
self.numer == other.numer and
self.denom == other.denom)
def __ne__(self, other): # Term
return not self == other
def _gcd_terms(terms, isprimitive=False, fraction=True):
"""Helper function for :func:`gcd_terms`.
Parameters
==========
isprimitive : boolean, optional
If ``isprimitive`` is True then the call to primitive
for an Add will be skipped. This is useful when the
content has already been extrated.
fraction : boolean, optional
If ``fraction`` is True then the expression will appear over a common
denominator, the lcm of all term denominators.
"""
if isinstance(terms, Basic) and not isinstance(terms, Tuple):
terms = Add.make_args(terms)
terms = list(map(Term, [t for t in terms if t]))
# there is some simplification that may happen if we leave this
# here rather than duplicate it before the mapping of Term onto
# the terms
if len(terms) == 0:
return S.Zero, S.Zero, S.One
if len(terms) == 1:
cont = terms[0].coeff
numer = terms[0].numer.as_expr()
denom = terms[0].denom.as_expr()
else:
cont = terms[0]
for term in terms[1:]:
cont = cont.gcd(term)
for i, term in enumerate(terms):
terms[i] = term.quo(cont)
if fraction:
denom = terms[0].denom
for term in terms[1:]:
denom = denom.lcm(term.denom)
numers = []
for term in terms:
numer = term.numer.mul(denom.quo(term.denom))
numers.append(term.coeff*numer.as_expr())
else:
numers = [t.as_expr() for t in terms]
denom = Term(S.One).numer
cont = cont.as_expr()
numer = Add(*numers)
denom = denom.as_expr()
if not isprimitive and numer.is_Add:
_cont, numer = numer.primitive()
cont *= _cont
return cont, numer, denom
def gcd_terms(terms, isprimitive=False, clear=True, fraction=True):
"""Compute the GCD of ``terms`` and put them together.
Parameters
==========
terms : Expr
Can be an expression or a non-Basic sequence of expressions
which will be handled as though they are terms from a sum.
isprimitive : bool, optional
If ``isprimitive`` is True the _gcd_terms will not run the primitive
method on the terms.
clear : bool, optional
It controls the removal of integers from the denominator of an Add
expression. When True (default), all numerical denominator will be cleared;
when False the denominators will be cleared only if all terms had numerical
denominators other than 1.
fraction : bool, optional
When True (default), will put the expression over a common
denominator.
Examples
========
>>> from sympy import gcd_terms
>>> from sympy.abc import x, y
>>> gcd_terms((x + 1)**2*y + (x + 1)*y**2)
y*(x + 1)*(x + y + 1)
>>> gcd_terms(x/2 + 1)
(x + 2)/2
>>> gcd_terms(x/2 + 1, clear=False)
x/2 + 1
>>> gcd_terms(x/2 + y/2, clear=False)
(x + y)/2
>>> gcd_terms(x/2 + 1/x)
(x**2 + 2)/(2*x)
>>> gcd_terms(x/2 + 1/x, fraction=False)
(x + 2/x)/2
>>> gcd_terms(x/2 + 1/x, fraction=False, clear=False)
x/2 + 1/x
>>> gcd_terms(x/2/y + 1/x/y)
(x**2 + 2)/(2*x*y)
>>> gcd_terms(x/2/y + 1/x/y, clear=False)
(x**2/2 + 1)/(x*y)
>>> gcd_terms(x/2/y + 1/x/y, clear=False, fraction=False)
(x/2 + 1/x)/y
The ``clear`` flag was ignored in this case because the returned
expression was a rational expression, not a simple sum.
See Also
========
factor_terms, sympy.polys.polytools.terms_gcd
"""
def mask(terms):
"""replace nc portions of each term with a unique Dummy symbols
and return the replacements to restore them"""
args = [(a, []) if a.is_commutative else a.args_cnc() for a in terms]
reps = []
for i, (c, nc) in enumerate(args):
if nc:
nc = Mul(*nc)
d = Dummy()
reps.append((d, nc))
c.append(d)
args[i] = Mul(*c)
else:
args[i] = c
return args, dict(reps)
isadd = isinstance(terms, Add)
addlike = isadd or not isinstance(terms, Basic) and \
is_sequence(terms, include=set) and \
not isinstance(terms, Dict)
if addlike:
if isadd: # i.e. an Add
terms = list(terms.args)
else:
terms = sympify(terms)
terms, reps = mask(terms)
cont, numer, denom = _gcd_terms(terms, isprimitive, fraction)
numer = numer.xreplace(reps)
coeff, factors = cont.as_coeff_Mul()
if not clear:
c, _coeff = coeff.as_coeff_Mul()
if not c.is_Integer and not clear and numer.is_Add:
n, d = c.as_numer_denom()
_numer = numer/d
if any(a.as_coeff_Mul()[0].is_Integer
for a in _numer.args):
numer = _numer
coeff = n*_coeff
return _keep_coeff(coeff, factors*numer/denom, clear=clear)
if not isinstance(terms, Basic):
return terms
if terms.is_Atom:
return terms
if terms.is_Mul:
c, args = terms.as_coeff_mul()
return _keep_coeff(c, Mul(*[gcd_terms(i, isprimitive, clear, fraction)
for i in args]), clear=clear)
def handle(a):
# don't treat internal args like terms of an Add
if not isinstance(a, Expr):
if isinstance(a, Basic):
if not a.args:
return a
return a.func(*[handle(i) for i in a.args])
return type(a)([handle(i) for i in a])
return gcd_terms(a, isprimitive, clear, fraction)
if isinstance(terms, Dict):
return Dict(*[(k, handle(v)) for k, v in terms.args])
return terms.func(*[handle(i) for i in terms.args])
def _factor_sum_int(expr, **kwargs):
"""Return Sum or Integral object with factors that are not
in the wrt variables removed. In cases where there are additive
terms in the function of the object that are independent, the
object will be separated into two objects.
Examples
========
>>> from sympy import Sum, factor_terms
>>> from sympy.abc import x, y
>>> factor_terms(Sum(x + y, (x, 1, 3)))
y*Sum(1, (x, 1, 3)) + Sum(x, (x, 1, 3))
>>> factor_terms(Sum(x*y, (x, 1, 3)))
y*Sum(x, (x, 1, 3))
Notes
=====
If a function in the summand or integrand is replaced
with a symbol, then this simplification should not be
done or else an incorrect result will be obtained when
the symbol is replaced with an expression that depends
on the variables of summation/integration:
>>> eq = Sum(y, (x, 1, 3))
>>> factor_terms(eq).subs(y, x).doit()
3*x
>>> eq.subs(y, x).doit()
6
"""
result = expr.function
if result == 0:
return S.Zero
limits = expr.limits
# get the wrt variables
wrt = {i.args[0] for i in limits}
# factor out any common terms that are independent of wrt
f = factor_terms(result, **kwargs)
i, d = f.as_independent(*wrt)
if isinstance(f, Add):
return i * expr.func(1, *limits) + expr.func(d, *limits)
else:
return i * expr.func(d, *limits)
def factor_terms(expr, radical=False, clear=False, fraction=False, sign=True):
"""Remove common factors from terms in all arguments without
changing the underlying structure of the expr. No expansion or
simplification (and no processing of non-commutatives) is performed.
Parameters
==========
radical: bool, optional
If radical=True then a radical common to all terms will be factored
out of any Add sub-expressions of the expr.
clear : bool, optional
If clear=False (default) then coefficients will not be separated
from a single Add if they can be distributed to leave one or more
terms with integer coefficients.
fraction : bool, optional
If fraction=True (default is False) then a common denominator will be
constructed for the expression.
sign : bool, optional
If sign=True (default) then even if the only factor in common is a -1,
it will be factored out of the expression.
Examples
========
>>> from sympy import factor_terms, Symbol
>>> from sympy.abc import x, y
>>> factor_terms(x + x*(2 + 4*y)**3)
x*(8*(2*y + 1)**3 + 1)
>>> A = Symbol('A', commutative=False)
>>> factor_terms(x*A + x*A + x*y*A)
x*(y*A + 2*A)
When ``clear`` is False, a rational will only be factored out of an
Add expression if all terms of the Add have coefficients that are
fractions:
>>> factor_terms(x/2 + 1, clear=False)
x/2 + 1
>>> factor_terms(x/2 + 1, clear=True)
(x + 2)/2
If a -1 is all that can be factored out, to *not* factor it out, the
flag ``sign`` must be False:
>>> factor_terms(-x - y)
-(x + y)
>>> factor_terms(-x - y, sign=False)
-x - y
>>> factor_terms(-2*x - 2*y, sign=False)
-2*(x + y)
See Also
========
gcd_terms, sympy.polys.polytools.terms_gcd
"""
def do(expr):
from sympy.concrete.summations import Sum
from sympy.integrals.integrals import Integral
is_iterable = iterable(expr)
if not isinstance(expr, Basic) or expr.is_Atom:
if is_iterable:
return type(expr)([do(i) for i in expr])
return expr
if expr.is_Pow or expr.is_Function or \
is_iterable or not hasattr(expr, 'args_cnc'):
args = expr.args
newargs = tuple([do(i) for i in args])
if newargs == args:
return expr
return expr.func(*newargs)
if isinstance(expr, (Sum, Integral)):
return _factor_sum_int(expr,
radical=radical, clear=clear,
fraction=fraction, sign=sign)
cont, p = expr.as_content_primitive(radical=radical, clear=clear)
if p.is_Add:
list_args = [do(a) for a in Add.make_args(p)]
# get a common negative (if there) which gcd_terms does not remove
if not any(a.as_coeff_Mul()[0].extract_multiplicatively(-1) is None
for a in list_args):
cont = -cont
list_args = [-a for a in list_args]
# watch out for exp(-(x+2)) which gcd_terms will change to exp(-x-2)
special = {}
for i, a in enumerate(list_args):
b, e = a.as_base_exp()
if e.is_Mul and e != Mul(*e.args):
list_args[i] = Dummy()
special[list_args[i]] = a
# rebuild p not worrying about the order which gcd_terms will fix
p = Add._from_args(list_args)
p = gcd_terms(p,
isprimitive=True,
clear=clear,
fraction=fraction).xreplace(special)
elif p.args:
p = p.func(
*[do(a) for a in p.args])
rv = _keep_coeff(cont, p, clear=clear, sign=sign)
return rv
expr = sympify(expr)
return do(expr)
def _mask_nc(eq, name=None):
"""
Return ``eq`` with non-commutative objects replaced with Dummy
symbols. A dictionary that can be used to restore the original
values is returned: if it is None, the expression is noncommutative
and cannot be made commutative. The third value returned is a list
of any non-commutative symbols that appear in the returned equation.
Explanation
===========
All non-commutative objects other than Symbols are replaced with
a non-commutative Symbol. Identical objects will be identified
by identical symbols.
If there is only 1 non-commutative object in an expression it will
be replaced with a commutative symbol. Otherwise, the non-commutative
entities are retained and the calling routine should handle
replacements in this case since some care must be taken to keep
track of the ordering of symbols when they occur within Muls.
Parameters
==========
name : str
``name``, if given, is the name that will be used with numbered Dummy
variables that will replace the non-commutative objects and is mainly
used for doctesting purposes.
Examples
========
>>> from sympy.physics.secondquant import Commutator, NO, F, Fd
>>> from sympy import symbols
>>> from sympy.core.exprtools import _mask_nc
>>> from sympy.abc import x, y
>>> A, B, C = symbols('A,B,C', commutative=False)
One nc-symbol:
>>> _mask_nc(A**2 - x**2, 'd')
(_d0**2 - x**2, {_d0: A}, [])
Multiple nc-symbols:
>>> _mask_nc(A**2 - B**2, 'd')
(A**2 - B**2, {}, [A, B])
An nc-object with nc-symbols but no others outside of it:
>>> _mask_nc(1 + x*Commutator(A, B), 'd')
(_d0*x + 1, {_d0: Commutator(A, B)}, [])
>>> _mask_nc(NO(Fd(x)*F(y)), 'd')
(_d0, {_d0: NO(CreateFermion(x)*AnnihilateFermion(y))}, [])
Multiple nc-objects:
>>> eq = x*Commutator(A, B) + x*Commutator(A, C)*Commutator(A, B)
>>> _mask_nc(eq, 'd')
(x*_d0 + x*_d1*_d0, {_d0: Commutator(A, B), _d1: Commutator(A, C)}, [_d0, _d1])
Multiple nc-objects and nc-symbols:
>>> eq = A*Commutator(A, B) + B*Commutator(A, C)
>>> _mask_nc(eq, 'd')
(A*_d0 + B*_d1, {_d0: Commutator(A, B), _d1: Commutator(A, C)}, [_d0, _d1, A, B])
"""
name = name or 'mask'
# Make Dummy() append sequential numbers to the name
def numbered_names():
i = 0
while True:
yield name + str(i)
i += 1
names = numbered_names()
def Dummy(*args, **kwargs):
from .symbol import Dummy
return Dummy(next(names), *args, **kwargs)
expr = eq
if expr.is_commutative:
return eq, {}, []
# identify nc-objects; symbols and other
rep = []
nc_obj = set()
nc_syms = set()
pot = preorder_traversal(expr, keys=default_sort_key)
for i, a in enumerate(pot):
if any(a == r[0] for r in rep):
pot.skip()
elif not a.is_commutative:
if a.is_symbol:
nc_syms.add(a)
pot.skip()
elif not (a.is_Add or a.is_Mul or a.is_Pow):
nc_obj.add(a)
pot.skip()
# If there is only one nc symbol or object, it can be factored regularly
# but polys is going to complain, so replace it with a Dummy.
if len(nc_obj) == 1 and not nc_syms:
rep.append((nc_obj.pop(), Dummy()))
elif len(nc_syms) == 1 and not nc_obj:
rep.append((nc_syms.pop(), Dummy()))
# Any remaining nc-objects will be replaced with an nc-Dummy and
# identified as an nc-Symbol to watch out for
nc_obj = sorted(nc_obj, key=default_sort_key)
for n in nc_obj:
nc = Dummy(commutative=False)
rep.append((n, nc))
nc_syms.add(nc)
expr = expr.subs(rep)
nc_syms = list(nc_syms)
nc_syms.sort(key=default_sort_key)
return expr, {v: k for k, v in rep}, nc_syms
def factor_nc(expr):
"""Return the factored form of ``expr`` while handling non-commutative
expressions.
Examples
========
>>> from sympy import factor_nc, Symbol
>>> from sympy.abc import x
>>> A = Symbol('A', commutative=False)
>>> B = Symbol('B', commutative=False)
>>> factor_nc((x**2 + 2*A*x + A**2).expand())
(x + A)**2
>>> factor_nc(((x + A)*(x + B)).expand())
(x + A)*(x + B)
"""
expr = sympify(expr)
if not isinstance(expr, Expr) or not expr.args:
return expr
if not expr.is_Add:
return expr.func(*[factor_nc(a) for a in expr.args])
from sympy.polys.polytools import gcd, factor
expr, rep, nc_symbols = _mask_nc(expr)
if rep:
return factor(expr).subs(rep)
else:
args = [a.args_cnc() for a in Add.make_args(expr)]
c = g = l = r = S.One
hit = False
# find any commutative gcd term
for i, a in enumerate(args):
if i == 0:
c = Mul._from_args(a[0])
elif a[0]:
c = gcd(c, Mul._from_args(a[0]))
else:
c = S.One
if c is not S.One:
hit = True
c, g = c.as_coeff_Mul()
if g is not S.One:
for i, (cc, _) in enumerate(args):
cc = list(Mul.make_args(Mul._from_args(list(cc))/g))
args[i][0] = cc
for i, (cc, _) in enumerate(args):
if cc:
cc[0] = cc[0]/c
else:
cc = [1/c]
args[i][0] = cc
# find any noncommutative common prefix
for i, a in enumerate(args):
if i == 0:
n = a[1][:]
else:
n = common_prefix(n, a[1])
if not n:
# is there a power that can be extracted?
if not args[0][1]:
break
b, e = args[0][1][0].as_base_exp()
ok = False
if e.is_Integer:
for t in args:
if not t[1]:
break
bt, et = t[1][0].as_base_exp()
if et.is_Integer and bt == b:
e = min(e, et)
else:
break
else:
ok = hit = True
l = b**e
il = b**-e
for _ in args:
_[1][0] = il*_[1][0]
break
if not ok:
break
else:
hit = True
lenn = len(n)
l = Mul(*n)
for _ in args:
_[1] = _[1][lenn:]
# find any noncommutative common suffix
for i, a in enumerate(args):
if i == 0:
n = a[1][:]
else:
n = common_suffix(n, a[1])
if not n:
# is there a power that can be extracted?
if not args[0][1]:
break
b, e = args[0][1][-1].as_base_exp()
ok = False
if e.is_Integer:
for t in args:
if not t[1]:
break
bt, et = t[1][-1].as_base_exp()
if et.is_Integer and bt == b:
e = min(e, et)
else:
break
else:
ok = hit = True
r = b**e
il = b**-e
for _ in args:
_[1][-1] = _[1][-1]*il
break
if not ok:
break
else:
hit = True
lenn = len(n)
r = Mul(*n)
for _ in args:
_[1] = _[1][:len(_[1]) - lenn]
if hit:
mid = Add(*[Mul(*cc)*Mul(*nc) for cc, nc in args])
else:
mid = expr
from sympy.simplify.powsimp import powsimp
# sort the symbols so the Dummys would appear in the same
# order as the original symbols, otherwise you may introduce
# a factor of -1, e.g. A**2 - B**2) -- {A:y, B:x} --> y**2 - x**2
# and the former factors into two terms, (A - B)*(A + B) while the
# latter factors into 3 terms, (-1)*(x - y)*(x + y)
rep1 = [(n, Dummy()) for n in sorted(nc_symbols, key=default_sort_key)]
unrep1 = [(v, k) for k, v in rep1]
unrep1.reverse()
new_mid, r2, _ = _mask_nc(mid.subs(rep1))
new_mid = powsimp(factor(new_mid))
new_mid = new_mid.subs(r2).subs(unrep1)
if new_mid.is_Pow:
return _keep_coeff(c, g*l*new_mid*r)
if new_mid.is_Mul:
def _pemexpand(expr):
"Expand with the minimal set of hints necessary to check the result."
return expr.expand(deep=True, mul=True, power_exp=True,
power_base=False, basic=False, multinomial=True, log=False)
# XXX TODO there should be a way to inspect what order the terms
# must be in and just select the plausible ordering without
# checking permutations
cfac = []
ncfac = []
for f in new_mid.args:
if f.is_commutative:
cfac.append(f)
else:
b, e = f.as_base_exp()
if e.is_Integer:
ncfac.extend([b]*e)
else:
ncfac.append(f)
pre_mid = g*Mul(*cfac)*l
target = _pemexpand(expr/c)
for s in variations(ncfac, len(ncfac)):
ok = pre_mid*Mul(*s)*r
if _pemexpand(ok) == target:
return _keep_coeff(c, ok)
# mid was an Add that didn't factor successfully
return _keep_coeff(c, g*l*mid*r)
|
33e64cb7e0d1520c9979dfc8d608a4c446d260424bc22ee6c298ddfbf885282f | from .basic import Basic
from .sorting import ordered
from .sympify import sympify
from sympy.utilities.iterables import iterable
def iterargs(expr):
"""Yield the args of a Basic object in a breadth-first traversal.
Depth-traversal stops if `arg.args` is either empty or is not
an iterable.
Examples
========
>>> from sympy import Integral, Function
>>> from sympy.abc import x
>>> f = Function('f')
>>> from sympy.core.traversal import iterargs
>>> list(iterargs(Integral(f(x), (f(x), 1))))
[Integral(f(x), (f(x), 1)), f(x), (f(x), 1), x, f(x), 1, x]
See Also
========
iterfreeargs, preorder_traversal
"""
args = [expr]
for i in args:
yield i
try:
args.extend(i.args)
except TypeError:
pass # for cases like f being an arg
def iterfreeargs(expr, _first=True):
"""Yield the args of a Basic object in a breadth-first traversal.
Depth-traversal stops if `arg.args` is either empty or is not
an iterable. The bound objects of an expression will be returned
as canonical variables.
Examples
========
>>> from sympy import Integral, Function
>>> from sympy.abc import x
>>> f = Function('f')
>>> from sympy.core.traversal import iterfreeargs
>>> list(iterfreeargs(Integral(f(x), (f(x), 1))))
[Integral(f(x), (f(x), 1)), 1]
See Also
========
iterargs, preorder_traversal
"""
args = [expr]
for i in args:
yield i
if _first and hasattr(i, 'bound_symbols'):
void = i.canonical_variables.values()
for i in iterfreeargs(i.as_dummy(), _first=False):
if not i.has(*void):
yield i
try:
args.extend(i.args)
except TypeError:
pass # for cases like f being an arg
class preorder_traversal:
"""
Do a pre-order traversal of a tree.
This iterator recursively yields nodes that it has visited in a pre-order
fashion. That is, it yields the current node then descends through the
tree breadth-first to yield all of a node's children's pre-order
traversal.
For an expression, the order of the traversal depends on the order of
.args, which in many cases can be arbitrary.
Parameters
==========
node : SymPy expression
The expression to traverse.
keys : (default None) sort key(s)
The key(s) used to sort args of Basic objects. When None, args of Basic
objects are processed in arbitrary order. If key is defined, it will
be passed along to ordered() as the only key(s) to use to sort the
arguments; if ``key`` is simply True then the default keys of ordered
will be used.
Yields
======
subtree : SymPy expression
All of the subtrees in the tree.
Examples
========
>>> from sympy import preorder_traversal, symbols
>>> x, y, z = symbols('x y z')
The nodes are returned in the order that they are encountered unless key
is given; simply passing key=True will guarantee that the traversal is
unique.
>>> list(preorder_traversal((x + y)*z, keys=None)) # doctest: +SKIP
[z*(x + y), z, x + y, y, x]
>>> list(preorder_traversal((x + y)*z, keys=True))
[z*(x + y), z, x + y, x, y]
"""
def __init__(self, node, keys=None):
self._skip_flag = False
self._pt = self._preorder_traversal(node, keys)
def _preorder_traversal(self, node, keys):
yield node
if self._skip_flag:
self._skip_flag = False
return
if isinstance(node, Basic):
if not keys and hasattr(node, '_argset'):
# LatticeOp keeps args as a set. We should use this if we
# don't care about the order, to prevent unnecessary sorting.
args = node._argset
else:
args = node.args
if keys:
if keys != True:
args = ordered(args, keys, default=False)
else:
args = ordered(args)
for arg in args:
yield from self._preorder_traversal(arg, keys)
elif iterable(node):
for item in node:
yield from self._preorder_traversal(item, keys)
def skip(self):
"""
Skip yielding current node's (last yielded node's) subtrees.
Examples
========
>>> from sympy import preorder_traversal, symbols
>>> x, y, z = symbols('x y z')
>>> pt = preorder_traversal((x + y*z)*z)
>>> for i in pt:
... print(i)
... if i == x + y*z:
... pt.skip()
z*(x + y*z)
z
x + y*z
"""
self._skip_flag = True
def __next__(self):
return next(self._pt)
def __iter__(self):
return self
def use(expr, func, level=0, args=(), kwargs={}):
"""
Use ``func`` to transform ``expr`` at the given level.
Examples
========
>>> from sympy import use, expand
>>> from sympy.abc import x, y
>>> f = (x + y)**2*x + 1
>>> use(f, expand, level=2)
x*(x**2 + 2*x*y + y**2) + 1
>>> expand(f)
x**3 + 2*x**2*y + x*y**2 + 1
"""
def _use(expr, level):
if not level:
return func(expr, *args, **kwargs)
else:
if expr.is_Atom:
return expr
else:
level -= 1
_args = [_use(arg, level) for arg in expr.args]
return expr.__class__(*_args)
return _use(sympify(expr), level)
def walk(e, *target):
"""Iterate through the args that are the given types (target) and
return a list of the args that were traversed; arguments
that are not of the specified types are not traversed.
Examples
========
>>> from sympy.core.traversal import walk
>>> from sympy import Min, Max
>>> from sympy.abc import x, y, z
>>> list(walk(Min(x, Max(y, Min(1, z))), Min))
[Min(x, Max(y, Min(1, z)))]
>>> list(walk(Min(x, Max(y, Min(1, z))), Min, Max))
[Min(x, Max(y, Min(1, z))), Max(y, Min(1, z)), Min(1, z)]
See Also
========
bottom_up
"""
if isinstance(e, target):
yield e
for i in e.args:
yield from walk(i, *target)
def bottom_up(rv, F, atoms=False, nonbasic=False):
"""Apply ``F`` to all expressions in an expression tree from the
bottom up. If ``atoms`` is True, apply ``F`` even if there are no args;
if ``nonbasic`` is True, try to apply ``F`` to non-Basic objects.
"""
args = getattr(rv, 'args', None)
if args is not None:
if args:
args = tuple([bottom_up(a, F, atoms, nonbasic) for a in args])
if args != rv.args:
rv = rv.func(*args)
rv = F(rv)
elif atoms:
rv = F(rv)
else:
if nonbasic:
try:
rv = F(rv)
except TypeError:
pass
return rv
def postorder_traversal(node, keys=None):
"""
Do a postorder traversal of a tree.
This generator recursively yields nodes that it has visited in a postorder
fashion. That is, it descends through the tree depth-first to yield all of
a node's children's postorder traversal before yielding the node itself.
Parameters
==========
node : SymPy expression
The expression to traverse.
keys : (default None) sort key(s)
The key(s) used to sort args of Basic objects. When None, args of Basic
objects are processed in arbitrary order. If key is defined, it will
be passed along to ordered() as the only key(s) to use to sort the
arguments; if ``key`` is simply True then the default keys of
``ordered`` will be used (node count and default_sort_key).
Yields
======
subtree : SymPy expression
All of the subtrees in the tree.
Examples
========
>>> from sympy import postorder_traversal
>>> from sympy.abc import w, x, y, z
The nodes are returned in the order that they are encountered unless key
is given; simply passing key=True will guarantee that the traversal is
unique.
>>> list(postorder_traversal(w + (x + y)*z)) # doctest: +SKIP
[z, y, x, x + y, z*(x + y), w, w + z*(x + y)]
>>> list(postorder_traversal(w + (x + y)*z, keys=True))
[w, z, x, y, x + y, z*(x + y), w + z*(x + y)]
"""
if isinstance(node, Basic):
args = node.args
if keys:
if keys != True:
args = ordered(args, keys, default=False)
else:
args = ordered(args)
for arg in args:
yield from postorder_traversal(arg, keys)
elif iterable(node):
for item in node:
yield from postorder_traversal(item, keys)
yield node
|
a3bfb774fbe2bd079d7447f794fd0631001fb8c9c7339cbf8ea6633697ab21e1 | from __future__ import annotations
from typing import TYPE_CHECKING
from collections.abc import Iterable
from functools import reduce
import re
from .sympify import sympify, _sympify
from .basic import Basic, Atom
from .singleton import S
from .evalf import EvalfMixin, pure_complex, DEFAULT_MAXPREC
from .decorators import call_highest_priority, sympify_method_args, sympify_return
from .cache import cacheit
from .sorting import default_sort_key
from .kind import NumberKind
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.misc import as_int, func_name, filldedent
from sympy.utilities.iterables import has_variety, sift
from mpmath.libmp import mpf_log, prec_to_dps
from mpmath.libmp.libintmath import giant_steps
if TYPE_CHECKING:
from .numbers import Number
from collections import defaultdict
def _corem(eq, c): # helper for extract_additively
# return co, diff from co*c + diff
co = []
non = []
for i in Add.make_args(eq):
ci = i.coeff(c)
if not ci:
non.append(i)
else:
co.append(ci)
return Add(*co), Add(*non)
@sympify_method_args
class Expr(Basic, EvalfMixin):
"""
Base class for algebraic expressions.
Explanation
===========
Everything that requires arithmetic operations to be defined
should subclass this class, instead of Basic (which should be
used only for argument storage and expression manipulation, i.e.
pattern matching, substitutions, etc).
If you want to override the comparisons of expressions:
Should use _eval_is_ge for inequality, or _eval_is_eq, with multiple dispatch.
_eval_is_ge return true if x >= y, false if x < y, and None if the two types
are not comparable or the comparison is indeterminate
See Also
========
sympy.core.basic.Basic
"""
__slots__: tuple[str, ...] = ()
is_scalar = True # self derivative is 1
@property
def _diff_wrt(self):
"""Return True if one can differentiate with respect to this
object, else False.
Explanation
===========
Subclasses such as Symbol, Function and Derivative return True
to enable derivatives wrt them. The implementation in Derivative
separates the Symbol and non-Symbol (_diff_wrt=True) variables and
temporarily converts the non-Symbols into Symbols when performing
the differentiation. By default, any object deriving from Expr
will behave like a scalar with self.diff(self) == 1. If this is
not desired then the object must also set `is_scalar = False` or
else define an _eval_derivative routine.
Note, see the docstring of Derivative for how this should work
mathematically. In particular, note that expr.subs(yourclass, Symbol)
should be well-defined on a structural level, or this will lead to
inconsistent results.
Examples
========
>>> from sympy import Expr
>>> e = Expr()
>>> e._diff_wrt
False
>>> class MyScalar(Expr):
... _diff_wrt = True
...
>>> MyScalar().diff(MyScalar())
1
>>> class MySymbol(Expr):
... _diff_wrt = True
... is_scalar = False
...
>>> MySymbol().diff(MySymbol())
Derivative(MySymbol(), MySymbol())
"""
return False
@cacheit
def sort_key(self, order=None):
coeff, expr = self.as_coeff_Mul()
if expr.is_Pow:
if expr.base is S.Exp1:
# If we remove this, many doctests will go crazy:
# (keeps E**x sorted like the exp(x) function,
# part of exp(x) to E**x transition)
expr, exp = Function("exp")(expr.exp), S.One
else:
expr, exp = expr.args
else:
exp = S.One
if expr.is_Dummy:
args = (expr.sort_key(),)
elif expr.is_Atom:
args = (str(expr),)
else:
if expr.is_Add:
args = expr.as_ordered_terms(order=order)
elif expr.is_Mul:
args = expr.as_ordered_factors(order=order)
else:
args = expr.args
args = tuple(
[ default_sort_key(arg, order=order) for arg in args ])
args = (len(args), tuple(args))
exp = exp.sort_key(order=order)
return expr.class_key(), args, exp, coeff
def _hashable_content(self):
"""Return a tuple of information about self that can be used to
compute the hash. If a class defines additional attributes,
like ``name`` in Symbol, then this method should be updated
accordingly to return such relevant attributes.
Defining more than _hashable_content is necessary if __eq__ has
been defined by a class. See note about this in Basic.__eq__."""
return self._args
# ***************
# * Arithmetics *
# ***************
# Expr and its sublcasses use _op_priority to determine which object
# passed to a binary special method (__mul__, etc.) will handle the
# operation. In general, the 'call_highest_priority' decorator will choose
# the object with the highest _op_priority to handle the call.
# Custom subclasses that want to define their own binary special methods
# should set an _op_priority value that is higher than the default.
#
# **NOTE**:
# This is a temporary fix, and will eventually be replaced with
# something better and more powerful. See issue 5510.
_op_priority = 10.0
@property
def _add_handler(self):
return Add
@property
def _mul_handler(self):
return Mul
def __pos__(self):
return self
def __neg__(self):
# Mul has its own __neg__ routine, so we just
# create a 2-args Mul with the -1 in the canonical
# slot 0.
c = self.is_commutative
return Mul._from_args((S.NegativeOne, self), c)
def __abs__(self) -> Expr:
from sympy.functions.elementary.complexes import Abs
return Abs(self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__radd__')
def __add__(self, other):
return Add(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__add__')
def __radd__(self, other):
return Add(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rsub__')
def __sub__(self, other):
return Add(self, -other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__sub__')
def __rsub__(self, other):
return Add(other, -self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rmul__')
def __mul__(self, other):
return Mul(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__mul__')
def __rmul__(self, other):
return Mul(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rpow__')
def _pow(self, other):
return Pow(self, other)
def __pow__(self, other, mod=None) -> Expr:
if mod is None:
return self._pow(other)
try:
_self, other, mod = as_int(self), as_int(other), as_int(mod)
if other >= 0:
return _sympify(pow(_self, other, mod))
else:
from .numbers import mod_inverse
return _sympify(mod_inverse(pow(_self, -other, mod), mod))
except ValueError:
power = self._pow(other)
try:
return power%mod
except TypeError:
return NotImplemented
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__pow__')
def __rpow__(self, other):
return Pow(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rtruediv__')
def __truediv__(self, other):
denom = Pow(other, S.NegativeOne)
if self is S.One:
return denom
else:
return Mul(self, denom)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__truediv__')
def __rtruediv__(self, other):
denom = Pow(self, S.NegativeOne)
if other is S.One:
return denom
else:
return Mul(other, denom)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rmod__')
def __mod__(self, other):
return Mod(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__mod__')
def __rmod__(self, other):
return Mod(other, self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rfloordiv__')
def __floordiv__(self, other):
from sympy.functions.elementary.integers import floor
return floor(self / other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__floordiv__')
def __rfloordiv__(self, other):
from sympy.functions.elementary.integers import floor
return floor(other / self)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__rdivmod__')
def __divmod__(self, other):
from sympy.functions.elementary.integers import floor
return floor(self / other), Mod(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
@call_highest_priority('__divmod__')
def __rdivmod__(self, other):
from sympy.functions.elementary.integers import floor
return floor(other / self), Mod(other, self)
def __int__(self):
# Although we only need to round to the units position, we'll
# get one more digit so the extra testing below can be avoided
# unless the rounded value rounded to an integer, e.g. if an
# expression were equal to 1.9 and we rounded to the unit position
# we would get a 2 and would not know if this rounded up or not
# without doing a test (as done below). But if we keep an extra
# digit we know that 1.9 is not the same as 1 and there is no
# need for further testing: our int value is correct. If the value
# were 1.99, however, this would round to 2.0 and our int value is
# off by one. So...if our round value is the same as the int value
# (regardless of how much extra work we do to calculate extra decimal
# places) we need to test whether we are off by one.
from .symbol import Dummy
if not self.is_number:
raise TypeError("Cannot convert symbols to int")
r = self.round(2)
if not r.is_Number:
raise TypeError("Cannot convert complex to int")
if r in (S.NaN, S.Infinity, S.NegativeInfinity):
raise TypeError("Cannot convert %s to int" % r)
i = int(r)
if not i:
return 0
# off-by-one check
if i == r and not (self - i).equals(0):
isign = 1 if i > 0 else -1
x = Dummy()
# in the following (self - i).evalf(2) will not always work while
# (self - r).evalf(2) and the use of subs does; if the test that
# was added when this comment was added passes, it might be safe
# to simply use sign to compute this rather than doing this by hand:
diff_sign = 1 if (self - x).evalf(2, subs={x: i}) > 0 else -1
if diff_sign != isign:
i -= isign
return i
def __float__(self):
# Don't bother testing if it's a number; if it's not this is going
# to fail, and if it is we still need to check that it evalf'ed to
# a number.
result = self.evalf()
if result.is_Number:
return float(result)
if result.is_number and result.as_real_imag()[1]:
raise TypeError("Cannot convert complex to float")
raise TypeError("Cannot convert expression to float")
def __complex__(self):
result = self.evalf()
re, im = result.as_real_imag()
return complex(float(re), float(im))
@sympify_return([('other', 'Expr')], NotImplemented)
def __ge__(self, other):
from .relational import GreaterThan
return GreaterThan(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
def __le__(self, other):
from .relational import LessThan
return LessThan(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
def __gt__(self, other):
from .relational import StrictGreaterThan
return StrictGreaterThan(self, other)
@sympify_return([('other', 'Expr')], NotImplemented)
def __lt__(self, other):
from .relational import StrictLessThan
return StrictLessThan(self, other)
def __trunc__(self):
if not self.is_number:
raise TypeError("Cannot truncate symbols and expressions")
else:
return Integer(self)
def __format__(self, format_spec: str):
if self.is_number:
mt = re.match(r'\+?\d*\.(\d+)f', format_spec)
if mt:
prec = int(mt.group(1))
rounded = self.round(prec)
if rounded.is_Integer:
return format(int(rounded), format_spec)
if rounded.is_Float:
return format(rounded, format_spec)
return super().__format__(format_spec)
@staticmethod
def _from_mpmath(x, prec):
if hasattr(x, "_mpf_"):
return Float._new(x._mpf_, prec)
elif hasattr(x, "_mpc_"):
re, im = x._mpc_
re = Float._new(re, prec)
im = Float._new(im, prec)*S.ImaginaryUnit
return re + im
else:
raise TypeError("expected mpmath number (mpf or mpc)")
@property
def is_number(self):
"""Returns True if ``self`` has no free symbols and no
undefined functions (AppliedUndef, to be precise). It will be
faster than ``if not self.free_symbols``, however, since
``is_number`` will fail as soon as it hits a free symbol
or undefined function.
Examples
========
>>> from sympy import Function, Integral, cos, sin, pi
>>> from sympy.abc import x
>>> f = Function('f')
>>> x.is_number
False
>>> f(1).is_number
False
>>> (2*x).is_number
False
>>> (2 + Integral(2, x)).is_number
False
>>> (2 + Integral(2, (x, 1, 2))).is_number
True
Not all numbers are Numbers in the SymPy sense:
>>> pi.is_number, pi.is_Number
(True, False)
If something is a number it should evaluate to a number with
real and imaginary parts that are Numbers; the result may not
be comparable, however, since the real and/or imaginary part
of the result may not have precision.
>>> cos(1).is_number and cos(1).is_comparable
True
>>> z = cos(1)**2 + sin(1)**2 - 1
>>> z.is_number
True
>>> z.is_comparable
False
See Also
========
sympy.core.basic.Basic.is_comparable
"""
return all(obj.is_number for obj in self.args)
def _random(self, n=None, re_min=-1, im_min=-1, re_max=1, im_max=1):
"""Return self evaluated, if possible, replacing free symbols with
random complex values, if necessary.
Explanation
===========
The random complex value for each free symbol is generated
by the random_complex_number routine giving real and imaginary
parts in the range given by the re_min, re_max, im_min, and im_max
values. The returned value is evaluated to a precision of n
(if given) else the maximum of 15 and the precision needed
to get more than 1 digit of precision. If the expression
could not be evaluated to a number, or could not be evaluated
to more than 1 digit of precision, then None is returned.
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import x, y
>>> x._random() # doctest: +SKIP
0.0392918155679172 + 0.916050214307199*I
>>> x._random(2) # doctest: +SKIP
-0.77 - 0.87*I
>>> (x + y/2)._random(2) # doctest: +SKIP
-0.57 + 0.16*I
>>> sqrt(2)._random(2)
1.4
See Also
========
sympy.core.random.random_complex_number
"""
free = self.free_symbols
prec = 1
if free:
from sympy.core.random import random_complex_number
a, c, b, d = re_min, re_max, im_min, im_max
reps = dict(list(zip(free, [random_complex_number(a, b, c, d, rational=True)
for zi in free])))
try:
nmag = abs(self.evalf(2, subs=reps))
except (ValueError, TypeError):
# if an out of range value resulted in evalf problems
# then return None -- XXX is there a way to know how to
# select a good random number for a given expression?
# e.g. when calculating n! negative values for n should not
# be used
return None
else:
reps = {}
nmag = abs(self.evalf(2))
if not hasattr(nmag, '_prec'):
# e.g. exp_polar(2*I*pi) doesn't evaluate but is_number is True
return None
if nmag._prec == 1:
# increase the precision up to the default maximum
# precision to see if we can get any significance
# evaluate
for prec in giant_steps(2, DEFAULT_MAXPREC):
nmag = abs(self.evalf(prec, subs=reps))
if nmag._prec != 1:
break
if nmag._prec != 1:
if n is None:
n = max(prec, 15)
return self.evalf(n, subs=reps)
# never got any significance
return None
def is_constant(self, *wrt, **flags):
"""Return True if self is constant, False if not, or None if
the constancy could not be determined conclusively.
Explanation
===========
If an expression has no free symbols then it is a constant. If
there are free symbols it is possible that the expression is a
constant, perhaps (but not necessarily) zero. To test such
expressions, a few strategies are tried:
1) numerical evaluation at two random points. If two such evaluations
give two different values and the values have a precision greater than
1 then self is not constant. If the evaluations agree or could not be
obtained with any precision, no decision is made. The numerical testing
is done only if ``wrt`` is different than the free symbols.
2) differentiation with respect to variables in 'wrt' (or all free
symbols if omitted) to see if the expression is constant or not. This
will not always lead to an expression that is zero even though an
expression is constant (see added test in test_expr.py). If
all derivatives are zero then self is constant with respect to the
given symbols.
3) finding out zeros of denominator expression with free_symbols.
It will not be constant if there are zeros. It gives more negative
answers for expression that are not constant.
If neither evaluation nor differentiation can prove the expression is
constant, None is returned unless two numerical values happened to be
the same and the flag ``failing_number`` is True -- in that case the
numerical value will be returned.
If flag simplify=False is passed, self will not be simplified;
the default is True since self should be simplified before testing.
Examples
========
>>> from sympy import cos, sin, Sum, S, pi
>>> from sympy.abc import a, n, x, y
>>> x.is_constant()
False
>>> S(2).is_constant()
True
>>> Sum(x, (x, 1, 10)).is_constant()
True
>>> Sum(x, (x, 1, n)).is_constant()
False
>>> Sum(x, (x, 1, n)).is_constant(y)
True
>>> Sum(x, (x, 1, n)).is_constant(n)
False
>>> Sum(x, (x, 1, n)).is_constant(x)
True
>>> eq = a*cos(x)**2 + a*sin(x)**2 - a
>>> eq.is_constant()
True
>>> eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
True
>>> (0**x).is_constant()
False
>>> x.is_constant()
False
>>> (x**x).is_constant()
False
>>> one = cos(x)**2 + sin(x)**2
>>> one.is_constant()
True
>>> ((one - 1)**(x + 1)).is_constant() in (True, False) # could be 0 or 1
True
"""
def check_denominator_zeros(expression):
from sympy.solvers.solvers import denoms
retNone = False
for den in denoms(expression):
z = den.is_zero
if z is True:
return True
if z is None:
retNone = True
if retNone:
return None
return False
simplify = flags.get('simplify', True)
if self.is_number:
return True
free = self.free_symbols
if not free:
return True # assume f(1) is some constant
# if we are only interested in some symbols and they are not in the
# free symbols then this expression is constant wrt those symbols
wrt = set(wrt)
if wrt and not wrt & free:
return True
wrt = wrt or free
# simplify unless this has already been done
expr = self
if simplify:
expr = expr.simplify()
# is_zero should be a quick assumptions check; it can be wrong for
# numbers (see test_is_not_constant test), giving False when it
# shouldn't, but hopefully it will never give True unless it is sure.
if expr.is_zero:
return True
# Don't attempt substitution or differentiation with non-number symbols
wrt_number = {sym for sym in wrt if sym.kind is NumberKind}
# try numerical evaluation to see if we get two different values
failing_number = None
if wrt_number == free:
# try 0 (for a) and 1 (for b)
try:
a = expr.subs(list(zip(free, [0]*len(free))),
simultaneous=True)
if a is S.NaN:
# evaluation may succeed when substitution fails
a = expr._random(None, 0, 0, 0, 0)
except ZeroDivisionError:
a = None
if a is not None and a is not S.NaN:
try:
b = expr.subs(list(zip(free, [1]*len(free))),
simultaneous=True)
if b is S.NaN:
# evaluation may succeed when substitution fails
b = expr._random(None, 1, 0, 1, 0)
except ZeroDivisionError:
b = None
if b is not None and b is not S.NaN and b.equals(a) is False:
return False
# try random real
b = expr._random(None, -1, 0, 1, 0)
if b is not None and b is not S.NaN and b.equals(a) is False:
return False
# try random complex
b = expr._random()
if b is not None and b is not S.NaN:
if b.equals(a) is False:
return False
failing_number = a if a.is_number else b
# now we will test each wrt symbol (or all free symbols) to see if the
# expression depends on them or not using differentiation. This is
# not sufficient for all expressions, however, so we don't return
# False if we get a derivative other than 0 with free symbols.
for w in wrt_number:
deriv = expr.diff(w)
if simplify:
deriv = deriv.simplify()
if deriv != 0:
if not (pure_complex(deriv, or_real=True)):
if flags.get('failing_number', False):
return failing_number
return False
cd = check_denominator_zeros(self)
if cd is True:
return False
elif cd is None:
return None
return True
def equals(self, other, failing_expression=False):
"""Return True if self == other, False if it does not, or None. If
failing_expression is True then the expression which did not simplify
to a 0 will be returned instead of None.
Explanation
===========
If ``self`` is a Number (or complex number) that is not zero, then
the result is False.
If ``self`` is a number and has not evaluated to zero, evalf will be
used to test whether the expression evaluates to zero. If it does so
and the result has significance (i.e. the precision is either -1, for
a Rational result, or is greater than 1) then the evalf value will be
used to return True or False.
"""
from sympy.simplify.simplify import nsimplify, simplify
from sympy.solvers.solvers import solve
from sympy.polys.polyerrors import NotAlgebraic
from sympy.polys.numberfields import minimal_polynomial
other = sympify(other)
if self == other:
return True
# they aren't the same so see if we can make the difference 0;
# don't worry about doing simplification steps one at a time
# because if the expression ever goes to 0 then the subsequent
# simplification steps that are done will be very fast.
diff = factor_terms(simplify(self - other), radical=True)
if not diff:
return True
if not diff.has(Add, Mod):
# if there is no expanding to be done after simplifying
# then this can't be a zero
return False
factors = diff.as_coeff_mul()[1]
if len(factors) > 1: # avoid infinity recursion
fac_zero = [fac.equals(0) for fac in factors]
if None not in fac_zero: # every part can be decided
return any(fac_zero)
constant = diff.is_constant(simplify=False, failing_number=True)
if constant is False:
return False
if not diff.is_number:
if constant is None:
# e.g. unless the right simplification is done, a symbolic
# zero is possible (see expression of issue 6829: without
# simplification constant will be None).
return
if constant is True:
# this gives a number whether there are free symbols or not
ndiff = diff._random()
# is_comparable will work whether the result is real
# or complex; it could be None, however.
if ndiff and ndiff.is_comparable:
return False
# sometimes we can use a simplified result to give a clue as to
# what the expression should be; if the expression is *not* zero
# then we should have been able to compute that and so now
# we can just consider the cases where the approximation appears
# to be zero -- we try to prove it via minimal_polynomial.
#
# removed
# ns = nsimplify(diff)
# if diff.is_number and (not ns or ns == diff):
#
# The thought was that if it nsimplifies to 0 that's a sure sign
# to try the following to prove it; or if it changed but wasn't
# zero that might be a sign that it's not going to be easy to
# prove. But tests seem to be working without that logic.
#
if diff.is_number:
# try to prove via self-consistency
surds = [s for s in diff.atoms(Pow) if s.args[0].is_Integer]
# it seems to work better to try big ones first
surds.sort(key=lambda x: -x.args[0])
for s in surds:
try:
# simplify is False here -- this expression has already
# been identified as being hard to identify as zero;
# we will handle the checking ourselves using nsimplify
# to see if we are in the right ballpark or not and if so
# *then* the simplification will be attempted.
sol = solve(diff, s, simplify=False)
if sol:
if s in sol:
# the self-consistent result is present
return True
if all(si.is_Integer for si in sol):
# perfect powers are removed at instantiation
# so surd s cannot be an integer
return False
if all(i.is_algebraic is False for i in sol):
# a surd is algebraic
return False
if any(si in surds for si in sol):
# it wasn't equal to s but it is in surds
# and different surds are not equal
return False
if any(nsimplify(s - si) == 0 and
simplify(s - si) == 0 for si in sol):
return True
if s.is_real:
if any(nsimplify(si, [s]) == s and simplify(si) == s
for si in sol):
return True
except NotImplementedError:
pass
# try to prove with minimal_polynomial but know when
# *not* to use this or else it can take a long time. e.g. issue 8354
if True: # change True to condition that assures non-hang
try:
mp = minimal_polynomial(diff)
if mp.is_Symbol:
return True
return False
except (NotAlgebraic, NotImplementedError):
pass
# diff has not simplified to zero; constant is either None, True
# or the number with significance (is_comparable) that was randomly
# calculated twice as the same value.
if constant not in (True, None) and constant != 0:
return False
if failing_expression:
return diff
return None
def _eval_is_extended_positive_negative(self, positive):
from sympy.polys.numberfields import minimal_polynomial
from sympy.polys.polyerrors import NotAlgebraic
if self.is_number:
# check to see that we can get a value
try:
n2 = self._eval_evalf(2)
# XXX: This shouldn't be caught here
# Catches ValueError: hypsum() failed to converge to the requested
# 34 bits of accuracy
except ValueError:
return None
if n2 is None:
return None
if getattr(n2, '_prec', 1) == 1: # no significance
return None
if n2 is S.NaN:
return None
f = self.evalf(2)
if f.is_Float:
match = f, S.Zero
else:
match = pure_complex(f)
if match is None:
return False
r, i = match
if not (i.is_Number and r.is_Number):
return False
if r._prec != 1 and i._prec != 1:
return bool(not i and ((r > 0) if positive else (r < 0)))
elif r._prec == 1 and (not i or i._prec == 1) and \
self._eval_is_algebraic() and not self.has(Function):
try:
if minimal_polynomial(self).is_Symbol:
return False
except (NotAlgebraic, NotImplementedError):
pass
def _eval_is_extended_positive(self):
return self._eval_is_extended_positive_negative(positive=True)
def _eval_is_extended_negative(self):
return self._eval_is_extended_positive_negative(positive=False)
def _eval_interval(self, x, a, b):
"""
Returns evaluation over an interval. For most functions this is:
self.subs(x, b) - self.subs(x, a),
possibly using limit() if NaN is returned from subs, or if
singularities are found between a and b.
If b or a is None, it only evaluates -self.subs(x, a) or self.subs(b, x),
respectively.
"""
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.exponential import log
from sympy.series.limits import limit, Limit
from sympy.sets.sets import Interval
from sympy.solvers.solveset import solveset
if (a is None and b is None):
raise ValueError('Both interval ends cannot be None.')
def _eval_endpoint(left):
c = a if left else b
if c is None:
return S.Zero
else:
C = self.subs(x, c)
if C.has(S.NaN, S.Infinity, S.NegativeInfinity,
S.ComplexInfinity, AccumBounds):
if (a < b) != False:
C = limit(self, x, c, "+" if left else "-")
else:
C = limit(self, x, c, "-" if left else "+")
if isinstance(C, Limit):
raise NotImplementedError("Could not compute limit")
return C
if a == b:
return S.Zero
A = _eval_endpoint(left=True)
if A is S.NaN:
return A
B = _eval_endpoint(left=False)
if (a and b) is None:
return B - A
value = B - A
if a.is_comparable and b.is_comparable:
if a < b:
domain = Interval(a, b)
else:
domain = Interval(b, a)
# check the singularities of self within the interval
# if singularities is a ConditionSet (not iterable), catch the exception and pass
singularities = solveset(self.cancel().as_numer_denom()[1], x,
domain=domain)
for logterm in self.atoms(log):
singularities = singularities | solveset(logterm.args[0], x,
domain=domain)
try:
for s in singularities:
if value is S.NaN:
# no need to keep adding, it will stay NaN
break
if not s.is_comparable:
continue
if (a < s) == (s < b) == True:
value += -limit(self, x, s, "+") + limit(self, x, s, "-")
elif (b < s) == (s < a) == True:
value += limit(self, x, s, "+") - limit(self, x, s, "-")
except TypeError:
pass
return value
def _eval_power(self, other):
# subclass to compute self**other for cases when
# other is not NaN, 0, or 1
return None
def _eval_conjugate(self):
if self.is_extended_real:
return self
elif self.is_imaginary:
return -self
def conjugate(self):
"""Returns the complex conjugate of 'self'."""
from sympy.functions.elementary.complexes import conjugate as c
return c(self)
def dir(self, x, cdir):
if self.is_zero:
return S.Zero
from sympy.functions.elementary.exponential import log
minexp = S.Zero
arg = self
while arg:
minexp += S.One
arg = arg.diff(x)
coeff = arg.subs(x, 0)
if coeff is S.NaN:
coeff = arg.limit(x, 0)
if coeff is S.ComplexInfinity:
try:
coeff, _ = arg.leadterm(x)
if coeff.has(log(x)):
raise ValueError()
except ValueError:
coeff = arg.limit(x, 0)
if coeff != S.Zero:
break
return coeff*cdir**minexp
def _eval_transpose(self):
from sympy.functions.elementary.complexes import conjugate
if (self.is_complex or self.is_infinite):
return self
elif self.is_hermitian:
return conjugate(self)
elif self.is_antihermitian:
return -conjugate(self)
def transpose(self):
from sympy.functions.elementary.complexes import transpose
return transpose(self)
def _eval_adjoint(self):
from sympy.functions.elementary.complexes import conjugate, transpose
if self.is_hermitian:
return self
elif self.is_antihermitian:
return -self
obj = self._eval_conjugate()
if obj is not None:
return transpose(obj)
obj = self._eval_transpose()
if obj is not None:
return conjugate(obj)
def adjoint(self):
from sympy.functions.elementary.complexes import adjoint
return adjoint(self)
@classmethod
def _parse_order(cls, order):
"""Parse and configure the ordering of terms. """
from sympy.polys.orderings import monomial_key
startswith = getattr(order, "startswith", None)
if startswith is None:
reverse = False
else:
reverse = startswith('rev-')
if reverse:
order = order[4:]
monom_key = monomial_key(order)
def neg(monom):
return tuple([neg(m) if isinstance(m, tuple) else -m for m in monom])
def key(term):
_, ((re, im), monom, ncpart) = term
monom = neg(monom_key(monom))
ncpart = tuple([e.sort_key(order=order) for e in ncpart])
coeff = ((bool(im), im), (re, im))
return monom, ncpart, coeff
return key, reverse
def as_ordered_factors(self, order=None):
"""Return list of ordered factors (if Mul) else [self]."""
return [self]
def as_poly(self, *gens, **args):
"""Converts ``self`` to a polynomial or returns ``None``.
Explanation
===========
>>> from sympy import sin
>>> from sympy.abc import x, y
>>> print((x**2 + x*y).as_poly())
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + x*y).as_poly(x, y))
Poly(x**2 + x*y, x, y, domain='ZZ')
>>> print((x**2 + sin(y)).as_poly(x, y))
None
"""
from sympy.polys.polyerrors import PolynomialError, GeneratorsNeeded
from sympy.polys.polytools import Poly
try:
poly = Poly(self, *gens, **args)
if not poly.is_Poly:
return None
else:
return poly
except (PolynomialError, GeneratorsNeeded):
# PolynomialError is caught for e.g. exp(x).as_poly(x)
# GeneratorsNeeded is caught for e.g. S(2).as_poly()
return None
def as_ordered_terms(self, order=None, data=False):
"""
Transform an expression to an ordered list of terms.
Examples
========
>>> from sympy import sin, cos
>>> from sympy.abc import x
>>> (sin(x)**2*cos(x) + sin(x)**2 + 1).as_ordered_terms()
[sin(x)**2*cos(x), sin(x)**2, 1]
"""
from .numbers import Number, NumberSymbol
if order is None and self.is_Add:
# Spot the special case of Add(Number, Mul(Number, expr)) with the
# first number positive and the second number negative
key = lambda x:not isinstance(x, (Number, NumberSymbol))
add_args = sorted(Add.make_args(self), key=key)
if (len(add_args) == 2
and isinstance(add_args[0], (Number, NumberSymbol))
and isinstance(add_args[1], Mul)):
mul_args = sorted(Mul.make_args(add_args[1]), key=key)
if (len(mul_args) == 2
and isinstance(mul_args[0], Number)
and add_args[0].is_positive
and mul_args[0].is_negative):
return add_args
key, reverse = self._parse_order(order)
terms, gens = self.as_terms()
if not any(term.is_Order for term, _ in terms):
ordered = sorted(terms, key=key, reverse=reverse)
else:
_terms, _order = [], []
for term, repr in terms:
if not term.is_Order:
_terms.append((term, repr))
else:
_order.append((term, repr))
ordered = sorted(_terms, key=key, reverse=True) \
+ sorted(_order, key=key, reverse=True)
if data:
return ordered, gens
else:
return [term for term, _ in ordered]
def as_terms(self):
"""Transform an expression to a list of terms. """
from .exprtools import decompose_power
gens, terms = set(), []
for term in Add.make_args(self):
coeff, _term = term.as_coeff_Mul()
coeff = complex(coeff)
cpart, ncpart = {}, []
if _term is not S.One:
for factor in Mul.make_args(_term):
if factor.is_number:
try:
coeff *= complex(factor)
except (TypeError, ValueError):
pass
else:
continue
if factor.is_commutative:
base, exp = decompose_power(factor)
cpart[base] = exp
gens.add(base)
else:
ncpart.append(factor)
coeff = coeff.real, coeff.imag
ncpart = tuple(ncpart)
terms.append((term, (coeff, cpart, ncpart)))
gens = sorted(gens, key=default_sort_key)
k, indices = len(gens), {}
for i, g in enumerate(gens):
indices[g] = i
result = []
for term, (coeff, cpart, ncpart) in terms:
monom = [0]*k
for base, exp in cpart.items():
monom[indices[base]] = exp
result.append((term, (coeff, tuple(monom), ncpart)))
return result, gens
def removeO(self):
"""Removes the additive O(..) symbol if there is one"""
return self
def getO(self):
"""Returns the additive O(..) symbol if there is one, else None."""
return None
def getn(self):
"""
Returns the order of the expression.
Explanation
===========
The order is determined either from the O(...) term. If there
is no O(...) term, it returns None.
Examples
========
>>> from sympy import O
>>> from sympy.abc import x
>>> (1 + x + O(x**2)).getn()
2
>>> (1 + x).getn()
"""
o = self.getO()
if o is None:
return None
elif o.is_Order:
o = o.expr
if o is S.One:
return S.Zero
if o.is_Symbol:
return S.One
if o.is_Pow:
return o.args[1]
if o.is_Mul: # x**n*log(x)**n or x**n/log(x)**n
for oi in o.args:
if oi.is_Symbol:
return S.One
if oi.is_Pow:
from .symbol import Dummy, Symbol
syms = oi.atoms(Symbol)
if len(syms) == 1:
x = syms.pop()
oi = oi.subs(x, Dummy('x', positive=True))
if oi.base.is_Symbol and oi.exp.is_Rational:
return abs(oi.exp)
raise NotImplementedError('not sure of order of %s' % o)
def count_ops(self, visual=None):
"""wrapper for count_ops that returns the operation count."""
from .function import count_ops
return count_ops(self, visual)
def args_cnc(self, cset=False, warn=True, split_1=True):
"""Return [commutative factors, non-commutative factors] of self.
Explanation
===========
self is treated as a Mul and the ordering of the factors is maintained.
If ``cset`` is True the commutative factors will be returned in a set.
If there were repeated factors (as may happen with an unevaluated Mul)
then an error will be raised unless it is explicitly suppressed by
setting ``warn`` to False.
Note: -1 is always separated from a Number unless split_1 is False.
Examples
========
>>> from sympy import symbols, oo
>>> A, B = symbols('A B', commutative=0)
>>> x, y = symbols('x y')
>>> (-2*x*y).args_cnc()
[[-1, 2, x, y], []]
>>> (-2.5*x).args_cnc()
[[-1, 2.5, x], []]
>>> (-2*x*A*B*y).args_cnc()
[[-1, 2, x, y], [A, B]]
>>> (-2*x*A*B*y).args_cnc(split_1=False)
[[-2, x, y], [A, B]]
>>> (-2*x*y).args_cnc(cset=True)
[{-1, 2, x, y}, []]
The arg is always treated as a Mul:
>>> (-2 + x + A).args_cnc()
[[], [x - 2 + A]]
>>> (-oo).args_cnc() # -oo is a singleton
[[-1, oo], []]
"""
if self.is_Mul:
args = list(self.args)
else:
args = [self]
for i, mi in enumerate(args):
if not mi.is_commutative:
c = args[:i]
nc = args[i:]
break
else:
c = args
nc = []
if c and split_1 and (
c[0].is_Number and
c[0].is_extended_negative and
c[0] is not S.NegativeOne):
c[:1] = [S.NegativeOne, -c[0]]
if cset:
clen = len(c)
c = set(c)
if clen and warn and len(c) != clen:
raise ValueError('repeated commutative arguments: %s' %
[ci for ci in c if list(self.args).count(ci) > 1])
return [c, nc]
def coeff(self, x, n=1, right=False, _first=True):
"""
Returns the coefficient from the term(s) containing ``x**n``. If ``n``
is zero then all terms independent of ``x`` will be returned.
Explanation
===========
When ``x`` is noncommutative, the coefficient to the left (default) or
right of ``x`` can be returned. The keyword 'right' is ignored when
``x`` is commutative.
Examples
========
>>> from sympy import symbols
>>> from sympy.abc import x, y, z
You can select terms that have an explicit negative in front of them:
>>> (-x + 2*y).coeff(-1)
x
>>> (x - 2*y).coeff(-1)
2*y
You can select terms with no Rational coefficient:
>>> (x + 2*y).coeff(1)
x
>>> (3 + 2*x + 4*x**2).coeff(1)
0
You can select terms independent of x by making n=0; in this case
expr.as_independent(x)[0] is returned (and 0 will be returned instead
of None):
>>> (3 + 2*x + 4*x**2).coeff(x, 0)
3
>>> eq = ((x + 1)**3).expand() + 1
>>> eq
x**3 + 3*x**2 + 3*x + 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 2]
>>> eq -= 2
>>> [eq.coeff(x, i) for i in reversed(range(4))]
[1, 3, 3, 0]
You can select terms that have a numerical term in front of them:
>>> (-x - 2*y).coeff(2)
-y
>>> from sympy import sqrt
>>> (x + sqrt(2)*x).coeff(sqrt(2))
x
The matching is exact:
>>> (3 + 2*x + 4*x**2).coeff(x)
2
>>> (3 + 2*x + 4*x**2).coeff(x**2)
4
>>> (3 + 2*x + 4*x**2).coeff(x**3)
0
>>> (z*(x + y)**2).coeff((x + y)**2)
z
>>> (z*(x + y)**2).coeff(x + y)
0
In addition, no factoring is done, so 1 + z*(1 + y) is not obtained
from the following:
>>> (x + z*(x + x*y)).coeff(x)
1
If such factoring is desired, factor_terms can be used first:
>>> from sympy import factor_terms
>>> factor_terms(x + z*(x + x*y)).coeff(x)
z*(y + 1) + 1
>>> n, m, o = symbols('n m o', commutative=False)
>>> n.coeff(n)
1
>>> (3*n).coeff(n)
3
>>> (n*m + m*n*m).coeff(n) # = (1 + m)*n*m
1 + m
>>> (n*m + m*n*m).coeff(n, right=True) # = (1 + m)*n*m
m
If there is more than one possible coefficient 0 is returned:
>>> (n*m + m*n).coeff(n)
0
If there is only one possible coefficient, it is returned:
>>> (n*m + x*m*n).coeff(m*n)
x
>>> (n*m + x*m*n).coeff(m*n, right=1)
1
See Also
========
as_coefficient: separate the expression into a coefficient and factor
as_coeff_Add: separate the additive constant from an expression
as_coeff_Mul: separate the multiplicative constant from an expression
as_independent: separate x-dependent terms/factors from others
sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used
"""
x = sympify(x)
if not isinstance(x, Basic):
return S.Zero
n = as_int(n)
if not x:
return S.Zero
if x == self:
if n == 1:
return S.One
return S.Zero
if x is S.One:
co = [a for a in Add.make_args(self)
if a.as_coeff_Mul()[0] is S.One]
if not co:
return S.Zero
return Add(*co)
if n == 0:
if x.is_Add and self.is_Add:
c = self.coeff(x, right=right)
if not c:
return S.Zero
if not right:
return self - Add(*[a*x for a in Add.make_args(c)])
return self - Add(*[x*a for a in Add.make_args(c)])
return self.as_independent(x, as_Add=True)[0]
# continue with the full method, looking for this power of x:
x = x**n
def incommon(l1, l2):
if not l1 or not l2:
return []
n = min(len(l1), len(l2))
for i in range(n):
if l1[i] != l2[i]:
return l1[:i]
return l1[:]
def find(l, sub, first=True):
""" Find where list sub appears in list l. When ``first`` is True
the first occurrence from the left is returned, else the last
occurrence is returned. Return None if sub is not in l.
Examples
========
>> l = range(5)*2
>> find(l, [2, 3])
2
>> find(l, [2, 3], first=0)
7
>> find(l, [2, 4])
None
"""
if not sub or not l or len(sub) > len(l):
return None
n = len(sub)
if not first:
l.reverse()
sub.reverse()
for i in range(len(l) - n + 1):
if all(l[i + j] == sub[j] for j in range(n)):
break
else:
i = None
if not first:
l.reverse()
sub.reverse()
if i is not None and not first:
i = len(l) - (i + n)
return i
co = []
args = Add.make_args(self)
self_c = self.is_commutative
x_c = x.is_commutative
if self_c and not x_c:
return S.Zero
if _first and self.is_Add and not self_c and not x_c:
# get the part that depends on x exactly
xargs = Mul.make_args(x)
d = Add(*[i for i in Add.make_args(self.as_independent(x)[1])
if all(xi in Mul.make_args(i) for xi in xargs)])
rv = d.coeff(x, right=right, _first=False)
if not rv.is_Add or not right:
return rv
c_part, nc_part = zip(*[i.args_cnc() for i in rv.args])
if has_variety(c_part):
return rv
return Add(*[Mul._from_args(i) for i in nc_part])
one_c = self_c or x_c
xargs, nx = x.args_cnc(cset=True, warn=bool(not x_c))
# find the parts that pass the commutative terms
for a in args:
margs, nc = a.args_cnc(cset=True, warn=bool(not self_c))
if nc is None:
nc = []
if len(xargs) > len(margs):
continue
resid = margs.difference(xargs)
if len(resid) + len(xargs) == len(margs):
if one_c:
co.append(Mul(*(list(resid) + nc)))
else:
co.append((resid, nc))
if one_c:
if co == []:
return S.Zero
elif co:
return Add(*co)
else: # both nc
# now check the non-comm parts
if not co:
return S.Zero
if all(n == co[0][1] for r, n in co):
ii = find(co[0][1], nx, right)
if ii is not None:
if not right:
return Mul(Add(*[Mul(*r) for r, c in co]), Mul(*co[0][1][:ii]))
else:
return Mul(*co[0][1][ii + len(nx):])
beg = reduce(incommon, (n[1] for n in co))
if beg:
ii = find(beg, nx, right)
if ii is not None:
if not right:
gcdc = co[0][0]
for i in range(1, len(co)):
gcdc = gcdc.intersection(co[i][0])
if not gcdc:
break
return Mul(*(list(gcdc) + beg[:ii]))
else:
m = ii + len(nx)
return Add(*[Mul(*(list(r) + n[m:])) for r, n in co])
end = list(reversed(
reduce(incommon, (list(reversed(n[1])) for n in co))))
if end:
ii = find(end, nx, right)
if ii is not None:
if not right:
return Add(*[Mul(*(list(r) + n[:-len(end) + ii])) for r, n in co])
else:
return Mul(*end[ii + len(nx):])
# look for single match
hit = None
for i, (r, n) in enumerate(co):
ii = find(n, nx, right)
if ii is not None:
if not hit:
hit = ii, r, n
else:
break
else:
if hit:
ii, r, n = hit
if not right:
return Mul(*(list(r) + n[:ii]))
else:
return Mul(*n[ii + len(nx):])
return S.Zero
def as_expr(self, *gens):
"""
Convert a polynomial to a SymPy expression.
Examples
========
>>> from sympy import sin
>>> from sympy.abc import x, y
>>> f = (x**2 + x*y).as_poly(x, y)
>>> f.as_expr()
x**2 + x*y
>>> sin(x).as_expr()
sin(x)
"""
return self
def as_coefficient(self, expr):
"""
Extracts symbolic coefficient at the given expression. In
other words, this functions separates 'self' into the product
of 'expr' and 'expr'-free coefficient. If such separation
is not possible it will return None.
Examples
========
>>> from sympy import E, pi, sin, I, Poly
>>> from sympy.abc import x
>>> E.as_coefficient(E)
1
>>> (2*E).as_coefficient(E)
2
>>> (2*sin(E)*E).as_coefficient(E)
Two terms have E in them so a sum is returned. (If one were
desiring the coefficient of the term exactly matching E then
the constant from the returned expression could be selected.
Or, for greater precision, a method of Poly can be used to
indicate the desired term from which the coefficient is
desired.)
>>> (2*E + x*E).as_coefficient(E)
x + 2
>>> _.args[0] # just want the exact match
2
>>> p = Poly(2*E + x*E); p
Poly(x*E + 2*E, x, E, domain='ZZ')
>>> p.coeff_monomial(E)
2
>>> p.nth(0, 1)
2
Since the following cannot be written as a product containing
E as a factor, None is returned. (If the coefficient ``2*x`` is
desired then the ``coeff`` method should be used.)
>>> (2*E*x + x).as_coefficient(E)
>>> (2*E*x + x).coeff(E)
2*x
>>> (E*(x + 1) + x).as_coefficient(E)
>>> (2*pi*I).as_coefficient(pi*I)
2
>>> (2*I).as_coefficient(pi*I)
See Also
========
coeff: return sum of terms have a given factor
as_coeff_Add: separate the additive constant from an expression
as_coeff_Mul: separate the multiplicative constant from an expression
as_independent: separate x-dependent terms/factors from others
sympy.polys.polytools.Poly.coeff_monomial: efficiently find the single coefficient of a monomial in Poly
sympy.polys.polytools.Poly.nth: like coeff_monomial but powers of monomial terms are used
"""
r = self.extract_multiplicatively(expr)
if r and not r.has(expr):
return r
def as_independent(self, *deps, **hint) -> tuple[Expr, Expr]:
"""
A mostly naive separation of a Mul or Add into arguments that are not
are dependent on deps. To obtain as complete a separation of variables
as possible, use a separation method first, e.g.:
* separatevars() to change Mul, Add and Pow (including exp) into Mul
* .expand(mul=True) to change Add or Mul into Add
* .expand(log=True) to change log expr into an Add
The only non-naive thing that is done here is to respect noncommutative
ordering of variables and to always return (0, 0) for `self` of zero
regardless of hints.
For nonzero `self`, the returned tuple (i, d) has the
following interpretation:
* i will has no variable that appears in deps
* d will either have terms that contain variables that are in deps, or
be equal to 0 (when self is an Add) or 1 (when self is a Mul)
* if self is an Add then self = i + d
* if self is a Mul then self = i*d
* otherwise (self, S.One) or (S.One, self) is returned.
To force the expression to be treated as an Add, use the hint as_Add=True
Examples
========
-- self is an Add
>>> from sympy import sin, cos, exp
>>> from sympy.abc import x, y, z
>>> (x + x*y).as_independent(x)
(0, x*y + x)
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> (2*x*sin(x) + y + x + z).as_independent(x)
(y + z, 2*x*sin(x) + x)
>>> (2*x*sin(x) + y + x + z).as_independent(x, y)
(z, 2*x*sin(x) + x + y)
-- self is a Mul
>>> (x*sin(x)*cos(y)).as_independent(x)
(cos(y), x*sin(x))
non-commutative terms cannot always be separated out when self is a Mul
>>> from sympy import symbols
>>> n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
>>> (n1 + n1*n2).as_independent(n2)
(n1, n1*n2)
>>> (n2*n1 + n1*n2).as_independent(n2)
(0, n1*n2 + n2*n1)
>>> (n1*n2*n3).as_independent(n1)
(1, n1*n2*n3)
>>> (n1*n2*n3).as_independent(n2)
(n1, n2*n3)
>>> ((x-n1)*(x-y)).as_independent(x)
(1, (x - y)*(x - n1))
-- self is anything else:
>>> (sin(x)).as_independent(x)
(1, sin(x))
>>> (sin(x)).as_independent(y)
(sin(x), 1)
>>> exp(x+y).as_independent(x)
(1, exp(x + y))
-- force self to be treated as an Add:
>>> (3*x).as_independent(x, as_Add=True)
(0, 3*x)
-- force self to be treated as a Mul:
>>> (3+x).as_independent(x, as_Add=False)
(1, x + 3)
>>> (-3+x).as_independent(x, as_Add=False)
(1, x - 3)
Note how the below differs from the above in making the
constant on the dep term positive.
>>> (y*(-3+x)).as_independent(x)
(y, x - 3)
-- use .as_independent() for true independence testing instead
of .has(). The former considers only symbols in the free
symbols while the latter considers all symbols
>>> from sympy import Integral
>>> I = Integral(x, (x, 1, 2))
>>> I.has(x)
True
>>> x in I.free_symbols
False
>>> I.as_independent(x) == (I, 1)
True
>>> (I + x).as_independent(x) == (I, x)
True
Note: when trying to get independent terms, a separation method
might need to be used first. In this case, it is important to keep
track of what you send to this routine so you know how to interpret
the returned values
>>> from sympy import separatevars, log
>>> separatevars(exp(x+y)).as_independent(x)
(exp(y), exp(x))
>>> (x + x*y).as_independent(y)
(x, x*y)
>>> separatevars(x + x*y).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).as_independent(y)
(x, y + 1)
>>> (x*(1 + y)).expand(mul=True).as_independent(y)
(x, x*y)
>>> a, b=symbols('a b', positive=True)
>>> (log(a*b).expand(log=True)).as_independent(b)
(log(a), log(b))
See Also
========
.separatevars(), .expand(log=True), sympy.core.add.Add.as_two_terms(),
sympy.core.mul.Mul.as_two_terms(), .as_coeff_add(), .as_coeff_mul()
"""
from .symbol import Symbol
from .add import _unevaluated_Add
from .mul import _unevaluated_Mul
if self is S.Zero:
return (self, self)
func = self.func
if hint.get('as_Add', isinstance(self, Add) ):
want = Add
else:
want = Mul
# sift out deps into symbolic and other and ignore
# all symbols but those that are in the free symbols
sym = set()
other = []
for d in deps:
if isinstance(d, Symbol): # Symbol.is_Symbol is True
sym.add(d)
else:
other.append(d)
def has(e):
"""return the standard has() if there are no literal symbols, else
check to see that symbol-deps are in the free symbols."""
has_other = e.has(*other)
if not sym:
return has_other
return has_other or e.has(*(e.free_symbols & sym))
if (want is not func or
func is not Add and func is not Mul):
if has(self):
return (want.identity, self)
else:
return (self, want.identity)
else:
if func is Add:
args = list(self.args)
else:
args, nc = self.args_cnc()
d = sift(args, has)
depend = d[True]
indep = d[False]
if func is Add: # all terms were treated as commutative
return (Add(*indep), _unevaluated_Add(*depend))
else: # handle noncommutative by stopping at first dependent term
for i, n in enumerate(nc):
if has(n):
depend.extend(nc[i:])
break
indep.append(n)
return Mul(*indep), (
Mul(*depend, evaluate=False) if nc else
_unevaluated_Mul(*depend))
def as_real_imag(self, deep=True, **hints):
"""Performs complex expansion on 'self' and returns a tuple
containing collected both real and imaginary parts. This
method cannot be confused with re() and im() functions,
which does not perform complex expansion at evaluation.
However it is possible to expand both re() and im()
functions and get exactly the same results as with
a single call to this function.
>>> from sympy import symbols, I
>>> x, y = symbols('x,y', real=True)
>>> (x + y*I).as_real_imag()
(x, y)
>>> from sympy.abc import z, w
>>> (z + w*I).as_real_imag()
(re(z) - im(w), re(w) + im(z))
"""
if hints.get('ignore') == self:
return None
else:
from sympy.functions.elementary.complexes import im, re
return (re(self), im(self))
def as_powers_dict(self):
"""Return self as a dictionary of factors with each factor being
treated as a power. The keys are the bases of the factors and the
values, the corresponding exponents. The resulting dictionary should
be used with caution if the expression is a Mul and contains non-
commutative factors since the order that they appeared will be lost in
the dictionary.
See Also
========
as_ordered_factors: An alternative for noncommutative applications,
returning an ordered list of factors.
args_cnc: Similar to as_ordered_factors, but guarantees separation
of commutative and noncommutative factors.
"""
d = defaultdict(int)
d.update(dict([self.as_base_exp()]))
return d
def as_coefficients_dict(self, *syms):
"""Return a dictionary mapping terms to their Rational coefficient.
Since the dictionary is a defaultdict, inquiries about terms which
were not present will return a coefficient of 0.
If symbols ``syms`` are provided, any multiplicative terms
independent of them will be considered a coefficient and a
regular dictionary of syms-dependent generators as keys and
their corresponding coefficients as values will be returned.
Examples
========
>>> from sympy.abc import a, x, y
>>> (3*x + a*x + 4).as_coefficients_dict()
{1: 4, x: 3, a*x: 1}
>>> _[a]
0
>>> (3*a*x).as_coefficients_dict()
{a*x: 3}
>>> (3*a*x).as_coefficients_dict(x)
{x: 3*a}
>>> (3*a*x).as_coefficients_dict(y)
{1: 3*a*x}
"""
d = defaultdict(list)
if not syms:
for ai in Add.make_args(self):
c, m = ai.as_coeff_Mul()
d[m].append(c)
for k, v in d.items():
if len(v) == 1:
d[k] = v[0]
else:
d[k] = Add(*v)
else:
ind, dep = self.as_independent(*syms, as_Add=True)
for i in Add.make_args(dep):
if i.is_Mul:
c, x = i.as_coeff_mul(*syms)
if c is S.One:
d[i].append(c)
else:
d[i._new_rawargs(*x)].append(c)
elif i:
d[i].append(S.One)
d = {k: Add(*d[k]) for k in d}
if ind is not S.Zero:
d.update({S.One: ind})
di = defaultdict(int)
di.update(d)
return di
def as_base_exp(self) -> tuple[Expr, Expr]:
# a -> b ** e
return self, S.One
def as_coeff_mul(self, *deps, **kwargs) -> tuple[Expr, tuple[Expr, ...]]:
"""Return the tuple (c, args) where self is written as a Mul, ``m``.
c should be a Rational multiplied by any factors of the Mul that are
independent of deps.
args should be a tuple of all other factors of m; args is empty
if self is a Number or if self is independent of deps (when given).
This should be used when you do not know if self is a Mul or not but
you want to treat self as a Mul or if you want to process the
individual arguments of the tail of self as a Mul.
- if you know self is a Mul and want only the head, use self.args[0];
- if you do not want to process the arguments of the tail but need the
tail then use self.as_two_terms() which gives the head and tail;
- if you want to split self into an independent and dependent parts
use ``self.as_independent(*deps)``
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_mul()
(3, ())
>>> (3*x*y).as_coeff_mul()
(3, (x, y))
>>> (3*x*y).as_coeff_mul(x)
(3*y, (x,))
>>> (3*y).as_coeff_mul(x)
(3*y, ())
"""
if deps:
if not self.has(*deps):
return self, tuple()
return S.One, (self,)
def as_coeff_add(self, *deps) -> tuple[Expr, tuple[Expr, ...]]:
"""Return the tuple (c, args) where self is written as an Add, ``a``.
c should be a Rational added to any terms of the Add that are
independent of deps.
args should be a tuple of all other terms of ``a``; args is empty
if self is a Number or if self is independent of deps (when given).
This should be used when you do not know if self is an Add or not but
you want to treat self as an Add or if you want to process the
individual arguments of the tail of self as an Add.
- if you know self is an Add and want only the head, use self.args[0];
- if you do not want to process the arguments of the tail but need the
tail then use self.as_two_terms() which gives the head and tail.
- if you want to split self into an independent and dependent parts
use ``self.as_independent(*deps)``
>>> from sympy import S
>>> from sympy.abc import x, y
>>> (S(3)).as_coeff_add()
(3, ())
>>> (3 + x).as_coeff_add()
(3, (x,))
>>> (3 + x + y).as_coeff_add(x)
(y + 3, (x,))
>>> (3 + y).as_coeff_add(x)
(y + 3, ())
"""
if deps:
if not self.has_free(*deps):
return self, tuple()
return S.Zero, (self,)
def primitive(self):
"""Return the positive Rational that can be extracted non-recursively
from every term of self (i.e., self is treated like an Add). This is
like the as_coeff_Mul() method but primitive always extracts a positive
Rational (never a negative or a Float).
Examples
========
>>> from sympy.abc import x
>>> (3*(x + 1)**2).primitive()
(3, (x + 1)**2)
>>> a = (6*x + 2); a.primitive()
(2, 3*x + 1)
>>> b = (x/2 + 3); b.primitive()
(1/2, x + 6)
>>> (a*b).primitive() == (1, a*b)
True
"""
if not self:
return S.One, S.Zero
c, r = self.as_coeff_Mul(rational=True)
if c.is_negative:
c, r = -c, -r
return c, r
def as_content_primitive(self, radical=False, clear=True):
"""This method should recursively remove a Rational from all arguments
and return that (content) and the new self (primitive). The content
should always be positive and ``Mul(*foo.as_content_primitive()) == foo``.
The primitive need not be in canonical form and should try to preserve
the underlying structure if possible (i.e. expand_mul should not be
applied to self).
Examples
========
>>> from sympy import sqrt
>>> from sympy.abc import x, y, z
>>> eq = 2 + 2*x + 2*y*(3 + 3*y)
The as_content_primitive function is recursive and retains structure:
>>> eq.as_content_primitive()
(2, x + 3*y*(y + 1) + 1)
Integer powers will have Rationals extracted from the base:
>>> ((2 + 6*x)**2).as_content_primitive()
(4, (3*x + 1)**2)
>>> ((2 + 6*x)**(2*y)).as_content_primitive()
(1, (2*(3*x + 1))**(2*y))
Terms may end up joining once their as_content_primitives are added:
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(11, x*(y + 1))
>>> ((3*(x*(1 + y)) + 2*x*(3 + 3*y))).as_content_primitive()
(9, x*(y + 1))
>>> ((3*(z*(1 + y)) + 2.0*x*(3 + 3*y))).as_content_primitive()
(1, 6.0*x*(y + 1) + 3*z*(y + 1))
>>> ((5*(x*(1 + y)) + 2*x*(3 + 3*y))**2).as_content_primitive()
(121, x**2*(y + 1)**2)
>>> ((x*(1 + y) + 0.4*x*(3 + 3*y))**2).as_content_primitive()
(1, 4.84*x**2*(y + 1)**2)
Radical content can also be factored out of the primitive:
>>> (2*sqrt(2) + 4*sqrt(10)).as_content_primitive(radical=True)
(2, sqrt(2)*(1 + 2*sqrt(5)))
If clear=False (default is True) then content will not be removed
from an Add if it can be distributed to leave one or more
terms with integer coefficients.
>>> (x/2 + y).as_content_primitive()
(1/2, x + 2*y)
>>> (x/2 + y).as_content_primitive(clear=False)
(1, x/2 + y)
"""
return S.One, self
def as_numer_denom(self):
""" expression -> a/b -> a, b
This is just a stub that should be defined by
an object's class methods to get anything else.
See Also
========
normal: return ``a/b`` instead of ``(a, b)``
"""
return self, S.One
def normal(self):
""" expression -> a/b
See Also
========
as_numer_denom: return ``(a, b)`` instead of ``a/b``
"""
from .mul import _unevaluated_Mul
n, d = self.as_numer_denom()
if d is S.One:
return n
if d.is_Number:
return _unevaluated_Mul(n, 1/d)
else:
return n/d
def extract_multiplicatively(self, c):
"""Return None if it's not possible to make self in the form
c * something in a nice way, i.e. preserving the properties
of arguments of self.
Examples
========
>>> from sympy import symbols, Rational
>>> x, y = symbols('x,y', real=True)
>>> ((x*y)**3).extract_multiplicatively(x**2 * y)
x*y**2
>>> ((x*y)**3).extract_multiplicatively(x**4 * y)
>>> (2*x).extract_multiplicatively(2)
x
>>> (2*x).extract_multiplicatively(3)
>>> (Rational(1, 2)*x).extract_multiplicatively(3)
x/6
"""
from sympy.functions.elementary.exponential import exp
from .add import _unevaluated_Add
c = sympify(c)
if self is S.NaN:
return None
if c is S.One:
return self
elif c == self:
return S.One
if c.is_Add:
cc, pc = c.primitive()
if cc is not S.One:
c = Mul(cc, pc, evaluate=False)
if c.is_Mul:
a, b = c.as_two_terms()
x = self.extract_multiplicatively(a)
if x is not None:
return x.extract_multiplicatively(b)
else:
return x
quotient = self / c
if self.is_Number:
if self is S.Infinity:
if c.is_positive:
return S.Infinity
elif self is S.NegativeInfinity:
if c.is_negative:
return S.Infinity
elif c.is_positive:
return S.NegativeInfinity
elif self is S.ComplexInfinity:
if not c.is_zero:
return S.ComplexInfinity
elif self.is_Integer:
if not quotient.is_Integer:
return None
elif self.is_positive and quotient.is_negative:
return None
else:
return quotient
elif self.is_Rational:
if not quotient.is_Rational:
return None
elif self.is_positive and quotient.is_negative:
return None
else:
return quotient
elif self.is_Float:
if not quotient.is_Float:
return None
elif self.is_positive and quotient.is_negative:
return None
else:
return quotient
elif self.is_NumberSymbol or self.is_Symbol or self is S.ImaginaryUnit:
if quotient.is_Mul and len(quotient.args) == 2:
if quotient.args[0].is_Integer and quotient.args[0].is_positive and quotient.args[1] == self:
return quotient
elif quotient.is_Integer and c.is_Number:
return quotient
elif self.is_Add:
cs, ps = self.primitive()
# assert cs >= 1
if c.is_Number and c is not S.NegativeOne:
# assert c != 1 (handled at top)
if cs is not S.One:
if c.is_negative:
xc = -(cs.extract_multiplicatively(-c))
else:
xc = cs.extract_multiplicatively(c)
if xc is not None:
return xc*ps # rely on 2-arg Mul to restore Add
return # |c| != 1 can only be extracted from cs
if c == ps:
return cs
# check args of ps
newargs = []
for arg in ps.args:
newarg = arg.extract_multiplicatively(c)
if newarg is None:
return # all or nothing
newargs.append(newarg)
if cs is not S.One:
args = [cs*t for t in newargs]
# args may be in different order
return _unevaluated_Add(*args)
else:
return Add._from_args(newargs)
elif self.is_Mul:
args = list(self.args)
for i, arg in enumerate(args):
newarg = arg.extract_multiplicatively(c)
if newarg is not None:
args[i] = newarg
return Mul(*args)
elif self.is_Pow or isinstance(self, exp):
sb, se = self.as_base_exp()
cb, ce = c.as_base_exp()
if cb == sb:
new_exp = se.extract_additively(ce)
if new_exp is not None:
return Pow(sb, new_exp)
elif c == sb:
new_exp = self.exp.extract_additively(1)
if new_exp is not None:
return Pow(sb, new_exp)
def extract_additively(self, c):
"""Return self - c if it's possible to subtract c from self and
make all matching coefficients move towards zero, else return None.
Examples
========
>>> from sympy.abc import x, y
>>> e = 2*x + 3
>>> e.extract_additively(x + 1)
x + 2
>>> e.extract_additively(3*x)
>>> e.extract_additively(4)
>>> (y*(x + 1)).extract_additively(x + 1)
>>> ((x + 1)*(x + 2*y + 1) + 3).extract_additively(x + 1)
(x + 1)*(x + 2*y) + 3
See Also
========
extract_multiplicatively
coeff
as_coefficient
"""
c = sympify(c)
if self is S.NaN:
return None
if c.is_zero:
return self
elif c == self:
return S.Zero
elif self == S.Zero:
return None
if self.is_Number:
if not c.is_Number:
return None
co = self
diff = co - c
# XXX should we match types? i.e should 3 - .1 succeed?
if (co > 0 and diff > 0 and diff < co or
co < 0 and diff < 0 and diff > co):
return diff
return None
if c.is_Number:
co, t = self.as_coeff_Add()
xa = co.extract_additively(c)
if xa is None:
return None
return xa + t
# handle the args[0].is_Number case separately
# since we will have trouble looking for the coeff of
# a number.
if c.is_Add and c.args[0].is_Number:
# whole term as a term factor
co = self.coeff(c)
xa0 = (co.extract_additively(1) or 0)*c
if xa0:
diff = self - co*c
return (xa0 + (diff.extract_additively(c) or diff)) or None
# term-wise
h, t = c.as_coeff_Add()
sh, st = self.as_coeff_Add()
xa = sh.extract_additively(h)
if xa is None:
return None
xa2 = st.extract_additively(t)
if xa2 is None:
return None
return xa + xa2
# whole term as a term factor
co, diff = _corem(self, c)
xa0 = (co.extract_additively(1) or 0)*c
if xa0:
return (xa0 + (diff.extract_additively(c) or diff)) or None
# term-wise
coeffs = []
for a in Add.make_args(c):
ac, at = a.as_coeff_Mul()
co = self.coeff(at)
if not co:
return None
coc, cot = co.as_coeff_Add()
xa = coc.extract_additively(ac)
if xa is None:
return None
self -= co*at
coeffs.append((cot + xa)*at)
coeffs.append(self)
return Add(*coeffs)
@property
def expr_free_symbols(self):
"""
Like ``free_symbols``, but returns the free symbols only if
they are contained in an expression node.
Examples
========
>>> from sympy.abc import x, y
>>> (x + y).expr_free_symbols # doctest: +SKIP
{x, y}
If the expression is contained in a non-expression object, do not return
the free symbols. Compare:
>>> from sympy import Tuple
>>> t = Tuple(x + y)
>>> t.expr_free_symbols # doctest: +SKIP
set()
>>> t.free_symbols
{x, y}
"""
sympy_deprecation_warning("""
The expr_free_symbols property is deprecated. Use free_symbols to get
the free symbols of an expression.
""",
deprecated_since_version="1.9",
active_deprecations_target="deprecated-expr-free-symbols")
return {j for i in self.args for j in i.expr_free_symbols}
def could_extract_minus_sign(self):
"""Return True if self has -1 as a leading factor or has
more literal negative signs than positive signs in a sum,
otherwise False.
Examples
========
>>> from sympy.abc import x, y
>>> e = x - y
>>> {i.could_extract_minus_sign() for i in (e, -e)}
{False, True}
Though the ``y - x`` is considered like ``-(x - y)``, since it
is in a product without a leading factor of -1, the result is
false below:
>>> (x*(y - x)).could_extract_minus_sign()
False
To put something in canonical form wrt to sign, use `signsimp`:
>>> from sympy import signsimp
>>> signsimp(x*(y - x))
-x*(x - y)
>>> _.could_extract_minus_sign()
True
"""
return False
def extract_branch_factor(self, allow_half=False):
"""
Try to write self as ``exp_polar(2*pi*I*n)*z`` in a nice way.
Return (z, n).
>>> from sympy import exp_polar, I, pi
>>> from sympy.abc import x, y
>>> exp_polar(I*pi).extract_branch_factor()
(exp_polar(I*pi), 0)
>>> exp_polar(2*I*pi).extract_branch_factor()
(1, 1)
>>> exp_polar(-pi*I).extract_branch_factor()
(exp_polar(I*pi), -1)
>>> exp_polar(3*pi*I + x).extract_branch_factor()
(exp_polar(x + I*pi), 1)
>>> (y*exp_polar(-5*pi*I)*exp_polar(3*pi*I + 2*pi*x)).extract_branch_factor()
(y*exp_polar(2*pi*x), -1)
>>> exp_polar(-I*pi/2).extract_branch_factor()
(exp_polar(-I*pi/2), 0)
If allow_half is True, also extract exp_polar(I*pi):
>>> exp_polar(I*pi).extract_branch_factor(allow_half=True)
(1, 1/2)
>>> exp_polar(2*I*pi).extract_branch_factor(allow_half=True)
(1, 1)
>>> exp_polar(3*I*pi).extract_branch_factor(allow_half=True)
(1, 3/2)
>>> exp_polar(-I*pi).extract_branch_factor(allow_half=True)
(1, -1/2)
"""
from sympy.functions.elementary.exponential import exp_polar
from sympy.functions.elementary.integers import ceiling
n = S.Zero
res = S.One
args = Mul.make_args(self)
exps = []
for arg in args:
if isinstance(arg, exp_polar):
exps += [arg.exp]
else:
res *= arg
piimult = S.Zero
extras = []
ipi = S.Pi*S.ImaginaryUnit
while exps:
exp = exps.pop()
if exp.is_Add:
exps += exp.args
continue
if exp.is_Mul:
coeff = exp.as_coefficient(ipi)
if coeff is not None:
piimult += coeff
continue
extras += [exp]
if piimult.is_number:
coeff = piimult
tail = ()
else:
coeff, tail = piimult.as_coeff_add(*piimult.free_symbols)
# round down to nearest multiple of 2
branchfact = ceiling(coeff/2 - S.Half)*2
n += branchfact/2
c = coeff - branchfact
if allow_half:
nc = c.extract_additively(1)
if nc is not None:
n += S.Half
c = nc
newexp = ipi*Add(*((c, ) + tail)) + Add(*extras)
if newexp != 0:
res *= exp_polar(newexp)
return res, n
def is_polynomial(self, *syms):
r"""
Return True if self is a polynomial in syms and False otherwise.
This checks if self is an exact polynomial in syms. This function
returns False for expressions that are "polynomials" with symbolic
exponents. Thus, you should be able to apply polynomial algorithms to
expressions for which this returns True, and Poly(expr, \*syms) should
work if and only if expr.is_polynomial(\*syms) returns True. The
polynomial does not have to be in expanded form. If no symbols are
given, all free symbols in the expression will be used.
This is not part of the assumptions system. You cannot do
Symbol('z', polynomial=True).
Examples
========
>>> from sympy import Symbol, Function
>>> x = Symbol('x')
>>> ((x**2 + 1)**4).is_polynomial(x)
True
>>> ((x**2 + 1)**4).is_polynomial()
True
>>> (2**x + 1).is_polynomial(x)
False
>>> (2**x + 1).is_polynomial(2**x)
True
>>> f = Function('f')
>>> (f(x) + 1).is_polynomial(x)
False
>>> (f(x) + 1).is_polynomial(f(x))
True
>>> (1/f(x) + 1).is_polynomial(f(x))
False
>>> n = Symbol('n', nonnegative=True, integer=True)
>>> (x**n + 1).is_polynomial(x)
False
This function does not attempt any nontrivial simplifications that may
result in an expression that does not appear to be a polynomial to
become one.
>>> from sympy import sqrt, factor, cancel
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)
>>> a.is_polynomial(y)
False
>>> factor(a)
y + 1
>>> factor(a).is_polynomial(y)
True
>>> b = (y**2 + 2*y + 1)/(y + 1)
>>> b.is_polynomial(y)
False
>>> cancel(b)
y + 1
>>> cancel(b).is_polynomial(y)
True
See also .is_rational_function()
"""
if syms:
syms = set(map(sympify, syms))
else:
syms = self.free_symbols
if not syms:
return True
return self._eval_is_polynomial(syms)
def _eval_is_polynomial(self, syms):
if self in syms:
return True
if not self.has_free(*syms):
# constant polynomial
return True
# subclasses should return True or False
def is_rational_function(self, *syms):
"""
Test whether function is a ratio of two polynomials in the given
symbols, syms. When syms is not given, all free symbols will be used.
The rational function does not have to be in expanded or in any kind of
canonical form.
This function returns False for expressions that are "rational
functions" with symbolic exponents. Thus, you should be able to call
.as_numer_denom() and apply polynomial algorithms to the result for
expressions for which this returns True.
This is not part of the assumptions system. You cannot do
Symbol('z', rational_function=True).
Examples
========
>>> from sympy import Symbol, sin
>>> from sympy.abc import x, y
>>> (x/y).is_rational_function()
True
>>> (x**2).is_rational_function()
True
>>> (x/sin(y)).is_rational_function(y)
False
>>> n = Symbol('n', integer=True)
>>> (x**n + 1).is_rational_function(x)
False
This function does not attempt any nontrivial simplifications that may
result in an expression that does not appear to be a rational function
to become one.
>>> from sympy import sqrt, factor
>>> y = Symbol('y', positive=True)
>>> a = sqrt(y**2 + 2*y + 1)/y
>>> a.is_rational_function(y)
False
>>> factor(a)
(y + 1)/y
>>> factor(a).is_rational_function(y)
True
See also is_algebraic_expr().
"""
if self in _illegal:
return False
if syms:
syms = set(map(sympify, syms))
else:
syms = self.free_symbols
if not syms:
return True
return self._eval_is_rational_function(syms)
def _eval_is_rational_function(self, syms):
if self in syms:
return True
if not self.has_free(*syms):
return True
# subclasses should return True or False
def is_meromorphic(self, x, a):
"""
This tests whether an expression is meromorphic as
a function of the given symbol ``x`` at the point ``a``.
This method is intended as a quick test that will return
None if no decision can be made without simplification or
more detailed analysis.
Examples
========
>>> from sympy import zoo, log, sin, sqrt
>>> from sympy.abc import x
>>> f = 1/x**2 + 1 - 2*x**3
>>> f.is_meromorphic(x, 0)
True
>>> f.is_meromorphic(x, 1)
True
>>> f.is_meromorphic(x, zoo)
True
>>> g = x**log(3)
>>> g.is_meromorphic(x, 0)
False
>>> g.is_meromorphic(x, 1)
True
>>> g.is_meromorphic(x, zoo)
False
>>> h = sin(1/x)*x**2
>>> h.is_meromorphic(x, 0)
False
>>> h.is_meromorphic(x, 1)
True
>>> h.is_meromorphic(x, zoo)
True
Multivalued functions are considered meromorphic when their
branches are meromorphic. Thus most functions are meromorphic
everywhere except at essential singularities and branch points.
In particular, they will be meromorphic also on branch cuts
except at their endpoints.
>>> log(x).is_meromorphic(x, -1)
True
>>> log(x).is_meromorphic(x, 0)
False
>>> sqrt(x).is_meromorphic(x, -1)
True
>>> sqrt(x).is_meromorphic(x, 0)
False
"""
if not x.is_symbol:
raise TypeError("{} should be of symbol type".format(x))
a = sympify(a)
return self._eval_is_meromorphic(x, a)
def _eval_is_meromorphic(self, x, a):
if self == x:
return True
if not self.has_free(x):
return True
# subclasses should return True or False
def is_algebraic_expr(self, *syms):
"""
This tests whether a given expression is algebraic or not, in the
given symbols, syms. When syms is not given, all free symbols
will be used. The rational function does not have to be in expanded
or in any kind of canonical form.
This function returns False for expressions that are "algebraic
expressions" with symbolic exponents. This is a simple extension to the
is_rational_function, including rational exponentiation.
Examples
========
>>> from sympy import Symbol, sqrt
>>> x = Symbol('x', real=True)
>>> sqrt(1 + x).is_rational_function()
False
>>> sqrt(1 + x).is_algebraic_expr()
True
This function does not attempt any nontrivial simplifications that may
result in an expression that does not appear to be an algebraic
expression to become one.
>>> from sympy import exp, factor
>>> a = sqrt(exp(x)**2 + 2*exp(x) + 1)/(exp(x) + 1)
>>> a.is_algebraic_expr(x)
False
>>> factor(a).is_algebraic_expr()
True
See Also
========
is_rational_function()
References
==========
.. [1] https://en.wikipedia.org/wiki/Algebraic_expression
"""
if syms:
syms = set(map(sympify, syms))
else:
syms = self.free_symbols
if not syms:
return True
return self._eval_is_algebraic_expr(syms)
def _eval_is_algebraic_expr(self, syms):
if self in syms:
return True
if not self.has_free(*syms):
return True
# subclasses should return True or False
###################################################################################
##################### SERIES, LEADING TERM, LIMIT, ORDER METHODS ##################
###################################################################################
def series(self, x=None, x0=0, n=6, dir="+", logx=None, cdir=0):
"""
Series expansion of "self" around ``x = x0`` yielding either terms of
the series one by one (the lazy series given when n=None), else
all the terms at once when n != None.
Returns the series expansion of "self" around the point ``x = x0``
with respect to ``x`` up to ``O((x - x0)**n, x, x0)`` (default n is 6).
If ``x=None`` and ``self`` is univariate, the univariate symbol will
be supplied, otherwise an error will be raised.
Parameters
==========
expr : Expression
The expression whose series is to be expanded.
x : Symbol
It is the variable of the expression to be calculated.
x0 : Value
The value around which ``x`` is calculated. Can be any value
from ``-oo`` to ``oo``.
n : Value
The value used to represent the order in terms of ``x**n``,
up to which the series is to be expanded.
dir : String, optional
The series-expansion can be bi-directional. If ``dir="+"``,
then (x->x0+). If ``dir="-", then (x->x0-). For infinite
``x0`` (``oo`` or ``-oo``), the ``dir`` argument is determined
from the direction of the infinity (i.e., ``dir="-"`` for
``oo``).
logx : optional
It is used to replace any log(x) in the returned series with a
symbolic value rather than evaluating the actual value.
cdir : optional
It stands for complex direction, and indicates the direction
from which the expansion needs to be evaluated.
Examples
========
>>> from sympy import cos, exp, tan
>>> from sympy.abc import x, y
>>> cos(x).series()
1 - x**2/2 + x**4/24 + O(x**6)
>>> cos(x).series(n=4)
1 - x**2/2 + O(x**4)
>>> cos(x).series(x, x0=1, n=2)
cos(1) - (x - 1)*sin(1) + O((x - 1)**2, (x, 1))
>>> e = cos(x + exp(y))
>>> e.series(y, n=2)
cos(x + 1) - y*sin(x + 1) + O(y**2)
>>> e.series(x, n=2)
cos(exp(y)) - x*sin(exp(y)) + O(x**2)
If ``n=None`` then a generator of the series terms will be returned.
>>> term=cos(x).series(n=None)
>>> [next(term) for i in range(2)]
[1, -x**2/2]
For ``dir=+`` (default) the series is calculated from the right and
for ``dir=-`` the series from the left. For smooth functions this
flag will not alter the results.
>>> abs(x).series(dir="+")
x
>>> abs(x).series(dir="-")
-x
>>> f = tan(x)
>>> f.series(x, 2, 6, "+")
tan(2) + (1 + tan(2)**2)*(x - 2) + (x - 2)**2*(tan(2)**3 + tan(2)) +
(x - 2)**3*(1/3 + 4*tan(2)**2/3 + tan(2)**4) + (x - 2)**4*(tan(2)**5 +
5*tan(2)**3/3 + 2*tan(2)/3) + (x - 2)**5*(2/15 + 17*tan(2)**2/15 +
2*tan(2)**4 + tan(2)**6) + O((x - 2)**6, (x, 2))
>>> f.series(x, 2, 3, "-")
tan(2) + (2 - x)*(-tan(2)**2 - 1) + (2 - x)**2*(tan(2)**3 + tan(2))
+ O((x - 2)**3, (x, 2))
For rational expressions this method may return original expression without the Order term.
>>> (1/x).series(x, n=8)
1/x
Returns
=======
Expr : Expression
Series expansion of the expression about x0
Raises
======
TypeError
If "n" and "x0" are infinity objects
PoleError
If "x0" is an infinity object
"""
if x is None:
syms = self.free_symbols
if not syms:
return self
elif len(syms) > 1:
raise ValueError('x must be given for multivariate functions.')
x = syms.pop()
from .symbol import Dummy, Symbol
if isinstance(x, Symbol):
dep = x in self.free_symbols
else:
d = Dummy()
dep = d in self.xreplace({x: d}).free_symbols
if not dep:
if n is None:
return (s for s in [self])
else:
return self
if len(dir) != 1 or dir not in '+-':
raise ValueError("Dir must be '+' or '-'")
x0 = sympify(x0)
cdir = sympify(cdir)
from sympy.functions.elementary.complexes import im, sign
if not cdir.is_zero:
if cdir.is_real:
dir = '+' if cdir.is_positive else '-'
else:
dir = '+' if im(cdir).is_positive else '-'
else:
if x0 and x0.is_infinite:
cdir = sign(x0).simplify()
elif str(dir) == "+":
cdir = S.One
elif str(dir) == "-":
cdir = S.NegativeOne
elif cdir == S.Zero:
cdir = S.One
cdir = cdir/abs(cdir)
if x0 and x0.is_infinite:
from .function import PoleError
try:
s = self.subs(x, cdir/x).series(x, n=n, dir='+', cdir=1)
if n is None:
return (si.subs(x, cdir/x) for si in s)
return s.subs(x, cdir/x)
except PoleError:
s = self.subs(x, cdir*x).aseries(x, n=n)
return s.subs(x, cdir*x)
# use rep to shift origin to x0 and change sign (if dir is negative)
# and undo the process with rep2
if x0 or cdir != 1:
s = self.subs({x: x0 + cdir*x}).series(x, x0=0, n=n, dir='+', logx=logx, cdir=1)
if n is None: # lseries...
return (si.subs({x: x/cdir - x0/cdir}) for si in s)
return s.subs({x: x/cdir - x0/cdir})
# from here on it's x0=0 and dir='+' handling
if x.is_positive is x.is_negative is None or x.is_Symbol is not True:
# replace x with an x that has a positive assumption
xpos = Dummy('x', positive=True)
rv = self.subs(x, xpos).series(xpos, x0, n, dir, logx=logx, cdir=cdir)
if n is None:
return (s.subs(xpos, x) for s in rv)
else:
return rv.subs(xpos, x)
from sympy.series.order import Order
if n is not None: # nseries handling
s1 = self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
o = s1.getO() or S.Zero
if o:
# make sure the requested order is returned
ngot = o.getn()
if ngot > n:
# leave o in its current form (e.g. with x*log(x)) so
# it eats terms properly, then replace it below
if n != 0:
s1 += o.subs(x, x**Rational(n, ngot))
else:
s1 += Order(1, x)
elif ngot < n:
# increase the requested number of terms to get the desired
# number keep increasing (up to 9) until the received order
# is different than the original order and then predict how
# many additional terms are needed
from sympy.functions.elementary.integers import ceiling
for more in range(1, 9):
s1 = self._eval_nseries(x, n=n + more, logx=logx, cdir=cdir)
newn = s1.getn()
if newn != ngot:
ndo = n + ceiling((n - ngot)*more/(newn - ngot))
s1 = self._eval_nseries(x, n=ndo, logx=logx, cdir=cdir)
while s1.getn() < n:
s1 = self._eval_nseries(x, n=ndo, logx=logx, cdir=cdir)
ndo += 1
break
else:
raise ValueError('Could not calculate %s terms for %s'
% (str(n), self))
s1 += Order(x**n, x)
o = s1.getO()
s1 = s1.removeO()
elif s1.has(Order):
# asymptotic expansion
return s1
else:
o = Order(x**n, x)
s1done = s1.doit()
try:
if (s1done + o).removeO() == s1done:
o = S.Zero
except NotImplementedError:
return s1
try:
from sympy.simplify.radsimp import collect
return collect(s1, x) + o
except NotImplementedError:
return s1 + o
else: # lseries handling
def yield_lseries(s):
"""Return terms of lseries one at a time."""
for si in s:
if not si.is_Add:
yield si
continue
# yield terms 1 at a time if possible
# by increasing order until all the
# terms have been returned
yielded = 0
o = Order(si, x)*x
ndid = 0
ndo = len(si.args)
while 1:
do = (si - yielded + o).removeO()
o *= x
if not do or do.is_Order:
continue
if do.is_Add:
ndid += len(do.args)
else:
ndid += 1
yield do
if ndid == ndo:
break
yielded += do
return yield_lseries(self.removeO()._eval_lseries(x, logx=logx, cdir=cdir))
def aseries(self, x=None, n=6, bound=0, hir=False):
"""Asymptotic Series expansion of self.
This is equivalent to ``self.series(x, oo, n)``.
Parameters
==========
self : Expression
The expression whose series is to be expanded.
x : Symbol
It is the variable of the expression to be calculated.
n : Value
The value used to represent the order in terms of ``x**n``,
up to which the series is to be expanded.
hir : Boolean
Set this parameter to be True to produce hierarchical series.
It stops the recursion at an early level and may provide nicer
and more useful results.
bound : Value, Integer
Use the ``bound`` parameter to give limit on rewriting
coefficients in its normalised form.
Examples
========
>>> from sympy import sin, exp
>>> from sympy.abc import x
>>> e = sin(1/x + exp(-x)) - sin(1/x)
>>> e.aseries(x)
(1/(24*x**4) - 1/(2*x**2) + 1 + O(x**(-6), (x, oo)))*exp(-x)
>>> e.aseries(x, n=3, hir=True)
-exp(-2*x)*sin(1/x)/2 + exp(-x)*cos(1/x) + O(exp(-3*x), (x, oo))
>>> e = exp(exp(x)/(1 - 1/x))
>>> e.aseries(x)
exp(exp(x)/(1 - 1/x))
>>> e.aseries(x, bound=3) # doctest: +SKIP
exp(exp(x)/x**2)*exp(exp(x)/x)*exp(-exp(x) + exp(x)/(1 - 1/x) - exp(x)/x - exp(x)/x**2)*exp(exp(x))
For rational expressions this method may return original expression without the Order term.
>>> (1/x).aseries(x, n=8)
1/x
Returns
=======
Expr
Asymptotic series expansion of the expression.
Notes
=====
This algorithm is directly induced from the limit computational algorithm provided by Gruntz.
It majorly uses the mrv and rewrite sub-routines. The overall idea of this algorithm is first
to look for the most rapidly varying subexpression w of a given expression f and then expands f
in a series in w. Then same thing is recursively done on the leading coefficient
till we get constant coefficients.
If the most rapidly varying subexpression of a given expression f is f itself,
the algorithm tries to find a normalised representation of the mrv set and rewrites f
using this normalised representation.
If the expansion contains an order term, it will be either ``O(x ** (-n))`` or ``O(w ** (-n))``
where ``w`` belongs to the most rapidly varying expression of ``self``.
References
==========
.. [1] Gruntz, Dominik. A new algorithm for computing asymptotic series.
In: Proc. 1993 Int. Symp. Symbolic and Algebraic Computation. 1993.
pp. 239-244.
.. [2] Gruntz thesis - p90
.. [3] http://en.wikipedia.org/wiki/Asymptotic_expansion
See Also
========
Expr.aseries: See the docstring of this function for complete details of this wrapper.
"""
from .symbol import Dummy
if x.is_positive is x.is_negative is None:
xpos = Dummy('x', positive=True)
return self.subs(x, xpos).aseries(xpos, n, bound, hir).subs(xpos, x)
from .function import PoleError
from sympy.series.gruntz import mrv, rewrite
try:
om, exps = mrv(self, x)
except PoleError:
return self
# We move one level up by replacing `x` by `exp(x)`, and then
# computing the asymptotic series for f(exp(x)). Then asymptotic series
# can be obtained by moving one-step back, by replacing x by ln(x).
from sympy.functions.elementary.exponential import exp, log
from sympy.series.order import Order
if x in om:
s = self.subs(x, exp(x)).aseries(x, n, bound, hir).subs(x, log(x))
if s.getO():
return s + Order(1/x**n, (x, S.Infinity))
return s
k = Dummy('k', positive=True)
# f is rewritten in terms of omega
func, logw = rewrite(exps, om, x, k)
if self in om:
if bound <= 0:
return self
s = (self.exp).aseries(x, n, bound=bound)
s = s.func(*[t.removeO() for t in s.args])
try:
res = exp(s.subs(x, 1/x).as_leading_term(x).subs(x, 1/x))
except PoleError:
res = self
func = exp(self.args[0] - res.args[0]) / k
logw = log(1/res)
s = func.series(k, 0, n)
# Hierarchical series
if hir:
return s.subs(k, exp(logw))
o = s.getO()
terms = sorted(Add.make_args(s.removeO()), key=lambda i: int(i.as_coeff_exponent(k)[1]))
s = S.Zero
has_ord = False
# Then we recursively expand these coefficients one by one into
# their asymptotic series in terms of their most rapidly varying subexpressions.
for t in terms:
coeff, expo = t.as_coeff_exponent(k)
if coeff.has(x):
# Recursive step
snew = coeff.aseries(x, n, bound=bound-1)
if has_ord and snew.getO():
break
elif snew.getO():
has_ord = True
s += (snew * k**expo)
else:
s += t
if not o or has_ord:
return s.subs(k, exp(logw))
return (s + o).subs(k, exp(logw))
def taylor_term(self, n, x, *previous_terms):
"""General method for the taylor term.
This method is slow, because it differentiates n-times. Subclasses can
redefine it to make it faster by using the "previous_terms".
"""
from .symbol import Dummy
from sympy.functions.combinatorial.factorials import factorial
x = sympify(x)
_x = Dummy('x')
return self.subs(x, _x).diff(_x, n).subs(_x, x).subs(x, 0) * x**n / factorial(n)
def lseries(self, x=None, x0=0, dir='+', logx=None, cdir=0):
"""
Wrapper for series yielding an iterator of the terms of the series.
Note: an infinite series will yield an infinite iterator. The following,
for exaxmple, will never terminate. It will just keep printing terms
of the sin(x) series::
for term in sin(x).lseries(x):
print term
The advantage of lseries() over nseries() is that many times you are
just interested in the next term in the series (i.e. the first term for
example), but you do not know how many you should ask for in nseries()
using the "n" parameter.
See also nseries().
"""
return self.series(x, x0, n=None, dir=dir, logx=logx, cdir=cdir)
def _eval_lseries(self, x, logx=None, cdir=0):
# default implementation of lseries is using nseries(), and adaptively
# increasing the "n". As you can see, it is not very efficient, because
# we are calculating the series over and over again. Subclasses should
# override this method and implement much more efficient yielding of
# terms.
n = 0
series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
while series.is_Order:
n += 1
series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
e = series.removeO()
yield e
if e is S.Zero:
return
while 1:
while 1:
n += 1
series = self._eval_nseries(x, n=n, logx=logx, cdir=cdir).removeO()
if e != series:
break
if (series - self).cancel() is S.Zero:
return
yield series - e
e = series
def nseries(self, x=None, x0=0, n=6, dir='+', logx=None, cdir=0):
"""
Wrapper to _eval_nseries if assumptions allow, else to series.
If x is given, x0 is 0, dir='+', and self has x, then _eval_nseries is
called. This calculates "n" terms in the innermost expressions and
then builds up the final series just by "cross-multiplying" everything
out.
The optional ``logx`` parameter can be used to replace any log(x) in the
returned series with a symbolic value to avoid evaluating log(x) at 0. A
symbol to use in place of log(x) should be provided.
Advantage -- it's fast, because we do not have to determine how many
terms we need to calculate in advance.
Disadvantage -- you may end up with less terms than you may have
expected, but the O(x**n) term appended will always be correct and
so the result, though perhaps shorter, will also be correct.
If any of those assumptions is not met, this is treated like a
wrapper to series which will try harder to return the correct
number of terms.
See also lseries().
Examples
========
>>> from sympy import sin, log, Symbol
>>> from sympy.abc import x, y
>>> sin(x).nseries(x, 0, 6)
x - x**3/6 + x**5/120 + O(x**6)
>>> log(x+1).nseries(x, 0, 5)
x - x**2/2 + x**3/3 - x**4/4 + O(x**5)
Handling of the ``logx`` parameter --- in the following example the
expansion fails since ``sin`` does not have an asymptotic expansion
at -oo (the limit of log(x) as x approaches 0):
>>> e = sin(log(x))
>>> e.nseries(x, 0, 6)
Traceback (most recent call last):
...
PoleError: ...
...
>>> logx = Symbol('logx')
>>> e.nseries(x, 0, 6, logx=logx)
sin(logx)
In the following example, the expansion works but only returns self
unless the ``logx`` parameter is used:
>>> e = x**y
>>> e.nseries(x, 0, 2)
x**y
>>> e.nseries(x, 0, 2, logx=logx)
exp(logx*y)
"""
if x and x not in self.free_symbols:
return self
if x is None or x0 or dir != '+': # {see XPOS above} or (x.is_positive == x.is_negative == None):
return self.series(x, x0, n, dir, cdir=cdir)
else:
return self._eval_nseries(x, n=n, logx=logx, cdir=cdir)
def _eval_nseries(self, x, n, logx, cdir):
"""
Return terms of series for self up to O(x**n) at x=0
from the positive direction.
This is a method that should be overridden in subclasses. Users should
never call this method directly (use .nseries() instead), so you do not
have to write docstrings for _eval_nseries().
"""
raise NotImplementedError(filldedent("""
The _eval_nseries method should be added to
%s to give terms up to O(x**n) at x=0
from the positive direction so it is available when
nseries calls it.""" % self.func)
)
def limit(self, x, xlim, dir='+'):
""" Compute limit x->xlim.
"""
from sympy.series.limits import limit
return limit(self, x, xlim, dir)
def compute_leading_term(self, x, logx=None):
"""
as_leading_term is only allowed for results of .series()
This is a wrapper to compute a series first.
"""
from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold
if self.has(Piecewise):
expr = piecewise_fold(self)
else:
expr = self
if self.removeO() == 0:
return self
from sympy.series.gruntz import calculate_series
if logx is None:
from .symbol import Dummy
from sympy.functions.elementary.exponential import log
d = Dummy('logx')
s = calculate_series(expr, x, d).subs(d, log(x))
else:
s = calculate_series(expr, x, logx)
return s.as_leading_term(x)
@cacheit
def as_leading_term(self, *symbols, logx=None, cdir=0):
"""
Returns the leading (nonzero) term of the series expansion of self.
The _eval_as_leading_term routines are used to do this, and they must
always return a non-zero value.
Examples
========
>>> from sympy.abc import x
>>> (1 + x + x**2).as_leading_term(x)
1
>>> (1/x**2 + x + x**2).as_leading_term(x)
x**(-2)
"""
if len(symbols) > 1:
c = self
for x in symbols:
c = c.as_leading_term(x, logx=logx, cdir=cdir)
return c
elif not symbols:
return self
x = sympify(symbols[0])
if not x.is_symbol:
raise ValueError('expecting a Symbol but got %s' % x)
if x not in self.free_symbols:
return self
obj = self._eval_as_leading_term(x, logx=logx, cdir=cdir)
if obj is not None:
from sympy.simplify.powsimp import powsimp
return powsimp(obj, deep=True, combine='exp')
raise NotImplementedError('as_leading_term(%s, %s)' % (self, x))
def _eval_as_leading_term(self, x, logx=None, cdir=0):
return self
def as_coeff_exponent(self, x) -> tuple[Expr, Expr]:
""" ``c*x**e -> c,e`` where x can be any symbolic expression.
"""
from sympy.simplify.radsimp import collect
s = collect(self, x)
c, p = s.as_coeff_mul(x)
if len(p) == 1:
b, e = p[0].as_base_exp()
if b == x:
return c, e
return s, S.Zero
def leadterm(self, x, logx=None, cdir=0):
"""
Returns the leading term a*x**b as a tuple (a, b).
Examples
========
>>> from sympy.abc import x
>>> (1+x+x**2).leadterm(x)
(1, 0)
>>> (1/x**2+x+x**2).leadterm(x)
(1, -2)
"""
from .symbol import Dummy
from sympy.functions.elementary.exponential import log
l = self.as_leading_term(x, logx=logx, cdir=cdir)
d = Dummy('logx')
if l.has(log(x)):
l = l.subs(log(x), d)
c, e = l.as_coeff_exponent(x)
if x in c.free_symbols:
raise ValueError(filldedent("""
cannot compute leadterm(%s, %s). The coefficient
should have been free of %s but got %s""" % (self, x, x, c)))
c = c.subs(d, log(x))
return c, e
def as_coeff_Mul(self, rational: bool = False) -> tuple['Number', Expr]:
"""Efficiently extract the coefficient of a product. """
return S.One, self
def as_coeff_Add(self, rational=False) -> tuple['Number', Expr]:
"""Efficiently extract the coefficient of a summation. """
return S.Zero, self
def fps(self, x=None, x0=0, dir=1, hyper=True, order=4, rational=True,
full=False):
"""
Compute formal power power series of self.
See the docstring of the :func:`fps` function in sympy.series.formal for
more information.
"""
from sympy.series.formal import fps
return fps(self, x, x0, dir, hyper, order, rational, full)
def fourier_series(self, limits=None):
"""Compute fourier sine/cosine series of self.
See the docstring of the :func:`fourier_series` in sympy.series.fourier
for more information.
"""
from sympy.series.fourier import fourier_series
return fourier_series(self, limits)
###################################################################################
##################### DERIVATIVE, INTEGRAL, FUNCTIONAL METHODS ####################
###################################################################################
def diff(self, *symbols, **assumptions):
assumptions.setdefault("evaluate", True)
return _derivative_dispatch(self, *symbols, **assumptions)
###########################################################################
###################### EXPRESSION EXPANSION METHODS #######################
###########################################################################
# Relevant subclasses should override _eval_expand_hint() methods. See
# the docstring of expand() for more info.
def _eval_expand_complex(self, **hints):
real, imag = self.as_real_imag(**hints)
return real + S.ImaginaryUnit*imag
@staticmethod
def _expand_hint(expr, hint, deep=True, **hints):
"""
Helper for ``expand()``. Recursively calls ``expr._eval_expand_hint()``.
Returns ``(expr, hit)``, where expr is the (possibly) expanded
``expr`` and ``hit`` is ``True`` if ``expr`` was truly expanded and
``False`` otherwise.
"""
hit = False
# XXX: Hack to support non-Basic args
# |
# V
if deep and getattr(expr, 'args', ()) and not expr.is_Atom:
sargs = []
for arg in expr.args:
arg, arghit = Expr._expand_hint(arg, hint, **hints)
hit |= arghit
sargs.append(arg)
if hit:
expr = expr.func(*sargs)
if hasattr(expr, hint):
newexpr = getattr(expr, hint)(**hints)
if newexpr != expr:
return (newexpr, True)
return (expr, hit)
@cacheit
def expand(self, deep=True, modulus=None, power_base=True, power_exp=True,
mul=True, log=True, multinomial=True, basic=True, **hints):
"""
Expand an expression using hints.
See the docstring of the expand() function in sympy.core.function for
more information.
"""
from sympy.simplify.radsimp import fraction
hints.update(power_base=power_base, power_exp=power_exp, mul=mul,
log=log, multinomial=multinomial, basic=basic)
expr = self
if hints.pop('frac', False):
n, d = [a.expand(deep=deep, modulus=modulus, **hints)
for a in fraction(self)]
return n/d
elif hints.pop('denom', False):
n, d = fraction(self)
return n/d.expand(deep=deep, modulus=modulus, **hints)
elif hints.pop('numer', False):
n, d = fraction(self)
return n.expand(deep=deep, modulus=modulus, **hints)/d
# Although the hints are sorted here, an earlier hint may get applied
# at a given node in the expression tree before another because of how
# the hints are applied. e.g. expand(log(x*(y + z))) -> log(x*y +
# x*z) because while applying log at the top level, log and mul are
# applied at the deeper level in the tree so that when the log at the
# upper level gets applied, the mul has already been applied at the
# lower level.
# Additionally, because hints are only applied once, the expression
# may not be expanded all the way. For example, if mul is applied
# before multinomial, x*(x + 1)**2 won't be expanded all the way. For
# now, we just use a special case to make multinomial run before mul,
# so that at least polynomials will be expanded all the way. In the
# future, smarter heuristics should be applied.
# TODO: Smarter heuristics
def _expand_hint_key(hint):
"""Make multinomial come before mul"""
if hint == 'mul':
return 'mulz'
return hint
for hint in sorted(hints.keys(), key=_expand_hint_key):
use_hint = hints[hint]
if use_hint:
hint = '_eval_expand_' + hint
expr, hit = Expr._expand_hint(expr, hint, deep=deep, **hints)
while True:
was = expr
if hints.get('multinomial', False):
expr, _ = Expr._expand_hint(
expr, '_eval_expand_multinomial', deep=deep, **hints)
if hints.get('mul', False):
expr, _ = Expr._expand_hint(
expr, '_eval_expand_mul', deep=deep, **hints)
if hints.get('log', False):
expr, _ = Expr._expand_hint(
expr, '_eval_expand_log', deep=deep, **hints)
if expr == was:
break
if modulus is not None:
modulus = sympify(modulus)
if not modulus.is_Integer or modulus <= 0:
raise ValueError(
"modulus must be a positive integer, got %s" % modulus)
terms = []
for term in Add.make_args(expr):
coeff, tail = term.as_coeff_Mul(rational=True)
coeff %= modulus
if coeff:
terms.append(coeff*tail)
expr = Add(*terms)
return expr
###########################################################################
################### GLOBAL ACTION VERB WRAPPER METHODS ####################
###########################################################################
def integrate(self, *args, **kwargs):
"""See the integrate function in sympy.integrals"""
from sympy.integrals.integrals import integrate
return integrate(self, *args, **kwargs)
def nsimplify(self, constants=(), tolerance=None, full=False):
"""See the nsimplify function in sympy.simplify"""
from sympy.simplify.simplify import nsimplify
return nsimplify(self, constants, tolerance, full)
def separate(self, deep=False, force=False):
"""See the separate function in sympy.simplify"""
from .function import expand_power_base
return expand_power_base(self, deep=deep, force=force)
def collect(self, syms, func=None, evaluate=True, exact=False, distribute_order_term=True):
"""See the collect function in sympy.simplify"""
from sympy.simplify.radsimp import collect
return collect(self, syms, func, evaluate, exact, distribute_order_term)
def together(self, *args, **kwargs):
"""See the together function in sympy.polys"""
from sympy.polys.rationaltools import together
return together(self, *args, **kwargs)
def apart(self, x=None, **args):
"""See the apart function in sympy.polys"""
from sympy.polys.partfrac import apart
return apart(self, x, **args)
def ratsimp(self):
"""See the ratsimp function in sympy.simplify"""
from sympy.simplify.ratsimp import ratsimp
return ratsimp(self)
def trigsimp(self, **args):
"""See the trigsimp function in sympy.simplify"""
from sympy.simplify.trigsimp import trigsimp
return trigsimp(self, **args)
def radsimp(self, **kwargs):
"""See the radsimp function in sympy.simplify"""
from sympy.simplify.radsimp import radsimp
return radsimp(self, **kwargs)
def powsimp(self, *args, **kwargs):
"""See the powsimp function in sympy.simplify"""
from sympy.simplify.powsimp import powsimp
return powsimp(self, *args, **kwargs)
def combsimp(self):
"""See the combsimp function in sympy.simplify"""
from sympy.simplify.combsimp import combsimp
return combsimp(self)
def gammasimp(self):
"""See the gammasimp function in sympy.simplify"""
from sympy.simplify.gammasimp import gammasimp
return gammasimp(self)
def factor(self, *gens, **args):
"""See the factor() function in sympy.polys.polytools"""
from sympy.polys.polytools import factor
return factor(self, *gens, **args)
def cancel(self, *gens, **args):
"""See the cancel function in sympy.polys"""
from sympy.polys.polytools import cancel
return cancel(self, *gens, **args)
def invert(self, g, *gens, **args):
"""Return the multiplicative inverse of ``self`` mod ``g``
where ``self`` (and ``g``) may be symbolic expressions).
See Also
========
sympy.core.numbers.mod_inverse, sympy.polys.polytools.invert
"""
if self.is_number and getattr(g, 'is_number', True):
from .numbers import mod_inverse
return mod_inverse(self, g)
from sympy.polys.polytools import invert
return invert(self, g, *gens, **args)
def round(self, n=None):
"""Return x rounded to the given decimal place.
If a complex number would results, apply round to the real
and imaginary components of the number.
Examples
========
>>> from sympy import pi, E, I, S, Number
>>> pi.round()
3
>>> pi.round(2)
3.14
>>> (2*pi + E*I).round()
6 + 3*I
The round method has a chopping effect:
>>> (2*pi + I/10).round()
6
>>> (pi/10 + 2*I).round()
2*I
>>> (pi/10 + E*I).round(2)
0.31 + 2.72*I
Notes
=====
The Python ``round`` function uses the SymPy ``round`` method so it
will always return a SymPy number (not a Python float or int):
>>> isinstance(round(S(123), -2), Number)
True
"""
x = self
if not x.is_number:
raise TypeError("Cannot round symbolic expression")
if not x.is_Atom:
if not pure_complex(x.n(2), or_real=True):
raise TypeError(
'Expected a number but got %s:' % func_name(x))
elif x in _illegal:
return x
if x.is_extended_real is False:
r, i = x.as_real_imag()
return r.round(n) + S.ImaginaryUnit*i.round(n)
if not x:
return S.Zero if n is None else x
p = as_int(n or 0)
if x.is_Integer:
return Integer(round(int(x), p))
digits_to_decimal = _mag(x) # _mag(12) = 2, _mag(.012) = -1
allow = digits_to_decimal + p
precs = [f._prec for f in x.atoms(Float)]
dps = prec_to_dps(max(precs)) if precs else None
if dps is None:
# assume everything is exact so use the Python
# float default or whatever was requested
dps = max(15, allow)
else:
allow = min(allow, dps)
# this will shift all digits to right of decimal
# and give us dps to work with as an int
shift = -digits_to_decimal + dps
extra = 1 # how far we look past known digits
# NOTE
# mpmath will calculate the binary representation to
# an arbitrary number of digits but we must base our
# answer on a finite number of those digits, e.g.
# .575 2589569785738035/2**52 in binary.
# mpmath shows us that the first 18 digits are
# >>> Float(.575).n(18)
# 0.574999999999999956
# The default precision is 15 digits and if we ask
# for 15 we get
# >>> Float(.575).n(15)
# 0.575000000000000
# mpmath handles rounding at the 15th digit. But we
# need to be careful since the user might be asking
# for rounding at the last digit and our semantics
# are to round toward the even final digit when there
# is a tie. So the extra digit will be used to make
# that decision. In this case, the value is the same
# to 15 digits:
# >>> Float(.575).n(16)
# 0.5750000000000000
# Now converting this to the 15 known digits gives
# 575000000000000.0
# which rounds to integer
# 5750000000000000
# And now we can round to the desired digt, e.g. at
# the second from the left and we get
# 5800000000000000
# and rescaling that gives
# 0.58
# as the final result.
# If the value is made slightly less than 0.575 we might
# still obtain the same value:
# >>> Float(.575-1e-16).n(16)*10**15
# 574999999999999.8
# What 15 digits best represents the known digits (which are
# to the left of the decimal? 5750000000000000, the same as
# before. The only way we will round down (in this case) is
# if we declared that we had more than 15 digits of precision.
# For example, if we use 16 digits of precision, the integer
# we deal with is
# >>> Float(.575-1e-16).n(17)*10**16
# 5749999999999998.4
# and this now rounds to 5749999999999998 and (if we round to
# the 2nd digit from the left) we get 5700000000000000.
#
xf = x.n(dps + extra)*Pow(10, shift)
xi = Integer(xf)
# use the last digit to select the value of xi
# nearest to x before rounding at the desired digit
sign = 1 if x > 0 else -1
dif2 = sign*(xf - xi).n(extra)
if dif2 < 0:
raise NotImplementedError(
'not expecting int(x) to round away from 0')
if dif2 > .5:
xi += sign # round away from 0
elif dif2 == .5:
xi += sign if xi%2 else -sign # round toward even
# shift p to the new position
ip = p - shift
# let Python handle the int rounding then rescale
xr = round(xi.p, ip)
# restore scale
rv = Rational(xr, Pow(10, shift))
# return Float or Integer
if rv.is_Integer:
if n is None: # the single-arg case
return rv
# use str or else it won't be a float
return Float(str(rv), dps) # keep same precision
else:
if not allow and rv > self:
allow += 1
return Float(rv, allow)
__round__ = round
def _eval_derivative_matrix_lines(self, x):
from sympy.matrices.expressions.matexpr import _LeftRightArgs
return [_LeftRightArgs([S.One, S.One], higher=self._eval_derivative(x))]
class AtomicExpr(Atom, Expr):
"""
A parent class for object which are both atoms and Exprs.
For example: Symbol, Number, Rational, Integer, ...
But not: Add, Mul, Pow, ...
"""
is_number = False
is_Atom = True
__slots__ = ()
def _eval_derivative(self, s):
if self == s:
return S.One
return S.Zero
def _eval_derivative_n_times(self, s, n):
from .containers import Tuple
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.matrices.common import MatrixCommon
if isinstance(s, (MatrixCommon, Tuple, Iterable, MatrixExpr)):
return super()._eval_derivative_n_times(s, n)
from .relational import Eq
from sympy.functions.elementary.piecewise import Piecewise
if self == s:
return Piecewise((self, Eq(n, 0)), (1, Eq(n, 1)), (0, True))
else:
return Piecewise((self, Eq(n, 0)), (0, True))
def _eval_is_polynomial(self, syms):
return True
def _eval_is_rational_function(self, syms):
return True
def _eval_is_meromorphic(self, x, a):
from sympy.calculus.accumulationbounds import AccumBounds
return (not self.is_Number or self.is_finite) and not isinstance(self, AccumBounds)
def _eval_is_algebraic_expr(self, syms):
return True
def _eval_nseries(self, x, n, logx, cdir=0):
return self
@property
def expr_free_symbols(self):
sympy_deprecation_warning("""
The expr_free_symbols property is deprecated. Use free_symbols to get
the free symbols of an expression.
""",
deprecated_since_version="1.9",
active_deprecations_target="deprecated-expr-free-symbols")
return {self}
def _mag(x):
r"""Return integer $i$ such that $0.1 \le x/10^i < 1$
Examples
========
>>> from sympy.core.expr import _mag
>>> from sympy import Float
>>> _mag(Float(.1))
0
>>> _mag(Float(.01))
-1
>>> _mag(Float(1234))
4
"""
from math import log10, ceil, log
xpos = abs(x.n())
if not xpos:
return S.Zero
try:
mag_first_dig = int(ceil(log10(xpos)))
except (ValueError, OverflowError):
mag_first_dig = int(ceil(Float(mpf_log(xpos._mpf_, 53))/log(10)))
# check that we aren't off by 1
if (xpos/10**mag_first_dig) >= 1:
assert 1 <= (xpos/10**mag_first_dig) < 10
mag_first_dig += 1
return mag_first_dig
class UnevaluatedExpr(Expr):
"""
Expression that is not evaluated unless released.
Examples
========
>>> from sympy import UnevaluatedExpr
>>> from sympy.abc import x
>>> x*(1/x)
1
>>> x*UnevaluatedExpr(1/x)
x*1/x
"""
def __new__(cls, arg, **kwargs):
arg = _sympify(arg)
obj = Expr.__new__(cls, arg, **kwargs)
return obj
def doit(self, **hints):
if hints.get("deep", True):
return self.args[0].doit(**hints)
else:
return self.args[0]
def unchanged(func, *args):
"""Return True if `func` applied to the `args` is unchanged.
Can be used instead of `assert foo == foo`.
Examples
========
>>> from sympy import Piecewise, cos, pi
>>> from sympy.core.expr import unchanged
>>> from sympy.abc import x
>>> unchanged(cos, 1) # instead of assert cos(1) == cos(1)
True
>>> unchanged(cos, pi)
False
Comparison of args uses the builtin capabilities of the object's
arguments to test for equality so args can be defined loosely. Here,
the ExprCondPair arguments of Piecewise compare as equal to the
tuples that can be used to create the Piecewise:
>>> unchanged(Piecewise, (x, x > 1), (0, True))
True
"""
f = func(*args)
return f.func == func and f.args == args
class ExprBuilder:
def __init__(self, op, args=None, validator=None, check=True):
if not hasattr(op, "__call__"):
raise TypeError("op {} needs to be callable".format(op))
self.op = op
if args is None:
self.args = []
else:
self.args = args
self.validator = validator
if (validator is not None) and check:
self.validate()
@staticmethod
def _build_args(args):
return [i.build() if isinstance(i, ExprBuilder) else i for i in args]
def validate(self):
if self.validator is None:
return
args = self._build_args(self.args)
self.validator(*args)
def build(self, check=True):
args = self._build_args(self.args)
if self.validator and check:
self.validator(*args)
return self.op(*args)
def append_argument(self, arg, check=True):
self.args.append(arg)
if self.validator and check:
self.validate(*self.args)
def __getitem__(self, item):
if item == 0:
return self.op
else:
return self.args[item-1]
def __repr__(self):
return str(self.build())
def search_element(self, elem):
for i, arg in enumerate(self.args):
if isinstance(arg, ExprBuilder):
ret = arg.search_index(elem)
if ret is not None:
return (i,) + ret
elif id(arg) == id(elem):
return (i,)
return None
from .mul import Mul
from .add import Add
from .power import Pow
from .function import Function, _derivative_dispatch
from .mod import Mod
from .exprtools import factor_terms
from .numbers import Float, Integer, Rational, _illegal
|
06bb4d2b3667eaf516786484a0b81e3bf957af5d9e68f7a3e5287ed9b299d30c | """sympify -- convert objects SymPy internal format"""
import typing
if typing.TYPE_CHECKING:
from typing import Any, Callable, Dict as tDict, Type
from inspect import getmro
import string
from sympy.core.random import choice
from .parameters import global_parameters
from sympy.utilities.exceptions import sympy_deprecation_warning
from sympy.utilities.iterables import iterable
class SympifyError(ValueError):
def __init__(self, expr, base_exc=None):
self.expr = expr
self.base_exc = base_exc
def __str__(self):
if self.base_exc is None:
return "SympifyError: %r" % (self.expr,)
return ("Sympify of expression '%s' failed, because of exception being "
"raised:\n%s: %s" % (self.expr, self.base_exc.__class__.__name__,
str(self.base_exc)))
converter = {} # type: tDict[Type[Any], Callable[[Any], Basic]]
#holds the conversions defined in SymPy itself, i.e. non-user defined conversions
_sympy_converter = {} # type: tDict[Type[Any], Callable[[Any], Basic]]
#alias for clearer use in the library
_external_converter = converter
class CantSympify:
"""
Mix in this trait to a class to disallow sympification of its instances.
Examples
========
>>> from sympy import sympify
>>> from sympy.core.sympify import CantSympify
>>> class Something(dict):
... pass
...
>>> sympify(Something())
{}
>>> class Something(dict, CantSympify):
... pass
...
>>> sympify(Something())
Traceback (most recent call last):
...
SympifyError: SympifyError: {}
"""
__slots__ = ()
def _is_numpy_instance(a):
"""
Checks if an object is an instance of a type from the numpy module.
"""
# This check avoids unnecessarily importing NumPy. We check the whole
# __mro__ in case any base type is a numpy type.
return any(type_.__module__ == 'numpy'
for type_ in type(a).__mro__)
def _convert_numpy_types(a, **sympify_args):
"""
Converts a numpy datatype input to an appropriate SymPy type.
"""
import numpy as np
if not isinstance(a, np.floating):
if np.iscomplex(a):
return _sympy_converter[complex](a.item())
else:
return sympify(a.item(), **sympify_args)
else:
try:
from .numbers import Float
prec = np.finfo(a).nmant + 1
# E.g. double precision means prec=53 but nmant=52
# Leading bit of mantissa is always 1, so is not stored
a = str(list(np.reshape(np.asarray(a),
(1, np.size(a)))[0]))[1:-1]
return Float(a, precision=prec)
except NotImplementedError:
raise SympifyError('Translation for numpy float : %s '
'is not implemented' % a)
def sympify(a, locals=None, convert_xor=True, strict=False, rational=False,
evaluate=None):
"""
Converts an arbitrary expression to a type that can be used inside SymPy.
Explanation
===========
It will convert Python ints into instances of :class:`~.Integer`, floats
into instances of :class:`~.Float`, etc. It is also able to coerce
symbolic expressions which inherit from :class:`~.Basic`. This can be
useful in cooperation with SAGE.
.. warning::
Note that this function uses ``eval``, and thus shouldn't be used on
unsanitized input.
If the argument is already a type that SymPy understands, it will do
nothing but return that value. This can be used at the beginning of a
function to ensure you are working with the correct type.
Examples
========
>>> from sympy import sympify
>>> sympify(2).is_integer
True
>>> sympify(2).is_real
True
>>> sympify(2.0).is_real
True
>>> sympify("2.0").is_real
True
>>> sympify("2e-45").is_real
True
If the expression could not be converted, a SympifyError is raised.
>>> sympify("x***2")
Traceback (most recent call last):
...
SympifyError: SympifyError: "could not parse 'x***2'"
Locals
------
The sympification happens with access to everything that is loaded
by ``from sympy import *``; anything used in a string that is not
defined by that import will be converted to a symbol. In the following,
the ``bitcount`` function is treated as a symbol and the ``O`` is
interpreted as the :class:`~.Order` object (used with series) and it raises
an error when used improperly:
>>> s = 'bitcount(42)'
>>> sympify(s)
bitcount(42)
>>> sympify("O(x)")
O(x)
>>> sympify("O + 1")
Traceback (most recent call last):
...
TypeError: unbound method...
In order to have ``bitcount`` be recognized it can be imported into a
namespace dictionary and passed as locals:
>>> ns = {}
>>> exec('from sympy.core.evalf import bitcount', ns)
>>> sympify(s, locals=ns)
6
In order to have the ``O`` interpreted as a Symbol, identify it as such
in the namespace dictionary. This can be done in a variety of ways; all
three of the following are possibilities:
>>> from sympy import Symbol
>>> ns["O"] = Symbol("O") # method 1
>>> exec('from sympy.abc import O', ns) # method 2
>>> ns.update(dict(O=Symbol("O"))) # method 3
>>> sympify("O + 1", locals=ns)
O + 1
If you want *all* single-letter and Greek-letter variables to be symbols
then you can use the clashing-symbols dictionaries that have been defined
there as private variables: ``_clash1`` (single-letter variables),
``_clash2`` (the multi-letter Greek names) or ``_clash`` (both single and
multi-letter names that are defined in ``abc``).
>>> from sympy.abc import _clash1
>>> set(_clash1) # if this fails, see issue #23903
{'E', 'I', 'N', 'O', 'Q', 'S'}
>>> sympify('I & Q', _clash1)
I & Q
Strict
------
If the option ``strict`` is set to ``True``, only the types for which an
explicit conversion has been defined are converted. In the other
cases, a SympifyError is raised.
>>> print(sympify(None))
None
>>> sympify(None, strict=True)
Traceback (most recent call last):
...
SympifyError: SympifyError: None
.. deprecated:: 1.6
``sympify(obj)`` automatically falls back to ``str(obj)`` when all
other conversion methods fail, but this is deprecated. ``strict=True``
will disable this deprecated behavior. See
:ref:`deprecated-sympify-string-fallback`.
Evaluation
----------
If the option ``evaluate`` is set to ``False``, then arithmetic and
operators will be converted into their SymPy equivalents and the
``evaluate=False`` option will be added. Nested ``Add`` or ``Mul`` will
be denested first. This is done via an AST transformation that replaces
operators with their SymPy equivalents, so if an operand redefines any
of those operations, the redefined operators will not be used. If
argument a is not a string, the mathematical expression is evaluated
before being passed to sympify, so adding ``evaluate=False`` will still
return the evaluated result of expression.
>>> sympify('2**2 / 3 + 5')
19/3
>>> sympify('2**2 / 3 + 5', evaluate=False)
2**2/3 + 5
>>> sympify('4/2+7', evaluate=True)
9
>>> sympify('4/2+7', evaluate=False)
4/2 + 7
>>> sympify(4/2+7, evaluate=False)
9.00000000000000
Extending
---------
To extend ``sympify`` to convert custom objects (not derived from ``Basic``),
just define a ``_sympy_`` method to your class. You can do that even to
classes that you do not own by subclassing or adding the method at runtime.
>>> from sympy import Matrix
>>> class MyList1(object):
... def __iter__(self):
... yield 1
... yield 2
... return
... def __getitem__(self, i): return list(self)[i]
... def _sympy_(self): return Matrix(self)
>>> sympify(MyList1())
Matrix([
[1],
[2]])
If you do not have control over the class definition you could also use the
``converter`` global dictionary. The key is the class and the value is a
function that takes a single argument and returns the desired SymPy
object, e.g. ``converter[MyList] = lambda x: Matrix(x)``.
>>> class MyList2(object): # XXX Do not do this if you control the class!
... def __iter__(self): # Use _sympy_!
... yield 1
... yield 2
... return
... def __getitem__(self, i): return list(self)[i]
>>> from sympy.core.sympify import converter
>>> converter[MyList2] = lambda x: Matrix(x)
>>> sympify(MyList2())
Matrix([
[1],
[2]])
Notes
=====
The keywords ``rational`` and ``convert_xor`` are only used
when the input is a string.
convert_xor
-----------
>>> sympify('x^y',convert_xor=True)
x**y
>>> sympify('x^y',convert_xor=False)
x ^ y
rational
--------
>>> sympify('0.1',rational=False)
0.1
>>> sympify('0.1',rational=True)
1/10
Sometimes autosimplification during sympification results in expressions
that are very different in structure than what was entered. Until such
autosimplification is no longer done, the ``kernS`` function might be of
some use. In the example below you can see how an expression reduces to
$-1$ by autosimplification, but does not do so when ``kernS`` is used.
>>> from sympy.core.sympify import kernS
>>> from sympy.abc import x
>>> -2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1
-1
>>> s = '-2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1'
>>> sympify(s)
-1
>>> kernS(s)
-2*(-(-x + 1/x)/(x*(x - 1/x)**2) - 1/(x*(x - 1/x))) - 1
Parameters
==========
a :
- any object defined in SymPy
- standard numeric Python types: ``int``, ``long``, ``float``, ``Decimal``
- strings (like ``"0.09"``, ``"2e-19"`` or ``'sin(x)'``)
- booleans, including ``None`` (will leave ``None`` unchanged)
- dicts, lists, sets or tuples containing any of the above
convert_xor : bool, optional
If true, treats ``^`` as exponentiation.
If False, treats ``^`` as XOR itself.
Used only when input is a string.
locals : any object defined in SymPy, optional
In order to have strings be recognized it can be imported
into a namespace dictionary and passed as locals.
strict : bool, optional
If the option strict is set to ``True``, only the types for which
an explicit conversion has been defined are converted. In the
other cases, a SympifyError is raised.
rational : bool, optional
If ``True``, converts floats into :class:`~.Rational`.
If ``False``, it lets floats remain as it is.
Used only when input is a string.
evaluate : bool, optional
If False, then arithmetic and operators will be converted into
their SymPy equivalents. If True the expression will be evaluated
and the result will be returned.
"""
# XXX: If a is a Basic subclass rather than instance (e.g. sin rather than
# sin(x)) then a.__sympy__ will be the property. Only on the instance will
# a.__sympy__ give the *value* of the property (True). Since sympify(sin)
# was used for a long time we allow it to pass. However if strict=True as
# is the case in internal calls to _sympify then we only allow
# is_sympy=True.
#
# https://github.com/sympy/sympy/issues/20124
is_sympy = getattr(a, '__sympy__', None)
if is_sympy is True:
return a
elif is_sympy is not None:
if not strict:
return a
else:
raise SympifyError(a)
if isinstance(a, CantSympify):
raise SympifyError(a)
cls = getattr(a, "__class__", None)
#Check if there exists a converter for any of the types in the mro
for superclass in getmro(cls):
#First check for user defined converters
conv = _external_converter.get(superclass)
if conv is None:
#if none exists, check for SymPy defined converters
conv = _sympy_converter.get(superclass)
if conv is not None:
return conv(a)
if cls is type(None):
if strict:
raise SympifyError(a)
else:
return a
if evaluate is None:
evaluate = global_parameters.evaluate
# Support for basic numpy datatypes
if _is_numpy_instance(a):
import numpy as np
if np.isscalar(a):
return _convert_numpy_types(a, locals=locals,
convert_xor=convert_xor, strict=strict, rational=rational,
evaluate=evaluate)
_sympy_ = getattr(a, "_sympy_", None)
if _sympy_ is not None:
try:
return a._sympy_()
# XXX: Catches AttributeError: 'SymPyConverter' object has no
# attribute 'tuple'
# This is probably a bug somewhere but for now we catch it here.
except AttributeError:
pass
if not strict:
# Put numpy array conversion _before_ float/int, see
# <https://github.com/sympy/sympy/issues/13924>.
flat = getattr(a, "flat", None)
if flat is not None:
shape = getattr(a, "shape", None)
if shape is not None:
from sympy.tensor.array import Array
return Array(a.flat, a.shape) # works with e.g. NumPy arrays
if not isinstance(a, str):
if _is_numpy_instance(a):
import numpy as np
assert not isinstance(a, np.number)
if isinstance(a, np.ndarray):
# Scalar arrays (those with zero dimensions) have sympify
# called on the scalar element.
if a.ndim == 0:
try:
return sympify(a.item(),
locals=locals,
convert_xor=convert_xor,
strict=strict,
rational=rational,
evaluate=evaluate)
except SympifyError:
pass
else:
# float and int can coerce size-one numpy arrays to their lone
# element. See issue https://github.com/numpy/numpy/issues/10404.
for coerce in (float, int):
try:
return sympify(coerce(a))
except (TypeError, ValueError, AttributeError, SympifyError):
continue
if strict:
raise SympifyError(a)
if iterable(a):
try:
return type(a)([sympify(x, locals=locals, convert_xor=convert_xor,
rational=rational, evaluate=evaluate) for x in a])
except TypeError:
# Not all iterables are rebuildable with their type.
pass
if not isinstance(a, str):
try:
a = str(a)
except Exception as exc:
raise SympifyError(a, exc)
sympy_deprecation_warning(
f"""
The string fallback in sympify() is deprecated.
To explicitly convert the string form of an object, use
sympify(str(obj)). To add define sympify behavior on custom
objects, use sympy.core.sympify.converter or define obj._sympy_
(see the sympify() docstring).
sympify() performed the string fallback resulting in the following string:
{a!r}
""",
deprecated_since_version='1.6',
active_deprecations_target="deprecated-sympify-string-fallback",
)
from sympy.parsing.sympy_parser import (parse_expr, TokenError,
standard_transformations)
from sympy.parsing.sympy_parser import convert_xor as t_convert_xor
from sympy.parsing.sympy_parser import rationalize as t_rationalize
transformations = standard_transformations
if rational:
transformations += (t_rationalize,)
if convert_xor:
transformations += (t_convert_xor,)
try:
a = a.replace('\n', '')
expr = parse_expr(a, local_dict=locals, transformations=transformations, evaluate=evaluate)
except (TokenError, SyntaxError) as exc:
raise SympifyError('could not parse %r' % a, exc)
return expr
def _sympify(a):
"""
Short version of :func:`~.sympify` for internal usage for ``__add__`` and
``__eq__`` methods where it is ok to allow some things (like Python
integers and floats) in the expression. This excludes things (like strings)
that are unwise to allow into such an expression.
>>> from sympy import Integer
>>> Integer(1) == 1
True
>>> Integer(1) == '1'
False
>>> from sympy.abc import x
>>> x + 1
x + 1
>>> x + '1'
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +: 'Symbol' and 'str'
see: sympify
"""
return sympify(a, strict=True)
def kernS(s):
"""Use a hack to try keep autosimplification from distributing a
a number into an Add; this modification does not
prevent the 2-arg Mul from becoming an Add, however.
Examples
========
>>> from sympy.core.sympify import kernS
>>> from sympy.abc import x, y
The 2-arg Mul distributes a number (or minus sign) across the terms
of an expression, but kernS will prevent that:
>>> 2*(x + y), -(x + 1)
(2*x + 2*y, -x - 1)
>>> kernS('2*(x + y)')
2*(x + y)
>>> kernS('-(x + 1)')
-(x + 1)
If use of the hack fails, the un-hacked string will be passed to sympify...
and you get what you get.
XXX This hack should not be necessary once issue 4596 has been resolved.
"""
hit = False
quoted = '"' in s or "'" in s
if '(' in s and not quoted:
if s.count('(') != s.count(")"):
raise SympifyError('unmatched left parenthesis')
# strip all space from s
s = ''.join(s.split())
olds = s
# now use space to represent a symbol that
# will
# step 1. turn potential 2-arg Muls into 3-arg versions
# 1a. *( -> * *(
s = s.replace('*(', '* *(')
# 1b. close up exponentials
s = s.replace('** *', '**')
# 2. handle the implied multiplication of a negated
# parenthesized expression in two steps
# 2a: -(...) --> -( *(...)
target = '-( *('
s = s.replace('-(', target)
# 2b: double the matching closing parenthesis
# -( *(...) --> -( *(...))
i = nest = 0
assert target.endswith('(') # assumption below
while True:
j = s.find(target, i)
if j == -1:
break
j += len(target) - 1
for j in range(j, len(s)):
if s[j] == "(":
nest += 1
elif s[j] == ")":
nest -= 1
if nest == 0:
break
s = s[:j] + ")" + s[j:]
i = j + 2 # the first char after 2nd )
if ' ' in s:
# get a unique kern
kern = '_'
while kern in s:
kern += choice(string.ascii_letters + string.digits)
s = s.replace(' ', kern)
hit = kern in s
else:
hit = False
for i in range(2):
try:
expr = sympify(s)
break
except TypeError: # the kern might cause unknown errors...
if hit:
s = olds # maybe it didn't like the kern; use un-kerned s
hit = False
continue
expr = sympify(s) # let original error raise
if not hit:
return expr
from .symbol import Symbol
rep = {Symbol(kern): 1}
def _clear(expr):
if isinstance(expr, (list, tuple, set)):
return type(expr)([_clear(e) for e in expr])
if hasattr(expr, 'subs'):
return expr.subs(rep, hack2=True)
return expr
expr = _clear(expr)
# hope that kern is not there anymore
return expr
# Avoid circular import
from .basic import Basic
|
943ac3c67c43dd6391b7c4539d91c23701c671ede804e7214de1fb5bc601d1b9 | """Algorithms for computing symbolic roots of polynomials. """
import math
from functools import reduce
from sympy.core import S, I, pi
from sympy.core.exprtools import factor_terms
from sympy.core.function import _mexpand
from sympy.core.logic import fuzzy_not
from sympy.core.mul import expand_2arg, Mul
from sympy.core.numbers import Rational, igcd, comp
from sympy.core.power import Pow
from sympy.core.relational import Eq
from sympy.core.sorting import ordered
from sympy.core.symbol import Dummy, Symbol, symbols
from sympy.core.sympify import sympify
from sympy.functions import exp, im, cos, acos, Piecewise
from sympy.functions.elementary.miscellaneous import root, sqrt
from sympy.ntheory import divisors, isprime, nextprime
from sympy.polys.domains import EX
from sympy.polys.polyerrors import (PolynomialError, GeneratorsNeeded,
DomainError, UnsolvableFactorError)
from sympy.polys.polyquinticconst import PolyQuintic
from sympy.polys.polytools import Poly, cancel, factor, gcd_list, discriminant
from sympy.polys.rationaltools import together
from sympy.polys.specialpolys import cyclotomic_poly
from sympy.utilities import public
from sympy.utilities.misc import filldedent
z = Symbol('z') # importing from abc cause O to be lost as clashing symbol
def roots_linear(f):
"""Returns a list of roots of a linear polynomial."""
r = -f.nth(0)/f.nth(1)
dom = f.get_domain()
if not dom.is_Numerical:
if dom.is_Composite:
r = factor(r)
else:
from sympy.simplify.simplify import simplify
r = simplify(r)
return [r]
def roots_quadratic(f):
"""Returns a list of roots of a quadratic polynomial. If the domain is ZZ
then the roots will be sorted with negatives coming before positives.
The ordering will be the same for any numerical coefficients as long as
the assumptions tested are correct, otherwise the ordering will not be
sorted (but will be canonical).
"""
a, b, c = f.all_coeffs()
dom = f.get_domain()
def _sqrt(d):
# remove squares from square root since both will be represented
# in the results; a similar thing is happening in roots() but
# must be duplicated here because not all quadratics are binomials
co = []
other = []
for di in Mul.make_args(d):
if di.is_Pow and di.exp.is_Integer and di.exp % 2 == 0:
co.append(Pow(di.base, di.exp//2))
else:
other.append(di)
if co:
d = Mul(*other)
co = Mul(*co)
return co*sqrt(d)
return sqrt(d)
def _simplify(expr):
if dom.is_Composite:
return factor(expr)
else:
from sympy.simplify.simplify import simplify
return simplify(expr)
if c is S.Zero:
r0, r1 = S.Zero, -b/a
if not dom.is_Numerical:
r1 = _simplify(r1)
elif r1.is_negative:
r0, r1 = r1, r0
elif b is S.Zero:
r = -c/a
if not dom.is_Numerical:
r = _simplify(r)
R = _sqrt(r)
r0 = -R
r1 = R
else:
d = b**2 - 4*a*c
A = 2*a
B = -b/A
if not dom.is_Numerical:
d = _simplify(d)
B = _simplify(B)
D = factor_terms(_sqrt(d)/A)
r0 = B - D
r1 = B + D
if a.is_negative:
r0, r1 = r1, r0
elif not dom.is_Numerical:
r0, r1 = [expand_2arg(i) for i in (r0, r1)]
return [r0, r1]
def roots_cubic(f, trig=False):
"""Returns a list of roots of a cubic polynomial.
References
==========
[1] https://en.wikipedia.org/wiki/Cubic_function, General formula for roots,
(accessed November 17, 2014).
"""
if trig:
a, b, c, d = f.all_coeffs()
p = (3*a*c - b**2)/(3*a**2)
q = (2*b**3 - 9*a*b*c + 27*a**2*d)/(27*a**3)
D = 18*a*b*c*d - 4*b**3*d + b**2*c**2 - 4*a*c**3 - 27*a**2*d**2
if (D > 0) == True:
rv = []
for k in range(3):
rv.append(2*sqrt(-p/3)*cos(acos(q/p*sqrt(-3/p)*Rational(3, 2))/3 - k*pi*Rational(2, 3)))
return [i - b/3/a for i in rv]
# a*x**3 + b*x**2 + c*x + d -> x**3 + a*x**2 + b*x + c
_, a, b, c = f.monic().all_coeffs()
if c is S.Zero:
x1, x2 = roots([1, a, b], multiple=True)
return [x1, S.Zero, x2]
# x**3 + a*x**2 + b*x + c -> u**3 + p*u + q
p = b - a**2/3
q = c - a*b/3 + 2*a**3/27
pon3 = p/3
aon3 = a/3
u1 = None
if p is S.Zero:
if q is S.Zero:
return [-aon3]*3
u1 = -root(q, 3) if q.is_positive else root(-q, 3)
elif q is S.Zero:
y1, y2 = roots([1, 0, p], multiple=True)
return [tmp - aon3 for tmp in [y1, S.Zero, y2]]
elif q.is_real and q.is_negative:
u1 = -root(-q/2 + sqrt(q**2/4 + pon3**3), 3)
coeff = I*sqrt(3)/2
if u1 is None:
u1 = S.One
u2 = Rational(-1, 2) + coeff
u3 = Rational(-1, 2) - coeff
b, c, d = a, b, c # a, b, c, d = S.One, a, b, c
D0 = b**2 - 3*c # b**2 - 3*a*c
D1 = 2*b**3 - 9*b*c + 27*d # 2*b**3 - 9*a*b*c + 27*a**2*d
C = root((D1 + sqrt(D1**2 - 4*D0**3))/2, 3)
return [-(b + uk*C + D0/C/uk)/3 for uk in [u1, u2, u3]] # -(b + uk*C + D0/C/uk)/3/a
u2 = u1*(Rational(-1, 2) + coeff)
u3 = u1*(Rational(-1, 2) - coeff)
if p is S.Zero:
return [u1 - aon3, u2 - aon3, u3 - aon3]
soln = [
-u1 + pon3/u1 - aon3,
-u2 + pon3/u2 - aon3,
-u3 + pon3/u3 - aon3
]
return soln
def _roots_quartic_euler(p, q, r, a):
"""
Descartes-Euler solution of the quartic equation
Parameters
==========
p, q, r: coefficients of ``x**4 + p*x**2 + q*x + r``
a: shift of the roots
Notes
=====
This is a helper function for ``roots_quartic``.
Look for solutions of the form ::
``x1 = sqrt(R) - sqrt(A + B*sqrt(R))``
``x2 = -sqrt(R) - sqrt(A - B*sqrt(R))``
``x3 = -sqrt(R) + sqrt(A - B*sqrt(R))``
``x4 = sqrt(R) + sqrt(A + B*sqrt(R))``
To satisfy the quartic equation one must have
``p = -2*(R + A); q = -4*B*R; r = (R - A)**2 - B**2*R``
so that ``R`` must satisfy the Descartes-Euler resolvent equation
``64*R**3 + 32*p*R**2 + (4*p**2 - 16*r)*R - q**2 = 0``
If the resolvent does not have a rational solution, return None;
in that case it is likely that the Ferrari method gives a simpler
solution.
Examples
========
>>> from sympy import S
>>> from sympy.polys.polyroots import _roots_quartic_euler
>>> p, q, r = -S(64)/5, -S(512)/125, -S(1024)/3125
>>> _roots_quartic_euler(p, q, r, S(0))[0]
-sqrt(32*sqrt(5)/125 + 16/5) + 4*sqrt(5)/5
"""
# solve the resolvent equation
x = Dummy('x')
eq = 64*x**3 + 32*p*x**2 + (4*p**2 - 16*r)*x - q**2
xsols = list(roots(Poly(eq, x), cubics=False).keys())
xsols = [sol for sol in xsols if sol.is_rational and sol.is_nonzero]
if not xsols:
return None
R = max(xsols)
c1 = sqrt(R)
B = -q*c1/(4*R)
A = -R - p/2
c2 = sqrt(A + B)
c3 = sqrt(A - B)
return [c1 - c2 - a, -c1 - c3 - a, -c1 + c3 - a, c1 + c2 - a]
def roots_quartic(f):
r"""
Returns a list of roots of a quartic polynomial.
There are many references for solving quartic expressions available [1-5].
This reviewer has found that many of them require one to select from among
2 or more possible sets of solutions and that some solutions work when one
is searching for real roots but do not work when searching for complex roots
(though this is not always stated clearly). The following routine has been
tested and found to be correct for 0, 2 or 4 complex roots.
The quasisymmetric case solution [6] looks for quartics that have the form
`x**4 + A*x**3 + B*x**2 + C*x + D = 0` where `(C/A)**2 = D`.
Although no general solution that is always applicable for all
coefficients is known to this reviewer, certain conditions are tested
to determine the simplest 4 expressions that can be returned:
1) `f = c + a*(a**2/8 - b/2) == 0`
2) `g = d - a*(a*(3*a**2/256 - b/16) + c/4) = 0`
3) if `f != 0` and `g != 0` and `p = -d + a*c/4 - b**2/12` then
a) `p == 0`
b) `p != 0`
Examples
========
>>> from sympy import Poly
>>> from sympy.polys.polyroots import roots_quartic
>>> r = roots_quartic(Poly('x**4-6*x**3+17*x**2-26*x+20'))
>>> # 4 complex roots: 1+-I*sqrt(3), 2+-I
>>> sorted(str(tmp.evalf(n=2)) for tmp in r)
['1.0 + 1.7*I', '1.0 - 1.7*I', '2.0 + 1.0*I', '2.0 - 1.0*I']
References
==========
1. http://mathforum.org/dr.math/faq/faq.cubic.equations.html
2. https://en.wikipedia.org/wiki/Quartic_function#Summary_of_Ferrari.27s_method
3. http://planetmath.org/encyclopedia/GaloisTheoreticDerivationOfTheQuarticFormula.html
4. http://staff.bath.ac.uk/masjhd/JHD-CA.pdf
5. http://www.albmath.org/files/Math_5713.pdf
6. http://www.statemaster.com/encyclopedia/Quartic-equation
7. eqworld.ipmnet.ru/en/solutions/ae/ae0108.pdf
"""
_, a, b, c, d = f.monic().all_coeffs()
if not d:
return [S.Zero] + roots([1, a, b, c], multiple=True)
elif (c/a)**2 == d:
x, m = f.gen, c/a
g = Poly(x**2 + a*x + b - 2*m, x)
z1, z2 = roots_quadratic(g)
h1 = Poly(x**2 - z1*x + m, x)
h2 = Poly(x**2 - z2*x + m, x)
r1 = roots_quadratic(h1)
r2 = roots_quadratic(h2)
return r1 + r2
else:
a2 = a**2
e = b - 3*a2/8
f = _mexpand(c + a*(a2/8 - b/2))
aon4 = a/4
g = _mexpand(d - aon4*(a*(3*a2/64 - b/4) + c))
if f.is_zero:
y1, y2 = [sqrt(tmp) for tmp in
roots([1, e, g], multiple=True)]
return [tmp - aon4 for tmp in [-y1, -y2, y1, y2]]
if g.is_zero:
y = [S.Zero] + roots([1, 0, e, f], multiple=True)
return [tmp - aon4 for tmp in y]
else:
# Descartes-Euler method, see [7]
sols = _roots_quartic_euler(e, f, g, aon4)
if sols:
return sols
# Ferrari method, see [1, 2]
p = -e**2/12 - g
q = -e**3/108 + e*g/3 - f**2/8
TH = Rational(1, 3)
def _ans(y):
w = sqrt(e + 2*y)
arg1 = 3*e + 2*y
arg2 = 2*f/w
ans = []
for s in [-1, 1]:
root = sqrt(-(arg1 + s*arg2))
for t in [-1, 1]:
ans.append((s*w - t*root)/2 - aon4)
return ans
# whether a Piecewise is returned or not
# depends on knowing p, so try to put
# in a simple form
p = _mexpand(p)
# p == 0 case
y1 = e*Rational(-5, 6) - q**TH
if p.is_zero:
return _ans(y1)
# if p != 0 then u below is not 0
root = sqrt(q**2/4 + p**3/27)
r = -q/2 + root # or -q/2 - root
u = r**TH # primary root of solve(x**3 - r, x)
y2 = e*Rational(-5, 6) + u - p/u/3
if fuzzy_not(p.is_zero):
return _ans(y2)
# sort it out once they know the values of the coefficients
return [Piecewise((a1, Eq(p, 0)), (a2, True))
for a1, a2 in zip(_ans(y1), _ans(y2))]
def roots_binomial(f):
"""Returns a list of roots of a binomial polynomial. If the domain is ZZ
then the roots will be sorted with negatives coming before positives.
The ordering will be the same for any numerical coefficients as long as
the assumptions tested are correct, otherwise the ordering will not be
sorted (but will be canonical).
"""
n = f.degree()
a, b = f.nth(n), f.nth(0)
base = -cancel(b/a)
alpha = root(base, n)
if alpha.is_number:
alpha = alpha.expand(complex=True)
# define some parameters that will allow us to order the roots.
# If the domain is ZZ this is guaranteed to return roots sorted
# with reals before non-real roots and non-real sorted according
# to real part and imaginary part, e.g. -1, 1, -1 + I, 2 - I
neg = base.is_negative
even = n % 2 == 0
if neg:
if even == True and (base + 1).is_positive:
big = True
else:
big = False
# get the indices in the right order so the computed
# roots will be sorted when the domain is ZZ
ks = []
imax = n//2
if even:
ks.append(imax)
imax -= 1
if not neg:
ks.append(0)
for i in range(imax, 0, -1):
if neg:
ks.extend([i, -i])
else:
ks.extend([-i, i])
if neg:
ks.append(0)
if big:
for i in range(0, len(ks), 2):
pair = ks[i: i + 2]
pair = list(reversed(pair))
# compute the roots
roots, d = [], 2*I*pi/n
for k in ks:
zeta = exp(k*d).expand(complex=True)
roots.append((alpha*zeta).expand(power_base=False))
return roots
def _inv_totient_estimate(m):
"""
Find ``(L, U)`` such that ``L <= phi^-1(m) <= U``.
Examples
========
>>> from sympy.polys.polyroots import _inv_totient_estimate
>>> _inv_totient_estimate(192)
(192, 840)
>>> _inv_totient_estimate(400)
(400, 1750)
"""
primes = [ d + 1 for d in divisors(m) if isprime(d + 1) ]
a, b = 1, 1
for p in primes:
a *= p
b *= p - 1
L = m
U = int(math.ceil(m*(float(a)/b)))
P = p = 2
primes = []
while P <= U:
p = nextprime(p)
primes.append(p)
P *= p
P //= p
b = 1
for p in primes[:-1]:
b *= p - 1
U = int(math.ceil(m*(float(P)/b)))
return L, U
def roots_cyclotomic(f, factor=False):
"""Compute roots of cyclotomic polynomials. """
L, U = _inv_totient_estimate(f.degree())
for n in range(L, U + 1):
g = cyclotomic_poly(n, f.gen, polys=True)
if f.expr == g.expr:
break
else: # pragma: no cover
raise RuntimeError("failed to find index of a cyclotomic polynomial")
roots = []
if not factor:
# get the indices in the right order so the computed
# roots will be sorted
h = n//2
ks = [i for i in range(1, n + 1) if igcd(i, n) == 1]
ks.sort(key=lambda x: (x, -1) if x <= h else (abs(x - n), 1))
d = 2*I*pi/n
for k in reversed(ks):
roots.append(exp(k*d).expand(complex=True))
else:
g = Poly(f, extension=root(-1, n))
for h, _ in ordered(g.factor_list()[1]):
roots.append(-h.TC())
return roots
def roots_quintic(f):
"""
Calculate exact roots of a solvable quintic
"""
result = []
coeff_5, coeff_4, p, q, r, s = f.all_coeffs()
# Eqn must be of the form x^5 + px^3 + qx^2 + rx + s
if coeff_4:
return result
if coeff_5 != 1:
l = [p/coeff_5, q/coeff_5, r/coeff_5, s/coeff_5]
if not all(coeff.is_Rational for coeff in l):
return result
f = Poly(f/coeff_5)
elif not all(coeff.is_Rational for coeff in (p, q, r, s)):
return result
quintic = PolyQuintic(f)
# Eqn standardized. Algo for solving starts here
if not f.is_irreducible:
return result
f20 = quintic.f20
# Check if f20 has linear factors over domain Z
if f20.is_irreducible:
return result
# Now, we know that f is solvable
for _factor in f20.factor_list()[1]:
if _factor[0].is_linear:
theta = _factor[0].root(0)
break
d = discriminant(f)
delta = sqrt(d)
# zeta = a fifth root of unity
zeta1, zeta2, zeta3, zeta4 = quintic.zeta
T = quintic.T(theta, d)
tol = S(1e-10)
alpha = T[1] + T[2]*delta
alpha_bar = T[1] - T[2]*delta
beta = T[3] + T[4]*delta
beta_bar = T[3] - T[4]*delta
disc = alpha**2 - 4*beta
disc_bar = alpha_bar**2 - 4*beta_bar
l0 = quintic.l0(theta)
Stwo = S(2)
l1 = _quintic_simplify((-alpha + sqrt(disc)) / Stwo)
l4 = _quintic_simplify((-alpha - sqrt(disc)) / Stwo)
l2 = _quintic_simplify((-alpha_bar + sqrt(disc_bar)) / Stwo)
l3 = _quintic_simplify((-alpha_bar - sqrt(disc_bar)) / Stwo)
order = quintic.order(theta, d)
test = (order*delta.n()) - ( (l1.n() - l4.n())*(l2.n() - l3.n()) )
# Comparing floats
if not comp(test, 0, tol):
l2, l3 = l3, l2
# Now we have correct order of l's
R1 = l0 + l1*zeta1 + l2*zeta2 + l3*zeta3 + l4*zeta4
R2 = l0 + l3*zeta1 + l1*zeta2 + l4*zeta3 + l2*zeta4
R3 = l0 + l2*zeta1 + l4*zeta2 + l1*zeta3 + l3*zeta4
R4 = l0 + l4*zeta1 + l3*zeta2 + l2*zeta3 + l1*zeta4
Res = [None, [None]*5, [None]*5, [None]*5, [None]*5]
Res_n = [None, [None]*5, [None]*5, [None]*5, [None]*5]
# Simplifying improves performance a lot for exact expressions
R1 = _quintic_simplify(R1)
R2 = _quintic_simplify(R2)
R3 = _quintic_simplify(R3)
R4 = _quintic_simplify(R4)
# hard-coded results for [factor(i) for i in _vsolve(x**5 - a - I*b, x)]
x0 = z**(S(1)/5)
x1 = sqrt(2)
x2 = sqrt(5)
x3 = sqrt(5 - x2)
x4 = I*x2
x5 = x4 + I
x6 = I*x0/4
x7 = x1*sqrt(x2 + 5)
sol = [x0, -x6*(x1*x3 - x5), x6*(x1*x3 + x5), -x6*(x4 + x7 - I), x6*(-x4 + x7 + I)]
R1 = R1.as_real_imag()
R2 = R2.as_real_imag()
R3 = R3.as_real_imag()
R4 = R4.as_real_imag()
for i, s in enumerate(sol):
Res[1][i] = _quintic_simplify(s.xreplace({z: R1[0] + I*R1[1]}))
Res[2][i] = _quintic_simplify(s.xreplace({z: R2[0] + I*R2[1]}))
Res[3][i] = _quintic_simplify(s.xreplace({z: R3[0] + I*R3[1]}))
Res[4][i] = _quintic_simplify(s.xreplace({z: R4[0] + I*R4[1]}))
for i in range(1, 5):
for j in range(5):
Res_n[i][j] = Res[i][j].n()
Res[i][j] = _quintic_simplify(Res[i][j])
r1 = Res[1][0]
r1_n = Res_n[1][0]
for i in range(5):
if comp(im(r1_n*Res_n[4][i]), 0, tol):
r4 = Res[4][i]
break
# Now we have various Res values. Each will be a list of five
# values. We have to pick one r value from those five for each Res
u, v = quintic.uv(theta, d)
testplus = (u + v*delta*sqrt(5)).n()
testminus = (u - v*delta*sqrt(5)).n()
# Evaluated numbers suffixed with _n
# We will use evaluated numbers for calculation. Much faster.
r4_n = r4.n()
r2 = r3 = None
for i in range(5):
r2temp_n = Res_n[2][i]
for j in range(5):
# Again storing away the exact number and using
# evaluated numbers in computations
r3temp_n = Res_n[3][j]
if (comp((r1_n*r2temp_n**2 + r4_n*r3temp_n**2 - testplus).n(), 0, tol) and
comp((r3temp_n*r1_n**2 + r2temp_n*r4_n**2 - testminus).n(), 0, tol)):
r2 = Res[2][i]
r3 = Res[3][j]
break
if r2:
break
else:
return [] # fall back to normal solve
# Now, we have r's so we can get roots
x1 = (r1 + r2 + r3 + r4)/5
x2 = (r1*zeta4 + r2*zeta3 + r3*zeta2 + r4*zeta1)/5
x3 = (r1*zeta3 + r2*zeta1 + r3*zeta4 + r4*zeta2)/5
x4 = (r1*zeta2 + r2*zeta4 + r3*zeta1 + r4*zeta3)/5
x5 = (r1*zeta1 + r2*zeta2 + r3*zeta3 + r4*zeta4)/5
result = [x1, x2, x3, x4, x5]
# Now check if solutions are distinct
saw = set()
for r in result:
r = r.n(2)
if r in saw:
# Roots were identical. Abort, return []
# and fall back to usual solve
return []
saw.add(r)
return result
def _quintic_simplify(expr):
from sympy.simplify.simplify import powsimp
expr = powsimp(expr)
expr = cancel(expr)
return together(expr)
def _integer_basis(poly):
"""Compute coefficient basis for a polynomial over integers.
Returns the integer ``div`` such that substituting ``x = div*y``
``p(x) = m*q(y)`` where the coefficients of ``q`` are smaller
than those of ``p``.
For example ``x**5 + 512*x + 1024 = 0``
with ``div = 4`` becomes ``y**5 + 2*y + 1 = 0``
Returns the integer ``div`` or ``None`` if there is no possible scaling.
Examples
========
>>> from sympy.polys import Poly
>>> from sympy.abc import x
>>> from sympy.polys.polyroots import _integer_basis
>>> p = Poly(x**5 + 512*x + 1024, x, domain='ZZ')
>>> _integer_basis(p)
4
"""
monoms, coeffs = list(zip(*poly.terms()))
monoms, = list(zip(*monoms))
coeffs = list(map(abs, coeffs))
if coeffs[0] < coeffs[-1]:
coeffs = list(reversed(coeffs))
n = monoms[0]
monoms = [n - i for i in reversed(monoms)]
else:
return None
monoms = monoms[:-1]
coeffs = coeffs[:-1]
# Special case for two-term polynominals
if len(monoms) == 1:
r = Pow(coeffs[0], S.One/monoms[0])
if r.is_Integer:
return int(r)
else:
return None
divs = reversed(divisors(gcd_list(coeffs))[1:])
try:
div = next(divs)
except StopIteration:
return None
while True:
for monom, coeff in zip(monoms, coeffs):
if coeff % div**monom != 0:
try:
div = next(divs)
except StopIteration:
return None
else:
break
else:
return div
def preprocess_roots(poly):
"""Try to get rid of symbolic coefficients from ``poly``. """
coeff = S.One
poly_func = poly.func
try:
_, poly = poly.clear_denoms(convert=True)
except DomainError:
return coeff, poly
poly = poly.primitive()[1]
poly = poly.retract()
# TODO: This is fragile. Figure out how to make this independent of construct_domain().
if poly.get_domain().is_Poly and all(c.is_term for c in poly.rep.coeffs()):
poly = poly.inject()
strips = list(zip(*poly.monoms()))
gens = list(poly.gens[1:])
base, strips = strips[0], strips[1:]
for gen, strip in zip(list(gens), strips):
reverse = False
if strip[0] < strip[-1]:
strip = reversed(strip)
reverse = True
ratio = None
for a, b in zip(base, strip):
if not a and not b:
continue
elif not a or not b:
break
elif b % a != 0:
break
else:
_ratio = b // a
if ratio is None:
ratio = _ratio
elif ratio != _ratio:
break
else:
if reverse:
ratio = -ratio
poly = poly.eval(gen, 1)
coeff *= gen**(-ratio)
gens.remove(gen)
if gens:
poly = poly.eject(*gens)
if poly.is_univariate and poly.get_domain().is_ZZ:
basis = _integer_basis(poly)
if basis is not None:
n = poly.degree()
def func(k, coeff):
return coeff//basis**(n - k[0])
poly = poly.termwise(func)
coeff *= basis
if not isinstance(poly, poly_func):
poly = poly_func(poly)
return coeff, poly
@public
def roots(f, *gens,
auto=True,
cubics=True,
trig=False,
quartics=True,
quintics=False,
multiple=False,
filter=None,
predicate=None,
strict=False,
**flags):
"""
Computes symbolic roots of a univariate polynomial.
Given a univariate polynomial f with symbolic coefficients (or
a list of the polynomial's coefficients), returns a dictionary
with its roots and their multiplicities.
Only roots expressible via radicals will be returned. To get
a complete set of roots use RootOf class or numerical methods
instead. By default cubic and quartic formulas are used in
the algorithm. To disable them because of unreadable output
set ``cubics=False`` or ``quartics=False`` respectively. If cubic
roots are real but are expressed in terms of complex numbers
(casus irreducibilis [1]) the ``trig`` flag can be set to True to
have the solutions returned in terms of cosine and inverse cosine
functions.
To get roots from a specific domain set the ``filter`` flag with
one of the following specifiers: Z, Q, R, I, C. By default all
roots are returned (this is equivalent to setting ``filter='C'``).
By default a dictionary is returned giving a compact result in
case of multiple roots. However to get a list containing all
those roots set the ``multiple`` flag to True; the list will
have identical roots appearing next to each other in the result.
(For a given Poly, the all_roots method will give the roots in
sorted numerical order.)
If the ``strict`` flag is True, ``UnsolvableFactorError`` will be
raised if the roots found are known to be incomplete (because
some roots are not expressible in radicals).
Examples
========
>>> from sympy import Poly, roots, degree
>>> from sympy.abc import x, y
>>> roots(x**2 - 1, x)
{-1: 1, 1: 1}
>>> p = Poly(x**2-1, x)
>>> roots(p)
{-1: 1, 1: 1}
>>> p = Poly(x**2-y, x, y)
>>> roots(Poly(p, x))
{-sqrt(y): 1, sqrt(y): 1}
>>> roots(x**2 - y, x)
{-sqrt(y): 1, sqrt(y): 1}
>>> roots([1, 0, -1])
{-1: 1, 1: 1}
``roots`` will only return roots expressible in radicals. If
the given polynomial has some or all of its roots inexpressible in
radicals, the result of ``roots`` will be incomplete or empty
respectively.
Example where result is incomplete:
>>> roots((x-1)*(x**5-x+1), x)
{1: 1}
In this case, the polynomial has an unsolvable quintic factor
whose roots cannot be expressed by radicals. The polynomial has a
rational root (due to the factor `(x-1)`), which is returned since
``roots`` always finds all rational roots.
Example where result is empty:
>>> roots(x**7-3*x**2+1, x)
{}
Here, the polynomial has no roots expressible in radicals, so
``roots`` returns an empty dictionary.
The result produced by ``roots`` is complete if and only if the
sum of the multiplicity of each root is equal to the degree of
the polynomial. If strict=True, UnsolvableFactorError will be
raised if the result is incomplete.
The result can be be checked for completeness as follows:
>>> f = x**3-2*x**2+1
>>> sum(roots(f, x).values()) == degree(f, x)
True
>>> f = (x-1)*(x**5-x+1)
>>> sum(roots(f, x).values()) == degree(f, x)
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
"""
from sympy.polys.polytools import to_rational_coeffs
flags = dict(flags)
if isinstance(f, list):
if gens:
raise ValueError('redundant generators given')
x = Dummy('x')
poly, i = {}, len(f) - 1
for coeff in f:
poly[i], i = sympify(coeff), i - 1
f = Poly(poly, x, field=True)
else:
try:
F = Poly(f, *gens, **flags)
if not isinstance(f, Poly) and not F.gen.is_Symbol:
raise PolynomialError("generator must be a Symbol")
f = F
except GeneratorsNeeded:
if multiple:
return []
else:
return {}
else:
n = f.degree()
if f.length() == 2 and n > 2:
# check for foo**n in constant if dep is c*gen**m
con, dep = f.as_expr().as_independent(*f.gens)
fcon = -(-con).factor()
if fcon != con:
con = fcon
bases = []
for i in Mul.make_args(con):
if i.is_Pow:
b, e = i.as_base_exp()
if e.is_Integer and b.is_Add:
bases.append((b, Dummy(positive=True)))
if bases:
rv = roots(Poly((dep + con).xreplace(dict(bases)),
*f.gens), *F.gens,
auto=auto,
cubics=cubics,
trig=trig,
quartics=quartics,
quintics=quintics,
multiple=multiple,
filter=filter,
predicate=predicate,
**flags)
return {factor_terms(k.xreplace(
{v: k for k, v in bases})
): v for k, v in rv.items()}
if f.is_multivariate:
raise PolynomialError('multivariate polynomials are not supported')
def _update_dict(result, zeros, currentroot, k):
if currentroot == S.Zero:
if S.Zero in zeros:
zeros[S.Zero] += k
else:
zeros[S.Zero] = k
if currentroot in result:
result[currentroot] += k
else:
result[currentroot] = k
def _try_decompose(f):
"""Find roots using functional decomposition. """
factors, roots = f.decompose(), []
for currentroot in _try_heuristics(factors[0]):
roots.append(currentroot)
for currentfactor in factors[1:]:
previous, roots = list(roots), []
for currentroot in previous:
g = currentfactor - Poly(currentroot, f.gen)
for currentroot in _try_heuristics(g):
roots.append(currentroot)
return roots
def _try_heuristics(f):
"""Find roots using formulas and some tricks. """
if f.is_ground:
return []
if f.is_monomial:
return [S.Zero]*f.degree()
if f.length() == 2:
if f.degree() == 1:
return list(map(cancel, roots_linear(f)))
else:
return roots_binomial(f)
result = []
for i in [-1, 1]:
if not f.eval(i):
f = f.quo(Poly(f.gen - i, f.gen))
result.append(i)
break
n = f.degree()
if n == 1:
result += list(map(cancel, roots_linear(f)))
elif n == 2:
result += list(map(cancel, roots_quadratic(f)))
elif f.is_cyclotomic:
result += roots_cyclotomic(f)
elif n == 3 and cubics:
result += roots_cubic(f, trig=trig)
elif n == 4 and quartics:
result += roots_quartic(f)
elif n == 5 and quintics:
result += roots_quintic(f)
return result
# Convert the generators to symbols
dumgens = symbols('x:%d' % len(f.gens), cls=Dummy)
f = f.per(f.rep, dumgens)
(k,), f = f.terms_gcd()
if not k:
zeros = {}
else:
zeros = {S.Zero: k}
coeff, f = preprocess_roots(f)
if auto and f.get_domain().is_Ring:
f = f.to_field()
# Use EX instead of ZZ_I or QQ_I
if f.get_domain().is_QQ_I:
f = f.per(f.rep.convert(EX))
rescale_x = None
translate_x = None
result = {}
if not f.is_ground:
dom = f.get_domain()
if not dom.is_Exact and dom.is_Numerical:
for r in f.nroots():
_update_dict(result, zeros, r, 1)
elif f.degree() == 1:
_update_dict(result, zeros, roots_linear(f)[0], 1)
elif f.length() == 2:
roots_fun = roots_quadratic if f.degree() == 2 else roots_binomial
for r in roots_fun(f):
_update_dict(result, zeros, r, 1)
else:
_, factors = Poly(f.as_expr()).factor_list()
if len(factors) == 1 and f.degree() == 2:
for r in roots_quadratic(f):
_update_dict(result, zeros, r, 1)
else:
if len(factors) == 1 and factors[0][1] == 1:
if f.get_domain().is_EX:
res = to_rational_coeffs(f)
if res:
if res[0] is None:
translate_x, f = res[2:]
else:
rescale_x, f = res[1], res[-1]
result = roots(f)
if not result:
for currentroot in _try_decompose(f):
_update_dict(result, zeros, currentroot, 1)
else:
for r in _try_heuristics(f):
_update_dict(result, zeros, r, 1)
else:
for currentroot in _try_decompose(f):
_update_dict(result, zeros, currentroot, 1)
else:
for currentfactor, k in factors:
for r in _try_heuristics(Poly(currentfactor, f.gen, field=True)):
_update_dict(result, zeros, r, k)
if coeff is not S.One:
_result, result, = result, {}
for currentroot, k in _result.items():
result[coeff*currentroot] = k
if filter not in [None, 'C']:
handlers = {
'Z': lambda r: r.is_Integer,
'Q': lambda r: r.is_Rational,
'R': lambda r: all(a.is_real for a in r.as_numer_denom()),
'I': lambda r: r.is_imaginary,
}
try:
query = handlers[filter]
except KeyError:
raise ValueError("Invalid filter: %s" % filter)
for zero in dict(result).keys():
if not query(zero):
del result[zero]
if predicate is not None:
for zero in dict(result).keys():
if not predicate(zero):
del result[zero]
if rescale_x:
result1 = {}
for k, v in result.items():
result1[k*rescale_x] = v
result = result1
if translate_x:
result1 = {}
for k, v in result.items():
result1[k + translate_x] = v
result = result1
# adding zero roots after non-trivial roots have been translated
result.update(zeros)
if strict and sum(result.values()) < f.degree():
raise UnsolvableFactorError(filldedent('''
Strict mode: some factors cannot be solved in radicals, so
a complete list of solutions cannot be returned. Call
roots with strict=False to get solutions expressible in
radicals (if there are any).
'''))
if not multiple:
return result
else:
zeros = []
for zero in ordered(result):
zeros.extend([zero]*result[zero])
return zeros
def root_factors(f, *gens, filter=None, **args):
"""
Returns all factors of a univariate polynomial.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.polys.polyroots import root_factors
>>> root_factors(x**2 - y, x)
[x - sqrt(y), x + sqrt(y)]
"""
args = dict(args)
F = Poly(f, *gens, **args)
if not F.is_Poly:
return [f]
if F.is_multivariate:
raise ValueError('multivariate polynomials are not supported')
x = F.gens[0]
zeros = roots(F, filter=filter)
if not zeros:
factors = [F]
else:
factors, N = [], 0
for r, n in ordered(zeros.items()):
factors, N = factors + [Poly(x - r, x)]*n, N + n
if N < F.degree():
G = reduce(lambda p, q: p*q, factors)
factors.append(F.quo(G))
if not isinstance(f, Poly):
factors = [ f.as_expr() for f in factors ]
return factors
|
5800929dbcaa3b329e710c27fac72814f19a641cba7ce665136afaca69705ff0 | """Parabolic geometrical entity.
Contains
* Parabola
"""
from sympy.core import S
from sympy.core.sorting import ordered
from sympy.core.symbol import _symbol, symbols
from sympy.geometry.entity import GeometryEntity, GeometrySet
from sympy.geometry.point import Point, Point2D
from sympy.geometry.line import Line, Line2D, Ray2D, Segment2D, LinearEntity3D
from sympy.geometry.ellipse import Ellipse
from sympy.functions import sign
from sympy.simplify import simplify
from sympy.solvers.solvers import solve
class Parabola(GeometrySet):
"""A parabolic GeometryEntity.
A parabola is declared with a point, that is called 'focus', and
a line, that is called 'directrix'.
Only vertical or horizontal parabolas are currently supported.
Parameters
==========
focus : Point
Default value is Point(0, 0)
directrix : Line
Attributes
==========
focus
directrix
axis of symmetry
focal length
p parameter
vertex
eccentricity
Raises
======
ValueError
When `focus` is not a two dimensional point.
When `focus` is a point of directrix.
NotImplementedError
When `directrix` is neither horizontal nor vertical.
Examples
========
>>> from sympy import Parabola, Point, Line
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7,8)))
>>> p1.focus
Point2D(0, 0)
>>> p1.directrix
Line2D(Point2D(5, 8), Point2D(7, 8))
"""
def __new__(cls, focus=None, directrix=None, **kwargs):
if focus:
focus = Point(focus, dim=2)
else:
focus = Point(0, 0)
directrix = Line(directrix)
if directrix.contains(focus):
raise ValueError('The focus must not be a point of directrix')
return GeometryEntity.__new__(cls, focus, directrix, **kwargs)
@property
def ambient_dimension(self):
"""Returns the ambient dimension of parabola.
Returns
=======
ambient_dimension : integer
Examples
========
>>> from sympy import Parabola, Point, Line
>>> f1 = Point(0, 0)
>>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8)))
>>> p1.ambient_dimension
2
"""
return 2
@property
def axis_of_symmetry(self):
"""Return the axis of symmetry of the parabola: a line
perpendicular to the directrix passing through the focus.
Returns
=======
axis_of_symmetry : Line
See Also
========
sympy.geometry.line.Line
Examples
========
>>> from sympy import Parabola, Point, Line
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
>>> p1.axis_of_symmetry
Line2D(Point2D(0, 0), Point2D(0, 1))
"""
return self.directrix.perpendicular_line(self.focus)
@property
def directrix(self):
"""The directrix of the parabola.
Returns
=======
directrix : Line
See Also
========
sympy.geometry.line.Line
Examples
========
>>> from sympy import Parabola, Point, Line
>>> l1 = Line(Point(5, 8), Point(7, 8))
>>> p1 = Parabola(Point(0, 0), l1)
>>> p1.directrix
Line2D(Point2D(5, 8), Point2D(7, 8))
"""
return self.args[1]
@property
def eccentricity(self):
"""The eccentricity of the parabola.
Returns
=======
eccentricity : number
A parabola may also be characterized as a conic section with an
eccentricity of 1. As a consequence of this, all parabolas are
similar, meaning that while they can be different sizes,
they are all the same shape.
See Also
========
https://en.wikipedia.org/wiki/Parabola
Examples
========
>>> from sympy import Parabola, Point, Line
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
>>> p1.eccentricity
1
Notes
-----
The eccentricity for every Parabola is 1 by definition.
"""
return S.One
def equation(self, x='x', y='y'):
"""The equation of the parabola.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
Returns
=======
equation : SymPy expression
Examples
========
>>> from sympy import Parabola, Point, Line
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
>>> p1.equation()
-x**2 - 16*y + 64
>>> p1.equation('f')
-f**2 - 16*y + 64
>>> p1.equation(y='z')
-x**2 - 16*z + 64
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
m = self.directrix.slope
if m is S.Infinity:
t1 = 4 * (self.p_parameter) * (x - self.vertex.x)
t2 = (y - self.vertex.y)**2
elif m == 0:
t1 = 4 * (self.p_parameter) * (y - self.vertex.y)
t2 = (x - self.vertex.x)**2
else:
a, b = self.focus
c, d = self.directrix.coefficients[:2]
t1 = (x - a)**2 + (y - b)**2
t2 = self.directrix.equation(x, y)**2/(c**2 + d**2)
return t1 - t2
@property
def focal_length(self):
"""The focal length of the parabola.
Returns
=======
focal_lenght : number or symbolic expression
Notes
=====
The distance between the vertex and the focus
(or the vertex and directrix), measured along the axis
of symmetry, is the "focal length".
See Also
========
https://en.wikipedia.org/wiki/Parabola
Examples
========
>>> from sympy import Parabola, Point, Line
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
>>> p1.focal_length
4
"""
distance = self.directrix.distance(self.focus)
focal_length = distance/2
return focal_length
@property
def focus(self):
"""The focus of the parabola.
Returns
=======
focus : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Parabola, Point, Line
>>> f1 = Point(0, 0)
>>> p1 = Parabola(f1, Line(Point(5, 8), Point(7, 8)))
>>> p1.focus
Point2D(0, 0)
"""
return self.args[0]
def intersection(self, o):
"""The intersection of the parabola and another geometrical entity `o`.
Parameters
==========
o : GeometryEntity, LinearEntity
Returns
=======
intersection : list of GeometryEntity objects
Examples
========
>>> from sympy import Parabola, Point, Ellipse, Line, Segment
>>> p1 = Point(0,0)
>>> l1 = Line(Point(1, -2), Point(-1,-2))
>>> parabola1 = Parabola(p1, l1)
>>> parabola1.intersection(Ellipse(Point(0, 0), 2, 5))
[Point2D(-2, 0), Point2D(2, 0)]
>>> parabola1.intersection(Line(Point(-7, 3), Point(12, 3)))
[Point2D(-4, 3), Point2D(4, 3)]
>>> parabola1.intersection(Segment((-12, -65), (14, -68)))
[]
"""
x, y = symbols('x y', real=True)
parabola_eq = self.equation()
if isinstance(o, Parabola):
if o in self:
return [o]
else:
return list(ordered([Point(i) for i in solve(
[parabola_eq, o.equation()], [x, y], set=True)[1]]))
elif isinstance(o, Point2D):
if simplify(parabola_eq.subs([(x, o._args[0]), (y, o._args[1])])) == 0:
return [o]
else:
return []
elif isinstance(o, (Segment2D, Ray2D)):
result = solve([parabola_eq,
Line2D(o.points[0], o.points[1]).equation()],
[x, y], set=True)[1]
return list(ordered([Point2D(i) for i in result if i in o]))
elif isinstance(o, (Line2D, Ellipse)):
return list(ordered([Point2D(i) for i in solve(
[parabola_eq, o.equation()], [x, y], set=True)[1]]))
elif isinstance(o, LinearEntity3D):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Wrong type of argument were put')
@property
def p_parameter(self):
"""P is a parameter of parabola.
Returns
=======
p : number or symbolic expression
Notes
=====
The absolute value of p is the focal length. The sign on p tells
which way the parabola faces. Vertical parabolas that open up
and horizontal that open right, give a positive value for p.
Vertical parabolas that open down and horizontal that open left,
give a negative value for p.
See Also
========
http://www.sparknotes.com/math/precalc/conicsections/section2.rhtml
Examples
========
>>> from sympy import Parabola, Point, Line
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
>>> p1.p_parameter
-4
"""
m = self.directrix.slope
if m is S.Infinity:
x = self.directrix.coefficients[2]
p = sign(self.focus.args[0] + x)
elif m == 0:
y = self.directrix.coefficients[2]
p = sign(self.focus.args[1] + y)
else:
d = self.directrix.projection(self.focus)
p = sign(self.focus.x - d.x)
return p * self.focal_length
@property
def vertex(self):
"""The vertex of the parabola.
Returns
=======
vertex : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Parabola, Point, Line
>>> p1 = Parabola(Point(0, 0), Line(Point(5, 8), Point(7, 8)))
>>> p1.vertex
Point2D(0, 4)
"""
focus = self.focus
m = self.directrix.slope
if m is S.Infinity:
vertex = Point(focus.args[0] - self.p_parameter, focus.args[1])
elif m == 0:
vertex = Point(focus.args[0], focus.args[1] - self.p_parameter)
else:
vertex = self.axis_of_symmetry.intersection(self)[0]
return vertex
|
ae599ed5a519a4f98791f07bd12fa6a0ab3263b9166cc56dee0fe5ff29a2d202 | """Elliptical geometrical entities.
Contains
* Ellipse
* Circle
"""
from sympy.core.expr import Expr
from sympy.core.relational import Eq
from sympy.core import S, pi, sympify
from sympy.core.evalf import N
from sympy.core.parameters import global_parameters
from sympy.core.logic import fuzzy_bool
from sympy.core.numbers import Rational, oo
from sympy.core.sorting import ordered
from sympy.core.symbol import Dummy, uniquely_named_symbol, _symbol
from sympy.simplify import simplify, trigsimp
from sympy.functions.elementary.miscellaneous import sqrt, Max
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.functions.special.elliptic_integrals import elliptic_e
from .entity import GeometryEntity, GeometrySet
from .exceptions import GeometryError
from .line import Line, Segment, Ray2D, Segment2D, Line2D, LinearEntity3D
from .point import Point, Point2D, Point3D
from .util import idiff, find
from sympy.polys import DomainError, Poly, PolynomialError
from sympy.polys.polyutils import _not_a_coeff, _nsort
from sympy.solvers import solve
from sympy.solvers.solveset import linear_coeffs
from sympy.utilities.misc import filldedent, func_name
from mpmath.libmp.libmpf import prec_to_dps
import random
class Ellipse(GeometrySet):
"""An elliptical GeometryEntity.
Parameters
==========
center : Point, optional
Default value is Point(0, 0)
hradius : number or SymPy expression, optional
vradius : number or SymPy expression, optional
eccentricity : number or SymPy expression, optional
Two of `hradius`, `vradius` and `eccentricity` must be supplied to
create an Ellipse. The third is derived from the two supplied.
Attributes
==========
center
hradius
vradius
area
circumference
eccentricity
periapsis
apoapsis
focus_distance
foci
Raises
======
GeometryError
When `hradius`, `vradius` and `eccentricity` are incorrectly supplied
as parameters.
TypeError
When `center` is not a Point.
See Also
========
Circle
Notes
-----
Constructed from a center and two radii, the first being the horizontal
radius (along the x-axis) and the second being the vertical radius (along
the y-axis).
When symbolic value for hradius and vradius are used, any calculation that
refers to the foci or the major or minor axis will assume that the ellipse
has its major radius on the x-axis. If this is not true then a manual
rotation is necessary.
Examples
========
>>> from sympy import Ellipse, Point, Rational
>>> e1 = Ellipse(Point(0, 0), 5, 1)
>>> e1.hradius, e1.vradius
(5, 1)
>>> e2 = Ellipse(Point(3, 1), hradius=3, eccentricity=Rational(4, 5))
>>> e2
Ellipse(Point2D(3, 1), 3, 9/5)
"""
def __contains__(self, o):
if isinstance(o, Point):
x = Dummy('x', real=True)
y = Dummy('y', real=True)
res = self.equation(x, y).subs({x: o.x, y: o.y})
return trigsimp(simplify(res)) is S.Zero
elif isinstance(o, Ellipse):
return self == o
return False
def __eq__(self, o):
"""Is the other GeometryEntity the same as this ellipse?"""
return isinstance(o, Ellipse) and (self.center == o.center and
self.hradius == o.hradius and
self.vradius == o.vradius)
def __hash__(self):
return super().__hash__()
def __new__(
cls, center=None, hradius=None, vradius=None, eccentricity=None, **kwargs):
hradius = sympify(hradius)
vradius = sympify(vradius)
if center is None:
center = Point(0, 0)
else:
if len(center) != 2:
raise ValueError('The center of "{}" must be a two dimensional point'.format(cls))
center = Point(center, dim=2)
if len(list(filter(lambda x: x is not None, (hradius, vradius, eccentricity)))) != 2:
raise ValueError(filldedent('''
Exactly two arguments of "hradius", "vradius", and
"eccentricity" must not be None.'''))
if eccentricity is not None:
eccentricity = sympify(eccentricity)
if eccentricity.is_negative:
raise GeometryError("Eccentricity of ellipse/circle should lie between [0, 1)")
elif hradius is None:
hradius = vradius / sqrt(1 - eccentricity**2)
elif vradius is None:
vradius = hradius * sqrt(1 - eccentricity**2)
if hradius == vradius:
return Circle(center, hradius, **kwargs)
if S.Zero in (hradius, vradius):
return Segment(Point(center[0] - hradius, center[1] - vradius), Point(center[0] + hradius, center[1] + vradius))
if hradius.is_real is False or vradius.is_real is False:
raise GeometryError("Invalid value encountered when computing hradius / vradius.")
return GeometryEntity.__new__(cls, center, hradius, vradius, **kwargs)
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG ellipse element for the Ellipse.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
c = N(self.center)
h, v = N(self.hradius), N(self.vradius)
return (
'<ellipse fill="{1}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" cx="{2}" cy="{3}" rx="{4}" ry="{5}"/>'
).format(2. * scale_factor, fill_color, c.x, c.y, h, v)
@property
def ambient_dimension(self):
return 2
@property
def apoapsis(self):
"""The apoapsis of the ellipse.
The greatest distance between the focus and the contour.
Returns
=======
apoapsis : number
See Also
========
periapsis : Returns shortest distance between foci and contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.apoapsis
2*sqrt(2) + 3
"""
return self.major * (1 + self.eccentricity)
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the functions.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.arbitrary_point()
Point2D(3*cos(t), 2*sin(t))
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError(filldedent('Symbol %s already appears in object '
'and cannot be used as a parameter.' % t.name))
return Point(self.center.x + self.hradius*cos(t),
self.center.y + self.vradius*sin(t))
@property
def area(self):
"""The area of the ellipse.
Returns
=======
area : number
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.area
3*pi
"""
return simplify(S.Pi * self.hradius * self.vradius)
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
h, v = self.hradius, self.vradius
return (self.center.x - h, self.center.y - v, self.center.x + h, self.center.y + v)
@property
def center(self):
"""The center of the ellipse.
Returns
=======
center : number
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.center
Point2D(0, 0)
"""
return self.args[0]
@property
def circumference(self):
"""The circumference of the ellipse.
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.circumference
12*elliptic_e(8/9)
"""
if self.eccentricity == 1:
# degenerate
return 4*self.major
elif self.eccentricity == 0:
# circle
return 2*pi*self.hradius
else:
return 4*self.major*elliptic_e(self.eccentricity**2)
@property
def eccentricity(self):
"""The eccentricity of the ellipse.
Returns
=======
eccentricity : number
Examples
========
>>> from sympy import Point, Ellipse, sqrt
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, sqrt(2))
>>> e1.eccentricity
sqrt(7)/3
"""
return self.focus_distance / self.major
def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.
Notes
-----
Being on the border of self is considered False.
Parameters
==========
p : Point
Returns
=======
encloses_point : True, False or None
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Ellipse, S
>>> from sympy.abc import t
>>> e = Ellipse((0, 0), 3, 2)
>>> e.encloses_point((0, 0))
True
>>> e.encloses_point(e.arbitrary_point(t).subs(t, S.Half))
False
>>> e.encloses_point((4, 0))
False
"""
p = Point(p, dim=2)
if p in self:
return False
if len(self.foci) == 2:
# if the combined distance from the foci to p (h1 + h2) is less
# than the combined distance from the foci to the minor axis
# (which is the same as the major axis length) then p is inside
# the ellipse
h1, h2 = [f.distance(p) for f in self.foci]
test = 2*self.major - (h1 + h2)
else:
test = self.radius - self.center.distance(p)
return fuzzy_bool(test.is_positive)
def equation(self, x='x', y='y', _slope=None):
"""
Returns the equation of an ellipse aligned with the x and y axes;
when slope is given, the equation returned corresponds to an ellipse
with a major axis having that slope.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
_slope : Expr, optional
The slope of the major axis. Ignored when 'None'.
Returns
=======
equation : SymPy expression
See Also
========
arbitrary_point : Returns parameterized point on ellipse
Examples
========
>>> from sympy import Point, Ellipse, pi
>>> from sympy.abc import x, y
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> eq1 = e1.equation(x, y); eq1
y**2/4 + (x/3 - 1/3)**2 - 1
>>> eq2 = e1.equation(x, y, _slope=1); eq2
(-x + y + 1)**2/8 + (x + y - 1)**2/18 - 1
A point on e1 satisfies eq1. Let's use one on the x-axis:
>>> p1 = e1.center + Point(e1.major, 0)
>>> assert eq1.subs(x, p1.x).subs(y, p1.y) == 0
When rotated the same as the rotated ellipse, about the center
point of the ellipse, it will satisfy the rotated ellipse's
equation, too:
>>> r1 = p1.rotate(pi/4, e1.center)
>>> assert eq2.subs(x, r1.x).subs(y, r1.y) == 0
References
==========
.. [1] https://math.stackexchange.com/questions/108270/what-is-the-equation-of-an-ellipse-that-is-not-aligned-with-the-axis
.. [2] https://en.wikipedia.org/wiki/Ellipse#Equation_of_a_shifted_ellipse
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
dx = x - self.center.x
dy = y - self.center.y
if _slope is not None:
L = (dy - _slope*dx)**2
l = (_slope*dy + dx)**2
h = 1 + _slope**2
b = h*self.major**2
a = h*self.minor**2
return l/b + L/a - 1
else:
t1 = (dx/self.hradius)**2
t2 = (dy/self.vradius)**2
return t1 + t2 - 1
def evolute(self, x='x', y='y'):
"""The equation of evolute of the ellipse.
Parameters
==========
x : str, optional
Label for the x-axis. Default value is 'x'.
y : str, optional
Label for the y-axis. Default value is 'y'.
Returns
=======
equation : SymPy expression
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(1, 0), 3, 2)
>>> e1.evolute()
2**(2/3)*y**(2/3) + (3*x - 3)**(2/3) - 5**(2/3)
"""
if len(self.args) != 3:
raise NotImplementedError('Evolute of arbitrary Ellipse is not supported.')
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (self.hradius*(x - self.center.x))**Rational(2, 3)
t2 = (self.vradius*(y - self.center.y))**Rational(2, 3)
return t1 + t2 - (self.hradius**2 - self.vradius**2)**Rational(2, 3)
@property
def foci(self):
"""The foci of the ellipse.
Notes
-----
The foci can only be calculated if the major/minor axes are known.
Raises
======
ValueError
When the major and minor axis cannot be determined.
See Also
========
sympy.geometry.point.Point
focus_distance : Returns the distance between focus and center
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.foci
(Point2D(-2*sqrt(2), 0), Point2D(2*sqrt(2), 0))
"""
c = self.center
hr, vr = self.hradius, self.vradius
if hr == vr:
return (c, c)
# calculate focus distance manually, since focus_distance calls this
# routine
fd = sqrt(self.major**2 - self.minor**2)
if hr == self.minor:
# foci on the y-axis
return (c + Point(0, -fd), c + Point(0, fd))
elif hr == self.major:
# foci on the x-axis
return (c + Point(-fd, 0), c + Point(fd, 0))
@property
def focus_distance(self):
"""The focal distance of the ellipse.
The distance between the center and one focus.
Returns
=======
focus_distance : number
See Also
========
foci
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.focus_distance
2*sqrt(2)
"""
return Point.distance(self.center, self.foci[0])
@property
def hradius(self):
"""The horizontal radius of the ellipse.
Returns
=======
hradius : number
See Also
========
vradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.hradius
3
"""
return self.args[1]
def intersection(self, o):
"""The intersection of this ellipse and another geometrical entity
`o`.
Parameters
==========
o : GeometryEntity
Returns
=======
intersection : list of GeometryEntity objects
Notes
-----
Currently supports intersections with Point, Line, Segment, Ray,
Circle and Ellipse types.
See Also
========
sympy.geometry.entity.GeometryEntity
Examples
========
>>> from sympy import Ellipse, Point, Line
>>> e = Ellipse(Point(0, 0), 5, 7)
>>> e.intersection(Point(0, 0))
[]
>>> e.intersection(Point(5, 0))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(0,0), Point(0, 1)))
[Point2D(0, -7), Point2D(0, 7)]
>>> e.intersection(Line(Point(5,0), Point(5, 1)))
[Point2D(5, 0)]
>>> e.intersection(Line(Point(6,0), Point(6, 1)))
[]
>>> e = Ellipse(Point(-1, 0), 4, 3)
>>> e.intersection(Ellipse(Point(1, 0), 4, 3))
[Point2D(0, -3*sqrt(15)/4), Point2D(0, 3*sqrt(15)/4)]
>>> e.intersection(Ellipse(Point(5, 0), 4, 3))
[Point2D(2, -3*sqrt(7)/4), Point2D(2, 3*sqrt(7)/4)]
>>> e.intersection(Ellipse(Point(100500, 0), 4, 3))
[]
>>> e.intersection(Ellipse(Point(0, 0), 3, 4))
[Point2D(3, 0), Point2D(-363/175, -48*sqrt(111)/175), Point2D(-363/175, 48*sqrt(111)/175)]
>>> e.intersection(Ellipse(Point(-1, 0), 3, 4))
[Point2D(-17/5, -12/5), Point2D(-17/5, 12/5), Point2D(7/5, -12/5), Point2D(7/5, 12/5)]
"""
# TODO: Replace solve with nonlinsolve, when nonlinsolve will be able to solve in real domain
x = Dummy('x', real=True)
y = Dummy('y', real=True)
if isinstance(o, Point):
if o in self:
return [o]
else:
return []
elif isinstance(o, (Segment2D, Ray2D)):
ellipse_equation = self.equation(x, y)
result = solve([ellipse_equation, Line(
o.points[0], o.points[1]).equation(x, y)], [x, y],
set=True)[1]
return list(ordered([Point(i) for i in result if i in o]))
elif isinstance(o, Polygon):
return o.intersection(self)
elif isinstance(o, (Ellipse, Line2D)):
if o == self:
return self
else:
ellipse_equation = self.equation(x, y)
return list(ordered([Point(i) for i in solve(
[ellipse_equation, o.equation(x, y)], [x, y],
set=True)[1]]))
elif isinstance(o, LinearEntity3D):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Intersection not handled for %s' % func_name(o))
def is_tangent(self, o):
"""Is `o` tangent to the ellipse?
Parameters
==========
o : GeometryEntity
An Ellipse, LinearEntity or Polygon
Raises
======
NotImplementedError
When the wrong type of argument is supplied.
Returns
=======
is_tangent: boolean
True if o is tangent to the ellipse, False otherwise.
See Also
========
tangent_lines
Examples
========
>>> from sympy import Point, Ellipse, Line
>>> p0, p1, p2 = Point(0, 0), Point(3, 0), Point(3, 3)
>>> e1 = Ellipse(p0, 3, 2)
>>> l1 = Line(p1, p2)
>>> e1.is_tangent(l1)
True
"""
if isinstance(o, Point2D):
return False
elif isinstance(o, Ellipse):
intersect = self.intersection(o)
if isinstance(intersect, Ellipse):
return True
elif intersect:
return all((self.tangent_lines(i)[0]).equals(o.tangent_lines(i)[0]) for i in intersect)
else:
return False
elif isinstance(o, Line2D):
hit = self.intersection(o)
if not hit:
return False
if len(hit) == 1:
return True
# might return None if it can't decide
return hit[0].equals(hit[1])
elif isinstance(o, Ray2D):
intersect = self.intersection(o)
if len(intersect) == 1:
return intersect[0] != o.source and not self.encloses_point(o.source)
else:
return False
elif isinstance(o, (Segment2D, Polygon)):
all_tangents = False
segments = o.sides if isinstance(o, Polygon) else [o]
for segment in segments:
intersect = self.intersection(segment)
if len(intersect) == 1:
if not any(intersect[0] in i for i in segment.points) \
and not any(self.encloses_point(i) for i in segment.points):
all_tangents = True
continue
else:
return False
else:
return all_tangents
return all_tangents
elif isinstance(o, (LinearEntity3D, Point3D)):
raise TypeError('Entity must be two dimensional, not three dimensional')
else:
raise TypeError('Is_tangent not handled for %s' % func_name(o))
@property
def major(self):
"""Longer axis of the ellipse (if it can be determined) else hradius.
Returns
=======
major : number or expression
See Also
========
hradius, vradius, minor
Examples
========
>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.major
3
>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).major
a
>>> Ellipse(p1, b, a).major
b
>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).major
m + 1
"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = b - a < 0
if o == True:
return a
elif o == False:
return b
return self.hradius
@property
def minor(self):
"""Shorter axis of the ellipse (if it can be determined) else vradius.
Returns
=======
minor : number or expression
See Also
========
hradius, vradius, major
Examples
========
>>> from sympy import Point, Ellipse, Symbol
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.minor
1
>>> a = Symbol('a')
>>> b = Symbol('b')
>>> Ellipse(p1, a, b).minor
b
>>> Ellipse(p1, b, a).minor
a
>>> m = Symbol('m')
>>> M = m + 1
>>> Ellipse(p1, m, M).minor
m
"""
ab = self.args[1:3]
if len(ab) == 1:
return ab[0]
a, b = ab
o = a - b < 0
if o == True:
return a
elif o == False:
return b
return self.vradius
def normal_lines(self, p, prec=None):
"""Normal lines between `p` and the ellipse.
Parameters
==========
p : Point
Returns
=======
normal_lines : list with 1, 2 or 4 Lines
Examples
========
>>> from sympy import Point, Ellipse
>>> e = Ellipse((0, 0), 2, 3)
>>> c = e.center
>>> e.normal_lines(c + Point(1, 0))
[Line2D(Point2D(0, 0), Point2D(1, 0))]
>>> e.normal_lines(c)
[Line2D(Point2D(0, 0), Point2D(0, 1)), Line2D(Point2D(0, 0), Point2D(1, 0))]
Off-axis points require the solution of a quartic equation. This
often leads to very large expressions that may be of little practical
use. An approximate solution of `prec` digits can be obtained by
passing in the desired value:
>>> e.normal_lines((3, 3), prec=2)
[Line2D(Point2D(-0.81, -2.7), Point2D(0.19, -1.2)),
Line2D(Point2D(1.5, -2.0), Point2D(2.5, -2.7))]
Whereas the above solution has an operation count of 12, the exact
solution has an operation count of 2020.
"""
p = Point(p, dim=2)
# XXX change True to something like self.angle == 0 if the arbitrarily
# rotated ellipse is introduced.
# https://github.com/sympy/sympy/issues/2815)
if True:
rv = []
if p.x == self.center.x:
rv.append(Line(self.center, slope=oo))
if p.y == self.center.y:
rv.append(Line(self.center, slope=0))
if rv:
# at these special orientations of p either 1 or 2 normals
# exist and we are done
return rv
# find the 4 normal points and construct lines through them with
# the corresponding slope
x, y = Dummy('x', real=True), Dummy('y', real=True)
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
norm = -1/dydx
slope = Line(p, (x, y)).slope
seq = slope - norm
# TODO: Replace solve with solveset, when this line is tested
yis = solve(seq, y)[0]
xeq = eq.subs(y, yis).as_numer_denom()[0].expand()
if len(xeq.free_symbols) == 1:
try:
# this is so much faster, it's worth a try
xsol = Poly(xeq, x).real_roots()
except (DomainError, PolynomialError, NotImplementedError):
# TODO: Replace solve with solveset, when these lines are tested
xsol = _nsort(solve(xeq, x), separated=True)[0]
points = [Point(i, solve(eq.subs(x, i), y)[0]) for i in xsol]
else:
raise NotImplementedError(
'intersections for the general ellipse are not supported')
slopes = [norm.subs(zip((x, y), pt.args)) for pt in points]
if prec is not None:
points = [pt.n(prec) for pt in points]
slopes = [i if _not_a_coeff(i) else i.n(prec) for i in slopes]
return [Line(pt, slope=s) for pt, s in zip(points, slopes)]
@property
def periapsis(self):
"""The periapsis of the ellipse.
The shortest distance between the focus and the contour.
Returns
=======
periapsis : number
See Also
========
apoapsis : Returns greatest distance between focus and contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.periapsis
3 - 2*sqrt(2)
"""
return self.major * (1 - self.eccentricity)
@property
def semilatus_rectum(self):
"""
Calculates the semi-latus rectum of the Ellipse.
Semi-latus rectum is defined as one half of the chord through a
focus parallel to the conic section directrix of a conic section.
Returns
=======
semilatus_rectum : number
See Also
========
apoapsis : Returns greatest distance between focus and contour
periapsis : The shortest distance between the focus and the contour
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.semilatus_rectum
1/3
References
==========
.. [1] http://mathworld.wolfram.com/SemilatusRectum.html
.. [2] https://en.wikipedia.org/wiki/Ellipse#Semi-latus_rectum
"""
return self.major * (1 - self.eccentricity ** 2)
def auxiliary_circle(self):
"""Returns a Circle whose diameter is the major axis of the ellipse.
Examples
========
>>> from sympy import Ellipse, Point, symbols
>>> c = Point(1, 2)
>>> Ellipse(c, 8, 7).auxiliary_circle()
Circle(Point2D(1, 2), 8)
>>> a, b = symbols('a b')
>>> Ellipse(c, a, b).auxiliary_circle()
Circle(Point2D(1, 2), Max(a, b))
"""
return Circle(self.center, Max(self.hradius, self.vradius))
def director_circle(self):
"""
Returns a Circle consisting of all points where two perpendicular
tangent lines to the ellipse cross each other.
Returns
=======
Circle
A director circle returned as a geometric object.
Examples
========
>>> from sympy import Ellipse, Point, symbols
>>> c = Point(3,8)
>>> Ellipse(c, 7, 9).director_circle()
Circle(Point2D(3, 8), sqrt(130))
>>> a, b = symbols('a b')
>>> Ellipse(c, a, b).director_circle()
Circle(Point2D(3, 8), sqrt(a**2 + b**2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Director_circle
"""
return Circle(self.center, sqrt(self.hradius**2 + self.vradius**2))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the Ellipse.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.plot_interval()
[t, -pi, pi]
"""
t = _symbol(parameter, real=True)
return [t, -S.Pi, S.Pi]
def random_point(self, seed=None):
"""A random point on the ellipse.
Returns
=======
point : Point
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.random_point() # gives some random point
Point2D(...)
>>> p1 = e1.random_point(seed=0); p1.n(2)
Point2D(2.1, 1.4)
Notes
=====
When creating a random point, one may simply replace the
parameter with a random number. When doing so, however, the
random number should be made a Rational or else the point
may not test as being in the ellipse:
>>> from sympy.abc import t
>>> from sympy import Rational
>>> arb = e1.arbitrary_point(t); arb
Point2D(3*cos(t), 2*sin(t))
>>> arb.subs(t, .1) in e1
False
>>> arb.subs(t, Rational(.1)) in e1
True
>>> arb.subs(t, Rational('.1')) in e1
True
See Also
========
sympy.geometry.point.Point
arbitrary_point : Returns parameterized point on ellipse
"""
t = _symbol('t', real=True)
x, y = self.arbitrary_point(t).args
# get a random value in [-1, 1) corresponding to cos(t)
# and confirm that it will test as being in the ellipse
if seed is not None:
rng = random.Random(seed)
else:
rng = random
# simplify this now or else the Float will turn s into a Float
r = Rational(rng.random())
c = 2*r - 1
s = sqrt(1 - c**2)
return Point(x.subs(cos(t), c), y.subs(sin(t), s))
def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
>>> from sympy import Ellipse, Line, Point
>>> Ellipse(Point(3, 4), 1, 3).reflect(Line(Point(0, -4), Point(5, 0)))
Traceback (most recent call last):
...
NotImplementedError:
General Ellipse is not supported but the equation of the reflected
Ellipse is given by the zeros of: f(x, y) = (9*x/41 + 40*y/41 +
37/41)**2 + (40*x/123 - 3*y/41 - 364/123)**2 - 1
Notes
=====
Until the general ellipse (with no axis parallel to the x-axis) is
supported a NotImplemented error is raised and the equation whose
zeros define the rotated ellipse is given.
"""
if line.slope in (0, oo):
c = self.center
c = c.reflect(line)
return self.func(c, -self.hradius, self.vradius)
else:
x, y = [uniquely_named_symbol(
name, (self, line), modify=lambda s: '_' + s, real=True)
for name in 'xy']
expr = self.equation(x, y)
p = Point(x, y).reflect(line)
result = expr.subs(zip((x, y), p.args
), simultaneous=True)
raise NotImplementedError(filldedent(
'General Ellipse is not supported but the equation '
'of the reflected Ellipse is given by the zeros of: ' +
"f(%s, %s) = %s" % (str(x), str(y), str(result))))
def rotate(self, angle=0, pt=None):
"""Rotate ``angle`` radians counterclockwise about Point ``pt``.
Note: since the general ellipse is not supported, only rotations that
are integer multiples of pi/2 are allowed.
Examples
========
>>> from sympy import Ellipse, pi
>>> Ellipse((1, 0), 2, 1).rotate(pi/2)
Ellipse(Point2D(0, 1), 1, 2)
>>> Ellipse((1, 0), 2, 1).rotate(pi)
Ellipse(Point2D(-1, 0), 2, 1)
"""
if self.hradius == self.vradius:
return self.func(self.center.rotate(angle, pt), self.hradius)
if (angle/S.Pi).is_integer:
return super().rotate(angle, pt)
if (2*angle/S.Pi).is_integer:
return self.func(self.center.rotate(angle, pt), self.vradius, self.hradius)
# XXX see https://github.com/sympy/sympy/issues/2815 for general ellipes
raise NotImplementedError('Only rotations of pi/2 are currently supported for Ellipse.')
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since it is the major and minor
axes which must be scaled and they are not GeometryEntities.
Examples
========
>>> from sympy import Ellipse
>>> Ellipse((0, 0), 2, 1).scale(2, 4)
Circle(Point2D(0, 0), 4)
>>> Ellipse((0, 0), 2, 1).scale(2)
Ellipse(Point2D(0, 0), 4, 1)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
h = self.hradius
v = self.vradius
return self.func(c.scale(x, y), hradius=h*x, vradius=v*y)
def tangent_lines(self, p):
"""Tangent lines between `p` and the ellipse.
If `p` is on the ellipse, returns the tangent line through point `p`.
Otherwise, returns the tangent line(s) from `p` to the ellipse, or
None if no tangent line is possible (e.g., `p` inside ellipse).
Parameters
==========
p : Point
Returns
=======
tangent_lines : list with 1 or 2 Lines
Raises
======
NotImplementedError
Can only find tangent lines for a point, `p`, on the ellipse.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Line
Examples
========
>>> from sympy import Point, Ellipse
>>> e1 = Ellipse(Point(0, 0), 3, 2)
>>> e1.tangent_lines(Point(3, 0))
[Line2D(Point2D(3, 0), Point2D(3, -12))]
"""
p = Point(p, dim=2)
if self.encloses_point(p):
return []
if p in self:
delta = self.center - p
rise = (self.vradius**2)*delta.x
run = -(self.hradius**2)*delta.y
p2 = Point(simplify(p.x + run),
simplify(p.y + rise))
return [Line(p, p2)]
else:
if len(self.foci) == 2:
f1, f2 = self.foci
maj = self.hradius
test = (2*maj -
Point.distance(f1, p) -
Point.distance(f2, p))
else:
test = self.radius - Point.distance(self.center, p)
if test.is_number and test.is_positive:
return []
# else p is outside the ellipse or we can't tell. In case of the
# latter, the solutions returned will only be valid if
# the point is not inside the ellipse; if it is, nan will result.
x, y = Dummy('x'), Dummy('y')
eq = self.equation(x, y)
dydx = idiff(eq, y, x)
slope = Line(p, Point(x, y)).slope
# TODO: Replace solve with solveset, when this line is tested
tangent_points = solve([slope - dydx, eq], [x, y])
# handle horizontal and vertical tangent lines
if len(tangent_points) == 1:
if tangent_points[0][
0] == p.x or tangent_points[0][1] == p.y:
return [Line(p, p + Point(1, 0)), Line(p, p + Point(0, 1))]
else:
return [Line(p, p + Point(0, 1)), Line(p, tangent_points[0])]
# others
return [Line(p, tangent_points[0]), Line(p, tangent_points[1])]
@property
def vradius(self):
"""The vertical radius of the ellipse.
Returns
=======
vradius : number
See Also
========
hradius, major, minor
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.vradius
1
"""
return self.args[2]
def second_moment_of_area(self, point=None):
"""Returns the second moment and product moment area of an ellipse.
Parameters
==========
point : Point, two-tuple of sympifiable objects, or None(default=None)
point is the point about which second moment of area is to be found.
If "point=None" it will be calculated about the axis passing through the
centroid of the ellipse.
Returns
=======
I_xx, I_yy, I_xy : number or SymPy expression
I_xx, I_yy are second moment of area of an ellise.
I_xy is product moment of area of an ellipse.
Examples
========
>>> from sympy import Point, Ellipse
>>> p1 = Point(0, 0)
>>> e1 = Ellipse(p1, 3, 1)
>>> e1.second_moment_of_area()
(3*pi/4, 27*pi/4, 0)
References
==========
.. [1] https://en.wikipedia.org/wiki/List_of_second_moments_of_area
"""
I_xx = (S.Pi*(self.hradius)*(self.vradius**3))/4
I_yy = (S.Pi*(self.hradius**3)*(self.vradius))/4
I_xy = 0
if point is None:
return I_xx, I_yy, I_xy
# parallel axis theorem
I_xx = I_xx + self.area*((point[1] - self.center.y)**2)
I_yy = I_yy + self.area*((point[0] - self.center.x)**2)
I_xy = I_xy + self.area*(point[0] - self.center.x)*(point[1] - self.center.y)
return I_xx, I_yy, I_xy
def polar_second_moment_of_area(self):
"""Returns the polar second moment of area of an Ellipse
It is a constituent of the second moment of area, linked through
the perpendicular axis theorem. While the planar second moment of
area describes an object's resistance to deflection (bending) when
subjected to a force applied to a plane parallel to the central
axis, the polar second moment of area describes an object's
resistance to deflection when subjected to a moment applied in a
plane perpendicular to the object's central axis (i.e. parallel to
the cross-section)
Examples
========
>>> from sympy import symbols, Circle, Ellipse
>>> c = Circle((5, 5), 4)
>>> c.polar_second_moment_of_area()
128*pi
>>> a, b = symbols('a, b')
>>> e = Ellipse((0, 0), a, b)
>>> e.polar_second_moment_of_area()
pi*a**3*b/4 + pi*a*b**3/4
References
==========
.. [1] https://en.wikipedia.org/wiki/Polar_moment_of_inertia
"""
second_moment = self.second_moment_of_area()
return second_moment[0] + second_moment[1]
def section_modulus(self, point=None):
"""Returns a tuple with the section modulus of an ellipse
Section modulus is a geometric property of an ellipse defined as the
ratio of second moment of area to the distance of the extreme end of
the ellipse from the centroidal axis.
Parameters
==========
point : Point, two-tuple of sympifyable objects, or None(default=None)
point is the point at which section modulus is to be found.
If "point=None" section modulus will be calculated for the
point farthest from the centroidal axis of the ellipse.
Returns
=======
S_x, S_y: numbers or SymPy expressions
S_x is the section modulus with respect to the x-axis
S_y is the section modulus with respect to the y-axis
A negative sign indicates that the section modulus is
determined for a point below the centroidal axis.
Examples
========
>>> from sympy import Symbol, Ellipse, Circle, Point2D
>>> d = Symbol('d', positive=True)
>>> c = Circle((0, 0), d/2)
>>> c.section_modulus()
(pi*d**3/32, pi*d**3/32)
>>> e = Ellipse(Point2D(0, 0), 2, 4)
>>> e.section_modulus()
(8*pi, 4*pi)
>>> e.section_modulus((2, 2))
(16*pi, 4*pi)
References
==========
.. [1] https://en.wikipedia.org/wiki/Section_modulus
"""
x_c, y_c = self.center
if point is None:
# taking x and y as maximum distances from centroid
x_min, y_min, x_max, y_max = self.bounds
y = max(y_c - y_min, y_max - y_c)
x = max(x_c - x_min, x_max - x_c)
else:
# taking x and y as distances of the given point from the center
point = Point2D(point)
y = point.y - y_c
x = point.x - x_c
second_moment = self.second_moment_of_area()
S_x = second_moment[0]/y
S_y = second_moment[1]/x
return S_x, S_y
class Circle(Ellipse):
"""A circle in space.
Constructed simply from a center and a radius, from three
non-collinear points, or the equation of a circle.
Parameters
==========
center : Point
radius : number or SymPy expression
points : sequence of three Points
equation : equation of a circle
Attributes
==========
radius (synonymous with hradius, vradius, major and minor)
circumference
equation
Raises
======
GeometryError
When the given equation is not that of a circle.
When trying to construct circle from incorrect parameters.
See Also
========
Ellipse, sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Circle, Eq
>>> from sympy.abc import x, y, a, b
A circle constructed from a center and radius:
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.hradius, c1.vradius, c1.radius
(5, 5, 5)
A circle constructed from three points:
>>> c2 = Circle(Point(0, 0), Point(1, 1), Point(1, 0))
>>> c2.hradius, c2.vradius, c2.radius, c2.center
(sqrt(2)/2, sqrt(2)/2, sqrt(2)/2, Point2D(1/2, 1/2))
A circle can be constructed from an equation in the form
`a*x**2 + by**2 + gx + hy + c = 0`, too:
>>> Circle(x**2 + y**2 - 25)
Circle(Point2D(0, 0), 5)
If the variables corresponding to x and y are named something
else, their name or symbol can be supplied:
>>> Circle(Eq(a**2 + b**2, 25), x='a', y=b)
Circle(Point2D(0, 0), 5)
"""
def __new__(cls, *args, **kwargs):
evaluate = kwargs.get('evaluate', global_parameters.evaluate)
if len(args) == 1 and isinstance(args[0], (Expr, Eq)):
x = kwargs.get('x', 'x')
y = kwargs.get('y', 'y')
equation = args[0].expand()
if isinstance(equation, Eq):
equation = equation.lhs - equation.rhs
x = find(x, equation)
y = find(y, equation)
try:
a, b, c, d, e = linear_coeffs(equation, x**2, y**2, x, y)
except ValueError:
raise GeometryError("The given equation is not that of a circle.")
if S.Zero in (a, b) or a != b:
raise GeometryError("The given equation is not that of a circle.")
center_x = -c/a/2
center_y = -d/b/2
r2 = (center_x**2) + (center_y**2) - e/a
return Circle((center_x, center_y), sqrt(r2), evaluate=evaluate)
else:
c, r = None, None
if len(args) == 3:
args = [Point(a, dim=2, evaluate=evaluate) for a in args]
t = Triangle(*args)
if not isinstance(t, Triangle):
return t
c = t.circumcenter
r = t.circumradius
elif len(args) == 2:
# Assume (center, radius) pair
c = Point(args[0], dim=2, evaluate=evaluate)
r = args[1]
# this will prohibit imaginary radius
try:
r = Point(r, 0, evaluate=evaluate).x
except ValueError:
raise GeometryError("Circle with imaginary radius is not permitted")
if not (c is None or r is None):
if r == 0:
return c
return GeometryEntity.__new__(cls, c, r, **kwargs)
raise GeometryError("Circle.__new__ received unknown arguments")
def _eval_evalf(self, prec=15, **options):
pt, r = self.args
dps = prec_to_dps(prec)
pt = pt.evalf(n=dps, **options)
r = r.evalf(n=dps, **options)
return self.func(pt, r, evaluate=False)
@property
def circumference(self):
"""The circumference of the circle.
Returns
=======
circumference : number or SymPy expression
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.circumference
12*pi
"""
return 2 * S.Pi * self.radius
def equation(self, x='x', y='y'):
"""The equation of the circle.
Parameters
==========
x : str or Symbol, optional
Default value is 'x'.
y : str or Symbol, optional
Default value is 'y'.
Returns
=======
equation : SymPy expression
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(0, 0), 5)
>>> c1.equation()
x**2 + y**2 - 25
"""
x = _symbol(x, real=True)
y = _symbol(y, real=True)
t1 = (x - self.center.x)**2
t2 = (y - self.center.y)**2
return t1 + t2 - self.major**2
def intersection(self, o):
"""The intersection of this circle with another geometrical entity.
Parameters
==========
o : GeometryEntity
Returns
=======
intersection : list of GeometryEntities
Examples
========
>>> from sympy import Point, Circle, Line, Ray
>>> p1, p2, p3 = Point(0, 0), Point(5, 5), Point(6, 0)
>>> p4 = Point(5, 0)
>>> c1 = Circle(p1, 5)
>>> c1.intersection(p2)
[]
>>> c1.intersection(p4)
[Point2D(5, 0)]
>>> c1.intersection(Ray(p1, p2))
[Point2D(5*sqrt(2)/2, 5*sqrt(2)/2)]
>>> c1.intersection(Line(p2, p3))
[]
"""
return Ellipse.intersection(self, o)
@property
def radius(self):
"""The radius of the circle.
Returns
=======
radius : number or SymPy expression
See Also
========
Ellipse.major, Ellipse.minor, Ellipse.hradius, Ellipse.vradius
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.radius
6
"""
return self.args[1]
def reflect(self, line):
"""Override GeometryEntity.reflect since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle, Line
>>> Circle((0, 1), 1).reflect(Line((0, 0), (1, 1)))
Circle(Point2D(1, 0), -1)
"""
c = self.center
c = c.reflect(line)
return self.func(c, -self.radius)
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since the radius
is not a GeometryEntity.
Examples
========
>>> from sympy import Circle
>>> Circle((0, 0), 1).scale(2, 2)
Circle(Point2D(0, 0), 2)
>>> Circle((0, 0), 1).scale(2, 4)
Ellipse(Point2D(0, 0), 2, 4)
"""
c = self.center
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
c = c.scale(x, y)
x, y = [abs(i) for i in (x, y)]
if x == y:
return self.func(c, x*self.radius)
h = v = self.radius
return Ellipse(c, hradius=h*x, vradius=v*y)
@property
def vradius(self):
"""
This Ellipse property is an alias for the Circle's radius.
Whereas hradius, major and minor can use Ellipse's conventions,
the vradius does not exist for a circle. It is always a positive
value in order that the Circle, like Polygons, will have an
area that can be positive or negative as determined by the sign
of the hradius.
Examples
========
>>> from sympy import Point, Circle
>>> c1 = Circle(Point(3, 4), 6)
>>> c1.vradius
6
"""
return abs(self.radius)
from .polygon import Polygon, Triangle
|
226e9a73820f28fb275753c23ce6ace448632f91aa4b21aaf939a9f2f7329c6d | from sympy.core import Expr, S, oo, pi, sympify
from sympy.core.evalf import N
from sympy.core.sorting import default_sort_key, ordered
from sympy.core.symbol import _symbol, Dummy, symbols, Symbol
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import cos, sin, tan
from .ellipse import Circle
from .entity import GeometryEntity, GeometrySet
from .exceptions import GeometryError
from .line import Line, Segment, Ray
from .point import Point
from sympy.logic import And
from sympy.matrices import Matrix
from sympy.simplify.simplify import simplify
from sympy.solvers.solvers import solve
from sympy.utilities.iterables import has_dups, has_variety, uniq, rotate_left, least_rotation
from sympy.utilities.misc import as_int, func_name
from mpmath.libmp.libmpf import prec_to_dps
import warnings
class Polygon(GeometrySet):
"""A two-dimensional polygon.
A simple polygon in space. Can be constructed from a sequence of points
or from a center, radius, number of sides and rotation angle.
Parameters
==========
vertices : sequence of Points
Optional parameters
==========
n : If > 0, an n-sided RegularPolygon is created. See below.
Default value is 0.
Attributes
==========
area
angles
perimeter
vertices
centroid
sides
Raises
======
GeometryError
If all parameters are not Points.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment, Triangle
Notes
=====
Polygons are treated as closed paths rather than 2D areas so
some calculations can be be negative or positive (e.g., area)
based on the orientation of the points.
Any consecutive identical points are reduced to a single point
and any points collinear and between two points will be removed
unless they are needed to define an explicit intersection (see examples).
A Triangle, Segment or Point will be returned when there are 3 or
fewer points provided.
Examples
========
>>> from sympy import Polygon, pi
>>> p1, p2, p3, p4, p5 = [(0, 0), (1, 0), (5, 1), (0, 1), (3, 0)]
>>> Polygon(p1, p2, p3, p4)
Polygon(Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1))
>>> Polygon(p1, p2)
Segment2D(Point2D(0, 0), Point2D(1, 0))
>>> Polygon(p1, p2, p5)
Segment2D(Point2D(0, 0), Point2D(3, 0))
The area of a polygon is calculated as positive when vertices are
traversed in a ccw direction. When the sides of a polygon cross the
area will have positive and negative contributions. The following
defines a Z shape where the bottom right connects back to the top
left.
>>> Polygon((0, 2), (2, 2), (0, 0), (2, 0)).area
0
When the keyword `n` is used to define the number of sides of the
Polygon then a RegularPolygon is created and the other arguments are
interpreted as center, radius and rotation. The unrotated RegularPolygon
will always have a vertex at Point(r, 0) where `r` is the radius of the
circle that circumscribes the RegularPolygon. Its method `spin` can be
used to increment that angle.
>>> p = Polygon((0,0), 1, n=3)
>>> p
RegularPolygon(Point2D(0, 0), 1, 3, 0)
>>> p.vertices[0]
Point2D(1, 0)
>>> p.args[0]
Point2D(0, 0)
>>> p.spin(pi/2)
>>> p.vertices[0]
Point2D(0, 1)
"""
__slots__ = ()
def __new__(cls, *args, n = 0, **kwargs):
if n:
args = list(args)
# return a virtual polygon with n sides
if len(args) == 2: # center, radius
args.append(n)
elif len(args) == 3: # center, radius, rotation
args.insert(2, n)
return RegularPolygon(*args, **kwargs)
vertices = [Point(a, dim=2, **kwargs) for a in args]
# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first
# remove collinear points
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = nodup[i], nodup[i + 1], nodup[i + 2]
if Point.is_collinear(a, b, c):
nodup.pop(i + 1)
if a == c:
nodup.pop(i)
else:
i += 1
vertices = list(nodup)
if len(vertices) > 3:
return GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 3:
return Triangle(*vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)
@property
def area(self):
"""
The area of the polygon.
Notes
=====
The area calculation can be positive or negative based on the
orientation of the points. If any side of the polygon crosses
any other side, there will be areas having opposite signs.
See Also
========
sympy.geometry.ellipse.Ellipse.area
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.area
3
In the Z shaped polygon (with the lower right connecting back
to the upper left) the areas cancel out:
>>> Z = Polygon((0, 1), (1, 1), (0, 0), (1, 0))
>>> Z.area
0
In the M shaped polygon, areas do not cancel because no side
crosses any other (though there is a point of contact).
>>> M = Polygon((0, 0), (0, 1), (2, 0), (3, 1), (3, 0))
>>> M.area
-3/2
"""
area = 0
args = self.args
for i in range(len(args)):
x1, y1 = args[i - 1].args
x2, y2 = args[i].args
area += x1*y2 - x2*y1
return simplify(area) / 2
@staticmethod
def _isright(a, b, c):
"""Return True/False for cw/ccw orientation.
Examples
========
>>> from sympy import Point, Polygon
>>> a, b, c = [Point(i) for i in [(0, 0), (1, 1), (1, 0)]]
>>> Polygon._isright(a, b, c)
True
>>> Polygon._isright(a, c, b)
False
"""
ba = b - a
ca = c - a
t_area = simplify(ba.x*ca.y - ca.x*ba.y)
res = t_area.is_nonpositive
if res is None:
raise ValueError("Can't determine orientation")
return res
@property
def angles(self):
"""The internal angle at each vertex.
Returns
=======
angles : dict
A dictionary where each key is a vertex and each value is the
internal angle at that vertex. The vertices are represented as
Points.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.LinearEntity.angle_between
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.angles[p1]
pi/2
>>> poly.angles[p2]
acos(-4*sqrt(17)/17)
"""
# Determine orientation of points
args = self.vertices
cw = self._isright(args[-1], args[0], args[1])
ret = {}
for i in range(len(args)):
a, b, c = args[i - 2], args[i - 1], args[i]
ang = Ray(b, a).angle_between(Ray(b, c))
if cw ^ self._isright(a, b, c):
ret[b] = 2*S.Pi - ang
else:
ret[b] = ang
return ret
@property
def ambient_dimension(self):
return self.vertices[0].ambient_dimension
@property
def perimeter(self):
"""The perimeter of the polygon.
Returns
=======
perimeter : number or Basic instance
See Also
========
sympy.geometry.line.Segment.length
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.perimeter
sqrt(17) + 7
"""
p = 0
args = self.vertices
for i in range(len(args)):
p += args[i - 1].distance(args[i])
return simplify(p)
@property
def vertices(self):
"""The vertices of the polygon.
Returns
=======
vertices : list of Points
Notes
=====
When iterating over the vertices, it is more efficient to index self
rather than to request the vertices and index them. Only use the
vertices when you want to process all of them at once. This is even
more important with RegularPolygons that calculate each vertex.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.vertices
[Point2D(0, 0), Point2D(1, 0), Point2D(5, 1), Point2D(0, 1)]
>>> poly.vertices[0]
Point2D(0, 0)
"""
return list(self.args)
@property
def centroid(self):
"""The centroid of the polygon.
Returns
=======
centroid : Point
See Also
========
sympy.geometry.point.Point, sympy.geometry.util.centroid
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.centroid
Point2D(31/18, 11/18)
"""
A = 1/(6*self.area)
cx, cy = 0, 0
args = self.args
for i in range(len(args)):
x1, y1 = args[i - 1].args
x2, y2 = args[i].args
v = x1*y2 - x2*y1
cx += v*(x1 + x2)
cy += v*(y1 + y2)
return Point(simplify(A*cx), simplify(A*cy))
def second_moment_of_area(self, point=None):
"""Returns the second moment and product moment of area of a two dimensional polygon.
Parameters
==========
point : Point, two-tuple of sympifyable objects, or None(default=None)
point is the point about which second moment of area is to be found.
If "point=None" it will be calculated about the axis passing through the
centroid of the polygon.
Returns
=======
I_xx, I_yy, I_xy : number or SymPy expression
I_xx, I_yy are second moment of area of a two dimensional polygon.
I_xy is product moment of area of a two dimensional polygon.
Examples
========
>>> from sympy import Polygon, symbols
>>> a, b = symbols('a, b')
>>> p1, p2, p3, p4, p5 = [(0, 0), (a, 0), (a, b), (0, b), (a/3, b/3)]
>>> rectangle = Polygon(p1, p2, p3, p4)
>>> rectangle.second_moment_of_area()
(a*b**3/12, a**3*b/12, 0)
>>> rectangle.second_moment_of_area(p5)
(a*b**3/9, a**3*b/9, a**2*b**2/36)
References
==========
.. [1] https://en.wikipedia.org/wiki/Second_moment_of_area
"""
I_xx, I_yy, I_xy = 0, 0, 0
args = self.vertices
for i in range(len(args)):
x1, y1 = args[i-1].args
x2, y2 = args[i].args
v = x1*y2 - x2*y1
I_xx += (y1**2 + y1*y2 + y2**2)*v
I_yy += (x1**2 + x1*x2 + x2**2)*v
I_xy += (x1*y2 + 2*x1*y1 + 2*x2*y2 + x2*y1)*v
A = self.area
c_x = self.centroid[0]
c_y = self.centroid[1]
# parallel axis theorem
I_xx_c = (I_xx/12) - (A*(c_y**2))
I_yy_c = (I_yy/12) - (A*(c_x**2))
I_xy_c = (I_xy/24) - (A*(c_x*c_y))
if point is None:
return I_xx_c, I_yy_c, I_xy_c
I_xx = (I_xx_c + A*((point[1]-c_y)**2))
I_yy = (I_yy_c + A*((point[0]-c_x)**2))
I_xy = (I_xy_c + A*((point[0]-c_x)*(point[1]-c_y)))
return I_xx, I_yy, I_xy
def first_moment_of_area(self, point=None):
"""
Returns the first moment of area of a two-dimensional polygon with
respect to a certain point of interest.
First moment of area is a measure of the distribution of the area
of a polygon in relation to an axis. The first moment of area of
the entire polygon about its own centroid is always zero. Therefore,
here it is calculated for an area, above or below a certain point
of interest, that makes up a smaller portion of the polygon. This
area is bounded by the point of interest and the extreme end
(top or bottom) of the polygon. The first moment for this area is
is then determined about the centroidal axis of the initial polygon.
References
==========
.. [1] https://skyciv.com/docs/tutorials/section-tutorials/calculating-the-statical-or-first-moment-of-area-of-beam-sections/?cc=BMD
.. [2] https://mechanicalc.com/reference/cross-sections
Parameters
==========
point: Point, two-tuple of sympifyable objects, or None (default=None)
point is the point above or below which the area of interest lies
If ``point=None`` then the centroid acts as the point of interest.
Returns
=======
Q_x, Q_y: number or SymPy expressions
Q_x is the first moment of area about the x-axis
Q_y is the first moment of area about the y-axis
A negative sign indicates that the section modulus is
determined for a section below (or left of) the centroidal axis
Examples
========
>>> from sympy import Point, Polygon
>>> a, b = 50, 10
>>> p1, p2, p3, p4 = [(0, b), (0, 0), (a, 0), (a, b)]
>>> p = Polygon(p1, p2, p3, p4)
>>> p.first_moment_of_area()
(625, 3125)
>>> p.first_moment_of_area(point=Point(30, 7))
(525, 3000)
"""
if point:
xc, yc = self.centroid
else:
point = self.centroid
xc, yc = point
h_line = Line(point, slope=0)
v_line = Line(point, slope=S.Infinity)
h_poly = self.cut_section(h_line)
v_poly = self.cut_section(v_line)
poly_1 = h_poly[0] if h_poly[0].area <= h_poly[1].area else h_poly[1]
poly_2 = v_poly[0] if v_poly[0].area <= v_poly[1].area else v_poly[1]
Q_x = (poly_1.centroid.y - yc)*poly_1.area
Q_y = (poly_2.centroid.x - xc)*poly_2.area
return Q_x, Q_y
def polar_second_moment_of_area(self):
"""Returns the polar modulus of a two-dimensional polygon
It is a constituent of the second moment of area, linked through
the perpendicular axis theorem. While the planar second moment of
area describes an object's resistance to deflection (bending) when
subjected to a force applied to a plane parallel to the central
axis, the polar second moment of area describes an object's
resistance to deflection when subjected to a moment applied in a
plane perpendicular to the object's central axis (i.e. parallel to
the cross-section)
Examples
========
>>> from sympy import Polygon, symbols
>>> a, b = symbols('a, b')
>>> rectangle = Polygon((0, 0), (a, 0), (a, b), (0, b))
>>> rectangle.polar_second_moment_of_area()
a**3*b/12 + a*b**3/12
References
==========
.. [1] https://en.wikipedia.org/wiki/Polar_moment_of_inertia
"""
second_moment = self.second_moment_of_area()
return second_moment[0] + second_moment[1]
def section_modulus(self, point=None):
"""Returns a tuple with the section modulus of a two-dimensional
polygon.
Section modulus is a geometric property of a polygon defined as the
ratio of second moment of area to the distance of the extreme end of
the polygon from the centroidal axis.
Parameters
==========
point : Point, two-tuple of sympifyable objects, or None(default=None)
point is the point at which section modulus is to be found.
If "point=None" it will be calculated for the point farthest from the
centroidal axis of the polygon.
Returns
=======
S_x, S_y: numbers or SymPy expressions
S_x is the section modulus with respect to the x-axis
S_y is the section modulus with respect to the y-axis
A negative sign indicates that the section modulus is
determined for a point below the centroidal axis
Examples
========
>>> from sympy import symbols, Polygon, Point
>>> a, b = symbols('a, b', positive=True)
>>> rectangle = Polygon((0, 0), (a, 0), (a, b), (0, b))
>>> rectangle.section_modulus()
(a*b**2/6, a**2*b/6)
>>> rectangle.section_modulus(Point(a/4, b/4))
(-a*b**2/3, -a**2*b/3)
References
==========
.. [1] https://en.wikipedia.org/wiki/Section_modulus
"""
x_c, y_c = self.centroid
if point is None:
# taking x and y as maximum distances from centroid
x_min, y_min, x_max, y_max = self.bounds
y = max(y_c - y_min, y_max - y_c)
x = max(x_c - x_min, x_max - x_c)
else:
# taking x and y as distances of the given point from the centroid
y = point.y - y_c
x = point.x - x_c
second_moment= self.second_moment_of_area()
S_x = second_moment[0]/y
S_y = second_moment[1]/x
return S_x, S_y
@property
def sides(self):
"""The directed line segments that form the sides of the polygon.
Returns
=======
sides : list of sides
Each side is a directed Segment.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.sides
[Segment2D(Point2D(0, 0), Point2D(1, 0)),
Segment2D(Point2D(1, 0), Point2D(5, 1)),
Segment2D(Point2D(5, 1), Point2D(0, 1)), Segment2D(Point2D(0, 1), Point2D(0, 0))]
"""
res = []
args = self.vertices
for i in range(-len(args), 0):
res.append(Segment(args[i], args[i + 1]))
return res
@property
def bounds(self):
"""Return a tuple (xmin, ymin, xmax, ymax) representing the bounding
rectangle for the geometric figure.
"""
verts = self.vertices
xs = [p.x for p in verts]
ys = [p.y for p in verts]
return (min(xs), min(ys), max(xs), max(ys))
def is_convex(self):
"""Is the polygon convex?
A polygon is convex if all its interior angles are less than 180
degrees and there are no intersections between sides.
Returns
=======
is_convex : boolean
True if this polygon is convex, False otherwise.
See Also
========
sympy.geometry.util.convex_hull
Examples
========
>>> from sympy import Point, Polygon
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly = Polygon(p1, p2, p3, p4)
>>> poly.is_convex()
True
"""
# Determine orientation of points
args = self.vertices
cw = self._isright(args[-2], args[-1], args[0])
for i in range(1, len(args)):
if cw ^ self._isright(args[i - 2], args[i - 1], args[i]):
return False
# check for intersecting sides
sides = self.sides
for i, si in enumerate(sides):
pts = si.args
# exclude the sides connected to si
for j in range(1 if i == len(sides) - 1 else 0, i - 1):
sj = sides[j]
if sj.p1 not in pts and sj.p2 not in pts:
hit = si.intersection(sj)
if hit:
return False
return True
def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.
Notes
=====
Being on the border of self is considered False.
Parameters
==========
p : Point
Returns
=======
encloses_point : True, False or None
See Also
========
sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.encloses_point
Examples
========
>>> from sympy import Polygon, Point
>>> p = Polygon((0, 0), (4, 0), (4, 4))
>>> p.encloses_point(Point(2, 1))
True
>>> p.encloses_point(Point(2, 2))
False
>>> p.encloses_point(Point(5, 5))
False
References
==========
.. [1] http://paulbourke.net/geometry/polygonmesh/#insidepoly
"""
p = Point(p, dim=2)
if p in self.vertices or any(p in s for s in self.sides):
return False
# move to p, checking that the result is numeric
lit = []
for v in self.vertices:
lit.append(v - p) # the difference is simplified
if lit[-1].free_symbols:
return None
poly = Polygon(*lit)
# polygon closure is assumed in the following test but Polygon removes duplicate pts so
# the last point has to be added so all sides are computed. Using Polygon.sides is
# not good since Segments are unordered.
args = poly.args
indices = list(range(-len(args), 1))
if poly.is_convex():
orientation = None
for i in indices:
a = args[i]
b = args[i + 1]
test = ((-a.y)*(b.x - a.x) - (-a.x)*(b.y - a.y)).is_negative
if orientation is None:
orientation = test
elif test is not orientation:
return False
return True
hit_odd = False
p1x, p1y = args[0].args
for i in indices[1:]:
p2x, p2y = args[i].args
if 0 > min(p1y, p2y):
if 0 <= max(p1y, p2y):
if 0 <= max(p1x, p2x):
if p1y != p2y:
xinters = (-p1y)*(p2x - p1x)/(p2y - p1y) + p1x
if p1x == p2x or 0 <= xinters:
hit_odd = not hit_odd
p1x, p1y = p2x, p2y
return hit_odd
def arbitrary_point(self, parameter='t'):
"""A parameterized point on the polygon.
The parameter, varying from 0 to 1, assigns points to the position on
the perimeter that is that fraction of the total perimeter. So the
point evaluated at t=1/2 would return the point from the first vertex
that is 1/2 way around the polygon.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
arbitrary_point : Point
Raises
======
ValueError
When `parameter` already appears in the Polygon's definition.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Polygon, Symbol
>>> t = Symbol('t', real=True)
>>> tri = Polygon((0, 0), (1, 0), (1, 1))
>>> p = tri.arbitrary_point('t')
>>> perimeter = tri.perimeter
>>> s1, s2 = [s.length for s in tri.sides[:2]]
>>> p.subs(t, (s1 + s2/2)/perimeter)
Point2D(1, 1/2)
"""
t = _symbol(parameter, real=True)
if t.name in (f.name for f in self.free_symbols):
raise ValueError('Symbol %s already appears in object and cannot be used as a parameter.' % t.name)
sides = []
perimeter = self.perimeter
perim_fraction_start = 0
for s in self.sides:
side_perim_fraction = s.length/perimeter
perim_fraction_end = perim_fraction_start + side_perim_fraction
pt = s.arbitrary_point(parameter).subs(
t, (t - perim_fraction_start)/side_perim_fraction)
sides.append(
(pt, (And(perim_fraction_start <= t, t < perim_fraction_end))))
perim_fraction_start = perim_fraction_end
return Piecewise(*sides)
def parameter_value(self, other, t):
if not isinstance(other,GeometryEntity):
other = Point(other, dim=self.ambient_dimension)
if not isinstance(other,Point):
raise ValueError("other must be a point")
if other.free_symbols:
raise NotImplementedError('non-numeric coordinates')
unknown = False
T = Dummy('t', real=True)
p = self.arbitrary_point(T)
for pt, cond in p.args:
sol = solve(pt - other, T, dict=True)
if not sol:
continue
value = sol[0][T]
if simplify(cond.subs(T, value)) == True:
return {t: value}
unknown = True
if unknown:
raise ValueError("Given point may not be on %s" % func_name(self))
raise ValueError("Given point is not on %s" % func_name(self))
def plot_interval(self, parameter='t'):
"""The plot interval for the default geometric plot of the polygon.
Parameters
==========
parameter : str, optional
Default value is 't'.
Returns
=======
plot_interval : list (plot interval)
[parameter, lower_bound, upper_bound]
Examples
========
>>> from sympy import Polygon
>>> p = Polygon((0, 0), (1, 0), (1, 1))
>>> p.plot_interval()
[t, 0, 1]
"""
t = Symbol(parameter, real=True)
return [t, 0, 1]
def intersection(self, o):
"""The intersection of polygon and geometry entity.
The intersection may be empty and can contain individual Points and
complete Line Segments.
Parameters
==========
other: GeometryEntity
Returns
=======
intersection : list
The list of Segments and Points
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment
Examples
========
>>> from sympy import Point, Polygon, Line
>>> p1, p2, p3, p4 = map(Point, [(0, 0), (1, 0), (5, 1), (0, 1)])
>>> poly1 = Polygon(p1, p2, p3, p4)
>>> p5, p6, p7 = map(Point, [(3, 2), (1, -1), (0, 2)])
>>> poly2 = Polygon(p5, p6, p7)
>>> poly1.intersection(poly2)
[Point2D(1/3, 1), Point2D(2/3, 0), Point2D(9/5, 1/5), Point2D(7/3, 1)]
>>> poly1.intersection(Line(p1, p2))
[Segment2D(Point2D(0, 0), Point2D(1, 0))]
>>> poly1.intersection(p1)
[Point2D(0, 0)]
"""
intersection_result = []
k = o.sides if isinstance(o, Polygon) else [o]
for side in self.sides:
for side1 in k:
intersection_result.extend(side.intersection(side1))
intersection_result = list(uniq(intersection_result))
points = [entity for entity in intersection_result if isinstance(entity, Point)]
segments = [entity for entity in intersection_result if isinstance(entity, Segment)]
if points and segments:
points_in_segments = list(uniq([point for point in points for segment in segments if point in segment]))
if points_in_segments:
for i in points_in_segments:
points.remove(i)
return list(ordered(segments + points))
else:
return list(ordered(intersection_result))
def cut_section(self, line):
"""
Returns a tuple of two polygon segments that lie above and below
the intersecting line respectively.
Parameters
==========
line: Line object of geometry module
line which cuts the Polygon. The part of the Polygon that lies
above and below this line is returned.
Returns
=======
upper_polygon, lower_polygon: Polygon objects or None
upper_polygon is the polygon that lies above the given line.
lower_polygon is the polygon that lies below the given line.
upper_polygon and lower polygon are ``None`` when no polygon
exists above the line or below the line.
Raises
======
ValueError: When the line does not intersect the polygon
Examples
========
>>> from sympy import Polygon, Line
>>> a, b = 20, 10
>>> p1, p2, p3, p4 = [(0, b), (0, 0), (a, 0), (a, b)]
>>> rectangle = Polygon(p1, p2, p3, p4)
>>> t = rectangle.cut_section(Line((0, 5), slope=0))
>>> t
(Polygon(Point2D(0, 10), Point2D(0, 5), Point2D(20, 5), Point2D(20, 10)),
Polygon(Point2D(0, 5), Point2D(0, 0), Point2D(20, 0), Point2D(20, 5)))
>>> upper_segment, lower_segment = t
>>> upper_segment.area
100
>>> upper_segment.centroid
Point2D(10, 15/2)
>>> lower_segment.centroid
Point2D(10, 5/2)
References
==========
.. [1] https://github.com/sympy/sympy/wiki/A-method-to-return-a-cut-section-of-any-polygon-geometry
"""
intersection_points = self.intersection(line)
if not intersection_points:
raise ValueError("This line does not intersect the polygon")
points = list(self.vertices)
points.append(points[0])
x, y = symbols('x, y', real=True, cls=Dummy)
eq = line.equation(x, y)
# considering equation of line to be `ax +by + c`
a = eq.coeff(x)
b = eq.coeff(y)
upper_vertices = []
lower_vertices = []
# prev is true when previous point is above the line
prev = True
prev_point = None
for point in points:
# when coefficient of y is 0, right side of the line is
# considered
compare = eq.subs({x: point.x, y: point.y})/b if b \
else eq.subs(x, point.x)/a
# if point lies above line
if compare > 0:
if not prev:
# if previous point lies below the line, the intersection
# point of the polygon egde and the line has to be included
edge = Line(point, prev_point)
new_point = edge.intersection(line)
upper_vertices.append(new_point[0])
lower_vertices.append(new_point[0])
upper_vertices.append(point)
prev = True
else:
if prev and prev_point:
edge = Line(point, prev_point)
new_point = edge.intersection(line)
upper_vertices.append(new_point[0])
lower_vertices.append(new_point[0])
lower_vertices.append(point)
prev = False
prev_point = point
upper_polygon, lower_polygon = None, None
if upper_vertices and isinstance(Polygon(*upper_vertices), Polygon):
upper_polygon = Polygon(*upper_vertices)
if lower_vertices and isinstance(Polygon(*lower_vertices), Polygon):
lower_polygon = Polygon(*lower_vertices)
return upper_polygon, lower_polygon
def distance(self, o):
"""
Returns the shortest distance between self and o.
If o is a point, then self does not need to be convex.
If o is another polygon self and o must be convex.
Examples
========
>>> from sympy import Point, Polygon, RegularPolygon
>>> p1, p2 = map(Point, [(0, 0), (7, 5)])
>>> poly = Polygon(*RegularPolygon(p1, 1, 3).vertices)
>>> poly.distance(p2)
sqrt(61)
"""
if isinstance(o, Point):
dist = oo
for side in self.sides:
current = side.distance(o)
if current == 0:
return S.Zero
elif current < dist:
dist = current
return dist
elif isinstance(o, Polygon) and self.is_convex() and o.is_convex():
return self._do_poly_distance(o)
raise NotImplementedError()
def _do_poly_distance(self, e2):
"""
Calculates the least distance between the exteriors of two
convex polygons e1 and e2. Does not check for the convexity
of the polygons as this is checked by Polygon.distance.
Notes
=====
- Prints a warning if the two polygons possibly intersect as the return
value will not be valid in such a case. For a more through test of
intersection use intersection().
See Also
========
sympy.geometry.point.Point.distance
Examples
========
>>> from sympy import Point, Polygon
>>> square = Polygon(Point(0, 0), Point(0, 1), Point(1, 1), Point(1, 0))
>>> triangle = Polygon(Point(1, 2), Point(2, 2), Point(2, 1))
>>> square._do_poly_distance(triangle)
sqrt(2)/2
Description of method used
==========================
Method:
[1] http://cgm.cs.mcgill.ca/~orm/mind2p.html
Uses rotating calipers:
[2] https://en.wikipedia.org/wiki/Rotating_calipers
and antipodal points:
[3] https://en.wikipedia.org/wiki/Antipodal_point
"""
e1 = self
'''Tests for a possible intersection between the polygons and outputs a warning'''
e1_center = e1.centroid
e2_center = e2.centroid
e1_max_radius = S.Zero
e2_max_radius = S.Zero
for vertex in e1.vertices:
r = Point.distance(e1_center, vertex)
if e1_max_radius < r:
e1_max_radius = r
for vertex in e2.vertices:
r = Point.distance(e2_center, vertex)
if e2_max_radius < r:
e2_max_radius = r
center_dist = Point.distance(e1_center, e2_center)
if center_dist <= e1_max_radius + e2_max_radius:
warnings.warn("Polygons may intersect producing erroneous output",
stacklevel=3)
'''
Find the upper rightmost vertex of e1 and the lowest leftmost vertex of e2
'''
e1_ymax = Point(0, -oo)
e2_ymin = Point(0, oo)
for vertex in e1.vertices:
if vertex.y > e1_ymax.y or (vertex.y == e1_ymax.y and vertex.x > e1_ymax.x):
e1_ymax = vertex
for vertex in e2.vertices:
if vertex.y < e2_ymin.y or (vertex.y == e2_ymin.y and vertex.x < e2_ymin.x):
e2_ymin = vertex
min_dist = Point.distance(e1_ymax, e2_ymin)
'''
Produce a dictionary with vertices of e1 as the keys and, for each vertex, the points
to which the vertex is connected as its value. The same is then done for e2.
'''
e1_connections = {}
e2_connections = {}
for side in e1.sides:
if side.p1 in e1_connections:
e1_connections[side.p1].append(side.p2)
else:
e1_connections[side.p1] = [side.p2]
if side.p2 in e1_connections:
e1_connections[side.p2].append(side.p1)
else:
e1_connections[side.p2] = [side.p1]
for side in e2.sides:
if side.p1 in e2_connections:
e2_connections[side.p1].append(side.p2)
else:
e2_connections[side.p1] = [side.p2]
if side.p2 in e2_connections:
e2_connections[side.p2].append(side.p1)
else:
e2_connections[side.p2] = [side.p1]
e1_current = e1_ymax
e2_current = e2_ymin
support_line = Line(Point(S.Zero, S.Zero), Point(S.One, S.Zero))
'''
Determine which point in e1 and e2 will be selected after e2_ymin and e1_ymax,
this information combined with the above produced dictionaries determines the
path that will be taken around the polygons
'''
point1 = e1_connections[e1_ymax][0]
point2 = e1_connections[e1_ymax][1]
angle1 = support_line.angle_between(Line(e1_ymax, point1))
angle2 = support_line.angle_between(Line(e1_ymax, point2))
if angle1 < angle2:
e1_next = point1
elif angle2 < angle1:
e1_next = point2
elif Point.distance(e1_ymax, point1) > Point.distance(e1_ymax, point2):
e1_next = point2
else:
e1_next = point1
point1 = e2_connections[e2_ymin][0]
point2 = e2_connections[e2_ymin][1]
angle1 = support_line.angle_between(Line(e2_ymin, point1))
angle2 = support_line.angle_between(Line(e2_ymin, point2))
if angle1 > angle2:
e2_next = point1
elif angle2 > angle1:
e2_next = point2
elif Point.distance(e2_ymin, point1) > Point.distance(e2_ymin, point2):
e2_next = point2
else:
e2_next = point1
'''
Loop which determines the distance between anti-podal pairs and updates the
minimum distance accordingly. It repeats until it reaches the starting position.
'''
while True:
e1_angle = support_line.angle_between(Line(e1_current, e1_next))
e2_angle = pi - support_line.angle_between(Line(
e2_current, e2_next))
if (e1_angle < e2_angle) is True:
support_line = Line(e1_current, e1_next)
e1_segment = Segment(e1_current, e1_next)
min_dist_current = e1_segment.distance(e2_current)
if min_dist_current.evalf() < min_dist.evalf():
min_dist = min_dist_current
if e1_connections[e1_next][0] != e1_current:
e1_current = e1_next
e1_next = e1_connections[e1_next][0]
else:
e1_current = e1_next
e1_next = e1_connections[e1_next][1]
elif (e1_angle > e2_angle) is True:
support_line = Line(e2_next, e2_current)
e2_segment = Segment(e2_current, e2_next)
min_dist_current = e2_segment.distance(e1_current)
if min_dist_current.evalf() < min_dist.evalf():
min_dist = min_dist_current
if e2_connections[e2_next][0] != e2_current:
e2_current = e2_next
e2_next = e2_connections[e2_next][0]
else:
e2_current = e2_next
e2_next = e2_connections[e2_next][1]
else:
support_line = Line(e1_current, e1_next)
e1_segment = Segment(e1_current, e1_next)
e2_segment = Segment(e2_current, e2_next)
min1 = e1_segment.distance(e2_next)
min2 = e2_segment.distance(e1_next)
min_dist_current = min(min1, min2)
if min_dist_current.evalf() < min_dist.evalf():
min_dist = min_dist_current
if e1_connections[e1_next][0] != e1_current:
e1_current = e1_next
e1_next = e1_connections[e1_next][0]
else:
e1_current = e1_next
e1_next = e1_connections[e1_next][1]
if e2_connections[e2_next][0] != e2_current:
e2_current = e2_next
e2_next = e2_connections[e2_next][0]
else:
e2_current = e2_next
e2_next = e2_connections[e2_next][1]
if e1_current == e1_ymax and e2_current == e2_ymin:
break
return min_dist
def _svg(self, scale_factor=1., fill_color="#66cc99"):
"""Returns SVG path element for the Polygon.
Parameters
==========
scale_factor : float
Multiplication factor for the SVG stroke-width. Default is 1.
fill_color : str, optional
Hex string for fill color. Default is "#66cc99".
"""
verts = map(N, self.vertices)
coords = ["{},{}".format(p.x, p.y) for p in verts]
path = "M {} L {} z".format(coords[0], " L ".join(coords[1:]))
return (
'<path fill-rule="evenodd" fill="{2}" stroke="#555555" '
'stroke-width="{0}" opacity="0.6" d="{1}" />'
).format(2. * scale_factor, path, fill_color)
def _hashable_content(self):
D = {}
def ref_list(point_list):
kee = {}
for i, p in enumerate(ordered(set(point_list))):
kee[p] = i
D[i] = p
return [kee[p] for p in point_list]
S1 = ref_list(self.args)
r_nor = rotate_left(S1, least_rotation(S1))
S2 = ref_list(list(reversed(self.args)))
r_rev = rotate_left(S2, least_rotation(S2))
if r_nor < r_rev:
r = r_nor
else:
r = r_rev
canonical_args = [ D[order] for order in r ]
return tuple(canonical_args)
def __contains__(self, o):
"""
Return True if o is contained within the boundary lines of self.altitudes
Parameters
==========
other : GeometryEntity
Returns
=======
contained in : bool
The points (and sides, if applicable) are contained in self.
See Also
========
sympy.geometry.entity.GeometryEntity.encloses
Examples
========
>>> from sympy import Line, Segment, Point
>>> p = Point(0, 0)
>>> q = Point(1, 1)
>>> s = Segment(p, q*2)
>>> l = Line(p, q)
>>> p in q
False
>>> p in s
True
>>> q*3 in s
False
>>> s in l
True
"""
if isinstance(o, Polygon):
return self == o
elif isinstance(o, Segment):
return any(o in s for s in self.sides)
elif isinstance(o, Point):
if o in self.vertices:
return True
for side in self.sides:
if o in side:
return True
return False
def bisectors(p, prec=None):
"""Returns angle bisectors of a polygon. If prec is given
then approximate the point defining the ray to that precision.
The distance between the points defining the bisector ray is 1.
Examples
========
>>> from sympy import Polygon, Point
>>> p = Polygon(Point(0, 0), Point(2, 0), Point(1, 1), Point(0, 3))
>>> p.bisectors(2)
{Point2D(0, 0): Ray2D(Point2D(0, 0), Point2D(0.71, 0.71)),
Point2D(0, 3): Ray2D(Point2D(0, 3), Point2D(0.23, 2.0)),
Point2D(1, 1): Ray2D(Point2D(1, 1), Point2D(0.19, 0.42)),
Point2D(2, 0): Ray2D(Point2D(2, 0), Point2D(1.1, 0.38))}
"""
b = {}
pts = list(p.args)
pts.append(pts[0]) # close it
cw = Polygon._isright(*pts[:3])
if cw:
pts = list(reversed(pts))
for v, a in p.angles.items():
i = pts.index(v)
p1, p2 = Point._normalize_dimension(pts[i], pts[i + 1])
ray = Ray(p1, p2).rotate(a/2, v)
dir = ray.direction
ray = Ray(ray.p1, ray.p1 + dir/dir.distance((0, 0)))
if prec is not None:
ray = Ray(ray.p1, ray.p2.n(prec))
b[v] = ray
return b
class RegularPolygon(Polygon):
"""
A regular polygon.
Such a polygon has all internal angles equal and all sides the same length.
Parameters
==========
center : Point
radius : number or Basic instance
The distance from the center to a vertex
n : int
The number of sides
Attributes
==========
vertices
center
radius
rotation
apothem
interior_angle
exterior_angle
circumcircle
incircle
angles
Raises
======
GeometryError
If the `center` is not a Point, or the `radius` is not a number or Basic
instance, or the number of sides, `n`, is less than three.
Notes
=====
A RegularPolygon can be instantiated with Polygon with the kwarg n.
Regular polygons are instantiated with a center, radius, number of sides
and a rotation angle. Whereas the arguments of a Polygon are vertices, the
vertices of the RegularPolygon must be obtained with the vertices method.
See Also
========
sympy.geometry.point.Point, Polygon
Examples
========
>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r
RegularPolygon(Point2D(0, 0), 5, 3, 0)
>>> r.vertices[0]
Point2D(5, 0)
"""
__slots__ = ('_n', '_center', '_radius', '_rot')
def __new__(self, c, r, n, rot=0, **kwargs):
r, n, rot = map(sympify, (r, n, rot))
c = Point(c, dim=2, **kwargs)
if not isinstance(r, Expr):
raise GeometryError("r must be an Expr object, not %s" % r)
if n.is_Number:
as_int(n) # let an error raise if necessary
if n < 3:
raise GeometryError("n must be a >= 3, not %s" % n)
obj = GeometryEntity.__new__(self, c, r, n, **kwargs)
obj._n = n
obj._center = c
obj._radius = r
obj._rot = rot % (2*S.Pi/n) if rot.is_number else rot
return obj
def _eval_evalf(self, prec=15, **options):
c, r, n, a = self.args
dps = prec_to_dps(prec)
c, r, a = [i.evalf(n=dps, **options) for i in (c, r, a)]
return self.func(c, r, n, a)
@property
def args(self):
"""
Returns the center point, the radius,
the number of sides, and the orientation angle.
Examples
========
>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.args
(Point2D(0, 0), 5, 3, 0)
"""
return self._center, self._radius, self._n, self._rot
def __str__(self):
return 'RegularPolygon(%s, %s, %s, %s)' % tuple(self.args)
def __repr__(self):
return 'RegularPolygon(%s, %s, %s, %s)' % tuple(self.args)
@property
def area(self):
"""Returns the area.
Examples
========
>>> from sympy import RegularPolygon
>>> square = RegularPolygon((0, 0), 1, 4)
>>> square.area
2
>>> _ == square.length**2
True
"""
c, r, n, rot = self.args
return sign(r)*n*self.length**2/(4*tan(pi/n))
@property
def length(self):
"""Returns the length of the sides.
The half-length of the side and the apothem form two legs
of a right triangle whose hypotenuse is the radius of the
regular polygon.
Examples
========
>>> from sympy import RegularPolygon
>>> from sympy import sqrt
>>> s = square_in_unit_circle = RegularPolygon((0, 0), 1, 4)
>>> s.length
sqrt(2)
>>> sqrt((_/2)**2 + s.apothem**2) == s.radius
True
"""
return self.radius*2*sin(pi/self._n)
@property
def center(self):
"""The center of the RegularPolygon
This is also the center of the circumscribing circle.
Returns
=======
center : Point
See Also
========
sympy.geometry.point.Point, sympy.geometry.ellipse.Ellipse.center
Examples
========
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.center
Point2D(0, 0)
"""
return self._center
centroid = center
@property
def circumcenter(self):
"""
Alias for center.
Examples
========
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.circumcenter
Point2D(0, 0)
"""
return self.center
@property
def radius(self):
"""Radius of the RegularPolygon
This is also the radius of the circumscribing circle.
Returns
=======
radius : number or instance of Basic
See Also
========
sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.radius
r
"""
return self._radius
@property
def circumradius(self):
"""
Alias for radius.
Examples
========
>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.circumradius
r
"""
return self.radius
@property
def rotation(self):
"""CCW angle by which the RegularPolygon is rotated
Returns
=======
rotation : number or instance of Basic
Examples
========
>>> from sympy import pi
>>> from sympy.abc import a
>>> from sympy import RegularPolygon, Point
>>> RegularPolygon(Point(0, 0), 3, 4, pi/4).rotation
pi/4
Numerical rotation angles are made canonical:
>>> RegularPolygon(Point(0, 0), 3, 4, a).rotation
a
>>> RegularPolygon(Point(0, 0), 3, 4, pi).rotation
0
"""
return self._rot
@property
def apothem(self):
"""The inradius of the RegularPolygon.
The apothem/inradius is the radius of the inscribed circle.
Returns
=======
apothem : number or instance of Basic
See Also
========
sympy.geometry.line.Segment.length, sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.apothem
sqrt(2)*r/2
"""
return self.radius * cos(S.Pi/self._n)
@property
def inradius(self):
"""
Alias for apothem.
Examples
========
>>> from sympy import Symbol
>>> from sympy import RegularPolygon, Point
>>> radius = Symbol('r')
>>> rp = RegularPolygon(Point(0, 0), radius, 4)
>>> rp.inradius
sqrt(2)*r/2
"""
return self.apothem
@property
def interior_angle(self):
"""Measure of the interior angles.
Returns
=======
interior_angle : number
See Also
========
sympy.geometry.line.LinearEntity.angle_between
Examples
========
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.interior_angle
3*pi/4
"""
return (self._n - 2)*S.Pi/self._n
@property
def exterior_angle(self):
"""Measure of the exterior angles.
Returns
=======
exterior_angle : number
See Also
========
sympy.geometry.line.LinearEntity.angle_between
Examples
========
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.exterior_angle
pi/4
"""
return 2*S.Pi/self._n
@property
def circumcircle(self):
"""The circumcircle of the RegularPolygon.
Returns
=======
circumcircle : Circle
See Also
========
circumcenter, sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 8)
>>> rp.circumcircle
Circle(Point2D(0, 0), 4)
"""
return Circle(self.center, self.radius)
@property
def incircle(self):
"""The incircle of the RegularPolygon.
Returns
=======
incircle : Circle
See Also
========
inradius, sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 4, 7)
>>> rp.incircle
Circle(Point2D(0, 0), 4*cos(pi/7))
"""
return Circle(self.center, self.apothem)
@property
def angles(self):
"""
Returns a dictionary with keys, the vertices of the Polygon,
and values, the interior angle at each vertex.
Examples
========
>>> from sympy import RegularPolygon, Point
>>> r = RegularPolygon(Point(0, 0), 5, 3)
>>> r.angles
{Point2D(-5/2, -5*sqrt(3)/2): pi/3,
Point2D(-5/2, 5*sqrt(3)/2): pi/3,
Point2D(5, 0): pi/3}
"""
ret = {}
ang = self.interior_angle
for v in self.vertices:
ret[v] = ang
return ret
def encloses_point(self, p):
"""
Return True if p is enclosed by (is inside of) self.
Notes
=====
Being on the border of self is considered False.
The general Polygon.encloses_point method is called only if
a point is not within or beyond the incircle or circumcircle,
respectively.
Parameters
==========
p : Point
Returns
=======
encloses_point : True, False or None
See Also
========
sympy.geometry.ellipse.Ellipse.encloses_point
Examples
========
>>> from sympy import RegularPolygon, S, Point, Symbol
>>> p = RegularPolygon((0, 0), 3, 4)
>>> p.encloses_point(Point(0, 0))
True
>>> r, R = p.inradius, p.circumradius
>>> p.encloses_point(Point((r + R)/2, 0))
True
>>> p.encloses_point(Point(R/2, R/2 + (R - r)/10))
False
>>> t = Symbol('t', real=True)
>>> p.encloses_point(p.arbitrary_point().subs(t, S.Half))
False
>>> p.encloses_point(Point(5, 5))
False
"""
c = self.center
d = Segment(c, p).length
if d >= self.radius:
return False
elif d < self.inradius:
return True
else:
# now enumerate the RegularPolygon like a general polygon.
return Polygon.encloses_point(self, p)
def spin(self, angle):
"""Increment *in place* the virtual Polygon's rotation by ccw angle.
See also: rotate method which moves the center.
>>> from sympy import Polygon, Point, pi
>>> r = Polygon(Point(0,0), 1, n=3)
>>> r.vertices[0]
Point2D(1, 0)
>>> r.spin(pi/6)
>>> r.vertices[0]
Point2D(sqrt(3)/2, 1/2)
See Also
========
rotation
rotate : Creates a copy of the RegularPolygon rotated about a Point
"""
self._rot += angle
def rotate(self, angle, pt=None):
"""Override GeometryEntity.rotate to first rotate the RegularPolygon
about its center.
>>> from sympy import Point, RegularPolygon, pi
>>> t = RegularPolygon(Point(1, 0), 1, 3)
>>> t.vertices[0] # vertex on x-axis
Point2D(2, 0)
>>> t.rotate(pi/2).vertices[0] # vertex on y axis now
Point2D(0, 2)
See Also
========
rotation
spin : Rotates a RegularPolygon in place
"""
r = type(self)(*self.args) # need a copy or else changes are in-place
r._rot += angle
return GeometryEntity.rotate(r, angle, pt)
def scale(self, x=1, y=1, pt=None):
"""Override GeometryEntity.scale since it is the radius that must be
scaled (if x == y) or else a new Polygon must be returned.
>>> from sympy import RegularPolygon
Symmetric scaling returns a RegularPolygon:
>>> RegularPolygon((0, 0), 1, 4).scale(2, 2)
RegularPolygon(Point2D(0, 0), 2, 4, 0)
Asymmetric scaling returns a kite as a Polygon:
>>> RegularPolygon((0, 0), 1, 4).scale(2, 1)
Polygon(Point2D(2, 0), Point2D(0, 1), Point2D(-2, 0), Point2D(0, -1))
"""
if pt:
pt = Point(pt, dim=2)
return self.translate(*(-pt).args).scale(x, y).translate(*pt.args)
if x != y:
return Polygon(*self.vertices).scale(x, y)
c, r, n, rot = self.args
r *= x
return self.func(c, r, n, rot)
def reflect(self, line):
"""Override GeometryEntity.reflect since this is not made of only
points.
Examples
========
>>> from sympy import RegularPolygon, Line
>>> RegularPolygon((0, 0), 1, 4).reflect(Line((0, 1), slope=-2))
RegularPolygon(Point2D(4/5, 2/5), -1, 4, atan(4/3))
"""
c, r, n, rot = self.args
v = self.vertices[0]
d = v - c
cc = c.reflect(line)
vv = v.reflect(line)
dd = vv - cc
# calculate rotation about the new center
# which will align the vertices
l1 = Ray((0, 0), dd)
l2 = Ray((0, 0), d)
ang = l1.closing_angle(l2)
rot += ang
# change sign of radius as point traversal is reversed
return self.func(cc, -r, n, rot)
@property
def vertices(self):
"""The vertices of the RegularPolygon.
Returns
=======
vertices : list
Each vertex is a Point.
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import RegularPolygon, Point
>>> rp = RegularPolygon(Point(0, 0), 5, 4)
>>> rp.vertices
[Point2D(5, 0), Point2D(0, 5), Point2D(-5, 0), Point2D(0, -5)]
"""
c = self._center
r = abs(self._radius)
rot = self._rot
v = 2*S.Pi/self._n
return [Point(c.x + r*cos(k*v + rot), c.y + r*sin(k*v + rot))
for k in range(self._n)]
def __eq__(self, o):
if not isinstance(o, Polygon):
return False
elif not isinstance(o, RegularPolygon):
return Polygon.__eq__(o, self)
return self.args == o.args
def __hash__(self):
return super().__hash__()
class Triangle(Polygon):
"""
A polygon with three vertices and three sides.
Parameters
==========
points : sequence of Points
keyword: asa, sas, or sss to specify sides/angles of the triangle
Attributes
==========
vertices
altitudes
orthocenter
circumcenter
circumradius
circumcircle
inradius
incircle
exradii
medians
medial
nine_point_circle
Raises
======
GeometryError
If the number of vertices is not equal to three, or one of the vertices
is not a Point, or a valid keyword is not given.
See Also
========
sympy.geometry.point.Point, Polygon
Examples
========
>>> from sympy import Triangle, Point
>>> Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
Triangle(Point2D(0, 0), Point2D(4, 0), Point2D(4, 3))
Keywords sss, sas, or asa can be used to give the desired
side lengths (in order) and interior angles (in degrees) that
define the triangle:
>>> Triangle(sss=(3, 4, 5))
Triangle(Point2D(0, 0), Point2D(3, 0), Point2D(3, 4))
>>> Triangle(asa=(30, 1, 30))
Triangle(Point2D(0, 0), Point2D(1, 0), Point2D(1/2, sqrt(3)/6))
>>> Triangle(sas=(1, 45, 2))
Triangle(Point2D(0, 0), Point2D(2, 0), Point2D(sqrt(2)/2, sqrt(2)/2))
"""
def __new__(cls, *args, **kwargs):
if len(args) != 3:
if 'sss' in kwargs:
return _sss(*[simplify(a) for a in kwargs['sss']])
if 'asa' in kwargs:
return _asa(*[simplify(a) for a in kwargs['asa']])
if 'sas' in kwargs:
return _sas(*[simplify(a) for a in kwargs['sas']])
msg = "Triangle instantiates with three points or a valid keyword."
raise GeometryError(msg)
vertices = [Point(a, dim=2, **kwargs) for a in args]
# remove consecutive duplicates
nodup = []
for p in vertices:
if nodup and p == nodup[-1]:
continue
nodup.append(p)
if len(nodup) > 1 and nodup[-1] == nodup[0]:
nodup.pop() # last point was same as first
# remove collinear points
i = -3
while i < len(nodup) - 3 and len(nodup) > 2:
a, b, c = sorted(
[nodup[i], nodup[i + 1], nodup[i + 2]], key=default_sort_key)
if Point.is_collinear(a, b, c):
nodup[i] = a
nodup[i + 1] = None
nodup.pop(i + 1)
i += 1
vertices = list(filter(lambda x: x is not None, nodup))
if len(vertices) == 3:
return GeometryEntity.__new__(cls, *vertices, **kwargs)
elif len(vertices) == 2:
return Segment(*vertices, **kwargs)
else:
return Point(*vertices, **kwargs)
@property
def vertices(self):
"""The triangle's vertices
Returns
=======
vertices : tuple
Each element in the tuple is a Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Triangle, Point
>>> t = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t.vertices
(Point2D(0, 0), Point2D(4, 0), Point2D(4, 3))
"""
return self.args
def is_similar(t1, t2):
"""Is another triangle similar to this one.
Two triangles are similar if one can be uniformly scaled to the other.
Parameters
==========
other: Triangle
Returns
=======
is_similar : boolean
See Also
========
sympy.geometry.entity.GeometryEntity.is_similar
Examples
========
>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -3))
>>> t1.is_similar(t2)
True
>>> t2 = Triangle(Point(0, 0), Point(-4, 0), Point(-4, -4))
>>> t1.is_similar(t2)
False
"""
if not isinstance(t2, Polygon):
return False
s1_1, s1_2, s1_3 = [side.length for side in t1.sides]
s2 = [side.length for side in t2.sides]
def _are_similar(u1, u2, u3, v1, v2, v3):
e1 = simplify(u1/v1)
e2 = simplify(u2/v2)
e3 = simplify(u3/v3)
return bool(e1 == e2) and bool(e2 == e3)
# There's only 6 permutations, so write them out
return _are_similar(s1_1, s1_2, s1_3, *s2) or \
_are_similar(s1_1, s1_3, s1_2, *s2) or \
_are_similar(s1_2, s1_1, s1_3, *s2) or \
_are_similar(s1_2, s1_3, s1_1, *s2) or \
_are_similar(s1_3, s1_1, s1_2, *s2) or \
_are_similar(s1_3, s1_2, s1_1, *s2)
def is_equilateral(self):
"""Are all the sides the same length?
Returns
=======
is_equilateral : boolean
See Also
========
sympy.geometry.entity.GeometryEntity.is_similar, RegularPolygon
is_isosceles, is_right, is_scalene
Examples
========
>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_equilateral()
False
>>> from sympy import sqrt
>>> t2 = Triangle(Point(0, 0), Point(10, 0), Point(5, 5*sqrt(3)))
>>> t2.is_equilateral()
True
"""
return not has_variety(s.length for s in self.sides)
def is_isosceles(self):
"""Are two or more of the sides the same length?
Returns
=======
is_isosceles : boolean
See Also
========
is_equilateral, is_right, is_scalene
Examples
========
>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(2, 4))
>>> t1.is_isosceles()
True
"""
return has_dups(s.length for s in self.sides)
def is_scalene(self):
"""Are all the sides of the triangle of different lengths?
Returns
=======
is_scalene : boolean
See Also
========
is_equilateral, is_isosceles, is_right
Examples
========
>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(1, 4))
>>> t1.is_scalene()
True
"""
return not has_dups(s.length for s in self.sides)
def is_right(self):
"""Is the triangle right-angled.
Returns
=======
is_right : boolean
See Also
========
sympy.geometry.line.LinearEntity.is_perpendicular
is_equilateral, is_isosceles, is_scalene
Examples
========
>>> from sympy import Triangle, Point
>>> t1 = Triangle(Point(0, 0), Point(4, 0), Point(4, 3))
>>> t1.is_right()
True
"""
s = self.sides
return Segment.is_perpendicular(s[0], s[1]) or \
Segment.is_perpendicular(s[1], s[2]) or \
Segment.is_perpendicular(s[0], s[2])
@property
def altitudes(self):
"""The altitudes of the triangle.
An altitude of a triangle is a segment through a vertex,
perpendicular to the opposite side, with length being the
height of the vertex measured from the line containing the side.
Returns
=======
altitudes : dict
The dictionary consists of keys which are vertices and values
which are Segments.
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment.length
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.altitudes[p1]
Segment2D(Point2D(0, 0), Point2D(1/2, 1/2))
"""
s = self.sides
v = self.vertices
return {v[0]: s[1].perpendicular_segment(v[0]),
v[1]: s[2].perpendicular_segment(v[1]),
v[2]: s[0].perpendicular_segment(v[2])}
@property
def orthocenter(self):
"""The orthocenter of the triangle.
The orthocenter is the intersection of the altitudes of a triangle.
It may lie inside, outside or on the triangle.
Returns
=======
orthocenter : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.orthocenter
Point2D(0, 0)
"""
a = self.altitudes
v = self.vertices
return Line(a[v[0]]).intersection(Line(a[v[1]]))[0]
@property
def circumcenter(self):
"""The circumcenter of the triangle
The circumcenter is the center of the circumcircle.
Returns
=======
circumcenter : Point
See Also
========
sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcenter
Point2D(1/2, 1/2)
"""
a, b, c = [x.perpendicular_bisector() for x in self.sides]
return a.intersection(b)[0]
@property
def circumradius(self):
"""The radius of the circumcircle of the triangle.
Returns
=======
circumradius : number of Basic instance
See Also
========
sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy import Symbol
>>> from sympy import Point, Triangle
>>> a = Symbol('a')
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, a)
>>> t = Triangle(p1, p2, p3)
>>> t.circumradius
sqrt(a**2/4 + 1/4)
"""
return Point.distance(self.circumcenter, self.vertices[0])
@property
def circumcircle(self):
"""The circle which passes through the three vertices of the triangle.
Returns
=======
circumcircle : Circle
See Also
========
sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.circumcircle
Circle(Point2D(1/2, 1/2), sqrt(2)/2)
"""
return Circle(self.circumcenter, self.circumradius)
def bisectors(self):
"""The angle bisectors of the triangle.
An angle bisector of a triangle is a straight line through a vertex
which cuts the corresponding angle in half.
Returns
=======
bisectors : dict
Each key is a vertex (Point) and each value is the corresponding
bisector (Segment).
See Also
========
sympy.geometry.point.Point, sympy.geometry.line.Segment
Examples
========
>>> from sympy import Point, Triangle, Segment
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> from sympy import sqrt
>>> t.bisectors()[p2] == Segment(Point(1, 0), Point(0, sqrt(2) - 1))
True
"""
# use lines containing sides so containment check during
# intersection calculation can be avoided, thus reducing
# the processing time for calculating the bisectors
s = [Line(l) for l in self.sides]
v = self.vertices
c = self.incenter
l1 = Segment(v[0], Line(v[0], c).intersection(s[1])[0])
l2 = Segment(v[1], Line(v[1], c).intersection(s[2])[0])
l3 = Segment(v[2], Line(v[2], c).intersection(s[0])[0])
return {v[0]: l1, v[1]: l2, v[2]: l3}
@property
def incenter(self):
"""The center of the incircle.
The incircle is the circle which lies inside the triangle and touches
all three sides.
Returns
=======
incenter : Point
See Also
========
incircle, sympy.geometry.point.Point
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.incenter
Point2D(1 - sqrt(2)/2, 1 - sqrt(2)/2)
"""
s = self.sides
l = Matrix([s[i].length for i in [1, 2, 0]])
p = sum(l)
v = self.vertices
x = simplify(l.dot(Matrix([vi.x for vi in v]))/p)
y = simplify(l.dot(Matrix([vi.y for vi in v]))/p)
return Point(x, y)
@property
def inradius(self):
"""The radius of the incircle.
Returns
=======
inradius : number of Basic instance
See Also
========
incircle, sympy.geometry.ellipse.Circle.radius
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(4, 0), Point(0, 3)
>>> t = Triangle(p1, p2, p3)
>>> t.inradius
1
"""
return simplify(2 * self.area / self.perimeter)
@property
def incircle(self):
"""The incircle of the triangle.
The incircle is the circle which lies inside the triangle and touches
all three sides.
Returns
=======
incircle : Circle
See Also
========
sympy.geometry.ellipse.Circle
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(2, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.incircle
Circle(Point2D(2 - sqrt(2), 2 - sqrt(2)), 2 - sqrt(2))
"""
return Circle(self.incenter, self.inradius)
@property
def exradii(self):
"""The radius of excircles of a triangle.
An excircle of the triangle is a circle lying outside the triangle,
tangent to one of its sides and tangent to the extensions of the
other two.
Returns
=======
exradii : dict
See Also
========
sympy.geometry.polygon.Triangle.inradius
Examples
========
The exradius touches the side of the triangle to which it is keyed, e.g.
the exradius touching side 2 is:
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.exradii[t.sides[2]]
-2 + sqrt(10)
References
==========
.. [1] http://mathworld.wolfram.com/Exradius.html
.. [2] http://mathworld.wolfram.com/Excircles.html
"""
side = self.sides
a = side[0].length
b = side[1].length
c = side[2].length
s = (a+b+c)/2
area = self.area
exradii = {self.sides[0]: simplify(area/(s-a)),
self.sides[1]: simplify(area/(s-b)),
self.sides[2]: simplify(area/(s-c))}
return exradii
@property
def excenters(self):
"""Excenters of the triangle.
An excenter is the center of a circle that is tangent to a side of the
triangle and the extensions of the other two sides.
Returns
=======
excenters : dict
Examples
========
The excenters are keyed to the side of the triangle to which their corresponding
excircle is tangent: The center is keyed, e.g. the excenter of a circle touching
side 0 is:
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(6, 0), Point(0, 2)
>>> t = Triangle(p1, p2, p3)
>>> t.excenters[t.sides[0]]
Point2D(12*sqrt(10), 2/3 + sqrt(10)/3)
See Also
========
sympy.geometry.polygon.Triangle.exradii
References
==========
.. [1] http://mathworld.wolfram.com/Excircles.html
"""
s = self.sides
v = self.vertices
a = s[0].length
b = s[1].length
c = s[2].length
x = [v[0].x, v[1].x, v[2].x]
y = [v[0].y, v[1].y, v[2].y]
exc_coords = {
"x1": simplify(-a*x[0]+b*x[1]+c*x[2]/(-a+b+c)),
"x2": simplify(a*x[0]-b*x[1]+c*x[2]/(a-b+c)),
"x3": simplify(a*x[0]+b*x[1]-c*x[2]/(a+b-c)),
"y1": simplify(-a*y[0]+b*y[1]+c*y[2]/(-a+b+c)),
"y2": simplify(a*y[0]-b*y[1]+c*y[2]/(a-b+c)),
"y3": simplify(a*y[0]+b*y[1]-c*y[2]/(a+b-c))
}
excenters = {
s[0]: Point(exc_coords["x1"], exc_coords["y1"]),
s[1]: Point(exc_coords["x2"], exc_coords["y2"]),
s[2]: Point(exc_coords["x3"], exc_coords["y3"])
}
return excenters
@property
def medians(self):
"""The medians of the triangle.
A median of a triangle is a straight line through a vertex and the
midpoint of the opposite side, and divides the triangle into two
equal areas.
Returns
=======
medians : dict
Each key is a vertex (Point) and each value is the median (Segment)
at that point.
See Also
========
sympy.geometry.point.Point.midpoint, sympy.geometry.line.Segment.midpoint
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medians[p1]
Segment2D(Point2D(0, 0), Point2D(1/2, 1/2))
"""
s = self.sides
v = self.vertices
return {v[0]: Segment(v[0], s[1].midpoint),
v[1]: Segment(v[1], s[2].midpoint),
v[2]: Segment(v[2], s[0].midpoint)}
@property
def medial(self):
"""The medial triangle of the triangle.
The triangle which is formed from the midpoints of the three sides.
Returns
=======
medial : Triangle
See Also
========
sympy.geometry.line.Segment.midpoint
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.medial
Triangle(Point2D(1/2, 0), Point2D(1/2, 1/2), Point2D(0, 1/2))
"""
s = self.sides
return Triangle(s[0].midpoint, s[1].midpoint, s[2].midpoint)
@property
def nine_point_circle(self):
"""The nine-point circle of the triangle.
Nine-point circle is the circumcircle of the medial triangle, which
passes through the feet of altitudes and the middle points of segments
connecting the vertices and the orthocenter.
Returns
=======
nine_point_circle : Circle
See also
========
sympy.geometry.line.Segment.midpoint
sympy.geometry.polygon.Triangle.medial
sympy.geometry.polygon.Triangle.orthocenter
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.nine_point_circle
Circle(Point2D(1/4, 1/4), sqrt(2)/4)
"""
return Circle(*self.medial.vertices)
@property
def eulerline(self):
"""The Euler line of the triangle.
The line which passes through circumcenter, centroid and orthocenter.
Returns
=======
eulerline : Line (or Point for equilateral triangles in which case all
centers coincide)
Examples
========
>>> from sympy import Point, Triangle
>>> p1, p2, p3 = Point(0, 0), Point(1, 0), Point(0, 1)
>>> t = Triangle(p1, p2, p3)
>>> t.eulerline
Line2D(Point2D(0, 0), Point2D(1/2, 1/2))
"""
if self.is_equilateral():
return self.orthocenter
return Line(self.orthocenter, self.circumcenter)
def rad(d):
"""Return the radian value for the given degrees (pi = 180 degrees)."""
return d*pi/180
def deg(r):
"""Return the degree value for the given radians (pi = 180 degrees)."""
return r/pi*180
def _slope(d):
rv = tan(rad(d))
return rv
def _asa(d1, l, d2):
"""Return triangle having side with length l on the x-axis."""
xy = Line((0, 0), slope=_slope(d1)).intersection(
Line((l, 0), slope=_slope(180 - d2)))[0]
return Triangle((0, 0), (l, 0), xy)
def _sss(l1, l2, l3):
"""Return triangle having side of length l1 on the x-axis."""
c1 = Circle((0, 0), l3)
c2 = Circle((l1, 0), l2)
inter = [a for a in c1.intersection(c2) if a.y.is_nonnegative]
if not inter:
return None
pt = inter[0]
return Triangle((0, 0), (l1, 0), pt)
def _sas(l1, d, l2):
"""Return triangle having side with length l2 on the x-axis."""
p1 = Point(0, 0)
p2 = Point(l2, 0)
p3 = Point(cos(rad(d))*l1, sin(rad(d))*l1)
return Triangle(p1, p2, p3)
|
3a10508479bf6fd3695357e9c0792745a75491e7e71ec5d0bbffab1dbad50390 | """Transform a string with Python-like source code into SymPy expression. """
from tokenize import (generate_tokens, untokenize, TokenError,
NUMBER, STRING, NAME, OP, ENDMARKER, ERRORTOKEN, NEWLINE)
from keyword import iskeyword
import ast
import unicodedata
from io import StringIO
import builtins
import types
from typing import Tuple as tTuple, Dict as tDict, Any, Callable, \
List, Optional, Union as tUnion
from sympy.assumptions.ask import AssumptionKeys
from sympy.core.basic import Basic
from sympy.core import Symbol
from sympy.core.function import Function
from sympy.utilities.misc import func_name
from sympy.functions.elementary.miscellaneous import Max, Min
null = ''
TOKEN = tTuple[int, str]
DICT = tDict[str, Any]
TRANS = Callable[[List[TOKEN], DICT, DICT], List[TOKEN]]
def _token_splittable(token_name: str) -> bool:
"""
Predicate for whether a token name can be split into multiple tokens.
A token is splittable if it does not contain an underscore character and
it is not the name of a Greek letter. This is used to implicitly convert
expressions like 'xyz' into 'x*y*z'.
"""
if '_' in token_name:
return False
try:
return not unicodedata.lookup('GREEK SMALL LETTER ' + token_name)
except KeyError:
return len(token_name) > 1
def _token_callable(token: TOKEN, local_dict: DICT, global_dict: DICT, nextToken=None):
"""
Predicate for whether a token name represents a callable function.
Essentially wraps ``callable``, but looks up the token name in the
locals and globals.
"""
func = local_dict.get(token[1])
if not func:
func = global_dict.get(token[1])
return callable(func) and not isinstance(func, Symbol)
def _add_factorial_tokens(name: str, result: List[TOKEN]) -> List[TOKEN]:
if result == [] or result[-1][1] == '(':
raise TokenError()
beginning = [(NAME, name), (OP, '(')]
end = [(OP, ')')]
diff = 0
length = len(result)
for index, token in enumerate(result[::-1]):
toknum, tokval = token
i = length - index - 1
if tokval == ')':
diff += 1
elif tokval == '(':
diff -= 1
if diff == 0:
if i - 1 >= 0 and result[i - 1][0] == NAME:
return result[:i - 1] + beginning + result[i - 1:] + end
else:
return result[:i] + beginning + result[i:] + end
return result
class ParenthesisGroup(List[TOKEN]):
"""List of tokens representing an expression in parentheses."""
pass
class AppliedFunction:
"""
A group of tokens representing a function and its arguments.
`exponent` is for handling the shorthand sin^2, ln^2, etc.
"""
def __init__(self, function: TOKEN, args: ParenthesisGroup, exponent=None):
if exponent is None:
exponent = []
self.function = function
self.args = args
self.exponent = exponent
self.items = ['function', 'args', 'exponent']
def expand(self) -> List[TOKEN]:
"""Return a list of tokens representing the function"""
return [self.function, *self.args]
def __getitem__(self, index):
return getattr(self, self.items[index])
def __repr__(self):
return "AppliedFunction(%s, %s, %s)" % (self.function, self.args,
self.exponent)
def _flatten(result: List[tUnion[TOKEN, AppliedFunction]]):
result2: List[TOKEN] = []
for tok in result:
if isinstance(tok, AppliedFunction):
result2.extend(tok.expand())
else:
result2.append(tok)
return result2
def _group_parentheses(recursor: TRANS):
def _inner(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Group tokens between parentheses with ParenthesisGroup.
Also processes those tokens recursively.
"""
result: List[tUnion[TOKEN, ParenthesisGroup]] = []
stacks: List[ParenthesisGroup] = []
stacklevel = 0
for token in tokens:
if token[0] == OP:
if token[1] == '(':
stacks.append(ParenthesisGroup([]))
stacklevel += 1
elif token[1] == ')':
stacks[-1].append(token)
stack = stacks.pop()
if len(stacks) > 0:
# We don't recurse here since the upper-level stack
# would reprocess these tokens
stacks[-1].extend(stack)
else:
# Recurse here to handle nested parentheses
# Strip off the outer parentheses to avoid an infinite loop
inner = stack[1:-1]
inner = recursor(inner,
local_dict,
global_dict)
parenGroup = [stack[0]] + inner + [stack[-1]]
result.append(ParenthesisGroup(parenGroup))
stacklevel -= 1
continue
if stacklevel:
stacks[-1].append(token)
else:
result.append(token)
if stacklevel:
raise TokenError("Mismatched parentheses")
return result
return _inner
def _apply_functions(tokens: List[tUnion[TOKEN, ParenthesisGroup]], local_dict: DICT, global_dict: DICT):
"""Convert a NAME token + ParenthesisGroup into an AppliedFunction.
Note that ParenthesisGroups, if not applied to any function, are
converted back into lists of tokens.
"""
result: List[tUnion[TOKEN, AppliedFunction]] = []
symbol = None
for tok in tokens:
if isinstance(tok, ParenthesisGroup):
if symbol and _token_callable(symbol, local_dict, global_dict):
result[-1] = AppliedFunction(symbol, tok)
symbol = None
else:
result.extend(tok)
elif tok[0] == NAME:
symbol = tok
result.append(tok)
else:
symbol = None
result.append(tok)
return result
def _implicit_multiplication(tokens: List[tUnion[TOKEN, AppliedFunction]], local_dict: DICT, global_dict: DICT):
"""Implicitly adds '*' tokens.
Cases:
- Two AppliedFunctions next to each other ("sin(x)cos(x)")
- AppliedFunction next to an open parenthesis ("sin x (cos x + 1)")
- A close parenthesis next to an AppliedFunction ("(x+2)sin x")\
- A close parenthesis next to an open parenthesis ("(x+2)(x+3)")
- AppliedFunction next to an implicitly applied function ("sin(x)cos x")
"""
result: List[tUnion[TOKEN, AppliedFunction]] = []
skip = False
for tok, nextTok in zip(tokens, tokens[1:]):
result.append(tok)
if skip:
skip = False
continue
if tok[0] == OP and tok[1] == '.' and nextTok[0] == NAME:
# Dotted name. Do not do implicit multiplication
skip = True
continue
if isinstance(tok, AppliedFunction):
if isinstance(nextTok, AppliedFunction):
result.append((OP, '*'))
elif nextTok == (OP, '('):
# Applied function followed by an open parenthesis
if tok.function[1] == "Function":
tok.function = (tok.function[0], 'Symbol')
result.append((OP, '*'))
elif nextTok[0] == NAME:
# Applied function followed by implicitly applied function
result.append((OP, '*'))
else:
if tok == (OP, ')'):
if isinstance(nextTok, AppliedFunction):
# Close parenthesis followed by an applied function
result.append((OP, '*'))
elif nextTok[0] == NAME:
# Close parenthesis followed by an implicitly applied function
result.append((OP, '*'))
elif nextTok == (OP, '('):
# Close parenthesis followed by an open parenthesis
result.append((OP, '*'))
elif tok[0] == NAME and not _token_callable(tok, local_dict, global_dict):
if isinstance(nextTok, AppliedFunction) or \
(nextTok[0] == NAME and _token_callable(nextTok, local_dict, global_dict)):
# Constant followed by (implicitly applied) function
result.append((OP, '*'))
elif nextTok == (OP, '('):
# Constant followed by parenthesis
result.append((OP, '*'))
elif nextTok[0] == NAME:
# Constant followed by constant
result.append((OP, '*'))
if tokens:
result.append(tokens[-1])
return result
def _implicit_application(tokens: List[tUnion[TOKEN, AppliedFunction]], local_dict: DICT, global_dict: DICT):
"""Adds parentheses as needed after functions."""
result: List[tUnion[TOKEN, AppliedFunction]] = []
appendParen = 0 # number of closing parentheses to add
skip = 0 # number of tokens to delay before adding a ')' (to
# capture **, ^, etc.)
exponentSkip = False # skipping tokens before inserting parentheses to
# work with function exponentiation
for tok, nextTok in zip(tokens, tokens[1:]):
result.append(tok)
if (tok[0] == NAME and nextTok[0] not in [OP, ENDMARKER, NEWLINE]):
if _token_callable(tok, local_dict, global_dict, nextTok): # type: ignore
result.append((OP, '('))
appendParen += 1
# name followed by exponent - function exponentiation
elif (tok[0] == NAME and nextTok[0] == OP and nextTok[1] == '**'):
if _token_callable(tok, local_dict, global_dict): # type: ignore
exponentSkip = True
elif exponentSkip:
# if the last token added was an applied function (i.e. the
# power of the function exponent) OR a multiplication (as
# implicit multiplication would have added an extraneous
# multiplication)
if (isinstance(tok, AppliedFunction)
or (tok[0] == OP and tok[1] == '*')):
# don't add anything if the next token is a multiplication
# or if there's already a parenthesis (if parenthesis, still
# stop skipping tokens)
if not (nextTok[0] == OP and nextTok[1] == '*'):
if not(nextTok[0] == OP and nextTok[1] == '('):
result.append((OP, '('))
appendParen += 1
exponentSkip = False
elif appendParen:
if nextTok[0] == OP and nextTok[1] in ('^', '**', '*'):
skip = 1
continue
if skip:
skip -= 1
continue
result.append((OP, ')'))
appendParen -= 1
if tokens:
result.append(tokens[-1])
if appendParen:
result.extend([(OP, ')')] * appendParen)
return result
def function_exponentiation(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Allows functions to be exponentiated, e.g. ``cos**2(x)``.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, function_exponentiation)
>>> transformations = standard_transformations + (function_exponentiation,)
>>> parse_expr('sin**4(x)', transformations=transformations)
sin(x)**4
"""
result: List[TOKEN] = []
exponent: List[TOKEN] = []
consuming_exponent = False
level = 0
for tok, nextTok in zip(tokens, tokens[1:]):
if tok[0] == NAME and nextTok[0] == OP and nextTok[1] == '**':
if _token_callable(tok, local_dict, global_dict):
consuming_exponent = True
elif consuming_exponent:
if tok[0] == NAME and tok[1] == 'Function':
tok = (NAME, 'Symbol')
exponent.append(tok)
# only want to stop after hitting )
if tok[0] == nextTok[0] == OP and tok[1] == ')' and nextTok[1] == '(':
consuming_exponent = False
# if implicit multiplication was used, we may have )*( instead
if tok[0] == nextTok[0] == OP and tok[1] == '*' and nextTok[1] == '(':
consuming_exponent = False
del exponent[-1]
continue
elif exponent and not consuming_exponent:
if tok[0] == OP:
if tok[1] == '(':
level += 1
elif tok[1] == ')':
level -= 1
if level == 0:
result.append(tok)
result.extend(exponent)
exponent = []
continue
result.append(tok)
if tokens:
result.append(tokens[-1])
if exponent:
result.extend(exponent)
return result
def split_symbols_custom(predicate: Callable[[str], bool]):
"""Creates a transformation that splits symbol names.
``predicate`` should return True if the symbol name is to be split.
For instance, to retain the default behavior but avoid splitting certain
symbol names, a predicate like this would work:
>>> from sympy.parsing.sympy_parser import (parse_expr, _token_splittable,
... standard_transformations, implicit_multiplication,
... split_symbols_custom)
>>> def can_split(symbol):
... if symbol not in ('list', 'of', 'unsplittable', 'names'):
... return _token_splittable(symbol)
... return False
...
>>> transformation = split_symbols_custom(can_split)
>>> parse_expr('unsplittable', transformations=standard_transformations +
... (transformation, implicit_multiplication))
unsplittable
"""
def _split_symbols(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
result: List[TOKEN] = []
split = False
split_previous=False
for tok in tokens:
if split_previous:
# throw out closing parenthesis of Symbol that was split
split_previous=False
continue
split_previous=False
if tok[0] == NAME and tok[1] in ['Symbol', 'Function']:
split = True
elif split and tok[0] == NAME:
symbol = tok[1][1:-1]
if predicate(symbol):
tok_type = result[-2][1] # Symbol or Function
del result[-2:] # Get rid of the call to Symbol
i = 0
while i < len(symbol):
char = symbol[i]
if char in local_dict or char in global_dict:
result.append((NAME, "%s" % char))
elif char.isdigit():
chars = [char]
for i in range(i + 1, len(symbol)):
if not symbol[i].isdigit():
i -= 1
break
chars.append(symbol[i])
char = ''.join(chars)
result.extend([(NAME, 'Number'), (OP, '('),
(NAME, "'%s'" % char), (OP, ')')])
else:
use = tok_type if i == len(symbol) else 'Symbol'
result.extend([(NAME, use), (OP, '('),
(NAME, "'%s'" % char), (OP, ')')])
i += 1
# Set split_previous=True so will skip
# the closing parenthesis of the original Symbol
split = False
split_previous = True
continue
else:
split = False
result.append(tok)
return result
return _split_symbols
#: Splits symbol names for implicit multiplication.
#:
#: Intended to let expressions like ``xyz`` be parsed as ``x*y*z``. Does not
#: split Greek character names, so ``theta`` will *not* become
#: ``t*h*e*t*a``. Generally this should be used with
#: ``implicit_multiplication``.
split_symbols = split_symbols_custom(_token_splittable)
def implicit_multiplication(tokens: List[TOKEN], local_dict: DICT,
global_dict: DICT) -> List[TOKEN]:
"""Makes the multiplication operator optional in most cases.
Use this before :func:`implicit_application`, otherwise expressions like
``sin 2x`` will be parsed as ``x * sin(2)`` rather than ``sin(2*x)``.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, implicit_multiplication)
>>> transformations = standard_transformations + (implicit_multiplication,)
>>> parse_expr('3 x y', transformations=transformations)
3*x*y
"""
# These are interdependent steps, so we don't expose them separately
res1 = _group_parentheses(implicit_multiplication)(tokens, local_dict, global_dict)
res2 = _apply_functions(res1, local_dict, global_dict)
res3 = _implicit_multiplication(res2, local_dict, global_dict)
result = _flatten(res3)
return result
def implicit_application(tokens: List[TOKEN], local_dict: DICT,
global_dict: DICT) -> List[TOKEN]:
"""Makes parentheses optional in some cases for function calls.
Use this after :func:`implicit_multiplication`, otherwise expressions
like ``sin 2x`` will be parsed as ``x * sin(2)`` rather than
``sin(2*x)``.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, implicit_application)
>>> transformations = standard_transformations + (implicit_application,)
>>> parse_expr('cot z + csc z', transformations=transformations)
cot(z) + csc(z)
"""
res1 = _group_parentheses(implicit_application)(tokens, local_dict, global_dict)
res2 = _apply_functions(res1, local_dict, global_dict)
res3 = _implicit_application(res2, local_dict, global_dict)
result = _flatten(res3)
return result
def implicit_multiplication_application(result: List[TOKEN], local_dict: DICT,
global_dict: DICT) -> List[TOKEN]:
"""Allows a slightly relaxed syntax.
- Parentheses for single-argument method calls are optional.
- Multiplication is implicit.
- Symbol names can be split (i.e. spaces are not needed between
symbols).
- Functions can be exponentiated.
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, implicit_multiplication_application)
>>> parse_expr("10sin**2 x**2 + 3xyz + tan theta",
... transformations=(standard_transformations +
... (implicit_multiplication_application,)))
3*x*y*z + 10*sin(x**2)**2 + tan(theta)
"""
for step in (split_symbols, implicit_multiplication,
implicit_application, function_exponentiation):
result = step(result, local_dict, global_dict)
return result
def auto_symbol(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Inserts calls to ``Symbol``/``Function`` for undefined variables."""
result: List[TOKEN] = []
prevTok = (-1, '')
tokens.append((-1, '')) # so zip traverses all tokens
for tok, nextTok in zip(tokens, tokens[1:]):
tokNum, tokVal = tok
nextTokNum, nextTokVal = nextTok
if tokNum == NAME:
name = tokVal
if (name in ['True', 'False', 'None']
or iskeyword(name)
# Don't convert attribute access
or (prevTok[0] == OP and prevTok[1] == '.')
# Don't convert keyword arguments
or (prevTok[0] == OP and prevTok[1] in ('(', ',')
and nextTokNum == OP and nextTokVal == '=')
# the name has already been defined
or name in local_dict and local_dict[name] is not null):
result.append((NAME, name))
continue
elif name in local_dict:
local_dict.setdefault(null, set()).add(name)
if nextTokVal == '(':
local_dict[name] = Function(name)
else:
local_dict[name] = Symbol(name)
result.append((NAME, name))
continue
elif name in global_dict:
obj = global_dict[name]
if isinstance(obj, (AssumptionKeys, Basic, type)) or callable(obj):
result.append((NAME, name))
continue
result.extend([
(NAME, 'Symbol' if nextTokVal != '(' else 'Function'),
(OP, '('),
(NAME, repr(str(name))),
(OP, ')'),
])
else:
result.append((tokNum, tokVal))
prevTok = (tokNum, tokVal)
return result
def lambda_notation(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Substitutes "lambda" with its SymPy equivalent Lambda().
However, the conversion does not take place if only "lambda"
is passed because that is a syntax error.
"""
result: List[TOKEN] = []
flag = False
toknum, tokval = tokens[0]
tokLen = len(tokens)
if toknum == NAME and tokval == 'lambda':
if tokLen == 2 or tokLen == 3 and tokens[1][0] == NEWLINE:
# In Python 3.6.7+, inputs without a newline get NEWLINE added to
# the tokens
result.extend(tokens)
elif tokLen > 2:
result.extend([
(NAME, 'Lambda'),
(OP, '('),
(OP, '('),
(OP, ')'),
(OP, ')'),
])
for tokNum, tokVal in tokens[1:]:
if tokNum == OP and tokVal == ':':
tokVal = ','
flag = True
if not flag and tokNum == OP and tokVal in ('*', '**'):
raise TokenError("Starred arguments in lambda not supported")
if flag:
result.insert(-1, (tokNum, tokVal))
else:
result.insert(-2, (tokNum, tokVal))
else:
result.extend(tokens)
return result
def factorial_notation(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Allows standard notation for factorial."""
result: List[TOKEN] = []
nfactorial = 0
for toknum, tokval in tokens:
if toknum == ERRORTOKEN:
op = tokval
if op == '!':
nfactorial += 1
else:
nfactorial = 0
result.append((OP, op))
else:
if nfactorial == 1:
result = _add_factorial_tokens('factorial', result)
elif nfactorial == 2:
result = _add_factorial_tokens('factorial2', result)
elif nfactorial > 2:
raise TokenError
nfactorial = 0
result.append((toknum, tokval))
return result
def convert_xor(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Treats XOR, ``^``, as exponentiation, ``**``."""
result: List[TOKEN] = []
for toknum, tokval in tokens:
if toknum == OP:
if tokval == '^':
result.append((OP, '**'))
else:
result.append((toknum, tokval))
else:
result.append((toknum, tokval))
return result
def repeated_decimals(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""
Allows 0.2[1] notation to represent the repeated decimal 0.2111... (19/90)
Run this before auto_number.
"""
result: List[TOKEN] = []
def is_digit(s):
return all(i in '0123456789_' for i in s)
# num will running match any DECIMAL [ INTEGER ]
num: List[TOKEN] = []
for toknum, tokval in tokens:
if toknum == NUMBER:
if (not num and '.' in tokval and 'e' not in tokval.lower() and
'j' not in tokval.lower()):
num.append((toknum, tokval))
elif is_digit(tokval)and len(num) == 2:
num.append((toknum, tokval))
elif is_digit(tokval) and len(num) == 3 and is_digit(num[-1][1]):
# Python 2 tokenizes 00123 as '00', '123'
# Python 3 tokenizes 01289 as '012', '89'
num.append((toknum, tokval))
else:
num = []
elif toknum == OP:
if tokval == '[' and len(num) == 1:
num.append((OP, tokval))
elif tokval == ']' and len(num) >= 3:
num.append((OP, tokval))
elif tokval == '.' and not num:
# handle .[1]
num.append((NUMBER, '0.'))
else:
num = []
else:
num = []
result.append((toknum, tokval))
if num and num[-1][1] == ']':
# pre.post[repetend] = a + b/c + d/e where a = pre, b/c = post,
# and d/e = repetend
result = result[:-len(num)]
pre, post = num[0][1].split('.')
repetend = num[2][1]
if len(num) == 5:
repetend += num[3][1]
pre = pre.replace('_', '')
post = post.replace('_', '')
repetend = repetend.replace('_', '')
zeros = '0'*len(post)
post, repetends = [w.lstrip('0') for w in [post, repetend]]
# or else interpreted as octal
a = pre or '0'
b, c = post or '0', '1' + zeros
d, e = repetends, ('9'*len(repetend)) + zeros
seq = [
(OP, '('),
(NAME, 'Integer'),
(OP, '('),
(NUMBER, a),
(OP, ')'),
(OP, '+'),
(NAME, 'Rational'),
(OP, '('),
(NUMBER, b),
(OP, ','),
(NUMBER, c),
(OP, ')'),
(OP, '+'),
(NAME, 'Rational'),
(OP, '('),
(NUMBER, d),
(OP, ','),
(NUMBER, e),
(OP, ')'),
(OP, ')'),
]
result.extend(seq)
num = []
return result
def auto_number(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""
Converts numeric literals to use SymPy equivalents.
Complex numbers use ``I``, integer literals use ``Integer``, and float
literals use ``Float``.
"""
result: List[TOKEN] = []
for toknum, tokval in tokens:
if toknum == NUMBER:
number = tokval
postfix = []
if number.endswith('j') or number.endswith('J'):
number = number[:-1]
postfix = [(OP, '*'), (NAME, 'I')]
if '.' in number or (('e' in number or 'E' in number) and
not (number.startswith('0x') or number.startswith('0X'))):
seq = [(NAME, 'Float'), (OP, '('),
(NUMBER, repr(str(number))), (OP, ')')]
else:
seq = [(NAME, 'Integer'), (OP, '('), (
NUMBER, number), (OP, ')')]
result.extend(seq + postfix)
else:
result.append((toknum, tokval))
return result
def rationalize(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Converts floats into ``Rational``. Run AFTER ``auto_number``."""
result: List[TOKEN] = []
passed_float = False
for toknum, tokval in tokens:
if toknum == NAME:
if tokval == 'Float':
passed_float = True
tokval = 'Rational'
result.append((toknum, tokval))
elif passed_float == True and toknum == NUMBER:
passed_float = False
result.append((STRING, tokval))
else:
result.append((toknum, tokval))
return result
def _transform_equals_sign(tokens: List[TOKEN], local_dict: DICT, global_dict: DICT):
"""Transforms the equals sign ``=`` to instances of Eq.
This is a helper function for ``convert_equals_signs``.
Works with expressions containing one equals sign and no
nesting. Expressions like ``(1=2)=False`` will not work with this
and should be used with ``convert_equals_signs``.
Examples: 1=2 to Eq(1,2)
1*2=x to Eq(1*2, x)
This does not deal with function arguments yet.
"""
result: List[TOKEN] = []
if (OP, "=") in tokens:
result.append((NAME, "Eq"))
result.append((OP, "("))
for token in tokens:
if token == (OP, "="):
result.append((OP, ","))
continue
result.append(token)
result.append((OP, ")"))
else:
result = tokens
return result
def convert_equals_signs(tokens: List[TOKEN], local_dict: DICT,
global_dict: DICT) -> List[TOKEN]:
""" Transforms all the equals signs ``=`` to instances of Eq.
Parses the equals signs in the expression and replaces them with
appropriate Eq instances. Also works with nested equals signs.
Does not yet play well with function arguments.
For example, the expression ``(x=y)`` is ambiguous and can be interpreted
as x being an argument to a function and ``convert_equals_signs`` will not
work for this.
See also
========
convert_equality_operators
Examples
========
>>> from sympy.parsing.sympy_parser import (parse_expr,
... standard_transformations, convert_equals_signs)
>>> parse_expr("1*2=x", transformations=(
... standard_transformations + (convert_equals_signs,)))
Eq(2, x)
>>> parse_expr("(1*2=x)=False", transformations=(
... standard_transformations + (convert_equals_signs,)))
Eq(Eq(2, x), False)
"""
res1 = _group_parentheses(convert_equals_signs)(tokens, local_dict, global_dict)
res2 = _apply_functions(res1, local_dict, global_dict)
res3 = _transform_equals_sign(res2, local_dict, global_dict)
result = _flatten(res3)
return result
#: Standard transformations for :func:`parse_expr`.
#: Inserts calls to :class:`~.Symbol`, :class:`~.Integer`, and other SymPy
#: datatypes and allows the use of standard factorial notation (e.g. ``x!``).
standard_transformations: tTuple[TRANS, ...] \
= (lambda_notation, auto_symbol, repeated_decimals, auto_number,
factorial_notation)
def stringify_expr(s: str, local_dict: DICT, global_dict: DICT,
transformations: tTuple[TRANS, ...]) -> str:
"""
Converts the string ``s`` to Python code, in ``local_dict``
Generally, ``parse_expr`` should be used.
"""
tokens = []
input_code = StringIO(s.strip())
for toknum, tokval, _, _, _ in generate_tokens(input_code.readline):
tokens.append((toknum, tokval))
for transform in transformations:
tokens = transform(tokens, local_dict, global_dict)
return untokenize(tokens)
def eval_expr(code, local_dict: DICT, global_dict: DICT):
"""
Evaluate Python code generated by ``stringify_expr``.
Generally, ``parse_expr`` should be used.
"""
expr = eval(
code, global_dict, local_dict) # take local objects in preference
return expr
def parse_expr(s: str, local_dict: Optional[DICT] = None,
transformations: tUnion[tTuple[TRANS, ...], str] \
= standard_transformations,
global_dict: Optional[DICT] = None, evaluate=True):
"""Converts the string ``s`` to a SymPy expression, in ``local_dict``
Parameters
==========
s : str
The string to parse.
local_dict : dict, optional
A dictionary of local variables to use when parsing.
global_dict : dict, optional
A dictionary of global variables. By default, this is initialized
with ``from sympy import *``; provide this parameter to override
this behavior (for instance, to parse ``"Q & S"``).
transformations : tuple or str
A tuple of transformation functions used to modify the tokens of the
parsed expression before evaluation. The default transformations
convert numeric literals into their SymPy equivalents, convert
undefined variables into SymPy symbols, and allow the use of standard
mathematical factorial notation (e.g. ``x!``). Selection via
string is available (see below).
evaluate : bool, optional
When False, the order of the arguments will remain as they were in the
string and automatic simplification that would normally occur is
suppressed. (see examples)
Examples
========
>>> from sympy.parsing.sympy_parser import parse_expr
>>> parse_expr("1/2")
1/2
>>> type(_)
<class 'sympy.core.numbers.Half'>
>>> from sympy.parsing.sympy_parser import standard_transformations,\\
... implicit_multiplication_application
>>> transformations = (standard_transformations +
... (implicit_multiplication_application,))
>>> parse_expr("2x", transformations=transformations)
2*x
When evaluate=False, some automatic simplifications will not occur:
>>> parse_expr("2**3"), parse_expr("2**3", evaluate=False)
(8, 2**3)
In addition the order of the arguments will not be made canonical.
This feature allows one to tell exactly how the expression was entered:
>>> a = parse_expr('1 + x', evaluate=False)
>>> b = parse_expr('x + 1', evaluate=0)
>>> a == b
False
>>> a.args
(1, x)
>>> b.args
(x, 1)
Note, however, that when these expressions are printed they will
appear the same:
>>> assert str(a) == str(b)
As a convenience, transformations can be seen by printing ``transformations``:
>>> from sympy.parsing.sympy_parser import transformations
>>> print(transformations)
0: lambda_notation
1: auto_symbol
2: repeated_decimals
3: auto_number
4: factorial_notation
5: implicit_multiplication_application
6: convert_xor
7: implicit_application
8: implicit_multiplication
9: convert_equals_signs
10: function_exponentiation
11: rationalize
The ``T`` object provides a way to select these transformations:
>>> from sympy.parsing.sympy_parser import T
If you print it, you will see the same list as shown above.
>>> str(T) == str(transformations)
True
Standard slicing will return a tuple of transformations:
>>> T[:5] == standard_transformations
True
So ``T`` can be used to specify the parsing transformations:
>>> parse_expr("2x", transformations=T[:5])
Traceback (most recent call last):
...
SyntaxError: invalid syntax
>>> parse_expr("2x", transformations=T[:6])
2*x
>>> parse_expr('.3', transformations=T[3, 11])
3/10
>>> parse_expr('.3x', transformations=T[:])
3*x/10
As a further convenience, strings 'implicit' and 'all' can be used
to select 0-5 and all the transformations, respectively.
>>> parse_expr('.3x', transformations='all')
3*x/10
See Also
========
stringify_expr, eval_expr, standard_transformations,
implicit_multiplication_application
"""
if local_dict is None:
local_dict = {}
elif not isinstance(local_dict, dict):
raise TypeError('expecting local_dict to be a dict')
elif null in local_dict:
raise ValueError('cannot use "" in local_dict')
if global_dict is None:
global_dict = {}
exec('from sympy import *', global_dict)
builtins_dict = vars(builtins)
for name, obj in builtins_dict.items():
if isinstance(obj, types.BuiltinFunctionType):
global_dict[name] = obj
global_dict['max'] = Max
global_dict['min'] = Min
elif not isinstance(global_dict, dict):
raise TypeError('expecting global_dict to be a dict')
transformations = transformations or ()
if isinstance(transformations, str):
if transformations == 'all':
_transformations = T[:]
elif transformations == 'implicit':
_transformations = T[:6]
else:
raise ValueError('unknown transformation group name')
else:
_transformations = transformations
code = stringify_expr(s, local_dict, global_dict, _transformations)
if not evaluate:
code = compile(evaluateFalse(code), '<string>', 'eval')
try:
rv = eval_expr(code, local_dict, global_dict)
# restore neutral definitions for names
for i in local_dict.pop(null, ()):
local_dict[i] = null
return rv
except Exception as e:
# restore neutral definitions for names
for i in local_dict.pop(null, ()):
local_dict[i] = null
raise e from ValueError(f"Error from parse_expr with transformed code: {code!r}")
def evaluateFalse(s: str):
"""
Replaces operators with the SymPy equivalent and sets evaluate=False.
"""
node = ast.parse(s)
transformed_node = EvaluateFalseTransformer().visit(node)
# node is a Module, we want an Expression
transformed_node = ast.Expression(transformed_node.body[0].value)
return ast.fix_missing_locations(transformed_node)
class EvaluateFalseTransformer(ast.NodeTransformer):
operators = {
ast.Add: 'Add',
ast.Mult: 'Mul',
ast.Pow: 'Pow',
ast.Sub: 'Add',
ast.Div: 'Mul',
ast.BitOr: 'Or',
ast.BitAnd: 'And',
ast.BitXor: 'Not',
}
functions = (
'Abs', 'im', 're', 'sign', 'arg', 'conjugate',
'acos', 'acot', 'acsc', 'asec', 'asin', 'atan',
'acosh', 'acoth', 'acsch', 'asech', 'asinh', 'atanh',
'cos', 'cot', 'csc', 'sec', 'sin', 'tan',
'cosh', 'coth', 'csch', 'sech', 'sinh', 'tanh',
'exp', 'ln', 'log', 'sqrt', 'cbrt',
)
def flatten(self, args, func):
result = []
for arg in args:
if isinstance(arg, ast.Call):
arg_func = arg.func
if isinstance(arg_func, ast.Call):
arg_func = arg_func.func
if arg_func.id == func:
result.extend(self.flatten(arg.args, func))
else:
result.append(arg)
else:
result.append(arg)
return result
def visit_BinOp(self, node):
if node.op.__class__ in self.operators:
sympy_class = self.operators[node.op.__class__]
right = self.visit(node.right)
left = self.visit(node.left)
rev = False
if isinstance(node.op, ast.Sub):
right = ast.Call(
func=ast.Name(id='Mul', ctx=ast.Load()),
args=[ast.UnaryOp(op=ast.USub(), operand=ast.Num(1)), right],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
elif isinstance(node.op, ast.Div):
if isinstance(node.left, ast.UnaryOp):
left, right = right, left
rev = True
left = ast.Call(
func=ast.Name(id='Pow', ctx=ast.Load()),
args=[left, ast.UnaryOp(op=ast.USub(), operand=ast.Num(1))],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
else:
right = ast.Call(
func=ast.Name(id='Pow', ctx=ast.Load()),
args=[right, ast.UnaryOp(op=ast.USub(), operand=ast.Num(1))],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
if rev: # undo reversal
left, right = right, left
new_node = ast.Call(
func=ast.Name(id=sympy_class, ctx=ast.Load()),
args=[left, right],
keywords=[ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load()))],
starargs=None,
kwargs=None
)
if sympy_class in ('Add', 'Mul'):
# Denest Add or Mul as appropriate
new_node.args = self.flatten(new_node.args, sympy_class)
return new_node
return node
def visit_Call(self, node):
new_node = self.generic_visit(node)
if isinstance(node.func, ast.Name) and node.func.id in self.functions:
new_node.keywords.append(ast.keyword(arg='evaluate', value=ast.NameConstant(value=False, ctx=ast.Load())))
return new_node
_transformation = { # items can be added but never re-ordered
0: lambda_notation,
1: auto_symbol,
2: repeated_decimals,
3: auto_number,
4: factorial_notation,
5: implicit_multiplication_application,
6: convert_xor,
7: implicit_application,
8: implicit_multiplication,
9: convert_equals_signs,
10: function_exponentiation,
11: rationalize}
transformations = '\n'.join('%s: %s' % (i, func_name(f)) for i, f in _transformation.items())
class _T():
"""class to retrieve transformations from a given slice
EXAMPLES
========
>>> from sympy.parsing.sympy_parser import T, standard_transformations
>>> assert T[:5] == standard_transformations
"""
def __init__(self):
self.N = len(_transformation)
def __str__(self):
return transformations
def __getitem__(self, t):
if not type(t) is tuple:
t = (t,)
i = []
for ti in t:
if type(ti) is int:
i.append(range(self.N)[ti])
elif type(ti) is slice:
i.extend(list(range(*ti.indices(self.N))))
else:
raise TypeError('unexpected slice arg')
return tuple([_transformation[_] for _ in i])
T = _T()
|
bd196143f511732cf5cbf6001f5f46b1744443c284fe8cf8173d553119ea8266 | """
Second quantization operators and states for bosons.
This follow the formulation of Fetter and Welecka, "Quantum Theory
of Many-Particle Systems."
"""
from collections import defaultdict
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.cache import cacheit
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import Function
from sympy.core.mul import Mul
from sympy.core.numbers import I
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Dummy, Symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.dense import zeros
from sympy.printing.str import StrPrinter
from sympy.utilities.iterables import has_dups
__all__ = [
'Dagger',
'KroneckerDelta',
'BosonicOperator',
'AnnihilateBoson',
'CreateBoson',
'AnnihilateFermion',
'CreateFermion',
'FockState',
'FockStateBra',
'FockStateKet',
'FockStateBosonKet',
'FockStateBosonBra',
'FockStateFermionKet',
'FockStateFermionBra',
'BBra',
'BKet',
'FBra',
'FKet',
'F',
'Fd',
'B',
'Bd',
'apply_operators',
'InnerProduct',
'BosonicBasis',
'VarBosonicBasis',
'FixedBosonicBasis',
'Commutator',
'matrix_rep',
'contraction',
'wicks',
'NO',
'evaluate_deltas',
'AntiSymmetricTensor',
'substitute_dummies',
'PermutationOperator',
'simplify_index_permutations',
]
class SecondQuantizationError(Exception):
pass
class AppliesOnlyToSymbolicIndex(SecondQuantizationError):
pass
class ContractionAppliesOnlyToFermions(SecondQuantizationError):
pass
class ViolationOfPauliPrinciple(SecondQuantizationError):
pass
class SubstitutionOfAmbigousOperatorFailed(SecondQuantizationError):
pass
class WicksTheoremDoesNotApply(SecondQuantizationError):
pass
class Dagger(Expr):
"""
Hermitian conjugate of creation/annihilation operators.
Examples
========
>>> from sympy import I
>>> from sympy.physics.secondquant import Dagger, B, Bd
>>> Dagger(2*I)
-2*I
>>> Dagger(B(0))
CreateBoson(0)
>>> Dagger(Bd(0))
AnnihilateBoson(0)
"""
def __new__(cls, arg):
arg = sympify(arg)
r = cls.eval(arg)
if isinstance(r, Basic):
return r
obj = Basic.__new__(cls, arg)
return obj
@classmethod
def eval(cls, arg):
"""
Evaluates the Dagger instance.
Examples
========
>>> from sympy import I
>>> from sympy.physics.secondquant import Dagger, B, Bd
>>> Dagger(2*I)
-2*I
>>> Dagger(B(0))
CreateBoson(0)
>>> Dagger(Bd(0))
AnnihilateBoson(0)
The eval() method is called automatically.
"""
dagger = getattr(arg, '_dagger_', None)
if dagger is not None:
return dagger()
if isinstance(arg, Basic):
if arg.is_Add:
return Add(*tuple(map(Dagger, arg.args)))
if arg.is_Mul:
return Mul(*tuple(map(Dagger, reversed(arg.args))))
if arg.is_Number:
return arg
if arg.is_Pow:
return Pow(Dagger(arg.args[0]), arg.args[1])
if arg == I:
return -arg
else:
return None
def _dagger_(self):
return self.args[0]
class TensorSymbol(Expr):
is_commutative = True
class AntiSymmetricTensor(TensorSymbol):
"""Stores upper and lower indices in separate Tuple's.
Each group of indices is assumed to be antisymmetric.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import AntiSymmetricTensor
>>> i, j = symbols('i j', below_fermi=True)
>>> a, b = symbols('a b', above_fermi=True)
>>> AntiSymmetricTensor('v', (a, i), (b, j))
AntiSymmetricTensor(v, (a, i), (b, j))
>>> AntiSymmetricTensor('v', (i, a), (b, j))
-AntiSymmetricTensor(v, (a, i), (b, j))
As you can see, the indices are automatically sorted to a canonical form.
"""
def __new__(cls, symbol, upper, lower):
try:
upper, signu = _sort_anticommuting_fermions(
upper, key=cls._sortkey)
lower, signl = _sort_anticommuting_fermions(
lower, key=cls._sortkey)
except ViolationOfPauliPrinciple:
return S.Zero
symbol = sympify(symbol)
upper = Tuple(*upper)
lower = Tuple(*lower)
if (signu + signl) % 2:
return -TensorSymbol.__new__(cls, symbol, upper, lower)
else:
return TensorSymbol.__new__(cls, symbol, upper, lower)
@classmethod
def _sortkey(cls, index):
"""Key for sorting of indices.
particle < hole < general
FIXME: This is a bottle-neck, can we do it faster?
"""
h = hash(index)
label = str(index)
if isinstance(index, Dummy):
if index.assumptions0.get('above_fermi'):
return (20, label, h)
elif index.assumptions0.get('below_fermi'):
return (21, label, h)
else:
return (22, label, h)
if index.assumptions0.get('above_fermi'):
return (10, label, h)
elif index.assumptions0.get('below_fermi'):
return (11, label, h)
else:
return (12, label, h)
def _latex(self, printer):
return "{%s^{%s}_{%s}}" % (
self.symbol,
"".join([ i.name for i in self.args[1]]),
"".join([ i.name for i in self.args[2]])
)
@property
def symbol(self):
"""
Returns the symbol of the tensor.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import AntiSymmetricTensor
>>> i, j = symbols('i,j', below_fermi=True)
>>> a, b = symbols('a,b', above_fermi=True)
>>> AntiSymmetricTensor('v', (a, i), (b, j))
AntiSymmetricTensor(v, (a, i), (b, j))
>>> AntiSymmetricTensor('v', (a, i), (b, j)).symbol
v
"""
return self.args[0]
@property
def upper(self):
"""
Returns the upper indices.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import AntiSymmetricTensor
>>> i, j = symbols('i,j', below_fermi=True)
>>> a, b = symbols('a,b', above_fermi=True)
>>> AntiSymmetricTensor('v', (a, i), (b, j))
AntiSymmetricTensor(v, (a, i), (b, j))
>>> AntiSymmetricTensor('v', (a, i), (b, j)).upper
(a, i)
"""
return self.args[1]
@property
def lower(self):
"""
Returns the lower indices.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import AntiSymmetricTensor
>>> i, j = symbols('i,j', below_fermi=True)
>>> a, b = symbols('a,b', above_fermi=True)
>>> AntiSymmetricTensor('v', (a, i), (b, j))
AntiSymmetricTensor(v, (a, i), (b, j))
>>> AntiSymmetricTensor('v', (a, i), (b, j)).lower
(b, j)
"""
return self.args[2]
def __str__(self):
return "%s(%s,%s)" % self.args
class SqOperator(Expr):
"""
Base class for Second Quantization operators.
"""
op_symbol = 'sq'
is_commutative = False
def __new__(cls, k):
obj = Basic.__new__(cls, sympify(k))
return obj
@property
def state(self):
"""
Returns the state index related to this operator.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F, Fd, B, Bd
>>> p = Symbol('p')
>>> F(p).state
p
>>> Fd(p).state
p
>>> B(p).state
p
>>> Bd(p).state
p
"""
return self.args[0]
@property
def is_symbolic(self):
"""
Returns True if the state is a symbol (as opposed to a number).
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> p = Symbol('p')
>>> F(p).is_symbolic
True
>>> F(1).is_symbolic
False
"""
if self.state.is_Integer:
return False
else:
return True
def __repr__(self):
return NotImplemented
def __str__(self):
return "%s(%r)" % (self.op_symbol, self.state)
def apply_operator(self, state):
"""
Applies an operator to itself.
"""
raise NotImplementedError('implement apply_operator in a subclass')
class BosonicOperator(SqOperator):
pass
class Annihilator(SqOperator):
pass
class Creator(SqOperator):
pass
class AnnihilateBoson(BosonicOperator, Annihilator):
"""
Bosonic annihilation operator.
Examples
========
>>> from sympy.physics.secondquant import B
>>> from sympy.abc import x
>>> B(x)
AnnihilateBoson(x)
"""
op_symbol = 'b'
def _dagger_(self):
return CreateBoson(self.state)
def apply_operator(self, state):
"""
Apply state to self if self is not symbolic and state is a FockStateKet, else
multiply self by state.
Examples
========
>>> from sympy.physics.secondquant import B, BKet
>>> from sympy.abc import x, y, n
>>> B(x).apply_operator(y)
y*AnnihilateBoson(x)
>>> B(0).apply_operator(BKet((n,)))
sqrt(n)*FockStateBosonKet((n - 1,))
"""
if not self.is_symbolic and isinstance(state, FockStateKet):
element = self.state
amp = sqrt(state[element])
return amp*state.down(element)
else:
return Mul(self, state)
def __repr__(self):
return "AnnihilateBoson(%s)" % self.state
def _latex(self, printer):
if self.state is S.Zero:
return "b_{0}"
else:
return "b_{%s}" % self.state.name
class CreateBoson(BosonicOperator, Creator):
"""
Bosonic creation operator.
"""
op_symbol = 'b+'
def _dagger_(self):
return AnnihilateBoson(self.state)
def apply_operator(self, state):
"""
Apply state to self if self is not symbolic and state is a FockStateKet, else
multiply self by state.
Examples
========
>>> from sympy.physics.secondquant import B, Dagger, BKet
>>> from sympy.abc import x, y, n
>>> Dagger(B(x)).apply_operator(y)
y*CreateBoson(x)
>>> B(0).apply_operator(BKet((n,)))
sqrt(n)*FockStateBosonKet((n - 1,))
"""
if not self.is_symbolic and isinstance(state, FockStateKet):
element = self.state
amp = sqrt(state[element] + 1)
return amp*state.up(element)
else:
return Mul(self, state)
def __repr__(self):
return "CreateBoson(%s)" % self.state
def _latex(self, printer):
if self.state is S.Zero:
return "{b^\\dagger_{0}}"
else:
return "{b^\\dagger_{%s}}" % self.state.name
B = AnnihilateBoson
Bd = CreateBoson
class FermionicOperator(SqOperator):
@property
def is_restricted(self):
"""
Is this FermionicOperator restricted with respect to fermi level?
Returns
=======
1 : restricted to orbits above fermi
0 : no restriction
-1 : restricted to orbits below fermi
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F, Fd
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_restricted
1
>>> Fd(a).is_restricted
1
>>> F(i).is_restricted
-1
>>> Fd(i).is_restricted
-1
>>> F(p).is_restricted
0
>>> Fd(p).is_restricted
0
"""
ass = self.args[0].assumptions0
if ass.get("below_fermi"):
return -1
if ass.get("above_fermi"):
return 1
return 0
@property
def is_above_fermi(self):
"""
Does the index of this FermionicOperator allow values above fermi?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_above_fermi
True
>>> F(i).is_above_fermi
False
>>> F(p).is_above_fermi
True
Note
====
The same applies to creation operators Fd
"""
return not self.args[0].assumptions0.get("below_fermi")
@property
def is_below_fermi(self):
"""
Does the index of this FermionicOperator allow values below fermi?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_below_fermi
False
>>> F(i).is_below_fermi
True
>>> F(p).is_below_fermi
True
The same applies to creation operators Fd
"""
return not self.args[0].assumptions0.get("above_fermi")
@property
def is_only_below_fermi(self):
"""
Is the index of this FermionicOperator restricted to values below fermi?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_only_below_fermi
False
>>> F(i).is_only_below_fermi
True
>>> F(p).is_only_below_fermi
False
The same applies to creation operators Fd
"""
return self.is_below_fermi and not self.is_above_fermi
@property
def is_only_above_fermi(self):
"""
Is the index of this FermionicOperator restricted to values above fermi?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_only_above_fermi
True
>>> F(i).is_only_above_fermi
False
>>> F(p).is_only_above_fermi
False
The same applies to creation operators Fd
"""
return self.is_above_fermi and not self.is_below_fermi
def _sortkey(self):
h = hash(self)
label = str(self.args[0])
if self.is_only_q_creator:
return 1, label, h
if self.is_only_q_annihilator:
return 4, label, h
if isinstance(self, Annihilator):
return 3, label, h
if isinstance(self, Creator):
return 2, label, h
class AnnihilateFermion(FermionicOperator, Annihilator):
"""
Fermionic annihilation operator.
"""
op_symbol = 'f'
def _dagger_(self):
return CreateFermion(self.state)
def apply_operator(self, state):
"""
Apply state to self if self is not symbolic and state is a FockStateKet, else
multiply self by state.
Examples
========
>>> from sympy.physics.secondquant import B, Dagger, BKet
>>> from sympy.abc import x, y, n
>>> Dagger(B(x)).apply_operator(y)
y*CreateBoson(x)
>>> B(0).apply_operator(BKet((n,)))
sqrt(n)*FockStateBosonKet((n - 1,))
"""
if isinstance(state, FockStateFermionKet):
element = self.state
return state.down(element)
elif isinstance(state, Mul):
c_part, nc_part = state.args_cnc()
if isinstance(nc_part[0], FockStateFermionKet):
element = self.state
return Mul(*(c_part + [nc_part[0].down(element)] + nc_part[1:]))
else:
return Mul(self, state)
else:
return Mul(self, state)
@property
def is_q_creator(self):
"""
Can we create a quasi-particle? (create hole or create particle)
If so, would that be above or below the fermi surface?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_q_creator
0
>>> F(i).is_q_creator
-1
>>> F(p).is_q_creator
-1
"""
if self.is_below_fermi:
return -1
return 0
@property
def is_q_annihilator(self):
"""
Can we destroy a quasi-particle? (annihilate hole or annihilate particle)
If so, would that be above or below the fermi surface?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=1)
>>> i = Symbol('i', below_fermi=1)
>>> p = Symbol('p')
>>> F(a).is_q_annihilator
1
>>> F(i).is_q_annihilator
0
>>> F(p).is_q_annihilator
1
"""
if self.is_above_fermi:
return 1
return 0
@property
def is_only_q_creator(self):
"""
Always create a quasi-particle? (create hole or create particle)
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_only_q_creator
False
>>> F(i).is_only_q_creator
True
>>> F(p).is_only_q_creator
False
"""
return self.is_only_below_fermi
@property
def is_only_q_annihilator(self):
"""
Always destroy a quasi-particle? (annihilate hole or annihilate particle)
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import F
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> F(a).is_only_q_annihilator
True
>>> F(i).is_only_q_annihilator
False
>>> F(p).is_only_q_annihilator
False
"""
return self.is_only_above_fermi
def __repr__(self):
return "AnnihilateFermion(%s)" % self.state
def _latex(self, printer):
if self.state is S.Zero:
return "a_{0}"
else:
return "a_{%s}" % self.state.name
class CreateFermion(FermionicOperator, Creator):
"""
Fermionic creation operator.
"""
op_symbol = 'f+'
def _dagger_(self):
return AnnihilateFermion(self.state)
def apply_operator(self, state):
"""
Apply state to self if self is not symbolic and state is a FockStateKet, else
multiply self by state.
Examples
========
>>> from sympy.physics.secondquant import B, Dagger, BKet
>>> from sympy.abc import x, y, n
>>> Dagger(B(x)).apply_operator(y)
y*CreateBoson(x)
>>> B(0).apply_operator(BKet((n,)))
sqrt(n)*FockStateBosonKet((n - 1,))
"""
if isinstance(state, FockStateFermionKet):
element = self.state
return state.up(element)
elif isinstance(state, Mul):
c_part, nc_part = state.args_cnc()
if isinstance(nc_part[0], FockStateFermionKet):
element = self.state
return Mul(*(c_part + [nc_part[0].up(element)] + nc_part[1:]))
return Mul(self, state)
@property
def is_q_creator(self):
"""
Can we create a quasi-particle? (create hole or create particle)
If so, would that be above or below the fermi surface?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import Fd
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> Fd(a).is_q_creator
1
>>> Fd(i).is_q_creator
0
>>> Fd(p).is_q_creator
1
"""
if self.is_above_fermi:
return 1
return 0
@property
def is_q_annihilator(self):
"""
Can we destroy a quasi-particle? (annihilate hole or annihilate particle)
If so, would that be above or below the fermi surface?
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import Fd
>>> a = Symbol('a', above_fermi=1)
>>> i = Symbol('i', below_fermi=1)
>>> p = Symbol('p')
>>> Fd(a).is_q_annihilator
0
>>> Fd(i).is_q_annihilator
-1
>>> Fd(p).is_q_annihilator
-1
"""
if self.is_below_fermi:
return -1
return 0
@property
def is_only_q_creator(self):
"""
Always create a quasi-particle? (create hole or create particle)
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import Fd
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> Fd(a).is_only_q_creator
True
>>> Fd(i).is_only_q_creator
False
>>> Fd(p).is_only_q_creator
False
"""
return self.is_only_above_fermi
@property
def is_only_q_annihilator(self):
"""
Always destroy a quasi-particle? (annihilate hole or annihilate particle)
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import Fd
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> Fd(a).is_only_q_annihilator
False
>>> Fd(i).is_only_q_annihilator
True
>>> Fd(p).is_only_q_annihilator
False
"""
return self.is_only_below_fermi
def __repr__(self):
return "CreateFermion(%s)" % self.state
def _latex(self, printer):
if self.state is S.Zero:
return "{a^\\dagger_{0}}"
else:
return "{a^\\dagger_{%s}}" % self.state.name
Fd = CreateFermion
F = AnnihilateFermion
class FockState(Expr):
"""
Many particle Fock state with a sequence of occupation numbers.
Anywhere you can have a FockState, you can also have S.Zero.
All code must check for this!
Base class to represent FockStates.
"""
is_commutative = False
def __new__(cls, occupations):
"""
occupations is a list with two possible meanings:
- For bosons it is a list of occupation numbers.
Element i is the number of particles in state i.
- For fermions it is a list of occupied orbits.
Element 0 is the state that was occupied first, element i
is the i'th occupied state.
"""
occupations = list(map(sympify, occupations))
obj = Basic.__new__(cls, Tuple(*occupations))
return obj
def __getitem__(self, i):
i = int(i)
return self.args[0][i]
def __repr__(self):
return ("FockState(%r)") % (self.args)
def __str__(self):
return "%s%r%s" % (getattr(self, 'lbracket', ""), self._labels(), getattr(self, 'rbracket', ""))
def _labels(self):
return self.args[0]
def __len__(self):
return len(self.args[0])
def _latex(self, printer):
return "%s%s%s" % (getattr(self, 'lbracket_latex', ""), printer._print(self._labels()), getattr(self, 'rbracket_latex', ""))
class BosonState(FockState):
"""
Base class for FockStateBoson(Ket/Bra).
"""
def up(self, i):
"""
Performs the action of a creation operator.
Examples
========
>>> from sympy.physics.secondquant import BBra
>>> b = BBra([1, 2])
>>> b
FockStateBosonBra((1, 2))
>>> b.up(1)
FockStateBosonBra((1, 3))
"""
i = int(i)
new_occs = list(self.args[0])
new_occs[i] = new_occs[i] + S.One
return self.__class__(new_occs)
def down(self, i):
"""
Performs the action of an annihilation operator.
Examples
========
>>> from sympy.physics.secondquant import BBra
>>> b = BBra([1, 2])
>>> b
FockStateBosonBra((1, 2))
>>> b.down(1)
FockStateBosonBra((1, 1))
"""
i = int(i)
new_occs = list(self.args[0])
if new_occs[i] == S.Zero:
return S.Zero
else:
new_occs[i] = new_occs[i] - S.One
return self.__class__(new_occs)
class FermionState(FockState):
"""
Base class for FockStateFermion(Ket/Bra).
"""
fermi_level = 0
def __new__(cls, occupations, fermi_level=0):
occupations = list(map(sympify, occupations))
if len(occupations) > 1:
try:
(occupations, sign) = _sort_anticommuting_fermions(
occupations, key=hash)
except ViolationOfPauliPrinciple:
return S.Zero
else:
sign = 0
cls.fermi_level = fermi_level
if cls._count_holes(occupations) > fermi_level:
return S.Zero
if sign % 2:
return S.NegativeOne*FockState.__new__(cls, occupations)
else:
return FockState.__new__(cls, occupations)
def up(self, i):
"""
Performs the action of a creation operator.
Explanation
===========
If below fermi we try to remove a hole,
if above fermi we try to create a particle.
If general index p we return ``Kronecker(p,i)*self``
where ``i`` is a new symbol with restriction above or below.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import FKet
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
>>> FKet([]).up(a)
FockStateFermionKet((a,))
A creator acting on vacuum below fermi vanishes
>>> FKet([]).up(i)
0
"""
present = i in self.args[0]
if self._only_above_fermi(i):
if present:
return S.Zero
else:
return self._add_orbit(i)
elif self._only_below_fermi(i):
if present:
return self._remove_orbit(i)
else:
return S.Zero
else:
if present:
hole = Dummy("i", below_fermi=True)
return KroneckerDelta(i, hole)*self._remove_orbit(i)
else:
particle = Dummy("a", above_fermi=True)
return KroneckerDelta(i, particle)*self._add_orbit(i)
def down(self, i):
"""
Performs the action of an annihilation operator.
Explanation
===========
If below fermi we try to create a hole,
If above fermi we try to remove a particle.
If general index p we return ``Kronecker(p,i)*self``
where ``i`` is a new symbol with restriction above or below.
Examples
========
>>> from sympy import Symbol
>>> from sympy.physics.secondquant import FKet
>>> a = Symbol('a', above_fermi=True)
>>> i = Symbol('i', below_fermi=True)
>>> p = Symbol('p')
An annihilator acting on vacuum above fermi vanishes
>>> FKet([]).down(a)
0
Also below fermi, it vanishes, unless we specify a fermi level > 0
>>> FKet([]).down(i)
0
>>> FKet([],4).down(i)
FockStateFermionKet((i,))
"""
present = i in self.args[0]
if self._only_above_fermi(i):
if present:
return self._remove_orbit(i)
else:
return S.Zero
elif self._only_below_fermi(i):
if present:
return S.Zero
else:
return self._add_orbit(i)
else:
if present:
hole = Dummy("i", below_fermi=True)
return KroneckerDelta(i, hole)*self._add_orbit(i)
else:
particle = Dummy("a", above_fermi=True)
return KroneckerDelta(i, particle)*self._remove_orbit(i)
@classmethod
def _only_below_fermi(cls, i):
"""
Tests if given orbit is only below fermi surface.
If nothing can be concluded we return a conservative False.
"""
if i.is_number:
return i <= cls.fermi_level
if i.assumptions0.get('below_fermi'):
return True
return False
@classmethod
def _only_above_fermi(cls, i):
"""
Tests if given orbit is only above fermi surface.
If fermi level has not been set we return True.
If nothing can be concluded we return a conservative False.
"""
if i.is_number:
return i > cls.fermi_level
if i.assumptions0.get('above_fermi'):
return True
return not cls.fermi_level
def _remove_orbit(self, i):
"""
Removes particle/fills hole in orbit i. No input tests performed here.
"""
new_occs = list(self.args[0])
pos = new_occs.index(i)
del new_occs[pos]
if (pos) % 2:
return S.NegativeOne*self.__class__(new_occs, self.fermi_level)
else:
return self.__class__(new_occs, self.fermi_level)
def _add_orbit(self, i):
"""
Adds particle/creates hole in orbit i. No input tests performed here.
"""
return self.__class__((i,) + self.args[0], self.fermi_level)
@classmethod
def _count_holes(cls, list):
"""
Returns the number of identified hole states in list.
"""
return len([i for i in list if cls._only_below_fermi(i)])
def _negate_holes(self, list):
return tuple([-i if i <= self.fermi_level else i for i in list])
def __repr__(self):
if self.fermi_level:
return "FockStateKet(%r, fermi_level=%s)" % (self.args[0], self.fermi_level)
else:
return "FockStateKet(%r)" % (self.args[0],)
def _labels(self):
return self._negate_holes(self.args[0])
class FockStateKet(FockState):
"""
Representation of a ket.
"""
lbracket = '|'
rbracket = '>'
lbracket_latex = r'\left|'
rbracket_latex = r'\right\rangle'
class FockStateBra(FockState):
"""
Representation of a bra.
"""
lbracket = '<'
rbracket = '|'
lbracket_latex = r'\left\langle'
rbracket_latex = r'\right|'
def __mul__(self, other):
if isinstance(other, FockStateKet):
return InnerProduct(self, other)
else:
return Expr.__mul__(self, other)
class FockStateBosonKet(BosonState, FockStateKet):
"""
Many particle Fock state with a sequence of occupation numbers.
Occupation numbers can be any integer >= 0.
Examples
========
>>> from sympy.physics.secondquant import BKet
>>> BKet([1, 2])
FockStateBosonKet((1, 2))
"""
def _dagger_(self):
return FockStateBosonBra(*self.args)
class FockStateBosonBra(BosonState, FockStateBra):
"""
Describes a collection of BosonBra particles.
Examples
========
>>> from sympy.physics.secondquant import BBra
>>> BBra([1, 2])
FockStateBosonBra((1, 2))
"""
def _dagger_(self):
return FockStateBosonKet(*self.args)
class FockStateFermionKet(FermionState, FockStateKet):
"""
Many-particle Fock state with a sequence of occupied orbits.
Explanation
===========
Each state can only have one particle, so we choose to store a list of
occupied orbits rather than a tuple with occupation numbers (zeros and ones).
states below fermi level are holes, and are represented by negative labels
in the occupation list.
For symbolic state labels, the fermi_level caps the number of allowed hole-
states.
Examples
========
>>> from sympy.physics.secondquant import FKet
>>> FKet([1, 2])
FockStateFermionKet((1, 2))
"""
def _dagger_(self):
return FockStateFermionBra(*self.args)
class FockStateFermionBra(FermionState, FockStateBra):
"""
See Also
========
FockStateFermionKet
Examples
========
>>> from sympy.physics.secondquant import FBra
>>> FBra([1, 2])
FockStateFermionBra((1, 2))
"""
def _dagger_(self):
return FockStateFermionKet(*self.args)
BBra = FockStateBosonBra
BKet = FockStateBosonKet
FBra = FockStateFermionBra
FKet = FockStateFermionKet
def _apply_Mul(m):
"""
Take a Mul instance with operators and apply them to states.
Explanation
===========
This method applies all operators with integer state labels
to the actual states. For symbolic state labels, nothing is done.
When inner products of FockStates are encountered (like <a|b>),
they are converted to instances of InnerProduct.
This does not currently work on double inner products like,
<a|b><c|d>.
If the argument is not a Mul, it is simply returned as is.
"""
if not isinstance(m, Mul):
return m
c_part, nc_part = m.args_cnc()
n_nc = len(nc_part)
if n_nc in (0, 1):
return m
else:
last = nc_part[-1]
next_to_last = nc_part[-2]
if isinstance(last, FockStateKet):
if isinstance(next_to_last, SqOperator):
if next_to_last.is_symbolic:
return m
else:
result = next_to_last.apply_operator(last)
if result == 0:
return S.Zero
else:
return _apply_Mul(Mul(*(c_part + nc_part[:-2] + [result])))
elif isinstance(next_to_last, Pow):
if isinstance(next_to_last.base, SqOperator) and \
next_to_last.exp.is_Integer:
if next_to_last.base.is_symbolic:
return m
else:
result = last
for i in range(next_to_last.exp):
result = next_to_last.base.apply_operator(result)
if result == 0:
break
if result == 0:
return S.Zero
else:
return _apply_Mul(Mul(*(c_part + nc_part[:-2] + [result])))
else:
return m
elif isinstance(next_to_last, FockStateBra):
result = InnerProduct(next_to_last, last)
if result == 0:
return S.Zero
else:
return _apply_Mul(Mul(*(c_part + nc_part[:-2] + [result])))
else:
return m
else:
return m
def apply_operators(e):
"""
Take a SymPy expression with operators and states and apply the operators.
Examples
========
>>> from sympy.physics.secondquant import apply_operators
>>> from sympy import sympify
>>> apply_operators(sympify(3)+4)
7
"""
e = e.expand()
muls = e.atoms(Mul)
subs_list = [(m, _apply_Mul(m)) for m in iter(muls)]
return e.subs(subs_list)
class InnerProduct(Basic):
"""
An unevaluated inner product between a bra and ket.
Explanation
===========
Currently this class just reduces things to a product of
Kronecker Deltas. In the future, we could introduce abstract
states like ``|a>`` and ``|b>``, and leave the inner product unevaluated as
``<a|b>``.
"""
is_commutative = True
def __new__(cls, bra, ket):
if not isinstance(bra, FockStateBra):
raise TypeError("must be a bra")
if not isinstance(ket, FockStateKet):
raise TypeError("must be a ket")
return cls.eval(bra, ket)
@classmethod
def eval(cls, bra, ket):
result = S.One
for i, j in zip(bra.args[0], ket.args[0]):
result *= KroneckerDelta(i, j)
if result == 0:
break
return result
@property
def bra(self):
"""Returns the bra part of the state"""
return self.args[0]
@property
def ket(self):
"""Returns the ket part of the state"""
return self.args[1]
def __repr__(self):
sbra = repr(self.bra)
sket = repr(self.ket)
return "%s|%s" % (sbra[:-1], sket[1:])
def __str__(self):
return self.__repr__()
def matrix_rep(op, basis):
"""
Find the representation of an operator in a basis.
Examples
========
>>> from sympy.physics.secondquant import VarBosonicBasis, B, matrix_rep
>>> b = VarBosonicBasis(5)
>>> o = B(0)
>>> matrix_rep(o, b)
Matrix([
[0, 1, 0, 0, 0],
[0, 0, sqrt(2), 0, 0],
[0, 0, 0, sqrt(3), 0],
[0, 0, 0, 0, 2],
[0, 0, 0, 0, 0]])
"""
a = zeros(len(basis))
for i in range(len(basis)):
for j in range(len(basis)):
a[i, j] = apply_operators(Dagger(basis[i])*op*basis[j])
return a
class BosonicBasis:
"""
Base class for a basis set of bosonic Fock states.
"""
pass
class VarBosonicBasis:
"""
A single state, variable particle number basis set.
Examples
========
>>> from sympy.physics.secondquant import VarBosonicBasis
>>> b = VarBosonicBasis(5)
>>> b
[FockState((0,)), FockState((1,)), FockState((2,)),
FockState((3,)), FockState((4,))]
"""
def __init__(self, n_max):
self.n_max = n_max
self._build_states()
def _build_states(self):
self.basis = []
for i in range(self.n_max):
self.basis.append(FockStateBosonKet([i]))
self.n_basis = len(self.basis)
def index(self, state):
"""
Returns the index of state in basis.
Examples
========
>>> from sympy.physics.secondquant import VarBosonicBasis
>>> b = VarBosonicBasis(3)
>>> state = b.state(1)
>>> b
[FockState((0,)), FockState((1,)), FockState((2,))]
>>> state
FockStateBosonKet((1,))
>>> b.index(state)
1
"""
return self.basis.index(state)
def state(self, i):
"""
The state of a single basis.
Examples
========
>>> from sympy.physics.secondquant import VarBosonicBasis
>>> b = VarBosonicBasis(5)
>>> b.state(3)
FockStateBosonKet((3,))
"""
return self.basis[i]
def __getitem__(self, i):
return self.state(i)
def __len__(self):
return len(self.basis)
def __repr__(self):
return repr(self.basis)
class FixedBosonicBasis(BosonicBasis):
"""
Fixed particle number basis set.
Examples
========
>>> from sympy.physics.secondquant import FixedBosonicBasis
>>> b = FixedBosonicBasis(2, 2)
>>> state = b.state(1)
>>> b
[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]
>>> state
FockStateBosonKet((1, 1))
>>> b.index(state)
1
"""
def __init__(self, n_particles, n_levels):
self.n_particles = n_particles
self.n_levels = n_levels
self._build_particle_locations()
self._build_states()
def _build_particle_locations(self):
tup = ["i%i" % i for i in range(self.n_particles)]
first_loop = "for i0 in range(%i)" % self.n_levels
other_loops = ''
for cur, prev in zip(tup[1:], tup):
temp = "for %s in range(%s + 1) " % (cur, prev)
other_loops = other_loops + temp
tup_string = "(%s)" % ", ".join(tup)
list_comp = "[%s %s %s]" % (tup_string, first_loop, other_loops)
result = eval(list_comp)
if self.n_particles == 1:
result = [(item,) for item in result]
self.particle_locations = result
def _build_states(self):
self.basis = []
for tuple_of_indices in self.particle_locations:
occ_numbers = self.n_levels*[0]
for level in tuple_of_indices:
occ_numbers[level] += 1
self.basis.append(FockStateBosonKet(occ_numbers))
self.n_basis = len(self.basis)
def index(self, state):
"""Returns the index of state in basis.
Examples
========
>>> from sympy.physics.secondquant import FixedBosonicBasis
>>> b = FixedBosonicBasis(2, 3)
>>> b.index(b.state(3))
3
"""
return self.basis.index(state)
def state(self, i):
"""Returns the state that lies at index i of the basis
Examples
========
>>> from sympy.physics.secondquant import FixedBosonicBasis
>>> b = FixedBosonicBasis(2, 3)
>>> b.state(3)
FockStateBosonKet((1, 0, 1))
"""
return self.basis[i]
def __getitem__(self, i):
return self.state(i)
def __len__(self):
return len(self.basis)
def __repr__(self):
return repr(self.basis)
class Commutator(Function):
"""
The Commutator: [A, B] = A*B - B*A
The arguments are ordered according to .__cmp__()
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import Commutator
>>> A, B = symbols('A,B', commutative=False)
>>> Commutator(B, A)
-Commutator(A, B)
Evaluate the commutator with .doit()
>>> comm = Commutator(A,B); comm
Commutator(A, B)
>>> comm.doit()
A*B - B*A
For two second quantization operators the commutator is evaluated
immediately:
>>> from sympy.physics.secondquant import Fd, F
>>> a = symbols('a', above_fermi=True)
>>> i = symbols('i', below_fermi=True)
>>> p,q = symbols('p,q')
>>> Commutator(Fd(a),Fd(i))
2*NO(CreateFermion(a)*CreateFermion(i))
But for more complicated expressions, the evaluation is triggered by
a call to .doit()
>>> comm = Commutator(Fd(p)*Fd(q),F(i)); comm
Commutator(CreateFermion(p)*CreateFermion(q), AnnihilateFermion(i))
>>> comm.doit(wicks=True)
-KroneckerDelta(i, p)*CreateFermion(q) +
KroneckerDelta(i, q)*CreateFermion(p)
"""
is_commutative = False
@classmethod
def eval(cls, a, b):
"""
The Commutator [A,B] is on canonical form if A < B.
Examples
========
>>> from sympy.physics.secondquant import Commutator, F, Fd
>>> from sympy.abc import x
>>> c1 = Commutator(F(x), Fd(x))
>>> c2 = Commutator(Fd(x), F(x))
>>> Commutator.eval(c1, c2)
0
"""
if not (a and b):
return S.Zero
if a == b:
return S.Zero
if a.is_commutative or b.is_commutative:
return S.Zero
#
# [A+B,C] -> [A,C] + [B,C]
#
a = a.expand()
if isinstance(a, Add):
return Add(*[cls(term, b) for term in a.args])
b = b.expand()
if isinstance(b, Add):
return Add(*[cls(a, term) for term in b.args])
#
# [xA,yB] -> xy*[A,B]
#
ca, nca = a.args_cnc()
cb, ncb = b.args_cnc()
c_part = list(ca) + list(cb)
if c_part:
return Mul(Mul(*c_part), cls(Mul._from_args(nca), Mul._from_args(ncb)))
#
# single second quantization operators
#
if isinstance(a, BosonicOperator) and isinstance(b, BosonicOperator):
if isinstance(b, CreateBoson) and isinstance(a, AnnihilateBoson):
return KroneckerDelta(a.state, b.state)
if isinstance(a, CreateBoson) and isinstance(b, AnnihilateBoson):
return S.NegativeOne*KroneckerDelta(a.state, b.state)
else:
return S.Zero
if isinstance(a, FermionicOperator) and isinstance(b, FermionicOperator):
return wicks(a*b) - wicks(b*a)
#
# Canonical ordering of arguments
#
if a.sort_key() > b.sort_key():
return S.NegativeOne*cls(b, a)
def doit(self, **hints):
"""
Enables the computation of complex expressions.
Examples
========
>>> from sympy.physics.secondquant import Commutator, F, Fd
>>> from sympy import symbols
>>> i, j = symbols('i,j', below_fermi=True)
>>> a, b = symbols('a,b', above_fermi=True)
>>> c = Commutator(Fd(a)*F(i),Fd(b)*F(j))
>>> c.doit(wicks=True)
0
"""
a = self.args[0]
b = self.args[1]
if hints.get("wicks"):
a = a.doit(**hints)
b = b.doit(**hints)
try:
return wicks(a*b) - wicks(b*a)
except ContractionAppliesOnlyToFermions:
pass
except WicksTheoremDoesNotApply:
pass
return (a*b - b*a).doit(**hints)
def __repr__(self):
return "Commutator(%s,%s)" % (self.args[0], self.args[1])
def __str__(self):
return "[%s,%s]" % (self.args[0], self.args[1])
def _latex(self, printer):
return "\\left[%s,%s\\right]" % tuple([
printer._print(arg) for arg in self.args])
class NO(Expr):
"""
This Object is used to represent normal ordering brackets.
i.e. {abcd} sometimes written :abcd:
Explanation
===========
Applying the function NO(arg) to an argument means that all operators in
the argument will be assumed to anticommute, and have vanishing
contractions. This allows an immediate reordering to canonical form
upon object creation.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import NO, F, Fd
>>> p,q = symbols('p,q')
>>> NO(Fd(p)*F(q))
NO(CreateFermion(p)*AnnihilateFermion(q))
>>> NO(F(q)*Fd(p))
-NO(CreateFermion(p)*AnnihilateFermion(q))
Note
====
If you want to generate a normal ordered equivalent of an expression, you
should use the function wicks(). This class only indicates that all
operators inside the brackets anticommute, and have vanishing contractions.
Nothing more, nothing less.
"""
is_commutative = False
def __new__(cls, arg):
"""
Use anticommutation to get canonical form of operators.
Explanation
===========
Employ associativity of normal ordered product: {ab{cd}} = {abcd}
but note that {ab}{cd} /= {abcd}.
We also employ distributivity: {ab + cd} = {ab} + {cd}.
Canonical form also implies expand() {ab(c+d)} = {abc} + {abd}.
"""
# {ab + cd} = {ab} + {cd}
arg = sympify(arg)
arg = arg.expand()
if arg.is_Add:
return Add(*[ cls(term) for term in arg.args])
if arg.is_Mul:
# take coefficient outside of normal ordering brackets
c_part, seq = arg.args_cnc()
if c_part:
coeff = Mul(*c_part)
if not seq:
return coeff
else:
coeff = S.One
# {ab{cd}} = {abcd}
newseq = []
foundit = False
for fac in seq:
if isinstance(fac, NO):
newseq.extend(fac.args)
foundit = True
else:
newseq.append(fac)
if foundit:
return coeff*cls(Mul(*newseq))
# We assume that the user don't mix B and F operators
if isinstance(seq[0], BosonicOperator):
raise NotImplementedError
try:
newseq, sign = _sort_anticommuting_fermions(seq)
except ViolationOfPauliPrinciple:
return S.Zero
if sign % 2:
return (S.NegativeOne*coeff)*cls(Mul(*newseq))
elif sign:
return coeff*cls(Mul(*newseq))
else:
pass # since sign==0, no permutations was necessary
# if we couldn't do anything with Mul object, we just
# mark it as normal ordered
if coeff != S.One:
return coeff*cls(Mul(*newseq))
return Expr.__new__(cls, Mul(*newseq))
if isinstance(arg, NO):
return arg
# if object was not Mul or Add, normal ordering does not apply
return arg
@property
def has_q_creators(self):
"""
Return 0 if the leftmost argument of the first argument is a not a
q_creator, else 1 if it is above fermi or -1 if it is below fermi.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import NO, F, Fd
>>> a = symbols('a', above_fermi=True)
>>> i = symbols('i', below_fermi=True)
>>> NO(Fd(a)*Fd(i)).has_q_creators
1
>>> NO(F(i)*F(a)).has_q_creators
-1
>>> NO(Fd(i)*F(a)).has_q_creators #doctest: +SKIP
0
"""
return self.args[0].args[0].is_q_creator
@property
def has_q_annihilators(self):
"""
Return 0 if the rightmost argument of the first argument is a not a
q_annihilator, else 1 if it is above fermi or -1 if it is below fermi.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import NO, F, Fd
>>> a = symbols('a', above_fermi=True)
>>> i = symbols('i', below_fermi=True)
>>> NO(Fd(a)*Fd(i)).has_q_annihilators
-1
>>> NO(F(i)*F(a)).has_q_annihilators
1
>>> NO(Fd(a)*F(i)).has_q_annihilators
0
"""
return self.args[0].args[-1].is_q_annihilator
def doit(self, **hints):
"""
Either removes the brackets or enables complex computations
in its arguments.
Examples
========
>>> from sympy.physics.secondquant import NO, Fd, F
>>> from textwrap import fill
>>> from sympy import symbols, Dummy
>>> p,q = symbols('p,q', cls=Dummy)
>>> print(fill(str(NO(Fd(p)*F(q)).doit())))
KroneckerDelta(_a, _p)*KroneckerDelta(_a,
_q)*CreateFermion(_a)*AnnihilateFermion(_a) + KroneckerDelta(_a,
_p)*KroneckerDelta(_i, _q)*CreateFermion(_a)*AnnihilateFermion(_i) -
KroneckerDelta(_a, _q)*KroneckerDelta(_i,
_p)*AnnihilateFermion(_a)*CreateFermion(_i) - KroneckerDelta(_i,
_p)*KroneckerDelta(_i, _q)*AnnihilateFermion(_i)*CreateFermion(_i)
"""
if hints.get("remove_brackets", True):
return self._remove_brackets()
else:
return self.__new__(type(self), self.args[0].doit(**hints))
def _remove_brackets(self):
"""
Returns the sorted string without normal order brackets.
The returned string have the property that no nonzero
contractions exist.
"""
# check if any creator is also an annihilator
subslist = []
for i in self.iter_q_creators():
if self[i].is_q_annihilator:
assume = self[i].state.assumptions0
# only operators with a dummy index can be split in two terms
if isinstance(self[i].state, Dummy):
# create indices with fermi restriction
assume.pop("above_fermi", None)
assume["below_fermi"] = True
below = Dummy('i', **assume)
assume.pop("below_fermi", None)
assume["above_fermi"] = True
above = Dummy('a', **assume)
cls = type(self[i])
split = (
self[i].__new__(cls, below)
* KroneckerDelta(below, self[i].state)
+ self[i].__new__(cls, above)
* KroneckerDelta(above, self[i].state)
)
subslist.append((self[i], split))
else:
raise SubstitutionOfAmbigousOperatorFailed(self[i])
if subslist:
result = NO(self.subs(subslist))
if isinstance(result, Add):
return Add(*[term.doit() for term in result.args])
else:
return self.args[0]
def _expand_operators(self):
"""
Returns a sum of NO objects that contain no ambiguous q-operators.
Explanation
===========
If an index q has range both above and below fermi, the operator F(q)
is ambiguous in the sense that it can be both a q-creator and a q-annihilator.
If q is dummy, it is assumed to be a summation variable and this method
rewrites it into a sum of NO terms with unambiguous operators:
{Fd(p)*F(q)} = {Fd(a)*F(b)} + {Fd(a)*F(i)} + {Fd(j)*F(b)} -{F(i)*Fd(j)}
where a,b are above and i,j are below fermi level.
"""
return NO(self._remove_brackets)
def __getitem__(self, i):
if isinstance(i, slice):
indices = i.indices(len(self))
return [self.args[0].args[i] for i in range(*indices)]
else:
return self.args[0].args[i]
def __len__(self):
return len(self.args[0].args)
def iter_q_annihilators(self):
"""
Iterates over the annihilation operators.
Examples
========
>>> from sympy import symbols
>>> i, j = symbols('i j', below_fermi=True)
>>> a, b = symbols('a b', above_fermi=True)
>>> from sympy.physics.secondquant import NO, F, Fd
>>> no = NO(Fd(a)*F(i)*F(b)*Fd(j))
>>> no.iter_q_creators()
<generator object... at 0x...>
>>> list(no.iter_q_creators())
[0, 1]
>>> list(no.iter_q_annihilators())
[3, 2]
"""
ops = self.args[0].args
iter = range(len(ops) - 1, -1, -1)
for i in iter:
if ops[i].is_q_annihilator:
yield i
else:
break
def iter_q_creators(self):
"""
Iterates over the creation operators.
Examples
========
>>> from sympy import symbols
>>> i, j = symbols('i j', below_fermi=True)
>>> a, b = symbols('a b', above_fermi=True)
>>> from sympy.physics.secondquant import NO, F, Fd
>>> no = NO(Fd(a)*F(i)*F(b)*Fd(j))
>>> no.iter_q_creators()
<generator object... at 0x...>
>>> list(no.iter_q_creators())
[0, 1]
>>> list(no.iter_q_annihilators())
[3, 2]
"""
ops = self.args[0].args
iter = range(0, len(ops))
for i in iter:
if ops[i].is_q_creator:
yield i
else:
break
def get_subNO(self, i):
"""
Returns a NO() without FermionicOperator at index i.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import F, NO
>>> p, q, r = symbols('p,q,r')
>>> NO(F(p)*F(q)*F(r)).get_subNO(1)
NO(AnnihilateFermion(p)*AnnihilateFermion(r))
"""
arg0 = self.args[0] # it's a Mul by definition of how it's created
mul = arg0._new_rawargs(*(arg0.args[:i] + arg0.args[i + 1:]))
return NO(mul)
def _latex(self, printer):
return "\\left\\{%s\\right\\}" % printer._print(self.args[0])
def __repr__(self):
return "NO(%s)" % self.args[0]
def __str__(self):
return ":%s:" % self.args[0]
def contraction(a, b):
"""
Calculates contraction of Fermionic operators a and b.
Examples
========
>>> from sympy import symbols
>>> from sympy.physics.secondquant import F, Fd, contraction
>>> p, q = symbols('p,q')
>>> a, b = symbols('a,b', above_fermi=True)
>>> i, j = symbols('i,j', below_fermi=True)
A contraction is non-zero only if a quasi-creator is to the right of a
quasi-annihilator:
>>> contraction(F(a),Fd(b))
KroneckerDelta(a, b)
>>> contraction(Fd(i),F(j))
KroneckerDelta(i, j)
For general indices a non-zero result restricts the indices to below/above
the fermi surface:
>>> contraction(Fd(p),F(q))
KroneckerDelta(_i, q)*KroneckerDelta(p, q)
>>> contraction(F(p),Fd(q))
KroneckerDelta(_a, q)*KroneckerDelta(p, q)
Two creators or two annihilators always vanishes:
>>> contraction(F(p),F(q))
0
>>> contraction(Fd(p),Fd(q))
0
"""
if isinstance(b, FermionicOperator) and isinstance(a, FermionicOperator):
if isinstance(a, AnnihilateFermion) and isinstance(b, CreateFermion):
if b.state.assumptions0.get("below_fermi"):
return S.Zero
if a.state.assumptions0.get("below_fermi"):
return S.Zero
if b.state.assumptions0.get("above_fermi"):
return KroneckerDelta(a.state, b.state)
if a.state.assumptions0.get("above_fermi"):
return KroneckerDelta(a.state, b.state)
return (KroneckerDelta(a.state, b.state)*
KroneckerDelta(b.state, Dummy('a', above_fermi=True)))
if isinstance(b, AnnihilateFermion) and isinstance(a, CreateFermion):
if b.state.assumptions0.get("above_fermi"):
return S.Zero
if a.state.assumptions0.get("above_fermi"):
return S.Zero
if b.state.assumptions0.get("below_fermi"):
return KroneckerDelta(a.state, b.state)
if a.state.assumptions0.get("below_fermi"):
return KroneckerDelta(a.state, b.state)
return (KroneckerDelta(a.state, b.state)*
KroneckerDelta(b.state, Dummy('i', below_fermi=True)))
# vanish if 2xAnnihilator or 2xCreator
return S.Zero
else:
#not fermion operators
t = ( isinstance(i, FermionicOperator) for i in (a, b) )
raise ContractionAppliesOnlyToFermions(*t)
def _sqkey(sq_operator):
"""Generates key for canonical sorting of SQ operators."""
return sq_operator._sortkey()
def _sort_anticommuting_fermions(string1, key=_sqkey):
"""Sort fermionic operators to canonical order, assuming all pairs anticommute.
Explanation
===========
Uses a bidirectional bubble sort. Items in string1 are not referenced
so in principle they may be any comparable objects. The sorting depends on the
operators '>' and '=='.
If the Pauli principle is violated, an exception is raised.
Returns
=======
tuple (sorted_str, sign)
sorted_str: list containing the sorted operators
sign: int telling how many times the sign should be changed
(if sign==0 the string was already sorted)
"""
verified = False
sign = 0
rng = list(range(len(string1) - 1))
rev = list(range(len(string1) - 3, -1, -1))
keys = list(map(key, string1))
key_val = dict(list(zip(keys, string1)))
while not verified:
verified = True
for i in rng:
left = keys[i]
right = keys[i + 1]
if left == right:
raise ViolationOfPauliPrinciple([left, right])
if left > right:
verified = False
keys[i:i + 2] = [right, left]
sign = sign + 1
if verified:
break
for i in rev:
left = keys[i]
right = keys[i + 1]
if left == right:
raise ViolationOfPauliPrinciple([left, right])
if left > right:
verified = False
keys[i:i + 2] = [right, left]
sign = sign + 1
string1 = [ key_val[k] for k in keys ]
return (string1, sign)
def evaluate_deltas(e):
"""
We evaluate KroneckerDelta symbols in the expression assuming Einstein summation.
Explanation
===========
If one index is repeated it is summed over and in effect substituted with
the other one. If both indices are repeated we substitute according to what
is the preferred index. this is determined by
KroneckerDelta.preferred_index and KroneckerDelta.killable_index.
In case there are no possible substitutions or if a substitution would
imply a loss of information, nothing is done.
In case an index appears in more than one KroneckerDelta, the resulting
substitution depends on the order of the factors. Since the ordering is platform
dependent, the literal expression resulting from this function may be hard to
predict.
Examples
========
We assume the following:
>>> from sympy import symbols, Function, Dummy, KroneckerDelta
>>> from sympy.physics.secondquant import evaluate_deltas
>>> i,j = symbols('i j', below_fermi=True, cls=Dummy)
>>> a,b = symbols('a b', above_fermi=True, cls=Dummy)
>>> p,q = symbols('p q', cls=Dummy)
>>> f = Function('f')
>>> t = Function('t')
The order of preference for these indices according to KroneckerDelta is
(a, b, i, j, p, q).
Trivial cases:
>>> evaluate_deltas(KroneckerDelta(i,j)*f(i)) # d_ij f(i) -> f(j)
f(_j)
>>> evaluate_deltas(KroneckerDelta(i,j)*f(j)) # d_ij f(j) -> f(i)
f(_i)
>>> evaluate_deltas(KroneckerDelta(i,p)*f(p)) # d_ip f(p) -> f(i)
f(_i)
>>> evaluate_deltas(KroneckerDelta(q,p)*f(p)) # d_qp f(p) -> f(q)
f(_q)
>>> evaluate_deltas(KroneckerDelta(q,p)*f(q)) # d_qp f(q) -> f(p)
f(_p)
More interesting cases:
>>> evaluate_deltas(KroneckerDelta(i,p)*t(a,i)*f(p,q))
f(_i, _q)*t(_a, _i)
>>> evaluate_deltas(KroneckerDelta(a,p)*t(a,i)*f(p,q))
f(_a, _q)*t(_a, _i)
>>> evaluate_deltas(KroneckerDelta(p,q)*f(p,q))
f(_p, _p)
Finally, here are some cases where nothing is done, because that would
imply a loss of information:
>>> evaluate_deltas(KroneckerDelta(i,p)*f(q))
f(_q)*KroneckerDelta(_i, _p)
>>> evaluate_deltas(KroneckerDelta(i,p)*f(i))
f(_i)*KroneckerDelta(_i, _p)
"""
# We treat Deltas only in mul objects
# for general function objects we don't evaluate KroneckerDeltas in arguments,
# but here we hard code exceptions to this rule
accepted_functions = (
Add,
)
if isinstance(e, accepted_functions):
return e.func(*[evaluate_deltas(arg) for arg in e.args])
elif isinstance(e, Mul):
# find all occurrences of delta function and count each index present in
# expression.
deltas = []
indices = {}
for i in e.args:
for s in i.free_symbols:
if s in indices:
indices[s] += 1
else:
indices[s] = 0 # geek counting simplifies logic below
if isinstance(i, KroneckerDelta):
deltas.append(i)
for d in deltas:
# If we do something, and there are more deltas, we should recurse
# to treat the resulting expression properly
if d.killable_index.is_Symbol and indices[d.killable_index]:
e = e.subs(d.killable_index, d.preferred_index)
if len(deltas) > 1:
return evaluate_deltas(e)
elif (d.preferred_index.is_Symbol and indices[d.preferred_index]
and d.indices_contain_equal_information):
e = e.subs(d.preferred_index, d.killable_index)
if len(deltas) > 1:
return evaluate_deltas(e)
else:
pass
return e
# nothing to do, maybe we hit a Symbol or a number
else:
return e
def substitute_dummies(expr, new_indices=False, pretty_indices={}):
"""
Collect terms by substitution of dummy variables.
Explanation
===========
This routine allows simplification of Add expressions containing terms
which differ only due to dummy variables.
The idea is to substitute all dummy variables consistently depending on
the structure of the term. For each term, we obtain a sequence of all
dummy variables, where the order is determined by the index range, what
factors the index belongs to and its position in each factor. See
_get_ordered_dummies() for more information about the sorting of dummies.
The index sequence is then substituted consistently in each term.
Examples
========
>>> from sympy import symbols, Function, Dummy
>>> from sympy.physics.secondquant import substitute_dummies
>>> a,b,c,d = symbols('a b c d', above_fermi=True, cls=Dummy)
>>> i,j = symbols('i j', below_fermi=True, cls=Dummy)
>>> f = Function('f')
>>> expr = f(a,b) + f(c,d); expr
f(_a, _b) + f(_c, _d)
Since a, b, c and d are equivalent summation indices, the expression can be
simplified to a single term (for which the dummy indices are still summed over)
>>> substitute_dummies(expr)
2*f(_a, _b)
Controlling output:
By default the dummy symbols that are already present in the expression
will be reused in a different permutation. However, if new_indices=True,
new dummies will be generated and inserted. The keyword 'pretty_indices'
can be used to control this generation of new symbols.
By default the new dummies will be generated on the form i_1, i_2, a_1,
etc. If you supply a dictionary with key:value pairs in the form:
{ index_group: string_of_letters }
The letters will be used as labels for the new dummy symbols. The
index_groups must be one of 'above', 'below' or 'general'.
>>> expr = f(a,b,i,j)
>>> my_dummies = { 'above':'st', 'below':'uv' }
>>> substitute_dummies(expr, new_indices=True, pretty_indices=my_dummies)
f(_s, _t, _u, _v)
If we run out of letters, or if there is no keyword for some index_group
the default dummy generator will be used as a fallback:
>>> p,q = symbols('p q', cls=Dummy) # general indices
>>> expr = f(p,q)
>>> substitute_dummies(expr, new_indices=True, pretty_indices=my_dummies)
f(_p_0, _p_1)
"""
# setup the replacing dummies
if new_indices:
letters_above = pretty_indices.get('above', "")
letters_below = pretty_indices.get('below', "")
letters_general = pretty_indices.get('general', "")
len_above = len(letters_above)
len_below = len(letters_below)
len_general = len(letters_general)
def _i(number):
try:
return letters_below[number]
except IndexError:
return 'i_' + str(number - len_below)
def _a(number):
try:
return letters_above[number]
except IndexError:
return 'a_' + str(number - len_above)
def _p(number):
try:
return letters_general[number]
except IndexError:
return 'p_' + str(number - len_general)
aboves = []
belows = []
generals = []
dummies = expr.atoms(Dummy)
if not new_indices:
dummies = sorted(dummies, key=default_sort_key)
# generate lists with the dummies we will insert
a = i = p = 0
for d in dummies:
assum = d.assumptions0
if assum.get("above_fermi"):
if new_indices:
sym = _a(a)
a += 1
l1 = aboves
elif assum.get("below_fermi"):
if new_indices:
sym = _i(i)
i += 1
l1 = belows
else:
if new_indices:
sym = _p(p)
p += 1
l1 = generals
if new_indices:
l1.append(Dummy(sym, **assum))
else:
l1.append(d)
expr = expr.expand()
terms = Add.make_args(expr)
new_terms = []
for term in terms:
i = iter(belows)
a = iter(aboves)
p = iter(generals)
ordered = _get_ordered_dummies(term)
subsdict = {}
for d in ordered:
if d.assumptions0.get('below_fermi'):
subsdict[d] = next(i)
elif d.assumptions0.get('above_fermi'):
subsdict[d] = next(a)
else:
subsdict[d] = next(p)
subslist = []
final_subs = []
for k, v in subsdict.items():
if k == v:
continue
if v in subsdict:
# We check if the sequence of substitutions end quickly. In
# that case, we can avoid temporary symbols if we ensure the
# correct substitution order.
if subsdict[v] in subsdict:
# (x, y) -> (y, x), we need a temporary variable
x = Dummy('x')
subslist.append((k, x))
final_subs.append((x, v))
else:
# (x, y) -> (y, a), x->y must be done last
# but before temporary variables are resolved
final_subs.insert(0, (k, v))
else:
subslist.append((k, v))
subslist.extend(final_subs)
new_terms.append(term.subs(subslist))
return Add(*new_terms)
class KeyPrinter(StrPrinter):
"""Printer for which only equal objects are equal in print"""
def _print_Dummy(self, expr):
return "(%s_%i)" % (expr.name, expr.dummy_index)
def __kprint(expr):
p = KeyPrinter()
return p.doprint(expr)
def _get_ordered_dummies(mul, verbose=False):
"""Returns all dummies in the mul sorted in canonical order.
Explanation
===========
The purpose of the canonical ordering is that dummies can be substituted
consistently across terms with the result that equivalent terms can be
simplified.
It is not possible to determine if two terms are equivalent based solely on
the dummy order. However, a consistent substitution guided by the ordered
dummies should lead to trivially (non-)equivalent terms, thereby revealing
the equivalence. This also means that if two terms have identical sequences of
dummies, the (non-)equivalence should already be apparent.
Strategy
--------
The canoncial order is given by an arbitrary sorting rule. A sort key
is determined for each dummy as a tuple that depends on all factors where
the index is present. The dummies are thereby sorted according to the
contraction structure of the term, instead of sorting based solely on the
dummy symbol itself.
After all dummies in the term has been assigned a key, we check for identical
keys, i.e. unorderable dummies. If any are found, we call a specialized
method, _determine_ambiguous(), that will determine a unique order based
on recursive calls to _get_ordered_dummies().
Key description
---------------
A high level description of the sort key:
1. Range of the dummy index
2. Relation to external (non-dummy) indices
3. Position of the index in the first factor
4. Position of the index in the second factor
The sort key is a tuple with the following components:
1. A single character indicating the range of the dummy (above, below
or general.)
2. A list of strings with fully masked string representations of all
factors where the dummy is present. By masked, we mean that dummies
are represented by a symbol to indicate either below fermi, above or
general. No other information is displayed about the dummies at
this point. The list is sorted stringwise.
3. An integer number indicating the position of the index, in the first
factor as sorted in 2.
4. An integer number indicating the position of the index, in the second
factor as sorted in 2.
If a factor is either of type AntiSymmetricTensor or SqOperator, the index
position in items 3 and 4 is indicated as 'upper' or 'lower' only.
(Creation operators are considered upper and annihilation operators lower.)
If the masked factors are identical, the two factors cannot be ordered
unambiguously in item 2. In this case, items 3, 4 are left out. If several
indices are contracted between the unorderable factors, it will be handled by
_determine_ambiguous()
"""
# setup dicts to avoid repeated calculations in key()
args = Mul.make_args(mul)
fac_dum = { fac: fac.atoms(Dummy) for fac in args }
fac_repr = { fac: __kprint(fac) for fac in args }
all_dums = set().union(*fac_dum.values())
mask = {}
for d in all_dums:
if d.assumptions0.get('below_fermi'):
mask[d] = '0'
elif d.assumptions0.get('above_fermi'):
mask[d] = '1'
else:
mask[d] = '2'
dum_repr = {d: __kprint(d) for d in all_dums}
def _key(d):
dumstruct = [ fac for fac in fac_dum if d in fac_dum[fac] ]
other_dums = set().union(*[fac_dum[fac] for fac in dumstruct])
fac = dumstruct[-1]
if other_dums is fac_dum[fac]:
other_dums = fac_dum[fac].copy()
other_dums.remove(d)
masked_facs = [ fac_repr[fac] for fac in dumstruct ]
for d2 in other_dums:
masked_facs = [ fac.replace(dum_repr[d2], mask[d2])
for fac in masked_facs ]
all_masked = [ fac.replace(dum_repr[d], mask[d])
for fac in masked_facs ]
masked_facs = dict(list(zip(dumstruct, masked_facs)))
# dummies for which the ordering cannot be determined
if has_dups(all_masked):
all_masked.sort()
return mask[d], tuple(all_masked) # positions are ambiguous
# sort factors according to fully masked strings
keydict = dict(list(zip(dumstruct, all_masked)))
dumstruct.sort(key=lambda x: keydict[x])
all_masked.sort()
pos_val = []
for fac in dumstruct:
if isinstance(fac, AntiSymmetricTensor):
if d in fac.upper:
pos_val.append('u')
if d in fac.lower:
pos_val.append('l')
elif isinstance(fac, Creator):
pos_val.append('u')
elif isinstance(fac, Annihilator):
pos_val.append('l')
elif isinstance(fac, NO):
ops = [ op for op in fac if op.has(d) ]
for op in ops:
if isinstance(op, Creator):
pos_val.append('u')
else:
pos_val.append('l')
else:
# fallback to position in string representation
facpos = -1
while 1:
facpos = masked_facs[fac].find(dum_repr[d], facpos + 1)
if facpos == -1:
break
pos_val.append(facpos)
return (mask[d], tuple(all_masked), pos_val[0], pos_val[-1])
dumkey = dict(list(zip(all_dums, list(map(_key, all_dums)))))
result = sorted(all_dums, key=lambda x: dumkey[x])
if has_dups(iter(dumkey.values())):
# We have ambiguities
unordered = defaultdict(set)
for d, k in dumkey.items():
unordered[k].add(d)
for k in [ k for k in unordered if len(unordered[k]) < 2 ]:
del unordered[k]
unordered = [ unordered[k] for k in sorted(unordered) ]
result = _determine_ambiguous(mul, result, unordered)
return result
def _determine_ambiguous(term, ordered, ambiguous_groups):
# We encountered a term for which the dummy substitution is ambiguous.
# This happens for terms with 2 or more contractions between factors that
# cannot be uniquely ordered independent of summation indices. For
# example:
#
# Sum(p, q) v^{p, .}_{q, .}v^{q, .}_{p, .}
#
# Assuming that the indices represented by . are dummies with the
# same range, the factors cannot be ordered, and there is no
# way to determine a consistent ordering of p and q.
#
# The strategy employed here, is to relabel all unambiguous dummies with
# non-dummy symbols and call _get_ordered_dummies again. This procedure is
# applied to the entire term so there is a possibility that
# _determine_ambiguous() is called again from a deeper recursion level.
# break recursion if there are no ordered dummies
all_ambiguous = set()
for dummies in ambiguous_groups:
all_ambiguous |= dummies
all_ordered = set(ordered) - all_ambiguous
if not all_ordered:
# FIXME: If we arrive here, there are no ordered dummies. A method to
# handle this needs to be implemented. In order to return something
# useful nevertheless, we choose arbitrarily the first dummy and
# determine the rest from this one. This method is dependent on the
# actual dummy labels which violates an assumption for the
# canonicalization procedure. A better implementation is needed.
group = [ d for d in ordered if d in ambiguous_groups[0] ]
d = group[0]
all_ordered.add(d)
ambiguous_groups[0].remove(d)
stored_counter = _symbol_factory._counter
subslist = []
for d in [ d for d in ordered if d in all_ordered ]:
nondum = _symbol_factory._next()
subslist.append((d, nondum))
newterm = term.subs(subslist)
neworder = _get_ordered_dummies(newterm)
_symbol_factory._set_counter(stored_counter)
# update ordered list with new information
for group in ambiguous_groups:
ordered_group = [ d for d in neworder if d in group ]
ordered_group.reverse()
result = []
for d in ordered:
if d in group:
result.append(ordered_group.pop())
else:
result.append(d)
ordered = result
return ordered
class _SymbolFactory:
def __init__(self, label):
self._counterVar = 0
self._label = label
def _set_counter(self, value):
"""
Sets counter to value.
"""
self._counterVar = value
@property
def _counter(self):
"""
What counter is currently at.
"""
return self._counterVar
def _next(self):
"""
Generates the next symbols and increments counter by 1.
"""
s = Symbol("%s%i" % (self._label, self._counterVar))
self._counterVar += 1
return s
_symbol_factory = _SymbolFactory('_]"]_') # most certainly a unique label
@cacheit
def _get_contractions(string1, keep_only_fully_contracted=False):
"""
Returns Add-object with contracted terms.
Uses recursion to find all contractions. -- Internal helper function --
Will find nonzero contractions in string1 between indices given in
leftrange and rightrange.
"""
# Should we store current level of contraction?
if keep_only_fully_contracted and string1:
result = []
else:
result = [NO(Mul(*string1))]
for i in range(len(string1) - 1):
for j in range(i + 1, len(string1)):
c = contraction(string1[i], string1[j])
if c:
sign = (j - i + 1) % 2
if sign:
coeff = S.NegativeOne*c
else:
coeff = c
#
# Call next level of recursion
# ============================
#
# We now need to find more contractions among operators
#
# oplist = string1[:i]+ string1[i+1:j] + string1[j+1:]
#
# To prevent overcounting, we don't allow contractions
# we have already encountered. i.e. contractions between
# string1[:i] <---> string1[i+1:j]
# and string1[:i] <---> string1[j+1:].
#
# This leaves the case:
oplist = string1[i + 1:j] + string1[j + 1:]
if oplist:
result.append(coeff*NO(
Mul(*string1[:i])*_get_contractions( oplist,
keep_only_fully_contracted=keep_only_fully_contracted)))
else:
result.append(coeff*NO( Mul(*string1[:i])))
if keep_only_fully_contracted:
break # next iteration over i leaves leftmost operator string1[0] uncontracted
return Add(*result)
def wicks(e, **kw_args):
"""
Returns the normal ordered equivalent of an expression using Wicks Theorem.
Examples
========
>>> from sympy import symbols, Dummy
>>> from sympy.physics.secondquant import wicks, F, Fd
>>> p, q, r = symbols('p,q,r')
>>> wicks(Fd(p)*F(q))
KroneckerDelta(_i, q)*KroneckerDelta(p, q) + NO(CreateFermion(p)*AnnihilateFermion(q))
By default, the expression is expanded:
>>> wicks(F(p)*(F(q)+F(r)))
NO(AnnihilateFermion(p)*AnnihilateFermion(q)) + NO(AnnihilateFermion(p)*AnnihilateFermion(r))
With the keyword 'keep_only_fully_contracted=True', only fully contracted
terms are returned.
By request, the result can be simplified in the following order:
-- KroneckerDelta functions are evaluated
-- Dummy variables are substituted consistently across terms
>>> p, q, r = symbols('p q r', cls=Dummy)
>>> wicks(Fd(p)*(F(q)+F(r)), keep_only_fully_contracted=True)
KroneckerDelta(_i, _q)*KroneckerDelta(_p, _q) + KroneckerDelta(_i, _r)*KroneckerDelta(_p, _r)
"""
if not e:
return S.Zero
opts = {
'simplify_kronecker_deltas': False,
'expand': True,
'simplify_dummies': False,
'keep_only_fully_contracted': False
}
opts.update(kw_args)
# check if we are already normally ordered
if isinstance(e, NO):
if opts['keep_only_fully_contracted']:
return S.Zero
else:
return e
elif isinstance(e, FermionicOperator):
if opts['keep_only_fully_contracted']:
return S.Zero
else:
return e
# break up any NO-objects, and evaluate commutators
e = e.doit(wicks=True)
# make sure we have only one term to consider
e = e.expand()
if isinstance(e, Add):
if opts['simplify_dummies']:
return substitute_dummies(Add(*[ wicks(term, **kw_args) for term in e.args]))
else:
return Add(*[ wicks(term, **kw_args) for term in e.args])
# For Mul-objects we can actually do something
if isinstance(e, Mul):
# we don't want to mess around with commuting part of Mul
# so we factorize it out before starting recursion
c_part = []
string1 = []
for factor in e.args:
if factor.is_commutative:
c_part.append(factor)
else:
string1.append(factor)
n = len(string1)
# catch trivial cases
if n == 0:
result = e
elif n == 1:
if opts['keep_only_fully_contracted']:
return S.Zero
else:
result = e
else: # non-trivial
if isinstance(string1[0], BosonicOperator):
raise NotImplementedError
string1 = tuple(string1)
# recursion over higher order contractions
result = _get_contractions(string1,
keep_only_fully_contracted=opts['keep_only_fully_contracted'] )
result = Mul(*c_part)*result
if opts['expand']:
result = result.expand()
if opts['simplify_kronecker_deltas']:
result = evaluate_deltas(result)
return result
# there was nothing to do
return e
class PermutationOperator(Expr):
"""
Represents the index permutation operator P(ij).
P(ij)*f(i)*g(j) = f(i)*g(j) - f(j)*g(i)
"""
is_commutative = True
def __new__(cls, i, j):
i, j = sorted(map(sympify, (i, j)), key=default_sort_key)
obj = Basic.__new__(cls, i, j)
return obj
def get_permuted(self, expr):
"""
Returns -expr with permuted indices.
Explanation
===========
>>> from sympy import symbols, Function
>>> from sympy.physics.secondquant import PermutationOperator
>>> p,q = symbols('p,q')
>>> f = Function('f')
>>> PermutationOperator(p,q).get_permuted(f(p,q))
-f(q, p)
"""
i = self.args[0]
j = self.args[1]
if expr.has(i) and expr.has(j):
tmp = Dummy()
expr = expr.subs(i, tmp)
expr = expr.subs(j, i)
expr = expr.subs(tmp, j)
return S.NegativeOne*expr
else:
return expr
def _latex(self, printer):
return "P(%s%s)" % self.args
def simplify_index_permutations(expr, permutation_operators):
"""
Performs simplification by introducing PermutationOperators where appropriate.
Explanation
===========
Schematically:
[abij] - [abji] - [baij] + [baji] -> P(ab)*P(ij)*[abij]
permutation_operators is a list of PermutationOperators to consider.
If permutation_operators=[P(ab),P(ij)] we will try to introduce the
permutation operators P(ij) and P(ab) in the expression. If there are other
possible simplifications, we ignore them.
>>> from sympy import symbols, Function
>>> from sympy.physics.secondquant import simplify_index_permutations
>>> from sympy.physics.secondquant import PermutationOperator
>>> p,q,r,s = symbols('p,q,r,s')
>>> f = Function('f')
>>> g = Function('g')
>>> expr = f(p)*g(q) - f(q)*g(p); expr
f(p)*g(q) - f(q)*g(p)
>>> simplify_index_permutations(expr,[PermutationOperator(p,q)])
f(p)*g(q)*PermutationOperator(p, q)
>>> PermutList = [PermutationOperator(p,q),PermutationOperator(r,s)]
>>> expr = f(p,r)*g(q,s) - f(q,r)*g(p,s) + f(q,s)*g(p,r) - f(p,s)*g(q,r)
>>> simplify_index_permutations(expr,PermutList)
f(p, r)*g(q, s)*PermutationOperator(p, q)*PermutationOperator(r, s)
"""
def _get_indices(expr, ind):
"""
Collects indices recursively in predictable order.
"""
result = []
for arg in expr.args:
if arg in ind:
result.append(arg)
else:
if arg.args:
result.extend(_get_indices(arg, ind))
return result
def _choose_one_to_keep(a, b, ind):
# we keep the one where indices in ind are in order ind[0] < ind[1]
return min(a, b, key=lambda x: default_sort_key(_get_indices(x, ind)))
expr = expr.expand()
if isinstance(expr, Add):
terms = set(expr.args)
for P in permutation_operators:
new_terms = set()
on_hold = set()
while terms:
term = terms.pop()
permuted = P.get_permuted(term)
if permuted in terms | on_hold:
try:
terms.remove(permuted)
except KeyError:
on_hold.remove(permuted)
keep = _choose_one_to_keep(term, permuted, P.args)
new_terms.add(P*keep)
else:
# Some terms must get a second chance because the permuted
# term may already have canonical dummy ordering. Then
# substitute_dummies() does nothing. However, the other
# term, if it exists, will be able to match with us.
permuted1 = permuted
permuted = substitute_dummies(permuted)
if permuted1 == permuted:
on_hold.add(term)
elif permuted in terms | on_hold:
try:
terms.remove(permuted)
except KeyError:
on_hold.remove(permuted)
keep = _choose_one_to_keep(term, permuted, P.args)
new_terms.add(P*keep)
else:
new_terms.add(term)
terms = new_terms | on_hold
return Add(*terms)
return expr
|
c2c071642d4c7650144bea43c694d06ba83a88dc5e99a8b2560708e190c086b2 | from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.numbers import (Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
from sympy.functions.elementary.complexes import polar_lift
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.bessel import besselk
from sympy.functions.special.gamma_functions import gamma
from sympy.matrices.dense import eye
from sympy.matrices.expressions.determinant import Determinant
from sympy.sets.fancysets import Range
from sympy.sets.sets import (Interval, ProductSet)
from sympy.simplify.simplify import simplify
from sympy.tensor.indexed import (Indexed, IndexedBase)
from sympy.core.numbers import comp
from sympy.integrals.integrals import integrate
from sympy.matrices import Matrix, MatrixSymbol
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.stats import density, median, marginal_distribution, Normal, Laplace, E, sample
from sympy.stats.joint_rv_types import (JointRV, MultivariateNormalDistribution,
JointDistributionHandmade, MultivariateT, NormalGamma,
GeneralizedMultivariateLogGammaOmega as GMVLGO, MultivariateBeta,
GeneralizedMultivariateLogGamma as GMVLG, MultivariateEwens,
Multinomial, NegativeMultinomial, MultivariateNormal,
MultivariateLaplace)
from sympy.testing.pytest import raises, XFAIL, skip, slow
from sympy.external import import_module
from sympy.abc import x, y
def test_Normal():
m = Normal('A', [1, 2], [[1, 0], [0, 1]])
A = MultivariateNormal('A', [1, 2], [[1, 0], [0, 1]])
assert m == A
assert density(m)(1, 2) == 1/(2*pi)
assert m.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
raises (ValueError, lambda:m[2])
n = Normal('B', [1, 2, 3], [[1, 0, 0], [0, 1, 0], [0, 0, 1]])
p = Normal('C', Matrix([1, 2]), Matrix([[1, 0], [0, 1]]))
assert density(m)(x, y) == density(p)(x, y)
assert marginal_distribution(n, 0, 1)(1, 2) == 1/(2*pi)
raises(ValueError, lambda: marginal_distribution(m))
assert integrate(density(m)(x, y), (x, -oo, oo), (y, -oo, oo)).evalf() == 1
N = Normal('N', [1, 2], [[x, 0], [0, y]])
assert density(N)(0, 0) == exp(-((4*x + y)/(2*x*y)))/(2*pi*sqrt(x*y))
raises (ValueError, lambda: Normal('M', [1, 2], [[1, 1], [1, -1]]))
# symbolic
n = symbols('n', integer=True, positive=True)
mu = MatrixSymbol('mu', n, 1)
sigma = MatrixSymbol('sigma', n, n)
X = Normal('X', mu, sigma)
assert density(X) == MultivariateNormalDistribution(mu, sigma)
raises (NotImplementedError, lambda: median(m))
# Below tests should work after issue #17267 is resolved
# assert E(X) == mu
# assert variance(X) == sigma
# test symbolic multivariate normal densities
n = 3
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X = density(X)
eval_a = density_X(obs).subs({Sg: eye(3),
mu: Matrix([0, 0, 0]), obs: Matrix([0, 0, 0])}).doit()
eval_b = density_X(0, 0, 0).subs({Sg: eye(3), mu: Matrix([0, 0, 0])}).doit()
assert eval_a == sqrt(2)/(4*pi**Rational(3/2))
assert eval_b == sqrt(2)/(4*pi**Rational(3/2))
n = symbols('n', integer=True, positive=True)
Sg = MatrixSymbol('Sg', n, n)
mu = MatrixSymbol('mu', n, 1)
obs = MatrixSymbol('obs', n, 1)
X = MultivariateNormal('X', mu, Sg)
density_X_at_obs = density(X)(obs)
expected_density = MatrixElement(
exp((S(1)/2) * (mu.T - obs.T) * Sg**(-1) * (-mu + obs)) / \
sqrt((2*pi)**n * Determinant(Sg)), 0, 0)
assert density_X_at_obs == expected_density
def test_MultivariateTDist():
t1 = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
assert(density(t1))(1, 1) == 1/(8*pi)
assert t1.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
assert integrate(density(t1)(x, y), (x, -oo, oo), \
(y, -oo, oo)).evalf() == 1
raises(ValueError, lambda: MultivariateT('T', [1, 2], [[1, 1], [1, -1]], 1))
t2 = MultivariateT('t2', [1, 2], [[x, 0], [0, y]], 1)
assert density(t2)(1, 2) == 1/(2*pi*sqrt(x*y))
def test_multivariate_laplace():
raises(ValueError, lambda: Laplace('T', [1, 2], [[1, 2], [2, 1]]))
L = Laplace('L', [1, 0], [[1, 0], [0, 1]])
L2 = MultivariateLaplace('L2', [1, 0], [[1, 0], [0, 1]])
assert density(L)(2, 3) == exp(2)*besselk(0, sqrt(39))/pi
L1 = Laplace('L1', [1, 2], [[x, 0], [0, y]])
assert density(L1)(0, 1) == \
exp(2/y)*besselk(0, sqrt((2 + 4/y + 1/x)/y))/(pi*sqrt(x*y))
assert L.pspace.distribution.set == ProductSet(S.Reals, S.Reals)
assert L.pspace.distribution == L2.pspace.distribution
def test_NormalGamma():
ng = NormalGamma('G', 1, 2, 3, 4)
assert density(ng)(1, 1) == 32*exp(-4)/sqrt(pi)
assert ng.pspace.distribution.set == ProductSet(S.Reals, Interval(0, oo))
raises(ValueError, lambda:NormalGamma('G', 1, 2, 3, -1))
assert marginal_distribution(ng, 0)(1) == \
3*sqrt(10)*gamma(Rational(7, 4))/(10*sqrt(pi)*gamma(Rational(5, 4)))
assert marginal_distribution(ng, y)(1) == exp(Rational(-1, 4))/128
assert marginal_distribution(ng,[0,1])(x) == x**2*exp(-x/4)/128
def test_GeneralizedMultivariateLogGammaDistribution():
h = S.Half
omega = Matrix([[1, h, h, h],
[h, 1, h, h],
[h, h, 1, h],
[h, h, h, 1]])
v, l, mu = (4, [1, 2, 3, 4], [1, 2, 3, 4])
y_1, y_2, y_3, y_4 = symbols('y_1:5', real=True)
delta = symbols('d', positive=True)
G = GMVLGO('G', omega, v, l, mu)
Gd = GMVLG('Gd', delta, v, l, mu)
dend = ("d**4*Sum(4*24**(-n - 4)*(1 - d)**n*exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 "
"+ 4*y_4) - exp(y_1) - exp(2*y_2)/2 - exp(3*y_3)/3 - exp(4*y_4)/4)/"
"(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))")
assert str(density(Gd)(y_1, y_2, y_3, y_4)) == dend
den = ("5*2**(2/3)*5**(1/3)*Sum(4*24**(-n - 4)*(-2**(2/3)*5**(1/3)/4 + 1)**n*"
"exp((n + 4)*(y_1 + 2*y_2 + 3*y_3 + 4*y_4) - exp(y_1) - exp(2*y_2)/2 - "
"exp(3*y_3)/3 - exp(4*y_4)/4)/(gamma(n + 1)*gamma(n + 4)**3), (n, 0, oo))/64")
assert str(density(G)(y_1, y_2, y_3, y_4)) == den
marg = ("5*2**(2/3)*5**(1/3)*exp(4*y_1)*exp(-exp(y_1))*Integral(exp(-exp(4*G[3])"
"/4)*exp(16*G[3])*Integral(exp(-exp(3*G[2])/3)*exp(12*G[2])*Integral(exp("
"-exp(2*G[1])/2)*exp(8*G[1])*Sum((-1/4)**n*(-4 + 2**(2/3)*5**(1/3"
"))**n*exp(n*y_1)*exp(2*n*G[1])*exp(3*n*G[2])*exp(4*n*G[3])/(24**n*gamma(n + 1)"
"*gamma(n + 4)**3), (n, 0, oo)), (G[1], -oo, oo)), (G[2], -oo, oo)), (G[3]"
", -oo, oo))/5308416")
assert str(marginal_distribution(G, G[0])(y_1)) == marg
omega_f1 = Matrix([[1, h, h]])
omega_f2 = Matrix([[1, h, h, h],
[h, 1, 2, h],
[h, h, 1, h],
[h, h, h, 1]])
omega_f3 = Matrix([[6, h, h, h],
[h, 1, 2, h],
[h, h, 1, h],
[h, h, h, 1]])
v_f = symbols("v_f", positive=False, real=True)
l_f = [1, 2, v_f, 4]
m_f = [v_f, 2, 3, 4]
omega_f4 = Matrix([[1, h, h, h, h],
[h, 1, h, h, h],
[h, h, 1, h, h],
[h, h, h, 1, h],
[h, h, h, h, 1]])
l_f1 = [1, 2, 3, 4, 5]
omega_f5 = Matrix([[1]])
mu_f5 = l_f5 = [1]
raises(ValueError, lambda: GMVLGO('G', omega_f1, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega_f2, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega_f3, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v_f, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v, l_f, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v, l, m_f))
raises(ValueError, lambda: GMVLGO('G', omega_f4, v, l, mu))
raises(ValueError, lambda: GMVLGO('G', omega, v, l_f1, mu))
raises(ValueError, lambda: GMVLGO('G', omega_f5, v, l_f5, mu_f5))
raises(ValueError, lambda: GMVLG('G', Rational(3, 2), v, l, mu))
def test_MultivariateBeta():
a1, a2 = symbols('a1, a2', positive=True)
a1_f, a2_f = symbols('a1, a2', positive=False, real=True)
mb = MultivariateBeta('B', [a1, a2])
mb_c = MultivariateBeta('C', a1, a2)
assert density(mb)(1, 2) == S(2)**(a2 - 1)*gamma(a1 + a2)/\
(gamma(a1)*gamma(a2))
assert marginal_distribution(mb_c, 0)(3) == S(3)**(a1 - 1)*gamma(a1 + a2)/\
(a2*gamma(a1)*gamma(a2))
raises(ValueError, lambda: MultivariateBeta('b1', [a1_f, a2]))
raises(ValueError, lambda: MultivariateBeta('b2', [a1, a2_f]))
raises(ValueError, lambda: MultivariateBeta('b3', [0, 0]))
raises(ValueError, lambda: MultivariateBeta('b4', [a1_f, a2_f]))
assert mb.pspace.distribution.set == ProductSet(Interval(0, 1), Interval(0, 1))
def test_MultivariateEwens():
n, theta, i = symbols('n theta i', positive=True)
# tests for integer dimensions
theta_f = symbols('t_f', negative=True)
a = symbols('a_1:4', positive = True, integer = True)
ed = MultivariateEwens('E', 3, theta)
assert density(ed)(a[0], a[1], a[2]) == Piecewise((6*2**(-a[1])*3**(-a[2])*
theta**a[0]*theta**a[1]*theta**a[2]/
(theta*(theta + 1)*(theta + 2)*
factorial(a[0])*factorial(a[1])*
factorial(a[2])), Eq(a[0] + 2*a[1] +
3*a[2], 3)), (0, True))
assert marginal_distribution(ed, ed[1])(a[1]) == Piecewise((6*2**(-a[1])*
theta**a[1]/((theta + 1)*
(theta + 2)*factorial(a[1])),
Eq(2*a[1] + 1, 3)), (0, True))
raises(ValueError, lambda: MultivariateEwens('e1', 5, theta_f))
assert ed.pspace.distribution.set == ProductSet(Range(0, 4, 1),
Range(0, 2, 1), Range(0, 2, 1))
# tests for symbolic dimensions
eds = MultivariateEwens('E', n, theta)
a = IndexedBase('a')
j, k = symbols('j, k')
den = Piecewise((factorial(n)*Product(theta**a[j]*(j + 1)**(-a[j])/
factorial(a[j]), (j, 0, n - 1))/RisingFactorial(theta, n),
Eq(n, Sum((k + 1)*a[k], (k, 0, n - 1)))), (0, True))
assert density(eds)(a).dummy_eq(den)
def test_Multinomial():
n, x1, x2, x3, x4 = symbols('n, x1, x2, x3, x4', nonnegative=True, integer=True)
p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True)
p1_f, n_f = symbols('p1_f, n_f', negative=True)
M = Multinomial('M', n, [p1, p2, p3, p4])
C = Multinomial('C', 3, p1, p2, p3)
f = factorial
assert density(M)(x1, x2, x3, x4) == Piecewise((p1**x1*p2**x2*p3**x3*p4**x4*
f(n)/(f(x1)*f(x2)*f(x3)*f(x4)),
Eq(n, x1 + x2 + x3 + x4)), (0, True))
assert marginal_distribution(C, C[0])(x1).subs(x1, 1) ==\
3*p1*p2**2 +\
6*p1*p2*p3 +\
3*p1*p3**2
raises(ValueError, lambda: Multinomial('b1', 5, [p1, p2, p3, p1_f]))
raises(ValueError, lambda: Multinomial('b2', n_f, [p1, p2, p3, p4]))
raises(ValueError, lambda: Multinomial('b3', n, 0.5, 0.4, 0.3, 0.1))
def test_NegativeMultinomial():
k0, x1, x2, x3, x4 = symbols('k0, x1, x2, x3, x4', nonnegative=True, integer=True)
p1, p2, p3, p4 = symbols('p1, p2, p3, p4', positive=True)
p1_f = symbols('p1_f', negative=True)
N = NegativeMultinomial('N', 4, [p1, p2, p3, p4])
C = NegativeMultinomial('C', 4, 0.1, 0.2, 0.3)
g = gamma
f = factorial
assert simplify(density(N)(x1, x2, x3, x4) -
p1**x1*p2**x2*p3**x3*p4**x4*(-p1 - p2 - p3 - p4 + 1)**4*g(x1 + x2 +
x3 + x4 + 4)/(6*f(x1)*f(x2)*f(x3)*f(x4))) is S.Zero
assert comp(marginal_distribution(C, C[0])(1).evalf(), 0.33, .01)
raises(ValueError, lambda: NegativeMultinomial('b1', 5, [p1, p2, p3, p1_f]))
raises(ValueError, lambda: NegativeMultinomial('b2', k0, 0.5, 0.4, 0.3, 0.4))
assert N.pspace.distribution.set == ProductSet(Range(0, oo, 1),
Range(0, oo, 1), Range(0, oo, 1), Range(0, oo, 1))
@slow
def test_JointPSpace_marginal_distribution():
T = MultivariateT('T', [0, 0], [[1, 0], [0, 1]], 2)
got = marginal_distribution(T, T[1])(x)
ans = sqrt(2)*(x**2/2 + 1)/(4*polar_lift(x**2/2 + 1)**(S(5)/2))
assert got == ans, got
assert integrate(marginal_distribution(T, 1)(x), (x, -oo, oo)) == 1
t = MultivariateT('T', [0, 0, 0], [[1, 0, 0], [0, 1, 0], [0, 0, 1]], 3)
assert comp(marginal_distribution(t, 0)(1).evalf(), 0.2, .01)
def test_JointRV():
x1, x2 = (Indexed('x', i) for i in (1, 2))
pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(2*pi)
X = JointRV('x', pdf)
assert density(X)(1, 2) == exp(-2)/(2*pi)
assert isinstance(X.pspace.distribution, JointDistributionHandmade)
assert marginal_distribution(X, 0)(2) == sqrt(2)*exp(Rational(-1, 2))/(2*sqrt(pi))
def test_expectation():
m = Normal('A', [x, y], [[1, 0], [0, 1]])
assert simplify(E(m[1])) == y
@XFAIL
def test_joint_vector_expectation():
m = Normal('A', [x, y], [[1, 0], [0, 1]])
assert E(m) == (x, y)
def test_sample_numpy():
distribs_numpy = [
MultivariateNormal("M", [3, 4], [[2, 1], [1, 2]]),
MultivariateBeta("B", [0.4, 5, 15, 50, 203]),
Multinomial("N", 50, [0.3, 0.2, 0.1, 0.25, 0.15])
]
size = 3
numpy = import_module('numpy')
if not numpy:
skip('Numpy is not installed. Abort tests for _sample_numpy.')
else:
for X in distribs_numpy:
samps = sample(X, size=size, library='numpy')
for sam in samps:
assert tuple(sam) in X.pspace.distribution.set
N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
raises(NotImplementedError, lambda: sample(N_c, library='numpy'))
def test_sample_scipy():
distribs_scipy = [
MultivariateNormal("M", [0, 0], [[0.1, 0.025], [0.025, 0.1]]),
MultivariateBeta("B", [0.4, 5, 15]),
Multinomial("N", 8, [0.3, 0.2, 0.1, 0.4])
]
size = 3
scipy = import_module('scipy')
if not scipy:
skip('Scipy not installed. Abort tests for _sample_scipy.')
else:
for X in distribs_scipy:
samps = sample(X, size=size)
samps2 = sample(X, size=(2, 2))
for sam in samps:
assert tuple(sam) in X.pspace.distribution.set
for i in range(2):
for j in range(2):
assert tuple(samps2[i][j]) in X.pspace.distribution.set
N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
raises(NotImplementedError, lambda: sample(N_c))
def test_sample_pymc():
distribs_pymc = [
MultivariateNormal("M", [5, 2], [[1, 0], [0, 1]]),
MultivariateBeta("B", [0.4, 5, 15]),
Multinomial("N", 4, [0.3, 0.2, 0.1, 0.4])
]
size = 3
pymc = import_module('pymc')
if not pymc:
skip('PyMC is not installed. Abort tests for _sample_pymc.')
else:
for X in distribs_pymc:
samps = sample(X, size=size, library='pymc')
for sam in samps:
assert tuple(sam.flatten()) in X.pspace.distribution.set
N_c = NegativeMultinomial('N', 3, 0.1, 0.1, 0.1)
raises(NotImplementedError, lambda: sample(N_c, library='pymc'))
def test_sample_seed():
x1, x2 = (Indexed('x', i) for i in (1, 2))
pdf = exp(-x1**2/2 + x1 - x2**2/2 - S.Half)/(2*pi)
X = JointRV('x', pdf)
libraries = ['scipy', 'numpy', 'pymc']
for lib in libraries:
try:
imported_lib = import_module(lib)
if imported_lib:
s0, s1, s2 = [], [], []
s0 = sample(X, size=10, library=lib, seed=0)
s1 = sample(X, size=10, library=lib, seed=0)
s2 = sample(X, size=10, library=lib, seed=1)
assert all(s0 == s1)
assert all(s1 != s2)
except NotImplementedError:
continue
#
# XXX: This fails for pymc. Previously the test appeared to pass but that is
# just because the library argument was not passed so the test always used
# scipy.
#
def test_issue_21057():
m = Normal("x", [0, 0], [[0, 0], [0, 0]])
n = MultivariateNormal("x", [0, 0], [[0, 0], [0, 0]])
p = Normal("x", [0, 0], [[0, 0], [0, 1]])
assert m == n
libraries = ('scipy', 'numpy') # , 'pymc') # <-- pymc fails
for library in libraries:
try:
imported_lib = import_module(library)
if imported_lib:
s1 = sample(m, size=8, library=library)
s2 = sample(n, size=8, library=library)
s3 = sample(p, size=8, library=library)
assert tuple(s1.flatten()) == tuple(s2.flatten())
for s in s3:
assert tuple(s.flatten()) in p.pspace.distribution.set
except NotImplementedError:
continue
#
# When this passes the pymc part can be uncommented in test_issue_21057 above
# and this can be deleted.
#
@XFAIL
def test_issue_21057_pymc():
m = Normal("x", [0, 0], [[0, 0], [0, 0]])
n = MultivariateNormal("x", [0, 0], [[0, 0], [0, 0]])
p = Normal("x", [0, 0], [[0, 0], [0, 1]])
assert m == n
libraries = ('pymc',)
for library in libraries:
try:
imported_lib = import_module(library)
if imported_lib:
s1 = sample(m, size=8, library=library)
s2 = sample(n, size=8, library=library)
s3 = sample(p, size=8, library=library)
assert tuple(s1.flatten()) == tuple(s2.flatten())
for s in s3:
assert tuple(s.flatten()) in p.pspace.distribution.set
except NotImplementedError:
continue
|
92bdcc8df234d9c3ef42b4084b09ac203104876829327cd9b2331de533a1025d | from sympy.concrete.summations import Sum
from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.elementary.complexes import (im, re)
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.bessel import besseli
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.zeta_functions import zeta
from sympy.sets.sets import FiniteSet
from sympy.simplify.simplify import simplify
from sympy.utilities.lambdify import lambdify
from sympy.core.relational import Eq, Ne
from sympy.functions.elementary.exponential import exp
from sympy.logic.boolalg import Or
from sympy.sets.fancysets import Range
from sympy.stats import (P, E, variance, density, characteristic_function,
where, moment_generating_function, skewness, cdf,
kurtosis, coskewness)
from sympy.stats.drv_types import (PoissonDistribution, GeometricDistribution,
FlorySchulz, Poisson, Geometric, Hermite, Logarithmic,
NegativeBinomial, Skellam, YuleSimon, Zeta,
DiscreteRV)
from sympy.testing.pytest import slow, nocache_fail, raises
from sympy.stats.symbolic_probability import Expectation
x = Symbol('x')
def test_PoissonDistribution():
l = 3
p = PoissonDistribution(l)
assert abs(p.cdf(10).evalf() - 1) < .001
assert abs(p.cdf(10.4).evalf() - 1) < .001
assert p.expectation(x, x) == l
assert p.expectation(x**2, x) - p.expectation(x, x)**2 == l
def test_Poisson():
l = 3
x = Poisson('x', l)
assert E(x) == l
assert E(2*x) == 2*l
assert variance(x) == l
assert density(x) == PoissonDistribution(l)
assert isinstance(E(x, evaluate=False), Expectation)
assert isinstance(E(2*x, evaluate=False), Expectation)
# issue 8248
assert x.pspace.compute_expectation(1) == 1
def test_FlorySchulz():
a = Symbol("a")
z = Symbol("z")
x = FlorySchulz('x', a)
assert E(x) == (2 - a)/a
assert (variance(x) - 2*(1 - a)/a**2).simplify() == S(0)
assert density(x)(z) == a**2*z*(1 - a)**(z - 1)
@slow
def test_GeometricDistribution():
p = S.One / 5
d = GeometricDistribution(p)
assert d.expectation(x, x) == 1/p
assert d.expectation(x**2, x) - d.expectation(x, x)**2 == (1-p)/p**2
assert abs(d.cdf(20000).evalf() - 1) < .001
assert abs(d.cdf(20000.8).evalf() - 1) < .001
G = Geometric('G', p=S(1)/4)
assert cdf(G)(S(7)/2) == P(G <= S(7)/2)
X = Geometric('X', Rational(1, 5))
Y = Geometric('Y', Rational(3, 10))
assert coskewness(X, X + Y, X + 2*Y).simplify() == sqrt(230)*Rational(81, 1150)
def test_Hermite():
a1 = Symbol("a1", positive=True)
a2 = Symbol("a2", negative=True)
raises(ValueError, lambda: Hermite("H", a1, a2))
a1 = Symbol("a1", negative=True)
a2 = Symbol("a2", positive=True)
raises(ValueError, lambda: Hermite("H", a1, a2))
a1 = Symbol("a1", positive=True)
x = Symbol("x")
H = Hermite("H", a1, a2)
assert moment_generating_function(H)(x) == exp(a1*(exp(x) - 1)
+ a2*(exp(2*x) - 1))
assert characteristic_function(H)(x) == exp(a1*(exp(I*x) - 1)
+ a2*(exp(2*I*x) - 1))
assert E(H) == a1 + 2*a2
H = Hermite("H", a1=5, a2=4)
assert density(H)(2) == 33*exp(-9)/2
assert E(H) == 13
assert variance(H) == 21
assert kurtosis(H) == Rational(464,147)
assert skewness(H) == 37*sqrt(21)/441
def test_Logarithmic():
p = S.Half
x = Logarithmic('x', p)
assert E(x) == -p / ((1 - p) * log(1 - p))
assert variance(x) == -1/log(2)**2 + 2/log(2)
assert E(2*x**2 + 3*x + 4) == 4 + 7 / log(2)
assert isinstance(E(x, evaluate=False), Expectation)
@nocache_fail
def test_negative_binomial():
r = 5
p = S.One / 3
x = NegativeBinomial('x', r, p)
assert E(x) == p*r / (1-p)
# This hangs when run with the cache disabled:
assert variance(x) == p*r / (1-p)**2
assert E(x**5 + 2*x + 3) == Rational(9207, 4)
assert isinstance(E(x, evaluate=False), Expectation)
def test_skellam():
mu1 = Symbol('mu1')
mu2 = Symbol('mu2')
z = Symbol('z')
X = Skellam('x', mu1, mu2)
assert density(X)(z) == (mu1/mu2)**(z/2) * \
exp(-mu1 - mu2)*besseli(z, 2*sqrt(mu1*mu2))
assert skewness(X).expand() == mu1/(mu1*sqrt(mu1 + mu2) + mu2 *
sqrt(mu1 + mu2)) - mu2/(mu1*sqrt(mu1 + mu2) + mu2*sqrt(mu1 + mu2))
assert variance(X).expand() == mu1 + mu2
assert E(X) == mu1 - mu2
assert characteristic_function(X)(z) == exp(
mu1*exp(I*z) - mu1 - mu2 + mu2*exp(-I*z))
assert moment_generating_function(X)(z) == exp(
mu1*exp(z) - mu1 - mu2 + mu2*exp(-z))
def test_yule_simon():
from sympy.core.singleton import S
rho = S(3)
x = YuleSimon('x', rho)
assert simplify(E(x)) == rho / (rho - 1)
assert simplify(variance(x)) == rho**2 / ((rho - 1)**2 * (rho - 2))
assert isinstance(E(x, evaluate=False), Expectation)
# To test the cdf function
assert cdf(x)(x) == Piecewise((-beta(floor(x), 4)*floor(x) + 1, x >= 1), (0, True))
def test_zeta():
s = S(5)
x = Zeta('x', s)
assert E(x) == zeta(s-1) / zeta(s)
assert simplify(variance(x)) == (
zeta(s) * zeta(s-2) - zeta(s-1)**2) / zeta(s)**2
def test_discrete_probability():
X = Geometric('X', Rational(1, 5))
Y = Poisson('Y', 4)
G = Geometric('e', x)
assert P(Eq(X, 3)) == Rational(16, 125)
assert P(X < 3) == Rational(9, 25)
assert P(X > 3) == Rational(64, 125)
assert P(X >= 3) == Rational(16, 25)
assert P(X <= 3) == Rational(61, 125)
assert P(Ne(X, 3)) == Rational(109, 125)
assert P(Eq(Y, 3)) == 32*exp(-4)/3
assert P(Y < 3) == 13*exp(-4)
assert P(Y > 3).equals(32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3)
assert P(Y >= 3).equals(32*(Rational(-39, 32) + 3*exp(4)/32)*exp(-4)/3)
assert P(Y <= 3) == 71*exp(-4)/3
assert P(Ne(Y, 3)).equals(
13*exp(-4) + 32*(Rational(-71, 32) + 3*exp(4)/32)*exp(-4)/3)
assert P(X < S.Infinity) is S.One
assert P(X > S.Infinity) is S.Zero
assert P(G < 3) == x*(2-x)
assert P(Eq(G, 3)) == x*(-x + 1)**2
def test_DiscreteRV():
p = S(1)/2
x = Symbol('x', integer=True, positive=True)
pdf = p*(1 - p)**(x - 1) # pdf of Geometric Distribution
D = DiscreteRV(x, pdf, set=S.Naturals, check=True)
assert E(D) == E(Geometric('G', S(1)/2)) == 2
assert P(D > 3) == S(1)/8
assert D.pspace.domain.set == S.Naturals
raises(ValueError, lambda: DiscreteRV(x, x, FiniteSet(*range(4)), check=True))
# purposeful invalid pmf but it should not raise since check=False
# see test_drv_types.test_ContinuousRV for explanation
X = DiscreteRV(x, 1/x, S.Naturals)
assert P(X < 2) == 1
assert E(X) == oo
def test_precomputed_characteristic_functions():
import mpmath
def test_cf(dist, support_lower_limit, support_upper_limit):
pdf = density(dist)
t = S('t')
x = S('x')
# first function is the hardcoded CF of the distribution
cf1 = lambdify([t], characteristic_function(dist)(t), 'mpmath')
# second function is the Fourier transform of the density function
f = lambdify([x, t], pdf(x)*exp(I*x*t), 'mpmath')
cf2 = lambda t: mpmath.nsum(lambda x: f(x, t), [
support_lower_limit, support_upper_limit], maxdegree=10)
# compare the two functions at various points
for test_point in [2, 5, 8, 11]:
n1 = cf1(test_point)
n2 = cf2(test_point)
assert abs(re(n1) - re(n2)) < 1e-12
assert abs(im(n1) - im(n2)) < 1e-12
test_cf(Geometric('g', Rational(1, 3)), 1, mpmath.inf)
test_cf(Logarithmic('l', Rational(1, 5)), 1, mpmath.inf)
test_cf(NegativeBinomial('n', 5, Rational(1, 7)), 0, mpmath.inf)
test_cf(Poisson('p', 5), 0, mpmath.inf)
test_cf(YuleSimon('y', 5), 1, mpmath.inf)
test_cf(Zeta('z', 5), 1, mpmath.inf)
def test_moment_generating_functions():
t = S('t')
geometric_mgf = moment_generating_function(Geometric('g', S.Half))(t)
assert geometric_mgf.diff(t).subs(t, 0) == 2
logarithmic_mgf = moment_generating_function(Logarithmic('l', S.Half))(t)
assert logarithmic_mgf.diff(t).subs(t, 0) == 1/log(2)
negative_binomial_mgf = moment_generating_function(
NegativeBinomial('n', 5, Rational(1, 3)))(t)
assert negative_binomial_mgf.diff(t).subs(t, 0) == Rational(5, 2)
poisson_mgf = moment_generating_function(Poisson('p', 5))(t)
assert poisson_mgf.diff(t).subs(t, 0) == 5
skellam_mgf = moment_generating_function(Skellam('s', 1, 1))(t)
assert skellam_mgf.diff(t).subs(
t, 2) == (-exp(-2) + exp(2))*exp(-2 + exp(-2) + exp(2))
yule_simon_mgf = moment_generating_function(YuleSimon('y', 3))(t)
assert simplify(yule_simon_mgf.diff(t).subs(t, 0)) == Rational(3, 2)
zeta_mgf = moment_generating_function(Zeta('z', 5))(t)
assert zeta_mgf.diff(t).subs(t, 0) == pi**4/(90*zeta(5))
def test_Or():
X = Geometric('X', S.Half)
assert P(Or(X < 3, X > 4)) == Rational(13, 16)
assert P(Or(X > 2, X > 1)) == P(X > 1)
assert P(Or(X >= 3, X < 3)) == 1
def test_where():
X = Geometric('X', Rational(1, 5))
Y = Poisson('Y', 4)
assert where(X**2 > 4).set == Range(3, S.Infinity, 1)
assert where(X**2 >= 4).set == Range(2, S.Infinity, 1)
assert where(Y**2 < 9).set == Range(0, 3, 1)
assert where(Y**2 <= 9).set == Range(0, 4, 1)
def test_conditional():
X = Geometric('X', Rational(2, 3))
Y = Poisson('Y', 3)
assert P(X > 2, X > 3) == 1
assert P(X > 3, X > 2) == Rational(1, 3)
assert P(Y > 2, Y < 2) == 0
assert P(Eq(Y, 3), Y >= 0) == 9*exp(-3)/2
assert P(Eq(Y, 3), Eq(Y, 2)) == 0
assert P(X < 2, Eq(X, 2)) == 0
assert P(X > 2, Eq(X, 3)) == 1
def test_product_spaces():
X1 = Geometric('X1', S.Half)
X2 = Geometric('X2', Rational(1, 3))
assert str(P(X1 + X2 < 3).rewrite(Sum)) == (
"Sum(Piecewise((1/(4*2**n), n >= -1), (0, True)), (n, -oo, -1))/3")
assert str(P(X1 + X2 > 3).rewrite(Sum)) == (
'Sum(Piecewise((2**(X2 - n - 2)*(3/2)**(1 - X2)/6, '
'X2 - n <= 2), (0, True)), (X2, 1, oo), (n, 1, oo))')
assert P(Eq(X1 + X2, 3)) == Rational(1, 12)
|
11a0dd6b15aa2d7196d2586bf202f22a030091aefde4def01f34c4c67d309975 | from itertools import product
from sympy.concrete.summations import Sum
from sympy.core.function import (Function, diff)
from sympy.core import EulerGamma
from sympy.core.numbers import (E, I, Rational, oo, pi, zoo)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.combinatorial.factorials import (binomial, factorial, subfactorial)
from sympy.functions.elementary.complexes import (Abs, re, sign)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (acosh, acoth, acsch, asech, atanh, sinh)
from sympy.functions.elementary.integers import (ceiling, floor, frac)
from sympy.functions.elementary.miscellaneous import (cbrt, real_root, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, acot, acsc, asec, asin,
atan, cos, cot, csc, sec, sin, tan)
from sympy.functions.special.bessel import (besselj, besselk)
from sympy.functions.special.error_functions import (Ei, erf, erfc, erfi, fresnelc, fresnels)
from sympy.functions.special.gamma_functions import (digamma, gamma, uppergamma)
from sympy.integrals.integrals import (Integral, integrate)
from sympy.series.limits import (Limit, limit)
from sympy.simplify.simplify import (logcombine, simplify)
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.core.mul import Mul
from sympy.series.limits import heuristics
from sympy.series.order import Order
from sympy.testing.pytest import XFAIL, raises
from sympy.abc import x, y, z, k
n = Symbol('n', integer=True, positive=True)
def test_basic1():
assert limit(x, x, oo) is oo
assert limit(x, x, -oo) is -oo
assert limit(-x, x, oo) is -oo
assert limit(x**2, x, -oo) is oo
assert limit(-x**2, x, oo) is -oo
assert limit(x*log(x), x, 0, dir="+") == 0
assert limit(1/x, x, oo) == 0
assert limit(exp(x), x, oo) is oo
assert limit(-exp(x), x, oo) is -oo
assert limit(exp(x)/x, x, oo) is oo
assert limit(1/x - exp(-x), x, oo) == 0
assert limit(x + 1/x, x, oo) is oo
assert limit(x - x**2, x, oo) is -oo
assert limit((1 + x)**(1 + sqrt(2)), x, 0) == 1
assert limit((1 + x)**oo, x, 0) == Limit((x + 1)**oo, x, 0)
assert limit((1 + x)**oo, x, 0, dir='-') == Limit((x + 1)**oo, x, 0, dir='-')
assert limit((1 + x + y)**oo, x, 0, dir='-') == Limit((1 + x + y)**oo, x, 0, dir='-')
assert limit(y/x/log(x), x, 0) == -oo*sign(y)
assert limit(cos(x + y)/x, x, 0) == sign(cos(y))*oo
assert limit(gamma(1/x + 3), x, oo) == 2
assert limit(S.NaN, x, -oo) is S.NaN
assert limit(Order(2)*x, x, S.NaN) is S.NaN
assert limit(1/(x - 1), x, 1, dir="+") is oo
assert limit(1/(x - 1), x, 1, dir="-") is -oo
assert limit(1/(5 - x)**3, x, 5, dir="+") is -oo
assert limit(1/(5 - x)**3, x, 5, dir="-") is oo
assert limit(1/sin(x), x, pi, dir="+") is -oo
assert limit(1/sin(x), x, pi, dir="-") is oo
assert limit(1/cos(x), x, pi/2, dir="+") is -oo
assert limit(1/cos(x), x, pi/2, dir="-") is oo
assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="+") is oo
assert limit(1/tan(x**3), x, (2*pi)**Rational(1, 3), dir="-") is -oo
assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="+") is -oo
assert limit(1/cot(x)**3, x, (pi*Rational(3, 2)), dir="-") is oo
assert limit(tan(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(cot(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(sec(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(csc(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
# test bi-directional limits
assert limit(sin(x)/x, x, 0, dir="+-") == 1
assert limit(x**2, x, 0, dir="+-") == 0
assert limit(1/x**2, x, 0, dir="+-") is oo
# test failing bi-directional limits
assert limit(1/x, x, 0, dir="+-") is zoo
# approaching 0
# from dir="+"
assert limit(1 + 1/x, x, 0) is oo
# from dir='-'
# Add
assert limit(1 + 1/x, x, 0, dir='-') is -oo
# Pow
assert limit(x**(-2), x, 0, dir='-') is oo
assert limit(x**(-3), x, 0, dir='-') is -oo
assert limit(1/sqrt(x), x, 0, dir='-') == (-oo)*I
assert limit(x**2, x, 0, dir='-') == 0
assert limit(sqrt(x), x, 0, dir='-') == 0
assert limit(x**-pi, x, 0, dir='-') == oo/(-1)**pi
assert limit((1 + cos(x))**oo, x, 0) == Limit((cos(x) + 1)**oo, x, 0)
# test pull request 22491
assert limit(1/asin(x), x, 0, dir = '+') == oo
assert limit(1/asin(x), x, 0, dir = '-') == -oo
assert limit(1/sinh(x), x, 0, dir = '+') == oo
assert limit(1/sinh(x), x, 0, dir = '-') == -oo
assert limit(log(1/x) + 1/sin(x), x, 0, dir = '+') == oo
assert limit(log(1/x) + 1/x, x, 0, dir = '+') == oo
def test_basic2():
assert limit(x**x, x, 0, dir="+") == 1
assert limit((exp(x) - 1)/x, x, 0) == 1
assert limit(1 + 1/x, x, oo) == 1
assert limit(-exp(1/x), x, oo) == -1
assert limit(x + exp(-x), x, oo) is oo
assert limit(x + exp(-x**2), x, oo) is oo
assert limit(x + exp(-exp(x)), x, oo) is oo
assert limit(13 + 1/x - exp(-x), x, oo) == 13
def test_basic3():
assert limit(1/x, x, 0, dir="+") is oo
assert limit(1/x, x, 0, dir="-") is -oo
def test_basic4():
assert limit(2*x + y*x, x, 0) == 0
assert limit(2*x + y*x, x, 1) == 2 + y
assert limit(2*x**8 + y*x**(-3), x, -2) == 512 - y/8
assert limit(sqrt(x + 1) - sqrt(x), x, oo) == 0
assert integrate(1/(x**3 + 1), (x, 0, oo)) == 2*pi*sqrt(3)/9
def test_log():
# https://github.com/sympy/sympy/issues/21598
a, b, c = symbols('a b c', positive=True)
A = log(a/b) - (log(a) - log(b))
assert A.limit(a, oo) == 0
assert (A * c).limit(a, oo) == 0
tau, x = symbols('tau x', positive=True)
# The value of manualintegrate in the issue
expr = tau**2*((tau - 1)*(tau + 1)*log(x + 1)/(tau**2 + 1)**2 + 1/((tau**2\
+ 1)*(x + 1)) - (-2*tau*atan(x/tau) + (tau**2/2 - 1/2)*log(tau**2\
+ x**2))/(tau**2 + 1)**2)
assert limit(expr, x, oo) == pi*tau**3/(tau**2 + 1)**2
def test_piecewise():
# https://github.com/sympy/sympy/issues/18363
assert limit((real_root(x - 6, 3) + 2)/(x + 2), x, -2, '+') == Rational(1, 12)
def test_piecewise2():
func1 = 2*sqrt(x)*Piecewise(((4*x - 2)/Abs(sqrt(4 - 4*(2*x - 1)**2)), 4*x - 2\
>= 0), ((2 - 4*x)/Abs(sqrt(4 - 4*(2*x - 1)**2)), True))
func2 = Piecewise((x**2/2, x <= 0.5), (x/2 - 0.125, True))
func3 = Piecewise(((x - 9) / 5, x < -1), ((x - 9) / 5, x > 4), (sqrt(Abs(x - 3)), True))
assert limit(func1, x, 0) == 1
assert limit(func2, x, 0) == 0
assert limit(func3, x, -1) == 2
def test_basic5():
class my(Function):
@classmethod
def eval(cls, arg):
if arg is S.Infinity:
return S.NaN
assert limit(my(x), x, oo) == Limit(my(x), x, oo)
def test_issue_3885():
assert limit(x*y + x*z, z, 2) == x*y + 2*x
def test_Limit():
assert Limit(sin(x)/x, x, 0) != 1
assert Limit(sin(x)/x, x, 0).doit() == 1
assert Limit(x, x, 0, dir='+-').args == (x, x, 0, Symbol('+-'))
def test_floor():
assert limit(floor(x), x, -2, "+") == -2
assert limit(floor(x), x, -2, "-") == -3
assert limit(floor(x), x, -1, "+") == -1
assert limit(floor(x), x, -1, "-") == -2
assert limit(floor(x), x, 0, "+") == 0
assert limit(floor(x), x, 0, "-") == -1
assert limit(floor(x), x, 1, "+") == 1
assert limit(floor(x), x, 1, "-") == 0
assert limit(floor(x), x, 2, "+") == 2
assert limit(floor(x), x, 2, "-") == 1
assert limit(floor(x), x, 248, "+") == 248
assert limit(floor(x), x, 248, "-") == 247
# https://github.com/sympy/sympy/issues/14478
assert limit(x*floor(3/x)/2, x, 0, '+') == Rational(3, 2)
assert limit(floor(x + 1/2) - floor(x), x, oo) == AccumBounds(-S.Half, S(3)/2)
# test issue 9158
assert limit(floor(atan(x)), x, oo) == 1
assert limit(floor(atan(x)), x, -oo) == -2
assert limit(ceiling(atan(x)), x, oo) == 2
assert limit(ceiling(atan(x)), x, -oo) == -1
def test_floor_requires_robust_assumptions():
assert limit(floor(sin(x)), x, 0, "+") == 0
assert limit(floor(sin(x)), x, 0, "-") == -1
assert limit(floor(cos(x)), x, 0, "+") == 0
assert limit(floor(cos(x)), x, 0, "-") == 0
assert limit(floor(5 + sin(x)), x, 0, "+") == 5
assert limit(floor(5 + sin(x)), x, 0, "-") == 4
assert limit(floor(5 + cos(x)), x, 0, "+") == 5
assert limit(floor(5 + cos(x)), x, 0, "-") == 5
def test_ceiling():
assert limit(ceiling(x), x, -2, "+") == -1
assert limit(ceiling(x), x, -2, "-") == -2
assert limit(ceiling(x), x, -1, "+") == 0
assert limit(ceiling(x), x, -1, "-") == -1
assert limit(ceiling(x), x, 0, "+") == 1
assert limit(ceiling(x), x, 0, "-") == 0
assert limit(ceiling(x), x, 1, "+") == 2
assert limit(ceiling(x), x, 1, "-") == 1
assert limit(ceiling(x), x, 2, "+") == 3
assert limit(ceiling(x), x, 2, "-") == 2
assert limit(ceiling(x), x, 248, "+") == 249
assert limit(ceiling(x), x, 248, "-") == 248
# https://github.com/sympy/sympy/issues/14478
assert limit(x*ceiling(3/x)/2, x, 0, '+') == Rational(3, 2)
assert limit(ceiling(x + 1/2) - ceiling(x), x, oo) == AccumBounds(-S.Half, S(3)/2)
def test_ceiling_requires_robust_assumptions():
assert limit(ceiling(sin(x)), x, 0, "+") == 1
assert limit(ceiling(sin(x)), x, 0, "-") == 0
assert limit(ceiling(cos(x)), x, 0, "+") == 1
assert limit(ceiling(cos(x)), x, 0, "-") == 1
assert limit(ceiling(5 + sin(x)), x, 0, "+") == 6
assert limit(ceiling(5 + sin(x)), x, 0, "-") == 5
assert limit(ceiling(5 + cos(x)), x, 0, "+") == 6
assert limit(ceiling(5 + cos(x)), x, 0, "-") == 6
def test_issue_14355():
assert limit(floor(sin(x)/x), x, 0, '+') == 0
assert limit(floor(sin(x)/x), x, 0, '-') == 0
# test comment https://github.com/sympy/sympy/issues/14355#issuecomment-372121314
assert limit(floor(-tan(x)/x), x, 0, '+') == -2
assert limit(floor(-tan(x)/x), x, 0, '-') == -2
def test_atan():
x = Symbol("x", real=True)
assert limit(atan(x)*sin(1/x), x, 0) == 0
assert limit(atan(x) + sqrt(x + 1) - sqrt(x), x, oo) == pi/2
def test_set_signs():
assert limit(abs(x), x, 0) == 0
assert limit(abs(sin(x)), x, 0) == 0
assert limit(abs(cos(x)), x, 0) == 1
assert limit(abs(sin(x + 1)), x, 0) == sin(1)
# https://github.com/sympy/sympy/issues/9449
assert limit((Abs(x + y) - Abs(x - y))/(2*x), x, 0) == sign(y)
# https://github.com/sympy/sympy/issues/12398
assert limit(Abs(log(x)/x**3), x, oo) == 0
assert limit(x*(Abs(log(x)/x**3)/Abs(log(x + 1)/(x + 1)**3) - 1), x, oo) == 3
# https://github.com/sympy/sympy/issues/18501
assert limit(Abs(log(x - 1)**3 - 1), x, 1, '+') == oo
# https://github.com/sympy/sympy/issues/18997
assert limit(Abs(log(x)), x, 0) == oo
assert limit(Abs(log(Abs(x))), x, 0) == oo
# https://github.com/sympy/sympy/issues/19026
z = Symbol('z', positive=True)
assert limit(Abs(log(z) + 1)/log(z), z, oo) == 1
# https://github.com/sympy/sympy/issues/20704
assert limit(z*(Abs(1/z + y) - Abs(y - 1/z))/2, z, 0) == 0
# https://github.com/sympy/sympy/issues/21606
assert limit(cos(z)/sign(z), z, pi, '-') == -1
def test_heuristic():
x = Symbol("x", real=True)
assert heuristics(sin(1/x) + atan(x), x, 0, '+') == AccumBounds(-1, 1)
assert limit(log(2 + sqrt(atan(x))*sqrt(sin(1/x))), x, 0) == log(2)
def test_issue_3871():
z = Symbol("z", positive=True)
f = -1/z*exp(-z*x)
assert limit(f, x, oo) == 0
assert f.limit(x, oo) == 0
def test_exponential():
n = Symbol('n')
x = Symbol('x', real=True)
assert limit((1 + x/n)**n, n, oo) == exp(x)
assert limit((1 + x/(2*n))**n, n, oo) == exp(x/2)
assert limit((1 + x/(2*n + 1))**n, n, oo) == exp(x/2)
assert limit(((x - 1)/(x + 1))**x, x, oo) == exp(-2)
assert limit(1 + (1 + 1/x)**x, x, oo) == 1 + S.Exp1
assert limit((2 + 6*x)**x/(6*x)**x, x, oo) == exp(S('1/3'))
def test_exponential2():
n = Symbol('n')
assert limit((1 + x/(n + sin(n)))**n, n, oo) == exp(x)
def test_doit():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
assert l.doit() is oo
def test_series_AccumBounds():
assert limit(sin(k) - sin(k + 1), k, oo) == AccumBounds(-2, 2)
assert limit(cos(k) - cos(k + 1) + 1, k, oo) == AccumBounds(-1, 3)
# not the exact bound
assert limit(sin(k) - sin(k)*cos(k), k, oo) == AccumBounds(-2, 2)
# test for issue #9934
lo = (-3 + cos(1))/2
hi = (1 + cos(1))/2
t1 = Mul(AccumBounds(lo, hi), 1/(-1 + cos(1)), evaluate=False)
assert limit(simplify(Sum(cos(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t1
t2 = Mul(AccumBounds(-1 + sin(1)/2, sin(1)/2 + 1), 1/(1 - cos(1)))
assert limit(simplify(Sum(sin(n).rewrite(exp), (n, 0, k)).doit().rewrite(sin)), k, oo) == t2
assert limit(frac(x)**x, x, oo) == AccumBounds(0, oo) # wolfram gives (0, 1)
assert limit(((sin(x) + 1)/2)**x, x, oo) == AccumBounds(0, oo) # wolfram says 0
# https://github.com/sympy/sympy/issues/12312
e = 2**(-x)*(sin(x) + 1)**x
assert limit(e, x, oo) == AccumBounds(0, oo)
@XFAIL
def test_doit2():
f = Integral(2 * x, x)
l = Limit(f, x, oo)
# limit() breaks on the contained Integral.
assert l.doit(deep=False) == l
def test_issue_2929():
assert limit((x * exp(x))/(exp(x) - 1), x, -oo) == 0
def test_issue_3792():
assert limit((1 - cos(x))/x**2, x, S.Half) == 4 - 4*cos(S.Half)
assert limit(sin(sin(x + 1) + 1), x, 0) == sin(1 + sin(1))
assert limit(abs(sin(x + 1) + 1), x, 0) == 1 + sin(1)
def test_issue_4090():
assert limit(1/(x + 3), x, 2) == Rational(1, 5)
assert limit(1/(x + pi), x, 2) == S.One/(2 + pi)
assert limit(log(x)/(x**2 + 3), x, 2) == log(2)/7
assert limit(log(x)/(x**2 + pi), x, 2) == log(2)/(4 + pi)
def test_issue_4547():
assert limit(cot(x), x, 0, dir='+') is oo
assert limit(cot(x), x, pi/2, dir='+') == 0
def test_issue_5164():
assert limit(x**0.5, x, oo) == oo**0.5 is oo
assert limit(x**0.5, x, 16) == S(16)**0.5
assert limit(x**0.5, x, 0) == 0
assert limit(x**(-0.5), x, oo) == 0
assert limit(x**(-0.5), x, 4) == S(4)**(-0.5)
def test_issue_5383():
func = (1.0 * 1 + 1.0 * x)**(1.0 * 1 / x)
assert limit(func, x, 0) == E
def test_issue_14793():
expr = ((x + S(1)/2) * log(x) - x + log(2*pi)/2 - \
log(factorial(x)) + S(1)/(12*x))*x**3
assert limit(expr, x, oo) == S(1)/360
def test_issue_5183():
# using list(...) so py.test can recalculate values
tests = list(product([x, -x],
[-1, 1],
[2, 3, S.Half, Rational(2, 3)],
['-', '+']))
results = (oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3), oo,
0, 0, 0, 0, 0, 0, 0, 0,
oo, oo, oo, -oo, oo, -oo*I, oo, -oo*(-1)**Rational(1, 3),
0, 0, 0, 0, 0, 0, 0, 0)
assert len(tests) == len(results)
for i, (args, res) in enumerate(zip(tests, results)):
y, s, e, d = args
eq = y**(s*e)
try:
assert limit(eq, x, 0, dir=d) == res
except AssertionError:
if 0: # change to 1 if you want to see the failing tests
print()
print(i, res, eq, d, limit(eq, x, 0, dir=d))
else:
assert None
def test_issue_5184():
assert limit(sin(x)/x, x, oo) == 0
assert limit(atan(x), x, oo) == pi/2
assert limit(gamma(x), x, oo) is oo
assert limit(cos(x)/x, x, oo) == 0
assert limit(gamma(x), x, S.Half) == sqrt(pi)
r = Symbol('r', real=True)
assert limit(r*sin(1/r), r, 0) == 0
def test_issue_5229():
assert limit((1 + y)**(1/y) - S.Exp1, y, 0) == 0
def test_issue_4546():
# using list(...) so py.test can recalculate values
tests = list(product([cot, tan],
[-pi/2, 0, pi/2, pi, pi*Rational(3, 2)],
['-', '+']))
results = (0, 0, -oo, oo, 0, 0, -oo, oo, 0, 0,
oo, -oo, 0, 0, oo, -oo, 0, 0, oo, -oo)
assert len(tests) == len(results)
for i, (args, res) in enumerate(zip(tests, results)):
f, l, d = args
eq = f(x)
try:
assert limit(eq, x, l, dir=d) == res
except AssertionError:
if 0: # change to 1 if you want to see the failing tests
print()
print(i, res, eq, l, d, limit(eq, x, l, dir=d))
else:
assert None
def test_issue_3934():
assert limit((1 + x**log(3))**(1/x), x, 0) == 1
assert limit((5**(1/x) + 3**(1/x))**x, x, 0) == 5
def test_calculate_series():
# needs gruntz calculate_series to go to n = 32
assert limit(x**Rational(77, 3)/(1 + x**Rational(77, 3)), x, oo) == 1
# needs gruntz calculate_series to go to n = 128
assert limit(x**101.1/(1 + x**101.1), x, oo) == 1
def test_issue_5955():
assert limit((x**16)/(1 + x**16), x, oo) == 1
assert limit((x**100)/(1 + x**100), x, oo) == 1
assert limit((x**1885)/(1 + x**1885), x, oo) == 1
assert limit((x**1000/((x + 1)**1000 + exp(-x))), x, oo) == 1
def test_newissue():
assert limit(exp(1/sin(x))/exp(cot(x)), x, 0) == 1
def test_extended_real_line():
assert limit(x - oo, x, oo) == Limit(x - oo, x, oo)
assert limit(1/(x + sin(x)) - oo, x, 0) == Limit(1/(x + sin(x)) - oo, x, 0)
assert limit(oo/x, x, oo) == Limit(oo/x, x, oo)
assert limit(x - oo + 1/x, x, oo) == Limit(x - oo + 1/x, x, oo)
@XFAIL
def test_order_oo():
x = Symbol('x', positive=True)
assert Order(x)*oo != Order(1, x)
assert limit(oo/(x**2 - 4), x, oo) is oo
def test_issue_5436():
raises(NotImplementedError, lambda: limit(exp(x*y), x, oo))
raises(NotImplementedError, lambda: limit(exp(-x*y), x, oo))
def test_Limit_dir():
raises(TypeError, lambda: Limit(x, x, 0, dir=0))
raises(ValueError, lambda: Limit(x, x, 0, dir='0'))
def test_polynomial():
assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, oo) == 1
assert limit((x + 1)**1000/((x + 1)**1000 + 1), x, -oo) == 1
def test_rational():
assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, oo) == (z - 1)/(y*z)
assert limit(1/y - (1/(y + x) + x/(y + x)/y)/z, x, -oo) == (z - 1)/(y*z)
def test_issue_5740():
assert limit(log(x)*z - log(2*x)*y, x, 0) == oo*sign(y - z)
def test_issue_6366():
n = Symbol('n', integer=True, positive=True)
r = (n + 1)*x**(n + 1)/(x**(n + 1) - 1) - x/(x - 1)
assert limit(r, x, 1).cancel() == n/2
def test_factorial():
f = factorial(x)
assert limit(f, x, oo) is oo
assert limit(x/f, x, oo) == 0
# see Stirling's approximation:
# https://en.wikipedia.org/wiki/Stirling's_approximation
assert limit(f/(sqrt(2*pi*x)*(x/E)**x), x, oo) == 1
assert limit(f, x, -oo) == factorial(-oo)
def test_issue_6560():
e = (5*x**3/4 - x*Rational(3, 4) + (y*(3*x**2/2 - S.Half) +
35*x**4/8 - 15*x**2/4 + Rational(3, 8))/(2*(y + 1)))
assert limit(e, y, oo) == 5*x**3/4 + 3*x**2/4 - 3*x/4 - Rational(1, 4)
@XFAIL
def test_issue_5172():
n = Symbol('n')
r = Symbol('r', positive=True)
c = Symbol('c')
p = Symbol('p', positive=True)
m = Symbol('m', negative=True)
expr = ((2*n*(n - r + 1)/(n + r*(n - r + 1)))**c +
(r - 1)*(n*(n - r + 2)/(n + r*(n - r + 1)))**c - n)/(n**c - n)
expr = expr.subs(c, c + 1)
raises(NotImplementedError, lambda: limit(expr, n, oo))
assert limit(expr.subs(c, m), n, oo) == 1
assert limit(expr.subs(c, p), n, oo).simplify() == \
(2**(p + 1) + r - 1)/(r + 1)**(p + 1)
def test_issue_7088():
a = Symbol('a')
assert limit(sqrt(x/(x + a)), x, oo) == 1
def test_branch_cuts():
assert limit(asin(I*x + 2), x, 0) == pi - asin(2)
assert limit(asin(I*x + 2), x, 0, '-') == asin(2)
assert limit(asin(I*x - 2), x, 0) == -asin(2)
assert limit(asin(I*x - 2), x, 0, '-') == -pi + asin(2)
assert limit(acos(I*x + 2), x, 0) == -acos(2)
assert limit(acos(I*x + 2), x, 0, '-') == acos(2)
assert limit(acos(I*x - 2), x, 0) == acos(-2)
assert limit(acos(I*x - 2), x, 0, '-') == 2*pi - acos(-2)
assert limit(atan(x + 2*I), x, 0) == I*atanh(2)
assert limit(atan(x + 2*I), x, 0, '-') == -pi + I*atanh(2)
assert limit(atan(x - 2*I), x, 0) == pi - I*atanh(2)
assert limit(atan(x - 2*I), x, 0, '-') == -I*atanh(2)
assert limit(atan(1/x), x, 0) == pi/2
assert limit(atan(1/x), x, 0, '-') == -pi/2
assert limit(atan(x), x, oo) == pi/2
assert limit(atan(x), x, -oo) == -pi/2
assert limit(acot(x + S(1)/2*I), x, 0) == pi - I*acoth(S(1)/2)
assert limit(acot(x + S(1)/2*I), x, 0, '-') == -I*acoth(S(1)/2)
assert limit(acot(x - S(1)/2*I), x, 0) == I*acoth(S(1)/2)
assert limit(acot(x - S(1)/2*I), x, 0, '-') == -pi + I*acoth(S(1)/2)
assert limit(acot(x), x, 0) == pi/2
assert limit(acot(x), x, 0, '-') == -pi/2
assert limit(asec(I*x + S(1)/2), x, 0) == asec(S(1)/2)
assert limit(asec(I*x + S(1)/2), x, 0, '-') == -asec(S(1)/2)
assert limit(asec(I*x - S(1)/2), x, 0) == 2*pi - asec(-S(1)/2)
assert limit(asec(I*x - S(1)/2), x, 0, '-') == asec(-S(1)/2)
assert limit(acsc(I*x + S(1)/2), x, 0) == acsc(S(1)/2)
assert limit(acsc(I*x + S(1)/2), x, 0, '-') == pi - acsc(S(1)/2)
assert limit(acsc(I*x - S(1)/2), x, 0) == -pi + acsc(S(1)/2)
assert limit(acsc(I*x - S(1)/2), x, 0, '-') == -acsc(S(1)/2)
assert limit(log(I*x - 1), x, 0) == I*pi
assert limit(log(I*x - 1), x, 0, '-') == -I*pi
assert limit(log(-I*x - 1), x, 0) == -I*pi
assert limit(log(-I*x - 1), x, 0, '-') == I*pi
assert limit(sqrt(I*x - 1), x, 0) == I
assert limit(sqrt(I*x - 1), x, 0, '-') == -I
assert limit(sqrt(-I*x - 1), x, 0) == -I
assert limit(sqrt(-I*x - 1), x, 0, '-') == I
assert limit(cbrt(I*x - 1), x, 0) == (-1)**(S(1)/3)
assert limit(cbrt(I*x - 1), x, 0, '-') == -(-1)**(S(2)/3)
assert limit(cbrt(-I*x - 1), x, 0) == -(-1)**(S(2)/3)
assert limit(cbrt(-I*x - 1), x, 0, '-') == (-1)**(S(1)/3)
def test_issue_6364():
a = Symbol('a')
e = z/(1 - sqrt(1 + z)*sin(a)**2 - sqrt(1 - z)*cos(a)**2)
assert limit(e, z, 0) == 1/(cos(a)**2 - S.Half)
def test_issue_6682():
assert limit(exp(2*Ei(-x))/x**2, x, 0) == exp(2*EulerGamma)
def test_issue_4099():
a = Symbol('a')
assert limit(a/x, x, 0) == oo*sign(a)
assert limit(-a/x, x, 0) == -oo*sign(a)
assert limit(-a*x, x, oo) == -oo*sign(a)
assert limit(a*x, x, oo) == oo*sign(a)
def test_issue_4503():
dx = Symbol('dx')
assert limit((sqrt(1 + exp(x + dx)) - sqrt(1 + exp(x)))/dx, dx, 0) == \
exp(x)/(2*sqrt(exp(x) + 1))
def test_issue_8208():
assert limit(n**(Rational(1, 1e9) - 1), n, oo) == 0
def test_issue_8229():
assert limit((x**Rational(1, 4) - 2)/(sqrt(x) - 4)**Rational(2, 3), x, 16) == 0
def test_issue_8433():
d, t = symbols('d t', positive=True)
assert limit(erf(1 - t/d), t, oo) == -1
def test_issue_8481():
k = Symbol('k', integer=True, nonnegative=True)
lamda = Symbol('lamda', positive=True)
assert limit(lamda**k * exp(-lamda) / factorial(k), k, oo) == 0
def test_issue_8635_18176():
x = Symbol('x', real=True)
k = Symbol('k', positive=True)
assert limit(x**n - x**(n - 0), x, oo) == 0
assert limit(x**n - x**(n - 5), x, oo) == oo
assert limit(x**n - x**(n - 2.5), x, oo) == oo
assert limit(x**n - x**(n - k - 1), x, oo) == oo
x = Symbol('x', positive=True)
assert limit(x**n - x**(n - 1), x, oo) == oo
assert limit(x**n - x**(n + 2), x, oo) == -oo
def test_issue_8730():
assert limit(subfactorial(x), x, oo) is oo
def test_issue_9252():
n = Symbol('n', integer=True)
c = Symbol('c', positive=True)
assert limit((log(n))**(n/log(n)) / (1 + c)**n, n, oo) == 0
# limit should depend on the value of c
raises(NotImplementedError, lambda: limit((log(n))**(n/log(n)) / c**n, n, oo))
def test_issue_9558():
assert limit(sin(x)**15, x, 0, '-') == 0
def test_issue_10801():
# make sure limits work with binomial
assert limit(16**k / (k * binomial(2*k, k)**2), k, oo) == pi
def test_issue_10976():
s, x = symbols('s x', real=True)
assert limit(erf(s*x)/erf(s), s, 0) == x
def test_issue_9041():
assert limit(factorial(n) / ((n/exp(1))**n * sqrt(2*pi*n)), n, oo) == 1
def test_issue_9205():
x, y, a = symbols('x, y, a')
assert Limit(x, x, a).free_symbols == {a}
assert Limit(x, x, a, '-').free_symbols == {a}
assert Limit(x + y, x + y, a).free_symbols == {a}
assert Limit(-x**2 + y, x**2, a).free_symbols == {y, a}
def test_issue_9471():
assert limit(((27**(log(n,3)))/n**3),n,oo) == 1
assert limit(((27**(log(n,3)+1))/n**3),n,oo) == 27
def test_issue_11496():
assert limit(erfc(log(1/x)), x, oo) == 2
def test_issue_11879():
assert simplify(limit(((x+y)**n-x**n)/y, y, 0)) == n*x**(n-1)
def test_limit_with_Float():
k = symbols("k")
assert limit(1.0 ** k, k, oo) == 1
assert limit(0.3*1.0**k, k, oo) == Rational(3, 10)
def test_issue_10610():
assert limit(3**x*3**(-x - 1)*(x + 1)**2/x**2, x, oo) == Rational(1, 3)
def test_issue_10868():
assert limit(log(x) + asech(x), x, 0, '+') == log(2)
assert limit(log(x) + asech(x), x, 0, '-') == log(2) + 2*I*pi
raises(ValueError, lambda: limit(log(x) + asech(x), x, 0, '+-'))
assert limit(log(x) + asech(x), x, oo) == oo
assert limit(log(x) + acsch(x), x, 0, '+') == log(2)
assert limit(log(x) + acsch(x), x, 0, '-') == -oo
raises(ValueError, lambda: limit(log(x) + acsch(x), x, 0, '+-'))
assert limit(log(x) + acsch(x), x, oo) == oo
def test_issue_6599():
assert limit((n + cos(n))/n, n, oo) == 1
def test_issue_12555():
assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, -oo) == 2
assert limit((3**x + 2* x**10) / (x**10 + exp(x)), x, oo) is oo
def test_issue_12769():
r, z, x = symbols('r z x', real=True)
a, b, s0, K, F0, s, T = symbols('a b s0 K F0 s T', positive=True, real=True)
fx = (F0**b*K**b*r*s0 - sqrt((F0**2*K**(2*b)*a**2*(b - 1) + \
F0**(2*b)*K**2*a**2*(b - 1) + F0**(2*b)*K**(2*b)*s0**2*(b - 1)*(b**2 - 2*b + 1) - \
2*F0**(2*b)*K**(b + 1)*a*r*s0*(b**2 - 2*b + 1) + \
2*F0**(b + 1)*K**(2*b)*a*r*s0*(b**2 - 2*b + 1) - \
2*F0**(b + 1)*K**(b + 1)*a**2*(b - 1))/((b - 1)*(b**2 - 2*b + 1))))*(b*r - b - r + 1)
assert fx.subs(K, F0).factor(deep=True) == limit(fx, K, F0).factor(deep=True)
def test_issue_13332():
assert limit(sqrt(30)*5**(-5*x - 1)*(46656*x)**x*(5*x + 2)**(5*x + 5*S.Half) *
(6*x + 2)**(-6*x - 5*S.Half), x, oo) == Rational(25, 36)
def test_issue_12564():
assert limit(x**2 + x*sin(x) + cos(x), x, -oo) is oo
assert limit(x**2 + x*sin(x) + cos(x), x, oo) is oo
assert limit(((x + cos(x))**2).expand(), x, oo) is oo
assert limit(((x + sin(x))**2).expand(), x, oo) is oo
assert limit(((x + cos(x))**2).expand(), x, -oo) is oo
assert limit(((x + sin(x))**2).expand(), x, -oo) is oo
def test_issue_14456():
raises(NotImplementedError, lambda: Limit(exp(x), x, zoo).doit())
raises(NotImplementedError, lambda: Limit(x**2/(x+1), x, zoo).doit())
def test_issue_14411():
assert limit(3*sec(4*pi*x - x/3), x, 3*pi/(24*pi - 2)) is -oo
def test_issue_13382():
assert limit(x*(((x + 1)**2 + 1)/(x**2 + 1) - 1), x, oo) == 2
def test_issue_13403():
assert limit(x*(-1 + (x + log(x + 1) + 1)/(x + log(x))), x, oo) == 1
def test_issue_13416():
assert limit((-x**3*log(x)**3 + (x - 1)*(x + 1)**2*log(x + 1)**3)/(x**2*log(x)**3), x, oo) == 1
def test_issue_13462():
assert limit(n**2*(2*n*(-(1 - 1/(2*n))**x + 1) - x - (-x**2/4 + x/4)/n), n, oo) == x**3/24 - x**2/8 + x/12
def test_issue_13750():
a = Symbol('a')
assert limit(erf(a - x), x, oo) == -1
assert limit(erf(sqrt(x) - x), x, oo) == -1
def test_issue_14514():
assert limit((1/(log(x)**log(x)))**(1/x), x, oo) == 1
def test_issues_14525():
assert limit(sin(x)**2 - cos(x) + tan(x)*csc(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(sin(x)**2 - cos(x) + sin(x)*cot(x), x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(cot(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert limit(cos(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One)
assert limit(sin(x) - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One)
assert limit(cos(x)**2 - tan(x)**2, x, oo) == AccumBounds(S.NegativeInfinity, S.One)
assert limit(tan(x)**2 + sin(x)**2 - cos(x), x, oo) == AccumBounds(-S.One, S.Infinity)
def test_issue_14574():
assert limit(sqrt(x)*cos(x - x**2) / (x + 1), x, oo) == 0
def test_issue_10102():
assert limit(fresnels(x), x, oo) == S.Half
assert limit(3 + fresnels(x), x, oo) == 3 + S.Half
assert limit(5*fresnels(x), x, oo) == Rational(5, 2)
assert limit(fresnelc(x), x, oo) == S.Half
assert limit(fresnels(x), x, -oo) == Rational(-1, 2)
assert limit(4*fresnelc(x), x, -oo) == -2
def test_issue_14377():
raises(NotImplementedError, lambda: limit(exp(I*x)*sin(pi*x), x, oo))
def test_issue_15146():
e = (x/2) * (-2*x**3 - 2*(x**3 - 1) * x**2 * digamma(x**3 + 1) + \
2*(x**3 - 1) * x**2 * digamma(x**3 + x + 1) + x + 3)
assert limit(e, x, oo) == S(1)/3
def test_issue_15202():
e = (2**x*(2 + 2**(-x)*(-2*2**x + x + 2))/(x + 1))**(x + 1)
assert limit(e, x, oo) == exp(1)
e = (log(x, 2)**7 + 10*x*factorial(x) + 5**x) / (factorial(x + 1) + 3*factorial(x) + 10**x)
assert limit(e, x, oo) == 10
def test_issue_15282():
assert limit((x**2000 - (x + 1)**2000) / x**1999, x, oo) == -2000
def test_issue_15984():
assert limit((-x + log(exp(x) + 1))/x, x, oo, dir='-') == 0
def test_issue_13571():
assert limit(uppergamma(x, 1) / gamma(x), x, oo) == 1
def test_issue_13575():
assert limit(acos(erfi(x)), x, 1) == acos(erfi(S.One))
def test_issue_17325():
assert Limit(sin(x)/x, x, 0, dir="+-").doit() == 1
assert Limit(x**2, x, 0, dir="+-").doit() == 0
assert Limit(1/x**2, x, 0, dir="+-").doit() is oo
assert Limit(1/x, x, 0, dir="+-").doit() is zoo
def test_issue_10978():
assert LambertW(x).limit(x, 0) == 0
def test_issue_14313_comment():
assert limit(floor(n/2), n, oo) is oo
@XFAIL
def test_issue_15323():
d = ((1 - 1/x)**x).diff(x)
assert limit(d, x, 1, dir='+') == 1
def test_issue_12571():
assert limit(-LambertW(-log(x))/log(x), x, 1) == 1
def test_issue_14590():
assert limit((x**3*((x + 1)/x)**x)/((x + 1)*(x + 2)*(x + 3)), x, oo) == exp(1)
def test_issue_14393():
a, b = symbols('a b')
assert limit((x**b - y**b)/(x**a - y**a), x, y) == b*y**(-a + b)/a
def test_issue_14556():
assert limit(factorial(n + 1)**(1/(n + 1)) - factorial(n)**(1/n), n, oo) == exp(-1)
def test_issue_14811():
assert limit(((1 + ((S(2)/3)**(x + 1)))**(2**x))/(2**((S(4)/3)**(x - 1))), x, oo) == oo
def test_issue_14874():
assert limit(besselk(0, x), x, oo) == 0
def test_issue_16222():
assert limit(exp(x), x, 1000000000) == exp(1000000000)
def test_issue_16714():
assert limit(((x**(x + 1) + (x + 1)**x) / x**(x + 1))**x, x, oo) == exp(exp(1))
def test_issue_16722():
z = symbols('z', positive=True)
assert limit(binomial(n + z, n)*n**-z, n, oo) == 1/gamma(z + 1)
z = symbols('z', positive=True, integer=True)
assert limit(binomial(n + z, n)*n**-z, n, oo) == 1/gamma(z + 1)
def test_issue_17431():
assert limit(((n + 1) + 1) / (((n + 1) + 2) * factorial(n + 1)) *
(n + 2) * factorial(n) / (n + 1), n, oo) == 0
assert limit((n + 2)**2*factorial(n)/((n + 1)*(n + 3)*factorial(n + 1))
, n, oo) == 0
assert limit((n + 1) * factorial(n) / (n * factorial(n + 1)), n, oo) == 0
def test_issue_17671():
assert limit(Ei(-log(x)) - log(log(x))/x, x, 1) == EulerGamma
def test_issue_17751():
a, b, c, x = symbols('a b c x', positive=True)
assert limit((a + 1)*x - sqrt((a + 1)**2*x**2 + b*x + c), x, oo) == -b/(2*a + 2)
def test_issue_17792():
assert limit(factorial(n)/sqrt(n)*(exp(1)/n)**n, n, oo) == sqrt(2)*sqrt(pi)
def test_issue_18118():
assert limit(sign(sin(x)), x, 0, "-") == -1
assert limit(sign(sin(x)), x, 0, "+") == 1
def test_issue_18306():
assert limit(sin(sqrt(x))/sqrt(sin(x)), x, 0, '+') == 1
def test_issue_18378():
assert limit(log(exp(3*x) + x)/log(exp(x) + x**100), x, oo) == 3
def test_issue_18399():
assert limit((1 - S(1)/2*x)**(3*x), x, oo) is zoo
assert limit((-x)**x, x, oo) is zoo
def test_issue_18442():
assert limit(tan(x)**(2**(sqrt(pi))), x, oo, dir='-') == Limit(tan(x)**(2**(sqrt(pi))), x, oo, dir='-')
def test_issue_18452():
assert limit(abs(log(x))**x, x, 0) == 1
assert limit(abs(log(x))**x, x, 0, "-") == 1
def test_issue_18473():
assert limit(sin(x)**(1/x), x, oo) == Limit(sin(x)**(1/x), x, oo, dir='-')
assert limit(cos(x)**(1/x), x, oo) == Limit(cos(x)**(1/x), x, oo, dir='-')
assert limit(tan(x)**(1/x), x, oo) == Limit(tan(x)**(1/x), x, oo, dir='-')
assert limit((cos(x) + 2)**(1/x), x, oo) == 1
assert limit((sin(x) + 10)**(1/x), x, oo) == 1
assert limit((cos(x) - 2)**(1/x), x, oo) == Limit((cos(x) - 2)**(1/x), x, oo, dir='-')
assert limit((cos(x) + 1)**(1/x), x, oo) == AccumBounds(0, 1)
assert limit((tan(x)**2)**(2/x) , x, oo) == AccumBounds(0, oo)
assert limit((sin(x)**2)**(1/x), x, oo) == AccumBounds(0, 1)
# Tests for issue #23751
assert limit((cos(x) + 1)**(1/x), x, -oo) == AccumBounds(1, oo)
assert limit((sin(x)**2)**(1/x), x, -oo) == AccumBounds(1, oo)
assert limit((tan(x)**2)**(2/x) , x, -oo) == AccumBounds(0, oo)
def test_issue_18482():
assert limit((2*exp(3*x)/(exp(2*x) + 1))**(1/x), x, oo) == exp(1)
def test_issue_18508():
assert limit(sin(x)/sqrt(1-cos(x)), x, 0) == sqrt(2)
assert limit(sin(x)/sqrt(1-cos(x)), x, 0, dir='+') == sqrt(2)
assert limit(sin(x)/sqrt(1-cos(x)), x, 0, dir='-') == -sqrt(2)
def test_issue_18521():
raises(NotImplementedError, lambda: limit(exp((2 - n) * x), x, oo))
def test_issue_18969():
a, b = symbols('a b', positive=True)
assert limit(LambertW(a), a, b) == LambertW(b)
assert limit(exp(LambertW(a)), a, b) == exp(LambertW(b))
def test_issue_18992():
assert limit(n/(factorial(n)**(1/n)), n, oo) == exp(1)
def test_issue_19067():
x = Symbol('x')
assert limit(gamma(x)/(gamma(x - 1)*gamma(x + 2)), x, 0) == -1
def test_issue_19586():
assert limit(x**(2**x*3**(-x)), x, oo) == 1
def test_issue_13715():
n = Symbol('n')
p = Symbol('p', zero=True)
assert limit(n + p, n, 0) == 0
def test_issue_15055():
assert limit(n**3*((-n - 1)*sin(1/n) + (n + 2)*sin(1/(n + 1)))/(-n + 1), n, oo) == 1
def test_issue_16708():
m, vi = symbols('m vi', positive=True)
B, ti, d = symbols('B ti d')
assert limit((B*ti*vi - sqrt(m)*sqrt(-2*B*d*vi + m*(vi)**2) + m*vi)/(B*vi), B, 0) == (d + ti*vi)/vi
def test_issue_19453():
beta = Symbol("beta", positive=True)
h = Symbol("h", positive=True)
m = Symbol("m", positive=True)
w = Symbol("omega", positive=True)
g = Symbol("g", positive=True)
e = exp(1)
q = 3*h**2*beta*g*e**(0.5*h*beta*w)
p = m**2*w**2
s = e**(h*beta*w) - 1
Z = -q/(4*p*s) - q/(2*p*s**2) - q*(e**(h*beta*w) + 1)/(2*p*s**3)\
+ e**(0.5*h*beta*w)/s
E = -diff(log(Z), beta)
assert limit(E - 0.5*h*w, beta, oo) == 0
assert limit(E.simplify() - 0.5*h*w, beta, oo) == 0
def test_issue_19739():
assert limit((-S(1)/4)**x, x, oo) == 0
def test_issue_19766():
assert limit(2**(-x)*sqrt(4**(x + 1) + 1), x, oo) == 2
def test_issue_19770():
m = Symbol('m')
# the result is not 0 for non-real m
assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-')
m = Symbol('m', real=True)
# can be improved to give the correct result 0
assert limit(cos(m*x)/x, x, oo) == Limit(cos(m*x)/x, x, oo, dir='-')
m = Symbol('m', nonzero=True)
assert limit(cos(m*x), x, oo) == AccumBounds(-1, 1)
assert limit(cos(m*x)/x, x, oo) == 0
def test_issue_7535():
assert limit(tan(x)/sin(tan(x)), x, pi/2) == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+')
assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='-')
assert limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-') == Limit(tan(x)/sin(tan(x)), x, pi/2, dir='+-')
assert limit(sin(tan(x)),x,pi/2) == AccumBounds(-1, 1)
assert -oo*(1/sin(-oo)) == AccumBounds(-oo, oo)
assert oo*(1/sin(oo)) == AccumBounds(-oo, oo)
assert oo*(1/sin(-oo)) == AccumBounds(-oo, oo)
assert -oo*(1/sin(oo)) == AccumBounds(-oo, oo)
def test_issue_20365():
assert limit(((x + 1)**(1/x) - E)/x, x, 0) == -E/2
def test_issue_21031():
assert limit(((1 + x)**(1/x) - (1 + 2*x)**(1/(2*x)))/asin(x), x, 0) == E/2
def test_issue_21038():
assert limit(sin(pi*x)/(3*x - 12), x, 4) == pi/3
def test_issue_20578():
expr = abs(x) * sin(1/x)
assert limit(expr,x,0,'+') == 0
assert limit(expr,x,0,'-') == 0
assert limit(expr,x,0,'+-') == 0
def test_issue_21227():
f = log(x)
assert f.nseries(x, logx=y) == y
assert f.nseries(x, logx=-x) == -x
f = log(-log(x))
assert f.nseries(x, logx=y) == log(-y)
assert f.nseries(x, logx=-x) == log(x)
f = log(log(x))
assert f.nseries(x, logx=y) == log(y)
assert f.nseries(x, logx=-x) == log(-x)
assert f.nseries(x, logx=x) == log(x)
f = log(log(log(1/x)))
assert f.nseries(x, logx=y) == log(log(-y))
assert f.nseries(x, logx=-y) == log(log(y))
assert f.nseries(x, logx=x) == log(log(-x))
assert f.nseries(x, logx=-x) == log(log(x))
def test_issue_21415():
exp = (x-1)*cos(1/(x-1))
assert exp.limit(x,1) == 0
assert exp.expand().limit(x,1) == 0
def test_issue_21530():
assert limit(sinh(n + 1)/sinh(n), n, oo) == E
def test_issue_21550():
r = (sqrt(5) - 1)/2
assert limit((x - r)/(x**2 + x - 1), x, r) == sqrt(5)/5
def test_issue_21661():
out = limit((x**(x + 1) * (log(x) + 1) + 1) / x, x, 11)
assert out == S(3138428376722)/11 + 285311670611*log(11)
def test_issue_21701():
assert limit((besselj(z, x)/x**z).subs(z, 7), x, 0) == S(1)/645120
def test_issue_21721():
a = Symbol('a', real=True)
I = integrate(1/(pi*(1 + (x - a)**2)), x)
assert I.limit(x, oo) == S.Half
def test_issue_21756():
term = (1 - exp(-2*I*pi*z))/(1 - exp(-2*I*pi*z/5))
assert term.limit(z, 0) == 5
assert re(term).limit(z, 0) == 5
def test_issue_21785():
a = Symbol('a')
assert sqrt((-a**2 + x**2)/(1 - x**2)).limit(a, 1, '-') == I
def test_issue_22181():
assert limit((-1)**x * 2**(-x), x, oo) == 0
def test_issue_22220():
e1 = sqrt(30)*atan(sqrt(30)*tan(x/2)/6)/30
e2 = sqrt(30)*I*(-log(sqrt(2)*tan(x/2) - 2*sqrt(15)*I/5) +
+log(sqrt(2)*tan(x/2) + 2*sqrt(15)*I/5))/60
assert limit(e1, x, -pi) == -sqrt(30)*pi/60
assert limit(e2, x, -pi) == -sqrt(30)*pi/30
assert limit(e1, x, -pi, '-') == sqrt(30)*pi/60
assert limit(e2, x, -pi, '-') == 0
# test https://github.com/sympy/sympy/issues/22220#issuecomment-972727694
expr = log(x - I) - log(-x - I)
expr2 = logcombine(expr, force=True)
assert limit(expr, x, oo) == limit(expr2, x, oo) == I*pi
# test https://github.com/sympy/sympy/issues/22220#issuecomment-1077618340
expr = expr = (-log(tan(x/2) - I) +log(tan(x/2) + I))
assert limit(expr, x, pi, '+') == 2*I*pi
assert limit(expr, x, pi, '-') == 0
def test_sympyissue_22986():
assert limit(acosh(1 + 1/x)*sqrt(x), x, oo) == sqrt(2)
def test_issue_23231():
f = (2**x - 2**(-x))/(2**x + 2**(-x))
assert limit(f, x, -oo) == -1
def test_issue_23596():
assert integrate(((1 + x)/x**2)*exp(-1/x), (x, 0, oo)) == oo
def test_issue_23752():
expr1 = sqrt(-I*x**2 + x - 3)
expr2 = sqrt(-I*x**2 + I*x - 3)
assert limit(expr1, x, 0, '+') == -sqrt(3)*I
assert limit(expr1, x, 0, '-') == -sqrt(3)*I
assert limit(expr2, x, 0, '+') == sqrt(3)*I
assert limit(expr2, x, 0, '-') == -sqrt(3)*I
|
dd75d5f1e92e8c3a27fab2c11d4071f8ca8f55d39a372fff50e6feef4e404380 | from sympy.core.function import (Derivative, PoleError)
from sympy.core.numbers import (E, I, Integer, Rational, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.hyperbolic import (acosh, acoth, asinh, atanh, cosh, coth, sinh, tanh)
from sympy.functions.elementary.integers import (ceiling, floor)
from sympy.functions.elementary.miscellaneous import (cbrt, sqrt)
from sympy.functions.elementary.trigonometric import (asin, cos, cot, sin, tan)
from sympy.series.limits import limit
from sympy.series.order import O
from sympy.abc import x, y, z
from sympy.testing.pytest import raises, XFAIL
def test_simple_1():
assert x.nseries(x, n=5) == x
assert y.nseries(x, n=5) == y
assert (1/(x*y)).nseries(y, n=5) == 1/(x*y)
assert Rational(3, 4).nseries(x, n=5) == Rational(3, 4)
assert x.nseries() == x
def test_mul_0():
assert (x*log(x)).nseries(x, n=5) == x*log(x)
def test_mul_1():
assert (x*log(2 + x)).nseries(x, n=5) == x*log(2) + x**2/2 - x**3/8 + \
x**4/24 + O(x**5)
assert (x*log(1 + x)).nseries(
x, n=5) == x**2 - x**3/2 + x**4/3 + O(x**5)
def test_pow_0():
assert (x**2).nseries(x, n=5) == x**2
assert (1/x).nseries(x, n=5) == 1/x
assert (1/x**2).nseries(x, n=5) == 1/x**2
assert (x**Rational(2, 3)).nseries(x, n=5) == (x**Rational(2, 3))
assert (sqrt(x)**3).nseries(x, n=5) == (sqrt(x)**3)
def test_pow_1():
assert ((1 + x)**2).nseries(x, n=5) == x**2 + 2*x + 1
# https://github.com/sympy/sympy/issues/21075
assert ((sqrt(x) + 1)**2).nseries(x) == 2*sqrt(x) + x + 1
assert ((sqrt(x) + cbrt(x))**2).nseries(x) == 2*x**Rational(5, 6)\
+ x**Rational(2, 3) + x
def test_geometric_1():
assert (1/(1 - x)).nseries(x, n=5) == 1 + x + x**2 + x**3 + x**4 + O(x**5)
assert (x/(1 - x)).nseries(x, n=6) == x + x**2 + x**3 + x**4 + x**5 + O(x**6)
assert (x**3/(1 - x)).nseries(x, n=8) == x**3 + x**4 + x**5 + x**6 + \
x**7 + O(x**8)
def test_sqrt_1():
assert sqrt(1 + x).nseries(x, n=5) == 1 + x/2 - x**2/8 + x**3/16 - 5*x**4/128 + O(x**5)
def test_exp_1():
assert exp(x).nseries(x, n=5) == 1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5)
assert exp(x).nseries(x, n=12) == 1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + \
x**6/720 + x**7/5040 + x**8/40320 + x**9/362880 + x**10/3628800 + \
x**11/39916800 + O(x**12)
assert exp(1/x).nseries(x, n=5) == exp(1/x)
assert exp(1/(1 + x)).nseries(x, n=4) == \
(E*(1 - x - 13*x**3/6 + 3*x**2/2)).expand() + O(x**4)
assert exp(2 + x).nseries(x, n=5) == \
(exp(2)*(1 + x + x**2/2 + x**3/6 + x**4/24)).expand() + O(x**5)
def test_exp_sqrt_1():
assert exp(1 + sqrt(x)).nseries(x, n=3) == \
(exp(1)*(1 + sqrt(x) + x/2 + sqrt(x)*x/6)).expand() + O(sqrt(x)**3)
def test_power_x_x1():
assert (exp(x*log(x))).nseries(x, n=4) == \
1 + x*log(x) + x**2*log(x)**2/2 + x**3*log(x)**3/6 + O(x**4*log(x)**4)
def test_power_x_x2():
assert (x**x).nseries(x, n=4) == \
1 + x*log(x) + x**2*log(x)**2/2 + x**3*log(x)**3/6 + O(x**4*log(x)**4)
def test_log_singular1():
assert log(1 + 1/x).nseries(x, n=5) == x - log(x) - x**2/2 + x**3/3 - \
x**4/4 + O(x**5)
def test_log_power1():
e = 1 / (1/x + x ** (log(3)/log(2)))
assert e.nseries(x, n=5) == -x**(log(3)/log(2) + 2) + x + O(x**5)
def test_log_series():
l = Symbol('l')
e = 1/(1 - log(x))
assert e.nseries(x, n=5, logx=l) == 1/(1 - l)
def test_log2():
e = log(-1/x)
assert e.nseries(x, n=5) == -log(x) + log(-1)
def test_log3():
l = Symbol('l')
e = 1/log(-1/x)
assert e.nseries(x, n=4, logx=l) == 1/(-l + log(-1))
def test_series1():
e = sin(x)
assert e.nseries(x, 0, 0) != 0
assert e.nseries(x, 0, 0) == O(1, x)
assert e.nseries(x, 0, 1) == O(x, x)
assert e.nseries(x, 0, 2) == x + O(x**2, x)
assert e.nseries(x, 0, 3) == x + O(x**3, x)
assert e.nseries(x, 0, 4) == x - x**3/6 + O(x**4, x)
e = (exp(x) - 1)/x
assert e.nseries(x, 0, 3) == 1 + x/2 + x**2/6 + O(x**3)
assert x.nseries(x, 0, 2) == x
@XFAIL
def test_series1_failing():
assert x.nseries(x, 0, 0) == O(1, x)
assert x.nseries(x, 0, 1) == O(x, x)
def test_seriesbug1():
assert (1/x).nseries(x, 0, 3) == 1/x
assert (x + 1/x).nseries(x, 0, 3) == x + 1/x
def test_series2x():
assert ((x + 1)**(-2)).nseries(x, 0, 4) == 1 - 2*x + 3*x**2 - 4*x**3 + O(x**4, x)
assert ((x + 1)**(-1)).nseries(x, 0, 4) == 1 - x + x**2 - x**3 + O(x**4, x)
assert ((x + 1)**0).nseries(x, 0, 3) == 1
assert ((x + 1)**1).nseries(x, 0, 3) == 1 + x
assert ((x + 1)**2).nseries(x, 0, 3) == x**2 + 2*x + 1
assert ((x + 1)**3).nseries(x, 0, 3) == 1 + 3*x + 3*x**2 + O(x**3)
assert (1/(1 + x)).nseries(x, 0, 4) == 1 - x + x**2 - x**3 + O(x**4, x)
assert (x + 3/(1 + 2*x)).nseries(x, 0, 4) == 3 - 5*x + 12*x**2 - 24*x**3 + O(x**4, x)
assert ((1/x + 1)**3).nseries(x, 0, 3) == 1 + 3/x + 3/x**2 + x**(-3)
assert (1/(1 + 1/x)).nseries(x, 0, 4) == x - x**2 + x**3 - O(x**4, x)
assert (1/(1 + 1/x**2)).nseries(x, 0, 6) == x**2 - x**4 + O(x**6, x)
def test_bug2(): # 1/log(0)*log(0) problem
w = Symbol("w")
e = (w**(-1) + w**(
-log(3)*log(2)**(-1)))**(-1)*(3*w**(-log(3)*log(2)**(-1)) + 2*w**(-1))
e = e.expand()
assert e.nseries(w, 0, 4).subs(w, 0) == 3
def test_exp():
e = (1 + x)**(1/x)
assert e.nseries(x, n=3) == exp(1) - x*exp(1)/2 + 11*exp(1)*x**2/24 + O(x**3)
def test_exp2():
w = Symbol("w")
e = w**(1 - log(x)/(log(2) + log(x)))
logw = Symbol("logw")
assert e.nseries(
w, 0, 1, logx=logw) == exp(logw*log(2)/(log(x) + log(2)))
def test_bug3():
e = (2/x + 3/x**2)/(1/x + 1/x**2)
assert e.nseries(x, n=3) == 3 - x + x**2 + O(x**3)
def test_generalexponent():
p = 2
e = (2/x + 3/x**p)/(1/x + 1/x**p)
assert e.nseries(x, 0, 3) == 3 - x + x**2 + O(x**3)
p = S.Half
e = (2/x + 3/x**p)/(1/x + 1/x**p)
assert e.nseries(x, 0, 2) == 2 - x + sqrt(x) + x**(S(3)/2) + O(x**2)
e = 1 + sqrt(x)
assert e.nseries(x, 0, 4) == 1 + sqrt(x)
# more complicated example
def test_genexp_x():
e = 1/(1 + sqrt(x))
assert e.nseries(x, 0, 2) == \
1 + x - sqrt(x) - sqrt(x)**3 + O(x**2, x)
# more complicated example
def test_genexp_x2():
p = Rational(3, 2)
e = (2/x + 3/x**p)/(1/x + 1/x**p)
assert e.nseries(x, 0, 3) == 3 + x + x**2 - sqrt(x) - x**(S(3)/2) - x**(S(5)/2) + O(x**3)
def test_seriesbug2():
w = Symbol("w")
#simple case (1):
e = ((2*w)/w)**(1 + w)
assert e.nseries(w, 0, 1) == 2 + O(w, w)
assert e.nseries(w, 0, 1).subs(w, 0) == 2
def test_seriesbug2b():
w = Symbol("w")
#test sin
e = sin(2*w)/w
assert e.nseries(w, 0, 3) == 2 - 4*w**2/3 + O(w**3)
def test_seriesbug2d():
w = Symbol("w", real=True)
e = log(sin(2*w)/w)
assert e.series(w, n=5) == log(2) - 2*w**2/3 - 4*w**4/45 + O(w**5)
def test_seriesbug2c():
w = Symbol("w", real=True)
#more complicated case, but sin(x)~x, so the result is the same as in (1)
e = (sin(2*w)/w)**(1 + w)
assert e.series(w, 0, 1) == 2 + O(w)
assert e.series(w, 0, 3) == 2 + 2*w*log(2) + \
w**2*(Rational(-4, 3) + log(2)**2) + O(w**3)
assert e.series(w, 0, 2).subs(w, 0) == 2
def test_expbug4():
x = Symbol("x", real=True)
assert (log(
sin(2*x)/x)*(1 + x)).series(x, 0, 2) == log(2) + x*log(2) + O(x**2, x)
assert exp(
log(sin(2*x)/x)*(1 + x)).series(x, 0, 2) == 2 + 2*x*log(2) + O(x**2)
assert exp(log(2) + O(x)).nseries(x, 0, 2) == 2 + O(x)
assert ((2 + O(x))**(1 + x)).nseries(x, 0, 2) == 2 + O(x)
def test_logbug4():
assert log(2 + O(x)).nseries(x, 0, 2) == log(2) + O(x, x)
def test_expbug5():
assert exp(log(1 + x)/x).nseries(x, n=3) == exp(1) + -exp(1)*x/2 + 11*exp(1)*x**2/24 + O(x**3)
assert exp(O(x)).nseries(x, 0, 2) == 1 + O(x)
def test_sinsinbug():
assert sin(sin(x)).nseries(x, 0, 8) == x - x**3/3 + x**5/10 - 8*x**7/315 + O(x**8)
def test_issue_3258():
a = x/(exp(x) - 1)
assert a.nseries(x, 0, 5) == 1 - x/2 - x**4/720 + x**2/12 + O(x**5)
def test_issue_3204():
x = Symbol("x", nonnegative=True)
f = sin(x**3)**Rational(1, 3)
assert f.nseries(x, 0, 17) == x - x**7/18 - x**13/3240 + O(x**17)
def test_issue_3224():
f = sqrt(1 - sqrt(y))
assert f.nseries(y, 0, 2) == 1 - sqrt(y)/2 - y/8 - sqrt(y)**3/16 + O(y**2)
def test_issue_3463():
w, i = symbols('w,i')
r = log(5)/log(3)
p = w**(-1 + r)
e = 1/x*(-log(w**(1 + r)) + log(w + w**r))
e_ser = -r*log(w)/x + p/x - p**2/(2*x) + O(w)
assert e.nseries(w, n=1) == e_ser
def test_sin():
assert sin(8*x).nseries(x, n=4) == 8*x - 256*x**3/3 + O(x**4)
assert sin(x + y).nseries(x, n=1) == sin(y) + O(x)
assert sin(x + y).nseries(x, n=2) == sin(y) + cos(y)*x + O(x**2)
assert sin(x + y).nseries(x, n=5) == sin(y) + cos(y)*x - sin(y)*x**2/2 - \
cos(y)*x**3/6 + sin(y)*x**4/24 + O(x**5)
def test_issue_3515():
e = sin(8*x)/x
assert e.nseries(x, n=6) == 8 - 256*x**2/3 + 4096*x**4/15 + O(x**6)
def test_issue_3505():
e = sin(x)**(-4)*(sqrt(cos(x))*sin(x)**2 -
cos(x)**Rational(1, 3)*sin(x)**2)
assert e.nseries(x, n=9) == Rational(-1, 12) - 7*x**2/288 - \
43*x**4/10368 - 1123*x**6/2488320 + 377*x**8/29859840 + O(x**9)
def test_issue_3501():
a = Symbol("a")
e = x**(-2)*(x*sin(a + x) - x*sin(a))
assert e.nseries(x, n=6) == cos(a) - sin(a)*x/2 - cos(a)*x**2/6 + \
x**3*sin(a)/24 + x**4*cos(a)/120 - x**5*sin(a)/720 + O(x**6)
e = x**(-2)*(x*cos(a + x) - x*cos(a))
assert e.nseries(x, n=6) == -sin(a) - cos(a)*x/2 + sin(a)*x**2/6 + \
cos(a)*x**3/24 - x**4*sin(a)/120 - x**5*cos(a)/720 + O(x**6)
def test_issue_3502():
e = sin(5*x)/sin(2*x)
assert e.nseries(x, n=2) == Rational(5, 2) + O(x**2)
assert e.nseries(x, n=6) == \
Rational(5, 2) - 35*x**2/4 + 329*x**4/48 + O(x**6)
def test_issue_3503():
e = sin(2 + x)/(2 + x)
assert e.nseries(x, n=2) == sin(2)/2 + x*cos(2)/2 - x*sin(2)/4 + O(x**2)
def test_issue_3506():
e = (x + sin(3*x))**(-2)*(x*(x + sin(3*x)) - (x + sin(3*x))*sin(2*x))
assert e.nseries(x, n=7) == \
Rational(-1, 4) + 5*x**2/96 + 91*x**4/768 + 11117*x**6/129024 + O(x**7)
def test_issue_3508():
x = Symbol("x", real=True)
assert log(sin(x)).series(x, n=5) == log(x) - x**2/6 - x**4/180 + O(x**5)
e = -log(x) + x*(-log(x) + log(sin(2*x))) + log(sin(2*x))
assert e.series(x, n=5) == \
log(2) + log(2)*x - 2*x**2/3 - 2*x**3/3 - 4*x**4/45 + O(x**5)
def test_issue_3507():
e = x**(-4)*(x**2 - x**2*sqrt(cos(x)))
assert e.nseries(x, n=9) == \
Rational(1, 4) + x**2/96 + 19*x**4/5760 + 559*x**6/645120 + 29161*x**8/116121600 + O(x**9)
def test_issue_3639():
assert sin(cos(x)).nseries(x, n=5) == \
sin(1) - x**2*cos(1)/2 - x**4*sin(1)/8 + x**4*cos(1)/24 + O(x**5)
def test_hyperbolic():
assert sinh(x).nseries(x, n=6) == x + x**3/6 + x**5/120 + O(x**6)
assert cosh(x).nseries(x, n=5) == 1 + x**2/2 + x**4/24 + O(x**5)
assert tanh(x).nseries(x, n=6) == x - x**3/3 + 2*x**5/15 + O(x**6)
assert coth(x).nseries(x, n=6) == \
1/x - x**3/45 + x/3 + 2*x**5/945 + O(x**6)
assert asinh(x).nseries(x, n=6) == x - x**3/6 + 3*x**5/40 + O(x**6)
assert acosh(x).nseries(x, n=6) == \
pi*I/2 - I*x - 3*I*x**5/40 - I*x**3/6 + O(x**6)
assert atanh(x).nseries(x, n=6) == x + x**3/3 + x**5/5 + O(x**6)
assert acoth(x).nseries(x, n=6) == -I*pi/2 + x + x**3/3 + x**5/5 + O(x**6)
def test_series2():
w = Symbol("w", real=True)
x = Symbol("x", real=True)
e = w**(-2)*(w*exp(1/x - w) - w*exp(1/x))
assert e.nseries(w, n=4) == -exp(1/x) + w*exp(1/x)/2 - w**2*exp(1/x)/6 + w**3*exp(1/x)/24 + O(w**4)
def test_series3():
w = Symbol("w", real=True)
e = w**(-6)*(w**3*tan(w) - w**3*sin(w))
assert e.nseries(w, n=8) == Integer(1)/2 + w**2/8 + 13*w**4/240 + 529*w**6/24192 + O(w**8)
def test_bug4():
w = Symbol("w")
e = x/(w**4 + x**2*w**4 + 2*x*w**4)*w**4
assert e.nseries(w, n=2).removeO().expand() in [x/(1 + 2*x + x**2),
1/(1 + x/2 + 1/x/2)/2, 1/x/(1 + 2/x + x**(-2))]
def test_bug5():
w = Symbol("w")
l = Symbol('l')
e = (-log(w) + log(1 + w*log(x)))**(-2)*w**(-2)*((-log(w) +
log(1 + x*w))*(-log(w) + log(1 + w*log(x)))*w - x*(-log(w) +
log(1 + w*log(x)))*w)
assert e.nseries(w, n=0, logx=l) == x/w/l + 1/w + O(1, w)
assert e.nseries(w, n=1, logx=l) == x/w/l + 1/w - x/l + 1/l*log(x) \
+ x*log(x)/l**2 + O(w)
def test_issue_4115():
assert (sin(x)/(1 - cos(x))).nseries(x, n=1) == 2/x + O(x)
assert (sin(x)**2/(1 - cos(x))).nseries(x, n=1) == 2 + O(x)
def test_pole():
raises(PoleError, lambda: sin(1/x).series(x, 0, 5))
raises(PoleError, lambda: sin(1 + 1/x).series(x, 0, 5))
raises(PoleError, lambda: (x*sin(1/x)).series(x, 0, 5))
def test_expsinbug():
assert exp(sin(x)).series(x, 0, 0) == O(1, x)
assert exp(sin(x)).series(x, 0, 1) == 1 + O(x)
assert exp(sin(x)).series(x, 0, 2) == 1 + x + O(x**2)
assert exp(sin(x)).series(x, 0, 3) == 1 + x + x**2/2 + O(x**3)
assert exp(sin(x)).series(x, 0, 4) == 1 + x + x**2/2 + O(x**4)
assert exp(sin(x)).series(x, 0, 5) == 1 + x + x**2/2 - x**4/8 + O(x**5)
def test_floor():
x = Symbol('x')
assert floor(x).series(x) == 0
assert floor(-x).series(x) == -1
assert floor(sin(x)).series(x) == 0
assert floor(sin(-x)).series(x) == -1
assert floor(x**3).series(x) == 0
assert floor(-x**3).series(x) == -1
assert floor(cos(x)).series(x) == 0
assert floor(cos(-x)).series(x) == 0
assert floor(5 + sin(x)).series(x) == 5
assert floor(5 + sin(-x)).series(x) == 4
assert floor(x).series(x, 2) == 2
assert floor(-x).series(x, 2) == -3
x = Symbol('x', negative=True)
assert floor(x + 1.5).series(x) == 1
def test_ceiling():
assert ceiling(x).series(x) == 1
assert ceiling(-x).series(x) == 0
assert ceiling(sin(x)).series(x) == 1
assert ceiling(sin(-x)).series(x) == 0
assert ceiling(1 - cos(x)).series(x) == 1
assert ceiling(1 - cos(-x)).series(x) == 1
assert ceiling(x).series(x, 2) == 3
assert ceiling(-x).series(x, 2) == -2
def test_abs():
a = Symbol('a')
assert abs(x).nseries(x, n=4) == x
assert abs(-x).nseries(x, n=4) == x
assert abs(x + 1).nseries(x, n=4) == x + 1
assert abs(sin(x)).nseries(x, n=4) == x - Rational(1, 6)*x**3 + O(x**4)
assert abs(sin(-x)).nseries(x, n=4) == x - Rational(1, 6)*x**3 + O(x**4)
assert abs(x - a).nseries(x, 1) == -a*sign(1 - a) + (x - 1)*sign(1 - a) + sign(1 - a)
def test_dir():
assert abs(x).series(x, 0, dir="+") == x
assert abs(x).series(x, 0, dir="-") == -x
assert floor(x + 2).series(x, 0, dir='+') == 2
assert floor(x + 2).series(x, 0, dir='-') == 1
assert floor(x + 2.2).series(x, 0, dir='-') == 2
assert ceiling(x + 2.2).series(x, 0, dir='-') == 3
assert sin(x + y).series(x, 0, dir='-') == sin(x + y).series(x, 0, dir='+')
def test_cdir():
assert abs(x).series(x, 0, cdir=1) == x
assert abs(x).series(x, 0, cdir=-1) == -x
assert floor(x + 2).series(x, 0, cdir=1) == 2
assert floor(x + 2).series(x, 0, cdir=-1) == 1
assert floor(x + 2.2).series(x, 0, cdir=1) == 2
assert ceiling(x + 2.2).series(x, 0, cdir=-1) == 3
assert sin(x + y).series(x, 0, cdir=-1) == sin(x + y).series(x, 0, cdir=1)
def test_issue_3504():
a = Symbol("a")
e = asin(a*x)/x
assert e.series(x, 4, n=2).removeO() == \
(x - 4)*(a/(4*sqrt(-16*a**2 + 1)) - asin(4*a)/16) + asin(4*a)/4
def test_issue_4441():
a, b = symbols('a,b')
f = 1/(1 + a*x)
assert f.series(x, 0, 5) == 1 - a*x + a**2*x**2 - a**3*x**3 + \
a**4*x**4 + O(x**5)
f = 1/(1 + (a + b)*x)
assert f.series(x, 0, 3) == 1 + x*(-a - b)\
+ x**2*(a + b)**2 + O(x**3)
def test_issue_4329():
assert tan(x).series(x, pi/2, n=3).removeO() == \
-pi/6 + x/3 - 1/(x - pi/2)
assert cot(x).series(x, pi, n=3).removeO() == \
-x/3 + pi/3 + 1/(x - pi)
assert limit(tan(x)**tan(2*x), x, pi/4) == exp(-1)
def test_issue_5183():
assert abs(x + x**2).series(n=1) == O(x)
assert abs(x + x**2).series(n=2) == x + O(x**2)
assert ((1 + x)**2).series(x, n=6) == x**2 + 2*x + 1
assert (1 + 1/x).series() == 1 + 1/x
assert Derivative(exp(x).series(), x).doit() == \
1 + x + x**2/2 + x**3/6 + x**4/24 + O(x**5)
def test_issue_5654():
a = Symbol('a')
assert (1/(x**2+a**2)**2).nseries(x, x0=I*a, n=0) == \
-I/(4*a**3*(-I*a + x)) - 1/(4*a**2*(-I*a + x)**2) + O(1, (x, I*a))
assert (1/(x**2+a**2)**2).nseries(x, x0=I*a, n=1) == 3/(16*a**4) \
-I/(4*a**3*(-I*a + x)) - 1/(4*a**2*(-I*a + x)**2) + O(-I*a + x, (x, I*a))
def test_issue_5925():
sx = sqrt(x + z).series(z, 0, 1)
sxy = sqrt(x + y + z).series(z, 0, 1)
s1, s2 = sx.subs(x, x + y), sxy
assert (s1 - s2).expand().removeO().simplify() == 0
sx = sqrt(x + z).series(z, 0, 1)
sxy = sqrt(x + y + z).series(z, 0, 1)
assert sxy.subs({x:1, y:2}) == sx.subs(x, 3)
def test_exp_2():
assert exp(x**3).nseries(x, 0, 14) == 1 + x**3 + x**6/2 + x**9/6 + x**12/24 + O(x**14)
|
361cf9180e794e56b44eb72b42303c25bad53b7d16bb55c53733612b2a4a5bff | from functools import reduce
import itertools
from operator import add
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import Function
from sympy.core.mul import Mul
from sympy.core.power import Pow
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.matrices.dense import Matrix
from sympy.polys.rootoftools import CRootOf
from sympy.series.order import O
from sympy.simplify.cse_main import cse
from sympy.simplify.simplify import signsimp
from sympy.tensor.indexed import (Idx, IndexedBase)
from sympy.core.function import count_ops
from sympy.simplify.cse_opts import sub_pre, sub_post
from sympy.functions.special.hyper import meijerg
from sympy.simplify import cse_main, cse_opts
from sympy.utilities.iterables import subsets
from sympy.testing.pytest import XFAIL, raises
from sympy.matrices import (MutableDenseMatrix, MutableSparseMatrix,
ImmutableDenseMatrix, ImmutableSparseMatrix)
from sympy.matrices.expressions import MatrixSymbol
w, x, y, z = symbols('w,x,y,z')
x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12 = symbols('x:13')
def test_numbered_symbols():
ns = cse_main.numbered_symbols(prefix='y')
assert list(itertools.islice(
ns, 0, 10)) == [Symbol('y%s' % i) for i in range(0, 10)]
ns = cse_main.numbered_symbols(prefix='y')
assert list(itertools.islice(
ns, 10, 20)) == [Symbol('y%s' % i) for i in range(10, 20)]
ns = cse_main.numbered_symbols()
assert list(itertools.islice(
ns, 0, 10)) == [Symbol('x%s' % i) for i in range(0, 10)]
# Dummy "optimization" functions for testing.
def opt1(expr):
return expr + y
def opt2(expr):
return expr*z
def test_preprocess_for_cse():
assert cse_main.preprocess_for_cse(x, [(opt1, None)]) == x + y
assert cse_main.preprocess_for_cse(x, [(None, opt1)]) == x
assert cse_main.preprocess_for_cse(x, [(None, None)]) == x
assert cse_main.preprocess_for_cse(x, [(opt1, opt2)]) == x + y
assert cse_main.preprocess_for_cse(
x, [(opt1, None), (opt2, None)]) == (x + y)*z
def test_postprocess_for_cse():
assert cse_main.postprocess_for_cse(x, [(opt1, None)]) == x
assert cse_main.postprocess_for_cse(x, [(None, opt1)]) == x + y
assert cse_main.postprocess_for_cse(x, [(None, None)]) == x
assert cse_main.postprocess_for_cse(x, [(opt1, opt2)]) == x*z
# Note the reverse order of application.
assert cse_main.postprocess_for_cse(
x, [(None, opt1), (None, opt2)]) == x*z + y
def test_cse_single():
# Simple substitution.
e = Add(Pow(x + y, 2), sqrt(x + y))
substs, reduced = cse([e])
assert substs == [(x0, x + y)]
assert reduced == [sqrt(x0) + x0**2]
subst42, (red42,) = cse([42]) # issue_15082
assert len(subst42) == 0 and red42 == 42
subst_half, (red_half,) = cse([0.5])
assert len(subst_half) == 0 and red_half == 0.5
def test_cse_single2():
# Simple substitution, test for being able to pass the expression directly
e = Add(Pow(x + y, 2), sqrt(x + y))
substs, reduced = cse(e)
assert substs == [(x0, x + y)]
assert reduced == [sqrt(x0) + x0**2]
substs, reduced = cse(Matrix([[1]]))
assert isinstance(reduced[0], Matrix)
subst42, (red42,) = cse(42) # issue 15082
assert len(subst42) == 0 and red42 == 42
subst_half, (red_half,) = cse(0.5) # issue 15082
assert len(subst_half) == 0 and red_half == 0.5
def test_cse_not_possible():
# No substitution possible.
e = Add(x, y)
substs, reduced = cse([e])
assert substs == []
assert reduced == [x + y]
# issue 6329
eq = (meijerg((1, 2), (y, 4), (5,), [], x) +
meijerg((1, 3), (y, 4), (5,), [], x))
assert cse(eq) == ([], [eq])
def test_nested_substitution():
# Substitution within a substitution.
e = Add(Pow(w*x + y, 2), sqrt(w*x + y))
substs, reduced = cse([e])
assert substs == [(x0, w*x + y)]
assert reduced == [sqrt(x0) + x0**2]
def test_subtraction_opt():
# Make sure subtraction is optimized.
e = (x - y)*(z - y) + exp((x - y)*(z - y))
substs, reduced = cse(
[e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)])
assert substs == [(x0, (x - y)*(y - z))]
assert reduced == [-x0 + exp(-x0)]
e = -(x - y)*(z - y) + exp(-(x - y)*(z - y))
substs, reduced = cse(
[e], optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)])
assert substs == [(x0, (x - y)*(y - z))]
assert reduced == [x0 + exp(x0)]
# issue 4077
n = -1 + 1/x
e = n/x/(-n)**2 - 1/n/x
assert cse(e, optimizations=[(cse_opts.sub_pre, cse_opts.sub_post)]) == \
([], [0])
assert cse(((w + x + y + z)*(w - y - z))/(w + x)**3) == \
([(x0, w + x), (x1, y + z)], [(w - x1)*(x0 + x1)/x0**3])
def test_multiple_expressions():
e1 = (x + y)*z
e2 = (x + y)*w
substs, reduced = cse([e1, e2])
assert substs == [(x0, x + y)]
assert reduced == [x0*z, x0*w]
l = [w*x*y + z, w*y]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == rsubsts
assert reduced == [z + x*x0, x0]
l = [w*x*y, w*x*y + z, w*y]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == rsubsts
assert reduced == [x1, x1 + z, x0]
l = [(x - z)*(y - z), x - z, y - z]
substs, reduced = cse(l)
rsubsts, _ = cse(reversed(l))
assert substs == [(x0, -z), (x1, x + x0), (x2, x0 + y)]
assert rsubsts == [(x0, -z), (x1, x0 + y), (x2, x + x0)]
assert reduced == [x1*x2, x1, x2]
l = [w*y + w + x + y + z, w*x*y]
assert cse(l) == ([(x0, w*y)], [w + x + x0 + y + z, x*x0])
assert cse([x + y, x + y + z]) == ([(x0, x + y)], [x0, z + x0])
assert cse([x + y, x + z]) == ([], [x + y, x + z])
assert cse([x*y, z + x*y, x*y*z + 3]) == \
([(x0, x*y)], [x0, z + x0, 3 + x0*z])
@XFAIL # CSE of non-commutative Mul terms is disabled
def test_non_commutative_cse():
A, B, C = symbols('A B C', commutative=False)
l = [A*B*C, A*C]
assert cse(l) == ([], l)
l = [A*B*C, A*B]
assert cse(l) == ([(x0, A*B)], [x0*C, x0])
# Test if CSE of non-commutative Mul terms is disabled
def test_bypass_non_commutatives():
A, B, C = symbols('A B C', commutative=False)
l = [A*B*C, A*C]
assert cse(l) == ([], l)
l = [A*B*C, A*B]
assert cse(l) == ([], l)
l = [B*C, A*B*C]
assert cse(l) == ([], l)
@XFAIL # CSE fails when replacing non-commutative sub-expressions
def test_non_commutative_order():
A, B, C = symbols('A B C', commutative=False)
x0 = symbols('x0', commutative=False)
l = [B+C, A*(B+C)]
assert cse(l) == ([(x0, B+C)], [x0, A*x0])
@XFAIL # Worked in gh-11232, but was reverted due to performance considerations
def test_issue_10228():
assert cse([x*y**2 + x*y]) == ([(x0, x*y)], [x0*y + x0])
assert cse([x + y, 2*x + y]) == ([(x0, x + y)], [x0, x + x0])
assert cse((w + 2*x + y + z, w + x + 1)) == (
[(x0, w + x)], [x0 + x + y + z, x0 + 1])
assert cse(((w + x + y + z)*(w - x))/(w + x)) == (
[(x0, w + x)], [(x0 + y + z)*(w - x)/x0])
a, b, c, d, f, g, j, m = symbols('a, b, c, d, f, g, j, m')
exprs = (d*g**2*j*m, 4*a*f*g*m, a*b*c*f**2)
assert cse(exprs) == (
[(x0, g*m), (x1, a*f)], [d*g*j*x0, 4*x0*x1, b*c*f*x1]
)
@XFAIL
def test_powers():
assert cse(x*y**2 + x*y) == ([(x0, x*y)], [x0*y + x0])
def test_issue_4498():
assert cse(w/(x - y) + z/(y - x), optimizations='basic') == \
([], [(w - z)/(x - y)])
def test_issue_4020():
assert cse(x**5 + x**4 + x**3 + x**2, optimizations='basic') \
== ([(x0, x**2)], [x0*(x**3 + x + x0 + 1)])
def test_issue_4203():
assert cse(sin(x**x)/x**x) == ([(x0, x**x)], [sin(x0)/x0])
def test_issue_6263():
e = Eq(x*(-x + 1) + x*(x - 1), 0)
assert cse(e, optimizations='basic') == ([], [True])
def test_dont_cse_tuples():
from sympy.core.function import Subs
f = Function("f")
g = Function("g")
name_val, (expr,) = cse(
Subs(f(x, y), (x, y), (0, 1))
+ Subs(g(x, y), (x, y), (0, 1)))
assert name_val == []
assert expr == (Subs(f(x, y), (x, y), (0, 1))
+ Subs(g(x, y), (x, y), (0, 1)))
name_val, (expr,) = cse(
Subs(f(x, y), (x, y), (0, x + y))
+ Subs(g(x, y), (x, y), (0, x + y)))
assert name_val == [(x0, x + y)]
assert expr == Subs(f(x, y), (x, y), (0, x0)) + \
Subs(g(x, y), (x, y), (0, x0))
def test_pow_invpow():
assert cse(1/x**2 + x**2) == \
([(x0, x**2)], [x0 + 1/x0])
assert cse(x**2 + (1 + 1/x**2)/x**2) == \
([(x0, x**2), (x1, 1/x0)], [x0 + x1*(x1 + 1)])
assert cse(1/x**2 + (1 + 1/x**2)*x**2) == \
([(x0, x**2), (x1, 1/x0)], [x0*(x1 + 1) + x1])
assert cse(cos(1/x**2) + sin(1/x**2)) == \
([(x0, x**(-2))], [sin(x0) + cos(x0)])
assert cse(cos(x**2) + sin(x**2)) == \
([(x0, x**2)], [sin(x0) + cos(x0)])
assert cse(y/(2 + x**2) + z/x**2/y) == \
([(x0, x**2)], [y/(x0 + 2) + z/(x0*y)])
assert cse(exp(x**2) + x**2*cos(1/x**2)) == \
([(x0, x**2)], [x0*cos(1/x0) + exp(x0)])
assert cse((1 + 1/x**2)/x**2) == \
([(x0, x**(-2))], [x0*(x0 + 1)])
assert cse(x**(2*y) + x**(-2*y)) == \
([(x0, x**(2*y))], [x0 + 1/x0])
def test_postprocess():
eq = (x + 1 + exp((x + 1)/(y + 1)) + cos(y + 1))
assert cse([eq, Eq(x, z + 1), z - 2, (z + 1)*(x + 1)],
postprocess=cse_main.cse_separate) == \
[[(x0, y + 1), (x2, z + 1), (x, x2), (x1, x + 1)],
[x1 + exp(x1/x0) + cos(x0), z - 2, x1*x2]]
def test_issue_4499():
# previously, this gave 16 constants
from sympy.abc import a, b
B = Function('B')
G = Function('G')
t = Tuple(*
(a, a + S.Half, 2*a, b, 2*a - b + 1, (sqrt(z)/2)**(-2*a + 1)*B(2*a -
b, sqrt(z))*B(b - 1, sqrt(z))*G(b)*G(2*a - b + 1),
sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b,
sqrt(z))*G(b)*G(2*a - b + 1), sqrt(z)*(sqrt(z)/2)**(-2*a + 1)*B(b - 1,
sqrt(z))*B(2*a - b + 1, sqrt(z))*G(b)*G(2*a - b + 1),
(sqrt(z)/2)**(-2*a + 1)*B(b, sqrt(z))*B(2*a - b + 1,
sqrt(z))*G(b)*G(2*a - b + 1), 1, 0, S.Half, z/2, -b + 1, -2*a + b,
-2*a))
c = cse(t)
ans = (
[(x0, 2*a), (x1, -b + x0), (x2, x1 + 1), (x3, b - 1), (x4, sqrt(z)),
(x5, B(x3, x4)), (x6, (x4/2)**(1 - x0)*G(b)*G(x2)), (x7, x6*B(x1, x4)),
(x8, B(b, x4)), (x9, x6*B(x2, x4))],
[(a, a + S.Half, x0, b, x2, x5*x7, x4*x7*x8, x4*x5*x9, x8*x9,
1, 0, S.Half, z/2, -x3, -x1, -x0)])
assert ans == c
def test_issue_6169():
r = CRootOf(x**6 - 4*x**5 - 2, 1)
assert cse(r) == ([], [r])
# and a check that the right thing is done with the new
# mechanism
assert sub_post(sub_pre((-x - y)*z - x - y)) == -z*(x + y) - x - y
def test_cse_Indexed():
len_y = 5
y = IndexedBase('y', shape=(len_y,))
x = IndexedBase('x', shape=(len_y,))
i = Idx('i', len_y-1)
expr1 = (y[i+1]-y[i])/(x[i+1]-x[i])
expr2 = 1/(x[i+1]-x[i])
replacements, reduced_exprs = cse([expr1, expr2])
assert len(replacements) > 0
def test_cse_MatrixSymbol():
# MatrixSymbols have non-Basic args, so make sure that works
A = MatrixSymbol("A", 3, 3)
assert cse(A) == ([], [A])
n = symbols('n', integer=True)
B = MatrixSymbol("B", n, n)
assert cse(B) == ([], [B])
assert cse(A[0] * A[0]) == ([], [A[0]*A[0]])
assert cse(A[0,0]*A[0,1] + A[0,0]*A[0,1]*A[0,2]) == ([(x0, A[0, 0]*A[0, 1])], [x0*A[0, 2] + x0])
def test_cse_MatrixExpr():
A = MatrixSymbol('A', 3, 3)
y = MatrixSymbol('y', 3, 1)
expr1 = (A.T*A).I * A * y
expr2 = (A.T*A) * A * y
replacements, reduced_exprs = cse([expr1, expr2])
assert len(replacements) > 0
replacements, reduced_exprs = cse([expr1 + expr2, expr1])
assert replacements
replacements, reduced_exprs = cse([A**2, A + A**2])
assert replacements
def test_Piecewise():
f = Piecewise((-z + x*y, Eq(y, 0)), (-z - x*y, True))
ans = cse(f)
actual_ans = ([(x0, x*y)],
[Piecewise((x0 - z, Eq(y, 0)), (-z - x0, True))])
assert ans == actual_ans
def test_ignore_order_terms():
eq = exp(x).series(x,0,3) + sin(y+x**3) - 1
assert cse(eq) == ([], [sin(x**3 + y) + x + x**2/2 + O(x**3)])
def test_name_conflict():
z1 = x0 + y
z2 = x2 + x3
l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
substs, reduced = cse(l)
assert [e.subs(reversed(substs)) for e in reduced] == l
def test_name_conflict_cust_symbols():
z1 = x0 + y
z2 = x2 + x3
l = [cos(z1) + z1, cos(z2) + z2, x0 + x2]
substs, reduced = cse(l, symbols("x:10"))
assert [e.subs(reversed(substs)) for e in reduced] == l
def test_symbols_exhausted_error():
l = cos(x+y)+x+y+cos(w+y)+sin(w+y)
sym = [x, y, z]
with raises(ValueError):
cse(l, symbols=sym)
def test_issue_7840():
# daveknippers' example
C393 = sympify( \
'Piecewise((C391 - 1.65, C390 < 0.5), (Piecewise((C391 - 1.65, \
C391 > 2.35), (C392, True)), True))'
)
C391 = sympify( \
'Piecewise((2.05*C390**(-1.03), C390 < 0.5), (2.5*C390**(-0.625), True))'
)
C393 = C393.subs('C391',C391)
# simple substitution
sub = {}
sub['C390'] = 0.703451854
sub['C392'] = 1.01417794
ss_answer = C393.subs(sub)
# cse
substitutions,new_eqn = cse(C393)
for pair in substitutions:
sub[pair[0].name] = pair[1].subs(sub)
cse_answer = new_eqn[0].subs(sub)
# both methods should be the same
assert ss_answer == cse_answer
# GitRay's example
expr = sympify(
"Piecewise((Symbol('ON'), Equality(Symbol('mode'), Symbol('ON'))), \
(Piecewise((Piecewise((Symbol('OFF'), StrictLessThan(Symbol('x'), \
Symbol('threshold'))), (Symbol('ON'), true)), Equality(Symbol('mode'), \
Symbol('AUTO'))), (Symbol('OFF'), true)), true))"
)
substitutions, new_eqn = cse(expr)
# this Piecewise should be exactly the same
assert new_eqn[0] == expr
# there should not be any replacements
assert len(substitutions) < 1
def test_issue_8891():
for cls in (MutableDenseMatrix, MutableSparseMatrix,
ImmutableDenseMatrix, ImmutableSparseMatrix):
m = cls(2, 2, [x + y, 0, 0, 0])
res = cse([x + y, m])
ans = ([(x0, x + y)], [x0, cls([[x0, 0], [0, 0]])])
assert res == ans
assert isinstance(res[1][-1], cls)
def test_issue_11230():
# a specific test that always failed
a, b, f, k, l, i = symbols('a b f k l i')
p = [a*b*f*k*l, a*i*k**2*l, f*i*k**2*l]
R, C = cse(p)
assert not any(i.is_Mul for a in C for i in a.args)
# random tests for the issue
from sympy.core.random import choice
from sympy.core.function import expand_mul
s = symbols('a:m')
# 35 Mul tests, none of which should ever fail
ex = [Mul(*[choice(s) for i in range(5)]) for i in range(7)]
for p in subsets(ex, 3):
p = list(p)
R, C = cse(p)
assert not any(i.is_Mul for a in C for i in a.args)
for ri in reversed(R):
for i in range(len(C)):
C[i] = C[i].subs(*ri)
assert p == C
# 35 Add tests, none of which should ever fail
ex = [Add(*[choice(s[:7]) for i in range(5)]) for i in range(7)]
for p in subsets(ex, 3):
p = list(p)
R, C = cse(p)
assert not any(i.is_Add for a in C for i in a.args)
for ri in reversed(R):
for i in range(len(C)):
C[i] = C[i].subs(*ri)
# use expand_mul to handle cases like this:
# p = [a + 2*b + 2*e, 2*b + c + 2*e, b + 2*c + 2*g]
# x0 = 2*(b + e) is identified giving a rebuilt p that
# is now `[a + 2*(b + e), c + 2*(b + e), b + 2*c + 2*g]`
assert p == [expand_mul(i) for i in C]
@XFAIL
def test_issue_11577():
def check(eq):
r, c = cse(eq)
assert eq.count_ops() >= \
len(r) + sum([i[1].count_ops() for i in r]) + \
count_ops(c)
eq = x**5*y**2 + x**5*y + x**5
assert cse(eq) == (
[(x0, x**4), (x1, x*y)], [x**5 + x0*x1*y + x0*x1])
# ([(x0, x**5*y)], [x0*y + x0 + x**5]) or
# ([(x0, x**5)], [x0*y**2 + x0*y + x0])
check(eq)
eq = x**2/(y + 1)**2 + x/(y + 1)
assert cse(eq) == (
[(x0, y + 1)], [x**2/x0**2 + x/x0])
# ([(x0, x/(y + 1))], [x0**2 + x0])
check(eq)
def test_hollow_rejection():
eq = [x + 3, x + 4]
assert cse(eq) == ([], eq)
def test_cse_ignore():
exprs = [exp(y)*(3*y + 3*sqrt(x+1)), exp(y)*(5*y + 5*sqrt(x+1))]
subst1, red1 = cse(exprs)
assert any(y in sub.free_symbols for _, sub in subst1), "cse failed to identify any term with y"
subst2, red2 = cse(exprs, ignore=(y,)) # y is not allowed in substitutions
assert not any(y in sub.free_symbols for _, sub in subst2), "Sub-expressions containing y must be ignored"
assert any(sub - sqrt(x + 1) == 0 for _, sub in subst2), "cse failed to identify sqrt(x + 1) as sub-expression"
def test_cse_ignore_issue_15002():
l = [
w*exp(x)*exp(-z),
exp(y)*exp(x)*exp(-z)
]
substs, reduced = cse(l, ignore=(x,))
rl = [e.subs(reversed(substs)) for e in reduced]
assert rl == l
def test_cse__performance():
nexprs, nterms = 3, 20
x = symbols('x:%d' % nterms)
exprs = [
reduce(add, [x[j]*(-1)**(i+j) for j in range(nterms)])
for i in range(nexprs)
]
assert (exprs[0] + exprs[1]).simplify() == 0
subst, red = cse(exprs)
assert len(subst) > 0, "exprs[0] == -exprs[2], i.e. a CSE"
for i, e in enumerate(red):
assert (e.subs(reversed(subst)) - exprs[i]).simplify() == 0
def test_issue_12070():
exprs = [x + y, 2 + x + y, x + y + z, 3 + x + y + z]
subst, red = cse(exprs)
assert 6 >= (len(subst) + sum([v.count_ops() for k, v in subst]) +
count_ops(red))
def test_issue_13000():
eq = x/(-4*x**2 + y**2)
cse_eq = cse(eq)[1][0]
assert cse_eq == eq
def test_issue_18203():
eq = CRootOf(x**5 + 11*x - 2, 0) + CRootOf(x**5 + 11*x - 2, 1)
assert cse(eq) == ([], [eq])
def test_unevaluated_mul():
eq = Mul(x + y, x + y, evaluate=False)
assert cse(eq) == ([(x0, x + y)], [x0**2])
def test_cse_release_variables():
from sympy.simplify.cse_main import cse_release_variables
_0, _1, _2, _3, _4 = symbols('_:5')
eqs = [(x + y - 1)**2, x,
x + y, (x + y)/(2*x + 1) + (x + y - 1)**2,
(2*x + 1)**(x + y)]
r, e = cse(eqs, postprocess=cse_release_variables)
# this can change in keeping with the intention of the function
assert r, e == ([
(x0, x + y), (x1, (x0 - 1)**2), (x2, 2*x + 1),
(_3, x0/x2 + x1), (_4, x2**x0), (x2, None), (_0, x1),
(x1, None), (_2, x0), (x0, None), (_1, x)], (_0, _1, _2, _3, _4))
r.reverse()
r = [(s, v) for s, v in r if v is not None]
assert eqs == [i.subs(r) for i in e]
def test_cse_list():
_cse = lambda x: cse(x, list=False)
assert _cse(x) == ([], x)
assert _cse('x') == ([], 'x')
it = [x]
for c in (list, tuple, set):
assert _cse(c(it)) == ([], c(it))
#Tuple works different from tuple:
assert _cse(Tuple(*it)) == ([], Tuple(*it))
d = {x: 1}
assert _cse(d) == ([], d)
def test_issue_18991():
A = MatrixSymbol('A', 2, 2)
assert signsimp(-A * A - A) == -A * A - A
def test_unevaluated_Mul():
m = [Mul(1, 2, evaluate=False)]
assert cse(m) == ([], m)
|
c856b806c9e5e4ef409c2ad143580c0e7574d75497f61b9142edde90bf959178 | import math
from sympy.core.containers import Tuple
from sympy.core.numbers import nan, oo, Float, Integer
from sympy.core.relational import Lt
from sympy.core.symbol import symbols, Symbol
from sympy.functions.elementary.trigonometric import sin
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.sets.fancysets import Range
from sympy.tensor.indexed import Idx, IndexedBase
from sympy.testing.pytest import raises
from sympy.codegen.ast import (
Assignment, Attribute, aug_assign, CodeBlock, For, Type, Variable, Pointer, Declaration,
AddAugmentedAssignment, SubAugmentedAssignment, MulAugmentedAssignment,
DivAugmentedAssignment, ModAugmentedAssignment, value_const, pointer_const,
integer, real, complex_, int8, uint8, float16 as f16, float32 as f32,
float64 as f64, float80 as f80, float128 as f128, complex64 as c64, complex128 as c128,
While, Scope, String, Print, QuotedString, FunctionPrototype, FunctionDefinition, Return,
FunctionCall, untyped, IntBaseType, intc, Node, none, NoneToken, Token, Comment
)
x, y, z, t, x0, x1, x2, a, b = symbols("x, y, z, t, x0, x1, x2, a, b")
n = symbols("n", integer=True)
A = MatrixSymbol('A', 3, 1)
mat = Matrix([1, 2, 3])
B = IndexedBase('B')
i = Idx("i", n)
A22 = MatrixSymbol('A22',2,2)
B22 = MatrixSymbol('B22',2,2)
def test_Assignment():
# Here we just do things to show they don't error
Assignment(x, y)
Assignment(x, 0)
Assignment(A, mat)
Assignment(A[1,0], 0)
Assignment(A[1,0], x)
Assignment(B[i], x)
Assignment(B[i], 0)
a = Assignment(x, y)
assert a.func(*a.args) == a
assert a.op == ':='
# Here we test things to show that they error
# Matrix to scalar
raises(ValueError, lambda: Assignment(B[i], A))
raises(ValueError, lambda: Assignment(B[i], mat))
raises(ValueError, lambda: Assignment(x, mat))
raises(ValueError, lambda: Assignment(x, A))
raises(ValueError, lambda: Assignment(A[1,0], mat))
# Scalar to matrix
raises(ValueError, lambda: Assignment(A, x))
raises(ValueError, lambda: Assignment(A, 0))
# Non-atomic lhs
raises(TypeError, lambda: Assignment(mat, A))
raises(TypeError, lambda: Assignment(0, x))
raises(TypeError, lambda: Assignment(x*x, 1))
raises(TypeError, lambda: Assignment(A + A, mat))
raises(TypeError, lambda: Assignment(B, 0))
def test_AugAssign():
# Here we just do things to show they don't error
aug_assign(x, '+', y)
aug_assign(x, '+', 0)
aug_assign(A, '+', mat)
aug_assign(A[1, 0], '+', 0)
aug_assign(A[1, 0], '+', x)
aug_assign(B[i], '+', x)
aug_assign(B[i], '+', 0)
# Check creation via aug_assign vs constructor
for binop, cls in [
('+', AddAugmentedAssignment),
('-', SubAugmentedAssignment),
('*', MulAugmentedAssignment),
('/', DivAugmentedAssignment),
('%', ModAugmentedAssignment),
]:
a = aug_assign(x, binop, y)
b = cls(x, y)
assert a.func(*a.args) == a == b
assert a.binop == binop
assert a.op == binop + '='
# Here we test things to show that they error
# Matrix to scalar
raises(ValueError, lambda: aug_assign(B[i], '+', A))
raises(ValueError, lambda: aug_assign(B[i], '+', mat))
raises(ValueError, lambda: aug_assign(x, '+', mat))
raises(ValueError, lambda: aug_assign(x, '+', A))
raises(ValueError, lambda: aug_assign(A[1, 0], '+', mat))
# Scalar to matrix
raises(ValueError, lambda: aug_assign(A, '+', x))
raises(ValueError, lambda: aug_assign(A, '+', 0))
# Non-atomic lhs
raises(TypeError, lambda: aug_assign(mat, '+', A))
raises(TypeError, lambda: aug_assign(0, '+', x))
raises(TypeError, lambda: aug_assign(x * x, '+', 1))
raises(TypeError, lambda: aug_assign(A + A, '+', mat))
raises(TypeError, lambda: aug_assign(B, '+', 0))
def test_Assignment_printing():
assignment_classes = [
Assignment,
AddAugmentedAssignment,
SubAugmentedAssignment,
MulAugmentedAssignment,
DivAugmentedAssignment,
ModAugmentedAssignment,
]
pairs = [
(x, 2 * y + 2),
(B[i], x),
(A22, B22),
(A[0, 0], x),
]
for cls in assignment_classes:
for lhs, rhs in pairs:
a = cls(lhs, rhs)
assert repr(a) == '%s(%s, %s)' % (cls.__name__, repr(lhs), repr(rhs))
def test_CodeBlock():
c = CodeBlock(Assignment(x, 1), Assignment(y, x + 1))
assert c.func(*c.args) == c
assert c.left_hand_sides == Tuple(x, y)
assert c.right_hand_sides == Tuple(1, x + 1)
def test_CodeBlock_topological_sort():
assignments = [
Assignment(x, y + z),
Assignment(z, 1),
Assignment(t, x),
Assignment(y, 2),
]
ordered_assignments = [
# Note that the unrelated z=1 and y=2 are kept in that order
Assignment(z, 1),
Assignment(y, 2),
Assignment(x, y + z),
Assignment(t, x),
]
c1 = CodeBlock.topological_sort(assignments)
assert c1 == CodeBlock(*ordered_assignments)
# Cycle
invalid_assignments = [
Assignment(x, y + z),
Assignment(z, 1),
Assignment(y, x),
Assignment(y, 2),
]
raises(ValueError, lambda: CodeBlock.topological_sort(invalid_assignments))
# Free symbols
free_assignments = [
Assignment(x, y + z),
Assignment(z, a * b),
Assignment(t, x),
Assignment(y, b + 3),
]
free_assignments_ordered = [
Assignment(z, a * b),
Assignment(y, b + 3),
Assignment(x, y + z),
Assignment(t, x),
]
c2 = CodeBlock.topological_sort(free_assignments)
assert c2 == CodeBlock(*free_assignments_ordered)
def test_CodeBlock_free_symbols():
c1 = CodeBlock(
Assignment(x, y + z),
Assignment(z, 1),
Assignment(t, x),
Assignment(y, 2),
)
assert c1.free_symbols == set()
c2 = CodeBlock(
Assignment(x, y + z),
Assignment(z, a * b),
Assignment(t, x),
Assignment(y, b + 3),
)
assert c2.free_symbols == {a, b}
def test_CodeBlock_cse():
c1 = CodeBlock(
Assignment(y, 1),
Assignment(x, sin(y)),
Assignment(z, sin(y)),
Assignment(t, x*z),
)
assert c1.cse() == CodeBlock(
Assignment(y, 1),
Assignment(x0, sin(y)),
Assignment(x, x0),
Assignment(z, x0),
Assignment(t, x*z),
)
# Multiple assignments to same symbol not supported
raises(NotImplementedError, lambda: CodeBlock(
Assignment(x, 1),
Assignment(y, 1), Assignment(y, 2)
).cse())
# Check auto-generated symbols do not collide with existing ones
c2 = CodeBlock(
Assignment(x0, sin(y) + 1),
Assignment(x1, 2 * sin(y)),
Assignment(z, x * y),
)
assert c2.cse() == CodeBlock(
Assignment(x2, sin(y)),
Assignment(x0, x2 + 1),
Assignment(x1, 2 * x2),
Assignment(z, x * y),
)
def test_CodeBlock_cse__issue_14118():
# see https://github.com/sympy/sympy/issues/14118
c = CodeBlock(
Assignment(A22, Matrix([[x, sin(y)],[3, 4]])),
Assignment(B22, Matrix([[sin(y), 2*sin(y)], [sin(y)**2, 7]]))
)
assert c.cse() == CodeBlock(
Assignment(x0, sin(y)),
Assignment(A22, Matrix([[x, x0],[3, 4]])),
Assignment(B22, Matrix([[x0, 2*x0], [x0**2, 7]]))
)
def test_For():
f = For(n, Range(0, 3), (Assignment(A[n, 0], x + n), aug_assign(x, '+', y)))
f = For(n, (1, 2, 3, 4, 5), (Assignment(A[n, 0], x + n),))
assert f.func(*f.args) == f
raises(TypeError, lambda: For(n, x, (x + y,)))
def test_none():
assert none.is_Atom
assert none == none
class Foo(Token):
pass
foo = Foo()
assert foo != none
assert none == None
assert none == NoneToken()
assert none.func(*none.args) == none
def test_String():
st = String('foobar')
assert st.is_Atom
assert st == String('foobar')
assert st.text == 'foobar'
assert st.func(**st.kwargs()) == st
assert st.func(*st.args) == st
class Signifier(String):
pass
si = Signifier('foobar')
assert si != st
assert si.text == st.text
s = String('foo')
assert str(s) == 'foo'
assert repr(s) == "String('foo')"
def test_Comment():
c = Comment('foobar')
assert c.text == 'foobar'
assert str(c) == 'foobar'
def test_Node():
n = Node()
assert n == Node()
assert n.func(*n.args) == n
def test_Type():
t = Type('MyType')
assert len(t.args) == 1
assert t.name == String('MyType')
assert str(t) == 'MyType'
assert repr(t) == "Type(String('MyType'))"
assert Type(t) == t
assert t.func(*t.args) == t
t1 = Type('t1')
t2 = Type('t2')
assert t1 != t2
assert t1 == t1 and t2 == t2
t1b = Type('t1')
assert t1 == t1b
assert t2 != t1b
def test_Type__from_expr():
assert Type.from_expr(i) == integer
u = symbols('u', real=True)
assert Type.from_expr(u) == real
assert Type.from_expr(n) == integer
assert Type.from_expr(3) == integer
assert Type.from_expr(3.0) == real
assert Type.from_expr(3+1j) == complex_
raises(ValueError, lambda: Type.from_expr(sum))
def test_Type__cast_check__integers():
# Rounding
raises(ValueError, lambda: integer.cast_check(3.5))
assert integer.cast_check('3') == 3
assert integer.cast_check(Float('3.0000000000000000000')) == 3
assert integer.cast_check(Float('3.0000000000000000001')) == 3 # unintuitive maybe?
# Range
assert int8.cast_check(127.0) == 127
raises(ValueError, lambda: int8.cast_check(128))
assert int8.cast_check(-128) == -128
raises(ValueError, lambda: int8.cast_check(-129))
assert uint8.cast_check(0) == 0
assert uint8.cast_check(128) == 128
raises(ValueError, lambda: uint8.cast_check(256.0))
raises(ValueError, lambda: uint8.cast_check(-1))
def test_Attribute():
noexcept = Attribute('noexcept')
assert noexcept == Attribute('noexcept')
alignas16 = Attribute('alignas', [16])
alignas32 = Attribute('alignas', [32])
assert alignas16 != alignas32
assert alignas16.func(*alignas16.args) == alignas16
def test_Variable():
v = Variable(x, type=real)
assert v == Variable(v)
assert v == Variable('x', type=real)
assert v.symbol == x
assert v.type == real
assert value_const not in v.attrs
assert v.func(*v.args) == v
assert str(v) == 'Variable(x, type=real)'
w = Variable(y, f32, attrs={value_const})
assert w.symbol == y
assert w.type == f32
assert value_const in w.attrs
assert w.func(*w.args) == w
v_n = Variable(n, type=Type.from_expr(n))
assert v_n.type == integer
assert v_n.func(*v_n.args) == v_n
v_i = Variable(i, type=Type.from_expr(n))
assert v_i.type == integer
assert v_i != v_n
a_i = Variable.deduced(i)
assert a_i.type == integer
assert Variable.deduced(Symbol('x', real=True)).type == real
assert a_i.func(*a_i.args) == a_i
v_n2 = Variable.deduced(n, value=3.5, cast_check=False)
assert v_n2.func(*v_n2.args) == v_n2
assert abs(v_n2.value - 3.5) < 1e-15
raises(ValueError, lambda: Variable.deduced(n, value=3.5, cast_check=True))
v_n3 = Variable.deduced(n)
assert v_n3.type == integer
assert str(v_n3) == 'Variable(n, type=integer)'
assert Variable.deduced(z, value=3).type == integer
assert Variable.deduced(z, value=3.0).type == real
assert Variable.deduced(z, value=3.0+1j).type == complex_
def test_Pointer():
p = Pointer(x)
assert p.symbol == x
assert p.type == untyped
assert value_const not in p.attrs
assert pointer_const not in p.attrs
assert p.func(*p.args) == p
u = symbols('u', real=True)
pu = Pointer(u, type=Type.from_expr(u), attrs={value_const, pointer_const})
assert pu.symbol is u
assert pu.type == real
assert value_const in pu.attrs
assert pointer_const in pu.attrs
assert pu.func(*pu.args) == pu
i = symbols('i', integer=True)
deref = pu[i]
assert deref.indices == (i,)
def test_Declaration():
u = symbols('u', real=True)
vu = Variable(u, type=Type.from_expr(u))
assert Declaration(vu).variable.type == real
vn = Variable(n, type=Type.from_expr(n))
assert Declaration(vn).variable.type == integer
# PR 19107, does not allow comparison between expressions and Basic
# lt = StrictLessThan(vu, vn)
# assert isinstance(lt, StrictLessThan)
vuc = Variable(u, Type.from_expr(u), value=3.0, attrs={value_const})
assert value_const in vuc.attrs
assert pointer_const not in vuc.attrs
decl = Declaration(vuc)
assert decl.variable == vuc
assert isinstance(decl.variable.value, Float)
assert decl.variable.value == 3.0
assert decl.func(*decl.args) == decl
assert vuc.as_Declaration() == decl
assert vuc.as_Declaration(value=None, attrs=None) == Declaration(vu)
vy = Variable(y, type=integer, value=3)
decl2 = Declaration(vy)
assert decl2.variable == vy
assert decl2.variable.value == Integer(3)
vi = Variable(i, type=Type.from_expr(i), value=3.0)
decl3 = Declaration(vi)
assert decl3.variable.type == integer
assert decl3.variable.value == 3.0
raises(ValueError, lambda: Declaration(vi, 42))
def test_IntBaseType():
assert intc.name == String('intc')
assert intc.args == (intc.name,)
assert str(IntBaseType('a').name) == 'a'
def test_FloatType():
assert f16.dig == 3
assert f32.dig == 6
assert f64.dig == 15
assert f80.dig == 18
assert f128.dig == 33
assert f16.decimal_dig == 5
assert f32.decimal_dig == 9
assert f64.decimal_dig == 17
assert f80.decimal_dig == 21
assert f128.decimal_dig == 36
assert f16.max_exponent == 16
assert f32.max_exponent == 128
assert f64.max_exponent == 1024
assert f80.max_exponent == 16384
assert f128.max_exponent == 16384
assert f16.min_exponent == -13
assert f32.min_exponent == -125
assert f64.min_exponent == -1021
assert f80.min_exponent == -16381
assert f128.min_exponent == -16381
assert abs(f16.eps / Float('0.00097656', precision=16) - 1) < 0.1*10**-f16.dig
assert abs(f32.eps / Float('1.1920929e-07', precision=32) - 1) < 0.1*10**-f32.dig
assert abs(f64.eps / Float('2.2204460492503131e-16', precision=64) - 1) < 0.1*10**-f64.dig
assert abs(f80.eps / Float('1.08420217248550443401e-19', precision=80) - 1) < 0.1*10**-f80.dig
assert abs(f128.eps / Float(' 1.92592994438723585305597794258492732e-34', precision=128) - 1) < 0.1*10**-f128.dig
assert abs(f16.max / Float('65504', precision=16) - 1) < .1*10**-f16.dig
assert abs(f32.max / Float('3.40282347e+38', precision=32) - 1) < 0.1*10**-f32.dig
assert abs(f64.max / Float('1.79769313486231571e+308', precision=64) - 1) < 0.1*10**-f64.dig # cf. np.finfo(np.float64).max
assert abs(f80.max / Float('1.18973149535723176502e+4932', precision=80) - 1) < 0.1*10**-f80.dig
assert abs(f128.max / Float('1.18973149535723176508575932662800702e+4932', precision=128) - 1) < 0.1*10**-f128.dig
# cf. np.finfo(np.float32).tiny
assert abs(f16.tiny / Float('6.1035e-05', precision=16) - 1) < 0.1*10**-f16.dig
assert abs(f32.tiny / Float('1.17549435e-38', precision=32) - 1) < 0.1*10**-f32.dig
assert abs(f64.tiny / Float('2.22507385850720138e-308', precision=64) - 1) < 0.1*10**-f64.dig
assert abs(f80.tiny / Float('3.36210314311209350626e-4932', precision=80) - 1) < 0.1*10**-f80.dig
assert abs(f128.tiny / Float('3.3621031431120935062626778173217526e-4932', precision=128) - 1) < 0.1*10**-f128.dig
assert f64.cast_check(0.5) == 0.5
assert abs(f64.cast_check(3.7) - 3.7) < 3e-17
assert isinstance(f64.cast_check(3), (Float, float))
assert f64.cast_nocheck(oo) == float('inf')
assert f64.cast_nocheck(-oo) == float('-inf')
assert f64.cast_nocheck(float(oo)) == float('inf')
assert f64.cast_nocheck(float(-oo)) == float('-inf')
assert math.isnan(f64.cast_nocheck(nan))
assert f32 != f64
assert f64 == f64.func(*f64.args)
def test_Type__cast_check__floating_point():
raises(ValueError, lambda: f32.cast_check(123.45678949))
raises(ValueError, lambda: f32.cast_check(12.345678949))
raises(ValueError, lambda: f32.cast_check(1.2345678949))
raises(ValueError, lambda: f32.cast_check(.12345678949))
assert abs(123.456789049 - f32.cast_check(123.456789049) - 4.9e-8) < 1e-8
assert abs(0.12345678904 - f32.cast_check(0.12345678904) - 4e-11) < 1e-11
dcm21 = Float('0.123456789012345670499') # 21 decimals
assert abs(dcm21 - f64.cast_check(dcm21) - 4.99e-19) < 1e-19
f80.cast_check(Float('0.12345678901234567890103', precision=88))
raises(ValueError, lambda: f80.cast_check(Float('0.12345678901234567890149', precision=88)))
v10 = 12345.67894
raises(ValueError, lambda: f32.cast_check(v10))
assert abs(Float(str(v10), precision=64+8) - f64.cast_check(v10)) < v10*1e-16
assert abs(f32.cast_check(2147483647) - 2147483650) < 1
def test_Type__cast_check__complex_floating_point():
val9_11 = 123.456789049 + 0.123456789049j
raises(ValueError, lambda: c64.cast_check(.12345678949 + .12345678949j))
assert abs(val9_11 - c64.cast_check(val9_11) - 4.9e-8) < 1e-8
dcm21 = Float('0.123456789012345670499') + 1e-20j # 21 decimals
assert abs(dcm21 - c128.cast_check(dcm21) - 4.99e-19) < 1e-19
v19 = Float('0.1234567890123456749') + 1j*Float('0.1234567890123456749')
raises(ValueError, lambda: c128.cast_check(v19))
def test_While():
xpp = AddAugmentedAssignment(x, 1)
whl1 = While(x < 2, [xpp])
assert whl1.condition.args[0] == x
assert whl1.condition.args[1] == 2
assert whl1.condition == Lt(x, 2, evaluate=False)
assert whl1.body.args == (xpp,)
assert whl1.func(*whl1.args) == whl1
cblk = CodeBlock(AddAugmentedAssignment(x, 1))
whl2 = While(x < 2, cblk)
assert whl1 == whl2
assert whl1 != While(x < 3, [xpp])
def test_Scope():
assign = Assignment(x, y)
incr = AddAugmentedAssignment(x, 1)
scp = Scope([assign, incr])
cblk = CodeBlock(assign, incr)
assert scp.body == cblk
assert scp == Scope(cblk)
assert scp != Scope([incr, assign])
assert scp.func(*scp.args) == scp
def test_Print():
fmt = "%d %.3f"
ps = Print([n, x], fmt)
assert str(ps.format_string) == fmt
assert ps.print_args == Tuple(n, x)
assert ps.args == (Tuple(n, x), QuotedString(fmt), none)
assert ps == Print((n, x), fmt)
assert ps != Print([x, n], fmt)
assert ps.func(*ps.args) == ps
ps2 = Print([n, x])
assert ps2 == Print([n, x])
assert ps2 != ps
assert ps2.format_string == None
def test_FunctionPrototype_and_FunctionDefinition():
vx = Variable(x, type=real)
vn = Variable(n, type=integer)
fp1 = FunctionPrototype(real, 'power', [vx, vn])
assert fp1.return_type == real
assert fp1.name == String('power')
assert fp1.parameters == Tuple(vx, vn)
assert fp1 == FunctionPrototype(real, 'power', [vx, vn])
assert fp1 != FunctionPrototype(real, 'power', [vn, vx])
assert fp1.func(*fp1.args) == fp1
body = [Assignment(x, x**n), Return(x)]
fd1 = FunctionDefinition(real, 'power', [vx, vn], body)
assert fd1.return_type == real
assert str(fd1.name) == 'power'
assert fd1.parameters == Tuple(vx, vn)
assert fd1.body == CodeBlock(*body)
assert fd1 == FunctionDefinition(real, 'power', [vx, vn], body)
assert fd1 != FunctionDefinition(real, 'power', [vx, vn], body[::-1])
assert fd1.func(*fd1.args) == fd1
fp2 = FunctionPrototype.from_FunctionDefinition(fd1)
assert fp2 == fp1
fd2 = FunctionDefinition.from_FunctionPrototype(fp1, body)
assert fd2 == fd1
def test_Return():
rs = Return(x)
assert rs.args == (x,)
assert rs == Return(x)
assert rs != Return(y)
assert rs.func(*rs.args) == rs
def test_FunctionCall():
fc = FunctionCall('power', (x, 3))
assert fc.function_args[0] == x
assert fc.function_args[1] == 3
assert len(fc.function_args) == 2
assert isinstance(fc.function_args[1], Integer)
assert fc == FunctionCall('power', (x, 3))
assert fc != FunctionCall('power', (3, x))
assert fc != FunctionCall('Power', (x, 3))
assert fc.func(*fc.args) == fc
fc2 = FunctionCall('fma', [2, 3, 4])
assert len(fc2.function_args) == 3
assert fc2.function_args[0] == 2
assert fc2.function_args[1] == 3
assert fc2.function_args[2] == 4
assert str(fc2) in ( # not sure if QuotedString is a better default...
'FunctionCall(fma, function_args=(2, 3, 4))',
'FunctionCall("fma", function_args=(2, 3, 4))',
)
def test_ast_replace():
x = Variable('x', real)
y = Variable('y', real)
n = Variable('n', integer)
pwer = FunctionDefinition(real, 'pwer', [x, n], [pow(x.symbol, n.symbol)])
pname = pwer.name
pcall = FunctionCall('pwer', [y, 3])
tree1 = CodeBlock(pwer, pcall)
assert str(tree1.args[0].name) == 'pwer'
assert str(tree1.args[1].name) == 'pwer'
for a, b in zip(tree1, [pwer, pcall]):
assert a == b
tree2 = tree1.replace(pname, String('power'))
assert str(tree1.args[0].name) == 'pwer'
assert str(tree1.args[1].name) == 'pwer'
assert str(tree2.args[0].name) == 'power'
assert str(tree2.args[1].name) == 'power'
|
3ed20b01e1037dce79012f312c81b4ab7692f7c34aaf3a650354f7aed085530f | from typing import Tuple as tTuple
from sympy.core.add import Add
from sympy.core.basic import sympify, cacheit
from sympy.core.expr import Expr
from sympy.core.function import Function, ArgumentIndexError, PoleError, expand_mul
from sympy.core.logic import fuzzy_not, fuzzy_or, FuzzyBool, fuzzy_and
from sympy.core.mod import Mod
from sympy.core.numbers import igcdex, Rational, pi, Integer, Float
from sympy.core.relational import Ne, Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, Dummy
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial
from sympy.functions.combinatorial.numbers import bernoulli, euler
from sympy.functions.elementary.complexes import arg as arg_f, im, re
from sympy.functions.elementary.exponential import log, exp
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.logic.boolalg import And
from sympy.ntheory import factorint
from sympy.polys.specialpolys import symmetric_poly
from sympy.utilities.iterables import numbered_symbols
###############################################################################
########################## UTILITIES ##########################################
###############################################################################
def _imaginary_unit_as_coefficient(arg):
""" Helper to extract symbolic coefficient for imaginary unit """
if isinstance(arg, Float):
return None
else:
return arg.as_coefficient(S.ImaginaryUnit)
###############################################################################
########################## TRIGONOMETRIC FUNCTIONS ############################
###############################################################################
class TrigonometricFunction(Function):
"""Base class for trigonometric functions. """
unbranched = True
_singularities = (S.ComplexInfinity,)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational and fuzzy_not(s.args[0].is_zero):
return False
else:
return s.is_rational
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic:
return False
pi_coeff = _pi_coeff(self.args[0])
if pi_coeff is not None and pi_coeff.is_rational:
return True
else:
return s.is_algebraic
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _as_real_imag(self, deep=True, **hints):
if self.args[0].is_extended_real:
if deep:
hints['complex'] = False
return (self.args[0].expand(deep, **hints), S.Zero)
else:
return (self.args[0], S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (re, im)
def _period(self, general_period, symbol=None):
f = expand_mul(self.args[0])
if symbol is None:
symbol = tuple(f.free_symbols)[0]
if not f.has(symbol):
return S.Zero
if f == symbol:
return general_period
if symbol in f.free_symbols:
if f.is_Mul:
g, h = f.as_independent(symbol)
if h == symbol:
return general_period/abs(g)
if f.is_Add:
a, h = f.as_independent(symbol)
g, h = h.as_independent(symbol, as_Add=False)
if h == symbol:
return general_period/abs(g)
raise NotImplementedError("Use the periodicity function instead.")
def _peeloff_pi(arg):
r"""
Split ARG into two parts, a "rest" and a multiple of $\pi$.
This assumes ARG to be an Add.
The multiple of $\pi$ returned in the second position is always a Rational.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _peeloff_pi
>>> from sympy import pi
>>> from sympy.abc import x, y
>>> _peeloff_pi(x + pi/2)
(x, 1/2)
>>> _peeloff_pi(x + 2*pi/3 + pi*y)
(x + pi*y + pi/6, 1/2)
"""
pi_coeff = S.Zero
rest_terms = []
for a in Add.make_args(arg):
K = a.coeff(pi)
if K and K.is_rational:
pi_coeff += K
else:
rest_terms.append(a)
if pi_coeff is S.Zero:
return arg, S.Zero
m1 = (pi_coeff % S.Half)
m2 = pi_coeff - m1
if m2.is_integer or ((2*m2).is_integer and m2.is_even is False):
return Add(*(rest_terms + [m1*pi])), m2
return arg, S.Zero
def _pi_coeff(arg, cycles=1):
r"""
When arg is a Number times $\pi$ (e.g. $3\pi/2$) then return the Number
normalized to be in the range $[0, 2]$, else `None`.
When an even multiple of $\pi$ is encountered, if it is multiplying
something with known parity then the multiple is returned as 0 otherwise
as 2.
Examples
========
>>> from sympy.functions.elementary.trigonometric import _pi_coeff
>>> from sympy import pi, Dummy
>>> from sympy.abc import x
>>> _pi_coeff(3*x*pi)
3*x
>>> _pi_coeff(11*pi/7)
11/7
>>> _pi_coeff(-11*pi/7)
3/7
>>> _pi_coeff(4*pi)
0
>>> _pi_coeff(5*pi)
1
>>> _pi_coeff(5.0*pi)
1
>>> _pi_coeff(5.5*pi)
3/2
>>> _pi_coeff(2 + pi)
>>> _pi_coeff(2*Dummy(integer=True)*pi)
2
>>> _pi_coeff(2*Dummy(even=True)*pi)
0
"""
if arg is pi:
return S.One
elif not arg:
return S.Zero
elif arg.is_Mul:
cx = arg.coeff(pi)
if cx:
c, x = cx.as_coeff_Mul() # pi is not included as coeff
if c.is_Float:
# recast exact binary fractions to Rationals
f = abs(c) % 1
if f != 0:
p = -int(round(log(f, 2).evalf()))
m = 2**p
cm = c*m
i = int(cm)
if i == cm:
c = Rational(i, m)
cx = c*x
else:
c = Rational(int(c))
cx = c*x
if x.is_integer:
c2 = c % 2
if c2 == 1:
return x
elif not c2:
if x.is_even is not None: # known parity
return S.Zero
return Integer(2)
else:
return c2*x
return cx
elif arg.is_zero:
return S.Zero
class sin(TrigonometricFunction):
r"""
The sine function.
Returns the sine of x (measured in radians).
Explanation
===========
This function will evaluate automatically in the
case $x/\pi$ is some rational number [4]_. For example,
if $x$ is a multiple of $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, and $\pi/6$.
Examples
========
>>> from sympy import sin, pi
>>> from sympy.abc import x
>>> sin(x**2).diff(x)
2*x*cos(x**2)
>>> sin(1).diff(x)
0
>>> sin(pi)
0
>>> sin(pi/2)
1
>>> sin(pi/6)
1/2
>>> sin(pi/12)
-sqrt(2)/4 + sqrt(6)/4
See Also
========
csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Sin
.. [4] http://mathworld.wolfram.com/TrigonometryAngles.html
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return cos(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.Zero
elif arg in (S.Infinity, S.NegativeInfinity):
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
from sympy.sets.sets import FiniteSet
min, max = arg.min, arg.max
d = floor(min/(2*pi))
if min is not S.NegativeInfinity:
min = min - d*2*pi
if max is not S.Infinity:
max = max - d*2*pi
if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \
is not S.EmptySet and \
AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2),
pi*Rational(7, 2))) is not S.EmptySet:
return AccumBounds(-1, 1)
elif AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \
is not S.EmptySet:
return AccumBounds(Min(sin(min), sin(max)), 1)
elif AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2), pi*Rational(8, 2))) \
is not S.EmptySet:
return AccumBounds(-1, Max(sin(min), sin(max)))
else:
return AccumBounds(Min(sin(min), sin(max)),
Max(sin(min), sin(max)))
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import sinh
return S.ImaginaryUnit*sinh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if (2*pi_coeff).is_integer:
# is_even-case handled above as then pi_coeff.is_integer,
# so check if known to be not even
if pi_coeff.is_even is False:
return S.NegativeOne**(pi_coeff - S.Half)
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
# https://github.com/sympy/sympy/issues/6048
# transform a sine to a cosine, to avoid redundant code
if pi_coeff.is_Rational:
x = pi_coeff % 2
if x > 1:
return -cls((x % 1)*pi)
if 2*x > 1:
return cls((1 - x)*pi)
narg = ((pi_coeff + Rational(3, 2)) % 2)*pi
result = cos(narg)
if not isinstance(result, cos):
return result
if pi_coeff*pi != arg:
return cls(pi_coeff*pi)
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
m = m*pi
return sin(m)*cos(x) + cos(m)*sin(x)
if arg.is_zero:
return S.Zero
if isinstance(arg, asin):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return x/sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return y/sqrt(x**2 + y**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2)
if isinstance(arg, acot):
x = arg.args[0]
return 1/(sqrt(1 + 1/x**2)*x)
if isinstance(arg, acsc):
x = arg.args[0]
return 1/x
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1/x**2)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p*x**2/(n*(n - 1))
else:
return S.NegativeOne**(n//2)*x**n/factorial(n)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
if logx is not None:
arg = arg.subs(log(x), logx)
if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity):
raise PoleError("Cannot expand %s around 0" % (self))
return Function._eval_nseries(self, x, n=n, logx=logx, cdir=cdir)
def _eval_rewrite_as_exp(self, arg, **kwargs):
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
I = S.ImaginaryUnit
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) - exp(-arg*I))/(2*I)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*x**-I/2 - I*x**I /2
def _eval_rewrite_as_cos(self, arg, **kwargs):
return cos(arg - pi/2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)
return 2*tan_half/(1 + tan_half**2)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)
return Piecewise((0, And(Eq(im(arg), 0), Eq(Mod(arg, pi), 0))),
(2*cot_half/(1 + cot_half**2), True))
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self.rewrite(cos).rewrite(pow)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self.rewrite(cos).rewrite(sqrt)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1/csc(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1/sec(arg - pi/2, evaluate=False)
def _eval_rewrite_as_sinc(self, arg, **kwargs):
return arg*sinc(arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy.functions.elementary.hyperbolic import cosh, sinh
re, im = self._as_real_imag(deep=deep, **hints)
return (sin(re)*cosh(im), cos(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy.functions.special.polynomials import chebyshevt, chebyshevu
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
# TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return sx*cy + sy*cx
elif arg.is_Mul:
n, x = arg.as_coeff_Mul(rational=True)
if n.is_Integer: # n will be positive because of .eval
# canonicalization
# See http://mathworld.wolfram.com/Multiple-AngleFormulas.html
if n.is_odd:
return S.NegativeOne**((n - 1)/2)*chebyshevt(n, sin(x))
else:
return expand_mul(S.NegativeOne**(n/2 - 1)*cos(x)*
chebyshevu(n - 1, sin(x)), deep=False)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return sin(arg)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = x0/pi
if n.is_integer:
lt = (arg - n*pi).as_leading_term(x)
return (S.NegativeOne**n)*lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in [S.Infinity, S.NegativeInfinity]:
return AccumBounds(-1, 1)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_extended_real:
return True
def _eval_is_zero(self):
rest, pi_mult = _peeloff_pi(self.args[0])
if rest.is_zero:
return pi_mult.is_integer
def _eval_is_complex(self):
if self.args[0].is_extended_real \
or self.args[0].is_complex:
return True
class cos(TrigonometricFunction):
"""
The cosine function.
Returns the cosine of x (measured in radians).
Explanation
===========
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cos, pi
>>> from sympy.abc import x
>>> cos(x**2).diff(x)
-2*x*sin(x**2)
>>> cos(1).diff(x)
0
>>> cos(pi)
-1
>>> cos(pi/2)
0
>>> cos(2*pi/3)
-1/2
>>> cos(pi/12)
sqrt(2)/4 + sqrt(6)/4
See Also
========
sin, csc, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Cos
"""
def period(self, symbol=None):
return self._period(2*pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return -sin(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.functions.special.polynomials import chebyshevt
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.sets.setexpr import SetExpr
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.One
elif arg in (S.Infinity, S.NegativeInfinity):
# In this case it is better to return AccumBounds(-1, 1)
# rather than returning S.NaN, since AccumBounds(-1, 1)
# preserves the information that sin(oo) is between
# -1 and 1, where S.NaN does not do that.
return AccumBounds(-1, 1)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return sin(arg + pi/2)
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.is_extended_real and arg.is_finite is False:
return AccumBounds(-1, 1)
if arg.could_extract_minus_sign():
return cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import cosh
return cosh(i_coeff)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
return (S.NegativeOne)**pi_coeff
if (2*pi_coeff).is_integer:
# is_even-case handled above as then pi_coeff.is_integer,
# so check if known to be not even
if pi_coeff.is_even is False:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
# cosine formula #####################
# https://github.com/sympy/sympy/issues/6048
# explicit calculations are performed for
# cos(k pi/n) for n = 8,10,12,15,20,24,30,40,60,120
# Some other exact values like cos(k pi/240) can be
# calculated using a partial-fraction decomposition
# by calling cos( X ).rewrite(sqrt)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1)/4,
}
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*pi
return -cls(narg)
# If nested sqrt's are worse than un-evaluation
# you can require q to be in (1, 2, 3, 4, 6, 12)
# q <= 12, q=15, q=20, q=24, q=30, q=40, q=60, q=120 return
# expressions with 2 or fewer sqrt nestings.
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
if q in table2:
a, b = p*pi/table2[q][0], p*pi/table2[q][1]
nvala, nvalb = cls(a), cls(b)
if None in (nvala, nvalb):
return None
return nvala*nvalb + cls(pi/2 - a)*cls(pi/2 - b)
if q > 12:
return None
if q in cst_table_some:
cts = cst_table_some[pi_coeff.q]
return chebyshevt(pi_coeff.p, cts).expand()
if 0 == q % 2:
narg = (pi_coeff*2)*pi
nval = cls(narg)
if None == nval:
return None
x = (2*pi_coeff + 1)/2
sign_cos = (-1)**((-1 if x < 0 else 1)*int(abs(x)))
return sign_cos*sqrt( (1 + nval)/2 )
return None
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
m = m*pi
return cos(m)*cos(x) - sin(m)*sin(x)
if arg.is_zero:
return S.One
if isinstance(arg, acos):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1/sqrt(1 + x**2)
if isinstance(arg, atan2):
y, x = arg.args
return x/sqrt(x**2 + y**2)
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x ** 2)
if isinstance(arg, acot):
x = arg.args[0]
return 1/sqrt(1 + 1/x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1/x**2)
if isinstance(arg, asec):
x = arg.args[0]
return 1/x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return -p*x**2/(n*(n - 1))
else:
return S.NegativeOne**(n//2)*x**n/factorial(n)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
if logx is not None:
arg = arg.subs(log(x), logx)
if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity):
raise PoleError("Cannot expand %s around 0" % (self))
return Function._eval_nseries(self, x, n=n, logx=logx, cdir=cdir)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
return (exp(arg*I) + exp(-arg*I))/2
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return x**I/2 + x**-I/2
def _eval_rewrite_as_sin(self, arg, **kwargs):
return sin(arg + pi/2, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
tan_half = tan(S.Half*arg)**2
return (1 - tan_half)/(1 + tan_half)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)*cos(arg)/sin(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(S.Half*arg)**2
return Piecewise((1, And(Eq(im(arg), 0), Eq(Mod(arg, 2*pi), 0))),
((cot_half - 1)/(cot_half + 1), True))
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._eval_rewrite_as_sqrt(arg)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
from sympy.functions.special.polynomials import chebyshevt
def migcdex(x):
# recursive calcuation of gcd and linear combination
# for a sequence of integers.
# Given (x1, x2, x3)
# Returns (y1, y1, y3, g)
# such that g is the gcd and x1*y1+x2*y2+x3*y3 - g = 0
# Note, that this is only one such linear combination.
if len(x) == 1:
return (1, x[0])
if len(x) == 2:
return igcdex(x[0], x[-1])
g = migcdex(x[1:])
u, v, h = igcdex(x[0], g[-1])
return tuple([u] + [v*i for i in g[0:-1] ] + [h])
def ipartfrac(r, factors=None):
if isinstance(r, int):
return r
if not isinstance(r, Rational):
raise TypeError("r is not rational")
n = r.q
if 2 > r.q*r.q:
return r.q
if None == factors:
a = [n//x**y for x, y in factorint(r.q).items()]
else:
a = [n//x for x in factors]
if len(a) == 1:
return [ r ]
h = migcdex(a)
ans = [ r.p*Rational(i*j, r.q) for i, j in zip(h[:-1], a) ]
assert r == sum(ans)
return ans
pi_coeff = _pi_coeff(arg)
if pi_coeff is None:
return None
if pi_coeff.is_integer:
# it was unevaluated
return self.func(pi_coeff*pi)
if not pi_coeff.is_Rational:
return None
def _cospi257():
""" Express cos(pi/257) explicitly as a function of radicals
Based upon the equations in
http://math.stackexchange.com/questions/516142/how-does-cos2-pi-257-look-like-in-real-radicals
See also http://www.susqu.edu/brakke/constructions/257-gon.m.txt
"""
def f1(a, b):
return (a + sqrt(a**2 + b))/2, (a - sqrt(a**2 + b))/2
def f2(a, b):
return (a - sqrt(a**2 + b))/2
t1, t2 = f1(-1, 256)
z1, z3 = f1(t1, 64)
z2, z4 = f1(t2, 64)
y1, y5 = f1(z1, 4*(5 + t1 + 2*z1))
y6, y2 = f1(z2, 4*(5 + t2 + 2*z2))
y3, y7 = f1(z3, 4*(5 + t1 + 2*z3))
y8, y4 = f1(z4, 4*(5 + t2 + 2*z4))
x1, x9 = f1(y1, -4*(t1 + y1 + y3 + 2*y6))
x2, x10 = f1(y2, -4*(t2 + y2 + y4 + 2*y7))
x3, x11 = f1(y3, -4*(t1 + y3 + y5 + 2*y8))
x4, x12 = f1(y4, -4*(t2 + y4 + y6 + 2*y1))
x5, x13 = f1(y5, -4*(t1 + y5 + y7 + 2*y2))
x6, x14 = f1(y6, -4*(t2 + y6 + y8 + 2*y3))
x15, x7 = f1(y7, -4*(t1 + y7 + y1 + 2*y4))
x8, x16 = f1(y8, -4*(t2 + y8 + y2 + 2*y5))
v1 = f2(x1, -4*(x1 + x2 + x3 + x6))
v2 = f2(x2, -4*(x2 + x3 + x4 + x7))
v3 = f2(x8, -4*(x8 + x9 + x10 + x13))
v4 = f2(x9, -4*(x9 + x10 + x11 + x14))
v5 = f2(x10, -4*(x10 + x11 + x12 + x15))
v6 = f2(x16, -4*(x16 + x1 + x2 + x5))
u1 = -f2(-v1, -4*(v2 + v3))
u2 = -f2(-v4, -4*(v5 + v6))
w1 = -2*f2(-u1, -4*u2)
return sqrt(sqrt(2)*sqrt(w1 + 4)/8 + S.Half)
cst_table_some = {
3: S.Half,
5: (sqrt(5) + 1)/4,
17: sqrt((15 + sqrt(17))/32 + sqrt(2)*(sqrt(17 - sqrt(17)) +
sqrt(sqrt(2)*(-8*sqrt(17 + sqrt(17)) - (1 - sqrt(17))
*sqrt(17 - sqrt(17))) + 6*sqrt(17) + 34))/32),
257: _cospi257()
# 65537 is the only other known Fermat prime and the very
# large expression is intentionally omitted from SymPy; see
# http://www.susqu.edu/brakke/constructions/65537-gon.m.txt
}
def _fermatCoords(n):
# if n can be factored in terms of Fermat primes with
# multiplicity of each being 1, return those primes, else
# False
primes = []
for p_i in cst_table_some:
quotient, remainder = divmod(n, p_i)
if remainder == 0:
n = quotient
primes.append(p_i)
if n == 1:
return tuple(primes)
return False
if pi_coeff.q in cst_table_some:
rv = chebyshevt(pi_coeff.p, cst_table_some[pi_coeff.q])
if pi_coeff.q < 257:
rv = rv.expand()
return rv
if not pi_coeff.q % 2: # recursively remove factors of 2
pico2 = pi_coeff*2
nval = cos(pico2*pi).rewrite(sqrt)
x = (pico2 + 1)/2
sign_cos = -1 if int(x) % 2 else 1
return sign_cos*sqrt( (1 + nval)/2 )
FC = _fermatCoords(pi_coeff.q)
if FC:
decomp = ipartfrac(pi_coeff, FC)
X = [(x[1], x[0]*pi) for x in zip(decomp, numbered_symbols('z'))]
pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)
return pcls.rewrite(sqrt)
else:
decomp = ipartfrac(pi_coeff)
X = [(x[1], x[0]*pi) for x in zip(decomp, numbered_symbols('z'))]
pcls = cos(sum([x[0] for x in X]))._eval_expand_trig().subs(X)
return pcls
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1/sec(arg)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return 1/sec(arg).rewrite(csc)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy.functions.elementary.hyperbolic import cosh, sinh
re, im = self._as_real_imag(deep=deep, **hints)
return (cos(re)*cosh(im), -sin(re)*sinh(im))
def _eval_expand_trig(self, **hints):
from sympy.functions.special.polynomials import chebyshevt
arg = self.args[0]
x = None
if arg.is_Add: # TODO: Do this more efficiently for more than two terms
x, y = arg.as_two_terms()
sx = sin(x, evaluate=False)._eval_expand_trig()
sy = sin(y, evaluate=False)._eval_expand_trig()
cx = cos(x, evaluate=False)._eval_expand_trig()
cy = cos(y, evaluate=False)._eval_expand_trig()
return cx*cy - sx*sy
elif arg.is_Mul:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer:
return chebyshevt(coeff, cos(terms))
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_Rational:
return self.rewrite(sqrt)
return cos(arg)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = (x0 + pi/2)/pi
if n.is_integer:
lt = (arg - n*pi + pi/2).as_leading_term(x)
return (S.NegativeOne**n)*lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in [S.Infinity, S.NegativeInfinity]:
return AccumBounds(-1, 1)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_extended_real:
return True
def _eval_is_complex(self):
if self.args[0].is_extended_real \
or self.args[0].is_complex:
return True
def _eval_is_zero(self):
rest, pi_mult = _peeloff_pi(self.args[0])
if pi_mult:
return fuzzy_and([(pi_mult - S.Half).is_integer, rest.is_zero])
else:
return rest.is_zero
class tan(TrigonometricFunction):
"""
The tangent function.
Returns the tangent of x (measured in radians).
Explanation
===========
See :class:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import tan, pi
>>> from sympy.abc import x
>>> tan(x**2).diff(x)
2*x*(tan(x**2)**2 + 1)
>>> tan(1).diff(x)
0
>>> tan(pi/8).expand()
-1 + sqrt(2)
See Also
========
sin, csc, cos, sec, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Tan
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.One + self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return atan
@classmethod
def eval(cls, arg):
from sympy.calculus.accumulationbounds import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.Zero
elif arg in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
min, max = arg.min, arg.max
d = floor(min/pi)
if min is not S.NegativeInfinity:
min = min - d*pi
if max is not S.Infinity:
max = max - d*pi
from sympy.sets.sets import FiniteSet
if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(3, 2))):
return AccumBounds(S.NegativeInfinity, S.Infinity)
else:
return AccumBounds(tan(min), tan(max))
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import tanh
return S.ImaginaryUnit*tanh(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.Zero
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
q = pi_coeff.q
p = pi_coeff.p % q
# ensure simplified results are returned for n*pi/5, n*pi/10
table10 = {
1: sqrt(1 - 2*sqrt(5)/5),
2: sqrt(5 - 2*sqrt(5)),
3: sqrt(1 + 2*sqrt(5)/5),
4: sqrt(5 + 2*sqrt(5))
}
if q in (5, 10):
n = 10*p/q
if n > 5:
n = 10 - n
return -table10[n]
else:
return table10[n]
if not pi_coeff.q % 2:
narg = pi_coeff*pi*2
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return 1/sresult - cresult/sresult
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
if q in table2:
nvala, nvalb = cls(p*pi/table2[q][0]), cls(p*pi/table2[q][1])
if None in (nvala, nvalb):
return None
return (nvala - nvalb)/(1 + nvala*nvalb)
narg = ((pi_coeff + S.Half) % 1 - S.Half)*pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if cresult == 0:
return S.ComplexInfinity
return (sresult/cresult)
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
tanm = tan(m*pi)
if tanm is S.ComplexInfinity:
return -cot(x)
else: # tanm == 0
return tan(x)
if arg.is_zero:
return S.Zero
if isinstance(arg, atan):
return arg.args[0]
if isinstance(arg, atan2):
y, x = arg.args
return y/x
if isinstance(arg, asin):
x = arg.args[0]
return x/sqrt(1 - x**2)
if isinstance(arg, acos):
x = arg.args[0]
return sqrt(1 - x**2)/x
if isinstance(arg, acot):
x = arg.args[0]
return 1/x
if isinstance(arg, acsc):
x = arg.args[0]
return 1/(sqrt(1 - 1/x**2)*x)
if isinstance(arg, asec):
x = arg.args[0]
return sqrt(1 - 1/x**2)*x
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a, b = ((n - 1)//2), 2**(n + 1)
B = bernoulli(n + 1)
F = factorial(n + 1)
return S.NegativeOne**a*b*(b - 1)*B/F*x**n
def _eval_nseries(self, x, n, logx, cdir=0):
i = self.args[0].limit(x, 0)*2/pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return Function._eval_nseries(self, x, n=n, logx=logx)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return I*(x**-I - x**I)/(x**-I + x**I)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
from sympy.functions.elementary.hyperbolic import cosh, sinh
denom = cos(2*re) + cosh(2*im)
return (sin(2*re)/denom, sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_expand_trig(self, **hints):
arg = self.args[0]
x = None
if arg.is_Add:
n = len(arg.args)
TX = []
for x in arg.args:
tx = tan(x, evaluate=False)._eval_expand_trig()
TX.append(tx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n + 1):
p[1 - i % 2] += symmetric_poly(i, Y)*(-1)**((i % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, TX)))
elif arg.is_Mul:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((1 + I*z)**coeff).expand()
return (im(P)/re(P)).subs([(z, tan(terms))])
return tan(arg)
def _eval_rewrite_as_exp(self, arg, **kwargs):
I = S.ImaginaryUnit
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
def _eval_rewrite_as_sin(self, x, **kwargs):
return 2*sin(x)**2/sin(2*x)
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x - pi/2, evaluate=False)/cos(x)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/cos(arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
return 1/cot(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
sin_in_sec_form = sin(arg).rewrite(sec)
cos_in_sec_form = cos(arg).rewrite(sec)
return sin_in_sec_form/cos_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
sin_in_csc_form = sin(arg).rewrite(csc)
cos_in_csc_form = cos(arg).rewrite(csc)
return sin_in_csc_form/cos_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = 2*x0/pi
if n.is_integer:
lt = (arg - n*pi/2).as_leading_term(x)
return lt if n.is_even else -1/lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
# FIXME: currently tan(pi/2) return zoo
return self.args[0].is_extended_real
def _eval_is_real(self):
arg = self.args[0]
if arg.is_real and (arg/pi - S.Half).is_integer is False:
return True
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real and (arg/pi - S.Half).is_integer is False:
return True
if arg.is_imaginary:
return True
def _eval_is_zero(self):
rest, pi_mult = _peeloff_pi(self.args[0])
if rest.is_zero:
return pi_mult.is_integer
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_real and (arg/pi - S.Half).is_integer is False:
return True
class cot(TrigonometricFunction):
"""
The cotangent function.
Returns the cotangent of x (measured in radians).
Explanation
===========
See :class:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import cot, pi
>>> from sympy.abc import x
>>> cot(x**2).diff(x)
2*x*(-cot(x**2)**2 - 1)
>>> cot(1).diff(x)
0
>>> cot(pi/12)
sqrt(3) + 2
See Also
========
sin, csc, cos, sec, tan
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Cot
"""
def period(self, symbol=None):
return self._period(pi, symbol)
def fdiff(self, argindex=1):
if argindex == 1:
return S.NegativeOne - self**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return acot
@classmethod
def eval(cls, arg):
from sympy.calculus.accumulationbounds import AccumBounds
if arg.is_Number:
if arg is S.NaN:
return S.NaN
if arg.is_zero:
return S.ComplexInfinity
elif arg in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
if arg is S.ComplexInfinity:
return S.NaN
if isinstance(arg, AccumBounds):
return -tan(arg + pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import coth
return -S.ImaginaryUnit*coth(i_coeff)
pi_coeff = _pi_coeff(arg, 2)
if pi_coeff is not None:
if pi_coeff.is_integer:
return S.ComplexInfinity
if not pi_coeff.is_Rational:
narg = pi_coeff*pi
if narg != arg:
return cls(narg)
return None
if pi_coeff.is_Rational:
if pi_coeff.q in (5, 10):
return tan(pi/2 - arg)
if pi_coeff.q > 2 and not pi_coeff.q % 2:
narg = pi_coeff*pi*2
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
return 1/sresult + cresult/sresult
table2 = {
12: (3, 4),
20: (4, 5),
30: (5, 6),
15: (6, 10),
24: (6, 8),
40: (8, 10),
60: (20, 30),
120: (40, 60)
}
q = pi_coeff.q
p = pi_coeff.p % q
if q in table2:
nvala, nvalb = cls(p*pi/table2[q][0]), cls(p*pi/table2[q][1])
if None in (nvala, nvalb):
return None
return (1 + nvala*nvalb)/(nvalb - nvala)
narg = (((pi_coeff + S.Half) % 1) - S.Half)*pi
# see cos() to specify which expressions should be
# expanded automatically in terms of radicals
cresult, sresult = cos(narg), cos(narg - pi/2)
if not isinstance(cresult, cos) \
and not isinstance(sresult, cos):
if sresult == 0:
return S.ComplexInfinity
return cresult/sresult
if narg != arg:
return cls(narg)
if arg.is_Add:
x, m = _peeloff_pi(arg)
if m:
cotm = cot(m*pi)
if cotm is S.ComplexInfinity:
return cot(x)
else: # cotm == 0
return -tan(x)
if arg.is_zero:
return S.ComplexInfinity
if isinstance(arg, acot):
return arg.args[0]
if isinstance(arg, atan):
x = arg.args[0]
return 1/x
if isinstance(arg, atan2):
y, x = arg.args
return x/y
if isinstance(arg, asin):
x = arg.args[0]
return sqrt(1 - x**2)/x
if isinstance(arg, acos):
x = arg.args[0]
return x/sqrt(1 - x**2)
if isinstance(arg, acsc):
x = arg.args[0]
return sqrt(1 - 1/x**2)*x
if isinstance(arg, asec):
x = arg.args[0]
return 1/(sqrt(1 - 1/x**2)*x)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return S.NegativeOne**((n + 1)//2)*2**(n + 1)*B/F*x**n
def _eval_nseries(self, x, n, logx, cdir=0):
i = self.args[0].limit(x, 0)/pi
if i and i.is_Integer:
return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
return self.rewrite(tan)._eval_nseries(x, n=n, logx=logx)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
re, im = self._as_real_imag(deep=deep, **hints)
if im:
from sympy.functions.elementary.hyperbolic import cosh, sinh
denom = cos(2*re) - cosh(2*im)
return (-sin(2*re)/denom, sinh(2*im)/denom)
else:
return (self.func(re), S.Zero)
def _eval_rewrite_as_exp(self, arg, **kwargs):
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
I = S.ImaginaryUnit
if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
arg = arg.func(arg.args[0]).rewrite(exp)
neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
return I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if isinstance(arg, log):
I = S.ImaginaryUnit
x = arg.args[0]
return -I*(x**-I + x**I)/(x**-I - x**I)
def _eval_rewrite_as_sin(self, x, **kwargs):
return sin(2*x)/(2*(sin(x)**2))
def _eval_rewrite_as_cos(self, x, **kwargs):
return cos(x)/cos(x - pi/2, evaluate=False)
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/sin(arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return 1/tan(arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
cos_in_sec_form = cos(arg).rewrite(sec)
sin_in_sec_form = sin(arg).rewrite(sec)
return cos_in_sec_form/sin_in_sec_form
def _eval_rewrite_as_csc(self, arg, **kwargs):
cos_in_csc_form = cos(arg).rewrite(csc)
sin_in_csc_form = sin(arg).rewrite(csc)
return cos_in_csc_form/sin_in_csc_form
def _eval_rewrite_as_pow(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(pow)
if y.has(cos):
return None
return y
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
y = self.rewrite(cos).rewrite(sqrt)
if y.has(cos):
return None
return y
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = 2*x0/pi
if n.is_integer:
lt = (arg - n*pi/2).as_leading_term(x)
return 1/lt if n.is_even else -lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
def _eval_is_extended_real(self):
return self.args[0].is_extended_real
def _eval_expand_trig(self, **hints):
arg = self.args[0]
x = None
if arg.is_Add:
n = len(arg.args)
CX = []
for x in arg.args:
cx = cot(x, evaluate=False)._eval_expand_trig()
CX.append(cx)
Yg = numbered_symbols('Y')
Y = [ next(Yg) for i in range(n) ]
p = [0, 0]
for i in range(n, -1, -1):
p[(n - i) % 2] += symmetric_poly(i, Y)*(-1)**(((n - i) % 4)//2)
return (p[0]/p[1]).subs(list(zip(Y, CX)))
elif arg.is_Mul:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
I = S.ImaginaryUnit
z = Symbol('dummy', real=True)
P = ((z + I)**coeff).expand()
return (re(P)/im(P)).subs([(z, cot(terms))])
return cot(arg) # XXX sec and csc return 1/cos and 1/sin
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
if arg.is_imaginary:
return True
def _eval_is_real(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
def _eval_is_zero(self):
rest, pimult = _peeloff_pi(self.args[0])
if pimult and rest.is_zero:
return (pimult - S.Half).is_integer
def _eval_subs(self, old, new):
arg = self.args[0]
argnew = arg.subs(old, new)
if arg != argnew and (argnew/pi).is_integer:
return S.ComplexInfinity
return cot(argnew)
class ReciprocalTrigonometricFunction(TrigonometricFunction):
"""Base class for reciprocal functions of trigonometric functions. """
_reciprocal_of = None # mandatory, to be defined in subclass
_singularities = (S.ComplexInfinity,)
# _is_even and _is_odd are used for correct evaluation of csc(-x), sec(-x)
# TODO refactor into TrigonometricFunction common parts of
# trigonometric functions eval() like even/odd, func(x+2*k*pi), etc.
# optional, to be defined in subclasses:
_is_even = None # type: FuzzyBool
_is_odd = None # type: FuzzyBool
@classmethod
def eval(cls, arg):
if arg.could_extract_minus_sign():
if cls._is_even:
return cls(-arg)
if cls._is_odd:
return -cls(-arg)
pi_coeff = _pi_coeff(arg)
if (pi_coeff is not None
and not (2*pi_coeff).is_integer
and pi_coeff.is_Rational):
q = pi_coeff.q
p = pi_coeff.p % (2*q)
if p > q:
narg = (pi_coeff - 1)*pi
return -cls(narg)
if 2*p > q:
narg = (1 - pi_coeff)*pi
if cls._is_odd:
return cls(narg)
elif cls._is_even:
return -cls(narg)
if hasattr(arg, 'inverse') and arg.inverse() == cls:
return arg.args[0]
t = cls._reciprocal_of.eval(arg)
if t is None:
return t
elif any(isinstance(i, cos) for i in (t, -t)):
return (1/t).rewrite(sec)
elif any(isinstance(i, sin) for i in (t, -t)):
return (1/t).rewrite(csc)
else:
return 1/t
def _call_reciprocal(self, method_name, *args, **kwargs):
# Calls method_name on _reciprocal_of
o = self._reciprocal_of(self.args[0])
return getattr(o, method_name)(*args, **kwargs)
def _calculate_reciprocal(self, method_name, *args, **kwargs):
# If calling method_name on _reciprocal_of returns a value != None
# then return the reciprocal of that value
t = self._call_reciprocal(method_name, *args, **kwargs)
return 1/t if t is not None else t
def _rewrite_reciprocal(self, method_name, arg):
# Special handling for rewrite functions. If reciprocal rewrite returns
# unmodified expression, then return None
t = self._call_reciprocal(method_name, arg)
if t is not None and t != self._reciprocal_of(arg):
return 1/t
def _period(self, symbol):
f = expand_mul(self.args[0])
return self._reciprocal_of(f).period(symbol)
def fdiff(self, argindex=1):
return -self._calculate_reciprocal("fdiff", argindex)/self**2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)
def _eval_rewrite_as_Pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_Pow", arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sin", arg)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_cos", arg)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tan", arg)
def _eval_rewrite_as_pow(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_pow", arg)
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_sqrt", arg)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
return (1/self._reciprocal_of(self.args[0])).as_real_imag(deep,
**hints)
def _eval_expand_trig(self, **hints):
return self._calculate_reciprocal("_eval_expand_trig", **hints)
def _eval_is_extended_real(self):
return self._reciprocal_of(self.args[0])._eval_is_extended_real()
def _eval_as_leading_term(self, x, logx=None, cdir=0):
return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)
def _eval_is_finite(self):
return (1/self._reciprocal_of(self.args[0])).is_finite
def _eval_nseries(self, x, n, logx, cdir=0):
return (1/self._reciprocal_of(self.args[0]))._eval_nseries(x, n, logx)
class sec(ReciprocalTrigonometricFunction):
"""
The secant function.
Returns the secant of x (measured in radians).
Explanation
===========
See :class:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import sec
>>> from sympy.abc import x
>>> sec(x**2).diff(x)
2*x*tan(x**2)*sec(x**2)
>>> sec(1).diff(x)
0
See Also
========
sin, csc, cos, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Sec
"""
_reciprocal_of = cos
_is_even = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half_sq = cot(arg/2)**2
return (cot_half_sq + 1)/(cot_half_sq - 1)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return (1/cos(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return sin(arg)/(cos(arg)*sin(arg))
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1/cos(arg).rewrite(sin))
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1/cos(arg).rewrite(tan))
def _eval_rewrite_as_csc(self, arg, **kwargs):
return csc(pi/2 - arg, evaluate=False)
def fdiff(self, argindex=1):
if argindex == 1:
return tan(self.args[0])*sec(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_complex and (arg/pi - S.Half).is_integer is False:
return True
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
# Reference Formula:
# http://functions.wolfram.com/ElementaryFunctions/Sec/06/01/02/01/
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
k = n//2
return S.NegativeOne**k*euler(2*k)/factorial(2*k)*x**(2*k)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = (x0 + pi/2)/pi
if n.is_integer:
lt = (arg - n*pi + pi/2).as_leading_term(x)
return (S.NegativeOne**n)/lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
class csc(ReciprocalTrigonometricFunction):
"""
The cosecant function.
Returns the cosecant of x (measured in radians).
Explanation
===========
See :func:`sin` for notes about automatic evaluation.
Examples
========
>>> from sympy import csc
>>> from sympy.abc import x
>>> csc(x**2).diff(x)
-2*x*cot(x**2)*csc(x**2)
>>> csc(1).diff(x)
0
See Also
========
sin, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
.. [2] http://dlmf.nist.gov/4.14
.. [3] http://functions.wolfram.com/ElementaryFunctions/Csc
"""
_reciprocal_of = sin
_is_odd = True
def period(self, symbol=None):
return self._period(symbol)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return (1/sin(arg))
def _eval_rewrite_as_sincos(self, arg, **kwargs):
return cos(arg)/(sin(arg)*cos(arg))
def _eval_rewrite_as_cot(self, arg, **kwargs):
cot_half = cot(arg/2)
return (1 + cot_half**2)/(2*cot_half)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return 1/sin(arg).rewrite(cos)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return sec(pi/2 - arg, evaluate=False)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return (1/sin(arg).rewrite(tan))
def fdiff(self, argindex=1):
if argindex == 1:
return -cot(self.args[0])*csc(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_complex(self):
arg = self.args[0]
if arg.is_real and (arg/pi).is_integer is False:
return True
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
k = n//2 + 1
return (S.NegativeOne**(k - 1)*2*(2**(2*k - 1) - 1)*
bernoulli(2*k)*x**(2*k - 1)/factorial(2*k))
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.functions.elementary.complexes import re
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
n = x0/pi
if n.is_integer:
lt = (arg - n*pi).as_leading_term(x)
return (S.NegativeOne**n)/lt
if x0 is S.ComplexInfinity:
x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
if x0 in (S.Infinity, S.NegativeInfinity):
return AccumBounds(S.NegativeInfinity, S.Infinity)
return self.func(x0) if x0.is_finite else self
class sinc(Function):
r"""
Represents an unnormalized sinc function:
.. math::
\operatorname{sinc}(x) =
\begin{cases}
\frac{\sin x}{x} & \qquad x \neq 0 \\
1 & \qquad x = 0
\end{cases}
Examples
========
>>> from sympy import sinc, oo, jn
>>> from sympy.abc import x
>>> sinc(x)
sinc(x)
* Automated Evaluation
>>> sinc(0)
1
>>> sinc(oo)
0
* Differentiation
>>> sinc(x).diff()
cos(x)/x - sin(x)/x**2
* Series Expansion
>>> sinc(x).series()
1 - x**2/6 + x**4/120 + O(x**6)
* As zero'th order spherical Bessel Function
>>> sinc(x).rewrite(jn)
jn(0, x)
See also
========
sin
References
==========
.. [1] https://en.wikipedia.org/wiki/Sinc_function
"""
_singularities = (S.ComplexInfinity,)
def fdiff(self, argindex=1):
x = self.args[0]
if argindex == 1:
# We would like to return the Piecewise here, but Piecewise.diff
# currently can't handle removable singularities, meaning things
# like sinc(x).diff(x, 2) give the wrong answer at x = 0. See
# https://github.com/sympy/sympy/issues/11402.
#
# return Piecewise(((x*cos(x) - sin(x))/x**2, Ne(x, S.Zero)), (S.Zero, S.true))
return cos(x)/x - sin(x)/x**2
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.One
if arg.is_Number:
if arg in [S.Infinity, S.NegativeInfinity]:
return S.Zero
elif arg is S.NaN:
return S.NaN
if arg is S.ComplexInfinity:
return S.NaN
if arg.could_extract_minus_sign():
return cls(-arg)
pi_coeff = _pi_coeff(arg)
if pi_coeff is not None:
if pi_coeff.is_integer:
if fuzzy_not(arg.is_zero):
return S.Zero
elif (2*pi_coeff).is_integer:
return S.NegativeOne**(pi_coeff - S.Half)/arg
def _eval_nseries(self, x, n, logx, cdir=0):
x = self.args[0]
return (sin(x)/x)._eval_nseries(x, n, logx)
def _eval_rewrite_as_jn(self, arg, **kwargs):
from sympy.functions.special.bessel import jn
return jn(0, arg)
def _eval_rewrite_as_sin(self, arg, **kwargs):
return Piecewise((sin(arg)/arg, Ne(arg, S.Zero)), (S.One, S.true))
def _eval_is_zero(self):
if self.args[0].is_infinite:
return True
rest, pi_mult = _peeloff_pi(self.args[0])
if rest.is_zero:
return fuzzy_and([pi_mult.is_integer, pi_mult.is_nonzero])
if rest.is_Number and pi_mult.is_integer:
return False
def _eval_is_real(self):
if self.args[0].is_extended_real or self.args[0].is_imaginary:
return True
_eval_is_finite = _eval_is_real
###############################################################################
########################### TRIGONOMETRIC INVERSES ############################
###############################################################################
class InverseTrigonometricFunction(Function):
"""Base class for inverse trigonometric functions."""
_singularities = (S.One, S.NegativeOne, S.Zero, S.ComplexInfinity) # type: tTuple[Expr, ...]
@staticmethod
@cacheit
def _asin_table():
# Only keys with could_extract_minus_sign() == False
# are actually needed.
return {
sqrt(3)/2: pi/3,
sqrt(2)/2: pi/4,
1/sqrt(2): pi/4,
sqrt((5 - sqrt(5))/8): pi/5,
sqrt(2)*sqrt(5 - sqrt(5))/4: pi/5,
sqrt((5 + sqrt(5))/8): pi*Rational(2, 5),
sqrt(2)*sqrt(5 + sqrt(5))/4: pi*Rational(2, 5),
S.Half: pi/6,
sqrt(2 - sqrt(2))/2: pi/8,
sqrt(S.Half - sqrt(2)/4): pi/8,
sqrt(2 + sqrt(2))/2: pi*Rational(3, 8),
sqrt(S.Half + sqrt(2)/4): pi*Rational(3, 8),
(sqrt(5) - 1)/4: pi/10,
(1 - sqrt(5))/4: -pi/10,
(sqrt(5) + 1)/4: pi*Rational(3, 10),
sqrt(6)/4 - sqrt(2)/4: pi/12,
-sqrt(6)/4 + sqrt(2)/4: -pi/12,
(sqrt(3) - 1)/sqrt(8): pi/12,
(1 - sqrt(3))/sqrt(8): -pi/12,
sqrt(6)/4 + sqrt(2)/4: pi*Rational(5, 12),
(1 + sqrt(3))/sqrt(8): pi*Rational(5, 12)
}
@staticmethod
@cacheit
def _atan_table():
# Only keys with could_extract_minus_sign() == False
# are actually needed.
return {
sqrt(3)/3: pi/6,
1/sqrt(3): pi/6,
sqrt(3): pi/3,
sqrt(2) - 1: pi/8,
1 - sqrt(2): -pi/8,
1 + sqrt(2): pi*Rational(3, 8),
sqrt(5 - 2*sqrt(5)): pi/5,
sqrt(5 + 2*sqrt(5)): pi*Rational(2, 5),
sqrt(1 - 2*sqrt(5)/5): pi/10,
sqrt(1 + 2*sqrt(5)/5): pi*Rational(3, 10),
2 - sqrt(3): pi/12,
-2 + sqrt(3): -pi/12,
2 + sqrt(3): pi*Rational(5, 12)
}
@staticmethod
@cacheit
def _acsc_table():
# Keys for which could_extract_minus_sign()
# will obviously return True are omitted.
return {
2*sqrt(3)/3: pi/3,
sqrt(2): pi/4,
sqrt(2 + 2*sqrt(5)/5): pi/5,
1/sqrt(Rational(5, 8) - sqrt(5)/8): pi/5,
sqrt(2 - 2*sqrt(5)/5): pi*Rational(2, 5),
1/sqrt(Rational(5, 8) + sqrt(5)/8): pi*Rational(2, 5),
2: pi/6,
sqrt(4 + 2*sqrt(2)): pi/8,
2/sqrt(2 - sqrt(2)): pi/8,
sqrt(4 - 2*sqrt(2)): pi*Rational(3, 8),
2/sqrt(2 + sqrt(2)): pi*Rational(3, 8),
1 + sqrt(5): pi/10,
sqrt(5) - 1: pi*Rational(3, 10),
-(sqrt(5) - 1): pi*Rational(-3, 10),
sqrt(6) + sqrt(2): pi/12,
sqrt(6) - sqrt(2): pi*Rational(5, 12),
-(sqrt(6) - sqrt(2)): pi*Rational(-5, 12)
}
class asin(InverseTrigonometricFunction):
r"""
The inverse sine function.
Returns the arcsine of x in radians.
Explanation
===========
``asin(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
result is a rational multiple of $\pi$ (see the ``eval`` class method).
A purely imaginary argument will lead to an asinh expression.
Examples
========
>>> from sympy import asin, oo
>>> asin(1)
pi/2
>>> asin(-1)
-pi/2
>>> asin(-oo)
oo*I
>>> asin(oo)
-oo*I
See Also
========
sin, csc, cos, sec, tan, cot
acsc, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self._eval_is_extended_real() and self.args[0].is_positive
def _eval_is_negative(self):
return self._eval_is_extended_real() and self.args[0].is_negative
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.NegativeInfinity*S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.Infinity*S.ImaginaryUnit
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return pi/2
elif arg is S.NegativeOne:
return -pi/2
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
asin_table = cls._asin_table()
if arg in asin_table:
return asin_table[arg]
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import asinh
return S.ImaginaryUnit*asinh(i_coeff)
if arg.is_zero:
return S.Zero
if isinstance(arg, sin):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, cos): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acos(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p*(n - 2)**2/(n*(n - 1))*x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return R/F*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # asin
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0.is_zero:
return arg.as_leading_term(x)
# Handling branch points
if x0 in (-S.One, S.One, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_negative:
return -pi - self.func(x0)
elif im(ndir).is_positive:
if x0.is_positive:
return pi - self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # asin
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = asin(S.One - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else pi/2 + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = asin(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else -pi/2 + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - arg0**2).is_negative:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_negative:
return -pi - res
elif im(ndir).is_positive:
if arg0.is_positive:
return pi - res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_acos(self, x, **kwargs):
return pi/2 - acos(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return 2*atan(x/(1 + sqrt(1 - x**2)))
def _eval_rewrite_as_log(self, x, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit*x + sqrt(1 - x**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_acot(self, arg, **kwargs):
return 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return pi/2 - asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return acsc(1/arg)
def _eval_is_extended_real(self):
x = self.args[0]
return x.is_extended_real and (1 - abs(x)).is_nonnegative
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sin
class acos(InverseTrigonometricFunction):
r"""
The inverse cosine function.
Returns the arc cosine of x (measured in radians).
Examples
========
``acos(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when
the result is a rational multiple of $\pi$ (see the eval class method).
``acos(zoo)`` evaluates to ``zoo``
(see note in :class:`sympy.functions.elementary.trigonometric.asec`)
A purely imaginary argument will be rewritten to asinh.
Examples
========
>>> from sympy import acos, oo
>>> acos(1)
0
>>> acos(0)
pi/2
>>> acos(oo)
oo*I
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1/sqrt(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity*S.ImaginaryUnit
elif arg is S.NegativeInfinity:
return S.NegativeInfinity*S.ImaginaryUnit
elif arg.is_zero:
return pi/2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return pi
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.is_number:
asin_table = cls._asin_table()
if arg in asin_table:
return pi/2 - asin_table[arg]
elif -arg in asin_table:
return pi/2 + asin_table[-arg]
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
return pi/2 - asin(arg)
if isinstance(arg, cos):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, sin): # acos(x) + asin(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asin(arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return pi/2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p*(n - 2)**2/(n*(n - 1))*x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return -R/F*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acos
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 == 1:
return sqrt(2)*sqrt((S.One - arg).as_leading_term(x))
if x0 in (-S.One, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_negative:
return 2*pi - self.func(x0)
elif im(ndir).is_positive:
if x0.is_positive:
return -self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_is_extended_real(self):
x = self.args[0]
return x.is_extended_real and (1 - abs(x)).is_nonnegative
def _eval_is_nonnegative(self):
return self._eval_is_extended_real()
def _eval_nseries(self, x, n, logx, cdir=0): # acos
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = acos(S.One - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = acos(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else pi + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-oo, -1) U (1, oo)
if (1 - arg0**2).is_negative:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_negative:
return 2*pi - res
elif im(ndir).is_positive:
if arg0.is_positive:
return -res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return pi/2 + S.ImaginaryUnit*\
log(S.ImaginaryUnit*x + sqrt(1 - x**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asin(self, x, **kwargs):
return pi/2 - asin(x)
def _eval_rewrite_as_atan(self, x, **kwargs):
return atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cos
def _eval_rewrite_as_acot(self, arg, **kwargs):
return pi/2 - 2*acot((1 + sqrt(1 - arg**2))/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return asec(1/arg)
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return pi/2 - acsc(1/arg)
def _eval_conjugate(self):
z = self.args[0]
r = self.func(self.args[0].conjugate())
if z.is_extended_real is False:
return r
elif z.is_extended_real and (z + 1).is_nonnegative and (z - 1).is_nonpositive:
return r
class atan(InverseTrigonometricFunction):
r"""
The inverse tangent function.
Returns the arc tangent of x (measured in radians).
Explanation
===========
``atan(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
result is a rational multiple of $\pi$ (see the eval class method).
Examples
========
>>> from sympy import atan, oo
>>> atan(0)
0
>>> atan(1)
pi/4
>>> atan(oo)
pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan
"""
args: tTuple[Expr]
_singularities = (S.ImaginaryUnit, -S.ImaginaryUnit)
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_extended_positive
def _eval_is_nonnegative(self):
return self.args[0].is_extended_nonnegative
def _eval_is_zero(self):
return self.args[0].is_zero
def _eval_is_real(self):
return self.args[0].is_extended_real
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return pi/2
elif arg is S.NegativeInfinity:
return -pi/2
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return pi/4
elif arg is S.NegativeOne:
return -pi/4
if arg is S.ComplexInfinity:
from sympy.calculus.accumulationbounds import AccumBounds
return AccumBounds(-pi/2, pi/2)
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
atan_table = cls._atan_table()
if arg in atan_table:
return atan_table[arg]
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import atanh
return S.ImaginaryUnit*atanh(i_coeff)
if arg.is_zero:
return S.Zero
if isinstance(arg, tan):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
if isinstance(arg, cot): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - acot(arg)
if ang > pi/2: # restrict to [-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return S.NegativeOne**((n - 1)//2)*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # atan
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0.is_zero:
return arg.as_leading_term(x)
# Handling branch points
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
# Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
if (1 + x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if re(ndir).is_negative:
if im(x0).is_positive:
return self.func(x0) - pi
elif re(ndir).is_positive:
if im(x0).is_negative:
return self.func(x0) + pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # atan
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
ndir = self.args[0].dir(x, cdir if cdir else 1)
if arg0 is S.ComplexInfinity:
if re(ndir) > 0:
return res - pi
return res
# Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
if (1 + arg0**2).is_negative:
if re(ndir).is_negative:
if im(arg0).is_positive:
return res - pi
elif re(ndir).is_positive:
if im(arg0).is_negative:
return res + pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2*(log(S.One - S.ImaginaryUnit*x)
- log(S.One + S.ImaginaryUnit*x))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_aseries(self, n, args0, x, logx):
if args0[0] is S.Infinity:
return (pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] is S.NegativeInfinity:
return (-pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super()._eval_aseries(n, args0, x, logx)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return tan
def _eval_rewrite_as_asin(self, arg, **kwargs):
return sqrt(arg**2)/arg*(pi/2 - asin(1/sqrt(1 + arg**2)))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return sqrt(arg**2)/arg*acos(1/sqrt(1 + arg**2))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return acot(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return sqrt(arg**2)/arg*asec(sqrt(1 + arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return sqrt(arg**2)/arg*(pi/2 - acsc(sqrt(1 + arg**2)))
class acot(InverseTrigonometricFunction):
r"""
The inverse cotangent function.
Returns the arc cotangent of x (measured in radians).
Explanation
===========
``acot(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, \tilde{\infty}, 0, 1, -1\}$
and for some instances when the result is a rational multiple of $\pi$
(see the eval class method).
A purely imaginary argument will lead to an ``acoth`` expression.
``acot(x)`` has a branch cut along $(-i, i)$, hence it is discontinuous
at 0. Its range for real $x$ is $(-\frac{\pi}{2}, \frac{\pi}{2}]$.
Examples
========
>>> from sympy import acot, sqrt
>>> acot(0)
pi/2
>>> acot(1)
pi/4
>>> acot(sqrt(3) - 2)
-5*pi/12
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, atan2
References
==========
.. [1] http://dlmf.nist.gov/4.23
.. [2] http://functions.wolfram.com/ElementaryFunctions/ArcCot
"""
_singularities = (S.ImaginaryUnit, -S.ImaginaryUnit)
def fdiff(self, argindex=1):
if argindex == 1:
return -1/(1 + self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if s.args[0].is_rational:
return False
else:
return s.is_rational
def _eval_is_positive(self):
return self.args[0].is_nonnegative
def _eval_is_negative(self):
return self.args[0].is_negative
def _eval_is_extended_real(self):
return self.args[0].is_extended_real
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg.is_zero:
return pi/ 2
elif arg is S.One:
return pi/4
elif arg is S.NegativeOne:
return -pi/4
if arg is S.ComplexInfinity:
return S.Zero
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_number:
atan_table = cls._atan_table()
if arg in atan_table:
ang = pi/2 - atan_table[arg]
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi
return ang
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
from sympy.functions.elementary.hyperbolic import acoth
return -S.ImaginaryUnit*acoth(i_coeff)
if arg.is_zero:
return pi*S.Half
if isinstance(arg, cot):
ang = arg.args[0]
if ang.is_comparable:
ang %= pi # restrict to [0,pi)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi;
return ang
if isinstance(arg, tan): # atan(x) + acot(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
ang = pi/2 - atan(arg)
if ang > pi/2: # restrict to (-pi/2,pi/2]
ang -= pi
return ang
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return pi/2 # FIX THIS
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return S.NegativeOne**((n + 1)//2)*x**n/n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acot
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0 is S.ComplexInfinity:
return (1/arg).as_leading_term(x)
# Handling branch points
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
# Handling points lying on branch cuts [-I, I]
if x0.is_imaginary and (1 + x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if re(ndir).is_positive:
if im(x0).is_positive:
return self.func(x0) + pi
elif re(ndir).is_negative:
if im(x0).is_negative:
return self.func(x0) - pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # acot
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
ndir = self.args[0].dir(x, cdir if cdir else 1)
if arg0.is_zero:
if re(ndir) < 0:
return res - pi
return res
# Handling points lying on branch cuts [-I, I]
if arg0.is_imaginary and (1 + arg0**2).is_positive:
if re(ndir).is_positive:
if im(arg0).is_positive:
return res + pi
elif re(ndir).is_negative:
if im(arg0).is_negative:
return res - pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_aseries(self, n, args0, x, logx):
if args0[0] is S.Infinity:
return (pi/2 - acot(1/self.args[0]))._eval_nseries(x, n, logx)
elif args0[0] is S.NegativeInfinity:
return (pi*Rational(3, 2) - acot(1/self.args[0]))._eval_nseries(x, n, logx)
else:
return super(atan, self)._eval_aseries(n, args0, x, logx)
def _eval_rewrite_as_log(self, x, **kwargs):
return S.ImaginaryUnit/2*(log(1 - S.ImaginaryUnit/x)
- log(1 + S.ImaginaryUnit/x))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cot
def _eval_rewrite_as_asin(self, arg, **kwargs):
return (arg*sqrt(1/arg**2)*
(pi/2 - asin(sqrt(-arg**2)/sqrt(-arg**2 - 1))))
def _eval_rewrite_as_acos(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*acos(sqrt(-arg**2)/sqrt(-arg**2 - 1))
def _eval_rewrite_as_atan(self, arg, **kwargs):
return atan(1/arg)
def _eval_rewrite_as_asec(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*asec(sqrt((1 + arg**2)/arg**2))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return arg*sqrt(1/arg**2)*(pi/2 - acsc(sqrt((1 + arg**2)/arg**2)))
class asec(InverseTrigonometricFunction):
r"""
The inverse secant function.
Returns the arc secant of x (measured in radians).
Explanation
===========
``asec(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
result is a rational multiple of $\pi$ (see the eval class method).
``asec(x)`` has branch cut in the interval $[-1, 1]$. For complex arguments,
it can be defined [4]_ as
.. math::
\operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z}
At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For
negative branch cut, the limit
.. math::
\lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z}
simplifies to :math:`-i\log\left(z/2 + O\left(z^3\right)\right)` which
ultimately evaluates to ``zoo``.
As ``acos(x) = asec(1/x)``, a similar argument can be given for
``acos(x)``.
Examples
========
>>> from sympy import asec, oo
>>> asec(1)
0
>>> asec(-1)
pi
>>> asec(0)
zoo
>>> asec(-oo)
pi/2
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec
.. [4] http://reference.wolfram.com/language/ref/ArcSec.html
"""
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.ComplexInfinity
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return pi
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return pi/2
if arg.is_number:
acsc_table = cls._acsc_table()
if arg in acsc_table:
return pi/2 - acsc_table[arg]
elif -arg in acsc_table:
return pi/2 + acsc_table[-arg]
if arg.is_infinite:
return S.Pi/2
if isinstance(arg, sec):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to [0,pi]
ang = 2*pi - ang
return ang
if isinstance(arg, csc): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - acsc(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sec
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return S.ImaginaryUnit*log(2 / x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
else:
k = n // 2
R = RisingFactorial(S.Half, k) * n
F = factorial(k) * n // 2 * n // 2
return -S.ImaginaryUnit * R / F * x**n / 4
def _eval_as_leading_term(self, x, logx=None, cdir=0): # asec
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 == 1:
return sqrt(2)*sqrt((arg - S.One).as_leading_term(x))
if x0 in (-S.One, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-1, 1)
if x0.is_real and (1 - x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_positive:
return -self.func(x0)
elif im(ndir).is_positive:
if x0.is_negative:
return 2*pi - self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # asec
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = asec(S.One + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = asec(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-1, 1)
if arg0.is_real and (1 - arg0**2).is_positive:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_positive:
return -res
elif im(ndir).is_positive:
if arg0.is_negative:
return 2*pi - res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_is_extended_real(self):
x = self.args[0]
if x.is_extended_real is False:
return False
return fuzzy_or(((x - 1).is_nonnegative, (-x - 1).is_nonnegative))
def _eval_rewrite_as_log(self, arg, **kwargs):
return pi/2 + S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asin(self, arg, **kwargs):
return pi/2 - asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return acos(1/arg)
def _eval_rewrite_as_atan(self, x, **kwargs):
sx2x = sqrt(x**2)/x
return pi/2*(1 - sx2x) + sx2x*atan(sqrt(x**2 - 1))
def _eval_rewrite_as_acot(self, x, **kwargs):
sx2x = sqrt(x**2)/x
return pi/2*(1 - sx2x) + sx2x*acot(1/sqrt(x**2 - 1))
def _eval_rewrite_as_acsc(self, arg, **kwargs):
return pi/2 - acsc(arg)
class acsc(InverseTrigonometricFunction):
r"""
The inverse cosecant function.
Returns the arc cosecant of x (measured in radians).
Explanation
===========
``acsc(x)`` will evaluate automatically in the cases
$x \in \{\infty, -\infty, 0, 1, -1\}$` and for some instances when the
result is a rational multiple of $\pi$ (see the ``eval`` class method).
Examples
========
>>> from sympy import acsc, oo
>>> acsc(1)
pi/2
>>> acsc(-1)
-pi/2
>>> acsc(oo)
0
>>> acsc(-oo) == acsc(oo)
True
>>> acsc(0)
zoo
See Also
========
sin, csc, cos, sec, tan, cot
asin, acos, asec, atan, acot, atan2
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] http://dlmf.nist.gov/4.23
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsc
"""
@classmethod
def eval(cls, arg):
if arg.is_zero:
return S.ComplexInfinity
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.One:
return pi/2
elif arg is S.NegativeOne:
return -pi/2
if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
return S.Zero
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_infinite:
return S.Zero
if arg.is_number:
acsc_table = cls._acsc_table()
if arg in acsc_table:
return acsc_table[arg]
if isinstance(arg, csc):
ang = arg.args[0]
if ang.is_comparable:
ang %= 2*pi # restrict to [0,2*pi)
if ang > pi: # restrict to (-pi,pi]
ang = pi - ang
# restrict to [-pi/2,pi/2]
if ang > pi/2:
ang = pi - ang
if ang < -pi/2:
ang = -pi - ang
return ang
if isinstance(arg, sec): # asec(x) + acsc(x) = pi/2
ang = arg.args[0]
if ang.is_comparable:
return pi/2 - asec(arg)
def fdiff(self, argindex=1):
if argindex == 1:
return -1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return csc
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return pi/2 - S.ImaginaryUnit*log(2) + S.ImaginaryUnit*log(x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
else:
k = n // 2
R = RisingFactorial(S.Half, k) * n
F = factorial(k) * n // 2 * n // 2
return S.ImaginaryUnit * R / F * x**n / 4
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acsc
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 in (-S.One, S.One, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
if x0 is S.ComplexInfinity:
return (1/arg).as_leading_term(x)
# Handling points lying on branch cuts (-1, 1)
if x0.is_real and (1 - x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_positive:
return pi - self.func(x0)
elif im(ndir).is_positive:
if x0.is_negative:
return -pi - self.func(x0)
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # acsc
from sympy.series.order import O
arg0 = self.args[0].subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = acsc(S.One + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = acsc(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.NegativeOne - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-1, 1)
if arg0.is_real and (1 - arg0**2).is_positive:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_positive:
return pi - res
elif im(ndir).is_positive:
if arg0.is_negative:
return -pi - res
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, arg, **kwargs):
return -S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asin(self, arg, **kwargs):
return asin(1/arg)
def _eval_rewrite_as_acos(self, arg, **kwargs):
return pi/2 - acos(1/arg)
def _eval_rewrite_as_atan(self, x, **kwargs):
return sqrt(x**2)/x*(pi/2 - atan(sqrt(x**2 - 1)))
def _eval_rewrite_as_acot(self, arg, **kwargs):
return sqrt(arg**2)/arg*(pi/2 - acot(1/sqrt(arg**2 - 1)))
def _eval_rewrite_as_asec(self, arg, **kwargs):
return pi/2 - asec(arg)
class atan2(InverseTrigonometricFunction):
r"""
The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking
two arguments `y` and `x`. Signs of both `y` and `x` are considered to
determine the appropriate quadrant of `\operatorname{atan}(y/x)`.
The range is `(-\pi, \pi]`. The complete definition reads as follows:
.. math::
\operatorname{atan2}(y, x) =
\begin{cases}
\arctan\left(\frac y x\right) & \qquad x > 0 \\
\arctan\left(\frac y x\right) + \pi& \qquad y \ge 0, x < 0 \\
\arctan\left(\frac y x\right) - \pi& \qquad y < 0, x < 0 \\
+\frac{\pi}{2} & \qquad y > 0, x = 0 \\
-\frac{\pi}{2} & \qquad y < 0, x = 0 \\
\text{undefined} & \qquad y = 0, x = 0
\end{cases}
Attention: Note the role reversal of both arguments. The `y`-coordinate
is the first argument and the `x`-coordinate the second.
If either `x` or `y` is complex:
.. math::
\operatorname{atan2}(y, x) =
-i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right)
Examples
========
Going counter-clock wise around the origin we find the
following angles:
>>> from sympy import atan2
>>> atan2(0, 1)
0
>>> atan2(1, 1)
pi/4
>>> atan2(1, 0)
pi/2
>>> atan2(1, -1)
3*pi/4
>>> atan2(0, -1)
pi
>>> atan2(-1, -1)
-3*pi/4
>>> atan2(-1, 0)
-pi/2
>>> atan2(-1, 1)
-pi/4
which are all correct. Compare this to the results of the ordinary
`\operatorname{atan}` function for the point `(x, y) = (-1, 1)`
>>> from sympy import atan, S
>>> atan(S(1)/-1)
-pi/4
>>> atan2(1, -1)
3*pi/4
where only the `\operatorname{atan2}` function reurns what we expect.
We can differentiate the function with respect to both arguments:
>>> from sympy import diff
>>> from sympy.abc import x, y
>>> diff(atan2(y, x), x)
-y/(x**2 + y**2)
>>> diff(atan2(y, x), y)
x/(x**2 + y**2)
We can express the `\operatorname{atan2}` function in terms of
complex logarithms:
>>> from sympy import log
>>> atan2(y, x).rewrite(log)
-I*log((x + I*y)/sqrt(x**2 + y**2))
and in terms of `\operatorname(atan)`:
>>> from sympy import atan
>>> atan2(y, x).rewrite(atan)
Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True))
but note that this form is undefined on the negative real axis.
See Also
========
sin, csc, cos, sec, tan, cot
asin, acsc, acos, asec, atan, acot
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
.. [2] https://en.wikipedia.org/wiki/Atan2
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan2
"""
@classmethod
def eval(cls, y, x):
from sympy.functions.special.delta_functions import Heaviside
if x is S.NegativeInfinity:
if y.is_zero:
# Special case y = 0 because we define Heaviside(0) = 1/2
return pi
return 2*pi*(Heaviside(re(y))) - pi
elif x is S.Infinity:
return S.Zero
elif x.is_imaginary and y.is_imaginary and x.is_number and y.is_number:
x = im(x)
y = im(y)
if x.is_extended_real and y.is_extended_real:
if x.is_positive:
return atan(y/x)
elif x.is_negative:
if y.is_negative:
return atan(y/x) - pi
elif y.is_nonnegative:
return atan(y/x) + pi
elif x.is_zero:
if y.is_positive:
return pi/2
elif y.is_negative:
return -pi/2
elif y.is_zero:
return S.NaN
if y.is_zero:
if x.is_extended_nonzero:
return pi*(S.One - Heaviside(x))
if x.is_number:
return Piecewise((pi, re(x) < 0),
(0, Ne(x, 0)),
(S.NaN, True))
if x.is_number and y.is_number:
return -S.ImaginaryUnit*log(
(x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))
def _eval_rewrite_as_log(self, y, x, **kwargs):
return -S.ImaginaryUnit*log((x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))
def _eval_rewrite_as_atan(self, y, x, **kwargs):
return Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)),
(pi, re(x) < 0),
(0, Ne(x, 0)),
(S.NaN, True))
def _eval_rewrite_as_arg(self, y, x, **kwargs):
if x.is_extended_real and y.is_extended_real:
return arg_f(x + y*S.ImaginaryUnit)
n = x + S.ImaginaryUnit*y
d = x**2 + y**2
return arg_f(n/sqrt(d)) - S.ImaginaryUnit*log(abs(n)/sqrt(abs(d)))
def _eval_is_extended_real(self):
return self.args[0].is_extended_real and self.args[1].is_extended_real
def _eval_conjugate(self):
return self.func(self.args[0].conjugate(), self.args[1].conjugate())
def fdiff(self, argindex):
y, x = self.args
if argindex == 1:
# Diff wrt y
return x/(x**2 + y**2)
elif argindex == 2:
# Diff wrt x
return -y/(x**2 + y**2)
else:
raise ArgumentIndexError(self, argindex)
def _eval_evalf(self, prec):
y, x = self.args
if x.is_extended_real and y.is_extended_real:
return super()._eval_evalf(prec)
|
49ea1180c4d2918b1ff385790885e3fc3d41c64c82db5a8f0d3a8fcd745c18a5 | from sympy.core import S, Function, diff, Tuple, Dummy, Mul
from sympy.core.basic import Basic, as_Basic
from sympy.core.numbers import Rational, NumberSymbol, _illegal
from sympy.core.parameters import global_parameters
from sympy.core.relational import (Lt, Gt, Eq, Ne, Relational,
_canonical, _canonical_coeff)
from sympy.core.sorting import ordered
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.logic.boolalg import (And, Boolean, distribute_and_over_or, Not,
true, false, Or, ITE, simplify_logic, to_cnf, distribute_or_over_and)
from sympy.utilities.iterables import uniq, sift, common_prefix
from sympy.utilities.misc import filldedent, func_name
from itertools import product
Undefined = S.NaN # Piecewise()
class ExprCondPair(Tuple):
"""Represents an expression, condition pair."""
def __new__(cls, expr, cond):
expr = as_Basic(expr)
if cond == True:
return Tuple.__new__(cls, expr, true)
elif cond == False:
return Tuple.__new__(cls, expr, false)
elif isinstance(cond, Basic) and cond.has(Piecewise):
cond = piecewise_fold(cond)
if isinstance(cond, Piecewise):
cond = cond.rewrite(ITE)
if not isinstance(cond, Boolean):
raise TypeError(filldedent('''
Second argument must be a Boolean,
not `%s`''' % func_name(cond)))
return Tuple.__new__(cls, expr, cond)
@property
def expr(self):
"""
Returns the expression of this pair.
"""
return self.args[0]
@property
def cond(self):
"""
Returns the condition of this pair.
"""
return self.args[1]
@property
def is_commutative(self):
return self.expr.is_commutative
def __iter__(self):
yield self.expr
yield self.cond
def _eval_simplify(self, **kwargs):
return self.func(*[a.simplify(**kwargs) for a in self.args])
class Piecewise(Function):
"""
Represents a piecewise function.
Usage:
Piecewise( (expr,cond), (expr,cond), ... )
- Each argument is a 2-tuple defining an expression and condition
- The conds are evaluated in turn returning the first that is True.
If any of the evaluated conds are not explicitly False,
e.g. ``x < 1``, the function is returned in symbolic form.
- If the function is evaluated at a place where all conditions are False,
nan will be returned.
- Pairs where the cond is explicitly False, will be removed and no pair
appearing after a True condition will ever be retained. If a single
pair with a True condition remains, it will be returned, even when
evaluation is False.
Examples
========
>>> from sympy import Piecewise, log, piecewise_fold
>>> from sympy.abc import x, y
>>> f = x**2
>>> g = log(x)
>>> p = Piecewise((0, x < -1), (f, x <= 1), (g, True))
>>> p.subs(x,1)
1
>>> p.subs(x,5)
log(5)
Booleans can contain Piecewise elements:
>>> cond = (x < y).subs(x, Piecewise((2, x < 0), (3, True))); cond
Piecewise((2, x < 0), (3, True)) < y
The folded version of this results in a Piecewise whose
expressions are Booleans:
>>> folded_cond = piecewise_fold(cond); folded_cond
Piecewise((2 < y, x < 0), (3 < y, True))
When a Boolean containing Piecewise (like cond) or a Piecewise
with Boolean expressions (like folded_cond) is used as a condition,
it is converted to an equivalent :class:`~.ITE` object:
>>> Piecewise((1, folded_cond))
Piecewise((1, ITE(x < 0, y > 2, y > 3)))
When a condition is an ``ITE``, it will be converted to a simplified
Boolean expression:
>>> piecewise_fold(_)
Piecewise((1, ((x >= 0) | (y > 2)) & ((y > 3) | (x < 0))))
See Also
========
piecewise_fold
piecewise_exclusive
ITE
"""
nargs = None
is_Piecewise = True
def __new__(cls, *args, **options):
if len(args) == 0:
raise TypeError("At least one (expr, cond) pair expected.")
# (Try to) sympify args first
newargs = []
for ec in args:
# ec could be a ExprCondPair or a tuple
pair = ExprCondPair(*getattr(ec, 'args', ec))
cond = pair.cond
if cond is false:
continue
newargs.append(pair)
if cond is true:
break
eval = options.pop('evaluate', global_parameters.evaluate)
if eval:
r = cls.eval(*newargs)
if r is not None:
return r
elif len(newargs) == 1 and newargs[0].cond == True:
return newargs[0].expr
return Basic.__new__(cls, *newargs, **options)
@classmethod
def eval(cls, *_args):
"""Either return a modified version of the args or, if no
modifications were made, return None.
Modifications that are made here:
1. relationals are made canonical
2. any False conditions are dropped
3. any repeat of a previous condition is ignored
4. any args past one with a true condition are dropped
If there are no args left, nan will be returned.
If there is a single arg with a True condition, its
corresponding expression will be returned.
EXAMPLES
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> cond = -x < -1
>>> args = [(1, cond), (4, cond), (3, False), (2, True), (5, x < 1)]
>>> Piecewise(*args, evaluate=False)
Piecewise((1, -x < -1), (4, -x < -1), (2, True))
>>> Piecewise(*args)
Piecewise((1, x > 1), (2, True))
"""
if not _args:
return Undefined
if len(_args) == 1 and _args[0][-1] == True:
return _args[0][0]
newargs = [] # the unevaluated conditions
current_cond = set() # the conditions up to a given e, c pair
for expr, cond in _args:
cond = cond.replace(
lambda _: _.is_Relational, _canonical_coeff)
# Check here if expr is a Piecewise and collapse if one of
# the conds in expr matches cond. This allows the collapsing
# of Piecewise((Piecewise((x,x<0)),x<0)) to Piecewise((x,x<0)).
# This is important when using piecewise_fold to simplify
# multiple Piecewise instances having the same conds.
# Eventually, this code should be able to collapse Piecewise's
# having different intervals, but this will probably require
# using the new assumptions.
if isinstance(expr, Piecewise):
unmatching = []
for i, (e, c) in enumerate(expr.args):
if c in current_cond:
# this would already have triggered
continue
if c == cond:
if c != True:
# nothing past this condition will ever
# trigger and only those args before this
# that didn't match a previous condition
# could possibly trigger
if unmatching:
expr = Piecewise(*(
unmatching + [(e, c)]))
else:
expr = e
break
else:
unmatching.append((e, c))
# check for condition repeats
got = False
# -- if an And contains a condition that was
# already encountered, then the And will be
# False: if the previous condition was False
# then the And will be False and if the previous
# condition is True then then we wouldn't get to
# this point. In either case, we can skip this condition.
for i in ([cond] +
(list(cond.args) if isinstance(cond, And) else
[])):
if i in current_cond:
got = True
break
if got:
continue
# -- if not(c) is already in current_cond then c is
# a redundant condition in an And. This does not
# apply to Or, however: (e1, c), (e2, Or(~c, d))
# is not (e1, c), (e2, d) because if c and d are
# both False this would give no results when the
# true answer should be (e2, True)
if isinstance(cond, And):
nonredundant = []
for c in cond.args:
if isinstance(c, Relational):
if c.negated.canonical in current_cond:
continue
# if a strict inequality appears after
# a non-strict one, then the condition is
# redundant
if isinstance(c, (Lt, Gt)) and (
c.weak in current_cond):
cond = False
break
nonredundant.append(c)
else:
cond = cond.func(*nonredundant)
elif isinstance(cond, Relational):
if cond.negated.canonical in current_cond:
cond = S.true
current_cond.add(cond)
# collect successive e,c pairs when exprs or cond match
if newargs:
if newargs[-1].expr == expr:
orcond = Or(cond, newargs[-1].cond)
if isinstance(orcond, (And, Or)):
orcond = distribute_and_over_or(orcond)
newargs[-1] = ExprCondPair(expr, orcond)
continue
elif newargs[-1].cond == cond:
newargs[-1] = ExprCondPair(expr, cond)
continue
newargs.append(ExprCondPair(expr, cond))
# some conditions may have been redundant
missing = len(newargs) != len(_args)
# some conditions may have changed
same = all(a == b for a, b in zip(newargs, _args))
# if either change happened we return the expr with the
# updated args
if not newargs:
raise ValueError(filldedent('''
There are no conditions (or none that
are not trivially false) to define an
expression.'''))
if missing or not same:
return cls(*newargs)
def doit(self, **hints):
"""
Evaluate this piecewise function.
"""
newargs = []
for e, c in self.args:
if hints.get('deep', True):
if isinstance(e, Basic):
newe = e.doit(**hints)
if newe != self:
e = newe
if isinstance(c, Basic):
c = c.doit(**hints)
newargs.append((e, c))
return self.func(*newargs)
def _eval_simplify(self, **kwargs):
return piecewise_simplify(self, **kwargs)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
for e, c in self.args:
if c == True or c.subs(x, 0) == True:
return e.as_leading_term(x)
def _eval_adjoint(self):
return self.func(*[(e.adjoint(), c) for e, c in self.args])
def _eval_conjugate(self):
return self.func(*[(e.conjugate(), c) for e, c in self.args])
def _eval_derivative(self, x):
return self.func(*[(diff(e, x), c) for e, c in self.args])
def _eval_evalf(self, prec):
return self.func(*[(e._evalf(prec), c) for e, c in self.args])
def piecewise_integrate(self, x, **kwargs):
"""Return the Piecewise with each expression being
replaced with its antiderivative. To obtain a continuous
antiderivative, use the :func:`~.integrate` function or method.
Examples
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> p = Piecewise((0, x < 0), (1, x < 1), (2, True))
>>> p.piecewise_integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x, True))
Note that this does not give a continuous function, e.g.
at x = 1 the 3rd condition applies and the antiderivative
there is 2*x so the value of the antiderivative is 2:
>>> anti = _
>>> anti.subs(x, 1)
2
The continuous derivative accounts for the integral *up to*
the point of interest, however:
>>> p.integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x - 1, True))
>>> _.subs(x, 1)
1
See Also
========
Piecewise._eval_integral
"""
from sympy.integrals import integrate
return self.func(*[(integrate(e, x, **kwargs), c) for e, c in self.args])
def _handle_irel(self, x, handler):
"""Return either None (if the conditions of self depend only on x) else
a Piecewise expression whose expressions (handled by the handler that
was passed) are paired with the governing x-independent relationals,
e.g. Piecewise((A, a(x) & b(y)), (B, c(x) | c(y)) ->
Piecewise(
(handler(Piecewise((A, a(x) & True), (B, c(x) | True)), b(y) & c(y)),
(handler(Piecewise((A, a(x) & True), (B, c(x) | False)), b(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | True)), c(y)),
(handler(Piecewise((A, a(x) & False), (B, c(x) | False)), True))
"""
# identify governing relationals
rel = self.atoms(Relational)
irel = list(ordered([r for r in rel if x not in r.free_symbols
and r not in (S.true, S.false)]))
if irel:
args = {}
exprinorder = []
for truth in product((1, 0), repeat=len(irel)):
reps = dict(zip(irel, truth))
# only store the true conditions since the false are implied
# when they appear lower in the Piecewise args
if 1 not in truth:
cond = None # flag this one so it doesn't get combined
else:
andargs = Tuple(*[i for i in reps if reps[i]])
free = list(andargs.free_symbols)
if len(free) == 1:
from sympy.solvers.inequalities import (
reduce_inequalities, _solve_inequality)
try:
t = reduce_inequalities(andargs, free[0])
# ValueError when there are potentially
# nonvanishing imaginary parts
except (ValueError, NotImplementedError):
# at least isolate free symbol on left
t = And(*[_solve_inequality(
a, free[0], linear=True)
for a in andargs])
else:
t = And(*andargs)
if t is S.false:
continue # an impossible combination
cond = t
expr = handler(self.xreplace(reps))
if isinstance(expr, self.func) and len(expr.args) == 1:
expr, econd = expr.args[0]
cond = And(econd, True if cond is None else cond)
# the ec pairs are being collected since all possibilities
# are being enumerated, but don't put the last one in since
# its expr might match a previous expression and it
# must appear last in the args
if cond is not None:
args.setdefault(expr, []).append(cond)
# but since we only store the true conditions we must maintain
# the order so that the expression with the most true values
# comes first
exprinorder.append(expr)
# convert collected conditions as args of Or
for k in args:
args[k] = Or(*args[k])
# take them in the order obtained
args = [(e, args[e]) for e in uniq(exprinorder)]
# add in the last arg
args.append((expr, True))
return Piecewise(*args)
def _eval_integral(self, x, _first=True, **kwargs):
"""Return the indefinite integral of the
Piecewise such that subsequent substitution of x with a
value will give the value of the integral (not including
the constant of integration) up to that point. To only
integrate the individual parts of Piecewise, use the
``piecewise_integrate`` method.
Examples
========
>>> from sympy import Piecewise
>>> from sympy.abc import x
>>> p = Piecewise((0, x < 0), (1, x < 1), (2, True))
>>> p.integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x - 1, True))
>>> p.piecewise_integrate(x)
Piecewise((0, x < 0), (x, x < 1), (2*x, True))
See Also
========
Piecewise.piecewise_integrate
"""
from sympy.integrals.integrals import integrate
if _first:
def handler(ipw):
if isinstance(ipw, self.func):
return ipw._eval_integral(x, _first=False, **kwargs)
else:
return ipw.integrate(x, **kwargs)
irv = self._handle_irel(x, handler)
if irv is not None:
return irv
# handle a Piecewise from -oo to oo with and no x-independent relationals
# -----------------------------------------------------------------------
ok, abei = self._intervals(x)
if not ok:
from sympy.integrals.integrals import Integral
return Integral(self, x) # unevaluated
pieces = [(a, b) for a, b, _, _ in abei]
oo = S.Infinity
done = [(-oo, oo, -1)]
for k, p in enumerate(pieces):
if p == (-oo, oo):
# all undone intervals will get this key
for j, (a, b, i) in enumerate(done):
if i == -1:
done[j] = a, b, k
break # nothing else to consider
N = len(done) - 1
for j, (a, b, i) in enumerate(reversed(done)):
if i == -1:
j = N - j
done[j: j + 1] = _clip(p, (a, b), k)
done = [(a, b, i) for a, b, i in done if a != b]
# append an arg if there is a hole so a reference to
# argument -1 will give Undefined
if any(i == -1 for (a, b, i) in done):
abei.append((-oo, oo, Undefined, -1))
# return the sum of the intervals
args = []
sum = None
for a, b, i in done:
anti = integrate(abei[i][-2], x, **kwargs)
if sum is None:
sum = anti
else:
sum = sum.subs(x, a)
e = anti._eval_interval(x, a, x)
if sum.has(*_illegal) or e.has(*_illegal):
sum = anti
else:
sum += e
# see if we know whether b is contained in original
# condition
if b is S.Infinity:
cond = True
elif self.args[abei[i][-1]].cond.subs(x, b) == False:
cond = (x < b)
else:
cond = (x <= b)
args.append((sum, cond))
return Piecewise(*args)
def _eval_interval(self, sym, a, b, _first=True):
"""Evaluates the function along the sym in a given interval [a, b]"""
# FIXME: Currently complex intervals are not supported. A possible
# replacement algorithm, discussed in issue 5227, can be found in the
# following papers;
# http://portal.acm.org/citation.cfm?id=281649
# http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.4127&rep=rep1&type=pdf
if a is None or b is None:
# In this case, it is just simple substitution
return super()._eval_interval(sym, a, b)
else:
x, lo, hi = map(as_Basic, (sym, a, b))
if _first: # get only x-dependent relationals
def handler(ipw):
if isinstance(ipw, self.func):
return ipw._eval_interval(x, lo, hi, _first=None)
else:
return ipw._eval_interval(x, lo, hi)
irv = self._handle_irel(x, handler)
if irv is not None:
return irv
if (lo < hi) is S.false or (
lo is S.Infinity or hi is S.NegativeInfinity):
rv = self._eval_interval(x, hi, lo, _first=False)
if isinstance(rv, Piecewise):
rv = Piecewise(*[(-e, c) for e, c in rv.args])
else:
rv = -rv
return rv
if (lo < hi) is S.true or (
hi is S.Infinity or lo is S.NegativeInfinity):
pass
else:
_a = Dummy('lo')
_b = Dummy('hi')
a = lo if lo.is_comparable else _a
b = hi if hi.is_comparable else _b
pos = self._eval_interval(x, a, b, _first=False)
if a == _a and b == _b:
# it's purely symbolic so just swap lo and hi and
# change the sign to get the value for when lo > hi
neg, pos = (-pos.xreplace({_a: hi, _b: lo}),
pos.xreplace({_a: lo, _b: hi}))
else:
# at least one of the bounds was comparable, so allow
# _eval_interval to use that information when computing
# the interval with lo and hi reversed
neg, pos = (-self._eval_interval(x, hi, lo, _first=False),
pos.xreplace({_a: lo, _b: hi}))
# allow simplification based on ordering of lo and hi
p = Dummy('', positive=True)
if lo.is_Symbol:
pos = pos.xreplace({lo: hi - p}).xreplace({p: hi - lo})
neg = neg.xreplace({lo: hi + p}).xreplace({p: lo - hi})
elif hi.is_Symbol:
pos = pos.xreplace({hi: lo + p}).xreplace({p: hi - lo})
neg = neg.xreplace({hi: lo - p}).xreplace({p: lo - hi})
# evaluate limits that may have unevaluate Min/Max
touch = lambda _: _.replace(
lambda x: isinstance(x, (Min, Max)),
lambda x: x.func(*x.args))
neg = touch(neg)
pos = touch(pos)
# assemble return expression; make the first condition be Lt
# b/c then the first expression will look the same whether
# the lo or hi limit is symbolic
if a == _a: # the lower limit was symbolic
rv = Piecewise(
(pos,
lo < hi),
(neg,
True))
else:
rv = Piecewise(
(neg,
hi < lo),
(pos,
True))
if rv == Undefined:
raise ValueError("Can't integrate across undefined region.")
if any(isinstance(i, Piecewise) for i in (pos, neg)):
rv = piecewise_fold(rv)
return rv
# handle a Piecewise with lo <= hi and no x-independent relationals
# -----------------------------------------------------------------
ok, abei = self._intervals(x)
if not ok:
from sympy.integrals.integrals import Integral
# not being able to do the interval of f(x) can
# be stated as not being able to do the integral
# of f'(x) over the same range
return Integral(self.diff(x), (x, lo, hi)) # unevaluated
pieces = [(a, b) for a, b, _, _ in abei]
done = [(lo, hi, -1)]
oo = S.Infinity
for k, p in enumerate(pieces):
if p[:2] == (-oo, oo):
# all undone intervals will get this key
for j, (a, b, i) in enumerate(done):
if i == -1:
done[j] = a, b, k
break # nothing else to consider
N = len(done) - 1
for j, (a, b, i) in enumerate(reversed(done)):
if i == -1:
j = N - j
done[j: j + 1] = _clip(p, (a, b), k)
done = [(a, b, i) for a, b, i in done if a != b]
# return the sum of the intervals
sum = S.Zero
upto = None
for a, b, i in done:
if i == -1:
if upto is None:
return Undefined
# TODO simplify hi <= upto
return Piecewise((sum, hi <= upto), (Undefined, True))
sum += abei[i][-2]._eval_interval(x, a, b)
upto = b
return sum
def _intervals(self, sym, err_on_Eq=False):
r"""Return a bool and a message (when bool is False), else a
list of unique tuples, (a, b, e, i), where a and b
are the lower and upper bounds in which the expression e of
argument i in self is defined and $a < b$ (when involving
numbers) or $a \le b$ when involving symbols.
If there are any relationals not involving sym, or any
relational cannot be solved for sym, the bool will be False
a message be given as the second return value. The calling
routine should have removed such relationals before calling
this routine.
The evaluated conditions will be returned as ranges.
Discontinuous ranges will be returned separately with
identical expressions. The first condition that evaluates to
True will be returned as the last tuple with a, b = -oo, oo.
"""
from sympy.solvers.inequalities import _solve_inequality
assert isinstance(self, Piecewise)
def nonsymfail(cond):
return False, filldedent('''
A condition not involving
%s appeared: %s''' % (sym, cond))
def _solve_relational(r):
if sym not in r.free_symbols:
return nonsymfail(r)
try:
rv = _solve_inequality(r, sym)
except NotImplementedError:
return False, 'Unable to solve relational %s for %s.' % (r, sym)
if isinstance(rv, Relational):
free = rv.args[1].free_symbols
if rv.args[0] != sym or sym in free:
return False, 'Unable to solve relational %s for %s.' % (r, sym)
if rv.rel_op == '==':
# this equality has been affirmed to have the form
# Eq(sym, rhs) where rhs is sym-free; it represents
# a zero-width interval which will be ignored
# whether it is an isolated condition or contained
# within an And or an Or
rv = S.false
elif rv.rel_op == '!=':
try:
rv = Or(sym < rv.rhs, sym > rv.rhs)
except TypeError:
# e.g. x != I ==> all real x satisfy
rv = S.true
elif rv == (S.NegativeInfinity < sym) & (sym < S.Infinity):
rv = S.true
return True, rv
args = list(self.args)
# make self canonical wrt Relationals
keys = self.atoms(Relational)
reps = {}
for r in keys:
ok, s = _solve_relational(r)
if ok != True:
return False, ok
reps[r] = s
# process args individually so if any evaluate, their position
# in the original Piecewise will be known
args = [i.xreplace(reps) for i in self.args]
# precondition args
expr_cond = []
default = idefault = None
for i, (expr, cond) in enumerate(args):
if cond is S.false:
continue
if cond is S.true:
default = expr
idefault = i
break
if isinstance(cond, Eq):
# unanticipated condition, but it is here in case a
# replacement caused an Eq to appear
if err_on_Eq:
return False, 'encountered Eq condition: %s' % cond
continue # zero width interval
cond = to_cnf(cond)
if isinstance(cond, And):
cond = distribute_or_over_and(cond)
if isinstance(cond, Or):
expr_cond.extend(
[(i, expr, o) for o in cond.args
if not isinstance(o, Eq)])
elif cond is not S.false:
expr_cond.append((i, expr, cond))
elif cond is S.true:
default = expr
idefault = i
break
# determine intervals represented by conditions
int_expr = []
for iarg, expr, cond in expr_cond:
if isinstance(cond, And):
lower = S.NegativeInfinity
upper = S.Infinity
exclude = []
for cond2 in cond.args:
if not isinstance(cond2, Relational):
return False, 'expecting only Relationals'
if isinstance(cond2, Eq):
lower = upper # ignore
if err_on_Eq:
return False, 'encountered secondary Eq condition'
break
elif isinstance(cond2, Ne):
l, r = cond2.args
if l == sym:
exclude.append(r)
elif r == sym:
exclude.append(l)
else:
return nonsymfail(cond2)
continue
elif cond2.lts == sym:
upper = Min(cond2.gts, upper)
elif cond2.gts == sym:
lower = Max(cond2.lts, lower)
else:
return nonsymfail(cond2) # should never get here
if exclude:
exclude = list(ordered(exclude))
newcond = []
for i, e in enumerate(exclude):
if e < lower == True or e > upper == True:
continue
if not newcond:
newcond.append((None, lower)) # add a primer
newcond.append((newcond[-1][1], e))
newcond.append((newcond[-1][1], upper))
newcond.pop(0) # remove the primer
expr_cond.extend([(iarg, expr, And(i[0] < sym, sym < i[1])) for i in newcond])
continue
elif isinstance(cond, Relational) and cond.rel_op != '!=':
lower, upper = cond.lts, cond.gts # part 1: initialize with givens
if cond.lts == sym: # part 1a: expand the side ...
lower = S.NegativeInfinity # e.g. x <= 0 ---> -oo <= 0
elif cond.gts == sym: # part 1a: ... that can be expanded
upper = S.Infinity # e.g. x >= 0 ---> oo >= 0
else:
return nonsymfail(cond)
else:
return False, 'unrecognized condition: %s' % cond
lower, upper = lower, Max(lower, upper)
if err_on_Eq and lower == upper:
return False, 'encountered Eq condition'
if (lower >= upper) is not S.true:
int_expr.append((lower, upper, expr, iarg))
if default is not None:
int_expr.append(
(S.NegativeInfinity, S.Infinity, default, idefault))
return True, list(uniq(int_expr))
def _eval_nseries(self, x, n, logx, cdir=0):
args = [(ec.expr._eval_nseries(x, n, logx), ec.cond) for ec in self.args]
return self.func(*args)
def _eval_power(self, s):
return self.func(*[(e**s, c) for e, c in self.args])
def _eval_subs(self, old, new):
# this is strictly not necessary, but we can keep track
# of whether True or False conditions arise and be
# somewhat more efficient by avoiding other substitutions
# and avoiding invalid conditions that appear after a
# True condition
args = list(self.args)
args_exist = False
for i, (e, c) in enumerate(args):
c = c._subs(old, new)
if c != False:
args_exist = True
e = e._subs(old, new)
args[i] = (e, c)
if c == True:
break
if not args_exist:
args = ((Undefined, True),)
return self.func(*args)
def _eval_transpose(self):
return self.func(*[(e.transpose(), c) for e, c in self.args])
def _eval_template_is_attr(self, is_attr):
b = None
for expr, _ in self.args:
a = getattr(expr, is_attr)
if a is None:
return
if b is None:
b = a
elif b is not a:
return
return b
_eval_is_finite = lambda self: self._eval_template_is_attr(
'is_finite')
_eval_is_complex = lambda self: self._eval_template_is_attr('is_complex')
_eval_is_even = lambda self: self._eval_template_is_attr('is_even')
_eval_is_imaginary = lambda self: self._eval_template_is_attr(
'is_imaginary')
_eval_is_integer = lambda self: self._eval_template_is_attr('is_integer')
_eval_is_irrational = lambda self: self._eval_template_is_attr(
'is_irrational')
_eval_is_negative = lambda self: self._eval_template_is_attr('is_negative')
_eval_is_nonnegative = lambda self: self._eval_template_is_attr(
'is_nonnegative')
_eval_is_nonpositive = lambda self: self._eval_template_is_attr(
'is_nonpositive')
_eval_is_nonzero = lambda self: self._eval_template_is_attr(
'is_nonzero')
_eval_is_odd = lambda self: self._eval_template_is_attr('is_odd')
_eval_is_polar = lambda self: self._eval_template_is_attr('is_polar')
_eval_is_positive = lambda self: self._eval_template_is_attr('is_positive')
_eval_is_extended_real = lambda self: self._eval_template_is_attr(
'is_extended_real')
_eval_is_extended_positive = lambda self: self._eval_template_is_attr(
'is_extended_positive')
_eval_is_extended_negative = lambda self: self._eval_template_is_attr(
'is_extended_negative')
_eval_is_extended_nonzero = lambda self: self._eval_template_is_attr(
'is_extended_nonzero')
_eval_is_extended_nonpositive = lambda self: self._eval_template_is_attr(
'is_extended_nonpositive')
_eval_is_extended_nonnegative = lambda self: self._eval_template_is_attr(
'is_extended_nonnegative')
_eval_is_real = lambda self: self._eval_template_is_attr('is_real')
_eval_is_zero = lambda self: self._eval_template_is_attr(
'is_zero')
@classmethod
def __eval_cond(cls, cond):
"""Return the truth value of the condition."""
if cond == True:
return True
if isinstance(cond, Eq):
try:
diff = cond.lhs - cond.rhs
if diff.is_commutative:
return diff.is_zero
except TypeError:
pass
def as_expr_set_pairs(self, domain=None):
"""Return tuples for each argument of self that give
the expression and the interval in which it is valid
which is contained within the given domain.
If a condition cannot be converted to a set, an error
will be raised. The variable of the conditions is
assumed to be real; sets of real values are returned.
Examples
========
>>> from sympy import Piecewise, Interval
>>> from sympy.abc import x
>>> p = Piecewise(
... (1, x < 2),
... (2,(x > 0) & (x < 4)),
... (3, True))
>>> p.as_expr_set_pairs()
[(1, Interval.open(-oo, 2)),
(2, Interval.Ropen(2, 4)),
(3, Interval(4, oo))]
>>> p.as_expr_set_pairs(Interval(0, 3))
[(1, Interval.Ropen(0, 2)),
(2, Interval(2, 3))]
"""
if domain is None:
domain = S.Reals
exp_sets = []
U = domain
complex = not domain.is_subset(S.Reals)
cond_free = set()
for expr, cond in self.args:
cond_free |= cond.free_symbols
if len(cond_free) > 1:
raise NotImplementedError(filldedent('''
multivariate conditions are not handled.'''))
if complex:
for i in cond.atoms(Relational):
if not isinstance(i, (Eq, Ne)):
raise ValueError(filldedent('''
Inequalities in the complex domain are
not supported. Try the real domain by
setting domain=S.Reals'''))
cond_int = U.intersect(cond.as_set())
U = U - cond_int
if cond_int != S.EmptySet:
exp_sets.append((expr, cond_int))
return exp_sets
def _eval_rewrite_as_ITE(self, *args, **kwargs):
byfree = {}
args = list(args)
default = any(c == True for b, c in args)
for i, (b, c) in enumerate(args):
if not isinstance(b, Boolean) and b != True:
raise TypeError(filldedent('''
Expecting Boolean or bool but got `%s`
''' % func_name(b)))
if c == True:
break
# loop over independent conditions for this b
for c in c.args if isinstance(c, Or) else [c]:
free = c.free_symbols
x = free.pop()
try:
byfree[x] = byfree.setdefault(
x, S.EmptySet).union(c.as_set())
except NotImplementedError:
if not default:
raise NotImplementedError(filldedent('''
A method to determine whether a multivariate
conditional is consistent with a complete coverage
of all variables has not been implemented so the
rewrite is being stopped after encountering `%s`.
This error would not occur if a default expression
like `(foo, True)` were given.
''' % c))
if byfree[x] in (S.UniversalSet, S.Reals):
# collapse the ith condition to True and break
args[i] = list(args[i])
c = args[i][1] = True
break
if c == True:
break
if c != True:
raise ValueError(filldedent('''
Conditions must cover all reals or a final default
condition `(foo, True)` must be given.
'''))
last, _ = args[i] # ignore all past ith arg
for a, c in reversed(args[:i]):
last = ITE(c, a, last)
return _canonical(last)
def _eval_rewrite_as_KroneckerDelta(self, *args):
from sympy.functions.special.tensor_functions import KroneckerDelta
rules = {
And: [False, False],
Or: [True, True],
Not: [True, False],
Eq: [None, None],
Ne: [None, None]
}
class UnrecognizedCondition(Exception):
pass
def rewrite(cond):
if isinstance(cond, Eq):
return KroneckerDelta(*cond.args)
if isinstance(cond, Ne):
return 1 - KroneckerDelta(*cond.args)
cls, args = type(cond), cond.args
if cls not in rules:
raise UnrecognizedCondition(cls)
b1, b2 = rules[cls]
k = Mul(*[1 - rewrite(c) for c in args]) if b1 else Mul(*[rewrite(c) for c in args])
if b2:
return 1 - k
return k
conditions = []
true_value = None
for value, cond in args:
if type(cond) in rules:
conditions.append((value, cond))
elif cond is S.true:
if true_value is None:
true_value = value
else:
return
if true_value is not None:
result = true_value
for value, cond in conditions[::-1]:
try:
k = rewrite(cond)
result = k * value + (1 - k) * result
except UnrecognizedCondition:
return
return result
def piecewise_fold(expr, evaluate=True):
"""
Takes an expression containing a piecewise function and returns the
expression in piecewise form. In addition, any ITE conditions are
rewritten in negation normal form and simplified.
The final Piecewise is evaluated (default) but if the raw form
is desired, send ``evaluate=False``; if trivial evaluation is
desired, send ``evaluate=None`` and duplicate conditions and
processing of True and False will be handled.
Examples
========
>>> from sympy import Piecewise, piecewise_fold, S
>>> from sympy.abc import x
>>> p = Piecewise((x, x < 1), (1, S(1) <= x))
>>> piecewise_fold(x*p)
Piecewise((x**2, x < 1), (x, True))
See Also
========
Piecewise
piecewise_exclusive
"""
if not isinstance(expr, Basic) or not expr.has(Piecewise):
return expr
new_args = []
if isinstance(expr, (ExprCondPair, Piecewise)):
for e, c in expr.args:
if not isinstance(e, Piecewise):
e = piecewise_fold(e)
# we don't keep Piecewise in condition because
# it has to be checked to see that it's complete
# and we convert it to ITE at that time
assert not c.has(Piecewise) # pragma: no cover
if isinstance(c, ITE):
c = c.to_nnf()
c = simplify_logic(c, form='cnf')
if isinstance(e, Piecewise):
new_args.extend([(piecewise_fold(ei), And(ci, c))
for ei, ci in e.args])
else:
new_args.append((e, c))
else:
# Given
# P1 = Piecewise((e11, c1), (e12, c2), A)
# P2 = Piecewise((e21, c1), (e22, c2), B)
# ...
# the folding of f(P1, P2) is trivially
# Piecewise(
# (f(e11, e21), c1),
# (f(e12, e22), c2),
# (f(Piecewise(A), Piecewise(B)), True))
# Certain objects end up rewriting themselves as thus, so
# we do that grouping before the more generic folding.
# The following applies this idea when f = Add or f = Mul
# (and the expression is commutative).
if expr.is_Add or expr.is_Mul and expr.is_commutative:
p, args = sift(expr.args, lambda x: x.is_Piecewise, binary=True)
pc = sift(p, lambda x: tuple([c for e,c in x.args]))
for c in list(ordered(pc)):
if len(pc[c]) > 1:
pargs = [list(i.args) for i in pc[c]]
# the first one is the same; there may be more
com = common_prefix(*[
[i.cond for i in j] for j in pargs])
n = len(com)
collected = []
for i in range(n):
collected.append((
expr.func(*[ai[i].expr for ai in pargs]),
com[i]))
remains = []
for a in pargs:
if n == len(a): # no more args
continue
if a[n].cond == True: # no longer Piecewise
remains.append(a[n].expr)
else: # restore the remaining Piecewise
remains.append(
Piecewise(*a[n:], evaluate=False))
if remains:
collected.append((expr.func(*remains), True))
args.append(Piecewise(*collected, evaluate=False))
continue
args.extend(pc[c])
else:
args = expr.args
# fold
folded = list(map(piecewise_fold, args))
for ec in product(*[
(i.args if isinstance(i, Piecewise) else
[(i, true)]) for i in folded]):
e, c = zip(*ec)
new_args.append((expr.func(*e), And(*c)))
if evaluate is None:
# don't return duplicate conditions, otherwise don't evaluate
new_args = list(reversed([(e, c) for c, e in {
c: e for e, c in reversed(new_args)}.items()]))
rv = Piecewise(*new_args, evaluate=evaluate)
if evaluate is None and len(rv.args) == 1 and rv.args[0].cond == True:
return rv.args[0].expr
return rv
def _clip(A, B, k):
"""Return interval B as intervals that are covered by A (keyed
to k) and all other intervals of B not covered by A keyed to -1.
The reference point of each interval is the rhs; if the lhs is
greater than the rhs then an interval of zero width interval will
result, e.g. (4, 1) is treated like (1, 1).
Examples
========
>>> from sympy.functions.elementary.piecewise import _clip
>>> from sympy import Tuple
>>> A = Tuple(1, 3)
>>> B = Tuple(2, 4)
>>> _clip(A, B, 0)
[(2, 3, 0), (3, 4, -1)]
Interpretation: interval portion (2, 3) of interval (2, 4) is
covered by interval (1, 3) and is keyed to 0 as requested;
interval (3, 4) was not covered by (1, 3) and is keyed to -1.
"""
a, b = B
c, d = A
c, d = Min(Max(c, a), b), Min(Max(d, a), b)
a, b = Min(a, b), b
p = []
if a != c:
p.append((a, c, -1))
else:
pass
if c != d:
p.append((c, d, k))
else:
pass
if b != d:
if d == c and p and p[-1][-1] == -1:
p[-1] = p[-1][0], b, -1
else:
p.append((d, b, -1))
else:
pass
return p
def piecewise_simplify_arguments(expr, **kwargs):
from sympy.simplify.simplify import simplify
# simplify conditions
f1 = expr.args[0].cond.free_symbols
args = None
if len(f1) == 1 and not expr.atoms(Eq):
x = f1.pop()
# this won't return intervals involving Eq
# and it won't handle symbols treated as
# booleans
ok, abe_ = expr._intervals(x, err_on_Eq=True)
def include(c, x, a):
"return True if c.subs(x, a) is True, else False"
try:
return c.subs(x, a) == True
except TypeError:
return False
if ok:
args = []
covered = S.EmptySet
from sympy.sets.sets import Interval
for a, b, e, i in abe_:
c = expr.args[i].cond
incl_a = include(c, x, a)
incl_b = include(c, x, b)
iv = Interval(a, b, not incl_a, not incl_b)
cset = iv - covered
if not cset:
continue
if incl_a and incl_b:
if a.is_infinite and b.is_infinite:
c = S.true
elif b.is_infinite:
c = (x >= a)
elif a in covered or a.is_infinite:
c = (x <= b)
else:
c = And(a <= x, x <= b)
elif incl_a:
if a in covered or a.is_infinite:
c = (x < b)
else:
c = And(a <= x, x < b)
elif incl_b:
if b.is_infinite:
c = (x > a)
else:
c = (x <= b)
else:
if a in covered:
c = (x < b)
else:
c = And(a < x, x < b)
covered |= iv
if a is S.NegativeInfinity and incl_a:
covered |= {S.NegativeInfinity}
if b is S.Infinity and incl_b:
covered |= {S.Infinity}
args.append((e, c))
if not S.Reals.is_subset(covered):
args.append((Undefined, True))
if args is None:
args = list(expr.args)
for i in range(len(args)):
e, c = args[i]
if isinstance(c, Basic):
c = simplify(c, **kwargs)
args[i] = (e, c)
# simplify expressions
doit = kwargs.pop('doit', None)
for i in range(len(args)):
e, c = args[i]
if isinstance(e, Basic):
# Skip doit to avoid growth at every call for some integrals
# and sums, see sympy/sympy#17165
newe = simplify(e, doit=False, **kwargs)
if newe != e:
e = newe
args[i] = (e, c)
# restore kwargs flag
if doit is not None:
kwargs['doit'] = doit
return Piecewise(*args)
def piecewise_simplify(expr, **kwargs):
expr = piecewise_simplify_arguments(expr, **kwargs)
if not isinstance(expr, Piecewise):
return expr
args = list(expr.args)
_blessed = lambda e: getattr(e.lhs, '_diff_wrt', False) and (
getattr(e.rhs, '_diff_wrt', None) or
isinstance(e.rhs, (Rational, NumberSymbol)))
for i, (expr, cond) in enumerate(args):
# try to simplify conditions and the expression for
# equalities that are part of the condition, e.g.
# Piecewise((n, And(Eq(n,0), Eq(n + m, 0))), (1, True))
# -> Piecewise((0, And(Eq(n, 0), Eq(m, 0))), (1, True))
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Eq), binary=True)
elif isinstance(cond, Eq):
eqs, other = [cond], []
else:
eqs = other = []
if eqs:
eqs = list(ordered(eqs))
for j, e in enumerate(eqs):
# these blessed lhs objects behave like Symbols
# and the rhs are simple replacements for the "symbols"
if _blessed(e):
expr = expr.subs(*e.args)
eqs[j + 1:] = [ei.subs(*e.args) for ei in eqs[j + 1:]]
other = [ei.subs(*e.args) for ei in other]
cond = And(*(eqs + other))
args[i] = args[i].func(expr, cond)
# See if expressions valid for an Equal expression happens to evaluate
# to the same function as in the next piecewise segment, see:
# https://github.com/sympy/sympy/issues/8458
prevexpr = None
for i, (expr, cond) in reversed(list(enumerate(args))):
if prevexpr is not None:
if isinstance(cond, And):
eqs, other = sift(cond.args,
lambda i: isinstance(i, Eq), binary=True)
elif isinstance(cond, Eq):
eqs, other = [cond], []
else:
eqs = other = []
_prevexpr = prevexpr
_expr = expr
if eqs and not other:
eqs = list(ordered(eqs))
for e in eqs:
# allow 2 args to collapse into 1 for any e
# otherwise limit simplification to only simple-arg
# Eq instances
if len(args) == 2 or _blessed(e):
_prevexpr = _prevexpr.subs(*e.args)
_expr = _expr.subs(*e.args)
# Did it evaluate to the same?
if _prevexpr == _expr:
# Set the expression for the Not equal section to the same
# as the next. These will be merged when creating the new
# Piecewise
args[i] = args[i].func(args[i+1][0], cond)
else:
# Update the expression that we compare against
prevexpr = expr
else:
prevexpr = expr
return Piecewise(*args)
def piecewise_exclusive(expr, *, skip_nan=False, deep=True):
"""
Rewrite :class:`Piecewise` with mutually exclusive conditions.
Explanation
===========
SymPy represents the conditions of a :class:`Piecewise` in an
"if-elif"-fashion, allowing more than one condition to be simultaneously
True. The interpretation is that the first condition that is True is the
case that holds. While this is a useful representation computationally it
is not how a piecewise formula is typically shown in a mathematical text.
The :func:`piecewise_exclusive` function can be used to rewrite any
:class:`Piecewise` with more typical mutually exclusive conditions.
Note that further manipulation of the resulting :class:`Piecewise`, e.g.
simplifying it, will most likely make it non-exclusive. Hence, this is
primarily a function to be used in conjunction with printing the Piecewise
or if one would like to reorder the expression-condition pairs.
If it is not possible to determine that all possibilities are covered by
the different cases of the :class:`Piecewise` then a final
:class:`~sympy.core.numbers.NaN` case will be included explicitly. This
can be prevented by passing ``skip_nan=True``.
Examples
========
>>> from sympy import piecewise_exclusive, Symbol, Piecewise, S
>>> x = Symbol('x', real=True)
>>> p = Piecewise((0, x < 0), (S.Half, x <= 0), (1, True))
>>> piecewise_exclusive(p)
Piecewise((0, x < 0), (1/2, Eq(x, 0)), (1, x > 0))
>>> piecewise_exclusive(Piecewise((2, x > 1)))
Piecewise((2, x > 1), (nan, x <= 1))
>>> piecewise_exclusive(Piecewise((2, x > 1)), skip_nan=True)
Piecewise((2, x > 1))
Parameters
==========
expr: a SymPy expression.
Any :class:`Piecewise` in the expression will be rewritten.
skip_nan: ``bool`` (default ``False``)
If ``skip_nan`` is set to ``True`` then a final
:class:`~sympy.core.numbers.NaN` case will not be included.
deep: ``bool`` (default ``True``)
If ``deep`` is ``True`` then :func:`piecewise_exclusive` will rewrite
any :class:`Piecewise` subexpressions in ``expr`` rather than just
rewriting ``expr`` itself.
Returns
=======
An expression equivalent to ``expr`` but where all :class:`Piecewise` have
been rewritten with mutually exclusive conditions.
See Also
========
Piecewise
piecewise_fold
"""
def make_exclusive(*pwargs):
cumcond = false
newargs = []
# Handle the first n-1 cases
for expr_i, cond_i in pwargs[:-1]:
cancond = And(cond_i, Not(cumcond)).simplify()
cumcond = Or(cond_i, cumcond).simplify()
newargs.append((expr_i, cancond))
# For the nth case defer simplification of cumcond
expr_n, cond_n = pwargs[-1]
cancond_n = And(cond_n, Not(cumcond)).simplify()
newargs.append((expr_n, cancond_n))
if not skip_nan:
cumcond = Or(cond_n, cumcond).simplify()
if cumcond is not true:
newargs.append((Undefined, Not(cumcond).simplify()))
return Piecewise(*newargs, evaluate=False)
if deep:
return expr.replace(Piecewise, make_exclusive)
elif isinstance(expr, Piecewise):
return make_exclusive(*expr.args)
else:
return expr
|
6d3974aa7f5c74fc740dd44418f32cc65708bd041199fefd8fcf893e61d52af1 | from typing import Tuple as tTuple
from sympy.core.basic import Basic
from sympy.core.expr import Expr
from sympy.core import Add, S
from sympy.core.evalf import get_integer_part, PrecisionExhausted
from sympy.core.function import Function
from sympy.core.logic import fuzzy_or
from sympy.core.numbers import Integer
from sympy.core.relational import Gt, Lt, Ge, Le, Relational, is_eq
from sympy.core.symbol import Symbol
from sympy.core.sympify import _sympify
from sympy.functions.elementary.complexes import im, re
from sympy.multipledispatch import dispatch
###############################################################################
######################### FLOOR and CEILING FUNCTIONS #########################
###############################################################################
class RoundFunction(Function):
"""Abstract base class for rounding functions."""
args: tTuple[Expr]
@classmethod
def eval(cls, arg):
v = cls._eval_number(arg)
if v is not None:
return v
if arg.is_integer or arg.is_finite is False:
return arg
if arg.is_imaginary or (S.ImaginaryUnit*arg).is_real:
i = im(arg)
if not i.has(S.ImaginaryUnit):
return cls(i)*S.ImaginaryUnit
return cls(arg, evaluate=False)
# Integral, numerical, symbolic part
ipart = npart = spart = S.Zero
# Extract integral (or complex integral) terms
terms = Add.make_args(arg)
for t in terms:
if t.is_integer or (t.is_imaginary and im(t).is_integer):
ipart += t
elif t.has(Symbol):
spart += t
else:
npart += t
if not (npart or spart):
return ipart
# Evaluate npart numerically if independent of spart
if npart and (
not spart or
npart.is_real and (spart.is_imaginary or (S.ImaginaryUnit*spart).is_real) or
npart.is_imaginary and spart.is_real):
try:
r, i = get_integer_part(
npart, cls._dir, {}, return_ints=True)
ipart += Integer(r) + Integer(i)*S.ImaginaryUnit
npart = S.Zero
except (PrecisionExhausted, NotImplementedError):
pass
spart += npart
if not spart:
return ipart
elif spart.is_imaginary or (S.ImaginaryUnit*spart).is_real:
return ipart + cls(im(spart), evaluate=False)*S.ImaginaryUnit
elif isinstance(spart, (floor, ceiling)):
return ipart + spart
else:
return ipart + cls(spart, evaluate=False)
@classmethod
def _eval_number(cls, arg):
raise NotImplementedError()
def _eval_is_finite(self):
return self.args[0].is_finite
def _eval_is_real(self):
return self.args[0].is_real
def _eval_is_integer(self):
return self.args[0].is_real
class floor(RoundFunction):
"""
Floor is a univariate function which returns the largest integer
value not greater than its argument. This implementation
generalizes floor to complex numbers by taking the floor of the
real and imaginary parts separately.
Examples
========
>>> from sympy import floor, E, I, S, Float, Rational
>>> floor(17)
17
>>> floor(Rational(23, 10))
2
>>> floor(2*E)
5
>>> floor(-Float(0.567))
-1
>>> floor(-I/2)
-I
>>> floor(S(5)/2 + 5*I/2)
2 + 2*I
See Also
========
sympy.functions.elementary.integers.ceiling
References
==========
.. [1] "Concrete mathematics" by Graham, pp. 87
.. [2] http://mathworld.wolfram.com/FloorFunction.html
"""
_dir = -1
@classmethod
def _eval_number(cls, arg):
if arg.is_Number:
return arg.floor()
elif any(isinstance(i, j)
for i in (arg, -arg) for j in (floor, ceiling)):
return arg
if arg.is_NumberSymbol:
return arg.approximation_interval(Integer)[0]
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
arg = self.args[0]
arg0 = arg.subs(x, 0)
r = self.subs(x, 0)
if arg0 is S.NaN or isinstance(arg0, AccumBounds):
arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
r = floor(arg0)
if arg0.is_finite:
if arg0 == r:
if cdir == 0:
ndirl = arg.dir(x, cdir=-1)
ndir = arg.dir(x, cdir=1)
if ndir != ndirl:
raise ValueError("Two sided limit of %s around 0"
"does not exist" % self)
else:
ndir = arg.dir(x, cdir=cdir)
return r - 1 if ndir.is_negative else r
else:
return r
return arg.as_leading_term(x, logx=logx, cdir=cdir)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
arg0 = arg.subs(x, 0)
r = self.subs(x, 0)
if arg0 is S.NaN:
arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
r = floor(arg0)
if arg0.is_infinite:
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.series.order import Order
s = arg._eval_nseries(x, n, logx, cdir)
o = Order(1, (x, 0)) if n <= 0 else AccumBounds(-1, 0)
return s + o
if arg0 == r:
ndir = arg.dir(x, cdir=cdir if cdir != 0 else 1)
return r - 1 if ndir.is_negative else r
else:
return r
def _eval_is_negative(self):
return self.args[0].is_negative
def _eval_is_nonnegative(self):
return self.args[0].is_nonnegative
def _eval_rewrite_as_ceiling(self, arg, **kwargs):
return -ceiling(-arg)
def _eval_rewrite_as_frac(self, arg, **kwargs):
return arg - frac(arg)
def __le__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] < other + 1
if other.is_number and other.is_real:
return self.args[0] < ceiling(other)
if self.args[0] == other and other.is_real:
return S.true
if other is S.Infinity and self.is_finite:
return S.true
return Le(self, other, evaluate=False)
def __ge__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] >= other
if other.is_number and other.is_real:
return self.args[0] >= ceiling(other)
if self.args[0] == other and other.is_real:
return S.false
if other is S.NegativeInfinity and self.is_finite:
return S.true
return Ge(self, other, evaluate=False)
def __gt__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] >= other + 1
if other.is_number and other.is_real:
return self.args[0] >= ceiling(other)
if self.args[0] == other and other.is_real:
return S.false
if other is S.NegativeInfinity and self.is_finite:
return S.true
return Gt(self, other, evaluate=False)
def __lt__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] < other
if other.is_number and other.is_real:
return self.args[0] < ceiling(other)
if self.args[0] == other and other.is_real:
return S.false
if other is S.Infinity and self.is_finite:
return S.true
return Lt(self, other, evaluate=False)
@dispatch(floor, Expr)
def _eval_is_eq(lhs, rhs): # noqa:F811
return is_eq(lhs.rewrite(ceiling), rhs) or \
is_eq(lhs.rewrite(frac),rhs)
class ceiling(RoundFunction):
"""
Ceiling is a univariate function which returns the smallest integer
value not less than its argument. This implementation
generalizes ceiling to complex numbers by taking the ceiling of the
real and imaginary parts separately.
Examples
========
>>> from sympy import ceiling, E, I, S, Float, Rational
>>> ceiling(17)
17
>>> ceiling(Rational(23, 10))
3
>>> ceiling(2*E)
6
>>> ceiling(-Float(0.567))
0
>>> ceiling(I/2)
I
>>> ceiling(S(5)/2 + 5*I/2)
3 + 3*I
See Also
========
sympy.functions.elementary.integers.floor
References
==========
.. [1] "Concrete mathematics" by Graham, pp. 87
.. [2] http://mathworld.wolfram.com/CeilingFunction.html
"""
_dir = 1
@classmethod
def _eval_number(cls, arg):
if arg.is_Number:
return arg.ceiling()
elif any(isinstance(i, j)
for i in (arg, -arg) for j in (floor, ceiling)):
return arg
if arg.is_NumberSymbol:
return arg.approximation_interval(Integer)[1]
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.accumulationbounds import AccumBounds
arg = self.args[0]
arg0 = arg.subs(x, 0)
r = self.subs(x, 0)
if arg0 is S.NaN or isinstance(arg0, AccumBounds):
arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
r = ceiling(arg0)
if arg0.is_finite:
if arg0 == r:
if cdir == 0:
ndirl = arg.dir(x, cdir=-1)
ndir = arg.dir(x, cdir=1)
if ndir != ndirl:
raise ValueError("Two sided limit of %s around 0"
"does not exist" % self)
else:
ndir = arg.dir(x, cdir=cdir)
return r if ndir.is_negative else r + 1
else:
return r
return arg.as_leading_term(x, logx=logx, cdir=cdir)
def _eval_nseries(self, x, n, logx, cdir=0):
arg = self.args[0]
arg0 = arg.subs(x, 0)
r = self.subs(x, 0)
if arg0 is S.NaN:
arg0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
r = ceiling(arg0)
if arg0.is_infinite:
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.series.order import Order
s = arg._eval_nseries(x, n, logx, cdir)
o = Order(1, (x, 0)) if n <= 0 else AccumBounds(0, 1)
return s + o
if arg0 == r:
ndir = arg.dir(x, cdir=cdir if cdir != 0 else 1)
return r if ndir.is_negative else r + 1
else:
return r
def _eval_rewrite_as_floor(self, arg, **kwargs):
return -floor(-arg)
def _eval_rewrite_as_frac(self, arg, **kwargs):
return arg + frac(-arg)
def _eval_is_positive(self):
return self.args[0].is_positive
def _eval_is_nonpositive(self):
return self.args[0].is_nonpositive
def __lt__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] <= other - 1
if other.is_number and other.is_real:
return self.args[0] <= floor(other)
if self.args[0] == other and other.is_real:
return S.false
if other is S.Infinity and self.is_finite:
return S.true
return Lt(self, other, evaluate=False)
def __gt__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] > other
if other.is_number and other.is_real:
return self.args[0] > floor(other)
if self.args[0] == other and other.is_real:
return S.false
if other is S.NegativeInfinity and self.is_finite:
return S.true
return Gt(self, other, evaluate=False)
def __ge__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] > other - 1
if other.is_number and other.is_real:
return self.args[0] > floor(other)
if self.args[0] == other and other.is_real:
return S.true
if other is S.NegativeInfinity and self.is_finite:
return S.true
return Ge(self, other, evaluate=False)
def __le__(self, other):
other = S(other)
if self.args[0].is_real:
if other.is_integer:
return self.args[0] <= other
if other.is_number and other.is_real:
return self.args[0] <= floor(other)
if self.args[0] == other and other.is_real:
return S.false
if other is S.Infinity and self.is_finite:
return S.true
return Le(self, other, evaluate=False)
@dispatch(ceiling, Basic) # type:ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
return is_eq(lhs.rewrite(floor), rhs) or is_eq(lhs.rewrite(frac),rhs)
class frac(Function):
r"""Represents the fractional part of x
For real numbers it is defined [1]_ as
.. math::
x - \left\lfloor{x}\right\rfloor
Examples
========
>>> from sympy import Symbol, frac, Rational, floor, I
>>> frac(Rational(4, 3))
1/3
>>> frac(-Rational(4, 3))
2/3
returns zero for integer arguments
>>> n = Symbol('n', integer=True)
>>> frac(n)
0
rewrite as floor
>>> x = Symbol('x')
>>> frac(x).rewrite(floor)
x - floor(x)
for complex arguments
>>> r = Symbol('r', real=True)
>>> t = Symbol('t', real=True)
>>> frac(t + I*r)
I*frac(r) + frac(t)
See Also
========
sympy.functions.elementary.integers.floor
sympy.functions.elementary.integers.ceiling
References
===========
.. [1] https://en.wikipedia.org/wiki/Fractional_part
.. [2] http://mathworld.wolfram.com/FractionalPart.html
"""
@classmethod
def eval(cls, arg):
from sympy.calculus.accumulationbounds import AccumBounds
def _eval(arg):
if arg in (S.Infinity, S.NegativeInfinity):
return AccumBounds(0, 1)
if arg.is_integer:
return S.Zero
if arg.is_number:
if arg is S.NaN:
return S.NaN
elif arg is S.ComplexInfinity:
return S.NaN
else:
return arg - floor(arg)
return cls(arg, evaluate=False)
terms = Add.make_args(arg)
real, imag = S.Zero, S.Zero
for t in terms:
# Two checks are needed for complex arguments
# see issue-7649 for details
if t.is_imaginary or (S.ImaginaryUnit*t).is_real:
i = im(t)
if not i.has(S.ImaginaryUnit):
imag += i
else:
real += t
else:
real += t
real = _eval(real)
imag = _eval(imag)
return real + S.ImaginaryUnit*imag
def _eval_rewrite_as_floor(self, arg, **kwargs):
return arg - floor(arg)
def _eval_rewrite_as_ceiling(self, arg, **kwargs):
return arg + ceiling(-arg)
def _eval_is_finite(self):
return True
def _eval_is_real(self):
return self.args[0].is_extended_real
def _eval_is_imaginary(self):
return self.args[0].is_imaginary
def _eval_is_integer(self):
return self.args[0].is_integer
def _eval_is_zero(self):
return fuzzy_or([self.args[0].is_zero, self.args[0].is_integer])
def _eval_is_negative(self):
return False
def __ge__(self, other):
if self.is_extended_real:
other = _sympify(other)
# Check if other <= 0
if other.is_extended_nonpositive:
return S.true
# Check if other >= 1
res = self._value_one_or_more(other)
if res is not None:
return not(res)
return Ge(self, other, evaluate=False)
def __gt__(self, other):
if self.is_extended_real:
other = _sympify(other)
# Check if other < 0
res = self._value_one_or_more(other)
if res is not None:
return not(res)
# Check if other >= 1
if other.is_extended_negative:
return S.true
return Gt(self, other, evaluate=False)
def __le__(self, other):
if self.is_extended_real:
other = _sympify(other)
# Check if other < 0
if other.is_extended_negative:
return S.false
# Check if other >= 1
res = self._value_one_or_more(other)
if res is not None:
return res
return Le(self, other, evaluate=False)
def __lt__(self, other):
if self.is_extended_real:
other = _sympify(other)
# Check if other <= 0
if other.is_extended_nonpositive:
return S.false
# Check if other >= 1
res = self._value_one_or_more(other)
if res is not None:
return res
return Lt(self, other, evaluate=False)
def _value_one_or_more(self, other):
if other.is_extended_real:
if other.is_number:
res = other >= 1
if res and not isinstance(res, Relational):
return S.true
if other.is_integer and other.is_positive:
return S.true
@dispatch(frac, Basic) # type:ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
if (lhs.rewrite(floor) == rhs) or \
(lhs.rewrite(ceiling) == rhs):
return True
# Check if other < 0
if rhs.is_extended_negative:
return False
# Check if other >= 1
res = lhs._value_one_or_more(rhs)
if res is not None:
return False
|
618f900495333101ac891fdf9350762fcc69f430e6fe15cf0a9fd63869cab3f7 | from itertools import product
from typing import Tuple as tTuple
from sympy.core.expr import Expr
from sympy.core import sympify
from sympy.core.add import Add
from sympy.core.cache import cacheit
from sympy.core.function import (Function, ArgumentIndexError, expand_log,
expand_mul, FunctionClass, PoleError, expand_multinomial, expand_complex)
from sympy.core.logic import fuzzy_and, fuzzy_not, fuzzy_or
from sympy.core.mul import Mul
from sympy.core.numbers import Integer, Rational, pi, I, ImaginaryUnit
from sympy.core.parameters import global_parameters
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import Wild, Dummy
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.complexes import arg, unpolarify, im, re, Abs
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.ntheory import multiplicity, perfect_power
from sympy.ntheory.factor_ import factorint
# NOTE IMPORTANT
# The series expansion code in this file is an important part of the gruntz
# algorithm for determining limits. _eval_nseries has to return a generalized
# power series with coefficients in C(log(x), log).
# In more detail, the result of _eval_nseries(self, x, n) must be
# c_0*x**e_0 + ... (finitely many terms)
# where e_i are numbers (not necessarily integers) and c_i involve only
# numbers, the function log, and log(x). [This also means it must not contain
# log(x(1+p)), this *has* to be expanded to log(x)+log(1+p) if x.is_positive and
# p.is_positive.]
class ExpBase(Function):
unbranched = True
_singularities = (S.ComplexInfinity,)
@property
def kind(self):
return self.exp.kind
def inverse(self, argindex=1):
"""
Returns the inverse function of ``exp(x)``.
"""
return log
def as_numer_denom(self):
"""
Returns this with a positive exponent as a 2-tuple (a fraction).
Examples
========
>>> from sympy import exp
>>> from sympy.abc import x
>>> exp(-x).as_numer_denom()
(1, exp(x))
>>> exp(x).as_numer_denom()
(exp(x), 1)
"""
# this should be the same as Pow.as_numer_denom wrt
# exponent handling
exp = self.exp
neg_exp = exp.is_negative
if not neg_exp and not (-exp).is_negative:
neg_exp = exp.could_extract_minus_sign()
if neg_exp:
return S.One, self.func(-exp)
return self, S.One
@property
def exp(self):
"""
Returns the exponent of the function.
"""
return self.args[0]
def as_base_exp(self):
"""
Returns the 2-tuple (base, exponent).
"""
return self.func(1), Mul(*self.args)
def _eval_adjoint(self):
return self.func(self.exp.adjoint())
def _eval_conjugate(self):
return self.func(self.exp.conjugate())
def _eval_transpose(self):
return self.func(self.exp.transpose())
def _eval_is_finite(self):
arg = self.exp
if arg.is_infinite:
if arg.is_extended_negative:
return True
if arg.is_extended_positive:
return False
if arg.is_finite:
return True
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
z = s.exp.is_zero
if z:
return True
elif s.exp.is_rational and fuzzy_not(z):
return False
else:
return s.is_rational
def _eval_is_zero(self):
return self.exp is S.NegativeInfinity
def _eval_power(self, other):
"""exp(arg)**e -> exp(arg*e) if assumptions allow it.
"""
b, e = self.as_base_exp()
return Pow._eval_power(Pow(b, e, evaluate=False), other)
def _eval_expand_power_exp(self, **hints):
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
arg = self.args[0]
if arg.is_Add and arg.is_commutative:
return Mul.fromiter(self.func(x) for x in arg.args)
elif isinstance(arg, Sum) and arg.is_commutative:
return Product(self.func(arg.function), *arg.limits)
return self.func(arg)
class exp_polar(ExpBase):
r"""
Represent a *polar number* (see g-function Sphinx documentation).
Explanation
===========
``exp_polar`` represents the function
`Exp: \mathbb{C} \rightarrow \mathcal{S}`, sending the complex number
`z = a + bi` to the polar number `r = exp(a), \theta = b`. It is one of
the main functions to construct polar numbers.
Examples
========
>>> from sympy import exp_polar, pi, I, exp
The main difference is that polar numbers do not "wrap around" at `2 \pi`:
>>> exp(2*pi*I)
1
>>> exp_polar(2*pi*I)
exp_polar(2*I*pi)
apart from that they behave mostly like classical complex numbers:
>>> exp_polar(2)*exp_polar(3)
exp_polar(5)
See Also
========
sympy.simplify.powsimp.powsimp
polar_lift
periodic_argument
principal_branch
"""
is_polar = True
is_comparable = False # cannot be evalf'd
def _eval_Abs(self): # Abs is never a polar number
return exp(re(self.args[0]))
def _eval_evalf(self, prec):
""" Careful! any evalf of polar numbers is flaky """
i = im(self.args[0])
try:
bad = (i <= -pi or i > pi)
except TypeError:
bad = True
if bad:
return self # cannot evalf for this argument
res = exp(self.args[0])._eval_evalf(prec)
if i > 0 and im(res) < 0:
# i ~ pi, but exp(I*i) evaluated to argument slightly bigger than pi
return re(res)
return res
def _eval_power(self, other):
return self.func(self.args[0]*other)
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
def as_base_exp(self):
# XXX exp_polar(0) is special!
if self.args[0] == 0:
return self, S.One
return ExpBase.as_base_exp(self)
class ExpMeta(FunctionClass):
def __instancecheck__(cls, instance):
if exp in instance.__class__.__mro__:
return True
return isinstance(instance, Pow) and instance.base is S.Exp1
class exp(ExpBase, metaclass=ExpMeta):
"""
The exponential function, :math:`e^x`.
Examples
========
>>> from sympy import exp, I, pi
>>> from sympy.abc import x
>>> exp(x)
exp(x)
>>> exp(x).diff(x)
exp(x)
>>> exp(I*pi)
-1
Parameters
==========
arg : Expr
See Also
========
log
"""
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return self
else:
raise ArgumentIndexError(self, argindex)
def _eval_refine(self, assumptions):
from sympy.assumptions import ask, Q
arg = self.args[0]
if arg.is_Mul:
Ioo = S.ImaginaryUnit*S.Infinity
if arg in [Ioo, -Ioo]:
return S.NaN
coeff = arg.as_coefficient(S.Pi*S.ImaginaryUnit)
if coeff:
if ask(Q.integer(2*coeff)):
if ask(Q.even(coeff)):
return S.One
elif ask(Q.odd(coeff)):
return S.NegativeOne
elif ask(Q.even(coeff + S.Half)):
return -S.ImaginaryUnit
elif ask(Q.odd(coeff + S.Half)):
return S.ImaginaryUnit
@classmethod
def eval(cls, arg):
from sympy.calculus import AccumBounds
from sympy.matrices.matrices import MatrixBase
from sympy.sets.setexpr import SetExpr
from sympy.simplify.simplify import logcombine
if isinstance(arg, MatrixBase):
return arg.exp()
elif global_parameters.exp_is_pow:
return Pow(S.Exp1, arg)
elif arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.One
elif arg is S.One:
return S.Exp1
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Zero
elif arg is S.ComplexInfinity:
return S.NaN
elif isinstance(arg, log):
return arg.args[0]
elif isinstance(arg, AccumBounds):
return AccumBounds(exp(arg.min), exp(arg.max))
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
elif arg.is_Mul:
coeff = arg.as_coefficient(S.Pi*S.ImaginaryUnit)
if coeff:
if (2*coeff).is_integer:
if coeff.is_even:
return S.One
elif coeff.is_odd:
return S.NegativeOne
elif (coeff + S.Half).is_even:
return -S.ImaginaryUnit
elif (coeff + S.Half).is_odd:
return S.ImaginaryUnit
elif coeff.is_Rational:
ncoeff = coeff % 2 # restrict to [0, 2pi)
if ncoeff > 1: # restrict to (-pi, pi]
ncoeff -= 2
if ncoeff != coeff:
return cls(ncoeff*S.Pi*S.ImaginaryUnit)
# Warning: code in risch.py will be very sensitive to changes
# in this (see DifferentialExtension).
# look for a single log factor
coeff, terms = arg.as_coeff_Mul()
# but it can't be multiplied by oo
if coeff in [S.NegativeInfinity, S.Infinity]:
if terms.is_number:
if coeff is S.NegativeInfinity:
terms = -terms
if re(terms).is_zero and terms is not S.Zero:
return S.NaN
if re(terms).is_positive and im(terms) is not S.Zero:
return S.ComplexInfinity
if re(terms).is_negative:
return S.Zero
return None
coeffs, log_term = [coeff], None
for term in Mul.make_args(terms):
term_ = logcombine(term)
if isinstance(term_, log):
if log_term is None:
log_term = term_.args[0]
else:
return None
elif term.is_comparable:
coeffs.append(term)
else:
return None
return log_term**Mul(*coeffs) if log_term else None
elif arg.is_Add:
out = []
add = []
argchanged = False
for a in arg.args:
if a is S.One:
add.append(a)
continue
newa = cls(a)
if isinstance(newa, cls):
if newa.args[0] != a:
add.append(newa.args[0])
argchanged = True
else:
add.append(a)
else:
out.append(newa)
if out or argchanged:
return Mul(*out)*cls(Add(*add), evaluate=False)
if arg.is_zero:
return S.One
@property
def base(self):
"""
Returns the base of the exponential function.
"""
return S.Exp1
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
"""
Calculates the next term in the Taylor series expansion.
"""
if n < 0:
return S.Zero
if n == 0:
return S.One
x = sympify(x)
if previous_terms:
p = previous_terms[-1]
if p is not None:
return p * x / n
return x**n/factorial(n)
def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a 2-tuple representing a complex number.
Examples
========
>>> from sympy import exp, I
>>> from sympy.abc import x
>>> exp(x).as_real_imag()
(exp(re(x))*cos(im(x)), exp(re(x))*sin(im(x)))
>>> exp(1).as_real_imag()
(E, 0)
>>> exp(I).as_real_imag()
(cos(1), sin(1))
>>> exp(1+I).as_real_imag()
(E*cos(1), E*sin(1))
See Also
========
sympy.functions.elementary.complexes.re
sympy.functions.elementary.complexes.im
"""
from sympy.functions.elementary.trigonometric import cos, sin
re, im = self.args[0].as_real_imag()
if deep:
re = re.expand(deep, **hints)
im = im.expand(deep, **hints)
cos, sin = cos(im), sin(im)
return (exp(re)*cos, exp(re)*sin)
def _eval_subs(self, old, new):
# keep processing of power-like args centralized in Pow
if old.is_Pow: # handle (exp(3*log(x))).subs(x**2, z) -> z**(3/2)
old = exp(old.exp*log(old.base))
elif old is S.Exp1 and new.is_Function:
old = exp
if isinstance(old, exp) or old is S.Exp1:
f = lambda a: Pow(*a.as_base_exp(), evaluate=False) if (
a.is_Pow or isinstance(a, exp)) else a
return Pow._eval_subs(f(self), f(old), new)
if old is exp and not new.is_Function:
return new**self.exp._subs(old, new)
return Function._eval_subs(self, old, new)
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
elif self.args[0].is_imaginary:
arg2 = -S(2) * S.ImaginaryUnit * self.args[0] / S.Pi
return arg2.is_even
def _eval_is_complex(self):
def complex_extended_negative(arg):
yield arg.is_complex
yield arg.is_extended_negative
return fuzzy_or(complex_extended_negative(self.args[0]))
def _eval_is_algebraic(self):
if (self.exp / S.Pi / S.ImaginaryUnit).is_rational:
return True
if fuzzy_not(self.exp.is_zero):
if self.exp.is_algebraic:
return False
elif (self.exp / S.Pi).is_rational:
return False
def _eval_is_extended_positive(self):
if self.exp.is_extended_real:
return self.args[0] is not S.NegativeInfinity
elif self.exp.is_imaginary:
arg2 = -S.ImaginaryUnit * self.args[0] / S.Pi
return arg2.is_even
def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.integers import ceiling
from sympy.series.limits import limit
from sympy.series.order import Order
from sympy.simplify.powsimp import powsimp
arg = self.exp
arg_series = arg._eval_nseries(x, n=n, logx=logx)
if arg_series.is_Order:
return 1 + arg_series
arg0 = limit(arg_series.removeO(), x, 0)
if arg0 is S.NegativeInfinity:
return Order(x**n, x)
if arg0 is S.Infinity:
return self
# checking for indecisiveness/ sign terms in arg0
if any(isinstance(arg, (sign, ImaginaryUnit)) for arg in arg0.args):
return self
t = Dummy("t")
nterms = n
try:
cf = Order(arg.as_leading_term(x, logx=logx), x).getn()
except (NotImplementedError, PoleError):
cf = 0
if cf and cf > 0:
nterms = ceiling(n/cf)
exp_series = exp(t)._taylor(t, nterms)
r = exp(arg0)*exp_series.subs(t, arg_series - arg0)
rep = {logx: log(x)} if logx is not None else {}
if r.subs(rep) == self:
return r
if cf and cf > 1:
r += Order((arg_series - arg0)**n, x)/x**((cf-1)*n)
else:
r += Order((arg_series - arg0)**n, x)
r = r.expand()
r = powsimp(r, deep=True, combine='exp')
# powsimp may introduce unexpanded (-1)**Rational; see PR #17201
simplerat = lambda x: x.is_Rational and x.q in [3, 4, 6]
w = Wild('w', properties=[simplerat])
r = r.replace(S.NegativeOne**w, expand_complex(S.NegativeOne**w))
return r
def _taylor(self, x, n):
l = []
g = None
for i in range(n):
g = self.taylor_term(i, self.args[0], g)
g = g.nseries(x, n=n)
l.append(g.removeO())
return Add(*l)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.calculus.util import AccumBounds
arg = self.args[0].cancel().as_leading_term(x, logx=logx)
arg0 = arg.subs(x, 0)
if arg is S.NaN:
return S.NaN
if isinstance(arg0, AccumBounds):
# This check addresses a corner case involving AccumBounds.
# if isinstance(arg, AccumBounds) is True, then arg0 can either be 0,
# AccumBounds(-oo, 0) or AccumBounds(-oo, oo).
# Check out function: test_issue_18473() in test_exponential.py and
# test_limits.py for more information.
if re(cdir) < S.Zero:
return exp(-arg0)
return exp(arg0)
if arg0 is S.NaN:
arg0 = arg.limit(x, 0)
if arg0.is_infinite is False:
return exp(arg0)
raise PoleError("Cannot expand %s around 0" % (self))
def _eval_rewrite_as_sin(self, arg, **kwargs):
from sympy.functions.elementary.trigonometric import sin
I = S.ImaginaryUnit
return sin(I*arg + S.Pi/2) - I*sin(I*arg)
def _eval_rewrite_as_cos(self, arg, **kwargs):
from sympy.functions.elementary.trigonometric import cos
I = S.ImaginaryUnit
return cos(I*arg) + I*cos(I*arg + S.Pi/2)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
from sympy.functions.elementary.hyperbolic import tanh
return (1 + tanh(arg/2))/(1 - tanh(arg/2))
def _eval_rewrite_as_sqrt(self, arg, **kwargs):
from sympy.functions.elementary.trigonometric import sin, cos
if arg.is_Mul:
coeff = arg.coeff(S.Pi*S.ImaginaryUnit)
if coeff and coeff.is_number:
cosine, sine = cos(S.Pi*coeff), sin(S.Pi*coeff)
if not isinstance(cosine, cos) and not isinstance (sine, sin):
return cosine + S.ImaginaryUnit*sine
def _eval_rewrite_as_Pow(self, arg, **kwargs):
if arg.is_Mul:
logs = [a for a in arg.args if isinstance(a, log) and len(a.args) == 1]
if logs:
return Pow(logs[0].args[0], arg.coeff(logs[0]))
def match_real_imag(expr):
r"""
Try to match expr with $a + Ib$ for real $a$ and $b$.
``match_real_imag`` returns a tuple containing the real and imaginary
parts of expr or ``(None, None)`` if direct matching is not possible. Contrary
to :func:`~.re()`, :func:`~.im()``, and ``as_real_imag()``, this helper will not force things
by returning expressions themselves containing ``re()`` or ``im()`` and it
does not expand its argument either.
"""
r_, i_ = expr.as_independent(S.ImaginaryUnit, as_Add=True)
if i_ == 0 and r_.is_real:
return (r_, i_)
i_ = i_.as_coefficient(S.ImaginaryUnit)
if i_ and i_.is_real and r_.is_real:
return (r_, i_)
else:
return (None, None) # simpler to check for than None
class log(Function):
r"""
The natural logarithm function `\ln(x)` or `\log(x)`.
Explanation
===========
Logarithms are taken with the natural base, `e`. To get
a logarithm of a different base ``b``, use ``log(x, b)``,
which is essentially short-hand for ``log(x)/log(b)``.
``log`` represents the principal branch of the natural
logarithm. As such it has a branch cut along the negative
real axis and returns values having a complex argument in
`(-\pi, \pi]`.
Examples
========
>>> from sympy import log, sqrt, S, I
>>> log(8, 2)
3
>>> log(S(8)/3, 2)
-log(3)/log(2) + 3
>>> log(-1 + I*sqrt(3))
log(2) + 2*I*pi/3
See Also
========
exp
"""
args: tTuple[Expr]
_singularities = (S.Zero, S.ComplexInfinity)
def fdiff(self, argindex=1):
"""
Returns the first derivative of the function.
"""
if argindex == 1:
return 1/self.args[0]
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
r"""
Returns `e^x`, the inverse function of `\log(x)`.
"""
return exp
@classmethod
def eval(cls, arg, base=None):
from sympy.calculus import AccumBounds
from sympy.sets.setexpr import SetExpr
arg = sympify(arg)
if base is not None:
base = sympify(base)
if base == 1:
if arg == 1:
return S.NaN
else:
return S.ComplexInfinity
try:
# handle extraction of powers of the base now
# or else expand_log in Mul would have to handle this
n = multiplicity(base, arg)
if n:
return n + log(arg / base**n) / log(base)
else:
return log(arg)/log(base)
except ValueError:
pass
if base is not S.Exp1:
return cls(arg)/cls(base)
else:
return cls(arg)
if arg.is_Number:
if arg.is_zero:
return S.ComplexInfinity
elif arg is S.One:
return S.Zero
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg is S.NaN:
return S.NaN
elif arg.is_Rational and arg.p == 1:
return -cls(arg.q)
if arg.is_Pow and arg.base is S.Exp1 and arg.exp.is_extended_real:
return arg.exp
I = S.ImaginaryUnit
if isinstance(arg, exp) and arg.exp.is_extended_real:
return arg.exp
elif isinstance(arg, exp) and arg.exp.is_number:
r_, i_ = match_real_imag(arg.exp)
if i_ and i_.is_comparable:
i_ %= 2*S.Pi
if i_ > S.Pi:
i_ -= 2*S.Pi
return r_ + expand_mul(i_ * I, deep=False)
elif isinstance(arg, exp_polar):
return unpolarify(arg.exp)
elif isinstance(arg, AccumBounds):
if arg.min.is_positive:
return AccumBounds(log(arg.min), log(arg.max))
elif arg.min.is_zero:
return AccumBounds(S.NegativeInfinity, log(arg.max))
else:
return S.NaN
elif isinstance(arg, SetExpr):
return arg._eval_func(cls)
if arg.is_number:
if arg.is_negative:
return S.Pi * I + cls(-arg)
elif arg is S.ComplexInfinity:
return S.ComplexInfinity
elif arg is S.Exp1:
return S.One
if arg.is_zero:
return S.ComplexInfinity
# don't autoexpand Pow or Mul (see the issue 3351):
if not arg.is_Add:
coeff = arg.as_coefficient(I)
if coeff is not None:
if coeff is S.Infinity:
return S.Infinity
elif coeff is S.NegativeInfinity:
return S.Infinity
elif coeff.is_Rational:
if coeff.is_nonnegative:
return S.Pi * I * S.Half + cls(coeff)
else:
return -S.Pi * I * S.Half + cls(-coeff)
if arg.is_number and arg.is_algebraic:
# Match arg = coeff*(r_ + i_*I) with coeff>0, r_ and i_ real.
coeff, arg_ = arg.as_independent(I, as_Add=False)
if coeff.is_negative:
coeff *= -1
arg_ *= -1
arg_ = expand_mul(arg_, deep=False)
r_, i_ = arg_.as_independent(I, as_Add=True)
i_ = i_.as_coefficient(I)
if coeff.is_real and i_ and i_.is_real and r_.is_real:
if r_.is_zero:
if i_.is_positive:
return S.Pi * I * S.Half + cls(coeff * i_)
elif i_.is_negative:
return -S.Pi * I * S.Half + cls(coeff * -i_)
else:
from sympy.simplify import ratsimp
# Check for arguments involving rational multiples of pi
t = (i_/r_).cancel()
t1 = (-t).cancel()
atan_table = {
# first quadrant only
sqrt(3): S.Pi/3,
1: S.Pi/4,
sqrt(5 - 2*sqrt(5)): S.Pi/5,
sqrt(2)*sqrt(5 - sqrt(5))/(1 + sqrt(5)): S.Pi/5,
sqrt(5 + 2*sqrt(5)): S.Pi*Rational(2, 5),
sqrt(2)*sqrt(sqrt(5) + 5)/(-1 + sqrt(5)): S.Pi*Rational(2, 5),
sqrt(3)/3: S.Pi/6,
sqrt(2) - 1: S.Pi/8,
sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2): S.Pi/8,
sqrt(2) + 1: S.Pi*Rational(3, 8),
sqrt(sqrt(2) + 2)/sqrt(2 - sqrt(2)): S.Pi*Rational(3, 8),
sqrt(1 - 2*sqrt(5)/5): S.Pi/10,
(-sqrt(2) + sqrt(10))/(2*sqrt(sqrt(5) + 5)): S.Pi/10,
sqrt(1 + 2*sqrt(5)/5): S.Pi*Rational(3, 10),
(sqrt(2) + sqrt(10))/(2*sqrt(5 - sqrt(5))): S.Pi*Rational(3, 10),
2 - sqrt(3): S.Pi/12,
(-1 + sqrt(3))/(1 + sqrt(3)): S.Pi/12,
2 + sqrt(3): S.Pi*Rational(5, 12),
(1 + sqrt(3))/(-1 + sqrt(3)): S.Pi*Rational(5, 12)
}
if t in atan_table:
modulus = ratsimp(coeff * Abs(arg_))
if r_.is_positive:
return cls(modulus) + I * atan_table[t]
else:
return cls(modulus) + I * (atan_table[t] - S.Pi)
elif t1 in atan_table:
modulus = ratsimp(coeff * Abs(arg_))
if r_.is_positive:
return cls(modulus) + I * (-atan_table[t1])
else:
return cls(modulus) + I * (S.Pi - atan_table[t1])
def as_base_exp(self):
"""
Returns this function in the form (base, exponent).
"""
return self, S.One
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms): # of log(1+x)
r"""
Returns the next term in the Taylor series expansion of `\log(1+x)`.
"""
from sympy.simplify.powsimp import powsimp
if n < 0:
return S.Zero
x = sympify(x)
if n == 0:
return x
if previous_terms:
p = previous_terms[-1]
if p is not None:
return powsimp((-n) * p * x / (n + 1), deep=True, combine='exp')
return (1 - 2*(n % 2)) * x**(n + 1)/(n + 1)
def _eval_expand_log(self, deep=True, **hints):
from sympy.concrete import Sum, Product
force = hints.get('force', False)
factor = hints.get('factor', False)
if (len(self.args) == 2):
return expand_log(self.func(*self.args), deep=deep, force=force)
arg = self.args[0]
if arg.is_Integer:
# remove perfect powers
p = perfect_power(arg)
logarg = None
coeff = 1
if p is not False:
arg, coeff = p
logarg = self.func(arg)
# expand as product of its prime factors if factor=True
if factor:
p = factorint(arg)
if arg not in p.keys():
logarg = sum(n*log(val) for val, n in p.items())
if logarg is not None:
return coeff*logarg
elif arg.is_Rational:
return log(arg.p) - log(arg.q)
elif arg.is_Mul:
expr = []
nonpos = []
for x in arg.args:
if force or x.is_positive or x.is_polar:
a = self.func(x)
if isinstance(a, log):
expr.append(self.func(x)._eval_expand_log(**hints))
else:
expr.append(a)
elif x.is_negative:
a = self.func(-x)
expr.append(a)
nonpos.append(S.NegativeOne)
else:
nonpos.append(x)
return Add(*expr) + log(Mul(*nonpos))
elif arg.is_Pow or isinstance(arg, exp):
if force or (arg.exp.is_extended_real and (arg.base.is_positive or ((arg.exp+1)
.is_positive and (arg.exp-1).is_nonpositive))) or arg.base.is_polar:
b = arg.base
e = arg.exp
a = self.func(b)
if isinstance(a, log):
return unpolarify(e) * a._eval_expand_log(**hints)
else:
return unpolarify(e) * a
elif isinstance(arg, Product):
if force or arg.function.is_positive:
return Sum(log(arg.function), *arg.limits)
return self.func(arg)
def _eval_simplify(self, **kwargs):
from sympy.simplify.simplify import expand_log, simplify, inversecombine
if len(self.args) == 2: # it's unevaluated
return simplify(self.func(*self.args), **kwargs)
expr = self.func(simplify(self.args[0], **kwargs))
if kwargs['inverse']:
expr = inversecombine(expr)
expr = expand_log(expr, deep=True)
return min([expr, self], key=kwargs['measure'])
def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a complex coordinate.
Examples
========
>>> from sympy import I, log
>>> from sympy.abc import x
>>> log(x).as_real_imag()
(log(Abs(x)), arg(x))
>>> log(I).as_real_imag()
(0, pi/2)
>>> log(1 + I).as_real_imag()
(log(sqrt(2)), pi/4)
>>> log(I*x).as_real_imag()
(log(Abs(x)), arg(I*x))
"""
sarg = self.args[0]
if deep:
sarg = self.args[0].expand(deep, **hints)
sarg_abs = Abs(sarg)
if sarg_abs == sarg:
return self, S.Zero
sarg_arg = arg(sarg)
if hints.get('log', False): # Expand the log
hints['complex'] = False
return (log(sarg_abs).expand(deep, **hints), sarg_arg)
else:
return log(sarg_abs), sarg_arg
def _eval_is_rational(self):
s = self.func(*self.args)
if s.func == self.func:
if (self.args[0] - 1).is_zero:
return True
if s.args[0].is_rational and fuzzy_not((self.args[0] - 1).is_zero):
return False
else:
return s.is_rational
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if (self.args[0] - 1).is_zero:
return True
elif fuzzy_not((self.args[0] - 1).is_zero):
if self.args[0].is_algebraic:
return False
else:
return s.is_algebraic
def _eval_is_extended_real(self):
return self.args[0].is_extended_positive
def _eval_is_complex(self):
z = self.args[0]
return fuzzy_and([z.is_complex, fuzzy_not(z.is_zero)])
def _eval_is_finite(self):
arg = self.args[0]
if arg.is_zero:
return False
return arg.is_finite
def _eval_is_extended_positive(self):
return (self.args[0] - 1).is_extended_positive
def _eval_is_zero(self):
return (self.args[0] - 1).is_zero
def _eval_is_extended_nonnegative(self):
return (self.args[0] - 1).is_extended_nonnegative
def _eval_nseries(self, x, n, logx, cdir=0):
# NOTE Please see the comment at the beginning of this file, labelled
# IMPORTANT.
from sympy.series.order import Order
from sympy.simplify.simplify import logcombine
from sympy.core.symbol import Dummy
if self.args[0] == x:
return log(x) if logx is None else logx
arg = self.args[0]
t = Dummy('t', positive=True)
if cdir == 0:
cdir = 1
z = arg.subs(x, cdir*t)
k, l = Wild("k"), Wild("l")
r = z.match(k*t**l)
if r is not None:
k, l = r[k], r[l]
if l != 0 and not l.has(t) and not k.has(t):
r = l*log(x) if logx is None else l*logx
r += log(k) - l*log(cdir) # XXX true regardless of assumptions?
return r
def coeff_exp(term, x):
coeff, exp = S.One, S.Zero
for factor in Mul.make_args(term):
if factor.has(x):
base, exp = factor.as_base_exp()
if base != x:
try:
return term.leadterm(x)
except ValueError:
return term, S.Zero
else:
coeff *= factor
return coeff, exp
# TODO new and probably slow
try:
a, b = z.leadterm(t, logx=logx, cdir=1)
except (ValueError, NotImplementedError, PoleError):
s = z._eval_nseries(t, n=n, logx=logx, cdir=1)
while s.is_Order:
n += 1
s = z._eval_nseries(t, n=n, logx=logx, cdir=1)
try:
a, b = s.removeO().leadterm(t, cdir=1)
except ValueError:
a, b = s.removeO().as_leading_term(t, cdir=1), S.Zero
p = (z/(a*t**b) - 1)._eval_nseries(t, n=n, logx=logx, cdir=1)
if p.has(exp):
p = logcombine(p)
if isinstance(p, Order):
n = p.getn()
_, d = coeff_exp(p, t)
logx = log(x) if logx is None else logx
if not d.is_positive:
res = log(a) - b*log(cdir) + b*logx
_res = res
logflags = dict(deep=True, log=True, mul=False, power_exp=False,
power_base=False, multinomial=False, basic=False, force=True,
factor=False)
expr = self.expand(**logflags)
if (not a.could_extract_minus_sign() and
logx.could_extract_minus_sign()):
_res = _res.subs(-logx, -log(x)).expand(**logflags)
else:
_res = _res.subs(logx, log(x)).expand(**logflags)
if _res == expr:
return res
return res + Order(x**n, x)
def mul(d1, d2):
res = {}
for e1, e2 in product(d1, d2):
ex = e1 + e2
if ex < n:
res[ex] = res.get(ex, S.Zero) + d1[e1]*d2[e2]
return res
pterms = {}
for term in Add.make_args(p.removeO()):
co1, e1 = coeff_exp(term, t)
pterms[e1] = pterms.get(e1, S.Zero) + co1
k = S.One
terms = {}
pk = pterms
while k*d < n:
coeff = -S.NegativeOne**k/k
for ex in pk:
_ = terms.get(ex, S.Zero) + coeff*pk[ex]
terms[ex] = _.nsimplify()
pk = mul(pk, pterms)
k += S.One
res = log(a) - b*log(cdir) + b*logx
for ex in terms:
res += terms[ex]*t**(ex)
if a.is_negative and im(z) != 0:
from sympy.functions.special.delta_functions import Heaviside
for i, term in enumerate(z.lseries(t)):
if not term.is_real or i == 5:
break
if i < 5:
coeff, _ = term.as_coeff_exponent(t)
res += -2*I*S.Pi*Heaviside(-im(coeff), 0)
res = res.subs(t, x/cdir)
return res + Order(x**n, x)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
# NOTE
# Refer https://github.com/sympy/sympy/pull/23592 for more information
# on each of the following steps involved in this method.
arg0 = self.args[0].together()
# STEP 1
t = Dummy('t', positive=True)
if cdir == 0:
cdir = 1
z = arg0.subs(x, cdir*t)
# STEP 2
try:
c, e = z.leadterm(t, logx=logx, cdir=1)
except ValueError:
arg = arg0.as_leading_term(x, logx=logx, cdir=cdir)
return log(arg)
if c.has(t):
c = c.subs(t, x/cdir)
if e != 0:
raise PoleError("Cannot expand %s around 0" % (self))
return log(c)
# STEP 3
if c == S.One and e == S.Zero:
return (arg0 - S.One).as_leading_term(x, logx=logx)
# STEP 4
res = log(c) - e*log(cdir)
logx = log(x) if logx is None else logx
res += e*logx
# STEP 5
if c.is_negative and im(z) != 0:
from sympy.functions.special.delta_functions import Heaviside
for i, term in enumerate(z.lseries(t)):
if not term.is_real or i == 5:
break
if i < 5:
coeff, _ = term.as_coeff_exponent(t)
res += -2*I*S.Pi*Heaviside(-im(coeff), 0)
return res
class LambertW(Function):
r"""
The Lambert W function $W(z)$ is defined as the inverse
function of $w \exp(w)$ [1]_.
Explanation
===========
In other words, the value of $W(z)$ is such that $z = W(z) \exp(W(z))$
for any complex number $z$. The Lambert W function is a multivalued
function with infinitely many branches $W_k(z)$, indexed by
$k \in \mathbb{Z}$. Each branch gives a different solution $w$
of the equation $z = w \exp(w)$.
The Lambert W function has two partially real branches: the
principal branch ($k = 0$) is real for real $z > -1/e$, and the
$k = -1$ branch is real for $-1/e < z < 0$. All branches except
$k = 0$ have a logarithmic singularity at $z = 0$.
Examples
========
>>> from sympy import LambertW
>>> LambertW(1.2)
0.635564016364870
>>> LambertW(1.2, -1).n()
-1.34747534407696 - 4.41624341514535*I
>>> LambertW(-1).is_real
False
References
==========
.. [1] https://en.wikipedia.org/wiki/Lambert_W_function
"""
_singularities = (-Pow(S.Exp1, -1, evaluate=False), S.ComplexInfinity)
@classmethod
def eval(cls, x, k=None):
if k == S.Zero:
return cls(x)
elif k is None:
k = S.Zero
if k.is_zero:
if x.is_zero:
return S.Zero
if x is S.Exp1:
return S.One
if x == -1/S.Exp1:
return S.NegativeOne
if x == -log(2)/2:
return -log(2)
if x == 2*log(2):
return log(2)
if x == -S.Pi/2:
return S.ImaginaryUnit*S.Pi/2
if x == exp(1 + S.Exp1):
return S.Exp1
if x is S.Infinity:
return S.Infinity
if x.is_zero:
return S.Zero
if fuzzy_not(k.is_zero):
if x.is_zero:
return S.NegativeInfinity
if k is S.NegativeOne:
if x == -S.Pi/2:
return -S.ImaginaryUnit*S.Pi/2
elif x == -1/S.Exp1:
return S.NegativeOne
elif x == -2*exp(-2):
return -Integer(2)
def fdiff(self, argindex=1):
"""
Return the first derivative of this function.
"""
x = self.args[0]
if len(self.args) == 1:
if argindex == 1:
return LambertW(x)/(x*(1 + LambertW(x)))
else:
k = self.args[1]
if argindex == 1:
return LambertW(x, k)/(x*(1 + LambertW(x, k)))
raise ArgumentIndexError(self, argindex)
def _eval_is_extended_real(self):
x = self.args[0]
if len(self.args) == 1:
k = S.Zero
else:
k = self.args[1]
if k.is_zero:
if (x + 1/S.Exp1).is_positive:
return True
elif (x + 1/S.Exp1).is_nonpositive:
return False
elif (k + 1).is_zero:
if x.is_negative and (x + 1/S.Exp1).is_positive:
return True
elif x.is_nonpositive or (x + 1/S.Exp1).is_nonnegative:
return False
elif fuzzy_not(k.is_zero) and fuzzy_not((k + 1).is_zero):
if x.is_extended_real:
return False
def _eval_is_finite(self):
return self.args[0].is_finite
def _eval_is_algebraic(self):
s = self.func(*self.args)
if s.func == self.func:
if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic:
return False
else:
return s.is_algebraic
def _eval_as_leading_term(self, x, logx=None, cdir=0):
if len(self.args) == 1:
arg = self.args[0]
arg0 = arg.subs(x, 0).cancel()
if not arg0.is_zero:
return self.func(arg0)
return arg.as_leading_term(x)
def _eval_nseries(self, x, n, logx, cdir=0):
if len(self.args) == 1:
from sympy.functions.elementary.integers import ceiling
from sympy.series.order import Order
arg = self.args[0].nseries(x, n=n, logx=logx)
lt = arg.compute_leading_term(x, logx=logx)
lte = 1
if lt.is_Pow:
lte = lt.exp
if ceiling(n/lte) >= 1:
s = Add(*[(-S.One)**(k - 1)*Integer(k)**(k - 2)/
factorial(k - 1)*arg**k for k in range(1, ceiling(n/lte))])
s = expand_multinomial(s)
else:
s = S.Zero
return s + Order(x**n, x)
return super()._eval_nseries(x, n, logx)
def _eval_is_zero(self):
x = self.args[0]
if len(self.args) == 1:
return x.is_zero
else:
return fuzzy_and([x.is_zero, self.args[1].is_zero])
|
e9d4c631529797292b4e567a99f28eec3f40041b05649da7d19cfe5a77fbdfad | from sympy.core import S, sympify, cacheit, pi, I, Rational
from sympy.core.add import Add
from sympy.core.function import Function, ArgumentIndexError
from sympy.core.logic import fuzzy_or, fuzzy_and, FuzzyBool
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import (binomial, factorial,
RisingFactorial)
from sympy.functions.combinatorial.numbers import bernoulli, euler, nC
from sympy.functions.elementary.complexes import Abs, im, re
from sympy.functions.elementary.exponential import exp, log, match_real_imag
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (
acos, acot, asin, atan, cos, cot, csc, sec, sin, tan,
_imaginary_unit_as_coefficient)
from sympy.polys.specialpolys import symmetric_poly
def _rewrite_hyperbolics_as_exp(expr):
return expr.xreplace({h: h.rewrite(exp)
for h in expr.atoms(HyperbolicFunction)})
###############################################################################
########################### HYPERBOLIC FUNCTIONS ##############################
###############################################################################
class HyperbolicFunction(Function):
"""
Base class for hyperbolic functions.
See Also
========
sinh, cosh, tanh, coth
"""
unbranched = True
def _peeloff_ipi(arg):
r"""
Split ARG into two parts, a "rest" and a multiple of $I\pi$.
This assumes ARG to be an ``Add``.
The multiple of $I\pi$ returned in the second position is always a ``Rational``.
Examples
========
>>> from sympy.functions.elementary.hyperbolic import _peeloff_ipi as peel
>>> from sympy import pi, I
>>> from sympy.abc import x, y
>>> peel(x + I*pi/2)
(x, 1/2)
>>> peel(x + I*2*pi/3 + I*pi*y)
(x + I*pi*y + I*pi/6, 1/2)
"""
ipi = S.Pi*S.ImaginaryUnit
for a in Add.make_args(arg):
if a == ipi:
K = S.One
break
elif a.is_Mul:
K, p = a.as_two_terms()
if p == ipi and K.is_Rational:
break
else:
return arg, S.Zero
m1 = (K % S.Half)
m2 = K - m1
return arg - m2*ipi, m2
class sinh(HyperbolicFunction):
r"""
``sinh(x)`` is the hyperbolic sine of ``x``.
The hyperbolic sine function is $\frac{e^x - e^{-x}}{2}$.
Examples
========
>>> from sympy import sinh
>>> from sympy.abc import x
>>> sinh(x)
sinh(x)
See Also
========
cosh, tanh, asinh
"""
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function.
"""
if argindex == 1:
return cosh(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return asinh
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg.is_zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
return S.ImaginaryUnit * sin(i_coeff)
else:
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
m = m*S.Pi*S.ImaginaryUnit
return sinh(m)*cosh(x) + cosh(m)*sinh(x)
if arg.is_zero:
return S.Zero
if arg.func == asinh:
return arg.args[0]
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1)
if arg.func == atanh:
x = arg.args[0]
return x/sqrt(1 - x**2)
if arg.func == acoth:
x = arg.args[0]
return 1/(sqrt(x - 1) * sqrt(x + 1))
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
"""
Returns the next term in the Taylor series expansion.
"""
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return p * x**2 / (n*(n - 1))
else:
return x**(n) / factorial(n)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
"""
Returns this function as a complex coordinate.
"""
if self.args[0].is_extended_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (sinh(re)*cos(im), cosh(re)*sin(im))
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _eval_expand_trig(self, deep=True, **hints):
if deep:
arg = self.args[0].expand(deep, **hints)
else:
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
x, y = arg.as_two_terms()
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One and coeff.is_Integer and terms is not S.One:
x = terms
y = (coeff - 1)*x
if x is not None:
return (sinh(x)*cosh(y) + sinh(y)*cosh(x)).expand(trig=True)
return sinh(arg)
def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
return (exp(arg) - exp(-arg)) / 2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return (exp(arg) - exp(-arg)) / 2
def _eval_rewrite_as_sin(self, arg, **kwargs):
return -I * sin(I * arg)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return -I / csc(I * arg)
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return -S.ImaginaryUnit*cosh(arg + S.Pi*S.ImaginaryUnit/2)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
tanh_half = tanh(S.Half*arg)
return 2*tanh_half/(1 - tanh_half**2)
def _eval_rewrite_as_coth(self, arg, **kwargs):
coth_half = coth(S.Half*arg)
return 2*coth_half/(coth_half**2 - 1)
def _eval_rewrite_as_csch(self, arg, **kwargs):
return 1 / csch(arg)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
arg0 = arg.subs(x, 0)
if arg0 is S.NaN:
arg0 = arg.limit(x, 0, dir='-' if cdir.is_negative else '+')
if arg0.is_zero:
return arg
elif arg0.is_finite:
return self.func(arg0)
else:
return self
def _eval_is_real(self):
arg = self.args[0]
if arg.is_real:
return True
# if `im` is of the form n*pi
# else, check if it is a number
re, im = arg.as_real_imag()
return (im%pi).is_zero
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
def _eval_is_positive(self):
if self.args[0].is_extended_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_extended_real:
return self.args[0].is_negative
def _eval_is_finite(self):
arg = self.args[0]
return arg.is_finite
def _eval_is_zero(self):
rest, ipi_mult = _peeloff_ipi(self.args[0])
if rest.is_zero:
return ipi_mult.is_integer
class cosh(HyperbolicFunction):
r"""
``cosh(x)`` is the hyperbolic cosine of ``x``.
The hyperbolic cosine function is $\frac{e^x + e^{-x}}{2}$.
Examples
========
>>> from sympy import cosh
>>> from sympy.abc import x
>>> cosh(x)
cosh(x)
See Also
========
sinh, tanh, acosh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return sinh(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
from sympy.functions.elementary.trigonometric import cos
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg.is_zero:
return S.One
elif arg.is_negative:
return cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
return cos(i_coeff)
else:
if arg.could_extract_minus_sign():
return cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
m = m*S.Pi*S.ImaginaryUnit
return cosh(m)*cosh(x) + sinh(m)*sinh(x)
if arg.is_zero:
return S.One
if arg.func == asinh:
return sqrt(1 + arg.args[0]**2)
if arg.func == acosh:
return arg.args[0]
if arg.func == atanh:
return 1/sqrt(1 - arg.args[0]**2)
if arg.func == acoth:
x = arg.args[0]
return x/(sqrt(x - 1) * sqrt(x + 1))
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2:
p = previous_terms[-2]
return p * x**2 / (n*(n - 1))
else:
return x**(n)/factorial(n)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_extended_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
return (cosh(re)*cos(im), sinh(re)*sin(im))
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=deep, **hints)
return re_part + im_part*S.ImaginaryUnit
def _eval_expand_trig(self, deep=True, **hints):
if deep:
arg = self.args[0].expand(deep, **hints)
else:
arg = self.args[0]
x = None
if arg.is_Add: # TODO, implement more if deep stuff here
x, y = arg.as_two_terms()
else:
coeff, terms = arg.as_coeff_Mul(rational=True)
if coeff is not S.One and coeff.is_Integer and terms is not S.One:
x = terms
y = (coeff - 1)*x
if x is not None:
return (cosh(x)*cosh(y) + sinh(x)*sinh(y)).expand(trig=True)
return cosh(arg)
def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
return (exp(arg) + exp(-arg)) / 2
def _eval_rewrite_as_exp(self, arg, **kwargs):
return (exp(arg) + exp(-arg)) / 2
def _eval_rewrite_as_cos(self, arg, **kwargs):
return cos(I * arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return 1 / sec(I * arg)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return -S.ImaginaryUnit*sinh(arg + S.Pi*S.ImaginaryUnit/2)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
tanh_half = tanh(S.Half*arg)**2
return (1 + tanh_half)/(1 - tanh_half)
def _eval_rewrite_as_coth(self, arg, **kwargs):
coth_half = coth(S.Half*arg)**2
return (coth_half + 1)/(coth_half - 1)
def _eval_rewrite_as_sech(self, arg, **kwargs):
return 1 / sech(arg)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
arg = self.args[0].as_leading_term(x, logx=logx, cdir=cdir)
arg0 = arg.subs(x, 0)
if arg0 is S.NaN:
arg0 = arg.limit(x, 0, dir='-' if cdir.is_negative else '+')
if arg0.is_zero:
return S.One
elif arg0.is_finite:
return self.func(arg0)
else:
return self
def _eval_is_real(self):
arg = self.args[0]
# `cosh(x)` is real for real OR purely imaginary `x`
if arg.is_real or arg.is_imaginary:
return True
# cosh(a+ib) = cos(b)*cosh(a) + i*sin(b)*sinh(a)
# the imaginary part can be an expression like n*pi
# if not, check if the imaginary part is a number
re, im = arg.as_real_imag()
return (im%pi).is_zero
def _eval_is_positive(self):
# cosh(x+I*y) = cos(y)*cosh(x) + I*sin(y)*sinh(x)
# cosh(z) is positive iff it is real and the real part is positive.
# So we need sin(y)*sinh(x) = 0 which gives x=0 or y=n*pi
# Case 1 (y=n*pi): cosh(z) = (-1)**n * cosh(x) -> positive for n even
# Case 2 (x=0): cosh(z) = cos(y) -> positive when cos(y) is positive
z = self.args[0]
x, y = z.as_real_imag()
ymod = y % (2*pi)
yzero = ymod.is_zero
# shortcut if ymod is zero
if yzero:
return True
xzero = x.is_zero
# shortcut x is not zero
if xzero is False:
return yzero
return fuzzy_or([
# Case 1:
yzero,
# Case 2:
fuzzy_and([
xzero,
fuzzy_or([ymod < pi/2, ymod > 3*pi/2])
])
])
def _eval_is_nonnegative(self):
z = self.args[0]
x, y = z.as_real_imag()
ymod = y % (2*pi)
yzero = ymod.is_zero
# shortcut if ymod is zero
if yzero:
return True
xzero = x.is_zero
# shortcut x is not zero
if xzero is False:
return yzero
return fuzzy_or([
# Case 1:
yzero,
# Case 2:
fuzzy_and([
xzero,
fuzzy_or([ymod <= pi/2, ymod >= 3*pi/2])
])
])
def _eval_is_finite(self):
arg = self.args[0]
return arg.is_finite
def _eval_is_zero(self):
rest, ipi_mult = _peeloff_ipi(self.args[0])
if ipi_mult and rest.is_zero:
return (ipi_mult - S.Half).is_integer
class tanh(HyperbolicFunction):
r"""
``tanh(x)`` is the hyperbolic tangent of ``x``.
The hyperbolic tangent function is $\frac{\sinh(x)}{\cosh(x)}$.
Examples
========
>>> from sympy import tanh
>>> from sympy.abc import x
>>> tanh(x)
tanh(x)
See Also
========
sinh, cosh, atanh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return S.One - tanh(self.args[0])**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return atanh
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg.is_zero:
return S.Zero
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
if i_coeff.could_extract_minus_sign():
return -S.ImaginaryUnit * tan(-i_coeff)
return S.ImaginaryUnit * tan(i_coeff)
else:
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
tanhm = tanh(m*S.Pi*S.ImaginaryUnit)
if tanhm is S.ComplexInfinity:
return coth(x)
else: # tanhm == 0
return tanh(x)
if arg.is_zero:
return S.Zero
if arg.func == asinh:
x = arg.args[0]
return x/sqrt(1 + x**2)
if arg.func == acosh:
x = arg.args[0]
return sqrt(x - 1) * sqrt(x + 1) / x
if arg.func == atanh:
return arg.args[0]
if arg.func == acoth:
return 1/arg.args[0]
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
a = 2**(n + 1)
B = bernoulli(n + 1)
F = factorial(n + 1)
return a*(a - 1) * B/F * x**n
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
if self.args[0].is_extended_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + cos(im)**2
return (sinh(re)*cosh(re)/denom, sin(im)*cos(im)/denom)
def _eval_expand_trig(self, **hints):
arg = self.args[0]
if arg.is_Add:
n = len(arg.args)
TX = [tanh(x, evaluate=False)._eval_expand_trig()
for x in arg.args]
p = [0, 0] # [den, num]
for i in range(n + 1):
p[i % 2] += symmetric_poly(i, TX)
return p[1]/p[0]
elif arg.is_Mul:
coeff, terms = arg.as_coeff_Mul()
if coeff.is_Integer and coeff > 1:
T = tanh(terms)
n = [nC(range(coeff), k)*T**k for k in range(1, coeff + 1, 2)]
d = [nC(range(coeff), k)*T**k for k in range(0, coeff + 1, 2)]
return Add(*n)/Add(*d)
return tanh(arg)
def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp - neg_exp)/(pos_exp + neg_exp)
def _eval_rewrite_as_exp(self, arg, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp - neg_exp)/(pos_exp + neg_exp)
def _eval_rewrite_as_tan(self, arg, **kwargs):
return -I * tan(I * arg)
def _eval_rewrite_as_cot(self, arg, **kwargs):
return -I / cot(I * arg)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return S.ImaginaryUnit*sinh(arg)/sinh(S.Pi*S.ImaginaryUnit/2 - arg)
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return S.ImaginaryUnit*cosh(S.Pi*S.ImaginaryUnit/2 - arg)/cosh(arg)
def _eval_rewrite_as_coth(self, arg, **kwargs):
return 1/coth(arg)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.series.order import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return arg
else:
return self.func(arg)
def _eval_is_real(self):
arg = self.args[0]
if arg.is_real:
return True
re, im = arg.as_real_imag()
# if denom = 0, tanh(arg) = zoo
if re == 0 and im % pi == pi/2:
return None
# check if im is of the form n*pi/2 to make sin(2*im) = 0
# if not, im could be a number, return False in that case
return (im % (pi/2)).is_zero
def _eval_is_extended_real(self):
if self.args[0].is_extended_real:
return True
def _eval_is_positive(self):
if self.args[0].is_extended_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_extended_real:
return self.args[0].is_negative
def _eval_is_finite(self):
arg = self.args[0]
re, im = arg.as_real_imag()
denom = cos(im)**2 + sinh(re)**2
if denom == 0:
return False
elif denom.is_number:
return True
if arg.is_extended_real:
return True
def _eval_is_zero(self):
arg = self.args[0]
if arg.is_zero:
return True
class coth(HyperbolicFunction):
r"""
``coth(x)`` is the hyperbolic cotangent of ``x``.
The hyperbolic cotangent function is $\frac{\cosh(x)}{\sinh(x)}$.
Examples
========
>>> from sympy import coth
>>> from sympy.abc import x
>>> coth(x)
coth(x)
See Also
========
sinh, cosh, acoth
"""
def fdiff(self, argindex=1):
if argindex == 1:
return -1/sinh(self.args[0])**2
else:
raise ArgumentIndexError(self, argindex)
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return acoth
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.One
elif arg is S.NegativeInfinity:
return S.NegativeOne
elif arg.is_zero:
return S.ComplexInfinity
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.NaN
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
if i_coeff.could_extract_minus_sign():
return S.ImaginaryUnit * cot(-i_coeff)
return -S.ImaginaryUnit * cot(i_coeff)
else:
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_Add:
x, m = _peeloff_ipi(arg)
if m:
cothm = coth(m*S.Pi*S.ImaginaryUnit)
if cothm is S.ComplexInfinity:
return coth(x)
else: # cothm == 0
return tanh(x)
if arg.is_zero:
return S.ComplexInfinity
if arg.func == asinh:
x = arg.args[0]
return sqrt(1 + x**2)/x
if arg.func == acosh:
x = arg.args[0]
return x/(sqrt(x - 1) * sqrt(x + 1))
if arg.func == atanh:
return 1/arg.args[0]
if arg.func == acoth:
return arg.args[0]
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return 1 / sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2**(n + 1) * B/F * x**n
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def as_real_imag(self, deep=True, **hints):
from sympy.functions.elementary.trigonometric import (cos, sin)
if self.args[0].is_extended_real:
if deep:
hints['complex'] = False
return (self.expand(deep, **hints), S.Zero)
else:
return (self, S.Zero)
if deep:
re, im = self.args[0].expand(deep, **hints).as_real_imag()
else:
re, im = self.args[0].as_real_imag()
denom = sinh(re)**2 + sin(im)**2
return (sinh(re)*cosh(re)/denom, -sin(im)*cos(im)/denom)
def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_exp(self, arg, **kwargs):
neg_exp, pos_exp = exp(-arg), exp(arg)
return (pos_exp + neg_exp)/(pos_exp - neg_exp)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return -S.ImaginaryUnit*sinh(S.Pi*S.ImaginaryUnit/2 - arg)/sinh(arg)
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return -S.ImaginaryUnit*cosh(arg)/cosh(S.Pi*S.ImaginaryUnit/2 - arg)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
return 1/tanh(arg)
def _eval_is_positive(self):
if self.args[0].is_extended_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_extended_real:
return self.args[0].is_negative
def _eval_as_leading_term(self, x, logx=None, cdir=0):
from sympy.series.order import Order
arg = self.args[0].as_leading_term(x)
if x in arg.free_symbols and Order(1, x).contains(arg):
return 1/arg
else:
return self.func(arg)
def _eval_expand_trig(self, **hints):
arg = self.args[0]
if arg.is_Add:
CX = [coth(x, evaluate=False)._eval_expand_trig() for x in arg.args]
p = [[], []]
n = len(arg.args)
for i in range(n, -1, -1):
p[(n - i) % 2].append(symmetric_poly(i, CX))
return Add(*p[0])/Add(*p[1])
elif arg.is_Mul:
coeff, x = arg.as_coeff_Mul(rational=True)
if coeff.is_Integer and coeff > 1:
c = coth(x, evaluate=False)
p = [[], []]
for i in range(coeff, -1, -1):
p[(coeff - i) % 2].append(binomial(coeff, i)*c**i)
return Add(*p[0])/Add(*p[1])
return coth(arg)
class ReciprocalHyperbolicFunction(HyperbolicFunction):
"""Base class for reciprocal functions of hyperbolic functions. """
#To be defined in class
_reciprocal_of = None
_is_even = None # type: FuzzyBool
_is_odd = None # type: FuzzyBool
@classmethod
def eval(cls, arg):
if arg.could_extract_minus_sign():
if cls._is_even:
return cls(-arg)
if cls._is_odd:
return -cls(-arg)
t = cls._reciprocal_of.eval(arg)
if hasattr(arg, 'inverse') and arg.inverse() == cls:
return arg.args[0]
return 1/t if t is not None else t
def _call_reciprocal(self, method_name, *args, **kwargs):
# Calls method_name on _reciprocal_of
o = self._reciprocal_of(self.args[0])
return getattr(o, method_name)(*args, **kwargs)
def _calculate_reciprocal(self, method_name, *args, **kwargs):
# If calling method_name on _reciprocal_of returns a value != None
# then return the reciprocal of that value
t = self._call_reciprocal(method_name, *args, **kwargs)
return 1/t if t is not None else t
def _rewrite_reciprocal(self, method_name, arg):
# Special handling for rewrite functions. If reciprocal rewrite returns
# unmodified expression, then return None
t = self._call_reciprocal(method_name, arg)
if t is not None and t != self._reciprocal_of(arg):
return 1/t
def _eval_rewrite_as_exp(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)
def _eval_rewrite_as_tractable(self, arg, limitvar=None, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tractable", arg)
def _eval_rewrite_as_tanh(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_tanh", arg)
def _eval_rewrite_as_coth(self, arg, **kwargs):
return self._rewrite_reciprocal("_eval_rewrite_as_coth", arg)
def as_real_imag(self, deep = True, **hints):
return (1 / self._reciprocal_of(self.args[0])).as_real_imag(deep, **hints)
def _eval_conjugate(self):
return self.func(self.args[0].conjugate())
def _eval_expand_complex(self, deep=True, **hints):
re_part, im_part = self.as_real_imag(deep=True, **hints)
return re_part + S.ImaginaryUnit*im_part
def _eval_expand_trig(self, **hints):
return self._calculate_reciprocal("_eval_expand_trig", **hints)
def _eval_as_leading_term(self, x, logx=None, cdir=0):
return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)
def _eval_is_extended_real(self):
return self._reciprocal_of(self.args[0]).is_extended_real
def _eval_is_finite(self):
return (1/self._reciprocal_of(self.args[0])).is_finite
class csch(ReciprocalHyperbolicFunction):
r"""
``csch(x)`` is the hyperbolic cosecant of ``x``.
The hyperbolic cosecant function is $\frac{2}{e^x - e^{-x}}$
Examples
========
>>> from sympy import csch
>>> from sympy.abc import x
>>> csch(x)
csch(x)
See Also
========
sinh, cosh, tanh, sech, asinh, acosh
"""
_reciprocal_of = sinh
_is_odd = True
def fdiff(self, argindex=1):
"""
Returns the first derivative of this function
"""
if argindex == 1:
return -coth(self.args[0]) * csch(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
"""
Returns the next term in the Taylor series expansion
"""
if n == 0:
return 1/sympify(x)
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
B = bernoulli(n + 1)
F = factorial(n + 1)
return 2 * (1 - 2**n) * B/F * x**n
def _eval_rewrite_as_sin(self, arg, **kwargs):
return I / sin(I * arg)
def _eval_rewrite_as_csc(self, arg, **kwargs):
return I * csc(I * arg)
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return S.ImaginaryUnit / cosh(arg + S.ImaginaryUnit * S.Pi / 2)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return 1 / sinh(arg)
def _eval_is_positive(self):
if self.args[0].is_extended_real:
return self.args[0].is_positive
def _eval_is_negative(self):
if self.args[0].is_extended_real:
return self.args[0].is_negative
class sech(ReciprocalHyperbolicFunction):
r"""
``sech(x)`` is the hyperbolic secant of ``x``.
The hyperbolic secant function is $\frac{2}{e^x + e^{-x}}$
Examples
========
>>> from sympy import sech
>>> from sympy.abc import x
>>> sech(x)
sech(x)
See Also
========
sinh, cosh, tanh, coth, csch, asinh, acosh
"""
_reciprocal_of = cosh
_is_even = True
def fdiff(self, argindex=1):
if argindex == 1:
return - tanh(self.args[0])*sech(self.args[0])
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
return euler(n) / factorial(n) * x**(n)
def _eval_rewrite_as_cos(self, arg, **kwargs):
return 1 / cos(I * arg)
def _eval_rewrite_as_sec(self, arg, **kwargs):
return sec(I * arg)
def _eval_rewrite_as_sinh(self, arg, **kwargs):
return S.ImaginaryUnit / sinh(arg + S.ImaginaryUnit * S.Pi /2)
def _eval_rewrite_as_cosh(self, arg, **kwargs):
return 1 / cosh(arg)
def _eval_is_positive(self):
if self.args[0].is_extended_real:
return True
###############################################################################
############################# HYPERBOLIC INVERSES #############################
###############################################################################
class InverseHyperbolicFunction(Function):
"""Base class for inverse hyperbolic functions."""
pass
class asinh(InverseHyperbolicFunction):
"""
``asinh(x)`` is the inverse hyperbolic sine of ``x``.
The inverse hyperbolic sine function.
Examples
========
>>> from sympy import asinh
>>> from sympy.abc import x
>>> asinh(x).diff(x)
1/sqrt(x**2 + 1)
>>> asinh(1)
log(1 + sqrt(2))
See Also
========
acosh, atanh, sinh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/sqrt(self.args[0]**2 + 1)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.NegativeInfinity
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return log(sqrt(2) + 1)
elif arg is S.NegativeOne:
return log(sqrt(2) - 1)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg.is_zero:
return S.Zero
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
return S.ImaginaryUnit * asin(i_coeff)
else:
if arg.could_extract_minus_sign():
return -cls(-arg)
if isinstance(arg, sinh) and arg.args[0].is_number:
z = arg.args[0]
if z.is_real:
return z
r, i = match_real_imag(z)
if r is not None and i is not None:
f = floor((i + pi/2)/pi)
m = z - I*pi*f
even = f.is_even
if even is True:
return m
elif even is False:
return -m
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return -p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return S.NegativeOne**k * R / F * x**n / n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # asinh
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0.is_zero:
return arg.as_leading_term(x)
# Handling branch points
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
if (1 + x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if re(ndir).is_positive:
if im(x0).is_negative:
return -self.func(x0) - I*pi
elif re(ndir).is_negative:
if im(x0).is_positive:
return -self.func(x0) + I*pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # asinh
arg = self.args[0]
arg0 = arg.subs(x, 0)
# Handling branch points
if arg0 in (S.ImaginaryUnit, -S.ImaginaryUnit):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
if (1 + arg0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if re(ndir).is_positive:
if im(arg0).is_negative:
return -res - I*pi
elif re(ndir).is_negative:
if im(arg0).is_positive:
return -res + I*pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return log(x + sqrt(x**2 + 1))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_atanh(self, x, **kwargs):
return atanh(x/sqrt(1 + x**2))
def _eval_rewrite_as_acosh(self, x, **kwargs):
ix = I*x
return I*(sqrt(1 - ix)/sqrt(ix - 1) * acosh(ix) - pi/2)
def _eval_rewrite_as_asin(self, x, **kwargs):
return -I * asin(I * x)
def _eval_rewrite_as_acos(self, x, **kwargs):
return I * acos(I * x) - I*pi/2
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sinh
def _eval_is_zero(self):
return self.args[0].is_zero
class acosh(InverseHyperbolicFunction):
"""
``acosh(x)`` is the inverse hyperbolic cosine of ``x``.
The inverse hyperbolic cosine function.
Examples
========
>>> from sympy import acosh
>>> from sympy.abc import x
>>> acosh(x).diff(x)
1/(sqrt(x - 1)*sqrt(x + 1))
>>> acosh(1)
0
See Also
========
asinh, atanh, cosh
"""
def fdiff(self, argindex=1):
if argindex == 1:
arg = self.args[0]
return 1/(sqrt(arg - 1)*sqrt(arg + 1))
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def _acosh_table():
return {
S.ImaginaryUnit: log(S.ImaginaryUnit*(1 + sqrt(2))),
-S.ImaginaryUnit: log(-S.ImaginaryUnit*(1 + sqrt(2))),
S.Half: S.Pi/3,
Rational(-1, 2): S.Pi*Rational(2, 3),
sqrt(2)/2: S.Pi/4,
-sqrt(2)/2: S.Pi*Rational(3, 4),
1/sqrt(2): S.Pi/4,
-1/sqrt(2): S.Pi*Rational(3, 4),
sqrt(3)/2: S.Pi/6,
-sqrt(3)/2: S.Pi*Rational(5, 6),
(sqrt(3) - 1)/sqrt(2**3): S.Pi*Rational(5, 12),
-(sqrt(3) - 1)/sqrt(2**3): S.Pi*Rational(7, 12),
sqrt(2 + sqrt(2))/2: S.Pi/8,
-sqrt(2 + sqrt(2))/2: S.Pi*Rational(7, 8),
sqrt(2 - sqrt(2))/2: S.Pi*Rational(3, 8),
-sqrt(2 - sqrt(2))/2: S.Pi*Rational(5, 8),
(1 + sqrt(3))/(2*sqrt(2)): S.Pi/12,
-(1 + sqrt(3))/(2*sqrt(2)): S.Pi*Rational(11, 12),
(sqrt(5) + 1)/4: S.Pi/5,
-(sqrt(5) + 1)/4: S.Pi*Rational(4, 5)
}
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Infinity
elif arg is S.NegativeInfinity:
return S.Infinity
elif arg.is_zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi*S.ImaginaryUnit
if arg.is_number:
cst_table = cls._acosh_table()
if arg in cst_table:
if arg.is_extended_real:
return cst_table[arg]*S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
return S.ComplexInfinity
if arg == S.ImaginaryUnit*S.Infinity:
return S.Infinity + S.ImaginaryUnit*S.Pi/2
if arg == -S.ImaginaryUnit*S.Infinity:
return S.Infinity - S.ImaginaryUnit*S.Pi/2
if arg.is_zero:
return S.Pi*S.ImaginaryUnit*S.Half
if isinstance(arg, cosh) and arg.args[0].is_number:
z = arg.args[0]
if z.is_real:
return Abs(z)
r, i = match_real_imag(z)
if r is not None and i is not None:
f = floor(i/pi)
m = z - I*pi*f
even = f.is_even
if even is True:
if r.is_nonnegative:
return m
elif r.is_negative:
return -m
elif even is False:
m -= I*pi
if r.is_nonpositive:
return -m
elif r.is_positive:
return m
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return I*pi/2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) >= 2 and n > 2:
p = previous_terms[-2]
return p * (n - 2)**2/(n*(n - 1)) * x**2
else:
k = (n - 1) // 2
R = RisingFactorial(S.Half, k)
F = factorial(k)
return -R / F * S.ImaginaryUnit * x**n / n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acosh
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 in (-S.One, S.Zero, S.One, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-oo, 1)
if (x0 - 1).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if (x0 + 1).is_negative:
return self.func(x0) - 2*I*pi
return -self.func(x0)
elif not im(ndir).is_positive:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # acosh
arg = self.args[0]
arg0 = arg.subs(x, 0)
# Handling branch points
if arg0 in (S.One, S.NegativeOne):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-oo, 1)
if (arg0 - 1).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if (arg0 + 1).is_negative:
return res - 2*I*pi
return -res
elif not im(ndir).is_positive:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return log(x + sqrt(x + 1) * sqrt(x - 1))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_acos(self, x, **kwargs):
return sqrt(x - 1)/sqrt(1 - x) * acos(x)
def _eval_rewrite_as_asin(self, x, **kwargs):
return sqrt(x - 1)/sqrt(1 - x) * (pi/2 - asin(x))
def _eval_rewrite_as_asinh(self, x, **kwargs):
return sqrt(x - 1)/sqrt(1 - x) * (pi/2 + I*asinh(I*x))
def _eval_rewrite_as_atanh(self, x, **kwargs):
sxm1 = sqrt(x - 1)
s1mx = sqrt(1 - x)
sx2m1 = sqrt(x**2 - 1)
return (pi/2*sxm1/s1mx*(1 - x * sqrt(1/x**2)) +
sxm1*sqrt(x + 1)/sx2m1 * atanh(sx2m1/x))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return cosh
def _eval_is_zero(self):
if (self.args[0] - 1).is_zero:
return True
class atanh(InverseHyperbolicFunction):
"""
``atanh(x)`` is the inverse hyperbolic tangent of ``x``.
The inverse hyperbolic tangent function.
Examples
========
>>> from sympy import atanh
>>> from sympy.abc import x
>>> atanh(x).diff(x)
1/(1 - x**2)
See Also
========
asinh, acosh, tanh
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg.is_zero:
return S.Zero
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg is S.Infinity:
return -S.ImaginaryUnit * atan(arg)
elif arg is S.NegativeInfinity:
return S.ImaginaryUnit * atan(-arg)
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
from sympy.calculus.accumulationbounds import AccumBounds
return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
return S.ImaginaryUnit * atan(i_coeff)
else:
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_zero:
return S.Zero
if isinstance(arg, tanh) and arg.args[0].is_number:
z = arg.args[0]
if z.is_real:
return z
r, i = match_real_imag(z)
if r is not None and i is not None:
f = floor(2*i/pi)
even = f.is_even
m = z - I*f*pi/2
if even is True:
return m
elif even is False:
return m - I*pi/2
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return x**n / n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # atanh
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0.is_zero:
return arg.as_leading_term(x)
# Handling branch points
if x0 in (-S.One, S.One, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-oo, -1] U [1, oo)
if (1 - x0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_negative:
return self.func(x0) - I*pi
elif im(ndir).is_positive:
if x0.is_positive:
return self.func(x0) + I*pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # atanh
arg = self.args[0]
arg0 = arg.subs(x, 0)
# Handling branch points
if arg0 in (S.One, S.NegativeOne):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-oo, -1] U [1, oo)
if (1 - arg0**2).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_negative:
return res - I*pi
elif im(ndir).is_positive:
if arg0.is_positive:
return res + I*pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return (log(1 + x) - log(1 - x)) / 2
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asinh(self, x, **kwargs):
f = sqrt(1/(x**2 - 1))
return (pi*x/(2*sqrt(-x**2)) -
sqrt(-x)*sqrt(1 - x**2)/sqrt(x)*f*asinh(f))
def _eval_is_zero(self):
if self.args[0].is_zero:
return True
def _eval_is_imaginary(self):
return self.args[0].is_imaginary
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return tanh
class acoth(InverseHyperbolicFunction):
"""
``acoth(x)`` is the inverse hyperbolic cotangent of ``x``.
The inverse hyperbolic cotangent function.
Examples
========
>>> from sympy import acoth
>>> from sympy.abc import x
>>> acoth(x).diff(x)
1/(1 - x**2)
See Also
========
asinh, acosh, coth
"""
def fdiff(self, argindex=1):
if argindex == 1:
return 1/(1 - self.args[0]**2)
else:
raise ArgumentIndexError(self, argindex)
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg.is_zero:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.One:
return S.Infinity
elif arg is S.NegativeOne:
return S.NegativeInfinity
elif arg.is_negative:
return -cls(-arg)
else:
if arg is S.ComplexInfinity:
return S.Zero
i_coeff = _imaginary_unit_as_coefficient(arg)
if i_coeff is not None:
return -S.ImaginaryUnit * acot(i_coeff)
else:
if arg.could_extract_minus_sign():
return -cls(-arg)
if arg.is_zero:
return S.Pi*S.ImaginaryUnit*S.Half
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return -I*pi/2
elif n < 0 or n % 2 == 0:
return S.Zero
else:
x = sympify(x)
return x**n / n
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acoth
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
if x0 is S.ComplexInfinity:
return (1/arg).as_leading_term(x)
# Handling branch points
if x0 in (-S.One, S.One, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts [-1, 1]
if x0.is_real and (1 - x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if x0.is_positive:
return self.func(x0) + I*pi
elif im(ndir).is_positive:
if x0.is_negative:
return self.func(x0) - I*pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # acoth
arg = self.args[0]
arg0 = arg.subs(x, 0)
# Handling branch points
if arg0 in (S.One, S.NegativeOne):
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts [-1, 1]
if arg0.is_real and (1 - arg0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_negative:
if arg0.is_positive:
return res + I*pi
elif im(ndir).is_positive:
if arg0.is_negative:
return res - I*pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def _eval_rewrite_as_log(self, x, **kwargs):
return (log(1 + 1/x) - log(1 - 1/x)) / 2
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_atanh(self, x, **kwargs):
return atanh(1/x)
def _eval_rewrite_as_asinh(self, x, **kwargs):
return (pi*I/2*(sqrt((x - 1)/x)*sqrt(x/(x - 1)) - sqrt(1 + 1/x)*sqrt(x/(x + 1))) +
x*sqrt(1/x**2)*asinh(sqrt(1/(x**2 - 1))))
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return coth
class asech(InverseHyperbolicFunction):
"""
``asech(x)`` is the inverse hyperbolic secant of ``x``.
The inverse hyperbolic secant function.
Examples
========
>>> from sympy import asech, sqrt, S
>>> from sympy.abc import x
>>> asech(x).diff(x)
-1/(x*sqrt(1 - x**2))
>>> asech(1).diff(x)
0
>>> asech(1)
0
>>> asech(S(2))
I*pi/3
>>> asech(-sqrt(2))
3*I*pi/4
>>> asech((sqrt(6) - sqrt(2)))
I*pi/12
See Also
========
asinh, atanh, cosh, acoth
References
==========
.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
.. [2] http://dlmf.nist.gov/4.37
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSech/
"""
def fdiff(self, argindex=1):
if argindex == 1:
z = self.args[0]
return -1/(z*sqrt(1 - z**2))
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def _asech_table():
return {
S.ImaginaryUnit: - (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
-S.ImaginaryUnit: (S.Pi*S.ImaginaryUnit / 2) + log(1 + sqrt(2)),
(sqrt(6) - sqrt(2)): S.Pi / 12,
(sqrt(2) - sqrt(6)): 11*S.Pi / 12,
sqrt(2 - 2/sqrt(5)): S.Pi / 10,
-sqrt(2 - 2/sqrt(5)): 9*S.Pi / 10,
2 / sqrt(2 + sqrt(2)): S.Pi / 8,
-2 / sqrt(2 + sqrt(2)): 7*S.Pi / 8,
2 / sqrt(3): S.Pi / 6,
-2 / sqrt(3): 5*S.Pi / 6,
(sqrt(5) - 1): S.Pi / 5,
(1 - sqrt(5)): 4*S.Pi / 5,
sqrt(2): S.Pi / 4,
-sqrt(2): 3*S.Pi / 4,
sqrt(2 + 2/sqrt(5)): 3*S.Pi / 10,
-sqrt(2 + 2/sqrt(5)): 7*S.Pi / 10,
S(2): S.Pi / 3,
-S(2): 2*S.Pi / 3,
sqrt(2*(2 + sqrt(2))): 3*S.Pi / 8,
-sqrt(2*(2 + sqrt(2))): 5*S.Pi / 8,
(1 + sqrt(5)): 2*S.Pi / 5,
(-1 - sqrt(5)): 3*S.Pi / 5,
(sqrt(6) + sqrt(2)): 5*S.Pi / 12,
(-sqrt(6) - sqrt(2)): 7*S.Pi / 12,
S.ImaginaryUnit*S.Infinity: -S.Pi*S.ImaginaryUnit / 2,
S.ImaginaryUnit*S.NegativeInfinity: S.Pi*S.ImaginaryUnit / 2,
}
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Pi*S.ImaginaryUnit / 2
elif arg is S.NegativeInfinity:
return S.Pi*S.ImaginaryUnit / 2
elif arg.is_zero:
return S.Infinity
elif arg is S.One:
return S.Zero
elif arg is S.NegativeOne:
return S.Pi*S.ImaginaryUnit
if arg.is_number:
cst_table = cls._asech_table()
if arg in cst_table:
if arg.is_extended_real:
return cst_table[arg]*S.ImaginaryUnit
return cst_table[arg]
if arg is S.ComplexInfinity:
from sympy.calculus.accumulationbounds import AccumBounds
return S.ImaginaryUnit*AccumBounds(-S.Pi/2, S.Pi/2)
if arg.is_zero:
return S.Infinity
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return log(2 / x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
else:
k = n // 2
R = RisingFactorial(S.Half, k) * n
F = factorial(k) * n // 2 * n // 2
return -1 * R / F * x**n / 4
def _eval_as_leading_term(self, x, logx=None, cdir=0): # asech
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 in (-S.One, S.Zero, S.One, S.ComplexInfinity):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
# Handling points lying on branch cuts (-oo, 0] U (1, oo)
if x0.is_negative or (1 - x0).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_positive:
if x0.is_positive or (x0 + 1).is_negative:
return -self.func(x0)
return self.func(x0) - 2*I*pi
elif not im(ndir).is_negative:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # asech
from sympy.series.order import O
arg = self.args[0]
arg0 = arg.subs(x, 0)
# Handling branch points
if arg0 is S.One:
t = Dummy('t', positive=True)
ser = asech(S.One - t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One - self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
if arg0 is S.NegativeOne:
t = Dummy('t', positive=True)
ser = asech(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.One + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else I*pi + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-oo, 0] U (1, oo)
if arg0.is_negative or (1 - arg0).is_negative:
ndir = arg.dir(x, cdir if cdir else 1)
if im(ndir).is_positive:
if arg0.is_positive or (arg0 + 1).is_negative:
return -res
return res - 2*I*pi
elif not im(ndir).is_negative:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return sech
def _eval_rewrite_as_log(self, arg, **kwargs):
return log(1/arg + sqrt(1/arg - 1) * sqrt(1/arg + 1))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_acosh(self, arg, **kwargs):
return acosh(1/arg)
def _eval_rewrite_as_asinh(self, arg, **kwargs):
return sqrt(1/arg - 1)/sqrt(1 - 1/arg)*(S.ImaginaryUnit*asinh(S.ImaginaryUnit/arg)
+ S.Pi*S.Half)
def _eval_rewrite_as_atanh(self, x, **kwargs):
return (I*pi*(1 - sqrt(x)*sqrt(1/x) - I/2*sqrt(-x)/sqrt(x) - I/2*sqrt(x**2)/sqrt(-x**2))
+ sqrt(1/(x + 1))*sqrt(x + 1)*atanh(sqrt(1 - x**2)))
def _eval_rewrite_as_acsch(self, x, **kwargs):
return sqrt(1/x - 1)/sqrt(1 - 1/x)*(pi/2 - I*acsch(I*x))
class acsch(InverseHyperbolicFunction):
"""
``acsch(x)`` is the inverse hyperbolic cosecant of ``x``.
The inverse hyperbolic cosecant function.
Examples
========
>>> from sympy import acsch, sqrt, S
>>> from sympy.abc import x
>>> acsch(x).diff(x)
-1/(x**2*sqrt(1 + x**(-2)))
>>> acsch(1).diff(x)
0
>>> acsch(1)
log(1 + sqrt(2))
>>> acsch(S.ImaginaryUnit)
-I*pi/2
>>> acsch(-2*S.ImaginaryUnit)
I*pi/6
>>> acsch(S.ImaginaryUnit*(sqrt(6) - sqrt(2)))
-5*I*pi/12
See Also
========
asinh
References
==========
.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
.. [2] http://dlmf.nist.gov/4.37
.. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCsch/
"""
def fdiff(self, argindex=1):
if argindex == 1:
z = self.args[0]
return -1/(z**2*sqrt(1 + 1/z**2))
else:
raise ArgumentIndexError(self, argindex)
@staticmethod
@cacheit
def _acsch_table():
return {
S.ImaginaryUnit: -S.Pi / 2,
S.ImaginaryUnit*(sqrt(2) + sqrt(6)): -S.Pi / 12,
S.ImaginaryUnit*(1 + sqrt(5)): -S.Pi / 10,
S.ImaginaryUnit*2 / sqrt(2 - sqrt(2)): -S.Pi / 8,
S.ImaginaryUnit*2: -S.Pi / 6,
S.ImaginaryUnit*sqrt(2 + 2/sqrt(5)): -S.Pi / 5,
S.ImaginaryUnit*sqrt(2): -S.Pi / 4,
S.ImaginaryUnit*(sqrt(5)-1): -3*S.Pi / 10,
S.ImaginaryUnit*2 / sqrt(3): -S.Pi / 3,
S.ImaginaryUnit*2 / sqrt(2 + sqrt(2)): -3*S.Pi / 8,
S.ImaginaryUnit*sqrt(2 - 2/sqrt(5)): -2*S.Pi / 5,
S.ImaginaryUnit*(sqrt(6) - sqrt(2)): -5*S.Pi / 12,
S(2): -S.ImaginaryUnit*log((1+sqrt(5))/2),
}
@classmethod
def eval(cls, arg):
if arg.is_Number:
if arg is S.NaN:
return S.NaN
elif arg is S.Infinity:
return S.Zero
elif arg is S.NegativeInfinity:
return S.Zero
elif arg.is_zero:
return S.ComplexInfinity
elif arg is S.One:
return log(1 + sqrt(2))
elif arg is S.NegativeOne:
return - log(1 + sqrt(2))
if arg.is_number:
cst_table = cls._acsch_table()
if arg in cst_table:
return cst_table[arg]*S.ImaginaryUnit
if arg is S.ComplexInfinity:
return S.Zero
if arg.is_infinite:
return S.Zero
if arg.is_zero:
return S.ComplexInfinity
if arg.could_extract_minus_sign():
return -cls(-arg)
@staticmethod
@cacheit
def taylor_term(n, x, *previous_terms):
if n == 0:
return log(2 / x)
elif n < 0 or n % 2 == 1:
return S.Zero
else:
x = sympify(x)
if len(previous_terms) > 2 and n > 2:
p = previous_terms[-2]
return -p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
else:
k = n // 2
R = RisingFactorial(S.Half, k) * n
F = factorial(k) * n // 2 * n // 2
return S.NegativeOne**(k +1) * R / F * x**n / 4
def _eval_as_leading_term(self, x, logx=None, cdir=0): # acsch
arg = self.args[0]
x0 = arg.subs(x, 0).cancel()
# Handling branch points
if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.Zero):
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
if x0 is S.ComplexInfinity:
return (1/arg).as_leading_term(x)
# Handling points lying on branch cuts (-I, I)
if x0.is_imaginary and (1 + x0**2).is_positive:
ndir = arg.dir(x, cdir if cdir else 1)
if re(ndir).is_positive:
if im(x0).is_positive:
return -self.func(x0) - I*pi
elif re(ndir).is_negative:
if im(x0).is_negative:
return -self.func(x0) + I*pi
else:
return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
return self.func(x0)
def _eval_nseries(self, x, n, logx, cdir=0): # acsch
from sympy.series.order import O
arg = self.args[0]
arg0 = arg.subs(x, 0)
# Handling branch points
if arg0 is S.ImaginaryUnit:
t = Dummy('t', positive=True)
ser = acsch(S.ImaginaryUnit + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = -S.ImaginaryUnit + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else -I*pi/2 + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
res = ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
return res
if arg0 == S.NegativeOne*S.ImaginaryUnit:
t = Dummy('t', positive=True)
ser = acsch(-S.ImaginaryUnit + t**2).rewrite(log).nseries(t, 0, 2*n)
arg1 = S.ImaginaryUnit + self.args[0]
f = arg1.as_leading_term(x)
g = (arg1 - f)/ f
if not g.is_meromorphic(x, 0): # cannot be expanded
return O(1) if n == 0 else I*pi/2 + O(sqrt(x))
res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
res = (res1.removeO()*sqrt(f)).expand()
return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)
res = Function._eval_nseries(self, x, n=n, logx=logx)
if arg0 is S.ComplexInfinity:
return res
# Handling points lying on branch cuts (-I, I)
if arg0.is_imaginary and (1 + arg0**2).is_positive:
ndir = self.args[0].dir(x, cdir if cdir else 1)
if re(ndir).is_positive:
if im(arg0).is_positive:
return -res - I*pi
elif re(ndir).is_negative:
if im(arg0).is_negative:
return -res + I*pi
else:
return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
return res
def inverse(self, argindex=1):
"""
Returns the inverse of this function.
"""
return csch
def _eval_rewrite_as_log(self, arg, **kwargs):
return log(1/arg + sqrt(1/arg**2 + 1))
_eval_rewrite_as_tractable = _eval_rewrite_as_log
def _eval_rewrite_as_asinh(self, arg, **kwargs):
return asinh(1/arg)
def _eval_rewrite_as_acosh(self, arg, **kwargs):
return S.ImaginaryUnit*(sqrt(1 - S.ImaginaryUnit/arg)/sqrt(S.ImaginaryUnit/arg - 1)*
acosh(S.ImaginaryUnit/arg) - S.Pi*S.Half)
def _eval_rewrite_as_atanh(self, arg, **kwargs):
arg2 = arg**2
arg2p1 = arg2 + 1
return sqrt(-arg2)/arg*(S.Pi*S.Half -
sqrt(-arg2p1**2)/arg2p1*atanh(sqrt(arg2p1)))
def _eval_is_zero(self):
return self.args[0].is_infinite
|
68ceeac879e8dd31d3a02c72d87328d5226520d9b83e3ed28e82ed00b97a0c9d | from sympy.concrete.products import Product
from sympy.core.function import expand_func
from sympy.core.mod import Mod
from sympy.core.mul import Mul
from sympy.core import EulerGamma
from sympy.core.numbers import (Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.combinatorial.factorials import (ff, rf, binomial, factorial, factorial2)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.gamma_functions import (gamma, polygamma)
from sympy.polys.polytools import Poly
from sympy.series.order import O
from sympy.simplify.simplify import simplify
from sympy.core.expr import unchanged
from sympy.core.function import ArgumentIndexError
from sympy.functions.combinatorial.factorials import subfactorial
from sympy.functions.special.gamma_functions import uppergamma
from sympy.testing.pytest import XFAIL, raises, slow
#Solves and Fixes Issue #10388 - This is the updated test for the same solved issue
def test_rf_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert rf(nan, y) is nan
assert rf(x, nan) is nan
assert unchanged(rf, x, y)
assert rf(oo, 0) == 1
assert rf(-oo, 0) == 1
assert rf(oo, 6) is oo
assert rf(-oo, 7) is -oo
assert rf(-oo, 6) is oo
assert rf(oo, -6) is oo
assert rf(-oo, -7) is oo
assert rf(-1, pi) == 0
assert rf(-5, 1 + I) == 0
assert unchanged(rf, -3, k)
assert unchanged(rf, x, Symbol('k', integer=False))
assert rf(-3, Symbol('k', integer=False)) == 0
assert rf(Symbol('x', negative=True, integer=True), Symbol('k', integer=False)) == 0
assert rf(x, 0) == 1
assert rf(x, 1) == x
assert rf(x, 2) == x*(x + 1)
assert rf(x, 3) == x*(x + 1)*(x + 2)
assert rf(x, 5) == x*(x + 1)*(x + 2)*(x + 3)*(x + 4)
assert rf(x, -1) == 1/(x - 1)
assert rf(x, -2) == 1/((x - 1)*(x - 2))
assert rf(x, -3) == 1/((x - 1)*(x - 2)*(x - 3))
assert rf(1, 100) == factorial(100)
assert rf(x**2 + 3*x, 2) == (x**2 + 3*x)*(x**2 + 3*x + 1)
assert isinstance(rf(x**2 + 3*x, 2), Mul)
assert rf(x**3 + x, -2) == 1/((x**3 + x - 1)*(x**3 + x - 2))
assert rf(Poly(x**2 + 3*x, x), 2) == Poly(x**4 + 8*x**3 + 19*x**2 + 12*x, x)
assert isinstance(rf(Poly(x**2 + 3*x, x), 2), Poly)
raises(ValueError, lambda: rf(Poly(x**2 + 3*x, x, y), 2))
assert rf(Poly(x**3 + x, x), -2) == 1/(x**6 - 9*x**5 + 35*x**4 - 75*x**3 + 94*x**2 - 66*x + 20)
raises(ValueError, lambda: rf(Poly(x**3 + x, x, y), -2))
assert rf(x, m).is_integer is None
assert rf(n, k).is_integer is None
assert rf(n, m).is_integer is True
assert rf(n, k + pi).is_integer is False
assert rf(n, m + pi).is_integer is False
assert rf(pi, m).is_integer is False
def check(x, k, o, n):
a, b = Dummy(), Dummy()
r = lambda x, k: o(a, b).rewrite(n).subs({a:x,b:k})
for i in range(-5,5):
for j in range(-5,5):
assert o(i, j) == r(i, j), (o, n, i, j)
check(x, k, rf, ff)
check(x, k, rf, binomial)
check(n, k, rf, factorial)
check(x, y, rf, factorial)
check(x, y, rf, binomial)
assert rf(x, k).rewrite(ff) == ff(x + k - 1, k)
assert rf(x, k).rewrite(gamma) == Piecewise(
(gamma(k + x)/gamma(x), x > 0),
((-1)**k*gamma(1 - x)/gamma(-k - x + 1), True))
assert rf(5, k).rewrite(gamma) == gamma(k + 5)/24
assert rf(x, k).rewrite(binomial) == factorial(k)*binomial(x + k - 1, k)
assert rf(n, k).rewrite(factorial) == Piecewise(
(factorial(k + n - 1)/factorial(n - 1), n > 0),
((-1)**k*factorial(-n)/factorial(-k - n), True))
assert rf(5, k).rewrite(factorial) == factorial(k + 4)/24
assert rf(x, y).rewrite(factorial) == rf(x, y)
assert rf(x, y).rewrite(binomial) == rf(x, y)
import random
from mpmath import rf as mpmath_rf
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
assert (abs(mpmath_rf(x, k) - rf(x, k)) < 10**(-15))
def test_ff_eval_apply():
x, y = symbols('x,y')
n, k = symbols('n k', integer=True)
m = Symbol('m', integer=True, nonnegative=True)
assert ff(nan, y) is nan
assert ff(x, nan) is nan
assert unchanged(ff, x, y)
assert ff(oo, 0) == 1
assert ff(-oo, 0) == 1
assert ff(oo, 6) is oo
assert ff(-oo, 7) is -oo
assert ff(-oo, 6) is oo
assert ff(oo, -6) is oo
assert ff(-oo, -7) is oo
assert ff(x, 0) == 1
assert ff(x, 1) == x
assert ff(x, 2) == x*(x - 1)
assert ff(x, 3) == x*(x - 1)*(x - 2)
assert ff(x, 5) == x*(x - 1)*(x - 2)*(x - 3)*(x - 4)
assert ff(x, -1) == 1/(x + 1)
assert ff(x, -2) == 1/((x + 1)*(x + 2))
assert ff(x, -3) == 1/((x + 1)*(x + 2)*(x + 3))
assert ff(100, 100) == factorial(100)
assert ff(2*x**2 - 5*x, 2) == (2*x**2 - 5*x)*(2*x**2 - 5*x - 1)
assert isinstance(ff(2*x**2 - 5*x, 2), Mul)
assert ff(x**2 + 3*x, -2) == 1/((x**2 + 3*x + 1)*(x**2 + 3*x + 2))
assert ff(Poly(2*x**2 - 5*x, x), 2) == Poly(4*x**4 - 28*x**3 + 59*x**2 - 35*x, x)
assert isinstance(ff(Poly(2*x**2 - 5*x, x), 2), Poly)
raises(ValueError, lambda: ff(Poly(2*x**2 - 5*x, x, y), 2))
assert ff(Poly(x**2 + 3*x, x), -2) == 1/(x**4 + 12*x**3 + 49*x**2 + 78*x + 40)
raises(ValueError, lambda: ff(Poly(x**2 + 3*x, x, y), -2))
assert ff(x, m).is_integer is None
assert ff(n, k).is_integer is None
assert ff(n, m).is_integer is True
assert ff(n, k + pi).is_integer is False
assert ff(n, m + pi).is_integer is False
assert ff(pi, m).is_integer is False
assert isinstance(ff(x, x), ff)
assert ff(n, n) == factorial(n)
def check(x, k, o, n):
a, b = Dummy(), Dummy()
r = lambda x, k: o(a, b).rewrite(n).subs({a:x,b:k})
for i in range(-5,5):
for j in range(-5,5):
assert o(i, j) == r(i, j), (o, n)
check(x, k, ff, rf)
check(x, k, ff, gamma)
check(n, k, ff, factorial)
check(x, k, ff, binomial)
check(x, y, ff, factorial)
check(x, y, ff, binomial)
assert ff(x, k).rewrite(rf) == rf(x - k + 1, k)
assert ff(x, k).rewrite(gamma) == Piecewise(
(gamma(x + 1)/gamma(-k + x + 1), x >= 0),
((-1)**k*gamma(k - x)/gamma(-x), True))
assert ff(5, k).rewrite(gamma) == 120/gamma(6 - k)
assert ff(n, k).rewrite(factorial) == Piecewise(
(factorial(n)/factorial(-k + n), n >= 0),
((-1)**k*factorial(k - n - 1)/factorial(-n - 1), True))
assert ff(5, k).rewrite(factorial) == 120/factorial(5 - k)
assert ff(x, k).rewrite(binomial) == factorial(k) * binomial(x, k)
assert ff(x, y).rewrite(factorial) == ff(x, y)
assert ff(x, y).rewrite(binomial) == ff(x, y)
import random
from mpmath import ff as mpmath_ff
for i in range(100):
x = -500 + 500 * random.random()
k = -500 + 500 * random.random()
a = mpmath_ff(x, k)
b = ff(x, k)
assert (abs(a - b) < abs(a) * 10**(-15))
def test_rf_ff_eval_hiprec():
maple = Float('6.9109401292234329956525265438452')
us = ff(18, Rational(2, 3)).evalf(32)
assert abs(us - maple)/us < 1e-31
maple = Float('6.8261540131125511557924466355367')
us = rf(18, Rational(2, 3)).evalf(32)
assert abs(us - maple)/us < 1e-31
maple = Float('34.007346127440197150854651814225')
us = rf(Float('4.4', 32), Float('2.2', 32));
assert abs(us - maple)/us < 1e-31
def test_rf_lambdify_mpmath():
from sympy.utilities.lambdify import lambdify
x, y = symbols('x,y')
f = lambdify((x,y), rf(x, y), 'mpmath')
maple = Float('34.007346127440197')
us = f(4.4, 2.2)
assert abs(us - maple)/us < 1e-15
def test_factorial():
x = Symbol('x')
n = Symbol('n', integer=True)
k = Symbol('k', integer=True, nonnegative=True)
r = Symbol('r', integer=False)
s = Symbol('s', integer=False, negative=True)
t = Symbol('t', nonnegative=True)
u = Symbol('u', noninteger=True)
assert factorial(-2) is zoo
assert factorial(0) == 1
assert factorial(7) == 5040
assert factorial(19) == 121645100408832000
assert factorial(31) == 8222838654177922817725562880000000
assert factorial(n).func == factorial
assert factorial(2*n).func == factorial
assert factorial(x).is_integer is None
assert factorial(n).is_integer is None
assert factorial(k).is_integer
assert factorial(r).is_integer is None
assert factorial(n).is_positive is None
assert factorial(k).is_positive
assert factorial(x).is_real is None
assert factorial(n).is_real is None
assert factorial(k).is_real is True
assert factorial(r).is_real is None
assert factorial(s).is_real is True
assert factorial(t).is_real is True
assert factorial(u).is_real is True
assert factorial(x).is_composite is None
assert factorial(n).is_composite is None
assert factorial(k).is_composite is None
assert factorial(k + 3).is_composite is True
assert factorial(r).is_composite is None
assert factorial(s).is_composite is None
assert factorial(t).is_composite is None
assert factorial(u).is_composite is None
assert factorial(oo) is oo
def test_factorial_Mod():
pr = Symbol('pr', prime=True)
p, q = 10**9 + 9, 10**9 + 33 # prime modulo
r, s = 10**7 + 5, 33333333 # composite modulo
assert Mod(factorial(pr - 1), pr) == pr - 1
assert Mod(factorial(pr - 1), -pr) == -1
assert Mod(factorial(r - 1, evaluate=False), r) == 0
assert Mod(factorial(s - 1, evaluate=False), s) == 0
assert Mod(factorial(p - 1, evaluate=False), p) == p - 1
assert Mod(factorial(q - 1, evaluate=False), q) == q - 1
assert Mod(factorial(p - 50, evaluate=False), p) == 854928834
assert Mod(factorial(q - 1800, evaluate=False), q) == 905504050
assert Mod(factorial(153, evaluate=False), r) == Mod(factorial(153), r)
assert Mod(factorial(255, evaluate=False), s) == Mod(factorial(255), s)
assert Mod(factorial(4, evaluate=False), 3) == S.Zero
assert Mod(factorial(5, evaluate=False), 6) == S.Zero
def test_factorial_diff():
n = Symbol('n', integer=True)
assert factorial(n).diff(n) == \
gamma(1 + n)*polygamma(0, 1 + n)
assert factorial(n**2).diff(n) == \
2*n*gamma(1 + n**2)*polygamma(0, 1 + n**2)
raises(ArgumentIndexError, lambda: factorial(n**2).fdiff(2))
def test_factorial_series():
n = Symbol('n', integer=True)
assert factorial(n).series(n, 0, 3) == \
1 - n*EulerGamma + n**2*(EulerGamma**2/2 + pi**2/12) + O(n**3)
def test_factorial_rewrite():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True, nonnegative=True)
assert factorial(n).rewrite(gamma) == gamma(n + 1)
_i = Dummy('i')
assert factorial(k).rewrite(Product).dummy_eq(Product(_i, (_i, 1, k)))
assert factorial(n).rewrite(Product) == factorial(n)
def test_factorial2():
n = Symbol('n', integer=True)
assert factorial2(-1) == 1
assert factorial2(0) == 1
assert factorial2(7) == 105
assert factorial2(8) == 384
# The following is exhaustive
tt = Symbol('tt', integer=True, nonnegative=True)
tte = Symbol('tte', even=True, nonnegative=True)
tpe = Symbol('tpe', even=True, positive=True)
tto = Symbol('tto', odd=True, nonnegative=True)
tf = Symbol('tf', integer=True, nonnegative=False)
tfe = Symbol('tfe', even=True, nonnegative=False)
tfo = Symbol('tfo', odd=True, nonnegative=False)
ft = Symbol('ft', integer=False, nonnegative=True)
ff = Symbol('ff', integer=False, nonnegative=False)
fn = Symbol('fn', integer=False)
nt = Symbol('nt', nonnegative=True)
nf = Symbol('nf', nonnegative=False)
nn = Symbol('nn')
z = Symbol('z', zero=True)
#Solves and Fixes Issue #10388 - This is the updated test for the same solved issue
raises(ValueError, lambda: factorial2(oo))
raises(ValueError, lambda: factorial2(Rational(5, 2)))
raises(ValueError, lambda: factorial2(-4))
assert factorial2(n).is_integer is None
assert factorial2(tt - 1).is_integer
assert factorial2(tte - 1).is_integer
assert factorial2(tpe - 3).is_integer
assert factorial2(tto - 4).is_integer
assert factorial2(tto - 2).is_integer
assert factorial2(tf).is_integer is None
assert factorial2(tfe).is_integer is None
assert factorial2(tfo).is_integer is None
assert factorial2(ft).is_integer is None
assert factorial2(ff).is_integer is None
assert factorial2(fn).is_integer is None
assert factorial2(nt).is_integer is None
assert factorial2(nf).is_integer is None
assert factorial2(nn).is_integer is None
assert factorial2(n).is_positive is None
assert factorial2(tt - 1).is_positive is True
assert factorial2(tte - 1).is_positive is True
assert factorial2(tpe - 3).is_positive is True
assert factorial2(tpe - 1).is_positive is True
assert factorial2(tto - 2).is_positive is True
assert factorial2(tto - 1).is_positive is True
assert factorial2(tf).is_positive is None
assert factorial2(tfe).is_positive is None
assert factorial2(tfo).is_positive is None
assert factorial2(ft).is_positive is None
assert factorial2(ff).is_positive is None
assert factorial2(fn).is_positive is None
assert factorial2(nt).is_positive is None
assert factorial2(nf).is_positive is None
assert factorial2(nn).is_positive is None
assert factorial2(tt).is_even is None
assert factorial2(tt).is_odd is None
assert factorial2(tte).is_even is None
assert factorial2(tte).is_odd is None
assert factorial2(tte + 2).is_even is True
assert factorial2(tpe).is_even is True
assert factorial2(tpe).is_odd is False
assert factorial2(tto).is_odd is True
assert factorial2(tf).is_even is None
assert factorial2(tf).is_odd is None
assert factorial2(tfe).is_even is None
assert factorial2(tfe).is_odd is None
assert factorial2(tfo).is_even is False
assert factorial2(tfo).is_odd is None
assert factorial2(z).is_even is False
assert factorial2(z).is_odd is True
def test_factorial2_rewrite():
n = Symbol('n', integer=True)
assert factorial2(n).rewrite(gamma) == \
2**(n/2)*Piecewise((1, Eq(Mod(n, 2), 0)), (sqrt(2)/sqrt(pi), Eq(Mod(n, 2), 1)))*gamma(n/2 + 1)
assert factorial2(2*n).rewrite(gamma) == 2**n*gamma(n + 1)
assert factorial2(2*n + 1).rewrite(gamma) == \
sqrt(2)*2**(n + S.Half)*gamma(n + Rational(3, 2))/sqrt(pi)
def test_binomial():
x = Symbol('x')
n = Symbol('n', integer=True)
nz = Symbol('nz', integer=True, nonzero=True)
k = Symbol('k', integer=True)
kp = Symbol('kp', integer=True, positive=True)
kn = Symbol('kn', integer=True, negative=True)
u = Symbol('u', negative=True)
v = Symbol('v', nonnegative=True)
p = Symbol('p', positive=True)
z = Symbol('z', zero=True)
nt = Symbol('nt', integer=False)
kt = Symbol('kt', integer=False)
a = Symbol('a', integer=True, nonnegative=True)
b = Symbol('b', integer=True, nonnegative=True)
assert binomial(0, 0) == 1
assert binomial(1, 1) == 1
assert binomial(10, 10) == 1
assert binomial(n, z) == 1
assert binomial(1, 2) == 0
assert binomial(-1, 2) == 1
assert binomial(1, -1) == 0
assert binomial(-1, 1) == -1
assert binomial(-1, -1) == 0
assert binomial(S.Half, S.Half) == 1
assert binomial(-10, 1) == -10
assert binomial(-10, 7) == -11440
assert binomial(n, -1) == 0 # holds for all integers (negative, zero, positive)
assert binomial(kp, -1) == 0
assert binomial(nz, 0) == 1
assert expand_func(binomial(n, 1)) == n
assert expand_func(binomial(n, 2)) == n*(n - 1)/2
assert expand_func(binomial(n, n - 2)) == n*(n - 1)/2
assert expand_func(binomial(n, n - 1)) == n
assert binomial(n, 3).func == binomial
assert binomial(n, 3).expand(func=True) == n**3/6 - n**2/2 + n/3
assert expand_func(binomial(n, 3)) == n*(n - 2)*(n - 1)/6
assert binomial(n, n).func == binomial # e.g. (-1, -1) == 0, (2, 2) == 1
assert binomial(n, n + 1).func == binomial # e.g. (-1, 0) == 1
assert binomial(kp, kp + 1) == 0
assert binomial(kn, kn) == 0 # issue #14529
assert binomial(n, u).func == binomial
assert binomial(kp, u).func == binomial
assert binomial(n, p).func == binomial
assert binomial(n, k).func == binomial
assert binomial(n, n + p).func == binomial
assert binomial(kp, kp + p).func == binomial
assert expand_func(binomial(n, n - 3)) == n*(n - 2)*(n - 1)/6
assert binomial(n, k).is_integer
assert binomial(nt, k).is_integer is None
assert binomial(x, nt).is_integer is False
assert binomial(gamma(25), 6) == 79232165267303928292058750056084441948572511312165380965440075720159859792344339983120618959044048198214221915637090855535036339620413440000
assert binomial(1324, 47) == 906266255662694632984994480774946083064699457235920708992926525848438478406790323869952
assert binomial(1735, 43) == 190910140420204130794758005450919715396159959034348676124678207874195064798202216379800
assert binomial(2512, 53) == 213894469313832631145798303740098720367984955243020898718979538096223399813295457822575338958939834177325304000
assert binomial(3383, 52) == 27922807788818096863529701501764372757272890613101645521813434902890007725667814813832027795881839396839287659777235
assert binomial(4321, 51) == 124595639629264868916081001263541480185227731958274383287107643816863897851139048158022599533438936036467601690983780576
assert binomial(a, b).is_nonnegative is True
assert binomial(-1, 2, evaluate=False).is_nonnegative is True
assert binomial(10, 5, evaluate=False).is_nonnegative is True
assert binomial(10, -3, evaluate=False).is_nonnegative is True
assert binomial(-10, -3, evaluate=False).is_nonnegative is True
assert binomial(-10, 2, evaluate=False).is_nonnegative is True
assert binomial(-10, 1, evaluate=False).is_nonnegative is False
assert binomial(-10, 7, evaluate=False).is_nonnegative is False
# issue #14625
for _ in (pi, -pi, nt, v, a):
assert binomial(_, _) == 1
assert binomial(_, _ - 1) == _
assert isinstance(binomial(u, u), binomial)
assert isinstance(binomial(u, u - 1), binomial)
assert isinstance(binomial(x, x), binomial)
assert isinstance(binomial(x, x - 1), binomial)
#issue #18802
assert expand_func(binomial(x + 1, x)) == x + 1
assert expand_func(binomial(x, x - 1)) == x
assert expand_func(binomial(x + 1, x - 1)) == x*(x + 1)/2
assert expand_func(binomial(x**2 + 1, x**2)) == x**2 + 1
# issue #13980 and #13981
assert binomial(-7, -5) == 0
assert binomial(-23, -12) == 0
assert binomial(Rational(13, 2), -10) == 0
assert binomial(-49, -51) == 0
assert binomial(19, Rational(-7, 2)) == S(-68719476736)/(911337863661225*pi)
assert binomial(0, Rational(3, 2)) == S(-2)/(3*pi)
assert binomial(-3, Rational(-7, 2)) is zoo
assert binomial(kn, kt) is zoo
assert binomial(nt, kt).func == binomial
assert binomial(nt, Rational(15, 6)) == 8*gamma(nt + 1)/(15*sqrt(pi)*gamma(nt - Rational(3, 2)))
assert binomial(Rational(20, 3), Rational(-10, 8)) == gamma(Rational(23, 3))/(gamma(Rational(-1, 4))*gamma(Rational(107, 12)))
assert binomial(Rational(19, 2), Rational(-7, 2)) == Rational(-1615, 8388608)
assert binomial(Rational(-13, 5), Rational(-7, 8)) == gamma(Rational(-8, 5))/(gamma(Rational(-29, 40))*gamma(Rational(1, 8)))
assert binomial(Rational(-19, 8), Rational(-13, 5)) == gamma(Rational(-11, 8))/(gamma(Rational(-8, 5))*gamma(Rational(49, 40)))
# binomial for complexes
assert binomial(I, Rational(-89, 8)) == gamma(1 + I)/(gamma(Rational(-81, 8))*gamma(Rational(97, 8) + I))
assert binomial(I, 2*I) == gamma(1 + I)/(gamma(1 - I)*gamma(1 + 2*I))
assert binomial(-7, I) is zoo
assert binomial(Rational(-7, 6), I) == gamma(Rational(-1, 6))/(gamma(Rational(-1, 6) - I)*gamma(1 + I))
assert binomial((1+2*I), (1+3*I)) == gamma(2 + 2*I)/(gamma(1 - I)*gamma(2 + 3*I))
assert binomial(I, 5) == Rational(1, 3) - I/S(12)
assert binomial((2*I + 3), 7) == -13*I/S(63)
assert isinstance(binomial(I, n), binomial)
assert expand_func(binomial(3, 2, evaluate=False)) == 3
assert expand_func(binomial(n, 0, evaluate=False)) == 1
assert expand_func(binomial(n, -2, evaluate=False)) == 0
assert expand_func(binomial(n, k)) == binomial(n, k)
def test_binomial_Mod():
p, q = 10**5 + 3, 10**9 + 33 # prime modulo
r = 10**7 + 5 # composite modulo
# A few tests to get coverage
# Lucas Theorem
assert Mod(binomial(156675, 4433, evaluate=False), p) == Mod(binomial(156675, 4433), p)
# factorial Mod
assert Mod(binomial(1234, 432, evaluate=False), q) == Mod(binomial(1234, 432), q)
# binomial factorize
assert Mod(binomial(253, 113, evaluate=False), r) == Mod(binomial(253, 113), r)
@slow
def test_binomial_Mod_slow():
p, q = 10**5 + 3, 10**9 + 33 # prime modulo
r, s = 10**7 + 5, 33333333 # composite modulo
n, k, m = symbols('n k m')
assert (binomial(n, k) % q).subs({n: s, k: p}) == Mod(binomial(s, p), q)
assert (binomial(n, k) % m).subs({n: 8, k: 5, m: 13}) == 4
assert (binomial(9, k) % 7).subs(k, 2) == 1
# Lucas Theorem
assert Mod(binomial(123456, 43253, evaluate=False), p) == Mod(binomial(123456, 43253), p)
assert Mod(binomial(-178911, 237, evaluate=False), p) == Mod(-binomial(178911 + 237 - 1, 237), p)
assert Mod(binomial(-178911, 238, evaluate=False), p) == Mod(binomial(178911 + 238 - 1, 238), p)
# factorial Mod
assert Mod(binomial(9734, 451, evaluate=False), q) == Mod(binomial(9734, 451), q)
assert Mod(binomial(-10733, 4459, evaluate=False), q) == Mod(binomial(-10733, 4459), q)
assert Mod(binomial(-15733, 4458, evaluate=False), q) == Mod(binomial(-15733, 4458), q)
assert Mod(binomial(23, -38, evaluate=False), q) is S.Zero
assert Mod(binomial(23, 38, evaluate=False), q) is S.Zero
# binomial factorize
assert Mod(binomial(753, 119, evaluate=False), r) == Mod(binomial(753, 119), r)
assert Mod(binomial(3781, 948, evaluate=False), s) == Mod(binomial(3781, 948), s)
assert Mod(binomial(25773, 1793, evaluate=False), s) == Mod(binomial(25773, 1793), s)
assert Mod(binomial(-753, 118, evaluate=False), r) == Mod(binomial(-753, 118), r)
assert Mod(binomial(-25773, 1793, evaluate=False), s) == Mod(binomial(-25773, 1793), s)
def test_binomial_diff():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True)
assert binomial(n, k).diff(n) == \
(-polygamma(0, 1 + n - k) + polygamma(0, 1 + n))*binomial(n, k)
assert binomial(n**2, k**3).diff(n) == \
2*n*(-polygamma(
0, 1 + n**2 - k**3) + polygamma(0, 1 + n**2))*binomial(n**2, k**3)
assert binomial(n, k).diff(k) == \
(-polygamma(0, 1 + k) + polygamma(0, 1 + n - k))*binomial(n, k)
assert binomial(n**2, k**3).diff(k) == \
3*k**2*(-polygamma(
0, 1 + k**3) + polygamma(0, 1 + n**2 - k**3))*binomial(n**2, k**3)
raises(ArgumentIndexError, lambda: binomial(n, k).fdiff(3))
def test_binomial_rewrite():
n = Symbol('n', integer=True)
k = Symbol('k', integer=True)
x = Symbol('x')
assert binomial(n, k).rewrite(
factorial) == factorial(n)/(factorial(k)*factorial(n - k))
assert binomial(
n, k).rewrite(gamma) == gamma(n + 1)/(gamma(k + 1)*gamma(n - k + 1))
assert binomial(n, k).rewrite(ff) == ff(n, k) / factorial(k)
assert binomial(n, x).rewrite(ff) == binomial(n, x)
@XFAIL
def test_factorial_simplify_fail():
# simplify(factorial(x + 1).diff(x) - ((x + 1)*factorial(x)).diff(x))) == 0
from sympy.abc import x
assert simplify(x*polygamma(0, x + 1) - x*polygamma(0, x + 2) +
polygamma(0, x + 1) - polygamma(0, x + 2) + 1) == 0
def test_subfactorial():
assert all(subfactorial(i) == ans for i, ans in enumerate(
[1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496]))
assert subfactorial(oo) is oo
assert subfactorial(nan) is nan
assert subfactorial(23) == 9510425471055777937262
assert unchanged(subfactorial, 2.2)
x = Symbol('x')
assert subfactorial(x).rewrite(uppergamma) == uppergamma(x + 1, -1)/S.Exp1
tt = Symbol('tt', integer=True, nonnegative=True)
tf = Symbol('tf', integer=True, nonnegative=False)
tn = Symbol('tf', integer=True)
ft = Symbol('ft', integer=False, nonnegative=True)
ff = Symbol('ff', integer=False, nonnegative=False)
fn = Symbol('ff', integer=False)
nt = Symbol('nt', nonnegative=True)
nf = Symbol('nf', nonnegative=False)
nn = Symbol('nf')
te = Symbol('te', even=True, nonnegative=True)
to = Symbol('to', odd=True, nonnegative=True)
assert subfactorial(tt).is_integer
assert subfactorial(tf).is_integer is None
assert subfactorial(tn).is_integer is None
assert subfactorial(ft).is_integer is None
assert subfactorial(ff).is_integer is None
assert subfactorial(fn).is_integer is None
assert subfactorial(nt).is_integer is None
assert subfactorial(nf).is_integer is None
assert subfactorial(nn).is_integer is None
assert subfactorial(tt).is_nonnegative
assert subfactorial(tf).is_nonnegative is None
assert subfactorial(tn).is_nonnegative is None
assert subfactorial(ft).is_nonnegative is None
assert subfactorial(ff).is_nonnegative is None
assert subfactorial(fn).is_nonnegative is None
assert subfactorial(nt).is_nonnegative is None
assert subfactorial(nf).is_nonnegative is None
assert subfactorial(nn).is_nonnegative is None
assert subfactorial(tt).is_even is None
assert subfactorial(tt).is_odd is None
assert subfactorial(te).is_odd is True
assert subfactorial(to).is_even is True
|
afca6abdf7c812cbbc2bc54a797092b07e085d424acc36b6c1dc7ea4296c7669 | from sympy.assumptions.refine import refine
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.function import expand_log
from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (adjoint, conjugate, re, sign, transpose)
from sympy.functions.elementary.exponential import (LambertW, exp, exp_polar, log)
from sympy.functions.elementary.hyperbolic import (cosh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.polys.polytools import gcd
from sympy.series.order import O
from sympy.simplify.simplify import simplify
from sympy.core.parameters import global_parameters
from sympy.functions.elementary.exponential import match_real_imag
from sympy.abc import x, y, z
from sympy.core.expr import unchanged
from sympy.core.function import ArgumentIndexError
from sympy.testing.pytest import raises, XFAIL, _both_exp_pow
@_both_exp_pow
def test_exp_values():
if global_parameters.exp_is_pow:
assert type(exp(x)) is Pow
else:
assert type(exp(x)) is exp
k = Symbol('k', integer=True)
assert exp(nan) is nan
assert exp(oo) is oo
assert exp(-oo) == 0
assert exp(0) == 1
assert exp(1) == E
assert exp(-1 + x).as_base_exp() == (S.Exp1, x - 1)
assert exp(1 + x).as_base_exp() == (S.Exp1, x + 1)
assert exp(pi*I/2) == I
assert exp(pi*I) == -1
assert exp(pi*I*Rational(3, 2)) == -I
assert exp(2*pi*I) == 1
assert refine(exp(pi*I*2*k)) == 1
assert refine(exp(pi*I*2*(k + S.Half))) == -1
assert refine(exp(pi*I*2*(k + Rational(1, 4)))) == I
assert refine(exp(pi*I*2*(k + Rational(3, 4)))) == -I
assert exp(log(x)) == x
assert exp(2*log(x)) == x**2
assert exp(pi*log(x)) == x**pi
assert exp(17*log(x) + E*log(y)) == x**17 * y**E
assert exp(x*log(x)) != x**x
assert exp(sin(x)*log(x)) != x
assert exp(3*log(x) + oo*x) == exp(oo*x) * x**3
assert exp(4*log(x)*log(y) + 3*log(x)) == x**3 * exp(4*log(x)*log(y))
assert exp(-oo, evaluate=False).is_finite is True
assert exp(oo, evaluate=False).is_finite is False
@_both_exp_pow
def test_exp_period():
assert exp(I*pi*Rational(9, 4)) == exp(I*pi/4)
assert exp(I*pi*Rational(46, 18)) == exp(I*pi*Rational(5, 9))
assert exp(I*pi*Rational(25, 7)) == exp(I*pi*Rational(-3, 7))
assert exp(I*pi*Rational(-19, 3)) == exp(-I*pi/3)
assert exp(I*pi*Rational(37, 8)) - exp(I*pi*Rational(-11, 8)) == 0
assert exp(I*pi*Rational(-5, 3)) / exp(I*pi*Rational(11, 5)) * exp(I*pi*Rational(148, 15)) == 1
assert exp(2 - I*pi*Rational(17, 5)) == exp(2 + I*pi*Rational(3, 5))
assert exp(log(3) + I*pi*Rational(29, 9)) == 3 * exp(I*pi*Rational(-7, 9))
n = Symbol('n', integer=True)
e = Symbol('e', even=True)
assert exp(e*I*pi) == 1
assert exp((e + 1)*I*pi) == -1
assert exp((1 + 4*n)*I*pi/2) == I
assert exp((-1 + 4*n)*I*pi/2) == -I
@_both_exp_pow
def test_exp_log():
x = Symbol("x", real=True)
assert log(exp(x)) == x
assert exp(log(x)) == x
if not global_parameters.exp_is_pow:
assert log(x).inverse() == exp
assert exp(x).inverse() == log
y = Symbol("y", polar=True)
assert log(exp_polar(z)) == z
assert exp(log(y)) == y
@_both_exp_pow
def test_exp_expand():
e = exp(log(Rational(2))*(1 + x) - log(Rational(2))*x)
assert e.expand() == 2
assert exp(x + y) != exp(x)*exp(y)
assert exp(x + y).expand() == exp(x)*exp(y)
@_both_exp_pow
def test_exp__as_base_exp():
assert exp(x).as_base_exp() == (E, x)
assert exp(2*x).as_base_exp() == (E, 2*x)
assert exp(x*y).as_base_exp() == (E, x*y)
assert exp(-x).as_base_exp() == (E, -x)
# Pow( *expr.as_base_exp() ) == expr invariant should hold
assert E**x == exp(x)
assert E**(2*x) == exp(2*x)
assert E**(x*y) == exp(x*y)
assert exp(x).base is S.Exp1
assert exp(x).exp == x
@_both_exp_pow
def test_exp_infinity():
assert exp(I*y) != nan
assert refine(exp(I*oo)) is nan
assert refine(exp(-I*oo)) is nan
assert exp(y*I*oo) != nan
assert exp(zoo) is nan
x = Symbol('x', extended_real=True, finite=False)
assert exp(x).is_complex is None
@_both_exp_pow
def test_exp_subs():
x = Symbol('x')
e = (exp(3*log(x), evaluate=False)) # evaluates to x**3
assert e.subs(x**3, y**3) == e
assert e.subs(x**2, 5) == e
assert (x**3).subs(x**2, y) != y**Rational(3, 2)
assert exp(exp(x) + exp(x**2)).subs(exp(exp(x)), y) == y * exp(exp(x**2))
assert exp(x).subs(E, y) == y**x
x = symbols('x', real=True)
assert exp(5*x).subs(exp(7*x), y) == y**Rational(5, 7)
assert exp(2*x + 7).subs(exp(3*x), y) == y**Rational(2, 3) * exp(7)
x = symbols('x', positive=True)
assert exp(3*log(x)).subs(x**2, y) == y**Rational(3, 2)
# differentiate between E and exp
assert exp(exp(x + E)).subs(exp, 3) == 3**(3**(x + E))
assert exp(exp(x + E)).subs(exp, sin) == sin(sin(x + E))
assert exp(exp(x + E)).subs(E, 3) == 3**(3**(x + 3))
assert exp(3).subs(E, sin) == sin(3)
def test_exp_adjoint():
assert adjoint(exp(x)) == exp(adjoint(x))
def test_exp_conjugate():
assert conjugate(exp(x)) == exp(conjugate(x))
@_both_exp_pow
def test_exp_transpose():
assert transpose(exp(x)) == exp(transpose(x))
@_both_exp_pow
def test_exp_rewrite():
assert exp(x).rewrite(sin) == sinh(x) + cosh(x)
assert exp(x*I).rewrite(cos) == cos(x) + I*sin(x)
assert exp(1).rewrite(cos) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(1).rewrite(sin) == sinh(1) + cosh(1)
assert exp(x).rewrite(tanh) == (1 + tanh(x/2))/(1 - tanh(x/2))
assert exp(pi*I/4).rewrite(sqrt) == sqrt(2)/2 + sqrt(2)*I/2
assert exp(pi*I/3).rewrite(sqrt) == S.Half + sqrt(3)*I/2
if not global_parameters.exp_is_pow:
assert exp(x*log(y)).rewrite(Pow) == y**x
assert exp(log(x)*log(y)).rewrite(Pow) in [x**log(y), y**log(x)]
assert exp(log(log(x))*y).rewrite(Pow) == log(x)**y
n = Symbol('n', integer=True)
assert Sum((exp(pi*I/2)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == Rational(4, 5) + I*2/5
assert Sum((exp(pi*I/4)/2)**n, (n, 0, oo)).rewrite(sqrt).doit() == 1/(1 - sqrt(2)*(1 + I)/4)
assert (Sum((exp(pi*I/3)/2)**n, (n, 0, oo)).rewrite(sqrt).doit().cancel()
== 4*I/(sqrt(3) + 3*I))
@_both_exp_pow
def test_exp_leading_term():
assert exp(x).as_leading_term(x) == 1
assert exp(2 + x).as_leading_term(x) == exp(2)
assert exp((2*x + 3) / (x+1)).as_leading_term(x) == exp(3)
# The following tests are commented, since now SymPy returns the
# original function when the leading term in the series expansion does
# not exist.
# raises(NotImplementedError, lambda: exp(1/x).as_leading_term(x))
# raises(NotImplementedError, lambda: exp((x + 1) / x**2).as_leading_term(x))
# raises(NotImplementedError, lambda: exp(x + 1/x).as_leading_term(x))
@_both_exp_pow
def test_exp_taylor_term():
x = symbols('x')
assert exp(x).taylor_term(1, x) == x
assert exp(x).taylor_term(3, x) == x**3/6
assert exp(x).taylor_term(4, x) == x**4/24
assert exp(x).taylor_term(-1, x) is S.Zero
def test_exp_MatrixSymbol():
A = MatrixSymbol("A", 2, 2)
assert exp(A).has(exp)
def test_exp_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: exp(x).fdiff(2))
def test_log_values():
assert log(nan) is nan
assert log(oo) is oo
assert log(-oo) is oo
assert log(zoo) is zoo
assert log(-zoo) is zoo
assert log(0) is zoo
assert log(1) == 0
assert log(-1) == I*pi
assert log(E) == 1
assert log(-E).expand() == 1 + I*pi
assert unchanged(log, pi)
assert log(-pi).expand() == log(pi) + I*pi
assert unchanged(log, 17)
assert log(-17) == log(17) + I*pi
assert log(I) == I*pi/2
assert log(-I) == -I*pi/2
assert log(17*I) == I*pi/2 + log(17)
assert log(-17*I).expand() == -I*pi/2 + log(17)
assert log(oo*I) is oo
assert log(-oo*I) is oo
assert log(0, 2) is zoo
assert log(0, 5) is zoo
assert exp(-log(3))**(-1) == 3
assert log(S.Half) == -log(2)
assert log(2*3).func is log
assert log(2*3**2).func is log
def test_match_real_imag():
x, y = symbols('x,y', real=True)
i = Symbol('i', imaginary=True)
assert match_real_imag(S.One) == (1, 0)
assert match_real_imag(I) == (0, 1)
assert match_real_imag(3 - 5*I) == (3, -5)
assert match_real_imag(-sqrt(3) + S.Half*I) == (-sqrt(3), S.Half)
assert match_real_imag(x + y*I) == (x, y)
assert match_real_imag(x*I + y*I) == (0, x + y)
assert match_real_imag((x + y)*I) == (0, x + y)
assert match_real_imag(Rational(-2, 3)*i*I) == (None, None)
assert match_real_imag(1 - 2*i) == (None, None)
assert match_real_imag(sqrt(2)*(3 - 5*I)) == (None, None)
def test_log_exact():
# check for pi/2, pi/3, pi/4, pi/6, pi/8, pi/12; pi/5, pi/10:
for n in range(-23, 24):
if gcd(n, 24) != 1:
assert log(exp(n*I*pi/24).rewrite(sqrt)) == n*I*pi/24
for n in range(-9, 10):
assert log(exp(n*I*pi/10).rewrite(sqrt)) == n*I*pi/10
assert log(S.Half - I*sqrt(3)/2) == -I*pi/3
assert log(Rational(-1, 2) + I*sqrt(3)/2) == I*pi*Rational(2, 3)
assert log(-sqrt(2)/2 - I*sqrt(2)/2) == -I*pi*Rational(3, 4)
assert log(-sqrt(3)/2 - I*S.Half) == -I*pi*Rational(5, 6)
assert log(Rational(-1, 4) + sqrt(5)/4 - I*sqrt(sqrt(5)/8 + Rational(5, 8))) == -I*pi*Rational(2, 5)
assert log(sqrt(Rational(5, 8) - sqrt(5)/8) + I*(Rational(1, 4) + sqrt(5)/4)) == I*pi*Rational(3, 10)
assert log(-sqrt(sqrt(2)/4 + S.Half) + I*sqrt(S.Half - sqrt(2)/4)) == I*pi*Rational(7, 8)
assert log(-sqrt(6)/4 - sqrt(2)/4 + I*(-sqrt(6)/4 + sqrt(2)/4)) == -I*pi*Rational(11, 12)
assert log(-1 + I*sqrt(3)) == log(2) + I*pi*Rational(2, 3)
assert log(5 + 5*I) == log(5*sqrt(2)) + I*pi/4
assert log(sqrt(-12)) == log(2*sqrt(3)) + I*pi/2
assert log(-sqrt(6) + sqrt(2) - I*sqrt(6) - I*sqrt(2)) == log(4) - I*pi*Rational(7, 12)
assert log(-sqrt(6-3*sqrt(2)) - I*sqrt(6+3*sqrt(2))) == log(2*sqrt(3)) - I*pi*Rational(5, 8)
assert log(1 + I*sqrt(2-sqrt(2))/sqrt(2+sqrt(2))) == log(2/sqrt(sqrt(2) + 2)) + I*pi/8
assert log(cos(pi*Rational(7, 12)) + I*sin(pi*Rational(7, 12))) == I*pi*Rational(7, 12)
assert log(cos(pi*Rational(6, 5)) + I*sin(pi*Rational(6, 5))) == I*pi*Rational(-4, 5)
assert log(5*(1 + I)/sqrt(2)) == log(5) + I*pi/4
assert log(sqrt(2)*(-sqrt(3) + 1 - sqrt(3)*I - I)) == log(4) - I*pi*Rational(7, 12)
assert log(-sqrt(2)*(1 - I*sqrt(3))) == log(2*sqrt(2)) + I*pi*Rational(2, 3)
assert log(sqrt(3)*I*(-sqrt(6 - 3*sqrt(2)) - I*sqrt(3*sqrt(2) + 6))) == log(6) - I*pi/8
zero = (1 + sqrt(2))**2 - 3 - 2*sqrt(2)
assert log(zero - I*sqrt(3)) == log(sqrt(3)) - I*pi/2
assert unchanged(log, zero + I*zero) or log(zero + zero*I) is zoo
# bail quickly if no obvious simplification is possible:
assert unchanged(log, (sqrt(2)-1/sqrt(sqrt(3)+I))**1000)
# beware of non-real coefficients
assert unchanged(log, sqrt(2-sqrt(5))*(1 + I))
def test_log_base():
assert log(1, 2) == 0
assert log(2, 2) == 1
assert log(3, 2) == log(3)/log(2)
assert log(6, 2) == 1 + log(3)/log(2)
assert log(6, 3) == 1 + log(2)/log(3)
assert log(2**3, 2) == 3
assert log(3**3, 3) == 3
assert log(5, 1) is zoo
assert log(1, 1) is nan
assert log(Rational(2, 3), 10) == log(Rational(2, 3))/log(10)
assert log(Rational(2, 3), Rational(1, 3)) == -log(2)/log(3) + 1
assert log(Rational(2, 3), Rational(2, 5)) == \
log(Rational(2, 3))/log(Rational(2, 5))
# issue 17148
assert log(Rational(8, 3), 2) == -log(3)/log(2) + 3
def test_log_symbolic():
assert log(x, exp(1)) == log(x)
assert log(exp(x)) != x
assert log(x, exp(1)) == log(x)
assert log(x*y) != log(x) + log(y)
assert log(x/y).expand() != log(x) - log(y)
assert log(x/y).expand(force=True) == log(x) - log(y)
assert log(x**y).expand() != y*log(x)
assert log(x**y).expand(force=True) == y*log(x)
assert log(x, 2) == log(x)/log(2)
assert log(E, 2) == 1/log(2)
p, q = symbols('p,q', positive=True)
r = Symbol('r', real=True)
assert log(p**2) != 2*log(p)
assert log(p**2).expand() == 2*log(p)
assert log(x**2).expand() != 2*log(x)
assert log(p**q) != q*log(p)
assert log(exp(p)) == p
assert log(p*q) != log(p) + log(q)
assert log(p*q).expand() == log(p) + log(q)
assert log(-sqrt(3)) == log(sqrt(3)) + I*pi
assert log(-exp(p)) != p + I*pi
assert log(-exp(x)).expand() != x + I*pi
assert log(-exp(r)).expand() == r + I*pi
assert log(x**y) != y*log(x)
assert (log(x**-5)**-1).expand() != -1/log(x)/5
assert (log(p**-5)**-1).expand() == -1/log(p)/5
assert log(-x).func is log and log(-x).args[0] == -x
assert log(-p).func is log and log(-p).args[0] == -p
def test_log_exp():
assert log(exp(4*I*pi)) == 0 # exp evaluates
assert log(exp(-5*I*pi)) == I*pi # exp evaluates
assert log(exp(I*pi*Rational(19, 4))) == I*pi*Rational(3, 4)
assert log(exp(I*pi*Rational(25, 7))) == I*pi*Rational(-3, 7)
assert log(exp(-5*I)) == -5*I + 2*I*pi
@_both_exp_pow
def test_exp_assumptions():
r = Symbol('r', real=True)
i = Symbol('i', imaginary=True)
for e in exp, exp_polar:
assert e(x).is_real is None
assert e(x).is_imaginary is None
assert e(i).is_real is None
assert e(i).is_imaginary is None
assert e(r).is_real is True
assert e(r).is_imaginary is False
assert e(re(x)).is_extended_real is True
assert e(re(x)).is_imaginary is False
assert Pow(E, I*pi, evaluate=False).is_imaginary == False
assert Pow(E, 2*I*pi, evaluate=False).is_imaginary == False
assert Pow(E, I*pi/2, evaluate=False).is_imaginary == True
assert Pow(E, I*pi/3, evaluate=False).is_imaginary is None
assert exp(0, evaluate=False).is_algebraic
a = Symbol('a', algebraic=True)
an = Symbol('an', algebraic=True, nonzero=True)
r = Symbol('r', rational=True)
rn = Symbol('rn', rational=True, nonzero=True)
assert exp(a).is_algebraic is None
assert exp(an).is_algebraic is False
assert exp(pi*r).is_algebraic is None
assert exp(pi*rn).is_algebraic is False
assert exp(0, evaluate=False).is_algebraic is True
assert exp(I*pi/3, evaluate=False).is_algebraic is True
assert exp(I*pi*r, evaluate=False).is_algebraic is True
@_both_exp_pow
def test_exp_AccumBounds():
assert exp(AccumBounds(1, 2)) == AccumBounds(E, E**2)
def test_log_assumptions():
p = symbols('p', positive=True)
n = symbols('n', negative=True)
z = symbols('z', zero=True)
x = symbols('x', infinite=True, extended_positive=True)
assert log(z).is_positive is False
assert log(x).is_extended_positive is True
assert log(2) > 0
assert log(1, evaluate=False).is_zero
assert log(1 + z).is_zero
assert log(p).is_zero is None
assert log(n).is_zero is False
assert log(0.5).is_negative is True
assert log(exp(p) + 1).is_positive
assert log(1, evaluate=False).is_algebraic
assert log(42, evaluate=False).is_algebraic is False
assert log(1 + z).is_rational
def test_log_hashing():
assert x != log(log(x))
assert hash(x) != hash(log(log(x)))
assert log(x) != log(log(log(x)))
e = 1/log(log(x) + log(log(x)))
assert e.base.func is log
e = 1/log(log(x) + log(log(log(x))))
assert e.base.func is log
e = log(log(x))
assert e.func is log
assert x.func is not log
assert hash(log(log(x))) != hash(x)
assert e != x
def test_log_sign():
assert sign(log(2)) == 1
def test_log_expand_complex():
assert log(1 + I).expand(complex=True) == log(2)/2 + I*pi/4
assert log(1 - sqrt(2)).expand(complex=True) == log(sqrt(2) - 1) + I*pi
def test_log_apply_evalf():
value = (log(3)/log(2) - 1).evalf()
assert value.epsilon_eq(Float("0.58496250072115618145373"))
def test_log_leading_term():
p = Symbol('p')
# Test for STEP 3
assert log(1 + x + x**2).as_leading_term(x, cdir=1) == x
# Test for STEP 4
assert log(2*x).as_leading_term(x, cdir=1) == log(x) + log(2)
assert log(2*x).as_leading_term(x, cdir=-1) == log(x) + log(2)
assert log(-2*x).as_leading_term(x, cdir=1, logx=p) == p + log(2) + I*pi
assert log(-2*x).as_leading_term(x, cdir=-1, logx=p) == p + log(2) - I*pi
# Test for STEP 5
assert log(-2*x + (3 - I)*x**2).as_leading_term(x, cdir=1) == log(x) + log(2) - I*pi
assert log(-2*x + (3 - I)*x**2).as_leading_term(x, cdir=-1) == log(x) + log(2) - I*pi
assert log(2*x + (3 - I)*x**2).as_leading_term(x, cdir=1) == log(x) + log(2)
assert log(2*x + (3 - I)*x**2).as_leading_term(x, cdir=-1) == log(x) + log(2) - 2*I*pi
assert log(-1 + x - I*x**2 + I*x**3).as_leading_term(x, cdir=1) == -I*pi
assert log(-1 + x - I*x**2 + I*x**3).as_leading_term(x, cdir=-1) == -I*pi
assert log(-1/(1 - x)).as_leading_term(x, cdir=1) == I*pi
assert log(-1/(1 - x)).as_leading_term(x, cdir=-1) == I*pi
def test_log_nseries():
p = Symbol('p')
assert log(1/x)._eval_nseries(x, 4, logx=-p, cdir=1) == p
assert log(1/x)._eval_nseries(x, 4, logx=-p, cdir=-1) == p + 2*I*pi
assert log(x - 1)._eval_nseries(x, 4, None, I) == I*pi - x - x**2/2 - x**3/3 + O(x**4)
assert log(x - 1)._eval_nseries(x, 4, None, -I) == -I*pi - x - x**2/2 - x**3/3 + O(x**4)
assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x + x**2/2 + O(x**3)
assert log(I*x + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == -I*pi - I*x + x**2/2 + O(x**3)
assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, 1) == I*pi - I*x**2 + O(x**3)
assert log(I*x**2 + I*x**3 - 1)._eval_nseries(x, 3, None, -1) == I*pi - I*x**2 + O(x**3)
assert log(2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, 1) == log(2) + log(x) + \
x*(S(3)/2 - I/2) + x**2*(-1 + 3*I/4) + O(x**3)
assert log(2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, -1) == -2*I*pi + log(2) + \
log(x) - x*(-S(3)/2 + I/2) + x**2*(-1 + 3*I/4) + O(x**3)
assert log(-2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, 1) == -I*pi + log(2) + log(x) + \
x*(-S(3)/2 + I/2) + x**2*(-1 + 3*I/4) + O(x**3)
assert log(-2*x + (3 - I)*x**2)._eval_nseries(x, 3, None, -1) == -I*pi + log(2) + log(x) - \
x*(S(3)/2 - I/2) + x**2*(-1 + 3*I/4) + O(x**3)
assert log(sqrt(-I*x**2 - 3)*sqrt(-I*x**2 - 1) - 2)._eval_nseries(x, 3, None, 1) == -I*pi + \
log(sqrt(3) + 2) + I*x**2*(-2 + 4*sqrt(3)/3) + O(x**3)
assert log(-1/(1 - x))._eval_nseries(x, 3, None, 1) == I*pi + x + x**2/2 + O(x**3)
assert log(-1/(1 - x))._eval_nseries(x, 3, None, -1) == I*pi + x + x**2/2 + O(x**3)
def test_log_series():
# Note Series at infinities other than oo/-oo were introduced as a part of
# pull request 23798. Refer https://github.com/sympy/sympy/pull/23798 for
# more information.
expr1 = log(1 + x)
expr2 = log(x + sqrt(x**2 + 1))
assert expr1.series(x, x0=I*oo, n=4) == 1/(3*x**3) - 1/(2*x**2) + 1/x + \
I*pi/2 - log(I/x) + O(x**(-4), (x, oo*I))
assert expr1.series(x, x0=-I*oo, n=4) == 1/(3*x**3) - 1/(2*x**2) + 1/x - \
I*pi/2 - log(-I/x) + O(x**(-4), (x, -oo*I))
assert expr2.series(x, x0=I*oo, n=4) == 1/(4*x**2) + I*pi/2 + log(2) - \
log(I/x) + O(x**(-4), (x, oo*I))
assert expr2.series(x, x0=-I*oo, n=4) == -1/(4*x**2) - I*pi/2 - log(2) + \
log(-I/x) + O(x**(-4), (x, -oo*I))
def test_log_expand():
w = Symbol("w", positive=True)
e = log(w**(log(5)/log(3)))
assert e.expand() == log(5)/log(3) * log(w)
x, y, z = symbols('x,y,z', positive=True)
assert log(x*(y + z)).expand(mul=False) == log(x) + log(y + z)
assert log(log(x**2)*log(y*z)).expand() in [log(2*log(x)*log(y) +
2*log(x)*log(z)), log(log(x)*log(z) + log(y)*log(x)) + log(2),
log((log(y) + log(z))*log(x)) + log(2)]
assert log(x**log(x**2)).expand(deep=False) == log(x)*log(x**2)
assert log(x**log(x**2)).expand() == 2*log(x)**2
x, y = symbols('x,y')
assert log(x*y).expand(force=True) == log(x) + log(y)
assert log(x**y).expand(force=True) == y*log(x)
assert log(exp(x)).expand(force=True) == x
# there's generally no need to expand out logs since this requires
# factoring and if simplification is sought, it's cheaper to put
# logs together than it is to take them apart.
assert log(2*3**2).expand() != 2*log(3) + log(2)
@XFAIL
def test_log_expand_fail():
x, y, z = symbols('x,y,z', positive=True)
assert (log(x*(y + z))*(x + y)).expand(mul=True, log=True) == y*log(
x) + y*log(y + z) + z*log(x) + z*log(y + z)
def test_log_simplify():
x = Symbol("x", positive=True)
assert log(x**2).expand() == 2*log(x)
assert expand_log(log(x**(2 + log(2)))) == (2 + log(2))*log(x)
z = Symbol('z')
assert log(sqrt(z)).expand() == log(z)/2
assert expand_log(log(z**(log(2) - 1))) == (log(2) - 1)*log(z)
assert log(z**(-1)).expand() != -log(z)
assert log(z**(x/(x+1))).expand() == x*log(z)/(x + 1)
def test_log_AccumBounds():
assert log(AccumBounds(1, E)) == AccumBounds(0, 1)
assert log(AccumBounds(0, E)) == AccumBounds(-oo, 1)
assert log(AccumBounds(-1, E)) == S.NaN
assert log(AccumBounds(0, oo)) == AccumBounds(-oo, oo)
assert log(AccumBounds(-oo, 0)) == S.NaN
assert log(AccumBounds(-oo, oo)) == S.NaN
@_both_exp_pow
def test_lambertw():
k = Symbol('k')
assert LambertW(x, 0) == LambertW(x)
assert LambertW(x, 0, evaluate=False) != LambertW(x)
assert LambertW(0) == 0
assert LambertW(E) == 1
assert LambertW(-1/E) == -1
assert LambertW(-log(2)/2) == -log(2)
assert LambertW(oo) is oo
assert LambertW(0, 1) is -oo
assert LambertW(0, 42) is -oo
assert LambertW(-pi/2, -1) == -I*pi/2
assert LambertW(-1/E, -1) == -1
assert LambertW(-2*exp(-2), -1) == -2
assert LambertW(2*log(2)) == log(2)
assert LambertW(-pi/2) == I*pi/2
assert LambertW(exp(1 + E)) == E
assert LambertW(x**2).diff(x) == 2*LambertW(x**2)/x/(1 + LambertW(x**2))
assert LambertW(x, k).diff(x) == LambertW(x, k)/x/(1 + LambertW(x, k))
assert LambertW(sqrt(2)).evalf(30).epsilon_eq(
Float("0.701338383413663009202120278965", 30), 1e-29)
assert re(LambertW(2, -1)).evalf().epsilon_eq(Float("-0.834310366631110"))
assert LambertW(-1).is_real is False # issue 5215
assert LambertW(2, evaluate=False).is_real
p = Symbol('p', positive=True)
assert LambertW(p, evaluate=False).is_real
assert LambertW(p - 1, evaluate=False).is_real is None
assert LambertW(-p - 2/S.Exp1, evaluate=False).is_real is False
assert LambertW(S.Half, -1, evaluate=False).is_real is False
assert LambertW(Rational(-1, 10), -1, evaluate=False).is_real
assert LambertW(-10, -1, evaluate=False).is_real is False
assert LambertW(-2, 2, evaluate=False).is_real is False
assert LambertW(0, evaluate=False).is_algebraic
na = Symbol('na', nonzero=True, algebraic=True)
assert LambertW(na).is_algebraic is False
assert LambertW(p).is_zero is False
n = Symbol('n', negative=True)
assert LambertW(n).is_zero is False
def test_issue_5673():
e = LambertW(-1)
assert e.is_comparable is False
assert e.is_positive is not True
e2 = 1 - 1/(1 - exp(-1000))
assert e2.is_positive is not True
e3 = -2 + exp(exp(LambertW(log(2)))*LambertW(log(2)))
assert e3.is_nonzero is not True
def test_log_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: log(x).fdiff(2))
def test_log_taylor_term():
x = symbols('x')
assert log(x).taylor_term(0, x) == x
assert log(x).taylor_term(1, x) == -x**2/2
assert log(x).taylor_term(4, x) == x**5/5
assert log(x).taylor_term(-1, x) is S.Zero
def test_exp_expand_NC():
A, B, C = symbols('A,B,C', commutative=False)
assert exp(A + B).expand() == exp(A + B)
assert exp(A + B + C).expand() == exp(A + B + C)
assert exp(x + y).expand() == exp(x)*exp(y)
assert exp(x + y + z).expand() == exp(x)*exp(y)*exp(z)
@_both_exp_pow
def test_as_numer_denom():
n = symbols('n', negative=True)
assert exp(x).as_numer_denom() == (exp(x), 1)
assert exp(-x).as_numer_denom() == (1, exp(x))
assert exp(-2*x).as_numer_denom() == (1, exp(2*x))
assert exp(-2).as_numer_denom() == (1, exp(2))
assert exp(n).as_numer_denom() == (1, exp(-n))
assert exp(-n).as_numer_denom() == (exp(-n), 1)
assert exp(-I*x).as_numer_denom() == (1, exp(I*x))
assert exp(-I*n).as_numer_denom() == (1, exp(I*n))
assert exp(-n).as_numer_denom() == (exp(-n), 1)
@_both_exp_pow
def test_polar():
x, y = symbols('x y', polar=True)
assert abs(exp_polar(I*4)) == 1
assert abs(exp_polar(0)) == 1
assert abs(exp_polar(2 + 3*I)) == exp(2)
assert exp_polar(I*10).n() == exp_polar(I*10)
assert log(exp_polar(z)) == z
assert log(x*y).expand() == log(x) + log(y)
assert log(x**z).expand() == z*log(x)
assert exp_polar(3).exp == 3
# Compare exp(1.0*pi*I).
assert (exp_polar(1.0*pi*I).n(n=5)).as_real_imag()[1] >= 0
assert exp_polar(0).is_rational is True # issue 8008
def test_exp_summation():
w = symbols("w")
m, n, i, j = symbols("m n i j")
expr = exp(Sum(w*i, (i, 0, n), (j, 0, m)))
assert expr.expand() == Product(exp(w*i), (i, 0, n), (j, 0, m))
def test_log_product():
from sympy.abc import n, m
i, j = symbols('i,j', positive=True, integer=True)
x, y = symbols('x,y', positive=True)
z = symbols('z', real=True)
w = symbols('w')
expr = log(Product(x**i, (i, 1, n)))
assert simplify(expr) == expr
assert expr.expand() == Sum(i*log(x), (i, 1, n))
expr = log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))
assert simplify(expr) == expr
assert expr.expand() == Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
expr = log(Product(-2, (n, 0, 4)))
assert simplify(expr) == expr
assert expr.expand() == expr
assert expr.expand(force=True) == Sum(log(-2), (n, 0, 4))
expr = log(Product(exp(z*i), (i, 0, n)))
assert expr.expand() == Sum(z*i, (i, 0, n))
expr = log(Product(exp(w*i), (i, 0, n)))
assert expr.expand() == expr
assert expr.expand(force=True) == Sum(w*i, (i, 0, n))
expr = log(Product(i**2*abs(j), (i, 1, n), (j, 1, m)))
assert expr.expand() == Sum(2*log(i) + log(j), (i, 1, n), (j, 1, m))
@XFAIL
def test_log_product_simplify_to_sum():
from sympy.abc import n, m
i, j = symbols('i,j', positive=True, integer=True)
x, y = symbols('x,y', positive=True)
assert simplify(log(Product(x**i, (i, 1, n)))) == Sum(i*log(x), (i, 1, n))
assert simplify(log(Product(x**i*y**j, (i, 1, n), (j, 1, m)))) == \
Sum(i*log(x) + j*log(y), (i, 1, n), (j, 1, m))
def test_issue_8866():
assert simplify(log(x, 10, evaluate=False)) == simplify(log(x, 10))
assert expand_log(log(x, 10, evaluate=False)) == expand_log(log(x, 10))
y = Symbol('y', positive=True)
l1 = log(exp(y), exp(10))
b1 = log(exp(y), exp(5))
l2 = log(exp(y), exp(10), evaluate=False)
b2 = log(exp(y), exp(5), evaluate=False)
assert simplify(log(l1, b1)) == simplify(log(l2, b2))
assert expand_log(log(l1, b1)) == expand_log(log(l2, b2))
def test_log_expand_factor():
assert (log(18)/log(3) - 2).expand(factor=True) == log(2)/log(3)
assert (log(12)/log(2)).expand(factor=True) == log(3)/log(2) + 2
assert (log(15)/log(3)).expand(factor=True) == 1 + log(5)/log(3)
assert (log(2)/(-log(12) + log(24))).expand(factor=True) == 1
assert expand_log(log(12), factor=True) == log(3) + 2*log(2)
assert expand_log(log(21)/log(7), factor=False) == log(3)/log(7) + 1
assert expand_log(log(45)/log(5) + log(20), factor=False) == \
1 + 2*log(3)/log(5) + log(20)
assert expand_log(log(45)/log(5) + log(26), factor=True) == \
log(2) + log(13) + (log(5) + 2*log(3))/log(5)
def test_issue_9116():
n = Symbol('n', positive=True, integer=True)
assert log(n).is_nonnegative is True
def test_issue_18473():
assert exp(x*log(cos(1/x))).as_leading_term(x) == S.NaN
assert exp(x*log(tan(1/x))).as_leading_term(x) == S.NaN
assert log(cos(1/x)).as_leading_term(x) == S.NaN
assert log(tan(1/x)).as_leading_term(x) == S.NaN
assert log(cos(1/x) + 2).as_leading_term(x) == AccumBounds(0, log(3))
assert exp(x*log(cos(1/x) + 2)).as_leading_term(x) == 1
assert log(cos(1/x) - 2).as_leading_term(x) == S.NaN
assert exp(x*log(cos(1/x) - 2)).as_leading_term(x) == S.NaN
assert log(cos(1/x) + 1).as_leading_term(x) == AccumBounds(-oo, log(2))
assert exp(x*log(cos(1/x) + 1)).as_leading_term(x) == AccumBounds(0, 1)
assert log(sin(1/x)**2).as_leading_term(x) == AccumBounds(-oo, 0)
assert exp(x*log(sin(1/x)**2)).as_leading_term(x) == AccumBounds(0, 1)
assert log(tan(1/x)**2).as_leading_term(x) == AccumBounds(-oo, oo)
assert exp(2*x*(log(tan(1/x)**2))).as_leading_term(x) == AccumBounds(0, oo)
|
9f1258c2745a5b64c406ffcf044510eaf32f6b96d157e4a738195ecaf23dbcd2 | from sympy.calculus.accumulationbounds import AccumBounds
from sympy.core.add import Add
from sympy.core.function import (Lambda, diff)
from sympy.core.mod import Mod
from sympy.core.mul import Mul
from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (arg, conjugate, im, re)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.hyperbolic import (acoth, asinh, atanh, cosh, coth, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, acot, acsc, asec, asin, atan, atan2,
cos, cot, csc, sec, sin, sinc, tan)
from sympy.functions.special.bessel import (besselj, jn)
from sympy.functions.special.delta_functions import Heaviside
from sympy.matrices.dense import Matrix
from sympy.polys.polytools import (cancel, gcd)
from sympy.series.limits import limit
from sympy.series.order import O
from sympy.series.series import series
from sympy.sets.fancysets import ImageSet
from sympy.sets.sets import (FiniteSet, Interval)
from sympy.simplify.simplify import simplify
from sympy.core.expr import unchanged
from sympy.core.function import ArgumentIndexError
from sympy.core.relational import Ne, Eq
from sympy.functions.elementary.piecewise import Piecewise
from sympy.sets.setexpr import SetExpr
from sympy.testing.pytest import XFAIL, slow, raises
x, y, z = symbols('x y z')
r = Symbol('r', real=True)
k, m = symbols('k m', integer=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
np = Symbol('p', nonpositive=True)
nn = Symbol('n', nonnegative=True)
nz = Symbol('nz', nonzero=True)
ep = Symbol('ep', extended_positive=True)
en = Symbol('en', extended_negative=True)
enp = Symbol('ep', extended_nonpositive=True)
enn = Symbol('en', extended_nonnegative=True)
enz = Symbol('enz', extended_nonzero=True)
a = Symbol('a', algebraic=True)
na = Symbol('na', nonzero=True, algebraic=True)
def test_sin():
x, y = symbols('x y')
z = symbols('z', imaginary=True)
assert sin.nargs == FiniteSet(1)
assert sin(nan) is nan
assert sin(zoo) is nan
assert sin(oo) == AccumBounds(-1, 1)
assert sin(oo) - sin(oo) == AccumBounds(-2, 2)
assert sin(oo*I) == oo*I
assert sin(-oo*I) == -oo*I
assert 0*sin(oo) is S.Zero
assert 0/sin(oo) is S.Zero
assert 0 + sin(oo) == AccumBounds(-1, 1)
assert 5 + sin(oo) == AccumBounds(4, 6)
assert sin(0) == 0
assert sin(z*I) == I*sinh(z)
assert sin(asin(x)) == x
assert sin(atan(x)) == x / sqrt(1 + x**2)
assert sin(acos(x)) == sqrt(1 - x**2)
assert sin(acot(x)) == 1 / (sqrt(1 + 1 / x**2) * x)
assert sin(acsc(x)) == 1 / x
assert sin(asec(x)) == sqrt(1 - 1 / x**2)
assert sin(atan2(y, x)) == y / sqrt(x**2 + y**2)
assert sin(pi*I) == sinh(pi)*I
assert sin(-pi*I) == -sinh(pi)*I
assert sin(-2*I) == -sinh(2)*I
assert sin(pi) == 0
assert sin(-pi) == 0
assert sin(2*pi) == 0
assert sin(-2*pi) == 0
assert sin(-3*10**73*pi) == 0
assert sin(7*10**103*pi) == 0
assert sin(pi/2) == 1
assert sin(-pi/2) == -1
assert sin(pi*Rational(5, 2)) == 1
assert sin(pi*Rational(7, 2)) == -1
ne = symbols('ne', integer=True, even=False)
e = symbols('e', even=True)
assert sin(pi*ne/2) == (-1)**(ne/2 - S.Half)
assert sin(pi*k/2).func == sin
assert sin(pi*e/2) == 0
assert sin(pi*k) == 0
assert sin(pi*k).subs(k, 3) == sin(pi*k/2).subs(k, 6) # issue 8298
assert sin(pi/3) == S.Half*sqrt(3)
assert sin(pi*Rational(-2, 3)) == Rational(-1, 2)*sqrt(3)
assert sin(pi/4) == S.Half*sqrt(2)
assert sin(-pi/4) == Rational(-1, 2)*sqrt(2)
assert sin(pi*Rational(17, 4)) == S.Half*sqrt(2)
assert sin(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
assert sin(pi/6) == S.Half
assert sin(-pi/6) == Rational(-1, 2)
assert sin(pi*Rational(7, 6)) == Rational(-1, 2)
assert sin(pi*Rational(-5, 6)) == Rational(-1, 2)
assert sin(pi*Rational(1, 5)) == sqrt((5 - sqrt(5)) / 8)
assert sin(pi*Rational(2, 5)) == sqrt((5 + sqrt(5)) / 8)
assert sin(pi*Rational(3, 5)) == sin(pi*Rational(2, 5))
assert sin(pi*Rational(4, 5)) == sin(pi*Rational(1, 5))
assert sin(pi*Rational(6, 5)) == -sin(pi*Rational(1, 5))
assert sin(pi*Rational(8, 5)) == -sin(pi*Rational(2, 5))
assert sin(pi*Rational(-1273, 5)) == -sin(pi*Rational(2, 5))
assert sin(pi/8) == sqrt((2 - sqrt(2))/4)
assert sin(pi/10) == Rational(-1, 4) + sqrt(5)/4
assert sin(pi/12) == -sqrt(2)/4 + sqrt(6)/4
assert sin(pi*Rational(5, 12)) == sqrt(2)/4 + sqrt(6)/4
assert sin(pi*Rational(-7, 12)) == -sqrt(2)/4 - sqrt(6)/4
assert sin(pi*Rational(-11, 12)) == sqrt(2)/4 - sqrt(6)/4
assert sin(pi*Rational(104, 105)) == sin(pi/105)
assert sin(pi*Rational(106, 105)) == -sin(pi/105)
assert sin(pi*Rational(-104, 105)) == -sin(pi/105)
assert sin(pi*Rational(-106, 105)) == sin(pi/105)
assert sin(x*I) == sinh(x)*I
assert sin(k*pi) == 0
assert sin(17*k*pi) == 0
assert sin(2*k*pi + 4) == sin(4)
assert sin(2*k*pi + m*pi + 1) == (-1)**(m + 2*k)*sin(1)
assert sin(k*pi*I) == sinh(k*pi)*I
assert sin(r).is_real is True
assert sin(0, evaluate=False).is_algebraic
assert sin(a).is_algebraic is None
assert sin(na).is_algebraic is False
q = Symbol('q', rational=True)
assert sin(pi*q).is_algebraic
qn = Symbol('qn', rational=True, nonzero=True)
assert sin(qn).is_rational is False
assert sin(q).is_rational is None # issue 8653
assert isinstance(sin( re(x) - im(y)), sin) is True
assert isinstance(sin(-re(x) + im(y)), sin) is False
assert sin(SetExpr(Interval(0, 1))) == SetExpr(ImageSet(Lambda(x, sin(x)),
Interval(0, 1)))
for d in list(range(1, 22)) + [60, 85]:
for n in range(d*2 + 1):
x = n*pi/d
e = abs( float(sin(x)) - sin(float(x)) )
assert e < 1e-12
assert sin(0, evaluate=False).is_zero is True
assert sin(k*pi, evaluate=False).is_zero is True
assert sin(Add(1, -1, evaluate=False), evaluate=False).is_zero is True
def test_sin_cos():
for d in [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 24, 30, 40, 60, 120]: # list is not exhaustive...
for n in range(-2*d, d*2):
x = n*pi/d
assert sin(x + pi/2) == cos(x), "fails for %d*pi/%d" % (n, d)
assert sin(x - pi/2) == -cos(x), "fails for %d*pi/%d" % (n, d)
assert sin(x) == cos(x - pi/2), "fails for %d*pi/%d" % (n, d)
assert -sin(x) == cos(x + pi/2), "fails for %d*pi/%d" % (n, d)
def test_sin_series():
assert sin(x).series(x, 0, 9) == \
x - x**3/6 + x**5/120 - x**7/5040 + O(x**9)
def test_sin_rewrite():
assert sin(x).rewrite(exp) == -I*(exp(I*x) - exp(-I*x))/2
assert sin(x).rewrite(tan) == 2*tan(x/2)/(1 + tan(x/2)**2)
assert sin(x).rewrite(cot) == \
Piecewise((0, Eq(im(x), 0) & Eq(Mod(x, pi), 0)),
(2*cot(x/2)/(cot(x/2)**2 + 1), True))
assert sin(sinh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sinh(3)).n()
assert sin(cosh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cosh(3)).n()
assert sin(tanh(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tanh(3)).n()
assert sin(coth(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, coth(3)).n()
assert sin(sin(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, sin(3)).n()
assert sin(cos(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cos(3)).n()
assert sin(tan(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, tan(3)).n()
assert sin(cot(x)).rewrite(
exp).subs(x, 3).n() == sin(x).rewrite(exp).subs(x, cot(3)).n()
assert sin(log(x)).rewrite(Pow) == I*x**-I / 2 - I*x**I /2
assert sin(x).rewrite(csc) == 1/csc(x)
assert sin(x).rewrite(cos) == cos(x - pi / 2, evaluate=False)
assert sin(x).rewrite(sec) == 1 / sec(x - pi / 2, evaluate=False)
assert sin(cos(x)).rewrite(Pow) == sin(cos(x))
def _test_extrig(f, i, e):
from sympy.core.function import expand_trig
assert unchanged(f, i)
assert expand_trig(f(i)) == f(i)
# testing directly instead of with .expand(trig=True)
# because the other expansions undo the unevaluated Mul
assert expand_trig(f(Mul(i, 1, evaluate=False))) == e
assert abs(f(i) - e).n() < 1e-10
def test_sin_expansion():
# Note: these formulas are not unique. The ones here come from the
# Chebyshev formulas.
assert sin(x + y).expand(trig=True) == sin(x)*cos(y) + cos(x)*sin(y)
assert sin(x - y).expand(trig=True) == sin(x)*cos(y) - cos(x)*sin(y)
assert sin(y - x).expand(trig=True) == cos(x)*sin(y) - sin(x)*cos(y)
assert sin(2*x).expand(trig=True) == 2*sin(x)*cos(x)
assert sin(3*x).expand(trig=True) == -4*sin(x)**3 + 3*sin(x)
assert sin(4*x).expand(trig=True) == -8*sin(x)**3*cos(x) + 4*sin(x)*cos(x)
_test_extrig(sin, 2, 2*sin(1)*cos(1))
_test_extrig(sin, 3, -4*sin(1)**3 + 3*sin(1))
def test_sin_AccumBounds():
assert sin(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, oo)) == AccumBounds(-1, 1)
assert sin(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
assert sin(AccumBounds(0, S.Pi*Rational(3, 4))) == AccumBounds(0, 1)
assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(7, 4))) == AccumBounds(-1, sin(S.Pi*Rational(3, 4)))
assert sin(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(sin(S.Pi/4), sin(S.Pi/3))
assert sin(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 6))) == AccumBounds(sin(S.Pi*Rational(5, 6)), sin(S.Pi*Rational(3, 4)))
def test_sin_fdiff():
assert sin(x).fdiff() == cos(x)
raises(ArgumentIndexError, lambda: sin(x).fdiff(2))
def test_trig_symmetry():
assert sin(-x) == -sin(x)
assert cos(-x) == cos(x)
assert tan(-x) == -tan(x)
assert cot(-x) == -cot(x)
assert sin(x + pi) == -sin(x)
assert sin(x + 2*pi) == sin(x)
assert sin(x + 3*pi) == -sin(x)
assert sin(x + 4*pi) == sin(x)
assert sin(x - 5*pi) == -sin(x)
assert cos(x + pi) == -cos(x)
assert cos(x + 2*pi) == cos(x)
assert cos(x + 3*pi) == -cos(x)
assert cos(x + 4*pi) == cos(x)
assert cos(x - 5*pi) == -cos(x)
assert tan(x + pi) == tan(x)
assert tan(x - 3*pi) == tan(x)
assert cot(x + pi) == cot(x)
assert cot(x - 3*pi) == cot(x)
assert sin(pi/2 - x) == cos(x)
assert sin(pi*Rational(3, 2) - x) == -cos(x)
assert sin(pi*Rational(5, 2) - x) == cos(x)
assert cos(pi/2 - x) == sin(x)
assert cos(pi*Rational(3, 2) - x) == -sin(x)
assert cos(pi*Rational(5, 2) - x) == sin(x)
assert tan(pi/2 - x) == cot(x)
assert tan(pi*Rational(3, 2) - x) == cot(x)
assert tan(pi*Rational(5, 2) - x) == cot(x)
assert cot(pi/2 - x) == tan(x)
assert cot(pi*Rational(3, 2) - x) == tan(x)
assert cot(pi*Rational(5, 2) - x) == tan(x)
assert sin(pi/2 + x) == cos(x)
assert cos(pi/2 + x) == -sin(x)
assert tan(pi/2 + x) == -cot(x)
assert cot(pi/2 + x) == -tan(x)
def test_cos():
x, y = symbols('x y')
assert cos.nargs == FiniteSet(1)
assert cos(nan) is nan
assert cos(oo) == AccumBounds(-1, 1)
assert cos(oo) - cos(oo) == AccumBounds(-2, 2)
assert cos(oo*I) is oo
assert cos(-oo*I) is oo
assert cos(zoo) is nan
assert cos(0) == 1
assert cos(acos(x)) == x
assert cos(atan(x)) == 1 / sqrt(1 + x**2)
assert cos(asin(x)) == sqrt(1 - x**2)
assert cos(acot(x)) == 1 / sqrt(1 + 1 / x**2)
assert cos(acsc(x)) == sqrt(1 - 1 / x**2)
assert cos(asec(x)) == 1 / x
assert cos(atan2(y, x)) == x / sqrt(x**2 + y**2)
assert cos(pi*I) == cosh(pi)
assert cos(-pi*I) == cosh(pi)
assert cos(-2*I) == cosh(2)
assert cos(pi/2) == 0
assert cos(-pi/2) == 0
assert cos(pi/2) == 0
assert cos(-pi/2) == 0
assert cos((-3*10**73 + 1)*pi/2) == 0
assert cos((7*10**103 + 1)*pi/2) == 0
n = symbols('n', integer=True, even=False)
e = symbols('e', even=True)
assert cos(pi*n/2) == 0
assert cos(pi*e/2) == (-1)**(e/2)
assert cos(pi) == -1
assert cos(-pi) == -1
assert cos(2*pi) == 1
assert cos(5*pi) == -1
assert cos(8*pi) == 1
assert cos(pi/3) == S.Half
assert cos(pi*Rational(-2, 3)) == Rational(-1, 2)
assert cos(pi/4) == S.Half*sqrt(2)
assert cos(-pi/4) == S.Half*sqrt(2)
assert cos(pi*Rational(11, 4)) == Rational(-1, 2)*sqrt(2)
assert cos(pi*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
assert cos(pi/6) == S.Half*sqrt(3)
assert cos(-pi/6) == S.Half*sqrt(3)
assert cos(pi*Rational(7, 6)) == Rational(-1, 2)*sqrt(3)
assert cos(pi*Rational(-5, 6)) == Rational(-1, 2)*sqrt(3)
assert cos(pi*Rational(1, 5)) == (sqrt(5) + 1)/4
assert cos(pi*Rational(2, 5)) == (sqrt(5) - 1)/4
assert cos(pi*Rational(3, 5)) == -cos(pi*Rational(2, 5))
assert cos(pi*Rational(4, 5)) == -cos(pi*Rational(1, 5))
assert cos(pi*Rational(6, 5)) == -cos(pi*Rational(1, 5))
assert cos(pi*Rational(8, 5)) == cos(pi*Rational(2, 5))
assert cos(pi*Rational(-1273, 5)) == -cos(pi*Rational(2, 5))
assert cos(pi/8) == sqrt((2 + sqrt(2))/4)
assert cos(pi/12) == sqrt(2)/4 + sqrt(6)/4
assert cos(pi*Rational(5, 12)) == -sqrt(2)/4 + sqrt(6)/4
assert cos(pi*Rational(7, 12)) == sqrt(2)/4 - sqrt(6)/4
assert cos(pi*Rational(11, 12)) == -sqrt(2)/4 - sqrt(6)/4
assert cos(pi*Rational(104, 105)) == -cos(pi/105)
assert cos(pi*Rational(106, 105)) == -cos(pi/105)
assert cos(pi*Rational(-104, 105)) == -cos(pi/105)
assert cos(pi*Rational(-106, 105)) == -cos(pi/105)
assert cos(x*I) == cosh(x)
assert cos(k*pi*I) == cosh(k*pi)
assert cos(r).is_real is True
assert cos(0, evaluate=False).is_algebraic
assert cos(a).is_algebraic is None
assert cos(na).is_algebraic is False
q = Symbol('q', rational=True)
assert cos(pi*q).is_algebraic
assert cos(pi*Rational(2, 7)).is_algebraic
assert cos(k*pi) == (-1)**k
assert cos(2*k*pi) == 1
for d in list(range(1, 22)) + [60, 85]:
for n in range(2*d + 1):
x = n*pi/d
e = abs( float(cos(x)) - cos(float(x)) )
assert e < 1e-12
def test_issue_6190():
c = Float('123456789012345678901234567890.25', '')
for cls in [sin, cos, tan, cot]:
assert cls(c*pi) == cls(pi/4)
assert cls(4.125*pi) == cls(pi/8)
assert cls(4.7*pi) == cls((4.7 % 2)*pi)
def test_cos_series():
assert cos(x).series(x, 0, 9) == \
1 - x**2/2 + x**4/24 - x**6/720 + x**8/40320 + O(x**9)
def test_cos_rewrite():
assert cos(x).rewrite(exp) == exp(I*x)/2 + exp(-I*x)/2
assert cos(x).rewrite(tan) == (1 - tan(x/2)**2)/(1 + tan(x/2)**2)
assert cos(x).rewrite(cot) == \
Piecewise((1, Eq(im(x), 0) & Eq(Mod(x, 2*pi), 0)),
((cot(x/2)**2 - 1)/(cot(x/2)**2 + 1), True))
assert cos(sinh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sinh(3)).n()
assert cos(cosh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cosh(3)).n()
assert cos(tanh(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tanh(3)).n()
assert cos(coth(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, coth(3)).n()
assert cos(sin(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, sin(3)).n()
assert cos(cos(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cos(3)).n()
assert cos(tan(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, tan(3)).n()
assert cos(cot(x)).rewrite(
exp).subs(x, 3).n() == cos(x).rewrite(exp).subs(x, cot(3)).n()
assert cos(log(x)).rewrite(Pow) == x**I/2 + x**-I/2
assert cos(x).rewrite(sec) == 1/sec(x)
assert cos(x).rewrite(sin) == sin(x + pi/2, evaluate=False)
assert cos(x).rewrite(csc) == 1/csc(-x + pi/2, evaluate=False)
assert cos(sin(x)).rewrite(Pow) == cos(sin(x))
def test_cos_expansion():
assert cos(x + y).expand(trig=True) == cos(x)*cos(y) - sin(x)*sin(y)
assert cos(x - y).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
assert cos(y - x).expand(trig=True) == cos(x)*cos(y) + sin(x)*sin(y)
assert cos(2*x).expand(trig=True) == 2*cos(x)**2 - 1
assert cos(3*x).expand(trig=True) == 4*cos(x)**3 - 3*cos(x)
assert cos(4*x).expand(trig=True) == 8*cos(x)**4 - 8*cos(x)**2 + 1
_test_extrig(cos, 2, 2*cos(1)**2 - 1)
_test_extrig(cos, 3, 4*cos(1)**3 - 3*cos(1))
def test_cos_AccumBounds():
assert cos(AccumBounds(-oo, oo)) == AccumBounds(-1, 1)
assert cos(AccumBounds(0, oo)) == AccumBounds(-1, 1)
assert cos(AccumBounds(-oo, 0)) == AccumBounds(-1, 1)
assert cos(AccumBounds(0, 2*S.Pi)) == AccumBounds(-1, 1)
assert cos(AccumBounds(-S.Pi/3, S.Pi/4)) == AccumBounds(cos(-S.Pi/3), 1)
assert cos(AccumBounds(S.Pi*Rational(3, 4), S.Pi*Rational(5, 4))) == AccumBounds(-1, cos(S.Pi*Rational(3, 4)))
assert cos(AccumBounds(S.Pi*Rational(5, 4), S.Pi*Rational(4, 3))) == AccumBounds(cos(S.Pi*Rational(5, 4)), cos(S.Pi*Rational(4, 3)))
assert cos(AccumBounds(S.Pi/4, S.Pi/3)) == AccumBounds(cos(S.Pi/3), cos(S.Pi/4))
def test_cos_fdiff():
assert cos(x).fdiff() == -sin(x)
raises(ArgumentIndexError, lambda: cos(x).fdiff(2))
def test_tan():
assert tan(nan) is nan
assert tan(zoo) is nan
assert tan(oo) == AccumBounds(-oo, oo)
assert tan(oo) - tan(oo) == AccumBounds(-oo, oo)
assert tan.nargs == FiniteSet(1)
assert tan(oo*I) == I
assert tan(-oo*I) == -I
assert tan(0) == 0
assert tan(atan(x)) == x
assert tan(asin(x)) == x / sqrt(1 - x**2)
assert tan(acos(x)) == sqrt(1 - x**2) / x
assert tan(acot(x)) == 1 / x
assert tan(acsc(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
assert tan(asec(x)) == sqrt(1 - 1 / x**2) * x
assert tan(atan2(y, x)) == y/x
assert tan(pi*I) == tanh(pi)*I
assert tan(-pi*I) == -tanh(pi)*I
assert tan(-2*I) == -tanh(2)*I
assert tan(pi) == 0
assert tan(-pi) == 0
assert tan(2*pi) == 0
assert tan(-2*pi) == 0
assert tan(-3*10**73*pi) == 0
assert tan(pi/2) is zoo
assert tan(pi*Rational(3, 2)) is zoo
assert tan(pi/3) == sqrt(3)
assert tan(pi*Rational(-2, 3)) == sqrt(3)
assert tan(pi/4) is S.One
assert tan(-pi/4) is S.NegativeOne
assert tan(pi*Rational(17, 4)) is S.One
assert tan(pi*Rational(-3, 4)) is S.One
assert tan(pi/5) == sqrt(5 - 2*sqrt(5))
assert tan(pi*Rational(2, 5)) == sqrt(5 + 2*sqrt(5))
assert tan(pi*Rational(18, 5)) == -sqrt(5 + 2*sqrt(5))
assert tan(pi*Rational(-16, 5)) == -sqrt(5 - 2*sqrt(5))
assert tan(pi/6) == 1/sqrt(3)
assert tan(-pi/6) == -1/sqrt(3)
assert tan(pi*Rational(7, 6)) == 1/sqrt(3)
assert tan(pi*Rational(-5, 6)) == 1/sqrt(3)
assert tan(pi/8) == -1 + sqrt(2)
assert tan(pi*Rational(3, 8)) == 1 + sqrt(2) # issue 15959
assert tan(pi*Rational(5, 8)) == -1 - sqrt(2)
assert tan(pi*Rational(7, 8)) == 1 - sqrt(2)
assert tan(pi/10) == sqrt(1 - 2*sqrt(5)/5)
assert tan(pi*Rational(3, 10)) == sqrt(1 + 2*sqrt(5)/5)
assert tan(pi*Rational(17, 10)) == -sqrt(1 + 2*sqrt(5)/5)
assert tan(pi*Rational(-31, 10)) == -sqrt(1 - 2*sqrt(5)/5)
assert tan(pi/12) == -sqrt(3) + 2
assert tan(pi*Rational(5, 12)) == sqrt(3) + 2
assert tan(pi*Rational(7, 12)) == -sqrt(3) - 2
assert tan(pi*Rational(11, 12)) == sqrt(3) - 2
assert tan(pi/24).radsimp() == -2 - sqrt(3) + sqrt(2) + sqrt(6)
assert tan(pi*Rational(5, 24)).radsimp() == -2 + sqrt(3) - sqrt(2) + sqrt(6)
assert tan(pi*Rational(7, 24)).radsimp() == 2 - sqrt(3) - sqrt(2) + sqrt(6)
assert tan(pi*Rational(11, 24)).radsimp() == 2 + sqrt(3) + sqrt(2) + sqrt(6)
assert tan(pi*Rational(13, 24)).radsimp() == -2 - sqrt(3) - sqrt(2) - sqrt(6)
assert tan(pi*Rational(17, 24)).radsimp() == -2 + sqrt(3) + sqrt(2) - sqrt(6)
assert tan(pi*Rational(19, 24)).radsimp() == 2 - sqrt(3) + sqrt(2) - sqrt(6)
assert tan(pi*Rational(23, 24)).radsimp() == 2 + sqrt(3) - sqrt(2) - sqrt(6)
assert tan(x*I) == tanh(x)*I
assert tan(k*pi) == 0
assert tan(17*k*pi) == 0
assert tan(k*pi*I) == tanh(k*pi)*I
assert tan(r).is_real is None
assert tan(r).is_extended_real is True
assert tan(0, evaluate=False).is_algebraic
assert tan(a).is_algebraic is None
assert tan(na).is_algebraic is False
assert tan(pi*Rational(10, 7)) == tan(pi*Rational(3, 7))
assert tan(pi*Rational(11, 7)) == -tan(pi*Rational(3, 7))
assert tan(pi*Rational(-11, 7)) == tan(pi*Rational(3, 7))
assert tan(pi*Rational(15, 14)) == tan(pi/14)
assert tan(pi*Rational(-15, 14)) == -tan(pi/14)
assert tan(r).is_finite is None
assert tan(I*r).is_finite is True
# https://github.com/sympy/sympy/issues/21177
f = tan(pi*(x + S(3)/2))/(3*x)
assert f.as_leading_term(x) == -1/(3*pi*x**2)
def test_tan_series():
assert tan(x).series(x, 0, 9) == \
x + x**3/3 + 2*x**5/15 + 17*x**7/315 + O(x**9)
def test_tan_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert tan(x).rewrite(exp) == I*(neg_exp - pos_exp)/(neg_exp + pos_exp)
assert tan(x).rewrite(sin) == 2*sin(x)**2/sin(2*x)
assert tan(x).rewrite(cos) == cos(x - S.Pi/2, evaluate=False)/cos(x)
assert tan(x).rewrite(cot) == 1/cot(x)
assert tan(sinh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sinh(3)).n()
assert tan(cosh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cosh(3)).n()
assert tan(tanh(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tanh(3)).n()
assert tan(coth(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, coth(3)).n()
assert tan(sin(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, sin(3)).n()
assert tan(cos(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cos(3)).n()
assert tan(tan(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, tan(3)).n()
assert tan(cot(x)).rewrite(
exp).subs(x, 3).n() == tan(x).rewrite(exp).subs(x, cot(3)).n()
assert tan(log(x)).rewrite(Pow) == I*(x**-I - x**I)/(x**-I + x**I)
assert 0 == (cos(pi/34)*tan(pi/34) - sin(pi/34)).rewrite(pow)
assert 0 == (cos(pi/17)*tan(pi/17) - sin(pi/17)).rewrite(pow)
assert tan(pi/19).rewrite(pow) == tan(pi/19)
assert tan(pi*Rational(8, 19)).rewrite(sqrt) == tan(pi*Rational(8, 19))
assert tan(x).rewrite(sec) == sec(x)/sec(x - pi/2, evaluate=False)
assert tan(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)/csc(x)
assert tan(sin(x)).rewrite(Pow) == tan(sin(x))
assert tan(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == sqrt(sqrt(5)/8 +
Rational(5, 8))/(Rational(-1, 4) + sqrt(5)/4)
def test_tan_subs():
assert tan(x).subs(tan(x), y) == y
assert tan(x).subs(x, y) == tan(y)
assert tan(x).subs(x, S.Pi/2) is zoo
assert tan(x).subs(x, S.Pi*Rational(3, 2)) is zoo
def test_tan_expansion():
assert tan(x + y).expand(trig=True) == ((tan(x) + tan(y))/(1 - tan(x)*tan(y))).expand()
assert tan(x - y).expand(trig=True) == ((tan(x) - tan(y))/(1 + tan(x)*tan(y))).expand()
assert tan(x + y + z).expand(trig=True) == (
(tan(x) + tan(y) + tan(z) - tan(x)*tan(y)*tan(z))/
(1 - tan(x)*tan(y) - tan(x)*tan(z) - tan(y)*tan(z))).expand()
assert 0 == tan(2*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 7))])*24 - 7
assert 0 == tan(3*x).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*55 - 37
assert 0 == tan(4*x - pi/4).expand(trig=True).rewrite(tan).subs([(tan(x), Rational(1, 5))])*239 - 1
_test_extrig(tan, 2, 2*tan(1)/(1 - tan(1)**2))
_test_extrig(tan, 3, (-tan(1)**3 + 3*tan(1))/(1 - 3*tan(1)**2))
def test_tan_AccumBounds():
assert tan(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
assert tan(AccumBounds(S.Pi/3, S.Pi*Rational(2, 3))) == AccumBounds(-oo, oo)
assert tan(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(tan(S.Pi/6), tan(S.Pi/3))
def test_tan_fdiff():
assert tan(x).fdiff() == tan(x)**2 + 1
raises(ArgumentIndexError, lambda: tan(x).fdiff(2))
def test_cot():
assert cot(nan) is nan
assert cot.nargs == FiniteSet(1)
assert cot(oo*I) == -I
assert cot(-oo*I) == I
assert cot(zoo) is nan
assert cot(0) is zoo
assert cot(2*pi) is zoo
assert cot(acot(x)) == x
assert cot(atan(x)) == 1 / x
assert cot(asin(x)) == sqrt(1 - x**2) / x
assert cot(acos(x)) == x / sqrt(1 - x**2)
assert cot(acsc(x)) == sqrt(1 - 1 / x**2) * x
assert cot(asec(x)) == 1 / (sqrt(1 - 1 / x**2) * x)
assert cot(atan2(y, x)) == x/y
assert cot(pi*I) == -coth(pi)*I
assert cot(-pi*I) == coth(pi)*I
assert cot(-2*I) == coth(2)*I
assert cot(pi) == cot(2*pi) == cot(3*pi)
assert cot(-pi) == cot(-2*pi) == cot(-3*pi)
assert cot(pi/2) == 0
assert cot(-pi/2) == 0
assert cot(pi*Rational(5, 2)) == 0
assert cot(pi*Rational(7, 2)) == 0
assert cot(pi/3) == 1/sqrt(3)
assert cot(pi*Rational(-2, 3)) == 1/sqrt(3)
assert cot(pi/4) is S.One
assert cot(-pi/4) is S.NegativeOne
assert cot(pi*Rational(17, 4)) is S.One
assert cot(pi*Rational(-3, 4)) is S.One
assert cot(pi/6) == sqrt(3)
assert cot(-pi/6) == -sqrt(3)
assert cot(pi*Rational(7, 6)) == sqrt(3)
assert cot(pi*Rational(-5, 6)) == sqrt(3)
assert cot(pi/8) == 1 + sqrt(2)
assert cot(pi*Rational(3, 8)) == -1 + sqrt(2)
assert cot(pi*Rational(5, 8)) == 1 - sqrt(2)
assert cot(pi*Rational(7, 8)) == -1 - sqrt(2)
assert cot(pi/12) == sqrt(3) + 2
assert cot(pi*Rational(5, 12)) == -sqrt(3) + 2
assert cot(pi*Rational(7, 12)) == sqrt(3) - 2
assert cot(pi*Rational(11, 12)) == -sqrt(3) - 2
assert cot(pi/24).radsimp() == sqrt(2) + sqrt(3) + 2 + sqrt(6)
assert cot(pi*Rational(5, 24)).radsimp() == -sqrt(2) - sqrt(3) + 2 + sqrt(6)
assert cot(pi*Rational(7, 24)).radsimp() == -sqrt(2) + sqrt(3) - 2 + sqrt(6)
assert cot(pi*Rational(11, 24)).radsimp() == sqrt(2) - sqrt(3) - 2 + sqrt(6)
assert cot(pi*Rational(13, 24)).radsimp() == -sqrt(2) + sqrt(3) + 2 - sqrt(6)
assert cot(pi*Rational(17, 24)).radsimp() == sqrt(2) - sqrt(3) + 2 - sqrt(6)
assert cot(pi*Rational(19, 24)).radsimp() == sqrt(2) + sqrt(3) - 2 - sqrt(6)
assert cot(pi*Rational(23, 24)).radsimp() == -sqrt(2) - sqrt(3) - 2 - sqrt(6)
assert cot(x*I) == -coth(x)*I
assert cot(k*pi*I) == -coth(k*pi)*I
assert cot(r).is_real is None
assert cot(r).is_extended_real is True
assert cot(a).is_algebraic is None
assert cot(na).is_algebraic is False
assert cot(pi*Rational(10, 7)) == cot(pi*Rational(3, 7))
assert cot(pi*Rational(11, 7)) == -cot(pi*Rational(3, 7))
assert cot(pi*Rational(-11, 7)) == cot(pi*Rational(3, 7))
assert cot(pi*Rational(39, 34)) == cot(pi*Rational(5, 34))
assert cot(pi*Rational(-41, 34)) == -cot(pi*Rational(7, 34))
assert cot(x).is_finite is None
assert cot(r).is_finite is None
i = Symbol('i', imaginary=True)
assert cot(i).is_finite is True
assert cot(x).subs(x, 3*pi) is zoo
# https://github.com/sympy/sympy/issues/21177
f = cot(pi*(x + 4))/(3*x)
assert f.as_leading_term(x) == 1/(3*pi*x**2)
def test_tan_cot_sin_cos_evalf():
assert abs((tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15)) - 1).evalf()) < 1e-14
assert abs((cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15)) - 1).evalf()) < 1e-14
@XFAIL
def test_tan_cot_sin_cos_ratsimp():
assert 1 == (tan(pi*Rational(8, 15))*cos(pi*Rational(8, 15))/sin(pi*Rational(8, 15))).ratsimp()
assert 1 == (cot(pi*Rational(4, 15))*sin(pi*Rational(4, 15))/cos(pi*Rational(4, 15))).ratsimp()
def test_cot_series():
assert cot(x).series(x, 0, 9) == \
1/x - x/3 - x**3/45 - 2*x**5/945 - x**7/4725 + O(x**9)
# issue 6210
assert cot(x**4 + x**5).series(x, 0, 1) == \
x**(-4) - 1/x**3 + x**(-2) - 1/x + 1 + O(x)
assert cot(pi*(1-x)).series(x, 0, 3) == -1/(pi*x) + pi*x/3 + O(x**3)
assert cot(x).taylor_term(0, x) == 1/x
assert cot(x).taylor_term(2, x) is S.Zero
assert cot(x).taylor_term(3, x) == -x**3/45
def test_cot_rewrite():
neg_exp, pos_exp = exp(-x*I), exp(x*I)
assert cot(x).rewrite(exp) == I*(pos_exp + neg_exp)/(pos_exp - neg_exp)
assert cot(x).rewrite(sin) == sin(2*x)/(2*(sin(x)**2))
assert cot(x).rewrite(cos) == cos(x)/cos(x - pi/2, evaluate=False)
assert cot(x).rewrite(tan) == 1/tan(x)
def check(func):
z = cot(func(x)).rewrite(exp
) - cot(x).rewrite(exp).subs(x, func(x))
assert z.rewrite(exp).expand() == 0
check(sinh)
check(cosh)
check(tanh)
check(coth)
check(sin)
check(cos)
check(tan)
assert cot(log(x)).rewrite(Pow) == -I*(x**-I + x**I)/(x**-I - x**I)
assert cot(pi*Rational(4, 34)).rewrite(pow).ratsimp() == (cos(pi*Rational(4, 34))/sin(pi*Rational(4, 34))).rewrite(pow).ratsimp()
assert cot(pi*Rational(4, 17)).rewrite(pow) == (cos(pi*Rational(4, 17))/sin(pi*Rational(4, 17))).rewrite(pow)
assert cot(pi/19).rewrite(pow) == cot(pi/19)
assert cot(pi/19).rewrite(sqrt) == cot(pi/19)
assert cot(x).rewrite(sec) == sec(x - pi / 2, evaluate=False) / sec(x)
assert cot(x).rewrite(csc) == csc(x) / csc(- x + pi / 2, evaluate=False)
assert cot(sin(x)).rewrite(Pow) == cot(sin(x))
assert cot(pi*Rational(2, 5), evaluate=False).rewrite(sqrt) == (Rational(-1, 4) + sqrt(5)/4)/\
sqrt(sqrt(5)/8 + Rational(5, 8))
def test_cot_subs():
assert cot(x).subs(cot(x), y) == y
assert cot(x).subs(x, y) == cot(y)
assert cot(x).subs(x, 0) is zoo
assert cot(x).subs(x, S.Pi) is zoo
def test_cot_expansion():
assert cot(x + y).expand(trig=True).together() == (
(cot(x)*cot(y) - 1)/(cot(x) + cot(y)))
assert cot(x - y).expand(trig=True).together() == (
cot(x)*cot(-y) - 1)/(cot(x) + cot(-y))
assert cot(x + y + z).expand(trig=True).together() == (
(cot(x)*cot(y)*cot(z) - cot(x) - cot(y) - cot(z))/
(-1 + cot(x)*cot(y) + cot(x)*cot(z) + cot(y)*cot(z)))
assert cot(3*x).expand(trig=True).together() == (
(cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1))
assert cot(2*x).expand(trig=True) == cot(x)/2 - 1/(2*cot(x))
assert cot(3*x).expand(trig=True).together() == (
cot(x)**2 - 3)*cot(x)/(3*cot(x)**2 - 1)
assert cot(4*x - pi/4).expand(trig=True).cancel() == (
-tan(x)**4 + 4*tan(x)**3 + 6*tan(x)**2 - 4*tan(x) - 1
)/(tan(x)**4 + 4*tan(x)**3 - 6*tan(x)**2 - 4*tan(x) + 1)
_test_extrig(cot, 2, (-1 + cot(1)**2)/(2*cot(1)))
_test_extrig(cot, 3, (-3*cot(1) + cot(1)**3)/(-1 + 3*cot(1)**2))
def test_cot_AccumBounds():
assert cot(AccumBounds(-oo, oo)) == AccumBounds(-oo, oo)
assert cot(AccumBounds(-S.Pi/3, S.Pi/3)) == AccumBounds(-oo, oo)
assert cot(AccumBounds(S.Pi/6, S.Pi/3)) == AccumBounds(cot(S.Pi/3), cot(S.Pi/6))
def test_cot_fdiff():
assert cot(x).fdiff() == -cot(x)**2 - 1
raises(ArgumentIndexError, lambda: cot(x).fdiff(2))
def test_sinc():
assert isinstance(sinc(x), sinc)
s = Symbol('s', zero=True)
assert sinc(s) is S.One
assert sinc(S.Infinity) is S.Zero
assert sinc(S.NegativeInfinity) is S.Zero
assert sinc(S.NaN) is S.NaN
assert sinc(S.ComplexInfinity) is S.NaN
n = Symbol('n', integer=True, nonzero=True)
assert sinc(n*pi) is S.Zero
assert sinc(-n*pi) is S.Zero
assert sinc(pi/2) == 2 / pi
assert sinc(-pi/2) == 2 / pi
assert sinc(pi*Rational(5, 2)) == 2 / (5*pi)
assert sinc(pi*Rational(7, 2)) == -2 / (7*pi)
assert sinc(-x) == sinc(x)
assert sinc(x).diff(x) == cos(x)/x - sin(x)/x**2
assert sinc(x).diff(x) == (sin(x)/x).diff(x)
assert sinc(x).diff(x, x) == (-sin(x) - 2*cos(x)/x + 2*sin(x)/x**2)/x
assert sinc(x).diff(x, x) == (sin(x)/x).diff(x, x)
assert limit(sinc(x).diff(x), x, 0) == 0
assert limit(sinc(x).diff(x, x), x, 0) == -S(1)/3
# https://github.com/sympy/sympy/issues/11402
#
# assert sinc(x).diff(x) == Piecewise(((x*cos(x) - sin(x)) / x**2, Ne(x, 0)), (0, True))
#
# assert sinc(x).diff(x).equals(sinc(x).rewrite(sin).diff(x))
#
# assert sinc(x).diff(x).subs(x, 0) is S.Zero
assert sinc(x).series() == 1 - x**2/6 + x**4/120 + O(x**6)
assert sinc(x).rewrite(jn) == jn(0, x)
assert sinc(x).rewrite(sin) == Piecewise((sin(x)/x, Ne(x, 0)), (1, True))
assert sinc(pi, evaluate=False).is_zero is True
assert sinc(0, evaluate=False).is_zero is False
assert sinc(n*pi, evaluate=False).is_zero is True
assert sinc(x).is_zero is None
xr = Symbol('xr', real=True, nonzero=True)
assert sinc(x).is_real is None
assert sinc(xr).is_real is True
assert sinc(I*xr).is_real is True
assert sinc(I*100).is_real is True
assert sinc(x).is_finite is None
assert sinc(xr).is_finite is True
def test_asin():
assert asin(nan) is nan
assert asin.nargs == FiniteSet(1)
assert asin(oo) == -I*oo
assert asin(-oo) == I*oo
assert asin(zoo) is zoo
# Note: asin(-x) = - asin(x)
assert asin(0) == 0
assert asin(1) == pi/2
assert asin(-1) == -pi/2
assert asin(sqrt(3)/2) == pi/3
assert asin(-sqrt(3)/2) == -pi/3
assert asin(sqrt(2)/2) == pi/4
assert asin(-sqrt(2)/2) == -pi/4
assert asin(sqrt((5 - sqrt(5))/8)) == pi/5
assert asin(-sqrt((5 - sqrt(5))/8)) == -pi/5
assert asin(S.Half) == pi/6
assert asin(Rational(-1, 2)) == -pi/6
assert asin((sqrt(2 - sqrt(2)))/2) == pi/8
assert asin(-(sqrt(2 - sqrt(2)))/2) == -pi/8
assert asin((sqrt(5) - 1)/4) == pi/10
assert asin(-(sqrt(5) - 1)/4) == -pi/10
assert asin((sqrt(3) - 1)/sqrt(2**3)) == pi/12
assert asin(-(sqrt(3) - 1)/sqrt(2**3)) == -pi/12
# check round-trip for exact values:
for d in [5, 6, 8, 10, 12]:
for n in range(-(d//2), d//2 + 1):
if gcd(n, d) == 1:
assert asin(sin(n*pi/d)) == n*pi/d
assert asin(x).diff(x) == 1/sqrt(1 - x**2)
assert asin(0.2, evaluate=False).is_real is True
assert asin(-2).is_real is False
assert asin(r).is_real is None
assert asin(-2*I) == -I*asinh(2)
assert asin(Rational(1, 7), evaluate=False).is_positive is True
assert asin(Rational(-1, 7), evaluate=False).is_positive is False
assert asin(p).is_positive is None
assert asin(sin(Rational(7, 2))) == Rational(-7, 2) + pi
assert asin(sin(Rational(-7, 4))) == Rational(7, 4) - pi
assert unchanged(asin, cos(x))
def test_asin_series():
assert asin(x).series(x, 0, 9) == \
x + x**3/6 + 3*x**5/40 + 5*x**7/112 + O(x**9)
t5 = asin(x).taylor_term(5, x)
assert t5 == 3*x**5/40
assert asin(x).taylor_term(7, x, t5, 0) == 5*x**7/112
def test_asin_leading_term():
assert asin(x).as_leading_term(x) == x
# Tests concerning branch points
assert asin(x + 1).as_leading_term(x) == pi/2
assert asin(x - 1).as_leading_term(x) == -pi/2
assert asin(1/x).as_leading_term(x, cdir=1) == I*log(x) + pi/2 - I*log(2)
assert asin(1/x).as_leading_term(x, cdir=-1) == -I*log(x) - 3*pi/2 + I*log(2)
# Tests concerning points lying on branch cuts
assert asin(I*x + 2).as_leading_term(x, cdir=1) == pi - asin(2)
assert asin(-I*x + 2).as_leading_term(x, cdir=1) == asin(2)
assert asin(I*x - 2).as_leading_term(x, cdir=1) == -asin(2)
assert asin(-I*x - 2).as_leading_term(x, cdir=1) == -pi + asin(2)
# Tests concerning im(ndir) == 0
assert asin(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == -pi/2 + I*log(2 - sqrt(3))
assert asin(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == -pi/2 + I*log(2 - sqrt(3))
def test_asin_rewrite():
assert asin(x).rewrite(log) == -I*log(I*x + sqrt(1 - x**2))
assert asin(x).rewrite(atan) == 2*atan(x/(1 + sqrt(1 - x**2)))
assert asin(x).rewrite(acos) == S.Pi/2 - acos(x)
assert asin(x).rewrite(acot) == 2*acot((sqrt(-x**2 + 1) + 1)/x)
assert asin(x).rewrite(asec) == -asec(1/x) + pi/2
assert asin(x).rewrite(acsc) == acsc(1/x)
def test_asin_fdiff():
assert asin(x).fdiff() == 1/sqrt(1 - x**2)
raises(ArgumentIndexError, lambda: asin(x).fdiff(2))
def test_acos():
assert acos(nan) is nan
assert acos(zoo) is zoo
assert acos.nargs == FiniteSet(1)
assert acos(oo) == I*oo
assert acos(-oo) == -I*oo
# Note: acos(-x) = pi - acos(x)
assert acos(0) == pi/2
assert acos(S.Half) == pi/3
assert acos(Rational(-1, 2)) == pi*Rational(2, 3)
assert acos(1) == 0
assert acos(-1) == pi
assert acos(sqrt(2)/2) == pi/4
assert acos(-sqrt(2)/2) == pi*Rational(3, 4)
# check round-trip for exact values:
for d in [5, 6, 8, 10, 12]:
for num in range(d):
if gcd(num, d) == 1:
assert acos(cos(num*pi/d)) == num*pi/d
assert acos(2*I) == pi/2 - asin(2*I)
assert acos(x).diff(x) == -1/sqrt(1 - x**2)
assert acos(0.2).is_real is True
assert acos(-2).is_real is False
assert acos(r).is_real is None
assert acos(Rational(1, 7), evaluate=False).is_positive is True
assert acos(Rational(-1, 7), evaluate=False).is_positive is True
assert acos(Rational(3, 2), evaluate=False).is_positive is False
assert acos(p).is_positive is None
assert acos(2 + p).conjugate() != acos(10 + p)
assert acos(-3 + n).conjugate() != acos(-3 + n)
assert acos(Rational(1, 3)).conjugate() == acos(Rational(1, 3))
assert acos(Rational(-1, 3)).conjugate() == acos(Rational(-1, 3))
assert acos(p + n*I).conjugate() == acos(p - n*I)
assert acos(z).conjugate() != acos(conjugate(z))
def test_acos_leading_term():
assert acos(x).as_leading_term(x) == pi/2
# Tests concerning branch points
assert acos(x + 1).as_leading_term(x) == sqrt(2)*sqrt(-x)
assert acos(x - 1).as_leading_term(x) == pi
assert acos(1/x).as_leading_term(x, cdir=1) == -I*log(x) + I*log(2)
assert acos(1/x).as_leading_term(x, cdir=-1) == I*log(x) + 2*pi - I*log(2)
# Tests concerning points lying on branch cuts
assert acos(I*x + 2).as_leading_term(x, cdir=1) == -acos(2)
assert acos(-I*x + 2).as_leading_term(x, cdir=1) == acos(2)
assert acos(I*x - 2).as_leading_term(x, cdir=1) == acos(-2)
assert acos(-I*x - 2).as_leading_term(x, cdir=1) == 2*pi - acos(-2)
# Tests concerning im(ndir) == 0
assert acos(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == pi + I*log(sqrt(3) + 2)
assert acos(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == pi + I*log(sqrt(3) + 2)
def test_acos_series():
assert acos(x).series(x, 0, 8) == \
pi/2 - x - x**3/6 - 3*x**5/40 - 5*x**7/112 + O(x**8)
assert acos(x).series(x, 0, 8) == pi/2 - asin(x).series(x, 0, 8)
t5 = acos(x).taylor_term(5, x)
assert t5 == -3*x**5/40
assert acos(x).taylor_term(7, x, t5, 0) == -5*x**7/112
assert acos(x).taylor_term(0, x) == pi/2
assert acos(x).taylor_term(2, x) is S.Zero
def test_acos_rewrite():
assert acos(x).rewrite(log) == pi/2 + I*log(I*x + sqrt(1 - x**2))
assert acos(x).rewrite(atan) == pi*(-x*sqrt(x**(-2)) + 1)/2 + atan(sqrt(1 - x**2)/x)
assert acos(0).rewrite(atan) == S.Pi/2
assert acos(0.5).rewrite(atan) == acos(0.5).rewrite(log)
assert acos(x).rewrite(asin) == S.Pi/2 - asin(x)
assert acos(x).rewrite(acot) == -2*acot((sqrt(-x**2 + 1) + 1)/x) + pi/2
assert acos(x).rewrite(asec) == asec(1/x)
assert acos(x).rewrite(acsc) == -acsc(1/x) + pi/2
def test_acos_fdiff():
assert acos(x).fdiff() == -1/sqrt(1 - x**2)
raises(ArgumentIndexError, lambda: acos(x).fdiff(2))
def test_atan():
assert atan(nan) is nan
assert atan.nargs == FiniteSet(1)
assert atan(oo) == pi/2
assert atan(-oo) == -pi/2
assert atan(zoo) == AccumBounds(-pi/2, pi/2)
assert atan(0) == 0
assert atan(1) == pi/4
assert atan(sqrt(3)) == pi/3
assert atan(-(1 + sqrt(2))) == pi*Rational(-3, 8)
assert atan(sqrt(5 - 2 * sqrt(5))) == pi/5
assert atan(-sqrt(1 - 2 * sqrt(5)/ 5)) == -pi/10
assert atan(sqrt(1 + 2 * sqrt(5) / 5)) == pi*Rational(3, 10)
assert atan(-2 + sqrt(3)) == -pi/12
assert atan(2 + sqrt(3)) == pi*Rational(5, 12)
assert atan(-2 - sqrt(3)) == pi*Rational(-5, 12)
# check round-trip for exact values:
for d in [5, 6, 8, 10, 12]:
for num in range(-(d//2), d//2 + 1):
if gcd(num, d) == 1:
assert atan(tan(num*pi/d)) == num*pi/d
assert atan(oo) == pi/2
assert atan(x).diff(x) == 1/(1 + x**2)
assert atan(r).is_real is True
assert atan(-2*I) == -I*atanh(2)
assert unchanged(atan, cot(x))
assert atan(cot(Rational(1, 4))) == Rational(-1, 4) + pi/2
assert acot(Rational(1, 4)).is_rational is False
for s in (x, p, n, np, nn, nz, ep, en, enp, enn, enz):
if s.is_real or s.is_extended_real is None:
assert s.is_nonzero is atan(s).is_nonzero
assert s.is_positive is atan(s).is_positive
assert s.is_negative is atan(s).is_negative
assert s.is_nonpositive is atan(s).is_nonpositive
assert s.is_nonnegative is atan(s).is_nonnegative
else:
assert s.is_extended_nonzero is atan(s).is_nonzero
assert s.is_extended_positive is atan(s).is_positive
assert s.is_extended_negative is atan(s).is_negative
assert s.is_extended_nonpositive is atan(s).is_nonpositive
assert s.is_extended_nonnegative is atan(s).is_nonnegative
assert s.is_extended_nonzero is atan(s).is_extended_nonzero
assert s.is_extended_positive is atan(s).is_extended_positive
assert s.is_extended_negative is atan(s).is_extended_negative
assert s.is_extended_nonpositive is atan(s).is_extended_nonpositive
assert s.is_extended_nonnegative is atan(s).is_extended_nonnegative
def test_atan_rewrite():
assert atan(x).rewrite(log) == I*(log(1 - I*x)-log(1 + I*x))/2
assert atan(x).rewrite(asin) == (-asin(1/sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
assert atan(x).rewrite(acos) == sqrt(x**2)*acos(1/sqrt(x**2 + 1))/x
assert atan(x).rewrite(acot) == acot(1/x)
assert atan(x).rewrite(asec) == sqrt(x**2)*asec(sqrt(x**2 + 1))/x
assert atan(x).rewrite(acsc) == (-acsc(sqrt(x**2 + 1)) + pi/2)*sqrt(x**2)/x
assert atan(-5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:-5*I})
assert atan(5*I).evalf() == atan(x).rewrite(log).evalf(subs={x:5*I})
def test_atan_fdiff():
assert atan(x).fdiff() == 1/(x**2 + 1)
raises(ArgumentIndexError, lambda: atan(x).fdiff(2))
def test_atan_leading_term():
assert atan(x).as_leading_term(x) == x
assert atan(1/x).as_leading_term(x, cdir=1) == pi/2
assert atan(1/x).as_leading_term(x, cdir=-1) == -pi/2
# Tests concerning branch points
assert atan(x + I).as_leading_term(x, cdir=1) == -I*log(x)/2 + pi/4 + I*log(2)/2
assert atan(x + I).as_leading_term(x, cdir=-1) == -I*log(x)/2 - 3*pi/4 + I*log(2)/2
assert atan(x - I).as_leading_term(x, cdir=1) == I*log(x)/2 + pi/4 - I*log(2)/2
assert atan(x - I).as_leading_term(x, cdir=-1) == I*log(x)/2 + pi/4 - I*log(2)/2
# Tests concerning points lying on branch cuts
assert atan(x + 2*I).as_leading_term(x, cdir=1) == I*atanh(2)
assert atan(x + 2*I).as_leading_term(x, cdir=-1) == -pi + I*atanh(2)
assert atan(x - 2*I).as_leading_term(x, cdir=1) == pi - I*atanh(2)
assert atan(x - 2*I).as_leading_term(x, cdir=-1) == -I*atanh(2)
# Tests concerning re(ndir) == 0
assert atan(2*I - I*x - x**2).as_leading_term(x, cdir=1) == -pi/2 + I*log(3)/2
assert atan(2*I - I*x - x**2).as_leading_term(x, cdir=-1) == -pi/2 + I*log(3)/2
def test_atan2():
assert atan2.nargs == FiniteSet(2)
assert atan2(0, 0) is S.NaN
assert atan2(0, 1) == 0
assert atan2(1, 1) == pi/4
assert atan2(1, 0) == pi/2
assert atan2(1, -1) == pi*Rational(3, 4)
assert atan2(0, -1) == pi
assert atan2(-1, -1) == pi*Rational(-3, 4)
assert atan2(-1, 0) == -pi/2
assert atan2(-1, 1) == -pi/4
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
eq = atan2(r, i)
ans = -I*log((i + I*r)/sqrt(i**2 + r**2))
reps = ((r, 2), (i, I))
assert eq.subs(reps) == ans.subs(reps)
x = Symbol('x', negative=True)
y = Symbol('y', negative=True)
assert atan2(y, x) == atan(y/x) - pi
y = Symbol('y', nonnegative=True)
assert atan2(y, x) == atan(y/x) + pi
y = Symbol('y')
assert atan2(y, x) == atan2(y, x, evaluate=False)
u = Symbol("u", positive=True)
assert atan2(0, u) == 0
u = Symbol("u", negative=True)
assert atan2(0, u) == pi
assert atan2(y, oo) == 0
assert atan2(y, -oo)== 2*pi*Heaviside(re(y), S.Half) - pi
assert atan2(y, x).rewrite(log) == -I*log((x + I*y)/sqrt(x**2 + y**2))
assert atan2(0, 0) is S.NaN
ex = atan2(y, x) - arg(x + I*y)
assert ex.subs({x:2, y:3}).rewrite(arg) == 0
assert ex.subs({x:2, y:3*I}).rewrite(arg) == -pi - I*log(sqrt(5)*I/5)
assert ex.subs({x:2*I, y:3}).rewrite(arg) == -pi/2 - I*log(sqrt(5)*I)
assert ex.subs({x:2*I, y:3*I}).rewrite(arg) == -pi + atan(Rational(2, 3)) + atan(Rational(3, 2))
i = symbols('i', imaginary=True)
r = symbols('r', real=True)
e = atan2(i, r)
rewrite = e.rewrite(arg)
reps = {i: I, r: -2}
assert rewrite == -I*log(abs(I*i + r)/sqrt(abs(i**2 + r**2))) + arg((I*i + r)/sqrt(i**2 + r**2))
assert (e - rewrite).subs(reps).equals(0)
assert atan2(0, x).rewrite(atan) == Piecewise((pi, re(x) < 0),
(0, Ne(x, 0)),
(nan, True))
assert atan2(0, r).rewrite(atan) == Piecewise((pi, r < 0), (0, Ne(r, 0)), (S.NaN, True))
assert atan2(0, i),rewrite(atan) == 0
assert atan2(0, r + i).rewrite(atan) == Piecewise((pi, r < 0), (0, True))
assert atan2(y, x).rewrite(atan) == Piecewise(
(2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)),
(pi, re(x) < 0),
(0, (re(x) > 0) | Ne(im(x), 0)),
(nan, True))
assert conjugate(atan2(x, y)) == atan2(conjugate(x), conjugate(y))
assert diff(atan2(y, x), x) == -y/(x**2 + y**2)
assert diff(atan2(y, x), y) == x/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), x)) == -y/(x**2 + y**2)
assert simplify(diff(atan2(y, x).rewrite(log), y)) == x/(x**2 + y**2)
assert str(atan2(1, 2).evalf(5)) == '0.46365'
raises(ArgumentIndexError, lambda: atan2(x, y).fdiff(3))
def test_issue_17461():
class A(Symbol):
is_extended_real = True
def _eval_evalf(self, prec):
return Float(5.0)
x = A('X')
y = A('Y')
assert abs(atan2(x, y).evalf() - 0.785398163397448) <= 1e-10
def test_acot():
assert acot(nan) is nan
assert acot.nargs == FiniteSet(1)
assert acot(-oo) == 0
assert acot(oo) == 0
assert acot(zoo) == 0
assert acot(1) == pi/4
assert acot(0) == pi/2
assert acot(sqrt(3)/3) == pi/3
assert acot(1/sqrt(3)) == pi/3
assert acot(-1/sqrt(3)) == -pi/3
assert acot(x).diff(x) == -1/(1 + x**2)
assert acot(r).is_extended_real is True
assert acot(I*pi) == -I*acoth(pi)
assert acot(-2*I) == I*acoth(2)
assert acot(x).is_positive is None
assert acot(n).is_positive is False
assert acot(p).is_positive is True
assert acot(I).is_positive is False
assert acot(Rational(1, 4)).is_rational is False
assert unchanged(acot, cot(x))
assert unchanged(acot, tan(x))
assert acot(cot(Rational(1, 4))) == Rational(1, 4)
assert acot(tan(Rational(-1, 4))) == Rational(1, 4) - pi/2
def test_acot_rewrite():
assert acot(x).rewrite(log) == I*(log(1 - I/x)-log(1 + I/x))/2
assert acot(x).rewrite(asin) == x*(-asin(sqrt(-x**2)/sqrt(-x**2 - 1)) + pi/2)*sqrt(x**(-2))
assert acot(x).rewrite(acos) == x*sqrt(x**(-2))*acos(sqrt(-x**2)/sqrt(-x**2 - 1))
assert acot(x).rewrite(atan) == atan(1/x)
assert acot(x).rewrite(asec) == x*sqrt(x**(-2))*asec(sqrt((x**2 + 1)/x**2))
assert acot(x).rewrite(acsc) == x*(-acsc(sqrt((x**2 + 1)/x**2)) + pi/2)*sqrt(x**(-2))
assert acot(-I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:-I/5})
assert acot(I/5).evalf() == acot(x).rewrite(log).evalf(subs={x:I/5})
def test_acot_fdiff():
assert acot(x).fdiff() == -1/(x**2 + 1)
raises(ArgumentIndexError, lambda: acot(x).fdiff(2))
def test_acot_leading_term():
assert acot(1/x).as_leading_term(x) == x
# Tests concerning branch points
assert acot(x + I).as_leading_term(x, cdir=1) == I*log(x)/2 + pi/4 - I*log(2)/2
assert acot(x + I).as_leading_term(x, cdir=-1) == I*log(x)/2 + pi/4 - I*log(2)/2
assert acot(x - I).as_leading_term(x, cdir=1) == -I*log(x)/2 + pi/4 + I*log(2)/2
assert acot(x - I).as_leading_term(x, cdir=-1) == -I*log(x)/2 - 3*pi/4 + I*log(2)/2
# Tests concerning points lying on branch cuts
assert acot(x).as_leading_term(x, cdir=1) == pi/2
assert acot(x).as_leading_term(x, cdir=-1) == -pi/2
assert acot(x + I/2).as_leading_term(x, cdir=1) == pi - I*acoth(S(1)/2)
assert acot(x + I/2).as_leading_term(x, cdir=-1) == -I*acoth(S(1)/2)
assert acot(x - I/2).as_leading_term(x, cdir=1) == I*acoth(S(1)/2)
assert acot(x - I/2).as_leading_term(x, cdir=-1) == -pi + I*acoth(S(1)/2)
# Tests concerning re(ndir) == 0
assert acot(I/2 - I*x - x**2).as_leading_term(x, cdir=1) == -pi/2 - I*log(3)/2
assert acot(I/2 - I*x - x**2).as_leading_term(x, cdir=-1) == -pi/2 - I*log(3)/2
def test_attributes():
assert sin(x).args == (x,)
def test_sincos_rewrite():
assert sin(pi/2 - x) == cos(x)
assert sin(pi - x) == sin(x)
assert cos(pi/2 - x) == sin(x)
assert cos(pi - x) == -cos(x)
def _check_even_rewrite(func, arg):
"""Checks that the expr has been rewritten using f(-x) -> f(x)
arg : -x
"""
return func(arg).args[0] == -arg
def _check_odd_rewrite(func, arg):
"""Checks that the expr has been rewritten using f(-x) -> -f(x)
arg : -x
"""
return func(arg).func.is_Mul
def _check_no_rewrite(func, arg):
"""Checks that the expr is not rewritten"""
return func(arg).args[0] == arg
def test_evenodd_rewrite():
a = cos(2) # negative
b = sin(1) # positive
even = [cos]
odd = [sin, tan, cot, asin, atan, acot]
with_minus = [-1, -2**1024 * E, -pi/105, -x*y, -x - y]
for func in even:
for expr in with_minus:
assert _check_even_rewrite(func, expr)
assert _check_no_rewrite(func, a*b)
assert func(
x - y) == func(y - x) # it doesn't matter which form is canonical
for func in odd:
for expr in with_minus:
assert _check_odd_rewrite(func, expr)
assert _check_no_rewrite(func, a*b)
assert func(
x - y) == -func(y - x) # it doesn't matter which form is canonical
def test_as_leading_term_issue_5272():
assert sin(x).as_leading_term(x) == x
assert cos(x).as_leading_term(x) == 1
assert tan(x).as_leading_term(x) == x
assert cot(x).as_leading_term(x) == 1/x
def test_leading_terms():
assert sin(1/x).as_leading_term(x) == AccumBounds(-1, 1)
assert sin(S.Half).as_leading_term(x) == sin(S.Half)
assert cos(1/x).as_leading_term(x) == AccumBounds(-1, 1)
assert cos(S.Half).as_leading_term(x) == cos(S.Half)
assert sec(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert csc(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert tan(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
assert cot(1/x).as_leading_term(x) == AccumBounds(S.NegativeInfinity, S.Infinity)
# https://github.com/sympy/sympy/issues/21038
f = sin(pi*(x + 4))/(3*x)
assert f.as_leading_term(x) == pi/3
def test_atan2_expansion():
assert cancel(atan2(x**2, x + 1).diff(x) - atan(x**2/(x + 1)).diff(x)) == 0
assert cancel(atan(y/x).series(y, 0, 5) - atan2(y, x).series(y, 0, 5)
+ atan2(0, x) - atan(0)) == O(y**5)
assert cancel(atan(y/x).series(x, 1, 4) - atan2(y, x).series(x, 1, 4)
+ atan2(y, 1) - atan(y)) == O((x - 1)**4, (x, 1))
assert cancel(atan((y + x)/x).series(x, 1, 3) - atan2(y + x, x).series(x, 1, 3)
+ atan2(1 + y, 1) - atan(1 + y)) == O((x - 1)**3, (x, 1))
assert Matrix([atan2(y, x)]).jacobian([y, x]) == \
Matrix([[x/(y**2 + x**2), -y/(y**2 + x**2)]])
def test_aseries():
def t(n, v, d, e):
assert abs(
n(1/v).evalf() - n(1/x).series(x, dir=d).removeO().subs(x, v)) < e
t(atan, 0.1, '+', 1e-5)
t(atan, -0.1, '-', 1e-5)
t(acot, 0.1, '+', 1e-5)
t(acot, -0.1, '-', 1e-5)
def test_issue_4420():
i = Symbol('i', integer=True)
e = Symbol('e', even=True)
o = Symbol('o', odd=True)
# unknown parity for variable
assert cos(4*i*pi) == 1
assert sin(4*i*pi) == 0
assert tan(4*i*pi) == 0
assert cot(4*i*pi) is zoo
assert cos(3*i*pi) == cos(pi*i) # +/-1
assert sin(3*i*pi) == 0
assert tan(3*i*pi) == 0
assert cot(3*i*pi) is zoo
assert cos(4.0*i*pi) == 1
assert sin(4.0*i*pi) == 0
assert tan(4.0*i*pi) == 0
assert cot(4.0*i*pi) is zoo
assert cos(3.0*i*pi) == cos(pi*i) # +/-1
assert sin(3.0*i*pi) == 0
assert tan(3.0*i*pi) == 0
assert cot(3.0*i*pi) is zoo
assert cos(4.5*i*pi) == cos(0.5*pi*i)
assert sin(4.5*i*pi) == sin(0.5*pi*i)
assert tan(4.5*i*pi) == tan(0.5*pi*i)
assert cot(4.5*i*pi) == cot(0.5*pi*i)
# parity of variable is known
assert cos(4*e*pi) == 1
assert sin(4*e*pi) == 0
assert tan(4*e*pi) == 0
assert cot(4*e*pi) is zoo
assert cos(3*e*pi) == 1
assert sin(3*e*pi) == 0
assert tan(3*e*pi) == 0
assert cot(3*e*pi) is zoo
assert cos(4.0*e*pi) == 1
assert sin(4.0*e*pi) == 0
assert tan(4.0*e*pi) == 0
assert cot(4.0*e*pi) is zoo
assert cos(3.0*e*pi) == 1
assert sin(3.0*e*pi) == 0
assert tan(3.0*e*pi) == 0
assert cot(3.0*e*pi) is zoo
assert cos(4.5*e*pi) == cos(0.5*pi*e)
assert sin(4.5*e*pi) == sin(0.5*pi*e)
assert tan(4.5*e*pi) == tan(0.5*pi*e)
assert cot(4.5*e*pi) == cot(0.5*pi*e)
assert cos(4*o*pi) == 1
assert sin(4*o*pi) == 0
assert tan(4*o*pi) == 0
assert cot(4*o*pi) is zoo
assert cos(3*o*pi) == -1
assert sin(3*o*pi) == 0
assert tan(3*o*pi) == 0
assert cot(3*o*pi) is zoo
assert cos(4.0*o*pi) == 1
assert sin(4.0*o*pi) == 0
assert tan(4.0*o*pi) == 0
assert cot(4.0*o*pi) is zoo
assert cos(3.0*o*pi) == -1
assert sin(3.0*o*pi) == 0
assert tan(3.0*o*pi) == 0
assert cot(3.0*o*pi) is zoo
assert cos(4.5*o*pi) == cos(0.5*pi*o)
assert sin(4.5*o*pi) == sin(0.5*pi*o)
assert tan(4.5*o*pi) == tan(0.5*pi*o)
assert cot(4.5*o*pi) == cot(0.5*pi*o)
# x could be imaginary
assert cos(4*x*pi) == cos(4*pi*x)
assert sin(4*x*pi) == sin(4*pi*x)
assert tan(4*x*pi) == tan(4*pi*x)
assert cot(4*x*pi) == cot(4*pi*x)
assert cos(3*x*pi) == cos(3*pi*x)
assert sin(3*x*pi) == sin(3*pi*x)
assert tan(3*x*pi) == tan(3*pi*x)
assert cot(3*x*pi) == cot(3*pi*x)
assert cos(4.0*x*pi) == cos(4.0*pi*x)
assert sin(4.0*x*pi) == sin(4.0*pi*x)
assert tan(4.0*x*pi) == tan(4.0*pi*x)
assert cot(4.0*x*pi) == cot(4.0*pi*x)
assert cos(3.0*x*pi) == cos(3.0*pi*x)
assert sin(3.0*x*pi) == sin(3.0*pi*x)
assert tan(3.0*x*pi) == tan(3.0*pi*x)
assert cot(3.0*x*pi) == cot(3.0*pi*x)
assert cos(4.5*x*pi) == cos(4.5*pi*x)
assert sin(4.5*x*pi) == sin(4.5*pi*x)
assert tan(4.5*x*pi) == tan(4.5*pi*x)
assert cot(4.5*x*pi) == cot(4.5*pi*x)
def test_inverses():
raises(AttributeError, lambda: sin(x).inverse())
raises(AttributeError, lambda: cos(x).inverse())
assert tan(x).inverse() == atan
assert cot(x).inverse() == acot
raises(AttributeError, lambda: csc(x).inverse())
raises(AttributeError, lambda: sec(x).inverse())
assert asin(x).inverse() == sin
assert acos(x).inverse() == cos
assert atan(x).inverse() == tan
assert acot(x).inverse() == cot
def test_real_imag():
a, b = symbols('a b', real=True)
z = a + b*I
for deep in [True, False]:
assert sin(
z).as_real_imag(deep=deep) == (sin(a)*cosh(b), cos(a)*sinh(b))
assert cos(
z).as_real_imag(deep=deep) == (cos(a)*cosh(b), -sin(a)*sinh(b))
assert tan(z).as_real_imag(deep=deep) == (sin(2*a)/(cos(2*a) +
cosh(2*b)), sinh(2*b)/(cos(2*a) + cosh(2*b)))
assert cot(z).as_real_imag(deep=deep) == (-sin(2*a)/(cos(2*a) -
cosh(2*b)), sinh(2*b)/(cos(2*a) - cosh(2*b)))
assert sin(a).as_real_imag(deep=deep) == (sin(a), 0)
assert cos(a).as_real_imag(deep=deep) == (cos(a), 0)
assert tan(a).as_real_imag(deep=deep) == (tan(a), 0)
assert cot(a).as_real_imag(deep=deep) == (cot(a), 0)
@XFAIL
def test_sin_cos_with_infinity():
# Test for issue 5196
# https://github.com/sympy/sympy/issues/5196
assert sin(oo) is S.NaN
assert cos(oo) is S.NaN
@slow
def test_sincos_rewrite_sqrt():
# equivalent to testing rewrite(pow)
for p in [1, 3, 5, 17]:
for t in [1, 8]:
n = t*p
# The vertices `exp(i*pi/n)` of a regular `n`-gon can
# be expressed by means of nested square roots if and
# only if `n` is a product of Fermat primes, `p`, and
# powers of 2, `t'. The code aims to check all vertices
# not belonging to an `m`-gon for `m < n`(`gcd(i, n) == 1`).
# For large `n` this makes the test too slow, therefore
# the vertices are limited to those of index `i < 10`.
for i in range(1, min((n + 1)//2 + 1, 10)):
if 1 == gcd(i, n):
x = i*pi/n
s1 = sin(x).rewrite(sqrt)
c1 = cos(x).rewrite(sqrt)
assert not s1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
assert not c1.has(cos, sin), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs(sin(x.evalf(5)) - s1.evalf(2)), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs(cos(x.evalf(5)) - c1.evalf(2)), "fails for %d*pi/%d" % (i, n)
assert cos(pi/14).rewrite(sqrt) == sqrt(cos(pi/7)/2 + S.Half)
assert cos(pi/257).rewrite(sqrt).evalf(64) == cos(pi/257).evalf(64)
assert cos(pi*Rational(-15, 2)/11, evaluate=False).rewrite(
sqrt) == -sqrt(-cos(pi*Rational(4, 11))/2 + S.Half)
assert cos(Mul(2, pi, S.Half, evaluate=False), evaluate=False).rewrite(
sqrt) == -1
e = cos(pi/3/17) # don't use pi/15 since that is caught at instantiation
a = (
-3*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17) + 17)/64 -
3*sqrt(34)*sqrt(sqrt(17) + 17)/128 - sqrt(sqrt(17) +
17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+ sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 - sqrt(-sqrt(17)
+ 17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 - Rational(1, 32) +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
3*sqrt(2)*sqrt(sqrt(17) + 17)/128 + sqrt(34)*sqrt(-sqrt(17) + 17)/128
+ 13*sqrt(2)*sqrt(-sqrt(17) + 17)/128 + sqrt(17)*sqrt(-sqrt(17) +
17)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) + 17)
+ sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/128 + 5*sqrt(17)/32
+ sqrt(3)*sqrt(-sqrt(2)*sqrt(sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/8 -
5*sqrt(2)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 -
3*sqrt(2)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/32
+ sqrt(34)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/64 +
sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 + Rational(15, 32))/2 +
S.Half + sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 + sqrt(2)*sqrt(-sqrt(17) +
17)/32 + sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
6*sqrt(17) + 34)/32 + Rational(15, 32))*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) -
sqrt(2)*sqrt(-sqrt(17) + 17) + sqrt(34)*sqrt(-sqrt(17) + 17) +
6*sqrt(17) + 34)/32 + sqrt(34)*sqrt(-sqrt(17) + 17)*sqrt(sqrt(17)/32 +
sqrt(2)*sqrt(-sqrt(17) + 17)/32 +
sqrt(2)*sqrt(-8*sqrt(2)*sqrt(sqrt(17) + 17) - sqrt(2)*sqrt(-sqrt(17) +
17) + sqrt(34)*sqrt(-sqrt(17) + 17) + 6*sqrt(17) + 34)/32 +
Rational(15, 32))/32)/2)
assert e.rewrite(sqrt) == a
assert e.n() == a.n()
# coverage of fermatCoords: multiplicity > 1; the following could be
# different but that portion of the code should be tested in some way
assert cos(pi/9/17).rewrite(sqrt) == \
sin(pi/9)*sin(pi*Rational(2, 17)) + cos(pi/9)*cos(pi*Rational(2, 17))
@slow
def test_tancot_rewrite_sqrt():
# equivalent to testing rewrite(pow)
for p in [1, 3, 5, 17]:
for t in [1, 8]:
n = t*p
for i in range(1, min((n + 1)//2 + 1, 10)):
if 1 == gcd(i, n):
x = i*pi/n
if 2*i != n and 3*i != 2*n:
t1 = tan(x).rewrite(sqrt)
assert not t1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs( tan(x.evalf(7)) - t1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
if i != 0 and i != n:
c1 = cot(x).rewrite(sqrt)
assert not c1.has(cot, tan), "fails for %d*pi/%d" % (i, n)
assert 1e-3 > abs( cot(x.evalf(7)) - c1.evalf(4) ), "fails for %d*pi/%d" % (i, n)
def test_sec():
x = symbols('x', real=True)
z = symbols('z')
assert sec.nargs == FiniteSet(1)
assert sec(zoo) is nan
assert sec(0) == 1
assert sec(pi) == -1
assert sec(pi/2) is zoo
assert sec(-pi/2) is zoo
assert sec(pi/6) == 2*sqrt(3)/3
assert sec(pi/3) == 2
assert sec(pi*Rational(5, 2)) is zoo
assert sec(pi*Rational(9, 7)) == -sec(pi*Rational(2, 7))
assert sec(pi*Rational(3, 4)) == -sqrt(2) # issue 8421
assert sec(I) == 1/cosh(1)
assert sec(x*I) == 1/cosh(x)
assert sec(-x) == sec(x)
assert sec(asec(x)) == x
assert sec(z).conjugate() == sec(conjugate(z))
assert (sec(z).as_real_imag() ==
(cos(re(z))*cosh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2),
sin(re(z))*sinh(im(z))/(sin(re(z))**2*sinh(im(z))**2 +
cos(re(z))**2*cosh(im(z))**2)))
assert sec(x).expand(trig=True) == 1/cos(x)
assert sec(2*x).expand(trig=True) == 1/(2*cos(x)**2 - 1)
assert sec(x).is_extended_real == True
assert sec(z).is_real == None
assert sec(a).is_algebraic is None
assert sec(na).is_algebraic is False
assert sec(x).as_leading_term() == sec(x)
assert sec(0, evaluate=False).is_finite == True
assert sec(x).is_finite == None
assert sec(pi/2, evaluate=False).is_finite == False
assert series(sec(x), x, x0=0, n=6) == 1 + x**2/2 + 5*x**4/24 + O(x**6)
# https://github.com/sympy/sympy/issues/7166
assert series(sqrt(sec(x))) == 1 + x**2/4 + 7*x**4/96 + O(x**6)
# https://github.com/sympy/sympy/issues/7167
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - pi*Rational(3, 2)) + (x - pi*Rational(3, 2))**Rational(3, 2)/12 +
(x - pi*Rational(3, 2))**Rational(7, 2)/160 + O((x - pi*Rational(3, 2))**4, (x, pi*Rational(3, 2))))
assert sec(x).diff(x) == tan(x)*sec(x)
# Taylor Term checks
assert sec(z).taylor_term(4, z) == 5*z**4/24
assert sec(z).taylor_term(6, z) == 61*z**6/720
assert sec(z).taylor_term(5, z) == 0
def test_sec_rewrite():
assert sec(x).rewrite(exp) == 1/(exp(I*x)/2 + exp(-I*x)/2)
assert sec(x).rewrite(cos) == 1/cos(x)
assert sec(x).rewrite(tan) == (tan(x/2)**2 + 1)/(-tan(x/2)**2 + 1)
assert sec(x).rewrite(pow) == sec(x)
assert sec(x).rewrite(sqrt) == sec(x)
assert sec(z).rewrite(cot) == (cot(z/2)**2 + 1)/(cot(z/2)**2 - 1)
assert sec(x).rewrite(sin) == 1 / sin(x + pi / 2, evaluate=False)
assert sec(x).rewrite(tan) == (tan(x / 2)**2 + 1) / (-tan(x / 2)**2 + 1)
assert sec(x).rewrite(csc) == csc(-x + pi/2, evaluate=False)
def test_sec_fdiff():
assert sec(x).fdiff() == tan(x)*sec(x)
raises(ArgumentIndexError, lambda: sec(x).fdiff(2))
def test_csc():
x = symbols('x', real=True)
z = symbols('z')
# https://github.com/sympy/sympy/issues/6707
cosecant = csc('x')
alternate = 1/sin('x')
assert cosecant.equals(alternate) == True
assert alternate.equals(cosecant) == True
assert csc.nargs == FiniteSet(1)
assert csc(0) is zoo
assert csc(pi) is zoo
assert csc(zoo) is nan
assert csc(pi/2) == 1
assert csc(-pi/2) == -1
assert csc(pi/6) == 2
assert csc(pi/3) == 2*sqrt(3)/3
assert csc(pi*Rational(5, 2)) == 1
assert csc(pi*Rational(9, 7)) == -csc(pi*Rational(2, 7))
assert csc(pi*Rational(3, 4)) == sqrt(2) # issue 8421
assert csc(I) == -I/sinh(1)
assert csc(x*I) == -I/sinh(x)
assert csc(-x) == -csc(x)
assert csc(acsc(x)) == x
assert csc(z).conjugate() == csc(conjugate(z))
assert (csc(z).as_real_imag() ==
(sin(re(z))*cosh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2),
-cos(re(z))*sinh(im(z))/(sin(re(z))**2*cosh(im(z))**2 +
cos(re(z))**2*sinh(im(z))**2)))
assert csc(x).expand(trig=True) == 1/sin(x)
assert csc(2*x).expand(trig=True) == 1/(2*sin(x)*cos(x))
assert csc(x).is_extended_real == True
assert csc(z).is_real == None
assert csc(a).is_algebraic is None
assert csc(na).is_algebraic is False
assert csc(x).as_leading_term() == csc(x)
assert csc(0, evaluate=False).is_finite == False
assert csc(x).is_finite == None
assert csc(pi/2, evaluate=False).is_finite == True
assert series(csc(x), x, x0=pi/2, n=6) == \
1 + (x - pi/2)**2/2 + 5*(x - pi/2)**4/24 + O((x - pi/2)**6, (x, pi/2))
assert series(csc(x), x, x0=0, n=6) == \
1/x + x/6 + 7*x**3/360 + 31*x**5/15120 + O(x**6)
assert csc(x).diff(x) == -cot(x)*csc(x)
assert csc(x).taylor_term(2, x) == 0
assert csc(x).taylor_term(3, x) == 7*x**3/360
assert csc(x).taylor_term(5, x) == 31*x**5/15120
raises(ArgumentIndexError, lambda: csc(x).fdiff(2))
def test_asec():
z = Symbol('z', zero=True)
assert asec(z) is zoo
assert asec(nan) is nan
assert asec(1) == 0
assert asec(-1) == pi
assert asec(oo) == pi/2
assert asec(-oo) == pi/2
assert asec(zoo) == pi/2
assert asec(sec(pi*Rational(13, 4))) == pi*Rational(3, 4)
assert asec(1 + sqrt(5)) == pi*Rational(2, 5)
assert asec(2/sqrt(3)) == pi/6
assert asec(sqrt(4 - 2*sqrt(2))) == pi/8
assert asec(-sqrt(4 + 2*sqrt(2))) == pi*Rational(5, 8)
assert asec(sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(3, 10)
assert asec(-sqrt(2 + 2*sqrt(5)/5)) == pi*Rational(7, 10)
assert asec(sqrt(2) - sqrt(6)) == pi*Rational(11, 12)
assert asec(x).diff(x) == 1/(x**2*sqrt(1 - 1/x**2))
assert asec(x).rewrite(log) == I*log(sqrt(1 - 1/x**2) + I/x) + pi/2
assert asec(x).rewrite(asin) == -asin(1/x) + pi/2
assert asec(x).rewrite(acos) == acos(1/x)
assert asec(x).rewrite(atan) == \
pi*(1 - sqrt(x**2)/x)/2 + sqrt(x**2)*atan(sqrt(x**2 - 1))/x
assert asec(x).rewrite(acot) == \
pi*(1 - sqrt(x**2)/x)/2 + sqrt(x**2)*acot(1/sqrt(x**2 - 1))/x
assert asec(x).rewrite(acsc) == -acsc(x) + pi/2
raises(ArgumentIndexError, lambda: asec(x).fdiff(2))
def test_asec_is_real():
assert asec(S.Half).is_real is False
n = Symbol('n', positive=True, integer=True)
assert asec(n).is_extended_real is True
assert asec(x).is_real is None
assert asec(r).is_real is None
t = Symbol('t', real=False, finite=True)
assert asec(t).is_real is False
def test_asec_leading_term():
assert asec(1/x).as_leading_term(x) == pi/2
# Tests concerning branch points
assert asec(x + 1).as_leading_term(x) == sqrt(2)*sqrt(x)
assert asec(x - 1).as_leading_term(x) == pi
# Tests concerning points lying on branch cuts
assert asec(x).as_leading_term(x, cdir=1) == -I*log(x) + I*log(2)
assert asec(x).as_leading_term(x, cdir=-1) == I*log(x) + 2*pi - I*log(2)
assert asec(I*x + 1/2).as_leading_term(x, cdir=1) == asec(1/2)
assert asec(-I*x + 1/2).as_leading_term(x, cdir=1) == -asec(1/2)
assert asec(I*x - 1/2).as_leading_term(x, cdir=1) == 2*pi - asec(-1/2)
assert asec(-I*x - 1/2).as_leading_term(x, cdir=1) == asec(-1/2)
# Tests concerning im(ndir) == 0
assert asec(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=1) == pi + I*log(2 - sqrt(3))
assert asec(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=-1) == pi + I*log(2 - sqrt(3))
def test_asec_series():
assert asec(x).series(x, 0, 9) == \
I*log(2) - I*log(x) - I*x**2/4 - 3*I*x**4/32 \
- 5*I*x**6/96 - 35*I*x**8/1024 + O(x**9)
t4 = asec(x).taylor_term(4, x)
assert t4 == -3*I*x**4/32
assert asec(x).taylor_term(6, x, t4, 0) == -5*I*x**6/96
def test_acsc():
assert acsc(nan) is nan
assert acsc(1) == pi/2
assert acsc(-1) == -pi/2
assert acsc(oo) == 0
assert acsc(-oo) == 0
assert acsc(zoo) == 0
assert acsc(0) is zoo
assert acsc(csc(3)) == -3 + pi
assert acsc(csc(4)) == -4 + pi
assert acsc(csc(6)) == 6 - 2*pi
assert unchanged(acsc, csc(x))
assert unchanged(acsc, sec(x))
assert acsc(2/sqrt(3)) == pi/3
assert acsc(csc(pi*Rational(13, 4))) == -pi/4
assert acsc(sqrt(2 + 2*sqrt(5)/5)) == pi/5
assert acsc(-sqrt(2 + 2*sqrt(5)/5)) == -pi/5
assert acsc(-2) == -pi/6
assert acsc(-sqrt(4 + 2*sqrt(2))) == -pi/8
assert acsc(sqrt(4 - 2*sqrt(2))) == pi*Rational(3, 8)
assert acsc(1 + sqrt(5)) == pi/10
assert acsc(sqrt(2) - sqrt(6)) == pi*Rational(-5, 12)
assert acsc(x).diff(x) == -1/(x**2*sqrt(1 - 1/x**2))
assert acsc(x).rewrite(log) == -I*log(sqrt(1 - 1/x**2) + I/x)
assert acsc(x).rewrite(asin) == asin(1/x)
assert acsc(x).rewrite(acos) == -acos(1/x) + pi/2
assert acsc(x).rewrite(atan) == \
(-atan(sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(acot) == (-acot(1/sqrt(x**2 - 1)) + pi/2)*sqrt(x**2)/x
assert acsc(x).rewrite(asec) == -asec(x) + pi/2
raises(ArgumentIndexError, lambda: acsc(x).fdiff(2))
def test_csc_rewrite():
assert csc(x).rewrite(pow) == csc(x)
assert csc(x).rewrite(sqrt) == csc(x)
assert csc(x).rewrite(exp) == 2*I/(exp(I*x) - exp(-I*x))
assert csc(x).rewrite(sin) == 1/sin(x)
assert csc(x).rewrite(tan) == (tan(x/2)**2 + 1)/(2*tan(x/2))
assert csc(x).rewrite(cot) == (cot(x/2)**2 + 1)/(2*cot(x/2))
assert csc(x).rewrite(cos) == 1/cos(x - pi/2, evaluate=False)
assert csc(x).rewrite(sec) == sec(-x + pi/2, evaluate=False)
# issue 17349
assert csc(1 - exp(-besselj(I, I))).rewrite(cos) == \
-1/cos(-pi/2 - 1 + cos(I*besselj(I, I)) +
I*cos(-pi/2 + I*besselj(I, I), evaluate=False), evaluate=False)
def test_acsc_leading_term():
assert acsc(1/x).as_leading_term(x) == x
# Tests concerning branch points
assert acsc(x + 1).as_leading_term(x) == pi/2
assert acsc(x - 1).as_leading_term(x) == -pi/2
# Tests concerning points lying on branch cuts
assert acsc(x).as_leading_term(x, cdir=1) == I*log(x) + pi/2 - I*log(2)
assert acsc(x).as_leading_term(x, cdir=-1) == -I*log(x) - 3*pi/2 + I*log(2)
assert acsc(I*x + 1/2).as_leading_term(x, cdir=1) == acsc(1/2)
assert acsc(-I*x + 1/2).as_leading_term(x, cdir=1) == pi - acsc(1/2)
assert acsc(I*x - 1/2).as_leading_term(x, cdir=1) == -pi - acsc(-1/2)
assert acsc(-I*x - 1/2).as_leading_term(x, cdir=1) == -acsc(1/2)
# Tests concerning im(ndir) == 0
assert acsc(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=1) == -pi/2 + I*log(sqrt(3) + 2)
assert acsc(-I*x**2 + x - S(1)/2).as_leading_term(x, cdir=-1) == -pi/2 + I*log(sqrt(3) + 2)
def test_acsc_series():
assert acsc(x).series(x, 0, 9) == \
-I*log(2) + pi/2 + I*log(x) + I*x**2/4 \
+ 3*I*x**4/32 + 5*I*x**6/96 + 35*I*x**8/1024 + O(x**9)
t6 = acsc(x).taylor_term(6, x)
assert t6 == 5*I*x**6/96
assert acsc(x).taylor_term(8, x, t6, 0) == 35*I*x**8/1024
def test_asin_nseries():
assert asin(x + 2)._eval_nseries(x, 4, None, I) == -asin(2) + pi + \
sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
assert asin(x + 2)._eval_nseries(x, 4, None, -I) == asin(2) - \
sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert asin(x - 2)._eval_nseries(x, 4, None, I) == -asin(2) - \
sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert asin(x - 2)._eval_nseries(x, 4, None, -I) == asin(2) - pi + \
sqrt(3)*I*x/3 + sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
# testing nseries for asin at branch points
assert asin(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) - \
sqrt(2)*(-x)**(S(3)/2)/12 - 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
assert asin(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) + \
sqrt(2)*x**(S(3)/2)/12 + 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
assert asin(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(-x) + \
sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
assert asin(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - \
sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
def test_acos_nseries():
assert acos(x + 2)._eval_nseries(x, 4, None, I) == -acos(2) - sqrt(3)*I*x/3 + \
sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert acos(x + 2)._eval_nseries(x, 4, None, -I) == acos(2) + sqrt(3)*I*x/3 - \
sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
assert acos(x - 2)._eval_nseries(x, 4, None, I) == acos(-2) + sqrt(3)*I*x/3 + \
sqrt(3)*I*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
assert acos(x - 2)._eval_nseries(x, 4, None, -I) == -acos(-2) + 2*pi - \
sqrt(3)*I*x/3 - sqrt(3)*I*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
# testing nseries for acos at branch points
assert acos(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) + \
sqrt(2)*(-x)**(S(3)/2)/12 + 3*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
assert acos(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) - \
sqrt(2)*x**(S(3)/2)/12 - 3*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
assert acos(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(-x) - \
sqrt(2)*(-x)**(S(3)/2)/6 + sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
assert acos(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + \
sqrt(2)*(-x)**(S(3)/2)/6 - sqrt(2)*(-x)**(S(5)/2)/120 + O(x**3)
def test_atan_nseries():
assert atan(x + 2*I)._eval_nseries(x, 4, None, 1) == I*atanh(2) - x/3 - \
2*I*x**2/9 + 13*x**3/81 + O(x**4)
assert atan(x + 2*I)._eval_nseries(x, 4, None, -1) == I*atanh(2) - pi - \
x/3 - 2*I*x**2/9 + 13*x**3/81 + O(x**4)
assert atan(x - 2*I)._eval_nseries(x, 4, None, 1) == -I*atanh(2) + pi - \
x/3 + 2*I*x**2/9 + 13*x**3/81 + O(x**4)
assert atan(x - 2*I)._eval_nseries(x, 4, None, -1) == -I*atanh(2) - x/3 + \
2*I*x**2/9 + 13*x**3/81 + O(x**4)
assert atan(1/x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2)
assert atan(1/x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2)
# testing nseries for atan at branch points
assert atan(x + I)._eval_nseries(x, 4, None) == I*log(2)/2 + pi/4 - \
I*log(x)/2 + x/4 + I*x**2/16 - x**3/48 + O(x**4)
assert atan(x - I)._eval_nseries(x, 4, None) == -I*log(2)/2 + pi/4 + \
I*log(x)/2 + x/4 - I*x**2/16 - x**3/48 + O(x**4)
def test_acot_nseries():
assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, 1) == -I*acoth(S(1)/2) + \
pi - 4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4)
assert acot(x + S(1)/2*I)._eval_nseries(x, 4, None, -1) == -I*acoth(S(1)/2) - \
4*x/3 + 8*I*x**2/9 + 112*x**3/81 + O(x**4)
assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, 1) == I*acoth(S(1)/2) - \
4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4)
assert acot(x - S(1)/2*I)._eval_nseries(x, 4, None, -1) == I*acoth(S(1)/2) - \
pi - 4*x/3 - 8*I*x**2/9 + 112*x**3/81 + O(x**4)
assert acot(x)._eval_nseries(x, 2, None, 1) == pi/2 - x + O(x**2)
assert acot(x)._eval_nseries(x, 2, None, -1) == -pi/2 - x + O(x**2)
# testing nseries for acot at branch points
assert acot(x + I)._eval_nseries(x, 4, None) == -I*log(2)/2 + pi/4 + \
I*log(x)/2 - x/4 - I*x**2/16 + x**3/48 + O(x**4)
assert acot(x - I)._eval_nseries(x, 4, None) == I*log(2)/2 + pi/4 - \
I*log(x)/2 - x/4 + I*x**2/16 + x**3/48 + O(x**4)
def test_asec_nseries():
assert asec(x + S(1)/2)._eval_nseries(x, 4, None, I) == asec(S(1)/2) - \
4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
assert asec(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -asec(S(1)/2) + \
4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert asec(x - S(1)/2)._eval_nseries(x, 4, None, I) == -asec(-S(1)/2) + \
2*pi + 4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert asec(x - S(1)/2)._eval_nseries(x, 4, None, -I) == asec(-S(1)/2) - \
4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
# testing nseries for asec at branch points
assert asec(1 + x)._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - \
5*sqrt(2)*x**(S(3)/2)/12 + 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
assert asec(-1 + x)._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(-x) + \
5*sqrt(2)*(-x)**(S(3)/2)/12 - 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
assert asec(exp(x))._eval_nseries(x, 3, None) == sqrt(2)*sqrt(x) - \
sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3)
assert asec(-exp(x))._eval_nseries(x, 3, None) == pi - sqrt(2)*sqrt(x) + \
sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3)
def test_acsc_nseries():
assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) + \
4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsc(x + S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + \
pi - 4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, I) == acsc(S(1)/2) - pi -\
4*sqrt(3)*I*x/3 - 8*sqrt(3)*I*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsc(x - S(1)/2)._eval_nseries(x, 4, None, -I) == -acsc(S(1)/2) + \
4*sqrt(3)*I*x/3 + 8*sqrt(3)*I*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
# testing nseries for acsc at branch points
assert acsc(1 + x)._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + \
5*sqrt(2)*x**(S(3)/2)/12 - 43*sqrt(2)*x**(S(5)/2)/160 + O(x**3)
assert acsc(-1 + x)._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(-x) - \
5*sqrt(2)*(-x)**(S(3)/2)/12 + 43*sqrt(2)*(-x)**(S(5)/2)/160 + O(x**3)
assert acsc(exp(x))._eval_nseries(x, 3, None) == pi/2 - sqrt(2)*sqrt(x) + \
sqrt(2)*x**(S(3)/2)/6 - sqrt(2)*x**(S(5)/2)/120 + O(x**3)
assert acsc(-exp(x))._eval_nseries(x, 3, None) == -pi/2 + sqrt(2)*sqrt(x) - \
sqrt(2)*x**(S(3)/2)/6 + sqrt(2)*x**(S(5)/2)/120 + O(x**3)
def test_issue_8653():
n = Symbol('n', integer=True)
assert sin(n).is_irrational is None
assert cos(n).is_irrational is None
assert tan(n).is_irrational is None
def test_issue_9157():
n = Symbol('n', integer=True, positive=True)
assert atan(n - 1).is_nonnegative is True
def test_trig_period():
x, y = symbols('x, y')
assert sin(x).period() == 2*pi
assert cos(x).period() == 2*pi
assert tan(x).period() == pi
assert cot(x).period() == pi
assert sec(x).period() == 2*pi
assert csc(x).period() == 2*pi
assert sin(2*x).period() == pi
assert cot(4*x - 6).period() == pi/4
assert cos((-3)*x).period() == pi*Rational(2, 3)
assert cos(x*y).period(x) == 2*pi/abs(y)
assert sin(3*x*y + 2*pi).period(y) == 2*pi/abs(3*x)
assert tan(3*x).period(y) is S.Zero
raises(NotImplementedError, lambda: sin(x**2).period(x))
def test_issue_7171():
assert sin(x).rewrite(sqrt) == sin(x)
assert sin(x).rewrite(pow) == sin(x)
def test_issue_11864():
w, k = symbols('w, k', real=True)
F = Piecewise((1, Eq(2*pi*k, 0)), (sin(pi*k)/(pi*k), True))
soln = Piecewise((1, Eq(2*pi*k, 0)), (sinc(pi*k), True))
assert F.rewrite(sinc) == soln
def test_real_assumptions():
z = Symbol('z', real=False, finite=True)
assert sin(z).is_real is None
assert cos(z).is_real is None
assert tan(z).is_real is False
assert sec(z).is_real is None
assert csc(z).is_real is None
assert cot(z).is_real is False
assert asin(p).is_real is None
assert asin(n).is_real is None
assert asec(p).is_real is None
assert asec(n).is_real is None
assert acos(p).is_real is None
assert acos(n).is_real is None
assert acsc(p).is_real is None
assert acsc(n).is_real is None
assert atan(p).is_positive is True
assert atan(n).is_negative is True
assert acot(p).is_positive is True
assert acot(n).is_negative is True
def test_issue_14320():
assert asin(sin(2)) == -2 + pi and (-pi/2 <= -2 + pi <= pi/2) and sin(2) == sin(-2 + pi)
assert asin(cos(2)) == -2 + pi/2 and (-pi/2 <= -2 + pi/2 <= pi/2) and cos(2) == sin(-2 + pi/2)
assert acos(sin(2)) == -pi/2 + 2 and (0 <= -pi/2 + 2 <= pi) and sin(2) == cos(-pi/2 + 2)
assert acos(cos(20)) == -6*pi + 20 and (0 <= -6*pi + 20 <= pi) and cos(20) == cos(-6*pi + 20)
assert acos(cos(30)) == -30 + 10*pi and (0 <= -30 + 10*pi <= pi) and cos(30) == cos(-30 + 10*pi)
assert atan(tan(17)) == -5*pi + 17 and (-pi/2 < -5*pi + 17 < pi/2) and tan(17) == tan(-5*pi + 17)
assert atan(tan(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 < pi/2) and tan(15) == tan(-5*pi + 15)
assert atan(cot(12)) == -12 + pi*Rational(7, 2) and (-pi/2 < -12 + pi*Rational(7, 2) < pi/2) and cot(12) == tan(-12 + pi*Rational(7, 2))
assert acot(cot(15)) == -5*pi + 15 and (-pi/2 < -5*pi + 15 <= pi/2) and cot(15) == cot(-5*pi + 15)
assert acot(tan(19)) == -19 + pi*Rational(13, 2) and (-pi/2 < -19 + pi*Rational(13, 2) <= pi/2) and tan(19) == cot(-19 + pi*Rational(13, 2))
assert asec(sec(11)) == -11 + 4*pi and (0 <= -11 + 4*pi <= pi) and cos(11) == cos(-11 + 4*pi)
assert asec(csc(13)) == -13 + pi*Rational(9, 2) and (0 <= -13 + pi*Rational(9, 2) <= pi) and sin(13) == cos(-13 + pi*Rational(9, 2))
assert acsc(csc(14)) == -4*pi + 14 and (-pi/2 <= -4*pi + 14 <= pi/2) and sin(14) == sin(-4*pi + 14)
assert acsc(sec(10)) == pi*Rational(-7, 2) + 10 and (-pi/2 <= pi*Rational(-7, 2) + 10 <= pi/2) and cos(10) == sin(pi*Rational(-7, 2) + 10)
def test_issue_14543():
assert sec(2*pi + 11) == sec(11)
assert sec(2*pi - 11) == sec(11)
assert sec(pi + 11) == -sec(11)
assert sec(pi - 11) == -sec(11)
assert csc(2*pi + 17) == csc(17)
assert csc(2*pi - 17) == -csc(17)
assert csc(pi + 17) == -csc(17)
assert csc(pi - 17) == csc(17)
x = Symbol('x')
assert csc(pi/2 + x) == sec(x)
assert csc(pi/2 - x) == sec(x)
assert csc(pi*Rational(3, 2) + x) == -sec(x)
assert csc(pi*Rational(3, 2) - x) == -sec(x)
assert sec(pi/2 - x) == csc(x)
assert sec(pi/2 + x) == -csc(x)
assert sec(pi*Rational(3, 2) + x) == csc(x)
assert sec(pi*Rational(3, 2) - x) == -csc(x)
def test_as_real_imag():
# This is for https://github.com/sympy/sympy/issues/17142
# If it start failing again in irrelevant builds or in the master
# please open up the issue again.
expr = atan(I/(I + I*tan(1)))
assert expr.as_real_imag() == (expr, 0)
def test_issue_18746():
e3 = cos(S.Pi*(x/4 + 1/4))
assert e3.period() == 8
|
779d21ba2a06cf2c61163316e94a7a5355a7314a018b42897908dc1be436af8b | from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import unchanged
from sympy.core.function import (Function, diff, expand)
from sympy.core.mul import Mul
from sympy.core.numbers import (Float, I, Rational, oo, pi, zoo)
from sympy.core.relational import (Eq, Ge, Gt, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.complexes import (Abs, adjoint, arg, conjugate, im, re, transpose)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import (Piecewise,
piecewise_fold, piecewise_exclusive, Undefined, ExprCondPair)
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.integrals.integrals import (Integral, integrate)
from sympy.logic.boolalg import (And, ITE, Not, Or)
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.printing import srepr
from sympy.sets.contains import Contains
from sympy.sets.sets import Interval
from sympy.solvers.solvers import solve
from sympy.testing.pytest import raises, slow
from sympy.utilities.lambdify import lambdify
a, b, c, d, x, y = symbols('a:d, x, y')
z = symbols('z', nonzero=True)
def test_piecewise1():
# Test canonicalization
assert unchanged(Piecewise, ExprCondPair(x, x < 1), ExprCondPair(0, True))
assert Piecewise((x, x < 1), (0, True)) == Piecewise(ExprCondPair(x, x < 1),
ExprCondPair(0, True))
assert Piecewise((x, x < 1), (0, True), (1, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, False), (-1, 1 > 2)) == \
Piecewise((x, x < 1))
assert Piecewise((x, x < 1), (0, x < 1), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (0, x < 2), (0, True)) == \
Piecewise((x, x < 1), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (0, True)) == \
Piecewise((x, Or(x < 1, x < 2)), (0, True))
assert Piecewise((x, x < 1), (x, x < 2), (x, True)) == x
assert Piecewise((x, True)) == x
# Explicitly constructed empty Piecewise not accepted
raises(TypeError, lambda: Piecewise())
# False condition is never retained
assert Piecewise((2*x, x < 0), (x, False)) == \
Piecewise((2*x, x < 0), (x, False), evaluate=False) == \
Piecewise((2*x, x < 0))
assert Piecewise((x, False)) == Undefined
raises(TypeError, lambda: Piecewise(x))
assert Piecewise((x, 1)) == x # 1 and 0 are accepted as True/False
raises(TypeError, lambda: Piecewise((x, 2)))
raises(TypeError, lambda: Piecewise((x, x**2)))
raises(TypeError, lambda: Piecewise(([1], True)))
assert Piecewise(((1, 2), True)) == Tuple(1, 2)
cond = (Piecewise((1, x < 0), (2, True)) < y)
assert Piecewise((1, cond)
) == Piecewise((1, ITE(x < 0, y > 1, y > 2)))
assert Piecewise((1, x > 0), (2, And(x <= 0, x > -1))
) == Piecewise((1, x > 0), (2, x > -1))
assert Piecewise((1, x <= 0), (2, (x < 0) & (x > -1))
) == Piecewise((1, x <= 0))
# test for supporting Contains in Piecewise
pwise = Piecewise(
(1, And(x <= 6, x > 1, Contains(x, S.Integers))),
(0, True))
assert pwise.subs(x, pi) == 0
assert pwise.subs(x, 2) == 1
assert pwise.subs(x, 7) == 0
# Test subs
p = Piecewise((-1, x < -1), (x**2, x < 0), (log(x), x >= 0))
p_x2 = Piecewise((-1, x**2 < -1), (x**4, x**2 < 0), (log(x**2), x**2 >= 0))
assert p.subs(x, x**2) == p_x2
assert p.subs(x, -5) == -1
assert p.subs(x, -1) == 1
assert p.subs(x, 1) == log(1)
# More subs tests
p2 = Piecewise((1, x < pi), (-1, x < 2*pi), (0, x > 2*pi))
p3 = Piecewise((1, Eq(x, 0)), (1/x, True))
p4 = Piecewise((1, Eq(x, 0)), (2, 1/x>2))
assert p2.subs(x, 2) == 1
assert p2.subs(x, 4) == -1
assert p2.subs(x, 10) == 0
assert p3.subs(x, 0.0) == 1
assert p4.subs(x, 0.0) == 1
f, g, h = symbols('f,g,h', cls=Function)
pf = Piecewise((f(x), x < -1), (f(x) + h(x) + 2, x <= 1))
pg = Piecewise((g(x), x < -1), (g(x) + h(x) + 2, x <= 1))
assert pg.subs(g, f) == pf
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 0) == 1
assert Piecewise((1, Eq(x, 0)), (0, True)).subs(x, 1) == 0
assert Piecewise((1, Eq(x, y)), (0, True)).subs(x, y) == 1
assert Piecewise((1, Eq(x, z)), (0, True)).subs(x, z) == 1
assert Piecewise((1, Eq(exp(x), cos(z))), (0, True)).subs(x, z) == \
Piecewise((1, Eq(exp(z), cos(z))), (0, True))
p5 = Piecewise( (0, Eq(cos(x) + y, 0)), (1, True))
assert p5.subs(y, 0) == Piecewise( (0, Eq(cos(x), 0)), (1, True))
assert Piecewise((-1, y < 1), (0, x < 0), (1, Eq(x, 0)), (2, True)
).subs(x, 1) == Piecewise((-1, y < 1), (2, True))
assert Piecewise((1, Eq(x**2, -1)), (2, x < 0)).subs(x, I) == 1
p6 = Piecewise((x, x > 0))
n = symbols('n', negative=True)
assert p6.subs(x, n) == Undefined
# Test evalf
assert p.evalf() == Piecewise((-1.0, x < -1), (x**2, x < 0), (log(x), True))
assert p.evalf(subs={x: -2}) == -1
assert p.evalf(subs={x: -1}) == 1
assert p.evalf(subs={x: 1}) == log(1)
assert p6.evalf(subs={x: -5}) == Undefined
# Test doit
f_int = Piecewise((Integral(x, (x, 0, 1)), x < 1))
assert f_int.doit() == Piecewise( (S.Half, x < 1) )
# Test differentiation
f = x
fp = x*p
dp = Piecewise((0, x < -1), (2*x, x < 0), (1/x, x >= 0))
fp_dx = x*dp + p
assert diff(p, x) == dp
assert diff(f*p, x) == fp_dx
# Test simple arithmetic
assert x*p == fp
assert x*p + p == p + x*p
assert p + f == f + p
assert p + dp == dp + p
assert p - dp == -(dp - p)
# Test power
dp2 = Piecewise((0, x < -1), (4*x**2, x < 0), (1/x**2, x >= 0))
assert dp**2 == dp2
# Test _eval_interval
f1 = x*y + 2
f2 = x*y**2 + 3
peval = Piecewise((f1, x < 0), (f2, x > 0))
peval_interval = f1.subs(
x, 0) - f1.subs(x, -1) + f2.subs(x, 1) - f2.subs(x, 0)
assert peval._eval_interval(x, 0, 0) == 0
assert peval._eval_interval(x, -1, 1) == peval_interval
peval2 = Piecewise((f1, x < 0), (f2, True))
assert peval2._eval_interval(x, 0, 0) == 0
assert peval2._eval_interval(x, 1, -1) == -peval_interval
assert peval2._eval_interval(x, -1, -2) == f1.subs(x, -2) - f1.subs(x, -1)
assert peval2._eval_interval(x, -1, 1) == peval_interval
assert peval2._eval_interval(x, None, 0) == peval2.subs(x, 0)
assert peval2._eval_interval(x, -1, None) == -peval2.subs(x, -1)
# Test integration
assert p.integrate() == Piecewise(
(-x, x < -1),
(x**3/3 + Rational(4, 3), x < 0),
(x*log(x) - x + Rational(4, 3), True))
p = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
assert integrate(p, (x, -2, 2)) == Rational(5, 6)
assert integrate(p, (x, 2, -2)) == Rational(-5, 6)
p = Piecewise((0, x < 0), (1, x < 1), (0, x < 2), (1, x < 3), (0, True))
assert integrate(p, (x, -oo, oo)) == 2
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
assert integrate(p, (x, -2, 2)) == Undefined
# Test commutativity
assert isinstance(p, Piecewise) and p.is_commutative is True
def test_piecewise_free_symbols():
f = Piecewise((x, a < 0), (y, True))
assert f.free_symbols == {x, y, a}
def test_piecewise_integrate1():
x, y = symbols('x y', real=True)
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert integrate(f, (x, -2, 2)) == Rational(14, 3)
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == Rational(43, 6)
assert g == Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
g = Piecewise(((x - 5)**5, 2 <= x), (f, x < 2))
assert integrate(g, (x, -2, 2)) == Rational(14, 3)
assert integrate(g, (x, -2, 5)) == Rational(-701, 6)
assert g == Piecewise(((x - 5)**5, 2 <= x), (f, True))
g = Piecewise(((x - 5)**5, 2 <= x), (2*f, True))
assert integrate(g, (x, -2, 2)) == Rational(28, 3)
assert integrate(g, (x, -2, 5)) == Rational(-673, 6)
def test_piecewise_integrate1b():
g = Piecewise((1, x > 0), (0, Eq(x, 0)), (-1, x < 0))
assert integrate(g, (x, -1, 1)) == 0
g = Piecewise((1, x - y < 0), (0, True))
assert integrate(g, (y, -oo, 0)) == -Min(0, x)
assert g.subs(x, -3).integrate((y, -oo, 0)) == 3
assert integrate(g, (y, 0, -oo)) == Min(0, x)
assert integrate(g, (y, 0, oo)) == -Max(0, x) + oo
assert integrate(g, (y, -oo, 42)) == -Min(42, x) + 42
assert integrate(g, (y, -oo, oo)) == -x + oo
g = Piecewise((0, x < 0), (x, x <= 1), (1, True))
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
for yy in (-1, S.Half, 2):
assert g.integrate((x, yy, 1)) == gy1.subs(y, yy)
assert g.integrate((x, 1, yy)) == g1y.subs(y, yy)
assert gy1 == Piecewise(
(-Min(1, Max(0, y))**2/2 + S.Half, y < 1),
(-y + 1, True))
assert g1y == Piecewise(
(Min(1, Max(0, y))**2/2 - S.Half, y < 1),
(y - 1, True))
@slow
def test_piecewise_integrate1ca():
y = symbols('y', real=True)
g = Piecewise(
(1 - x, Interval(0, 1).contains(x)),
(1 + x, Interval(-1, 0).contains(x)),
(0, True)
)
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
assert g.integrate((x, 2, 1)) == gy1.subs(y, 2)
assert g.integrate((x, 1, 2)) == g1y.subs(y, 2)
assert piecewise_fold(gy1.rewrite(Piecewise)
).simplify() == Piecewise(
(1, y <= -1),
(-y**2/2 - y + S.Half, y <= 0),
(y**2/2 - y + S.Half, y < 1),
(0, True))
assert piecewise_fold(g1y.rewrite(Piecewise)
).simplify() == Piecewise(
(-1, y <= -1),
(y**2/2 + y - S.Half, y <= 0),
(-y**2/2 + y - S.Half, y < 1),
(0, True))
assert gy1 == Piecewise(
(
-Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
Min(1, Max(0, y))**2 + S.Half, y < 1),
(0, True)
)
assert g1y == Piecewise(
(
Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
Min(1, Max(0, y))**2 - S.Half, y < 1),
(0, True))
@slow
def test_piecewise_integrate1cb():
y = symbols('y', real=True)
g = Piecewise(
(0, Or(x <= -1, x >= 1)),
(1 - x, x > 0),
(1 + x, True)
)
gy1 = g.integrate((x, y, 1))
g1y = g.integrate((x, 1, y))
assert g.integrate((x, -2, 1)) == gy1.subs(y, -2)
assert g.integrate((x, 1, -2)) == g1y.subs(y, -2)
assert g.integrate((x, 0, 1)) == gy1.subs(y, 0)
assert g.integrate((x, 1, 0)) == g1y.subs(y, 0)
assert g.integrate((x, 2, 1)) == gy1.subs(y, 2)
assert g.integrate((x, 1, 2)) == g1y.subs(y, 2)
assert piecewise_fold(gy1.rewrite(Piecewise)
).simplify() == Piecewise(
(1, y <= -1),
(-y**2/2 - y + S.Half, y <= 0),
(y**2/2 - y + S.Half, y < 1),
(0, True))
assert piecewise_fold(g1y.rewrite(Piecewise)
).simplify() == Piecewise(
(-1, y <= -1),
(y**2/2 + y - S.Half, y <= 0),
(-y**2/2 + y - S.Half, y < 1),
(0, True))
# g1y and gy1 should simplify if the condition that y < 1
# is applied, e.g. Min(1, Max(-1, y)) --> Max(-1, y)
assert gy1 == Piecewise(
(
-Min(1, Max(-1, y))**2/2 - Min(1, Max(-1, y)) +
Min(1, Max(0, y))**2 + S.Half, y < 1),
(0, True)
)
assert g1y == Piecewise(
(
Min(1, Max(-1, y))**2/2 + Min(1, Max(-1, y)) -
Min(1, Max(0, y))**2 - S.Half, y < 1),
(0, True))
def test_piecewise_integrate2():
from itertools import permutations
lim = Tuple(x, c, d)
p = Piecewise((1, x < a), (2, x > b), (3, True))
q = p.integrate(lim)
assert q == Piecewise(
(-c + 2*d - 2*Min(d, Max(a, c)) + Min(d, Max(a, b, c)), c < d),
(-2*c + d + 2*Min(c, Max(a, d)) - Min(c, Max(a, b, d)), True))
for v in permutations((1, 2, 3, 4)):
r = dict(zip((a, b, c, d), v))
assert p.subs(r).integrate(lim.subs(r)) == q.subs(r)
def test_meijer_bypass():
# totally bypass meijerg machinery when dealing
# with Piecewise in integrate
assert Piecewise((1, x < 4), (0, True)).integrate((x, oo, 1)) == -3
def test_piecewise_integrate3_inequality_conditions():
from sympy.utilities.iterables import cartes
lim = (x, 0, 5)
# set below includes two pts below range, 2 pts in range,
# 2 pts above range, and the boundaries
N = (-2, -1, 0, 1, 2, 5, 6, 7)
p = Piecewise((1, x > a), (2, x > b), (0, True))
ans = p.integrate(lim)
for i, j in cartes(N, repeat=2):
reps = dict(zip((a, b), (i, j)))
assert ans.subs(reps) == p.subs(reps).integrate(lim)
assert ans.subs(a, 4).subs(b, 1) == 0 + 2*3 + 1
p = Piecewise((1, x > a), (2, x < b), (0, True))
ans = p.integrate(lim)
for i, j in cartes(N, repeat=2):
reps = dict(zip((a, b), (i, j)))
assert ans.subs(reps) == p.subs(reps).integrate(lim)
# delete old tests that involved c1 and c2 since those
# reduce to the above except that a value of 0 was used
# for two expressions whereas the above uses 3 different
# values
@slow
def test_piecewise_integrate4_symbolic_conditions():
a = Symbol('a', real=True)
b = Symbol('b', real=True)
x = Symbol('x', real=True)
y = Symbol('y', real=True)
p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
p1 = Piecewise((0, x < a), (0, x > b), (1, True))
p2 = Piecewise((0, x > b), (0, x < a), (1, True))
p3 = Piecewise((0, x < a), (1, x < b), (0, True))
p4 = Piecewise((0, x > b), (1, x > a), (0, True))
p5 = Piecewise((1, And(a < x, x < b)), (0, True))
# check values of a=1, b=3 (and reversed) with values
# of y of 0, 1, 2, 3, 4
lim = Tuple(x, -oo, y)
for p in (p0, p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a: 3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
lim = Tuple(x, y, oo)
for p in (p0, p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a:3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
ans = Piecewise(
(0, x <= Min(a, b)),
(x - Min(a, b), x <= b),
(b - Min(a, b), True))
for i in (p0, p1, p2, p4):
assert i.integrate(x) == ans
assert p3.integrate(x) == Piecewise(
(0, x < a),
(-a + x, x <= Max(a, b)),
(-a + Max(a, b), True))
assert p5.integrate(x) == Piecewise(
(0, x <= a),
(-a + x, x <= Max(a, b)),
(-a + Max(a, b), True))
p1 = Piecewise((0, x < a), (0.5, x > b), (1, True))
p2 = Piecewise((0.5, x > b), (0, x < a), (1, True))
p3 = Piecewise((0, x < a), (1, x < b), (0.5, True))
p4 = Piecewise((0.5, x > b), (1, x > a), (0, True))
p5 = Piecewise((1, And(a < x, x < b)), (0.5, x > b), (0, True))
# check values of a=1, b=3 (and reversed) with values
# of y of 0, 1, 2, 3, 4
lim = Tuple(x, -oo, y)
for p in (p1, p2, p3, p4, p5):
ans = p.integrate(lim)
for i in range(5):
reps = {a:1, b:3, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
reps = {a: 3, b:1, y:i}
assert ans.subs(reps) == p.subs(reps).integrate(lim.subs(reps))
def test_piecewise_integrate5_independent_conditions():
p = Piecewise((0, Eq(y, 0)), (x*y, True))
assert integrate(p, (x, 1, 3)) == Piecewise((0, Eq(y, 0)), (4*y, True))
def test_piecewise_simplify():
p = Piecewise(((x**2 + 1)/x**2, Eq(x*(1 + x) - x**2, 0)),
((-1)**x*(-1), True))
assert p.simplify() == \
Piecewise((zoo, Eq(x, 0)), ((-1)**(x + 1), True))
# simplify when there are Eq in conditions
assert Piecewise(
(a, And(Eq(a, 0), Eq(a + b, 0))), (1, True)).simplify(
) == Piecewise(
(0, And(Eq(a, 0), Eq(b, 0))), (1, True))
assert Piecewise((2*x*factorial(a)/(factorial(y)*factorial(-y + a)),
Eq(y, 0) & Eq(-y + a, 0)), (2*factorial(a)/(factorial(y)*factorial(-y
+ a)), Eq(y, 0) & Eq(-y + a, 1)), (0, True)).simplify(
) == Piecewise(
(2*x, And(Eq(a, 0), Eq(y, 0))),
(2, And(Eq(a, 1), Eq(y, 0))),
(0, True))
args = (2, And(Eq(x, 2), Ge(y, 0))), (x, True)
assert Piecewise(*args).simplify() == Piecewise(*args)
args = (1, Eq(x, 0)), (sin(x)/x, True)
assert Piecewise(*args).simplify() == Piecewise(*args)
assert Piecewise((2 + y, And(Eq(x, 2), Eq(y, 0))), (x, True)
).simplify() == x
# check that x or f(x) are recognized as being Symbol-like for lhs
args = Tuple((1, Eq(x, 0)), (sin(x) + 1 + x, True))
ans = x + sin(x) + 1
f = Function('f')
assert Piecewise(*args).simplify() == ans
assert Piecewise(*args.subs(x, f(x))).simplify() == ans.subs(x, f(x))
# issue 18634
d = Symbol("d", integer=True)
n = Symbol("n", integer=True)
t = Symbol("t", positive=True)
expr = Piecewise((-d + 2*n, Eq(1/t, 1)), (t**(1 - 4*n)*t**(4*n - 1)*(-d + 2*n), True))
assert expr.simplify() == -d + 2*n
# issue 22747
p = Piecewise((0, (t < -2) & (t < -1) & (t < 0)), ((t/2 + 1)*(t +
1)*(t + 2), (t < -1) & (t < 0)), ((S.Half - t/2)*(1 - t)*(t + 1),
(t < -2) & (t < -1) & (t < 1)), ((t + 1)*(-t*(t/2 + 1) + (S.Half
- t/2)*(1 - t)), (t < -2) & (t < -1) & (t < 0) & (t < 1)), ((t +
1)*((S.Half - t/2)*(1 - t) + (t/2 + 1)*(t + 2)), (t < -1) & (t <
1)), ((t + 1)*(-t*(t/2 + 1) + (S.Half - t/2)*(1 - t)), (t < -1) &
(t < 0) & (t < 1)), (0, (t < -2) & (t < -1)), ((t/2 + 1)*(t +
1)*(t + 2), t < -1), ((t + 1)*(-t*(t/2 + 1) + (S.Half - t/2)*(t +
1)), (t < 0) & ((t < -2) | (t < 0))), ((S.Half - t/2)*(1 - t)*(t
+ 1), (t < 1) & ((t < -2) | (t < 1))), (0, True)) + Piecewise((0,
(t < -1) & (t < 0) & (t < 1)), ((1 - t)*(t/2 + S.Half)*(t + 1),
(t < 0) & (t < 1)), ((1 - t)*(1 - t/2)*(2 - t), (t < -1) & (t <
0) & (t < 2)), ((1 - t)*((1 - t)*(t/2 + S.Half) + (1 - t/2)*(2 -
t)), (t < -1) & (t < 0) & (t < 1) & (t < 2)), ((1 - t)*((1 -
t/2)*(2 - t) + (t/2 + S.Half)*(t + 1)), (t < 0) & (t < 2)), ((1 -
t)*((1 - t)*(t/2 + S.Half) + (1 - t/2)*(2 - t)), (t < 0) & (t <
1) & (t < 2)), (0, (t < -1) & (t < 0)), ((1 - t)*(t/2 +
S.Half)*(t + 1), t < 0), ((1 - t)*(t*(1 - t/2) + (1 - t)*(t/2 +
S.Half)), (t < 1) & ((t < -1) | (t < 1))), ((1 - t)*(1 - t/2)*(2
- t), (t < 2) & ((t < -1) | (t < 2))), (0, True))
assert p.simplify() == Piecewise(
(0, t < -2), ((t + 1)*(t + 2)**2/2, t < -1), (-3*t**3/2
- 5*t**2/2 + 1, t < 0), (3*t**3/2 - 5*t**2/2 + 1, t < 1), ((1 -
t)*(t - 2)**2/2, t < 2), (0, True))
# coverage
nan = Undefined
covered = Piecewise((1, x > 3), (2, x < 2), (3, x > 1))
assert covered.simplify().args == covered.args
assert Piecewise((1, x < 2), (2, x < 1), (3, True)).simplify(
) == Piecewise((1, x < 2), (3, True))
assert Piecewise((1, x > 2)).simplify() == Piecewise((1, x > 2),
(nan, True))
assert Piecewise((1, (x >= 2) & (x < oo))
).simplify() == Piecewise((1, (x >= 2) & (x < oo)), (nan, True))
assert Piecewise((1, x < 2), (2, (x > 1) & (x < 3)), (3, True)
). simplify() == Piecewise((1, x < 2), (2, x < 3), (3, True))
assert Piecewise((1, x < 2), (2, (x <= 3) & (x > 1)), (3, True)
).simplify() == Piecewise((1, x < 2), (2, x <= 3), (3, True))
assert Piecewise((1, x < 2), (2, (x > 2) & (x < 3)), (3, True)
).simplify() == Piecewise((1, x < 2), (2, (x > 2) & (x < 3)),
(3, True))
assert Piecewise((1, x < 2), (2, (x >= 1) & (x <= 3)), (3, True)
).simplify() == Piecewise((1, x < 2), (2, x <= 3), (3, True))
assert Piecewise((1, x < 1), (2, (x >= 2) & (x <= 3)), (3, True)
).simplify() == Piecewise((1, x < 1), (2, (x >= 2) & (x <= 3)),
(3, True))
def test_piecewise_solve():
abs2 = Piecewise((-x, x <= 0), (x, x > 0))
f = abs2.subs(x, x - 2)
assert solve(f, x) == [2]
assert solve(f - 1, x) == [1, 3]
f = Piecewise(((x - 2)**2, x >= 0), (1, True))
assert solve(f, x) == [2]
g = Piecewise(((x - 5)**5, x >= 4), (f, True))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 4), (f, x < 4))
assert solve(g, x) == [2, 5]
g = Piecewise(((x - 5)**5, x >= 2), (f, x < 2))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2), (f, True), (10, False))
assert solve(g, x) == [5]
g = Piecewise(((x - 5)**5, x >= 2),
(-x + 2, x - 2 <= 0), (x - 2, x - 2 > 0))
assert solve(g, x) == [5]
# if no symbol is given the piecewise detection must still work
assert solve(Piecewise((x - 2, x > 2), (2 - x, True)) - 3) == [-1, 5]
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
raises(NotImplementedError, lambda: solve(f, x))
def nona(ans):
return list(filter(lambda x: x is not S.NaN, ans))
p = Piecewise((x**2 - 4, x < y), (x - 2, True))
ans = solve(p, x)
assert nona([i.subs(y, -2) for i in ans]) == [2]
assert nona([i.subs(y, 2) for i in ans]) == [-2, 2]
assert nona([i.subs(y, 3) for i in ans]) == [-2, 2]
assert ans == [
Piecewise((-2, y > -2), (S.NaN, True)),
Piecewise((2, y <= 2), (S.NaN, True)),
Piecewise((2, y > 2), (S.NaN, True))]
# issue 6060
absxm3 = Piecewise(
(x - 3, 0 <= x - 3),
(3 - x, 0 > x - 3)
)
assert solve(absxm3 - y, x) == [
Piecewise((-y + 3, -y < 0), (S.NaN, True)),
Piecewise((y + 3, y >= 0), (S.NaN, True))]
p = Symbol('p', positive=True)
assert solve(absxm3 - p, x) == [-p + 3, p + 3]
# issue 6989
f = Function('f')
assert solve(Eq(-f(x), Piecewise((1, x > 0), (0, True))), f(x)) == \
[Piecewise((-1, x > 0), (0, True))]
# issue 8587
f = Piecewise((2*x**2, And(0 < x, x < 1)), (2, True))
assert solve(f - 1) == [1/sqrt(2)]
def test_piecewise_fold():
p = Piecewise((x, x < 1), (1, 1 <= x))
assert piecewise_fold(x*p) == Piecewise((x**2, x < 1), (x, 1 <= x))
assert piecewise_fold(p + p) == Piecewise((2*x, x < 1), (2, 1 <= x))
assert piecewise_fold(Piecewise((1, x < 0), (2, True))
+ Piecewise((10, x < 0), (-10, True))) == \
Piecewise((11, x < 0), (-8, True))
p1 = Piecewise((0, x < 0), (x, x <= 1), (0, True))
p2 = Piecewise((0, x < 0), (1 - x, x <= 1), (0, True))
p = 4*p1 + 2*p2
assert integrate(
piecewise_fold(p), (x, -oo, oo)) == integrate(2*x + 2, (x, 0, 1))
assert piecewise_fold(
Piecewise((1, y <= 0), (-Piecewise((2, y >= 0)), True)
)) == Piecewise((1, y <= 0), (-2, y >= 0))
assert piecewise_fold(Piecewise((x, ITE(x > 0, y < 1, y > 1)))
) == Piecewise((x, ((x <= 0) | (y < 1)) & ((x > 0) | (y > 1))))
a, b = (Piecewise((2, Eq(x, 0)), (0, True)),
Piecewise((x, Eq(-x + y, 0)), (1, Eq(-x + y, 1)), (0, True)))
assert piecewise_fold(Mul(a, b, evaluate=False)
) == piecewise_fold(Mul(b, a, evaluate=False))
def test_piecewise_fold_piecewise_in_cond():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (p1 / Abs(p1), True))
assert p2.subs(x, -pi/2) == 0
assert p2.subs(x, 1) == 0
assert p2.subs(x, -pi/4) == 1
p4 = Piecewise((0, Eq(p1, 0)), (1,True))
ans = piecewise_fold(p4)
for i in range(-1, 1):
assert ans.subs(x, i) == p4.subs(x, i)
r1 = 1 < Piecewise((1, x < 1), (3, True))
ans = piecewise_fold(r1)
for i in range(2):
assert ans.subs(x, i) == r1.subs(x, i)
p5 = Piecewise((1, x < 0), (3, True))
p6 = Piecewise((1, x < 1), (3, True))
p7 = Piecewise((1, p5 < p6), (0, True))
ans = piecewise_fold(p7)
for i in range(-1, 2):
assert ans.subs(x, i) == p7.subs(x, i)
def test_piecewise_fold_piecewise_in_cond_2():
p1 = Piecewise((cos(x), x < 0), (0, True))
p2 = Piecewise((0, Eq(p1, 0)), (1 / p1, True))
p3 = Piecewise(
(0, (x >= 0) | Eq(cos(x), 0)),
(1/cos(x), x < 0),
(zoo, True)) # redundant b/c all x are already covered
assert(piecewise_fold(p2) == p3)
def test_piecewise_fold_expand():
p1 = Piecewise((1, Interval(0, 1, False, True).contains(x)), (0, True))
p2 = piecewise_fold(expand((1 - x)*p1))
cond = ((x >= 0) & (x < 1))
assert piecewise_fold(expand((1 - x)*p1), evaluate=False
) == Piecewise((1 - x, cond), (-x, cond), (1, cond), (0, True), evaluate=False)
assert piecewise_fold(expand((1 - x)*p1), evaluate=None
) == Piecewise((1 - x, cond), (0, True))
assert p2 == Piecewise((1 - x, cond), (0, True))
assert p2 == expand(piecewise_fold((1 - x)*p1))
def test_piecewise_duplicate():
p = Piecewise((x, x < -10), (x**2, x <= -1), (x, 1 < x))
assert p == Piecewise(*p.args)
def test_doit():
p1 = Piecewise((x, x < 1), (x**2, -1 <= x), (x, 3 < x))
p2 = Piecewise((x, x < 1), (Integral(2 * x), -1 <= x), (x, 3 < x))
assert p2.doit() == p1
assert p2.doit(deep=False) == p2
# issue 17165
p1 = Sum(y**x, (x, -1, oo)).doit()
assert p1.doit() == p1
def test_piecewise_interval():
p1 = Piecewise((x, Interval(0, 1).contains(x)), (0, True))
assert p1.subs(x, -0.5) == 0
assert p1.subs(x, 0.5) == 0.5
assert p1.diff(x) == Piecewise((1, Interval(0, 1).contains(x)), (0, True))
assert integrate(p1, x) == Piecewise(
(0, x <= 0),
(x**2/2, x <= 1),
(S.Half, True))
def test_piecewise_exclusive():
p = Piecewise((0, x < 0), (S.Half, x <= 0), (1, True))
assert piecewise_exclusive(p) == Piecewise((0, x < 0), (S.Half, Eq(x, 0)),
(1, x > 0), evaluate=False)
assert piecewise_exclusive(p + 2) == Piecewise((0, x < 0), (S.Half, Eq(x, 0)),
(1, x > 0), evaluate=False) + 2
assert piecewise_exclusive(Piecewise((1, y <= 0),
(-Piecewise((2, y >= 0)), True))) == \
Piecewise((1, y <= 0),
(-Piecewise((2, y >= 0),
(S.NaN, y < 0), evaluate=False), y > 0), evaluate=False)
assert piecewise_exclusive(Piecewise((1, x > y))) == Piecewise((1, x > y),
(S.NaN, x <= y),
evaluate=False)
assert piecewise_exclusive(Piecewise((1, x > y)),
skip_nan=True) == Piecewise((1, x > y))
xr, yr = symbols('xr, yr', real=True)
p1 = Piecewise((1, xr < 0), (2, True), evaluate=False)
p1x = Piecewise((1, xr < 0), (2, xr >= 0), evaluate=False)
p2 = Piecewise((p1, yr < 0), (3, True), evaluate=False)
p2x = Piecewise((p1, yr < 0), (3, yr >= 0), evaluate=False)
p2xx = Piecewise((p1x, yr < 0), (3, yr >= 0), evaluate=False)
assert piecewise_exclusive(p2) == p2xx
assert piecewise_exclusive(p2, deep=False) == p2x
def test_piecewise_collapse():
assert Piecewise((x, True)) == x
a = x < 1
assert Piecewise((x, a), (x + 1, a)) == Piecewise((x, a))
assert Piecewise((x, a), (x + 1, a.reversed)) == Piecewise((x, a))
b = x < 5
def canonical(i):
if isinstance(i, Piecewise):
return Piecewise(*i.args)
return i
for args in [
((1, a), (Piecewise((2, a), (3, b)), b)),
((1, a), (Piecewise((2, a), (3, b.reversed)), b)),
((1, a), (Piecewise((2, a), (3, b)), b), (4, True)),
((1, a), (Piecewise((2, a), (3, b), (4, True)), b)),
((1, a), (Piecewise((2, a), (3, b), (4, True)), b), (5, True))]:
for i in (0, 2, 10):
assert canonical(
Piecewise(*args, evaluate=False).subs(x, i)
) == canonical(Piecewise(*args).subs(x, i))
r1, r2, r3, r4 = symbols('r1:5')
a = x < r1
b = x < r2
c = x < r3
d = x < r4
assert Piecewise((1, a), (Piecewise(
(2, a), (3, b), (4, c)), b), (5, c)
) == Piecewise((1, a), (3, b), (5, c))
assert Piecewise((1, a), (Piecewise(
(2, a), (3, b), (4, c), (6, True)), c), (5, d)
) == Piecewise((1, a), (Piecewise(
(3, b), (4, c)), c), (5, d))
assert Piecewise((1, Or(a, d)), (Piecewise(
(2, d), (3, b), (4, c)), b), (5, c)
) == Piecewise((1, Or(a, d)), (Piecewise(
(2, d), (3, b)), b), (5, c))
assert Piecewise((1, c), (2, ~c), (3, S.true)
) == Piecewise((1, c), (2, S.true))
assert Piecewise((1, c), (2, And(~c, b)), (3,True)
) == Piecewise((1, c), (2, b), (3, True))
assert Piecewise((1, c), (2, Or(~c, b)), (3,True)
).subs(dict(zip((r1, r2, r3, r4, x), (1, 2, 3, 4, 3.5)))) == 2
assert Piecewise((1, c), (2, ~c)) == Piecewise((1, c), (2, True))
def test_piecewise_lambdify():
p = Piecewise(
(x**2, x < 0),
(x, Interval(0, 1, False, True).contains(x)),
(2 - x, x >= 1),
(0, True)
)
f = lambdify(x, p)
assert f(-2.0) == 4.0
assert f(0.0) == 0.0
assert f(0.5) == 0.5
assert f(2.0) == 0.0
def test_piecewise_series():
from sympy.series.order import O
p1 = Piecewise((sin(x), x < 0), (cos(x), x > 0))
p2 = Piecewise((x + O(x**2), x < 0), (1 + O(x**2), x > 0))
assert p1.nseries(x, n=2) == p2
def test_piecewise_as_leading_term():
p1 = Piecewise((1/x, x > 1), (0, True))
p2 = Piecewise((x, x > 1), (0, True))
p3 = Piecewise((1/x, x > 1), (x, True))
p4 = Piecewise((x, x > 1), (1/x, True))
p5 = Piecewise((1/x, x > 1), (x, True))
p6 = Piecewise((1/x, x < 1), (x, True))
p7 = Piecewise((x, x < 1), (1/x, True))
p8 = Piecewise((x, x > 1), (1/x, True))
assert p1.as_leading_term(x) == 0
assert p2.as_leading_term(x) == 0
assert p3.as_leading_term(x) == x
assert p4.as_leading_term(x) == 1/x
assert p5.as_leading_term(x) == x
assert p6.as_leading_term(x) == 1/x
assert p7.as_leading_term(x) == x
assert p8.as_leading_term(x) == 1/x
def test_piecewise_complex():
p1 = Piecewise((2, x < 0), (1, 0 <= x))
p2 = Piecewise((2*I, x < 0), (I, 0 <= x))
p3 = Piecewise((I*x, x > 1), (1 + I, True))
p4 = Piecewise((-I*conjugate(x), x > 1), (1 - I, True))
assert conjugate(p1) == p1
assert conjugate(p2) == piecewise_fold(-p2)
assert conjugate(p3) == p4
assert p1.is_imaginary is False
assert p1.is_real is True
assert p2.is_imaginary is True
assert p2.is_real is False
assert p3.is_imaginary is None
assert p3.is_real is None
assert p1.as_real_imag() == (p1, 0)
assert p2.as_real_imag() == (0, -I*p2)
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
p = Piecewise((A*B**2, x > 0), (A**2*B, True))
assert p.adjoint() == \
Piecewise((adjoint(A*B**2), x > 0), (adjoint(A**2*B), True))
assert p.conjugate() == \
Piecewise((conjugate(A*B**2), x > 0), (conjugate(A**2*B), True))
assert p.transpose() == \
Piecewise((transpose(A*B**2), x > 0), (transpose(A**2*B), True))
def test_piecewise_evaluate():
assert Piecewise((x, True)) == x
assert Piecewise((x, True), evaluate=True) == x
assert Piecewise((1, Eq(1, x))).args == ((1, Eq(x, 1)),)
assert Piecewise((1, Eq(1, x)), evaluate=False).args == (
(1, Eq(1, x)),)
# like the additive and multiplicative identities that
# cannot be kept in Add/Mul, we also do not keep a single True
p = Piecewise((x, True), evaluate=False)
assert p == x
def test_as_expr_set_pairs():
assert Piecewise((x, x > 0), (-x, x <= 0)).as_expr_set_pairs() == \
[(x, Interval(0, oo, True, True)), (-x, Interval(-oo, 0))]
assert Piecewise(((x - 2)**2, x >= 0), (0, True)).as_expr_set_pairs() == \
[((x - 2)**2, Interval(0, oo)), (0, Interval(-oo, 0, True, True))]
def test_S_srepr_is_identity():
p = Piecewise((10, Eq(x, 0)), (12, True))
q = S(srepr(p))
assert p == q
def test_issue_12587():
# sort holes into intervals
p = Piecewise((1, x > 4), (2, Not((x <= 3) & (x > -1))), (3, True))
assert p.integrate((x, -5, 5)) == 23
p = Piecewise((1, x > 1), (2, x < y), (3, True))
lim = x, -3, 3
ans = p.integrate(lim)
for i in range(-1, 3):
assert ans.subs(y, i) == p.subs(y, i).integrate(lim)
def test_issue_11045():
assert integrate(1/(x*sqrt(x**2 - 1)), (x, 1, 2)) == pi/3
# handle And with Or arguments
assert Piecewise((1, And(Or(x < 1, x > 3), x < 2)), (0, True)
).integrate((x, 0, 3)) == 1
# hidden false
assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
).integrate((x, 0, 3)) == 5
# targetcond is Eq
assert Piecewise((1, x > 1), (2, Eq(1, x)), (3, True)
).integrate((x, 0, 4)) == 6
# And has Relational needing to be solved
assert Piecewise((1, And(2*x > x + 1, x < 2)), (0, True)
).integrate((x, 0, 3)) == 1
# Or has Relational needing to be solved
assert Piecewise((1, Or(2*x > x + 2, x < 1)), (0, True)
).integrate((x, 0, 3)) == 2
# ignore hidden false (handled in canonicalization)
assert Piecewise((1, x > 1), (2, x > x + 1), (3, True)
).integrate((x, 0, 3)) == 5
# watch for hidden True Piecewise
assert Piecewise((2, Eq(1 - x, x*(1/x - 1))), (0, True)
).integrate((x, 0, 3)) == 6
# overlapping conditions of targetcond are recognized and ignored;
# the condition x > 3 will be pre-empted by the first condition
assert Piecewise((1, Or(x < 1, x > 2)), (2, x > 3), (3, True)
).integrate((x, 0, 4)) == 6
# convert Ne to Or
assert Piecewise((1, Ne(x, 0)), (2, True)
).integrate((x, -1, 1)) == 2
# no default but well defined
assert Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4))
).integrate((x, 1, 4)) == 5
p = Piecewise((x, (x > 1) & (x < 3)), (1, (x < 4)))
nan = Undefined
i = p.integrate((x, 1, y))
assert i == Piecewise(
(y - 1, y < 1),
(Min(3, y)**2/2 - Min(3, y) + Min(4, y) - S.Half,
y <= Min(4, y)),
(nan, True))
assert p.integrate((x, 1, -1)) == i.subs(y, -1)
assert p.integrate((x, 1, 4)) == 5
assert p.integrate((x, 1, 5)) is nan
# handle Not
p = Piecewise((1, x > 1), (2, Not(And(x > 1, x< 3))), (3, True))
assert p.integrate((x, 0, 3)) == 4
# handle updating of int_expr when there is overlap
p = Piecewise(
(1, And(5 > x, x > 1)),
(2, Or(x < 3, x > 7)),
(4, x < 8))
assert p.integrate((x, 0, 10)) == 20
# And with Eq arg handling
assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1))
).integrate((x, 0, 3)) is S.NaN
assert Piecewise((1, x < 1), (2, And(Eq(x, 3), x > 1)), (3, True)
).integrate((x, 0, 3)) == 7
assert Piecewise((1, x < 0), (2, And(Eq(x, 3), x < 1)), (3, True)
).integrate((x, -1, 1)) == 4
# middle condition doesn't matter: it's a zero width interval
assert Piecewise((1, x < 1), (2, Eq(x, 3) & (y < x)), (3, True)
).integrate((x, 0, 3)) == 7
def test_holes():
nan = Undefined
assert Piecewise((1, x < 2)).integrate(x) == Piecewise(
(x, x < 2), (nan, True))
assert Piecewise((1, And(x > 1, x < 2))).integrate(x) == Piecewise(
(nan, x < 1), (x, x < 2), (nan, True))
assert Piecewise((1, And(x > 1, x < 2))).integrate((x, 0, 3)) is nan
assert Piecewise((1, And(x > 0, x < 4))).integrate((x, 1, 3)) == 2
# this also tests that the integrate method is used on non-Piecwise
# arguments in _eval_integral
A, B = symbols("A B")
a, b = symbols('a b', real=True)
assert Piecewise((A, And(x < 0, a < 1)), (B, Or(x < 1, a > 2))
).integrate(x) == Piecewise(
(B*x, (a > 2)),
(Piecewise((A*x, x < 0), (B*x, x < 1), (nan, True)), a < 1),
(Piecewise((B*x, x < 1), (nan, True)), True))
def test_issue_11922():
def f(x):
return Piecewise((0, x < -1), (1 - x**2, x < 1), (0, True))
autocorr = lambda k: (
f(x) * f(x + k)).integrate((x, -1, 1))
assert autocorr(1.9) > 0
k = symbols('k')
good_autocorr = lambda k: (
(1 - x**2) * f(x + k)).integrate((x, -1, 1))
a = good_autocorr(k)
assert a.subs(k, 3) == 0
k = symbols('k', positive=True)
a = good_autocorr(k)
assert a.subs(k, 3) == 0
assert Piecewise((0, x < 1), (10, (x >= 1))
).integrate() == Piecewise((0, x < 1), (10*x - 10, True))
def test_issue_5227():
f = 0.0032513612725229*Piecewise((0, x < -80.8461538461539),
(-0.0160799238820171*x + 1.33215984776403, x < 2),
(Piecewise((0.3, x > 123), (0.7, True)) +
Piecewise((0.4, x > 2), (0.6, True)), x <=
123), (-0.00817409766454352*x + 2.10541401273885, x <
380.571428571429), (0, True))
i = integrate(f, (x, -oo, oo))
assert i == Integral(f, (x, -oo, oo)).doit()
assert str(i) == '1.00195081676351'
assert Piecewise((1, x - y < 0), (0, True)
).integrate(y) == Piecewise((0, y <= x), (-x + y, True))
def test_issue_10137():
a = Symbol('a', real=True)
b = Symbol('b', real=True)
x = Symbol('x', real=True)
y = Symbol('y', real=True)
p0 = Piecewise((0, Or(x < a, x > b)), (1, True))
p1 = Piecewise((0, Or(a > x, b < x)), (1, True))
assert integrate(p0, (x, y, oo)) == integrate(p1, (x, y, oo))
p3 = Piecewise((1, And(0 < x, x < a)), (0, True))
p4 = Piecewise((1, And(a > x, x > 0)), (0, True))
ip3 = integrate(p3, x)
assert ip3 == Piecewise(
(0, x <= 0),
(x, x <= Max(0, a)),
(Max(0, a), True))
ip4 = integrate(p4, x)
assert ip4 == ip3
assert p3.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
assert p4.integrate((x, 2, 4)) == Min(4, Max(2, a)) - 2
def test_stackoverflow_43852159():
f = lambda x: Piecewise((1, (x >= -1) & (x <= 1)), (0, True))
Conv = lambda x: integrate(f(x - y)*f(y), (y, -oo, +oo))
cx = Conv(x)
assert cx.subs(x, -1.5) == cx.subs(x, 1.5)
assert cx.subs(x, 3) == 0
assert piecewise_fold(f(x - y)*f(y)) == Piecewise(
(1, (y >= -1) & (y <= 1) & (x - y >= -1) & (x - y <= 1)),
(0, True))
def test_issue_12557():
'''
# 3200 seconds to compute the fourier part of issue
import sympy as sym
x,y,z,t = sym.symbols('x y z t')
k = sym.symbols("k", integer=True)
fourier = sym.fourier_series(sym.cos(k*x)*sym.sqrt(x**2),
(x, -sym.pi, sym.pi))
assert fourier == FourierSeries(
sqrt(x**2)*cos(k*x), (x, -pi, pi), (Piecewise((pi**2,
Eq(k, 0)), (2*(-1)**k/k**2 - 2/k**2, True))/(2*pi),
SeqFormula(Piecewise((pi**2, (Eq(_n, 0) & Eq(k, 0)) | (Eq(_n, 0) &
Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) & Eq(k, 0) & Eq(_n, -k)) | (Eq(_n,
0) & Eq(_n, k) & Eq(k, 0) & Eq(_n, -k))), (pi**2/2, Eq(_n, k) | Eq(_n,
-k) | (Eq(_n, 0) & Eq(_n, k)) | (Eq(_n, k) & Eq(k, 0)) | (Eq(_n, 0) &
Eq(_n, -k)) | (Eq(_n, k) & Eq(_n, -k)) | (Eq(k, 0) & Eq(_n, -k)) |
(Eq(_n, 0) & Eq(_n, k) & Eq(_n, -k)) | (Eq(_n, k) & Eq(k, 0) & Eq(_n,
-k))), ((-1)**k*pi**2*_n**3*sin(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi**2*_n**3*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2
- pi*k**4) + (-1)**k*pi*_n**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi*_n**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) - (-1)**k*pi**2*_n*k**2*sin(pi*_n)/(pi*_n**4 -
2*pi*_n**2*k**2 + pi*k**4) +
(-1)**k*pi**2*_n*k**2*sin(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) + (-1)**k*pi*k**2*cos(pi*_n)/(pi*_n**4 - 2*pi*_n**2*k**2 +
pi*k**4) - (-1)**k*pi*k**2*cos(pi*_n)/(-pi*_n**4 + 2*pi*_n**2*k**2 -
pi*k**4) - (2*_n**2 + 2*k**2)/(_n**4 - 2*_n**2*k**2 + k**4),
True))*cos(_n*x)/pi, (_n, 1, oo)), SeqFormula(0, (_k, 1, oo))))
'''
x = symbols("x", real=True)
k = symbols('k', integer=True, finite=True)
abs2 = lambda x: Piecewise((-x, x <= 0), (x, x > 0))
assert integrate(abs2(x), (x, -pi, pi)) == pi**2
func = cos(k*x)*sqrt(x**2)
assert integrate(func, (x, -pi, pi)) == Piecewise(
(2*(-1)**k/k**2 - 2/k**2, Ne(k, 0)), (pi**2, True))
def test_issue_6900():
from itertools import permutations
t0, t1, T, t = symbols('t0, t1 T t')
f = Piecewise((0, t < t0), (x, And(t0 <= t, t < t1)), (0, t >= t1))
g = f.integrate(t)
assert g == Piecewise(
(0, t <= t0),
(t*x - t0*x, t <= Max(t0, t1)),
(-t0*x + x*Max(t0, t1), True))
for i in permutations(range(2)):
reps = dict(zip((t0,t1), i))
for tt in range(-1,3):
assert (g.xreplace(reps).subs(t,tt) ==
f.xreplace(reps).integrate(t).subs(t,tt))
lim = Tuple(t, t0, T)
g = f.integrate(lim)
ans = Piecewise(
(-t0*x + x*Min(T, Max(t0, t1)), T > t0),
(0, True))
for i in permutations(range(3)):
reps = dict(zip((t0,t1,T), i))
tru = f.xreplace(reps).integrate(lim.xreplace(reps))
assert tru == ans.xreplace(reps)
assert g == ans
def test_issue_10122():
assert solve(abs(x) + abs(x - 1) - 1 > 0, x
) == Or(And(-oo < x, x < S.Zero), And(S.One < x, x < oo))
def test_issue_4313():
u = Piecewise((0, x <= 0), (1, x >= a), (x/a, True))
e = (u - u.subs(x, y))**2/(x - y)**2
M = Max(0, a)
assert integrate(e, x).expand() == Piecewise(
(Piecewise(
(0, x <= 0),
(-y**2/(a**2*x - a**2*y) + x/a**2 - 2*y*log(-y)/a**2 +
2*y*log(x - y)/a**2 - y/a**2, x <= M),
(-y**2/(-a**2*y + a**2*M) + 1/(-y + M) -
1/(x - y) - 2*y*log(-y)/a**2 + 2*y*log(-y +
M)/a**2 - y/a**2 + M/a**2, True)),
((a <= y) & (y <= 0)) | ((y <= 0) & (y > -oo))),
(Piecewise(
(-1/(x - y), x <= 0),
(-a**2/(a**2*x - a**2*y) + 2*a*y/(a**2*x - a**2*y) -
y**2/(a**2*x - a**2*y) + 2*log(-y)/a - 2*log(x - y)/a +
2/a + x/a**2 - 2*y*log(-y)/a**2 + 2*y*log(x - y)/a**2 -
y/a**2, x <= M),
(-a**2/(-a**2*y + a**2*M) + 2*a*y/(-a**2*y +
a**2*M) - y**2/(-a**2*y + a**2*M) +
2*log(-y)/a - 2*log(-y + M)/a + 2/a -
2*y*log(-y)/a**2 + 2*y*log(-y + M)/a**2 -
y/a**2 + M/a**2, True)),
a <= y),
(Piecewise(
(-y**2/(a**2*x - a**2*y), x <= 0),
(x/a**2 + y/a**2, x <= M),
(a**2/(-a**2*y + a**2*M) -
a**2/(a**2*x - a**2*y) - 2*a*y/(-a**2*y + a**2*M) +
2*a*y/(a**2*x - a**2*y) + y**2/(-a**2*y + a**2*M) -
y**2/(a**2*x - a**2*y) + y/a**2 + M/a**2, True)),
True))
def test__intervals():
assert Piecewise((x + 2, Eq(x, 3)))._intervals(x) == (True, [])
assert Piecewise(
(1, x > x + 1),
(Piecewise((1, x < x + 1)), 2*x < 2*x + 1),
(1, True))._intervals(x) == (True, [(-oo, oo, 1, 1)])
assert Piecewise((1, Ne(x, I)), (0, True))._intervals(x) == (True,
[(-oo, oo, 1, 0)])
assert Piecewise((-cos(x), sin(x) >= 0), (cos(x), True)
)._intervals(x) == (True,
[(0, pi, -cos(x), 0), (-oo, oo, cos(x), 1)])
# the following tests that duplicates are removed and that non-Eq
# generated zero-width intervals are removed
assert Piecewise((1, Abs(x**(-2)) > 1), (0, True)
)._intervals(x) == (True,
[(-1, 0, 1, 0), (0, 1, 1, 0), (-oo, oo, 0, 1)])
def test_containment():
a, b, c, d, e = [1, 2, 3, 4, 5]
p = (Piecewise((d, x > 1), (e, True))*
Piecewise((a, Abs(x - 1) < 1), (b, Abs(x - 2) < 2), (c, True)))
assert p.integrate(x).diff(x) == Piecewise(
(c*e, x <= 0),
(a*e, x <= 1),
(a*d, x < 2), # this is what we want to get right
(b*d, x < 4),
(c*d, True))
def test_piecewise_with_DiracDelta():
d1 = DiracDelta(x - 1)
assert integrate(d1, (x, -oo, oo)) == 1
assert integrate(d1, (x, 0, 2)) == 1
assert Piecewise((d1, Eq(x, 2)), (0, True)).integrate(x) == 0
assert Piecewise((d1, x < 2), (0, True)).integrate(x) == Piecewise(
(Heaviside(x - 1), x < 2), (1, True))
# TODO raise error if function is discontinuous at limit of
# integration, e.g. integrate(d1, (x, -2, 1)) or Piecewise(
# (d1, Eq(x, 1)
def test_issue_10258():
assert Piecewise((0, x < 1), (1, True)).is_zero is None
assert Piecewise((-1, x < 1), (1, True)).is_zero is False
a = Symbol('a', zero=True)
assert Piecewise((0, x < 1), (a, True)).is_zero
assert Piecewise((1, x < 1), (a, x < 3)).is_zero is None
a = Symbol('a')
assert Piecewise((0, x < 1), (a, True)).is_zero is None
assert Piecewise((0, x < 1), (1, True)).is_nonzero is None
assert Piecewise((1, x < 1), (2, True)).is_nonzero
assert Piecewise((0, x < 1), (oo, True)).is_finite is None
assert Piecewise((0, x < 1), (1, True)).is_finite
b = Basic()
assert Piecewise((b, x < 1)).is_finite is None
# 10258
c = Piecewise((1, x < 0), (2, True)) < 3
assert c != True
assert piecewise_fold(c) == True
def test_issue_10087():
a, b = Piecewise((x, x > 1), (2, True)), Piecewise((x, x > 3), (3, True))
m = a*b
f = piecewise_fold(m)
for i in (0, 2, 4):
assert m.subs(x, i) == f.subs(x, i)
m = a + b
f = piecewise_fold(m)
for i in (0, 2, 4):
assert m.subs(x, i) == f.subs(x, i)
def test_issue_8919():
c = symbols('c:5')
x = symbols("x")
f1 = Piecewise((c[1], x < 1), (c[2], True))
f2 = Piecewise((c[3], x < Rational(1, 3)), (c[4], True))
assert integrate(f1*f2, (x, 0, 2)
) == c[1]*c[3]/3 + 2*c[1]*c[4]/3 + c[2]*c[4]
f1 = Piecewise((0, x < 1), (2, True))
f2 = Piecewise((3, x < 2), (0, True))
assert integrate(f1*f2, (x, 0, 3)) == 6
y = symbols("y", positive=True)
a, b, c, x, z = symbols("a,b,c,x,z", real=True)
I = Integral(Piecewise(
(0, (x >= y) | (x < 0) | (b > c)),
(a, True)), (x, 0, z))
ans = I.doit()
assert ans == Piecewise((0, b > c), (a*Min(y, z) - a*Min(0, z), True))
for cond in (True, False):
for yy in range(1, 3):
for zz in range(-yy, 0, yy):
reps = [(b > c, cond), (y, yy), (z, zz)]
assert ans.subs(reps) == I.subs(reps).doit()
def test_unevaluated_integrals():
f = Function('f')
p = Piecewise((1, Eq(f(x) - 1, 0)), (2, x - 10 < 0), (0, True))
assert p.integrate(x) == Integral(p, x)
assert p.integrate((x, 0, 5)) == Integral(p, (x, 0, 5))
# test it by replacing f(x) with x%2 which will not
# affect the answer: the integrand is essentially 2 over
# the domain of integration
assert Integral(p, (x, 0, 5)).subs(f(x), x%2).n() == 10
# this is a test of using _solve_inequality when
# solve_univariate_inequality fails
assert p.integrate(y) == Piecewise(
(y, Eq(f(x), 1) | ((x < 10) & Eq(f(x), 1))),
(2*y, (x > -oo) & (x < 10)), (0, True))
def test_conditions_as_alternate_booleans():
a, b, c = symbols('a:c')
assert Piecewise((x, Piecewise((y < 1, x > 0), (y > 1, True)))
) == Piecewise((x, ITE(x > 0, y < 1, y > 1)))
def test_Piecewise_rewrite_as_ITE():
a, b, c, d = symbols('a:d')
def _ITE(*args):
return Piecewise(*args).rewrite(ITE)
assert _ITE((a, x < 1), (b, x >= 1)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, x < oo)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, Or(y < 1, x < oo)), (c, y > 0)
) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, True)) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (b, x < 2), (c, True)
) == ITE(x < 1, a, ITE(x < 2, b, c))
assert _ITE((a, x < 1), (b, y < 2), (c, True)
) == ITE(x < 1, a, ITE(y < 2, b, c))
assert _ITE((a, x < 1), (b, x < oo), (c, y < 1)
) == ITE(x < 1, a, b)
assert _ITE((a, x < 1), (c, y < 1), (b, x < oo), (d, True)
) == ITE(x < 1, a, ITE(y < 1, c, b))
assert _ITE((a, x < 0), (b, Or(x < oo, y < 1))
) == ITE(x < 0, a, b)
raises(TypeError, lambda: _ITE((x + 1, x < 1), (x, True)))
# if `a` in the following were replaced with y then the coverage
# is complete but something other than as_set would need to be
# used to detect this
raises(NotImplementedError, lambda: _ITE((x, x < y), (y, x >= a)))
raises(ValueError, lambda: _ITE((a, x < 2), (b, x > 3)))
def test_issue_14052():
assert integrate(abs(sin(x)), (x, 0, 2*pi)) == 4
def test_issue_14240():
assert piecewise_fold(
Piecewise((1, a), (2, b), (4, True)) +
Piecewise((8, a), (16, True))
) == Piecewise((9, a), (18, b), (20, True))
assert piecewise_fold(
Piecewise((2, a), (3, b), (5, True)) *
Piecewise((7, a), (11, True))
) == Piecewise((14, a), (33, b), (55, True))
# these will hang if naive folding is used
assert piecewise_fold(Add(*[
Piecewise((i, a), (0, True)) for i in range(40)])
) == Piecewise((780, a), (0, True))
assert piecewise_fold(Mul(*[
Piecewise((i, a), (0, True)) for i in range(1, 41)])
) == Piecewise((factorial(40), a), (0, True))
def test_issue_14787():
x = Symbol('x')
f = Piecewise((x, x < 1), ((S(58) / 7), True))
assert str(f.evalf()) == "Piecewise((x, x < 1), (8.28571428571429, True))"
def test_issue_8458():
x, y = symbols('x y')
# Original issue
p1 = Piecewise((0, Eq(x, 0)), (sin(x), True))
assert p1.simplify() == sin(x)
# Slightly larger variant
p2 = Piecewise((x, Eq(x, 0)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
assert p2.simplify() == sin(x)
# Test for problem highlighted during review
p3 = Piecewise((x+1, Eq(x, -1)), (4*x + (y-2)**4, Eq(x, 0) & Eq(x+y, 2)), (sin(x), True))
assert p3.simplify() == Piecewise((0, Eq(x, -1)), (sin(x), True))
def test_issue_16417():
z = Symbol('z')
assert unchanged(Piecewise, (1, Or(Eq(im(z), 0), Gt(re(z), 0))), (2, True))
x = Symbol('x')
assert unchanged(Piecewise, (S.Pi, re(x) < 0),
(0, Or(re(x) > 0, Ne(im(x), 0))),
(S.NaN, True))
r = Symbol('r', real=True)
p = Piecewise((S.Pi, re(r) < 0),
(0, Or(re(r) > 0, Ne(im(r), 0))),
(S.NaN, True))
assert p == Piecewise((S.Pi, r < 0),
(0, r > 0),
(S.NaN, True), evaluate=False)
# Does not work since imaginary != 0...
#i = Symbol('i', imaginary=True)
#p = Piecewise((S.Pi, re(i) < 0),
# (0, Or(re(i) > 0, Ne(im(i), 0))),
# (S.NaN, True))
#assert p == Piecewise((0, Ne(im(i), 0)),
# (S.NaN, True), evaluate=False)
i = I*r
p = Piecewise((S.Pi, re(i) < 0),
(0, Or(re(i) > 0, Ne(im(i), 0))),
(S.NaN, True))
assert p == Piecewise((0, Ne(im(i), 0)),
(S.NaN, True), evaluate=False)
assert p == Piecewise((0, Ne(r, 0)),
(S.NaN, True), evaluate=False)
def test_eval_rewrite_as_KroneckerDelta():
x, y, z, n, t, m = symbols('x y z n t m')
K = KroneckerDelta
f = lambda p: expand(p.rewrite(K))
p1 = Piecewise((0, Eq(x, y)), (1, True))
assert f(p1) == 1 - K(x, y)
p2 = Piecewise((x, Eq(y,0)), (z, Eq(t,0)), (n, True))
assert f(p2) == n*K(0, t)*K(0, y) - n*K(0, t) - n*K(0, y) + n + \
x*K(0, y) - z*K(0, t)*K(0, y) + z*K(0, t)
p3 = Piecewise((1, Ne(x, y)), (0, True))
assert f(p3) == 1 - K(x, y)
p4 = Piecewise((1, Eq(x, 3)), (4, True), (5, True))
assert f(p4) == 4 - 3*K(3, x)
p5 = Piecewise((3, Ne(x, 2)), (4, Eq(y, 2)), (5, True))
assert f(p5) == -K(2, x)*K(2, y) + 2*K(2, x) + 3
p6 = Piecewise((0, Ne(x, 1) & Ne(y, 4)), (1, True))
assert f(p6) == -K(1, x)*K(4, y) + K(1, x) + K(4, y)
p7 = Piecewise((2, Eq(y, 3) & Ne(x, 2)), (1, True))
assert f(p7) == -K(2, x)*K(3, y) + K(3, y) + 1
p8 = Piecewise((4, Eq(x, 3) & Ne(y, 2)), (1, True))
assert f(p8) == -3*K(2, y)*K(3, x) + 3*K(3, x) + 1
p9 = Piecewise((6, Eq(x, 4) & Eq(y, 1)), (1, True))
assert f(p9) == 5 * K(1, y) * K(4, x) + 1
p10 = Piecewise((4, Ne(x, -4) | Ne(y, 1)), (1, True))
assert f(p10) == -3 * K(-4, x) * K(1, y) + 4
p11 = Piecewise((1, Eq(y, 2) | Ne(x, -3)), (2, True))
assert f(p11) == -K(-3, x)*K(2, y) + K(-3, x) + 1
p12 = Piecewise((-1, Eq(x, 1) | Ne(y, 3)), (1, True))
assert f(p12) == -2*K(1, x)*K(3, y) + 2*K(3, y) - 1
p13 = Piecewise((3, Eq(x, 2) | Eq(y, 4)), (1, True))
assert f(p13) == -2*K(2, x)*K(4, y) + 2*K(2, x) + 2*K(4, y) + 1
p14 = Piecewise((1, Ne(x, 0) | Ne(y, 1)), (3, True))
assert f(p14) == 2 * K(0, x) * K(1, y) + 1
p15 = Piecewise((2, Eq(x, 3) | Ne(y, 2)), (3, Eq(x, 4) & Eq(y, 5)), (1, True))
assert f(p15) == -2*K(2, y)*K(3, x)*K(4, x)*K(5, y) + K(2, y)*K(3, x) + \
2*K(2, y)*K(4, x)*K(5, y) - K(2, y) + 2
p16 = Piecewise((0, Ne(m, n)), (1, True))*Piecewise((0, Ne(n, t)), (1, True))\
*Piecewise((0, Ne(n, x)), (1, True)) - Piecewise((0, Ne(t, x)), (1, True))
assert f(p16) == K(m, n)*K(n, t)*K(n, x) - K(t, x)
p17 = Piecewise((0, Ne(t, x) & (Ne(m, n) | Ne(n, t) | Ne(n, x))),
(1, Ne(t, x)), (-1, Ne(m, n) | Ne(n, t) | Ne(n, x)), (0, True))
assert f(p17) == K(m, n)*K(n, t)*K(n, x) - K(t, x)
p18 = Piecewise((-4, Eq(y, 1) | (Eq(x, -5) & Eq(x, z))), (4, True))
assert f(p18) == 8*K(-5, x)*K(1, y)*K(x, z) - 8*K(-5, x)*K(x, z) - 8*K(1, y) + 4
p19 = Piecewise((0, x > 2), (1, True))
assert f(p19) == p19
p20 = Piecewise((0, And(x < 2, x > -5)), (1, True))
assert f(p20) == p20
p21 = Piecewise((0, Or(x > 1, x < 0)), (1, True))
assert f(p21) == p21
p22 = Piecewise((0, ~((Eq(y, -1) | Ne(x, 0)) & (Ne(x, 1) | Ne(y, -1)))), (1, True))
assert f(p22) == K(-1, y)*K(0, x) - K(-1, y)*K(1, x) - K(0, x) + 1
@slow
def test_identical_conds_issue():
from sympy.stats import Uniform, density
u1 = Uniform('u1', 0, 1)
u2 = Uniform('u2', 0, 1)
# Result is quite big, so not really important here (and should ideally be
# simpler). Should not give an exception though.
density(u1 + u2)
def test_issue_7370():
f = Piecewise((1, x <= 2400))
v = integrate(f, (x, 0, Float("252.4", 30)))
assert str(v) == '252.400000000000000000000000000'
def test_issue_14933():
x = Symbol('x')
y = Symbol('y')
inp = MatrixSymbol('inp', 1, 1)
rep_dict = {y: inp[0, 0], x: inp[0, 0]}
p = Piecewise((1, ITE(y > 0, x < 0, True)))
assert p.xreplace(rep_dict) == Piecewise((1, ITE(inp[0, 0] > 0, inp[0, 0] < 0, True)))
def test_issue_16715():
raises(NotImplementedError, lambda: Piecewise((x, x<0), (0, y>1)).as_expr_set_pairs())
def test_issue_20360():
t, tau = symbols("t tau", real=True)
n = symbols("n", integer=True)
lam = pi * (n - S.Half)
eq = integrate(exp(lam * tau), (tau, 0, t))
assert eq.simplify() == (2*exp(pi*t*(2*n - 1)/2) - 2)/(pi*(2*n - 1))
def test_piecewise_eval():
# XXX these tests might need modification if this
# simplification is moved out of eval and into
# boolalg or Piecewise simplification functions
f = lambda x: x.args[0].cond
# unsimplified
assert f(Piecewise((x, (x > -oo) & (x < 3)))
) == ((x > -oo) & (x < 3))
assert f(Piecewise((x, (x > -oo) & (x < oo)))
) == ((x > -oo) & (x < oo))
assert f(Piecewise((x, (x > -3) & (x < 3)))
) == ((x > -3) & (x < 3))
assert f(Piecewise((x, (x > -3) & (x < oo)))
) == ((x > -3) & (x < oo))
assert f(Piecewise((x, (x <= 3) & (x > -oo)))
) == ((x <= 3) & (x > -oo))
assert f(Piecewise((x, (x <= 3) & (x > -3)))
) == ((x <= 3) & (x > -3))
assert f(Piecewise((x, (x >= -3) & (x < 3)))
) == ((x >= -3) & (x < 3))
assert f(Piecewise((x, (x >= -3) & (x < oo)))
) == ((x >= -3) & (x < oo))
assert f(Piecewise((x, (x >= -3) & (x <= 3)))
) == ((x >= -3) & (x <= 3))
# could simplify by keeping only the first
# arg of result
assert f(Piecewise((x, (x <= oo) & (x > -oo)))
) == (x > -oo) & (x <= oo)
assert f(Piecewise((x, (x <= oo) & (x > -3)))
) == (x > -3) & (x <= oo)
assert f(Piecewise((x, (x >= -oo) & (x < 3)))
) == (x < 3) & (x >= -oo)
assert f(Piecewise((x, (x >= -oo) & (x < oo)))
) == (x < oo) & (x >= -oo)
assert f(Piecewise((x, (x >= -oo) & (x <= 3)))
) == (x <= 3) & (x >= -oo)
assert f(Piecewise((x, (x >= -oo) & (x <= oo)))
) == (x <= oo) & (x >= -oo) # but cannot be True unless x is real
assert f(Piecewise((x, (x >= -3) & (x <= oo)))
) == (x >= -3) & (x <= oo)
assert f(Piecewise((x, (Abs(arg(a)) <= 1) | (Abs(arg(a)) < 1)))
) == (Abs(arg(a)) <= 1) | (Abs(arg(a)) < 1)
def test_issue_22533():
x = Symbol('x', real=True)
f = Piecewise((-1 / x, x <= 0), (1 / x, True))
assert integrate(f, x) == Piecewise((-log(x), x <= 0), (log(x), True))
|
833ec8b3df16474842ad9c039287298b60985a7e85a97d753b3182b69cef6d92 | from sympy.calculus.accumulationbounds import AccumBounds
from sympy.core.function import (expand_mul, expand_trig)
from sympy.core.numbers import (E, I, Integer, Rational, nan, oo, pi, zoo)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import (im, re)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.hyperbolic import (acosh, acoth, acsch, asech, asinh, atanh, cosh, coth, csch, sech, sinh, tanh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, asin, cos, cot, sec, sin, tan)
from sympy.series.order import O
from sympy.core.expr import unchanged
from sympy.core.function import ArgumentIndexError
from sympy.testing.pytest import raises
def test_sinh():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert sinh(nan) is nan
assert sinh(zoo) is nan
assert sinh(oo) is oo
assert sinh(-oo) is -oo
assert sinh(0) == 0
assert unchanged(sinh, 1)
assert sinh(-1) == -sinh(1)
assert unchanged(sinh, x)
assert sinh(-x) == -sinh(x)
assert unchanged(sinh, pi)
assert sinh(-pi) == -sinh(pi)
assert unchanged(sinh, 2**1024 * E)
assert sinh(-2**1024 * E) == -sinh(2**1024 * E)
assert sinh(pi*I) == 0
assert sinh(-pi*I) == 0
assert sinh(2*pi*I) == 0
assert sinh(-2*pi*I) == 0
assert sinh(-3*10**73*pi*I) == 0
assert sinh(7*10**103*pi*I) == 0
assert sinh(pi*I/2) == I
assert sinh(-pi*I/2) == -I
assert sinh(pi*I*Rational(5, 2)) == I
assert sinh(pi*I*Rational(7, 2)) == -I
assert sinh(pi*I/3) == S.Half*sqrt(3)*I
assert sinh(pi*I*Rational(-2, 3)) == Rational(-1, 2)*sqrt(3)*I
assert sinh(pi*I/4) == S.Half*sqrt(2)*I
assert sinh(-pi*I/4) == Rational(-1, 2)*sqrt(2)*I
assert sinh(pi*I*Rational(17, 4)) == S.Half*sqrt(2)*I
assert sinh(pi*I*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)*I
assert sinh(pi*I/6) == S.Half*I
assert sinh(-pi*I/6) == Rational(-1, 2)*I
assert sinh(pi*I*Rational(7, 6)) == Rational(-1, 2)*I
assert sinh(pi*I*Rational(-5, 6)) == Rational(-1, 2)*I
assert sinh(pi*I/105) == sin(pi/105)*I
assert sinh(-pi*I/105) == -sin(pi/105)*I
assert unchanged(sinh, 2 + 3*I)
assert sinh(x*I) == sin(x)*I
assert sinh(k*pi*I) == 0
assert sinh(17*k*pi*I) == 0
assert sinh(k*pi*I/2) == sin(k*pi/2)*I
assert sinh(x).as_real_imag(deep=False) == (cos(im(x))*sinh(re(x)),
sin(im(x))*cosh(re(x)))
x = Symbol('x', extended_real=True)
assert sinh(x).as_real_imag(deep=False) == (sinh(x), 0)
x = Symbol('x', real=True)
assert sinh(I*x).is_finite is True
assert sinh(x).is_real is True
assert sinh(I).is_real is False
p = Symbol('p', positive=True)
assert sinh(p).is_zero is False
assert sinh(0, evaluate=False).is_zero is True
assert sinh(2*pi*I, evaluate=False).is_zero is True
def test_sinh_series():
x = Symbol('x')
assert sinh(x).series(x, 0, 10) == \
x + x**3/6 + x**5/120 + x**7/5040 + x**9/362880 + O(x**10)
def test_sinh_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: sinh(x).fdiff(2))
def test_cosh():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert cosh(nan) is nan
assert cosh(zoo) is nan
assert cosh(oo) is oo
assert cosh(-oo) is oo
assert cosh(0) == 1
assert unchanged(cosh, 1)
assert cosh(-1) == cosh(1)
assert unchanged(cosh, x)
assert cosh(-x) == cosh(x)
assert cosh(pi*I) == cos(pi)
assert cosh(-pi*I) == cos(pi)
assert unchanged(cosh, 2**1024 * E)
assert cosh(-2**1024 * E) == cosh(2**1024 * E)
assert cosh(pi*I/2) == 0
assert cosh(-pi*I/2) == 0
assert cosh((-3*10**73 + 1)*pi*I/2) == 0
assert cosh((7*10**103 + 1)*pi*I/2) == 0
assert cosh(pi*I) == -1
assert cosh(-pi*I) == -1
assert cosh(5*pi*I) == -1
assert cosh(8*pi*I) == 1
assert cosh(pi*I/3) == S.Half
assert cosh(pi*I*Rational(-2, 3)) == Rational(-1, 2)
assert cosh(pi*I/4) == S.Half*sqrt(2)
assert cosh(-pi*I/4) == S.Half*sqrt(2)
assert cosh(pi*I*Rational(11, 4)) == Rational(-1, 2)*sqrt(2)
assert cosh(pi*I*Rational(-3, 4)) == Rational(-1, 2)*sqrt(2)
assert cosh(pi*I/6) == S.Half*sqrt(3)
assert cosh(-pi*I/6) == S.Half*sqrt(3)
assert cosh(pi*I*Rational(7, 6)) == Rational(-1, 2)*sqrt(3)
assert cosh(pi*I*Rational(-5, 6)) == Rational(-1, 2)*sqrt(3)
assert cosh(pi*I/105) == cos(pi/105)
assert cosh(-pi*I/105) == cos(pi/105)
assert unchanged(cosh, 2 + 3*I)
assert cosh(x*I) == cos(x)
assert cosh(k*pi*I) == cos(k*pi)
assert cosh(17*k*pi*I) == cos(17*k*pi)
assert unchanged(cosh, k*pi)
assert cosh(x).as_real_imag(deep=False) == (cos(im(x))*cosh(re(x)),
sin(im(x))*sinh(re(x)))
x = Symbol('x', extended_real=True)
assert cosh(x).as_real_imag(deep=False) == (cosh(x), 0)
x = Symbol('x', real=True)
assert cosh(I*x).is_finite is True
assert cosh(I*x).is_real is True
assert cosh(I*2 + 1).is_real is False
assert cosh(5*I*S.Pi/2, evaluate=False).is_zero is True
assert cosh(x).is_zero is False
def test_cosh_series():
x = Symbol('x')
assert cosh(x).series(x, 0, 10) == \
1 + x**2/2 + x**4/24 + x**6/720 + x**8/40320 + O(x**10)
def test_cosh_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: cosh(x).fdiff(2))
def test_tanh():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert tanh(nan) is nan
assert tanh(zoo) is nan
assert tanh(oo) == 1
assert tanh(-oo) == -1
assert tanh(0) == 0
assert unchanged(tanh, 1)
assert tanh(-1) == -tanh(1)
assert unchanged(tanh, x)
assert tanh(-x) == -tanh(x)
assert unchanged(tanh, pi)
assert tanh(-pi) == -tanh(pi)
assert unchanged(tanh, 2**1024 * E)
assert tanh(-2**1024 * E) == -tanh(2**1024 * E)
assert tanh(pi*I) == 0
assert tanh(-pi*I) == 0
assert tanh(2*pi*I) == 0
assert tanh(-2*pi*I) == 0
assert tanh(-3*10**73*pi*I) == 0
assert tanh(7*10**103*pi*I) == 0
assert tanh(pi*I/2) is zoo
assert tanh(-pi*I/2) is zoo
assert tanh(pi*I*Rational(5, 2)) is zoo
assert tanh(pi*I*Rational(7, 2)) is zoo
assert tanh(pi*I/3) == sqrt(3)*I
assert tanh(pi*I*Rational(-2, 3)) == sqrt(3)*I
assert tanh(pi*I/4) == I
assert tanh(-pi*I/4) == -I
assert tanh(pi*I*Rational(17, 4)) == I
assert tanh(pi*I*Rational(-3, 4)) == I
assert tanh(pi*I/6) == I/sqrt(3)
assert tanh(-pi*I/6) == -I/sqrt(3)
assert tanh(pi*I*Rational(7, 6)) == I/sqrt(3)
assert tanh(pi*I*Rational(-5, 6)) == I/sqrt(3)
assert tanh(pi*I/105) == tan(pi/105)*I
assert tanh(-pi*I/105) == -tan(pi/105)*I
assert unchanged(tanh, 2 + 3*I)
assert tanh(x*I) == tan(x)*I
assert tanh(k*pi*I) == 0
assert tanh(17*k*pi*I) == 0
assert tanh(k*pi*I/2) == tan(k*pi/2)*I
assert tanh(x).as_real_imag(deep=False) == (sinh(re(x))*cosh(re(x))/(cos(im(x))**2
+ sinh(re(x))**2),
sin(im(x))*cos(im(x))/(cos(im(x))**2 + sinh(re(x))**2))
x = Symbol('x', extended_real=True)
assert tanh(x).as_real_imag(deep=False) == (tanh(x), 0)
assert tanh(I*pi/3 + 1).is_real is False
assert tanh(x).is_real is True
assert tanh(I*pi*x/2).is_real is None
def test_tanh_series():
x = Symbol('x')
assert tanh(x).series(x, 0, 10) == \
x - x**3/3 + 2*x**5/15 - 17*x**7/315 + 62*x**9/2835 + O(x**10)
def test_tanh_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: tanh(x).fdiff(2))
def test_coth():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
assert coth(nan) is nan
assert coth(zoo) is nan
assert coth(oo) == 1
assert coth(-oo) == -1
assert coth(0) is zoo
assert unchanged(coth, 1)
assert coth(-1) == -coth(1)
assert unchanged(coth, x)
assert coth(-x) == -coth(x)
assert coth(pi*I) == -I*cot(pi)
assert coth(-pi*I) == cot(pi)*I
assert unchanged(coth, 2**1024 * E)
assert coth(-2**1024 * E) == -coth(2**1024 * E)
assert coth(pi*I) == -I*cot(pi)
assert coth(-pi*I) == I*cot(pi)
assert coth(2*pi*I) == -I*cot(2*pi)
assert coth(-2*pi*I) == I*cot(2*pi)
assert coth(-3*10**73*pi*I) == I*cot(3*10**73*pi)
assert coth(7*10**103*pi*I) == -I*cot(7*10**103*pi)
assert coth(pi*I/2) == 0
assert coth(-pi*I/2) == 0
assert coth(pi*I*Rational(5, 2)) == 0
assert coth(pi*I*Rational(7, 2)) == 0
assert coth(pi*I/3) == -I/sqrt(3)
assert coth(pi*I*Rational(-2, 3)) == -I/sqrt(3)
assert coth(pi*I/4) == -I
assert coth(-pi*I/4) == I
assert coth(pi*I*Rational(17, 4)) == -I
assert coth(pi*I*Rational(-3, 4)) == -I
assert coth(pi*I/6) == -sqrt(3)*I
assert coth(-pi*I/6) == sqrt(3)*I
assert coth(pi*I*Rational(7, 6)) == -sqrt(3)*I
assert coth(pi*I*Rational(-5, 6)) == -sqrt(3)*I
assert coth(pi*I/105) == -cot(pi/105)*I
assert coth(-pi*I/105) == cot(pi/105)*I
assert unchanged(coth, 2 + 3*I)
assert coth(x*I) == -cot(x)*I
assert coth(k*pi*I) == -cot(k*pi)*I
assert coth(17*k*pi*I) == -cot(17*k*pi)*I
assert coth(k*pi*I) == -cot(k*pi)*I
assert coth(log(tan(2))) == coth(log(-tan(2)))
assert coth(1 + I*pi/2) == tanh(1)
assert coth(x).as_real_imag(deep=False) == (sinh(re(x))*cosh(re(x))/(sin(im(x))**2
+ sinh(re(x))**2),
-sin(im(x))*cos(im(x))/(sin(im(x))**2 + sinh(re(x))**2))
x = Symbol('x', extended_real=True)
assert coth(x).as_real_imag(deep=False) == (coth(x), 0)
assert expand_trig(coth(2*x)) == (coth(x)**2 + 1)/(2*coth(x))
assert expand_trig(coth(3*x)) == (coth(x)**3 + 3*coth(x))/(1 + 3*coth(x)**2)
assert expand_trig(coth(x + y)) == (1 + coth(x)*coth(y))/(coth(x) + coth(y))
def test_coth_series():
x = Symbol('x')
assert coth(x).series(x, 0, 8) == \
1/x + x/3 - x**3/45 + 2*x**5/945 - x**7/4725 + O(x**8)
def test_coth_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: coth(x).fdiff(2))
def test_csch():
x, y = symbols('x,y')
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert csch(nan) is nan
assert csch(zoo) is nan
assert csch(oo) == 0
assert csch(-oo) == 0
assert csch(0) is zoo
assert csch(-1) == -csch(1)
assert csch(-x) == -csch(x)
assert csch(-pi) == -csch(pi)
assert csch(-2**1024 * E) == -csch(2**1024 * E)
assert csch(pi*I) is zoo
assert csch(-pi*I) is zoo
assert csch(2*pi*I) is zoo
assert csch(-2*pi*I) is zoo
assert csch(-3*10**73*pi*I) is zoo
assert csch(7*10**103*pi*I) is zoo
assert csch(pi*I/2) == -I
assert csch(-pi*I/2) == I
assert csch(pi*I*Rational(5, 2)) == -I
assert csch(pi*I*Rational(7, 2)) == I
assert csch(pi*I/3) == -2/sqrt(3)*I
assert csch(pi*I*Rational(-2, 3)) == 2/sqrt(3)*I
assert csch(pi*I/4) == -sqrt(2)*I
assert csch(-pi*I/4) == sqrt(2)*I
assert csch(pi*I*Rational(7, 4)) == sqrt(2)*I
assert csch(pi*I*Rational(-3, 4)) == sqrt(2)*I
assert csch(pi*I/6) == -2*I
assert csch(-pi*I/6) == 2*I
assert csch(pi*I*Rational(7, 6)) == 2*I
assert csch(pi*I*Rational(-7, 6)) == -2*I
assert csch(pi*I*Rational(-5, 6)) == 2*I
assert csch(pi*I/105) == -1/sin(pi/105)*I
assert csch(-pi*I/105) == 1/sin(pi/105)*I
assert csch(x*I) == -1/sin(x)*I
assert csch(k*pi*I) is zoo
assert csch(17*k*pi*I) is zoo
assert csch(k*pi*I/2) == -1/sin(k*pi/2)*I
assert csch(n).is_real is True
assert expand_trig(csch(x + y)) == 1/(sinh(x)*cosh(y) + cosh(x)*sinh(y))
def test_csch_series():
x = Symbol('x')
assert csch(x).series(x, 0, 10) == \
1/ x - x/6 + 7*x**3/360 - 31*x**5/15120 + 127*x**7/604800 \
- 73*x**9/3421440 + O(x**10)
def test_csch_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: csch(x).fdiff(2))
def test_sech():
x, y = symbols('x, y')
k = Symbol('k', integer=True)
n = Symbol('n', positive=True)
assert sech(nan) is nan
assert sech(zoo) is nan
assert sech(oo) == 0
assert sech(-oo) == 0
assert sech(0) == 1
assert sech(-1) == sech(1)
assert sech(-x) == sech(x)
assert sech(pi*I) == sec(pi)
assert sech(-pi*I) == sec(pi)
assert sech(-2**1024 * E) == sech(2**1024 * E)
assert sech(pi*I/2) is zoo
assert sech(-pi*I/2) is zoo
assert sech((-3*10**73 + 1)*pi*I/2) is zoo
assert sech((7*10**103 + 1)*pi*I/2) is zoo
assert sech(pi*I) == -1
assert sech(-pi*I) == -1
assert sech(5*pi*I) == -1
assert sech(8*pi*I) == 1
assert sech(pi*I/3) == 2
assert sech(pi*I*Rational(-2, 3)) == -2
assert sech(pi*I/4) == sqrt(2)
assert sech(-pi*I/4) == sqrt(2)
assert sech(pi*I*Rational(5, 4)) == -sqrt(2)
assert sech(pi*I*Rational(-5, 4)) == -sqrt(2)
assert sech(pi*I/6) == 2/sqrt(3)
assert sech(-pi*I/6) == 2/sqrt(3)
assert sech(pi*I*Rational(7, 6)) == -2/sqrt(3)
assert sech(pi*I*Rational(-5, 6)) == -2/sqrt(3)
assert sech(pi*I/105) == 1/cos(pi/105)
assert sech(-pi*I/105) == 1/cos(pi/105)
assert sech(x*I) == 1/cos(x)
assert sech(k*pi*I) == 1/cos(k*pi)
assert sech(17*k*pi*I) == 1/cos(17*k*pi)
assert sech(n).is_real is True
assert expand_trig(sech(x + y)) == 1/(cosh(x)*cosh(y) + sinh(x)*sinh(y))
def test_sech_series():
x = Symbol('x')
assert sech(x).series(x, 0, 10) == \
1 - x**2/2 + 5*x**4/24 - 61*x**6/720 + 277*x**8/8064 + O(x**10)
def test_sech_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: sech(x).fdiff(2))
def test_asinh():
x, y = symbols('x,y')
assert unchanged(asinh, x)
assert asinh(-x) == -asinh(x)
#at specific points
assert asinh(nan) is nan
assert asinh( 0) == 0
assert asinh(+1) == log(sqrt(2) + 1)
assert asinh(-1) == log(sqrt(2) - 1)
assert asinh(I) == pi*I/2
assert asinh(-I) == -pi*I/2
assert asinh(I/2) == pi*I/6
assert asinh(-I/2) == -pi*I/6
# at infinites
assert asinh(oo) is oo
assert asinh(-oo) is -oo
assert asinh(I*oo) is oo
assert asinh(-I *oo) is -oo
assert asinh(zoo) is zoo
#properties
assert asinh(I *(sqrt(3) - 1)/(2**Rational(3, 2))) == pi*I/12
assert asinh(-I *(sqrt(3) - 1)/(2**Rational(3, 2))) == -pi*I/12
assert asinh(I*(sqrt(5) - 1)/4) == pi*I/10
assert asinh(-I*(sqrt(5) - 1)/4) == -pi*I/10
assert asinh(I*(sqrt(5) + 1)/4) == pi*I*Rational(3, 10)
assert asinh(-I*(sqrt(5) + 1)/4) == pi*I*Rational(-3, 10)
# Symmetry
assert asinh(Rational(-1, 2)) == -asinh(S.Half)
# inverse composition
assert unchanged(asinh, sinh(Symbol('v1')))
assert asinh(sinh(0, evaluate=False)) == 0
assert asinh(sinh(-3, evaluate=False)) == -3
assert asinh(sinh(2, evaluate=False)) == 2
assert asinh(sinh(I, evaluate=False)) == I
assert asinh(sinh(-I, evaluate=False)) == -I
assert asinh(sinh(5*I, evaluate=False)) == -2*I*pi + 5*I
assert asinh(sinh(15 + 11*I)) == 15 - 4*I*pi + 11*I
assert asinh(sinh(-73 + 97*I)) == 73 - 97*I + 31*I*pi
assert asinh(sinh(-7 - 23*I)) == 7 - 7*I*pi + 23*I
assert asinh(sinh(13 - 3*I)) == -13 - I*pi + 3*I
p = Symbol('p', positive=True)
assert asinh(p).is_zero is False
assert asinh(sinh(0, evaluate=False), evaluate=False).is_zero is True
def test_asinh_rewrite():
x = Symbol('x')
assert asinh(x).rewrite(log) == log(x + sqrt(x**2 + 1))
assert asinh(x).rewrite(atanh) == atanh(x/sqrt(1 + x**2))
assert asinh(x).rewrite(asin) == asinh(x)
assert asinh(x*(1 + I)).rewrite(asin) == -I*asin(I*x*(1+I))
assert asinh(x).rewrite(acos) == I*(-I*asinh(x) + pi/2) - I*pi/2
def test_asinh_leading_term():
x = Symbol('x')
assert asinh(x).as_leading_term(x, cdir=1) == x
# Tests concerning branch points
assert asinh(x + I).as_leading_term(x, cdir=1) == I*pi/2
assert asinh(x - I).as_leading_term(x, cdir=1) == -I*pi/2
assert asinh(1/x).as_leading_term(x, cdir=1) == -log(x) + log(2)
assert asinh(1/x).as_leading_term(x, cdir=-1) == log(x) - log(2) - I*pi
# Tests concerning points lying on branch cuts
assert asinh(x + 2*I).as_leading_term(x, cdir=1) == I*asin(2)
assert asinh(x + 2*I).as_leading_term(x, cdir=-1) == -I*asin(2) + I*pi
assert asinh(x - 2*I).as_leading_term(x, cdir=1) == -I*pi + I*asin(2)
assert asinh(x - 2*I).as_leading_term(x, cdir=-1) == -I*asin(2)
# Tests concerning re(ndir) == 0
assert asinh(2*I + I*x - x**2).as_leading_term(x, cdir=1) == log(2 - sqrt(3)) + I*pi/2
assert asinh(2*I + I*x - x**2).as_leading_term(x, cdir=-1) == log(2 - sqrt(3)) + I*pi/2
def test_asinh_series():
x = Symbol('x')
assert asinh(x).series(x, 0, 8) == \
x - x**3/6 + 3*x**5/40 - 5*x**7/112 + O(x**8)
t5 = asinh(x).taylor_term(5, x)
assert t5 == 3*x**5/40
assert asinh(x).taylor_term(7, x, t5, 0) == -5*x**7/112
def test_asinh_nseries():
x = Symbol('x')
# Tests concerning branch points
assert asinh(x + I)._eval_nseries(x, 4, None) == I*pi/2 + \
sqrt(x)*(1 - I) + x**(S(3)/2)*(S(1)/12 + I/12) + x**(S(5)/2)*(-S(3)/160 + 3*I/160) + \
x**(S(7)/2)*(-S(5)/896 - 5*I/896) + O(x**4)
assert asinh(x - I)._eval_nseries(x, 4, None) == -I*pi/2 + \
sqrt(x)*(1 + I) + x**(S(3)/2)*(S(1)/12 - I/12) + x**(S(5)/2)*(-S(3)/160 - 3*I/160) + \
x**(S(7)/2)*(-S(5)/896 + 5*I/896) + O(x**4)
# Tests concerning points lying on branch cuts
assert asinh(x + 2*I)._eval_nseries(x, 4, None, cdir=1) == I*asin(2) - \
sqrt(3)*I*x/3 + sqrt(3)*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
assert asinh(x + 2*I)._eval_nseries(x, 4, None, cdir=-1) == I*pi - I*asin(2) + \
sqrt(3)*I*x/3 - sqrt(3)*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert asinh(x - 2*I)._eval_nseries(x, 4, None, cdir=1) == I*asin(2) - I*pi + \
sqrt(3)*I*x/3 + sqrt(3)*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert asinh(x - 2*I)._eval_nseries(x, 4, None, cdir=-1) == -I*asin(2) - \
sqrt(3)*I*x/3 - sqrt(3)*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
# Tests concerning re(ndir) == 0
assert asinh(2*I + I*x - x**2)._eval_nseries(x, 4, None) == I*pi/2 + log(2 - sqrt(3)) - \
sqrt(3)*x/3 + x**2*(sqrt(3)/9 - sqrt(3)*I/3) + x**3*(-sqrt(3)/18 + 2*sqrt(3)*I/9) + O(x**4)
def test_asinh_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: asinh(x).fdiff(2))
def test_acosh():
x = Symbol('x')
assert unchanged(acosh, -x)
#at specific points
assert acosh(1) == 0
assert acosh(-1) == pi*I
assert acosh(0) == I*pi/2
assert acosh(S.Half) == I*pi/3
assert acosh(Rational(-1, 2)) == pi*I*Rational(2, 3)
assert acosh(nan) is nan
# at infinites
assert acosh(oo) is oo
assert acosh(-oo) is oo
assert acosh(I*oo) == oo + I*pi/2
assert acosh(-I*oo) == oo - I*pi/2
assert acosh(zoo) is zoo
assert acosh(I) == log(I*(1 + sqrt(2)))
assert acosh(-I) == log(-I*(1 + sqrt(2)))
assert acosh((sqrt(3) - 1)/(2*sqrt(2))) == pi*I*Rational(5, 12)
assert acosh(-(sqrt(3) - 1)/(2*sqrt(2))) == pi*I*Rational(7, 12)
assert acosh(sqrt(2)/2) == I*pi/4
assert acosh(-sqrt(2)/2) == I*pi*Rational(3, 4)
assert acosh(sqrt(3)/2) == I*pi/6
assert acosh(-sqrt(3)/2) == I*pi*Rational(5, 6)
assert acosh(sqrt(2 + sqrt(2))/2) == I*pi/8
assert acosh(-sqrt(2 + sqrt(2))/2) == I*pi*Rational(7, 8)
assert acosh(sqrt(2 - sqrt(2))/2) == I*pi*Rational(3, 8)
assert acosh(-sqrt(2 - sqrt(2))/2) == I*pi*Rational(5, 8)
assert acosh((1 + sqrt(3))/(2*sqrt(2))) == I*pi/12
assert acosh(-(1 + sqrt(3))/(2*sqrt(2))) == I*pi*Rational(11, 12)
assert acosh((sqrt(5) + 1)/4) == I*pi/5
assert acosh(-(sqrt(5) + 1)/4) == I*pi*Rational(4, 5)
assert str(acosh(5*I).n(6)) == '2.31244 + 1.5708*I'
assert str(acosh(-5*I).n(6)) == '2.31244 - 1.5708*I'
# inverse composition
assert unchanged(acosh, Symbol('v1'))
assert acosh(cosh(-3, evaluate=False)) == 3
assert acosh(cosh(3, evaluate=False)) == 3
assert acosh(cosh(0, evaluate=False)) == 0
assert acosh(cosh(I, evaluate=False)) == I
assert acosh(cosh(-I, evaluate=False)) == I
assert acosh(cosh(7*I, evaluate=False)) == -2*I*pi + 7*I
assert acosh(cosh(1 + I)) == 1 + I
assert acosh(cosh(3 - 3*I)) == 3 - 3*I
assert acosh(cosh(-3 + 2*I)) == 3 - 2*I
assert acosh(cosh(-5 - 17*I)) == 5 - 6*I*pi + 17*I
assert acosh(cosh(-21 + 11*I)) == 21 - 11*I + 4*I*pi
assert acosh(cosh(cosh(1) + I)) == cosh(1) + I
assert acosh(1, evaluate=False).is_zero is True
def test_acosh_rewrite():
x = Symbol('x')
assert acosh(x).rewrite(log) == log(x + sqrt(x - 1)*sqrt(x + 1))
assert acosh(x).rewrite(asin) == sqrt(x - 1)*(-asin(x) + pi/2)/sqrt(1 - x)
assert acosh(x).rewrite(asinh) == sqrt(x - 1)*(-asin(x) + pi/2)/sqrt(1 - x)
assert acosh(x).rewrite(atanh) == \
(sqrt(x - 1)*sqrt(x + 1)*atanh(sqrt(x**2 - 1)/x)/sqrt(x**2 - 1) +
pi*sqrt(x - 1)*(-x*sqrt(x**(-2)) + 1)/(2*sqrt(1 - x)))
x = Symbol('x', positive=True)
assert acosh(x).rewrite(atanh) == \
sqrt(x - 1)*sqrt(x + 1)*atanh(sqrt(x**2 - 1)/x)/sqrt(x**2 - 1)
def test_acosh_leading_term():
x = Symbol('x')
# Tests concerning branch points
assert acosh(x).as_leading_term(x) == I*pi/2
assert acosh(x + 1).as_leading_term(x) == sqrt(2)*sqrt(x)
assert acosh(x - 1).as_leading_term(x) == I*pi
assert acosh(1/x).as_leading_term(x, cdir=1) == -log(x) + log(2)
assert acosh(1/x).as_leading_term(x, cdir=-1) == -log(x) + log(2) + 2*I*pi
# Tests concerning points lying on branch cuts
assert acosh(I*x - 2).as_leading_term(x, cdir=1) == acosh(-2)
assert acosh(-I*x - 2).as_leading_term(x, cdir=1) == -2*I*pi + acosh(-2)
assert acosh(x**2 - I*x + S(1)/3).as_leading_term(x, cdir=1) == -acosh(S(1)/3)
assert acosh(x**2 - I*x + S(1)/3).as_leading_term(x, cdir=-1) == acosh(S(1)/3)
assert acosh(1/(I*x - 3)).as_leading_term(x, cdir=1) == -acosh(-S(1)/3)
assert acosh(1/(I*x - 3)).as_leading_term(x, cdir=-1) == acosh(-S(1)/3)
# Tests concerning im(ndir) == 0
assert acosh(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == log(sqrt(3) + 2) - I*pi
assert acosh(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == log(sqrt(3) + 2) - I*pi
def test_acosh_series():
x = Symbol('x')
assert acosh(x).series(x, 0, 8) == \
-I*x + pi*I/2 - I*x**3/6 - 3*I*x**5/40 - 5*I*x**7/112 + O(x**8)
t5 = acosh(x).taylor_term(5, x)
assert t5 == - 3*I*x**5/40
assert acosh(x).taylor_term(7, x, t5, 0) == - 5*I*x**7/112
def test_acosh_nseries():
x = Symbol('x')
# Tests concerning branch points
assert acosh(x + 1)._eval_nseries(x, 4, None) == sqrt(2)*sqrt(x) - \
sqrt(2)*x**(S(3)/2)/12 + 3*sqrt(2)*x**(S(5)/2)/160 - 5*sqrt(2)*x**(S(7)/2)/896 + O(x**4)
# Tests concerning points lying on branch cuts
assert acosh(x - 1)._eval_nseries(x, 4, None) == I*pi - \
sqrt(2)*I*sqrt(x) - sqrt(2)*I*x**(S(3)/2)/12 - 3*sqrt(2)*I*x**(S(5)/2)/160 - \
5*sqrt(2)*I*x**(S(7)/2)/896 + O(x**4)
assert acosh(I*x - 2)._eval_nseries(x, 4, None, cdir=1) == acosh(-2) - \
sqrt(3)*I*x/3 + sqrt(3)*x**2/9 + sqrt(3)*I*x**3/18 + O(x**4)
assert acosh(-I*x - 2)._eval_nseries(x, 4, None, cdir=1) == acosh(-2) - \
2*I*pi + sqrt(3)*I*x/3 + sqrt(3)*x**2/9 - sqrt(3)*I*x**3/18 + O(x**4)
assert acosh(1/(I*x - 3))._eval_nseries(x, 4, None, cdir=1) == -acosh(-S(1)/3) + \
sqrt(2)*x/12 + 17*sqrt(2)*I*x**2/576 - 443*sqrt(2)*x**3/41472 + O(x**4)
assert acosh(1/(I*x - 3))._eval_nseries(x, 4, None, cdir=-1) == acosh(-S(1)/3) - \
sqrt(2)*x/12 - 17*sqrt(2)*I*x**2/576 + 443*sqrt(2)*x**3/41472 + O(x**4)
# Tests concerning im(ndir) == 0
assert acosh(-I*x**2 + x - 2)._eval_nseries(x, 4, None) == -I*pi + log(sqrt(3) + 2) - \
sqrt(3)*x/3 + x**2*(-sqrt(3)/9 + sqrt(3)*I/3) + x**3*(-sqrt(3)/18 + 2*sqrt(3)*I/9) + O(x**4)
def test_acosh_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: acosh(x).fdiff(2))
def test_asech():
x = Symbol('x')
assert unchanged(asech, -x)
# values at fixed points
assert asech(1) == 0
assert asech(-1) == pi*I
assert asech(0) is oo
assert asech(2) == I*pi/3
assert asech(-2) == 2*I*pi / 3
assert asech(nan) is nan
# at infinites
assert asech(oo) == I*pi/2
assert asech(-oo) == I*pi/2
assert asech(zoo) == I*AccumBounds(-pi/2, pi/2)
assert asech(I) == log(1 + sqrt(2)) - I*pi/2
assert asech(-I) == log(1 + sqrt(2)) + I*pi/2
assert asech(sqrt(2) - sqrt(6)) == 11*I*pi / 12
assert asech(sqrt(2 - 2/sqrt(5))) == I*pi / 10
assert asech(-sqrt(2 - 2/sqrt(5))) == 9*I*pi / 10
assert asech(2 / sqrt(2 + sqrt(2))) == I*pi / 8
assert asech(-2 / sqrt(2 + sqrt(2))) == 7*I*pi / 8
assert asech(sqrt(5) - 1) == I*pi / 5
assert asech(1 - sqrt(5)) == 4*I*pi / 5
assert asech(-sqrt(2*(2 + sqrt(2)))) == 5*I*pi / 8
# properties
# asech(x) == acosh(1/x)
assert asech(sqrt(2)) == acosh(1/sqrt(2))
assert asech(2/sqrt(3)) == acosh(sqrt(3)/2)
assert asech(2/sqrt(2 + sqrt(2))) == acosh(sqrt(2 + sqrt(2))/2)
assert asech(2) == acosh(S.Half)
# asech(x) == I*acos(1/x)
# (Note: the exact formula is asech(x) == +/- I*acos(1/x))
assert asech(-sqrt(2)) == I*acos(-1/sqrt(2))
assert asech(-2/sqrt(3)) == I*acos(-sqrt(3)/2)
assert asech(-S(2)) == I*acos(Rational(-1, 2))
assert asech(-2/sqrt(2)) == I*acos(-sqrt(2)/2)
# sech(asech(x)) / x == 1
assert expand_mul(sech(asech(sqrt(6) - sqrt(2))) / (sqrt(6) - sqrt(2))) == 1
assert expand_mul(sech(asech(sqrt(6) + sqrt(2))) / (sqrt(6) + sqrt(2))) == 1
assert (sech(asech(sqrt(2 + 2/sqrt(5)))) / (sqrt(2 + 2/sqrt(5)))).simplify() == 1
assert (sech(asech(-sqrt(2 + 2/sqrt(5)))) / (-sqrt(2 + 2/sqrt(5)))).simplify() == 1
assert (sech(asech(sqrt(2*(2 + sqrt(2))))) / (sqrt(2*(2 + sqrt(2))))).simplify() == 1
assert expand_mul(sech(asech(1 + sqrt(5))) / (1 + sqrt(5))) == 1
assert expand_mul(sech(asech(-1 - sqrt(5))) / (-1 - sqrt(5))) == 1
assert expand_mul(sech(asech(-sqrt(6) - sqrt(2))) / (-sqrt(6) - sqrt(2))) == 1
# numerical evaluation
assert str(asech(5*I).n(6)) == '0.19869 - 1.5708*I'
assert str(asech(-5*I).n(6)) == '0.19869 + 1.5708*I'
def test_asech_leading_term():
x = Symbol('x')
# Tests concerning branch points
assert asech(x).as_leading_term(x, cdir=1) == -log(x) + log(2)
assert asech(x).as_leading_term(x, cdir=-1) == -log(x) + log(2) + 2*I*pi
assert asech(x + 1).as_leading_term(x, cdir=1) == sqrt(2)*I*sqrt(x)
assert asech(1/x).as_leading_term(x, cdir=1) == I*pi/2
# Tests concerning points lying on branch cuts
assert asech(x - 1).as_leading_term(x, cdir=1) == I*pi
assert asech(I*x + 3).as_leading_term(x, cdir=1) == -asech(3)
assert asech(-I*x + 3).as_leading_term(x, cdir=1) == asech(3)
assert asech(I*x - 3).as_leading_term(x, cdir=1) == -asech(-3)
assert asech(-I*x - 3).as_leading_term(x, cdir=1) == asech(-3)
assert asech(I*x - S(1)/3).as_leading_term(x, cdir=1) == -2*I*pi + asech(-S(1)/3)
assert asech(I*x - S(1)/3).as_leading_term(x, cdir=-1) == asech(-S(1)/3)
# Tests concerning im(ndir) == 0
assert asech(-I*x**2 + x - 3).as_leading_term(x, cdir=1) == log(-S(1)/3 + 2*sqrt(2)*I/3)
assert asech(-I*x**2 + x - 3).as_leading_term(x, cdir=-1) == log(-S(1)/3 + 2*sqrt(2)*I/3)
def test_asech_series():
x = Symbol('x')
assert asech(x).series(x, 0, 9, cdir=1) == log(2) - log(x) - x**2/4 - 3*x**4/32 \
- 5*x**6/96 - 35*x**8/1024 + O(x**9)
assert asech(x).series(x, 0, 9, cdir=-1) == I*pi + log(2) - log(-x) - x**2/4 - \
3*x**4/32 - 5*x**6/96 - 35*x**8/1024 + O(x**9)
t6 = asech(x).taylor_term(6, x)
assert t6 == -5*x**6/96
assert asech(x).taylor_term(8, x, t6, 0) == -35*x**8/1024
def test_asech_nseries():
x = Symbol('x')
# Tests concerning branch points
assert asech(x + 1)._eval_nseries(x, 4, None) == sqrt(2)*sqrt(-x) + 5*sqrt(2)*(-x)**(S(3)/2)/12 + \
43*sqrt(2)*(-x)**(S(5)/2)/160 + 177*sqrt(2)*(-x)**(S(7)/2)/896 + O(x**4)
# Tests concerning points lying on branch cuts
assert asech(x - 1)._eval_nseries(x, 4, None) == I*pi + sqrt(2)*sqrt(x) + \
5*sqrt(2)*x**(S(3)/2)/12 + 43*sqrt(2)*x**(S(5)/2)/160 + 177*sqrt(2)*x**(S(7)/2)/896 + O(x**4)
assert asech(I*x + 3)._eval_nseries(x, 4, None) == -asech(3) + sqrt(2)*x/12 - \
17*sqrt(2)*I*x**2/576 - 443*sqrt(2)*x**3/41472 + O(x**4)
assert asech(-I*x + 3)._eval_nseries(x, 4, None) == asech(3) + sqrt(2)*x/12 + \
17*sqrt(2)*I*x**2/576 - 443*sqrt(2)*x**3/41472 + O(x**4)
assert asech(I*x - 3)._eval_nseries(x, 4, None) == -asech(-3) - sqrt(2)*x/12 - \
17*sqrt(2)*I*x**2/576 + 443*sqrt(2)*x**3/41472 + O(x**4)
assert asech(-I*x - 3)._eval_nseries(x, 4, None) == asech(-3) - sqrt(2)*x/12 + \
17*sqrt(2)*I*x**2/576 + 443*sqrt(2)*x**3/41472 + O(x**4)
# Tests concerning im(ndir) == 0
assert asech(-I*x**2 + x - 2)._eval_nseries(x, 3, None) == 2*I*pi/3 + sqrt(3)*I*x/6 + \
x**2*(sqrt(3)/6 + 7*sqrt(3)*I/72) + O(x**3)
def test_asech_rewrite():
x = Symbol('x')
assert asech(x).rewrite(log) == log(1/x + sqrt(1/x - 1) * sqrt(1/x + 1))
assert asech(x).rewrite(acosh) == acosh(1/x)
assert asech(x).rewrite(asinh) == sqrt(-1 + 1/x)*(-asin(1/x) + pi/2)/sqrt(1 - 1/x)
assert asech(x).rewrite(atanh) == \
sqrt(x + 1)*sqrt(1/(x + 1))*atanh(sqrt(1 - x**2)) + I*pi*(-sqrt(x)*sqrt(1/x) + 1 - I*sqrt(x**2)/(2*sqrt(-x**2)) - I*sqrt(-x)/(2*sqrt(x)))
def test_asech_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: asech(x).fdiff(2))
def test_acsch():
x = Symbol('x')
assert unchanged(acsch, x)
assert acsch(-x) == -acsch(x)
# values at fixed points
assert acsch(1) == log(1 + sqrt(2))
assert acsch(-1) == - log(1 + sqrt(2))
assert acsch(0) is zoo
assert acsch(2) == log((1+sqrt(5))/2)
assert acsch(-2) == - log((1+sqrt(5))/2)
assert acsch(I) == - I*pi/2
assert acsch(-I) == I*pi/2
assert acsch(-I*(sqrt(6) + sqrt(2))) == I*pi / 12
assert acsch(I*(sqrt(2) + sqrt(6))) == -I*pi / 12
assert acsch(-I*(1 + sqrt(5))) == I*pi / 10
assert acsch(I*(1 + sqrt(5))) == -I*pi / 10
assert acsch(-I*2 / sqrt(2 - sqrt(2))) == I*pi / 8
assert acsch(I*2 / sqrt(2 - sqrt(2))) == -I*pi / 8
assert acsch(-I*2) == I*pi / 6
assert acsch(I*2) == -I*pi / 6
assert acsch(-I*sqrt(2 + 2/sqrt(5))) == I*pi / 5
assert acsch(I*sqrt(2 + 2/sqrt(5))) == -I*pi / 5
assert acsch(-I*sqrt(2)) == I*pi / 4
assert acsch(I*sqrt(2)) == -I*pi / 4
assert acsch(-I*(sqrt(5)-1)) == 3*I*pi / 10
assert acsch(I*(sqrt(5)-1)) == -3*I*pi / 10
assert acsch(-I*2 / sqrt(3)) == I*pi / 3
assert acsch(I*2 / sqrt(3)) == -I*pi / 3
assert acsch(-I*2 / sqrt(2 + sqrt(2))) == 3*I*pi / 8
assert acsch(I*2 / sqrt(2 + sqrt(2))) == -3*I*pi / 8
assert acsch(-I*sqrt(2 - 2/sqrt(5))) == 2*I*pi / 5
assert acsch(I*sqrt(2 - 2/sqrt(5))) == -2*I*pi / 5
assert acsch(-I*(sqrt(6) - sqrt(2))) == 5*I*pi / 12
assert acsch(I*(sqrt(6) - sqrt(2))) == -5*I*pi / 12
assert acsch(nan) is nan
# properties
# acsch(x) == asinh(1/x)
assert acsch(-I*sqrt(2)) == asinh(I/sqrt(2))
assert acsch(-I*2 / sqrt(3)) == asinh(I*sqrt(3) / 2)
# acsch(x) == -I*asin(I/x)
assert acsch(-I*sqrt(2)) == -I*asin(-1/sqrt(2))
assert acsch(-I*2 / sqrt(3)) == -I*asin(-sqrt(3)/2)
# csch(acsch(x)) / x == 1
assert expand_mul(csch(acsch(-I*(sqrt(6) + sqrt(2)))) / (-I*(sqrt(6) + sqrt(2)))) == 1
assert expand_mul(csch(acsch(I*(1 + sqrt(5)))) / (I*(1 + sqrt(5)))) == 1
assert (csch(acsch(I*sqrt(2 - 2/sqrt(5)))) / (I*sqrt(2 - 2/sqrt(5)))).simplify() == 1
assert (csch(acsch(-I*sqrt(2 - 2/sqrt(5)))) / (-I*sqrt(2 - 2/sqrt(5)))).simplify() == 1
# numerical evaluation
assert str(acsch(5*I+1).n(6)) == '0.0391819 - 0.193363*I'
assert str(acsch(-5*I+1).n(6)) == '0.0391819 + 0.193363*I'
def test_acsch_infinities():
assert acsch(oo) == 0
assert acsch(-oo) == 0
assert acsch(zoo) == 0
def test_acsch_leading_term():
x = Symbol('x')
assert acsch(1/x).as_leading_term(x) == x
# Tests concerning branch points
assert acsch(x + I).as_leading_term(x) == -I*pi/2
assert acsch(x - I).as_leading_term(x) == I*pi/2
# Tests concerning points lying on branch cuts
assert acsch(x).as_leading_term(x, cdir=1) == -log(x) + log(2)
assert acsch(x).as_leading_term(x, cdir=-1) == log(x) - log(2) - I*pi
assert acsch(x + I/2).as_leading_term(x, cdir=1) == -I*pi - acsch(I/2)
assert acsch(x + I/2).as_leading_term(x, cdir=-1) == acsch(I/2)
assert acsch(x - I/2).as_leading_term(x, cdir=1) == -acsch(I/2)
assert acsch(x - I/2).as_leading_term(x, cdir=-1) == acsch(I/2) + I*pi
# Tests concerning re(ndir) == 0
assert acsch(I/2 + I*x - x**2).as_leading_term(x, cdir=1) == log(2 - sqrt(3)) - I*pi/2
assert acsch(I/2 + I*x - x**2).as_leading_term(x, cdir=-1) == log(2 - sqrt(3)) - I*pi/2
def test_acsch_series():
x = Symbol('x')
assert acsch(x).series(x, 0, 9) == log(2) - log(x) + x**2/4 - 3*x**4/32 \
+ 5*x**6/96 - 35*x**8/1024 + O(x**9)
t4 = acsch(x).taylor_term(4, x)
assert t4 == -3*x**4/32
assert acsch(x).taylor_term(6, x, t4, 0) == 5*x**6/96
def test_acsch_nseries():
x = Symbol('x')
# Tests concerning branch points
assert acsch(x + I)._eval_nseries(x, 4, None) == -I*pi/2 + I*sqrt(x) + \
sqrt(x) + 5*I*x**(S(3)/2)/12 - 5*x**(S(3)/2)/12 - 43*I*x**(S(5)/2)/160 - \
43*x**(S(5)/2)/160 - 177*I*x**(S(7)/2)/896 + 177*x**(S(7)/2)/896 + O(x**4)
assert acsch(x - I)._eval_nseries(x, 4, None) == I*pi/2 - I*sqrt(x) + \
sqrt(x) - 5*I*x**(S(3)/2)/12 - 5*x**(S(3)/2)/12 + 43*I*x**(S(5)/2)/160 - \
43*x**(S(5)/2)/160 + 177*I*x**(S(7)/2)/896 + 177*x**(S(7)/2)/896 + O(x**4)
# Tests concerning points lying on branch cuts
assert acsch(x + I/2)._eval_nseries(x, 4, None, cdir=1) == -acsch(I/2) - \
I*pi + 4*sqrt(3)*I*x/3 - 8*sqrt(3)*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsch(x + I/2)._eval_nseries(x, 4, None, cdir=-1) == acsch(I/2) - \
4*sqrt(3)*I*x/3 + 8*sqrt(3)*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsch(x - I/2)._eval_nseries(x, 4, None, cdir=1) == -acsch(I/2) - \
4*sqrt(3)*I*x/3 - 8*sqrt(3)*x**2/9 + 16*sqrt(3)*I*x**3/9 + O(x**4)
assert acsch(x - I/2)._eval_nseries(x, 4, None, cdir=-1) == I*pi + \
acsch(I/2) + 4*sqrt(3)*I*x/3 + 8*sqrt(3)*x**2/9 - 16*sqrt(3)*I*x**3/9 + O(x**4)
# TODO: Tests concerning re(ndir) == 0
assert acsch(I/2 + I*x - x**2)._eval_nseries(x, 4, None) == -I*pi/2 + \
log(2 - sqrt(3)) + 4*sqrt(3)*x/3 + x**2*(-8*sqrt(3)/9 + 4*sqrt(3)*I/3) + \
x**3*(16*sqrt(3)/9 - 16*sqrt(3)*I/9) + O(x**4)
def test_acsch_rewrite():
x = Symbol('x')
assert acsch(x).rewrite(log) == log(1/x + sqrt(1/x**2 + 1))
assert acsch(x).rewrite(asinh) == asinh(1/x)
assert acsch(x).rewrite(atanh) == (sqrt(-x**2)*(-sqrt(-(x**2 + 1)**2)
*atanh(sqrt(x**2 + 1))/(x**2 + 1)
+ pi/2)/x)
def test_acsch_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: acsch(x).fdiff(2))
def test_atanh():
x = Symbol('x')
#at specific points
assert atanh(0) == 0
assert atanh(I) == I*pi/4
assert atanh(-I) == -I*pi/4
assert atanh(1) is oo
assert atanh(-1) is -oo
assert atanh(nan) is nan
# at infinites
assert atanh(oo) == -I*pi/2
assert atanh(-oo) == I*pi/2
assert atanh(I*oo) == I*pi/2
assert atanh(-I*oo) == -I*pi/2
assert atanh(zoo) == I*AccumBounds(-pi/2, pi/2)
#properties
assert atanh(-x) == -atanh(x)
assert atanh(I/sqrt(3)) == I*pi/6
assert atanh(-I/sqrt(3)) == -I*pi/6
assert atanh(I*sqrt(3)) == I*pi/3
assert atanh(-I*sqrt(3)) == -I*pi/3
assert atanh(I*(1 + sqrt(2))) == pi*I*Rational(3, 8)
assert atanh(I*(sqrt(2) - 1)) == pi*I/8
assert atanh(I*(1 - sqrt(2))) == -pi*I/8
assert atanh(-I*(1 + sqrt(2))) == pi*I*Rational(-3, 8)
assert atanh(I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(2, 5)
assert atanh(-I*sqrt(5 + 2*sqrt(5))) == I*pi*Rational(-2, 5)
assert atanh(I*(2 - sqrt(3))) == pi*I/12
assert atanh(I*(sqrt(3) - 2)) == -pi*I/12
assert atanh(oo) == -I*pi/2
# Symmetry
assert atanh(Rational(-1, 2)) == -atanh(S.Half)
# inverse composition
assert unchanged(atanh, tanh(Symbol('v1')))
assert atanh(tanh(-5, evaluate=False)) == -5
assert atanh(tanh(0, evaluate=False)) == 0
assert atanh(tanh(7, evaluate=False)) == 7
assert atanh(tanh(I, evaluate=False)) == I
assert atanh(tanh(-I, evaluate=False)) == -I
assert atanh(tanh(-11*I, evaluate=False)) == -11*I + 4*I*pi
assert atanh(tanh(3 + I)) == 3 + I
assert atanh(tanh(4 + 5*I)) == 4 - 2*I*pi + 5*I
assert atanh(tanh(pi/2)) == pi/2
assert atanh(tanh(pi)) == pi
assert atanh(tanh(-3 + 7*I)) == -3 - 2*I*pi + 7*I
assert atanh(tanh(9 - I*2/3)) == 9 - I*2/3
assert atanh(tanh(-32 - 123*I)) == -32 - 123*I + 39*I*pi
def test_atanh_rewrite():
x = Symbol('x')
assert atanh(x).rewrite(log) == (log(1 + x) - log(1 - x)) / 2
assert atanh(x).rewrite(asinh) == \
pi*x/(2*sqrt(-x**2)) - sqrt(-x)*sqrt(1 - x**2)*sqrt(1/(x**2 - 1))*asinh(sqrt(1/(x**2 - 1)))/sqrt(x)
def test_atanh_leading_term():
x = Symbol('x')
assert atanh(x).as_leading_term(x) == x
# Tests concerning branch points
assert atanh(x + 1).as_leading_term(x, cdir=1) == -log(x)/2 + log(2)/2 - I*pi/2
assert atanh(x + 1).as_leading_term(x, cdir=-1) == -log(x)/2 + log(2)/2 + I*pi/2
assert atanh(x - 1).as_leading_term(x, cdir=1) == log(x)/2 - log(2)/2
assert atanh(x - 1).as_leading_term(x, cdir=-1) == log(x)/2 - log(2)/2
assert atanh(1/x).as_leading_term(x, cdir=1) == -I*pi/2
assert atanh(1/x).as_leading_term(x, cdir=-1) == I*pi/2
# Tests concerning points lying on branch cuts
assert atanh(I*x + 2).as_leading_term(x, cdir=1) == atanh(2) + I*pi
assert atanh(-I*x + 2).as_leading_term(x, cdir=1) == atanh(2)
assert atanh(I*x - 2).as_leading_term(x, cdir=1) == -atanh(2)
assert atanh(-I*x - 2).as_leading_term(x, cdir=1) == -I*pi - atanh(2)
# Tests concerning im(ndir) == 0
assert atanh(-I*x**2 + x - 2).as_leading_term(x, cdir=1) == -log(3)/2 - I*pi/2
assert atanh(-I*x**2 + x - 2).as_leading_term(x, cdir=-1) == -log(3)/2 - I*pi/2
def test_atanh_series():
x = Symbol('x')
assert atanh(x).series(x, 0, 10) == \
x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
def test_atanh_nseries():
x = Symbol('x')
# Tests concerning branch points
assert atanh(x + 1)._eval_nseries(x, 4, None, cdir=1) == -I*pi/2 + log(2)/2 - \
log(x)/2 + x/4 - x**2/16 + x**3/48 + O(x**4)
assert atanh(x + 1)._eval_nseries(x, 4, None, cdir=-1) == I*pi/2 + log(2)/2 - \
log(x)/2 + x/4 - x**2/16 + x**3/48 + O(x**4)
assert atanh(x - 1)._eval_nseries(x, 4, None, cdir=1) == -log(2)/2 + log(x)/2 + \
x/4 + x**2/16 + x**3/48 + O(x**4)
assert atanh(x - 1)._eval_nseries(x, 4, None, cdir=-1) == -log(2)/2 + log(x)/2 + \
x/4 + x**2/16 + x**3/48 + O(x**4)
# Tests concerning points lying on branch cuts
assert atanh(I*x + 2)._eval_nseries(x, 4, None, cdir=1) == I*pi + atanh(2) - \
I*x/3 - 2*x**2/9 + 13*I*x**3/81 + O(x**4)
assert atanh(I*x + 2)._eval_nseries(x, 4, None, cdir=-1) == atanh(2) - I*x/3 - \
2*x**2/9 + 13*I*x**3/81 + O(x**4)
assert atanh(I*x - 2)._eval_nseries(x, 4, None, cdir=1) == -atanh(2) - I*x/3 + \
2*x**2/9 + 13*I*x**3/81 + O(x**4)
assert atanh(I*x - 2)._eval_nseries(x, 4, None, cdir=-1) == -atanh(2) - I*pi - \
I*x/3 + 2*x**2/9 + 13*I*x**3/81 + O(x**4)
# Tests concerning im(ndir) == 0
assert atanh(-I*x**2 + x - 2)._eval_nseries(x, 4, None) == -I*pi/2 - log(3)/2 - x/3 + \
x**2*(-S(1)/4 + I/2) + x**2*(S(1)/36 - I/6) + x**3*(-S(1)/6 + I/2) + x**3*(S(1)/162 - I/18) + O(x**4)
def test_atanh_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: atanh(x).fdiff(2))
def test_acoth():
x = Symbol('x')
#at specific points
assert acoth(0) == I*pi/2
assert acoth(I) == -I*pi/4
assert acoth(-I) == I*pi/4
assert acoth(1) is oo
assert acoth(-1) is -oo
assert acoth(nan) is nan
# at infinites
assert acoth(oo) == 0
assert acoth(-oo) == 0
assert acoth(I*oo) == 0
assert acoth(-I*oo) == 0
assert acoth(zoo) == 0
#properties
assert acoth(-x) == -acoth(x)
assert acoth(I/sqrt(3)) == -I*pi/3
assert acoth(-I/sqrt(3)) == I*pi/3
assert acoth(I*sqrt(3)) == -I*pi/6
assert acoth(-I*sqrt(3)) == I*pi/6
assert acoth(I*(1 + sqrt(2))) == -pi*I/8
assert acoth(-I*(sqrt(2) + 1)) == pi*I/8
assert acoth(I*(1 - sqrt(2))) == pi*I*Rational(3, 8)
assert acoth(I*(sqrt(2) - 1)) == pi*I*Rational(-3, 8)
assert acoth(I*sqrt(5 + 2*sqrt(5))) == -I*pi/10
assert acoth(-I*sqrt(5 + 2*sqrt(5))) == I*pi/10
assert acoth(I*(2 + sqrt(3))) == -pi*I/12
assert acoth(-I*(2 + sqrt(3))) == pi*I/12
assert acoth(I*(2 - sqrt(3))) == pi*I*Rational(-5, 12)
assert acoth(I*(sqrt(3) - 2)) == pi*I*Rational(5, 12)
# Symmetry
assert acoth(Rational(-1, 2)) == -acoth(S.Half)
def test_acoth_rewrite():
x = Symbol('x')
assert acoth(x).rewrite(log) == (log(1 + 1/x) - log(1 - 1/x)) / 2
assert acoth(x).rewrite(atanh) == atanh(1/x)
assert acoth(x).rewrite(asinh) == \
x*sqrt(x**(-2))*asinh(sqrt(1/(x**2 - 1))) + I*pi*(sqrt((x - 1)/x)*sqrt(x/(x - 1)) - sqrt(x/(x + 1))*sqrt(1 + 1/x))/2
def test_acoth_leading_term():
x = Symbol('x')
# Tests concerning branch points
assert acoth(x + 1).as_leading_term(x, cdir=1) == -log(x)/2 + log(2)/2
assert acoth(x + 1).as_leading_term(x, cdir=-1) == -log(x)/2 + log(2)/2
assert acoth(x - 1).as_leading_term(x, cdir=1) == log(x)/2 - log(2)/2 + I*pi/2
assert acoth(x - 1).as_leading_term(x, cdir=-1) == log(x)/2 - log(2)/2 - I*pi/2
# Tests concerning points lying on branch cuts
assert acoth(x).as_leading_term(x, cdir=-1) == I*pi/2
assert acoth(x).as_leading_term(x, cdir=1) == -I*pi/2
assert acoth(I*x + 1/2).as_leading_term(x, cdir=1) == acoth(1/2)
assert acoth(-I*x + 1/2).as_leading_term(x, cdir=1) == acoth(1/2) + I*pi
assert acoth(I*x - 1/2).as_leading_term(x, cdir=1) == -I*pi - acoth(1/2)
assert acoth(-I*x - 1/2).as_leading_term(x, cdir=1) == -acoth(1/2)
# Tests concerning im(ndir) == 0
assert acoth(-I*x**2 - x - S(1)/2).as_leading_term(x, cdir=1) == -log(3)/2 + I*pi/2
assert acoth(-I*x**2 - x - S(1)/2).as_leading_term(x, cdir=-1) == -log(3)/2 + I*pi/2
def test_acoth_series():
x = Symbol('x')
assert acoth(x).series(x, 0, 10) == \
-I*pi/2 + x + x**3/3 + x**5/5 + x**7/7 + x**9/9 + O(x**10)
def test_acoth_nseries():
x = Symbol('x')
# Tests concerning branch points
assert acoth(x + 1)._eval_nseries(x, 4, None) == log(2)/2 - log(x)/2 + x/4 - \
x**2/16 + x**3/48 + O(x**4)
assert acoth(x - 1)._eval_nseries(x, 4, None, cdir=1) == I*pi/2 - log(2)/2 + \
log(x)/2 + x/4 + x**2/16 + x**3/48 + O(x**4)
assert acoth(x - 1)._eval_nseries(x, 4, None, cdir=-1) == -I*pi/2 - log(2)/2 + \
log(x)/2 + x/4 + x**2/16 + x**3/48 + O(x**4)
# Tests concerning points lying on branch cuts
assert acoth(I*x + S(1)/2)._eval_nseries(x, 4, None, cdir=1) == acoth(S(1)/2) + \
4*I*x/3 - 8*x**2/9 - 112*I*x**3/81 + O(x**4)
assert acoth(I*x + S(1)/2)._eval_nseries(x, 4, None, cdir=-1) == I*pi + \
acoth(S(1)/2) + 4*I*x/3 - 8*x**2/9 - 112*I*x**3/81 + O(x**4)
assert acoth(I*x - S(1)/2)._eval_nseries(x, 4, None, cdir=1) == -acoth(S(1)/2) - \
I*pi + 4*I*x/3 + 8*x**2/9 - 112*I*x**3/81 + O(x**4)
assert acoth(I*x - S(1)/2)._eval_nseries(x, 4, None, cdir=-1) == -acoth(S(1)/2) + \
4*I*x/3 + 8*x**2/9 - 112*I*x**3/81 + O(x**4)
# Tests concerning im(ndir) == 0
assert acoth(-I*x**2 - x - S(1)/2)._eval_nseries(x, 4, None) == I*pi/2 - log(3)/2 - \
4*x/3 + x**2*(-S(8)/9 + 2*I/3) - 2*I*x**2 + x**3*(S(104)/81 - 16*I/9) - 8*x**3/3 + O(x**4)
def test_acoth_fdiff():
x = Symbol('x')
raises(ArgumentIndexError, lambda: acoth(x).fdiff(2))
def test_inverses():
x = Symbol('x')
assert sinh(x).inverse() == asinh
raises(AttributeError, lambda: cosh(x).inverse())
assert tanh(x).inverse() == atanh
assert coth(x).inverse() == acoth
assert asinh(x).inverse() == sinh
assert acosh(x).inverse() == cosh
assert atanh(x).inverse() == tanh
assert acoth(x).inverse() == coth
assert asech(x).inverse() == sech
assert acsch(x).inverse() == csch
def test_leading_term():
x = Symbol('x')
assert cosh(x).as_leading_term(x) == 1
assert coth(x).as_leading_term(x) == 1/x
for func in [sinh, tanh]:
assert func(x).as_leading_term(x) == x
for func in [sinh, cosh, tanh, coth]:
for ar in (1/x, S.Half):
eq = func(ar)
assert eq.as_leading_term(x) == eq
for func in [csch, sech]:
eq = func(S.Half)
assert eq.as_leading_term(x) == eq
def test_complex():
a, b = symbols('a,b', real=True)
z = a + b*I
for func in [sinh, cosh, tanh, coth, sech, csch]:
assert func(z).conjugate() == func(a - b*I)
for deep in [True, False]:
assert sinh(z).expand(
complex=True, deep=deep) == sinh(a)*cos(b) + I*cosh(a)*sin(b)
assert cosh(z).expand(
complex=True, deep=deep) == cosh(a)*cos(b) + I*sinh(a)*sin(b)
assert tanh(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
a)/(cos(b)**2 + sinh(a)**2) + I*sin(b)*cos(b)/(cos(b)**2 + sinh(a)**2)
assert coth(z).expand(complex=True, deep=deep) == sinh(a)*cosh(
a)/(sin(b)**2 + sinh(a)**2) - I*sin(b)*cos(b)/(sin(b)**2 + sinh(a)**2)
assert csch(z).expand(complex=True, deep=deep) == cos(b) * sinh(a) / (sin(b)**2\
*cosh(a)**2 + cos(b)**2 * sinh(a)**2) - I*sin(b) * cosh(a) / (sin(b)**2\
*cosh(a)**2 + cos(b)**2 * sinh(a)**2)
assert sech(z).expand(complex=True, deep=deep) == cos(b) * cosh(a) / (sin(b)**2\
*sinh(a)**2 + cos(b)**2 * cosh(a)**2) - I*sin(b) * sinh(a) / (sin(b)**2\
*sinh(a)**2 + cos(b)**2 * cosh(a)**2)
def test_complex_2899():
a, b = symbols('a,b', real=True)
for deep in [True, False]:
for func in [sinh, cosh, tanh, coth]:
assert func(a).expand(complex=True, deep=deep) == func(a)
def test_simplifications():
x = Symbol('x')
assert sinh(asinh(x)) == x
assert sinh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1)
assert sinh(atanh(x)) == x/sqrt(1 - x**2)
assert sinh(acoth(x)) == 1/(sqrt(x - 1) * sqrt(x + 1))
assert cosh(asinh(x)) == sqrt(1 + x**2)
assert cosh(acosh(x)) == x
assert cosh(atanh(x)) == 1/sqrt(1 - x**2)
assert cosh(acoth(x)) == x/(sqrt(x - 1) * sqrt(x + 1))
assert tanh(asinh(x)) == x/sqrt(1 + x**2)
assert tanh(acosh(x)) == sqrt(x - 1) * sqrt(x + 1) / x
assert tanh(atanh(x)) == x
assert tanh(acoth(x)) == 1/x
assert coth(asinh(x)) == sqrt(1 + x**2)/x
assert coth(acosh(x)) == x/(sqrt(x - 1) * sqrt(x + 1))
assert coth(atanh(x)) == 1/x
assert coth(acoth(x)) == x
assert csch(asinh(x)) == 1/x
assert csch(acosh(x)) == 1/(sqrt(x - 1) * sqrt(x + 1))
assert csch(atanh(x)) == sqrt(1 - x**2)/x
assert csch(acoth(x)) == sqrt(x - 1) * sqrt(x + 1)
assert sech(asinh(x)) == 1/sqrt(1 + x**2)
assert sech(acosh(x)) == 1/x
assert sech(atanh(x)) == sqrt(1 - x**2)
assert sech(acoth(x)) == sqrt(x - 1) * sqrt(x + 1)/x
def test_issue_4136():
assert cosh(asinh(Integer(3)/2)) == sqrt(Integer(13)/4)
def test_sinh_rewrite():
x = Symbol('x')
assert sinh(x).rewrite(exp) == (exp(x) - exp(-x))/2 \
== sinh(x).rewrite('tractable')
assert sinh(x).rewrite(cosh) == -I*cosh(x + I*pi/2)
tanh_half = tanh(S.Half*x)
assert sinh(x).rewrite(tanh) == 2*tanh_half/(1 - tanh_half**2)
coth_half = coth(S.Half*x)
assert sinh(x).rewrite(coth) == 2*coth_half/(coth_half**2 - 1)
def test_cosh_rewrite():
x = Symbol('x')
assert cosh(x).rewrite(exp) == (exp(x) + exp(-x))/2 \
== cosh(x).rewrite('tractable')
assert cosh(x).rewrite(sinh) == -I*sinh(x + I*pi/2)
tanh_half = tanh(S.Half*x)**2
assert cosh(x).rewrite(tanh) == (1 + tanh_half)/(1 - tanh_half)
coth_half = coth(S.Half*x)**2
assert cosh(x).rewrite(coth) == (coth_half + 1)/(coth_half - 1)
def test_tanh_rewrite():
x = Symbol('x')
assert tanh(x).rewrite(exp) == (exp(x) - exp(-x))/(exp(x) + exp(-x)) \
== tanh(x).rewrite('tractable')
assert tanh(x).rewrite(sinh) == I*sinh(x)/sinh(I*pi/2 - x)
assert tanh(x).rewrite(cosh) == I*cosh(I*pi/2 - x)/cosh(x)
assert tanh(x).rewrite(coth) == 1/coth(x)
def test_coth_rewrite():
x = Symbol('x')
assert coth(x).rewrite(exp) == (exp(x) + exp(-x))/(exp(x) - exp(-x)) \
== coth(x).rewrite('tractable')
assert coth(x).rewrite(sinh) == -I*sinh(I*pi/2 - x)/sinh(x)
assert coth(x).rewrite(cosh) == -I*cosh(x)/cosh(I*pi/2 - x)
assert coth(x).rewrite(tanh) == 1/tanh(x)
def test_csch_rewrite():
x = Symbol('x')
assert csch(x).rewrite(exp) == 1 / (exp(x)/2 - exp(-x)/2) \
== csch(x).rewrite('tractable')
assert csch(x).rewrite(cosh) == I/cosh(x + I*pi/2)
tanh_half = tanh(S.Half*x)
assert csch(x).rewrite(tanh) == (1 - tanh_half**2)/(2*tanh_half)
coth_half = coth(S.Half*x)
assert csch(x).rewrite(coth) == (coth_half**2 - 1)/(2*coth_half)
def test_sech_rewrite():
x = Symbol('x')
assert sech(x).rewrite(exp) == 1 / (exp(x)/2 + exp(-x)/2) \
== sech(x).rewrite('tractable')
assert sech(x).rewrite(sinh) == I/sinh(x + I*pi/2)
tanh_half = tanh(S.Half*x)**2
assert sech(x).rewrite(tanh) == (1 - tanh_half)/(1 + tanh_half)
coth_half = coth(S.Half*x)**2
assert sech(x).rewrite(coth) == (coth_half - 1)/(coth_half + 1)
def test_derivs():
x = Symbol('x')
assert coth(x).diff(x) == -sinh(x)**(-2)
assert sinh(x).diff(x) == cosh(x)
assert cosh(x).diff(x) == sinh(x)
assert tanh(x).diff(x) == -tanh(x)**2 + 1
assert csch(x).diff(x) == -coth(x)*csch(x)
assert sech(x).diff(x) == -tanh(x)*sech(x)
assert acoth(x).diff(x) == 1/(-x**2 + 1)
assert asinh(x).diff(x) == 1/sqrt(x**2 + 1)
assert acosh(x).diff(x) == 1/(sqrt(x - 1)*sqrt(x + 1))
assert acosh(x).diff(x) == acosh(x).rewrite(log).diff(x).together()
assert atanh(x).diff(x) == 1/(-x**2 + 1)
assert asech(x).diff(x) == -1/(x*sqrt(1 - x**2))
assert acsch(x).diff(x) == -1/(x**2*sqrt(1 + x**(-2)))
def test_sinh_expansion():
x, y = symbols('x,y')
assert sinh(x+y).expand(trig=True) == sinh(x)*cosh(y) + cosh(x)*sinh(y)
assert sinh(2*x).expand(trig=True) == 2*sinh(x)*cosh(x)
assert sinh(3*x).expand(trig=True).expand() == \
sinh(x)**3 + 3*sinh(x)*cosh(x)**2
def test_cosh_expansion():
x, y = symbols('x,y')
assert cosh(x+y).expand(trig=True) == cosh(x)*cosh(y) + sinh(x)*sinh(y)
assert cosh(2*x).expand(trig=True) == cosh(x)**2 + sinh(x)**2
assert cosh(3*x).expand(trig=True).expand() == \
3*sinh(x)**2*cosh(x) + cosh(x)**3
def test_cosh_positive():
# See issue 11721
# cosh(x) is positive for real values of x
k = symbols('k', real=True)
n = symbols('n', integer=True)
assert cosh(k, evaluate=False).is_positive is True
assert cosh(k + 2*n*pi*I, evaluate=False).is_positive is True
assert cosh(I*pi/4, evaluate=False).is_positive is True
assert cosh(3*I*pi/4, evaluate=False).is_positive is False
def test_cosh_nonnegative():
k = symbols('k', real=True)
n = symbols('n', integer=True)
assert cosh(k, evaluate=False).is_nonnegative is True
assert cosh(k + 2*n*pi*I, evaluate=False).is_nonnegative is True
assert cosh(I*pi/4, evaluate=False).is_nonnegative is True
assert cosh(3*I*pi/4, evaluate=False).is_nonnegative is False
assert cosh(S.Zero, evaluate=False).is_nonnegative is True
def test_real_assumptions():
z = Symbol('z', real=False)
assert sinh(z).is_real is None
assert cosh(z).is_real is None
assert tanh(z).is_real is None
assert sech(z).is_real is None
assert csch(z).is_real is None
assert coth(z).is_real is None
def test_sign_assumptions():
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
assert sinh(n).is_negative is True
assert sinh(p).is_positive is True
assert cosh(n).is_positive is True
assert cosh(p).is_positive is True
assert tanh(n).is_negative is True
assert tanh(p).is_positive is True
assert csch(n).is_negative is True
assert csch(p).is_positive is True
assert sech(n).is_positive is True
assert sech(p).is_positive is True
assert coth(n).is_negative is True
assert coth(p).is_positive is True
|
12ed97f17c1a44ff31e998edbd09e64c21a19b2a2df29600dce7bdc9c0d785ed | from sympy.core.add import Add
from sympy.core.assumptions import check_assumptions
from sympy.core.containers import Tuple
from sympy.core.exprtools import factor_terms
from sympy.core.function import _mexpand
from sympy.core.mul import Mul
from sympy.core.numbers import Rational
from sympy.core.numbers import igcdex, ilcm, igcd
from sympy.core.power import integer_nthroot, isqrt
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key, ordered
from sympy.core.symbol import Symbol, symbols
from sympy.core.sympify import _sympify
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.ntheory.factor_ import (
divisors, factorint, multiplicity, perfect_power)
from sympy.ntheory.generate import nextprime
from sympy.ntheory.primetest import is_square, isprime
from sympy.ntheory.residue_ntheory import sqrt_mod
from sympy.polys.polyerrors import GeneratorsNeeded
from sympy.polys.polytools import Poly, factor_list
from sympy.simplify.simplify import signsimp
from sympy.solvers.solveset import solveset_real
from sympy.utilities import numbered_symbols
from sympy.utilities.misc import as_int, filldedent
from sympy.utilities.iterables import (is_sequence, subsets, permute_signs,
signed_permutations, ordered_partitions)
# these are imported with 'from sympy.solvers.diophantine import *
__all__ = ['diophantine', 'classify_diop']
class DiophantineSolutionSet(set):
"""
Container for a set of solutions to a particular diophantine equation.
The base representation is a set of tuples representing each of the solutions.
Parameters
==========
symbols : list
List of free symbols in the original equation.
parameters: list
List of parameters to be used in the solution.
Examples
========
Adding solutions:
>>> from sympy.solvers.diophantine.diophantine import DiophantineSolutionSet
>>> from sympy.abc import x, y, t, u
>>> s1 = DiophantineSolutionSet([x, y], [t, u])
>>> s1
set()
>>> s1.add((2, 3))
>>> s1.add((-1, u))
>>> s1
{(-1, u), (2, 3)}
>>> s2 = DiophantineSolutionSet([x, y], [t, u])
>>> s2.add((3, 4))
>>> s1.update(*s2)
>>> s1
{(-1, u), (2, 3), (3, 4)}
Conversion of solutions into dicts:
>>> list(s1.dict_iterator())
[{x: -1, y: u}, {x: 2, y: 3}, {x: 3, y: 4}]
Substituting values:
>>> s3 = DiophantineSolutionSet([x, y], [t, u])
>>> s3.add((t**2, t + u))
>>> s3
{(t**2, t + u)}
>>> s3.subs({t: 2, u: 3})
{(4, 5)}
>>> s3.subs(t, -1)
{(1, u - 1)}
>>> s3.subs(t, 3)
{(9, u + 3)}
Evaluation at specific values. Positional arguments are given in the same order as the parameters:
>>> s3(-2, 3)
{(4, 1)}
>>> s3(5)
{(25, u + 5)}
>>> s3(None, 2)
{(t**2, t + 2)}
"""
def __init__(self, symbols_seq, parameters):
super().__init__()
if not is_sequence(symbols_seq):
raise ValueError("Symbols must be given as a sequence.")
if not is_sequence(parameters):
raise ValueError("Parameters must be given as a sequence.")
self.symbols = tuple(symbols_seq)
self.parameters = tuple(parameters)
def add(self, solution):
if len(solution) != len(self.symbols):
raise ValueError("Solution should have a length of %s, not %s" % (len(self.symbols), len(solution)))
super().add(Tuple(*solution))
def update(self, *solutions):
for solution in solutions:
self.add(solution)
def dict_iterator(self):
for solution in ordered(self):
yield dict(zip(self.symbols, solution))
def subs(self, *args, **kwargs):
result = DiophantineSolutionSet(self.symbols, self.parameters)
for solution in self:
result.add(solution.subs(*args, **kwargs))
return result
def __call__(self, *args):
if len(args) > len(self.parameters):
raise ValueError("Evaluation should have at most %s values, not %s" % (len(self.parameters), len(args)))
rep = {p: v for p, v in zip(self.parameters, args) if v is not None}
return self.subs(rep)
class DiophantineEquationType:
"""
Internal representation of a particular diophantine equation type.
Parameters
==========
equation :
The diophantine equation that is being solved.
free_symbols : list (optional)
The symbols being solved for.
Attributes
==========
total_degree :
The maximum of the degrees of all terms in the equation
homogeneous :
Does the equation contain a term of degree 0
homogeneous_order :
Does the equation contain any coefficient that is in the symbols being solved for
dimension :
The number of symbols being solved for
"""
name = None # type: str
def __init__(self, equation, free_symbols=None):
self.equation = _sympify(equation).expand(force=True)
if free_symbols is not None:
self.free_symbols = free_symbols
else:
self.free_symbols = list(self.equation.free_symbols)
self.free_symbols.sort(key=default_sort_key)
if not self.free_symbols:
raise ValueError('equation should have 1 or more free symbols')
self.coeff = self.equation.as_coefficients_dict()
if not all(_is_int(c) for c in self.coeff.values()):
raise TypeError("Coefficients should be Integers")
self.total_degree = Poly(self.equation).total_degree()
self.homogeneous = 1 not in self.coeff
self.homogeneous_order = not (set(self.coeff) & set(self.free_symbols))
self.dimension = len(self.free_symbols)
self._parameters = None
def matches(self):
"""
Determine whether the given equation can be matched to the particular equation type.
"""
return False
@property
def n_parameters(self):
return self.dimension
@property
def parameters(self):
if self._parameters is None:
self._parameters = symbols('t_:%i' % (self.n_parameters,), integer=True)
return self._parameters
def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet:
raise NotImplementedError('No solver has been written for %s.' % self.name)
def pre_solve(self, parameters=None):
if not self.matches():
raise ValueError("This equation does not match the %s equation type." % self.name)
if parameters is not None:
if len(parameters) != self.n_parameters:
raise ValueError("Expected %s parameter(s) but got %s" % (self.n_parameters, len(parameters)))
self._parameters = parameters
class Univariate(DiophantineEquationType):
"""
Representation of a univariate diophantine equation.
A univariate diophantine equation is an equation of the form
`a_{0} + a_{1}x + a_{2}x^2 + .. + a_{n}x^n = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x` is an integer variable.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import Univariate
>>> from sympy.abc import x
>>> Univariate((x - 2)*(x - 3)**2).solve() # solves equation (x - 2)*(x - 3)**2 == 0
{(2,), (3,)}
"""
name = 'univariate'
def matches(self):
return self.dimension == 1
def solve(self, parameters=None, limit=None):
self.pre_solve(parameters)
result = DiophantineSolutionSet(self.free_symbols, parameters=self.parameters)
for i in solveset_real(self.equation, self.free_symbols[0]).intersect(S.Integers):
result.add((i,))
return result
class Linear(DiophantineEquationType):
"""
Representation of a linear diophantine equation.
A linear diophantine equation is an equation of the form `a_{1}x_{1} +
a_{2}x_{2} + .. + a_{n}x_{n} = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x_{1}, x_{2}, ..x_{n}` are integer variables.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import Linear
>>> from sympy.abc import x, y, z
>>> l1 = Linear(2*x - 3*y - 5)
>>> l1.matches() # is this equation linear
True
>>> l1.solve() # solves equation 2*x - 3*y - 5 == 0
{(3*t_0 - 5, 2*t_0 - 5)}
Here x = -3*t_0 - 5 and y = -2*t_0 - 5
>>> Linear(2*x - 3*y - 4*z -3).solve()
{(t_0, 2*t_0 + 4*t_1 + 3, -t_0 - 3*t_1 - 3)}
"""
name = 'linear'
def matches(self):
return self.total_degree == 1
def solve(self, parameters=None, limit=None):
self.pre_solve(parameters)
coeff = self.coeff
var = self.free_symbols
if 1 in coeff:
# negate coeff[] because input is of the form: ax + by + c == 0
# but is used as: ax + by == -c
c = -coeff[1]
else:
c = 0
result = DiophantineSolutionSet(var, parameters=self.parameters)
params = result.parameters
if len(var) == 1:
q, r = divmod(c, coeff[var[0]])
if not r:
result.add((q,))
return result
else:
return result
'''
base_solution_linear() can solve diophantine equations of the form:
a*x + b*y == c
We break down multivariate linear diophantine equations into a
series of bivariate linear diophantine equations which can then
be solved individually by base_solution_linear().
Consider the following:
a_0*x_0 + a_1*x_1 + a_2*x_2 == c
which can be re-written as:
a_0*x_0 + g_0*y_0 == c
where
g_0 == gcd(a_1, a_2)
and
y == (a_1*x_1)/g_0 + (a_2*x_2)/g_0
This leaves us with two binary linear diophantine equations.
For the first equation:
a == a_0
b == g_0
c == c
For the second:
a == a_1/g_0
b == a_2/g_0
c == the solution we find for y_0 in the first equation.
The arrays A and B are the arrays of integers used for
'a' and 'b' in each of the n-1 bivariate equations we solve.
'''
A = [coeff[v] for v in var]
B = []
if len(var) > 2:
B.append(igcd(A[-2], A[-1]))
A[-2] = A[-2] // B[0]
A[-1] = A[-1] // B[0]
for i in range(len(A) - 3, 0, -1):
gcd = igcd(B[0], A[i])
B[0] = B[0] // gcd
A[i] = A[i] // gcd
B.insert(0, gcd)
B.append(A[-1])
'''
Consider the trivariate linear equation:
4*x_0 + 6*x_1 + 3*x_2 == 2
This can be re-written as:
4*x_0 + 3*y_0 == 2
where
y_0 == 2*x_1 + x_2
(Note that gcd(3, 6) == 3)
The complete integral solution to this equation is:
x_0 == 2 + 3*t_0
y_0 == -2 - 4*t_0
where 't_0' is any integer.
Now that we have a solution for 'x_0', find 'x_1' and 'x_2':
2*x_1 + x_2 == -2 - 4*t_0
We can then solve for '-2' and '-4' independently,
and combine the results:
2*x_1a + x_2a == -2
x_1a == 0 + t_0
x_2a == -2 - 2*t_0
2*x_1b + x_2b == -4*t_0
x_1b == 0*t_0 + t_1
x_2b == -4*t_0 - 2*t_1
==>
x_1 == t_0 + t_1
x_2 == -2 - 6*t_0 - 2*t_1
where 't_0' and 't_1' are any integers.
Note that:
4*(2 + 3*t_0) + 6*(t_0 + t_1) + 3*(-2 - 6*t_0 - 2*t_1) == 2
for any integral values of 't_0', 't_1'; as required.
This method is generalised for many variables, below.
'''
solutions = []
for Ai, Bi in zip(A, B):
tot_x, tot_y = [], []
for j, arg in enumerate(Add.make_args(c)):
if arg.is_Integer:
# example: 5 -> k = 5
k, p = arg, S.One
pnew = params[0]
else: # arg is a Mul or Symbol
# example: 3*t_1 -> k = 3
# example: t_0 -> k = 1
k, p = arg.as_coeff_Mul()
pnew = params[params.index(p) + 1]
sol = sol_x, sol_y = base_solution_linear(k, Ai, Bi, pnew)
if p is S.One:
if None in sol:
return result
else:
# convert a + b*pnew -> a*p + b*pnew
if isinstance(sol_x, Add):
sol_x = sol_x.args[0]*p + sol_x.args[1]
if isinstance(sol_y, Add):
sol_y = sol_y.args[0]*p + sol_y.args[1]
tot_x.append(sol_x)
tot_y.append(sol_y)
solutions.append(Add(*tot_x))
c = Add(*tot_y)
solutions.append(c)
result.add(solutions)
return result
class BinaryQuadratic(DiophantineEquationType):
"""
Representation of a binary quadratic diophantine equation.
A binary quadratic diophantine equation is an equation of the
form `Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`, where `A, B, C, D, E,
F` are integer constants and `x` and `y` are integer variables.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import BinaryQuadratic
>>> b1 = BinaryQuadratic(x**3 + y**2 + 1)
>>> b1.matches()
False
>>> b2 = BinaryQuadratic(x**2 + y**2 + 2*x + 2*y + 2)
>>> b2.matches()
True
>>> b2.solve()
{(-1, -1)}
References
==========
.. [1] Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, [online],
Available: http://www.alpertron.com.ar/METHODS.HTM
.. [2] Solving the equation ax^2+ bxy + cy^2 + dx + ey + f= 0, [online],
Available: https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
"""
name = 'binary_quadratic'
def matches(self):
return self.total_degree == 2 and self.dimension == 2
def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet:
self.pre_solve(parameters)
var = self.free_symbols
coeff = self.coeff
x, y = var
A = coeff[x**2]
B = coeff[x*y]
C = coeff[y**2]
D = coeff[x]
E = coeff[y]
F = coeff[S.One]
A, B, C, D, E, F = [as_int(i) for i in _remove_gcd(A, B, C, D, E, F)]
# (1) Simple-Hyperbolic case: A = C = 0, B != 0
# In this case equation can be converted to (Bx + E)(By + D) = DE - BF
# We consider two cases; DE - BF = 0 and DE - BF != 0
# More details, http://www.alpertron.com.ar/METHODS.HTM#SHyperb
result = DiophantineSolutionSet(var, self.parameters)
t, u = result.parameters
discr = B**2 - 4*A*C
if A == 0 and C == 0 and B != 0:
if D*E - B*F == 0:
q, r = divmod(E, B)
if not r:
result.add((-q, t))
q, r = divmod(D, B)
if not r:
result.add((t, -q))
else:
div = divisors(D*E - B*F)
div = div + [-term for term in div]
for d in div:
x0, r = divmod(d - E, B)
if not r:
q, r = divmod(D*E - B*F, d)
if not r:
y0, r = divmod(q - D, B)
if not r:
result.add((x0, y0))
# (2) Parabolic case: B**2 - 4*A*C = 0
# There are two subcases to be considered in this case.
# sqrt(c)D - sqrt(a)E = 0 and sqrt(c)D - sqrt(a)E != 0
# More Details, http://www.alpertron.com.ar/METHODS.HTM#Parabol
elif discr == 0:
if A == 0:
s = BinaryQuadratic(self.equation, free_symbols=[y, x]).solve(parameters=[t, u])
for soln in s:
result.add((soln[1], soln[0]))
else:
g = sign(A)*igcd(A, C)
a = A // g
c = C // g
e = sign(B / A)
sqa = isqrt(a)
sqc = isqrt(c)
_c = e*sqc*D - sqa*E
if not _c:
z = Symbol("z", real=True)
eq = sqa*g*z**2 + D*z + sqa*F
roots = solveset_real(eq, z).intersect(S.Integers)
for root in roots:
ans = diop_solve(sqa*x + e*sqc*y - root)
result.add((ans[0], ans[1]))
elif _is_int(c):
solve_x = lambda u: -e*sqc*g*_c*t**2 - (E + 2*e*sqc*g*u)*t \
- (e*sqc*g*u**2 + E*u + e*sqc*F) // _c
solve_y = lambda u: sqa*g*_c*t**2 + (D + 2*sqa*g*u)*t \
+ (sqa*g*u**2 + D*u + sqa*F) // _c
for z0 in range(0, abs(_c)):
# Check if the coefficients of y and x obtained are integers or not
if (divisible(sqa*g*z0**2 + D*z0 + sqa*F, _c) and
divisible(e*sqc*g*z0**2 + E*z0 + e*sqc*F, _c)):
result.add((solve_x(z0), solve_y(z0)))
# (3) Method used when B**2 - 4*A*C is a square, is described in p. 6 of the below paper
# by John P. Robertson.
# https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
elif is_square(discr):
if A != 0:
r = sqrt(discr)
u, v = symbols("u, v", integer=True)
eq = _mexpand(
4*A*r*u*v + 4*A*D*(B*v + r*u + r*v - B*u) +
2*A*4*A*E*(u - v) + 4*A*r*4*A*F)
solution = diop_solve(eq, t)
for s0, t0 in solution:
num = B*t0 + r*s0 + r*t0 - B*s0
x_0 = S(num) / (4*A*r)
y_0 = S(s0 - t0) / (2*r)
if isinstance(s0, Symbol) or isinstance(t0, Symbol):
if len(check_param(x_0, y_0, 4*A*r, parameters)) > 0:
ans = check_param(x_0, y_0, 4*A*r, parameters)
result.update(*ans)
elif x_0.is_Integer and y_0.is_Integer:
if is_solution_quad(var, coeff, x_0, y_0):
result.add((x_0, y_0))
else:
s = BinaryQuadratic(self.equation, free_symbols=var[::-1]).solve(parameters=[t, u]) # Interchange x and y
while s:
result.add(s.pop()[::-1]) # and solution <--------+
# (4) B**2 - 4*A*C > 0 and B**2 - 4*A*C not a square or B**2 - 4*A*C < 0
else:
P, Q = _transformation_to_DN(var, coeff)
D, N = _find_DN(var, coeff)
solns_pell = diop_DN(D, N)
if D < 0:
for x0, y0 in solns_pell:
for x in [-x0, x0]:
for y in [-y0, y0]:
s = P*Matrix([x, y]) + Q
try:
result.add([as_int(_) for _ in s])
except ValueError:
pass
else:
# In this case equation can be transformed into a Pell equation
solns_pell = set(solns_pell)
for X, Y in list(solns_pell):
solns_pell.add((-X, -Y))
a = diop_DN(D, 1)
T = a[0][0]
U = a[0][1]
if all(_is_int(_) for _ in P[:4] + Q[:2]):
for r, s in solns_pell:
_a = (r + s*sqrt(D))*(T + U*sqrt(D))**t
_b = (r - s*sqrt(D))*(T - U*sqrt(D))**t
x_n = _mexpand(S(_a + _b) / 2)
y_n = _mexpand(S(_a - _b) / (2*sqrt(D)))
s = P*Matrix([x_n, y_n]) + Q
result.add(s)
else:
L = ilcm(*[_.q for _ in P[:4] + Q[:2]])
k = 1
T_k = T
U_k = U
while (T_k - 1) % L != 0 or U_k % L != 0:
T_k, U_k = T_k*T + D*U_k*U, T_k*U + U_k*T
k += 1
for X, Y in solns_pell:
for i in range(k):
if all(_is_int(_) for _ in P*Matrix([X, Y]) + Q):
_a = (X + sqrt(D)*Y)*(T_k + sqrt(D)*U_k)**t
_b = (X - sqrt(D)*Y)*(T_k - sqrt(D)*U_k)**t
Xt = S(_a + _b) / 2
Yt = S(_a - _b) / (2*sqrt(D))
s = P*Matrix([Xt, Yt]) + Q
result.add(s)
X, Y = X*T + D*U*Y, X*U + Y*T
return result
class InhomogeneousTernaryQuadratic(DiophantineEquationType):
"""
Representation of an inhomogeneous ternary quadratic.
No solver is currently implemented for this equation type.
"""
name = 'inhomogeneous_ternary_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
return not self.homogeneous_order
class HomogeneousTernaryQuadraticNormal(DiophantineEquationType):
"""
Representation of a homogeneous ternary quadratic normal diophantine equation.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import HomogeneousTernaryQuadraticNormal
>>> HomogeneousTernaryQuadraticNormal(4*x**2 - 5*y**2 + z**2).solve()
{(1, 2, 4)}
"""
name = 'homogeneous_ternary_quadratic_normal'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
if not self.homogeneous_order:
return False
nonzero = [k for k in self.coeff if self.coeff[k]]
return len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols)
def solve(self, parameters=None, limit=None) -> DiophantineSolutionSet:
self.pre_solve(parameters)
var = self.free_symbols
coeff = self.coeff
x, y, z = var
a = coeff[x**2]
b = coeff[y**2]
c = coeff[z**2]
(sqf_of_a, sqf_of_b, sqf_of_c), (a_1, b_1, c_1), (a_2, b_2, c_2) = \
sqf_normal(a, b, c, steps=True)
A = -a_2*c_2
B = -b_2*c_2
result = DiophantineSolutionSet(var, parameters=self.parameters)
# If following two conditions are satisfied then there are no solutions
if A < 0 and B < 0:
return result
if (
sqrt_mod(-b_2*c_2, a_2) is None or
sqrt_mod(-c_2*a_2, b_2) is None or
sqrt_mod(-a_2*b_2, c_2) is None):
return result
z_0, x_0, y_0 = descent(A, B)
z_0, q = _rational_pq(z_0, abs(c_2))
x_0 *= q
y_0 *= q
x_0, y_0, z_0 = _remove_gcd(x_0, y_0, z_0)
# Holzer reduction
if sign(a) == sign(b):
x_0, y_0, z_0 = holzer(x_0, y_0, z_0, abs(a_2), abs(b_2), abs(c_2))
elif sign(a) == sign(c):
x_0, z_0, y_0 = holzer(x_0, z_0, y_0, abs(a_2), abs(c_2), abs(b_2))
else:
y_0, z_0, x_0 = holzer(y_0, z_0, x_0, abs(b_2), abs(c_2), abs(a_2))
x_0 = reconstruct(b_1, c_1, x_0)
y_0 = reconstruct(a_1, c_1, y_0)
z_0 = reconstruct(a_1, b_1, z_0)
sq_lcm = ilcm(sqf_of_a, sqf_of_b, sqf_of_c)
x_0 = abs(x_0*sq_lcm // sqf_of_a)
y_0 = abs(y_0*sq_lcm // sqf_of_b)
z_0 = abs(z_0*sq_lcm // sqf_of_c)
result.add(_remove_gcd(x_0, y_0, z_0))
return result
class HomogeneousTernaryQuadratic(DiophantineEquationType):
"""
Representation of a homogeneous ternary quadratic diophantine equation.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import HomogeneousTernaryQuadratic
>>> HomogeneousTernaryQuadratic(x**2 + y**2 - 3*z**2 + x*y).solve()
{(-1, 2, 1)}
>>> HomogeneousTernaryQuadratic(3*x**2 + y**2 - 3*z**2 + 5*x*y + y*z).solve()
{(3, 12, 13)}
"""
name = 'homogeneous_ternary_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension == 3):
return False
if not self.homogeneous:
return False
if not self.homogeneous_order:
return False
nonzero = [k for k in self.coeff if self.coeff[k]]
return not (len(nonzero) == 3 and all(i**2 in nonzero for i in self.free_symbols))
def solve(self, parameters=None, limit=None):
self.pre_solve(parameters)
_var = self.free_symbols
coeff = self.coeff
x, y, z = _var
var = [x, y, z]
# Equations of the form B*x*y + C*z*x + E*y*z = 0 and At least two of the
# coefficients A, B, C are non-zero.
# There are infinitely many solutions for the equation.
# Ex: (0, 0, t), (0, t, 0), (t, 0, 0)
# Equation can be re-written as y*(B*x + E*z) = -C*x*z and we can find rather
# unobvious solutions. Set y = -C and B*x + E*z = x*z. The latter can be solved by
# using methods for binary quadratic diophantine equations. Let's select the
# solution which minimizes |x| + |z|
result = DiophantineSolutionSet(var, parameters=self.parameters)
def unpack_sol(sol):
if len(sol) > 0:
return list(sol)[0]
return None, None, None
if not any(coeff[i**2] for i in var):
if coeff[x*z]:
sols = diophantine(coeff[x*y]*x + coeff[y*z]*z - x*z)
s = sols.pop()
min_sum = abs(s[0]) + abs(s[1])
for r in sols:
m = abs(r[0]) + abs(r[1])
if m < min_sum:
s = r
min_sum = m
result.add(_remove_gcd(s[0], -coeff[x*z], s[1]))
return result
else:
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
if x_0 is not None:
result.add((x_0, y_0, z_0))
return result
if coeff[x**2] == 0:
# If the coefficient of x is zero change the variables
if coeff[y**2] == 0:
var[0], var[2] = _var[2], _var[0]
z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
if coeff[x*y] or coeff[x*z]:
# Apply the transformation x --> X - (B*y + C*z)/(2*A)
A = coeff[x**2]
B = coeff[x*y]
C = coeff[x*z]
D = coeff[y**2]
E = coeff[y*z]
F = coeff[z**2]
_coeff = {}
_coeff[x**2] = 4*A**2
_coeff[y**2] = 4*A*D - B**2
_coeff[z**2] = 4*A*F - C**2
_coeff[y*z] = 4*A*E - 2*B*C
_coeff[x*y] = 0
_coeff[x*z] = 0
x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, _coeff))
if x_0 is None:
return result
p, q = _rational_pq(B*y_0 + C*z_0, 2*A)
x_0, y_0, z_0 = x_0*q - p, y_0*q, z_0*q
elif coeff[z*y] != 0:
if coeff[y**2] == 0:
if coeff[z**2] == 0:
# Equations of the form A*x**2 + E*yz = 0.
A = coeff[x**2]
E = coeff[y*z]
b, a = _rational_pq(-E, A)
x_0, y_0, z_0 = b, a, b
else:
# Ax**2 + E*y*z + F*z**2 = 0
var[0], var[2] = _var[2], _var[0]
z_0, y_0, x_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
# A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, C may be zero
var[0], var[1] = _var[1], _var[0]
y_0, x_0, z_0 = unpack_sol(_diop_ternary_quadratic(var, coeff))
else:
# Ax**2 + D*y**2 + F*z**2 = 0, C may be zero
x_0, y_0, z_0 = unpack_sol(_diop_ternary_quadratic_normal(var, coeff))
if x_0 is None:
return result
result.add(_remove_gcd(x_0, y_0, z_0))
return result
class InhomogeneousGeneralQuadratic(DiophantineEquationType):
"""
Representation of an inhomogeneous general quadratic.
No solver is currently implemented for this equation type.
"""
name = 'inhomogeneous_general_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return True
else:
# there may be Pow keys like x**2 or Mul keys like x*y
if any(k.is_Mul for k in self.coeff): # cross terms
return not self.homogeneous
return False
class HomogeneousGeneralQuadratic(DiophantineEquationType):
"""
Representation of a homogeneous general quadratic.
No solver is currently implemented for this equation type.
"""
name = 'homogeneous_general_quadratic'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
else:
# there may be Pow keys like x**2 or Mul keys like x*y
if any(k.is_Mul for k in self.coeff): # cross terms
return self.homogeneous
return False
class GeneralSumOfSquares(DiophantineEquationType):
r"""
Representation of the diophantine equation
`x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Details
=======
When `n = 3` if `k = 4^a(8m + 7)` for some `a, m \in Z` then there will be
no solutions. Refer [1]_ for more details.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import GeneralSumOfSquares
>>> from sympy.abc import a, b, c, d, e
>>> GeneralSumOfSquares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345).solve()
{(15, 22, 22, 24, 24)}
By default only 1 solution is returned. Use the `limit` keyword for more:
>>> sorted(GeneralSumOfSquares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345).solve(limit=3))
[(15, 22, 22, 24, 24), (16, 19, 24, 24, 24), (16, 20, 22, 23, 26)]
References
==========
.. [1] Representing an integer as a sum of three squares, [online],
Available:
http://www.proofwiki.org/wiki/Integer_as_Sum_of_Three_Squares
"""
name = 'general_sum_of_squares'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
if any(k.is_Mul for k in self.coeff):
return False
return all(self.coeff[k] == 1 for k in self.coeff if k != 1)
def solve(self, parameters=None, limit=1):
self.pre_solve(parameters)
var = self.free_symbols
k = -int(self.coeff[1])
n = self.dimension
result = DiophantineSolutionSet(var, parameters=self.parameters)
if k < 0 or limit < 1:
return result
signs = [-1 if x.is_nonpositive else 1 for x in var]
negs = signs.count(-1) != 0
took = 0
for t in sum_of_squares(k, n, zeros=True):
if negs:
result.add([signs[i]*j for i, j in enumerate(t)])
else:
result.add(t)
took += 1
if took == limit:
break
return result
class GeneralPythagorean(DiophantineEquationType):
"""
Representation of the general pythagorean equation,
`a_{1}^2x_{1}^2 + a_{2}^2x_{2}^2 + . . . + a_{n}^2x_{n}^2 - a_{n + 1}^2x_{n + 1}^2 = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import GeneralPythagorean
>>> from sympy.abc import a, b, c, d, e, x, y, z, t
>>> GeneralPythagorean(a**2 + b**2 + c**2 - d**2).solve()
{(t_0**2 + t_1**2 - t_2**2, 2*t_0*t_2, 2*t_1*t_2, t_0**2 + t_1**2 + t_2**2)}
>>> GeneralPythagorean(9*a**2 - 4*b**2 + 16*c**2 + 25*d**2 + e**2).solve(parameters=[x, y, z, t])
{(-10*t**2 + 10*x**2 + 10*y**2 + 10*z**2, 15*t**2 + 15*x**2 + 15*y**2 + 15*z**2, 15*t*x, 12*t*y, 60*t*z)}
"""
name = 'general_pythagorean'
def matches(self):
if not (self.total_degree == 2 and self.dimension >= 3):
return False
if not self.homogeneous_order:
return False
if any(k.is_Mul for k in self.coeff):
return False
if all(self.coeff[k] == 1 for k in self.coeff if k != 1):
return False
if not all(is_square(abs(self.coeff[k])) for k in self.coeff):
return False
# all but one has the same sign
# e.g. 4*x**2 + y**2 - 4*z**2
return abs(sum(sign(self.coeff[k]) for k in self.coeff)) == self.dimension - 2
@property
def n_parameters(self):
return self.dimension - 1
def solve(self, parameters=None, limit=1):
self.pre_solve(parameters)
coeff = self.coeff
var = self.free_symbols
n = self.dimension
if sign(coeff[var[0] ** 2]) + sign(coeff[var[1] ** 2]) + sign(coeff[var[2] ** 2]) < 0:
for key in coeff.keys():
coeff[key] = -coeff[key]
result = DiophantineSolutionSet(var, parameters=self.parameters)
index = 0
for i, v in enumerate(var):
if sign(coeff[v ** 2]) == -1:
index = i
m = result.parameters
ith = sum(m_i ** 2 for m_i in m)
L = [ith - 2 * m[n - 2] ** 2]
L.extend([2 * m[i] * m[n - 2] for i in range(n - 2)])
sol = L[:index] + [ith] + L[index:]
lcm = 1
for i, v in enumerate(var):
if i == index or (index > 0 and i == 0) or (index == 0 and i == 1):
lcm = ilcm(lcm, sqrt(abs(coeff[v ** 2])))
else:
s = sqrt(coeff[v ** 2])
lcm = ilcm(lcm, s if _odd(s) else s // 2)
for i, v in enumerate(var):
sol[i] = (lcm * sol[i]) / sqrt(abs(coeff[v ** 2]))
result.add(sol)
return result
class CubicThue(DiophantineEquationType):
"""
Representation of a cubic Thue diophantine equation.
A cubic Thue diophantine equation is a polynomial of the form
`f(x, y) = r` of degree 3, where `x` and `y` are integers
and `r` is a rational number.
No solver is currently implemented for this equation type.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import CubicThue
>>> c1 = CubicThue(x**3 + y**2 + 1)
>>> c1.matches()
True
"""
name = 'cubic_thue'
def matches(self):
return self.total_degree == 3 and self.dimension == 2
class GeneralSumOfEvenPowers(DiophantineEquationType):
"""
Representation of the diophantine equation
`x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`
where `e` is an even, integer power.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import GeneralSumOfEvenPowers
>>> from sympy.abc import a, b
>>> GeneralSumOfEvenPowers(a**4 + b**4 - (2**4 + 3**4)).solve()
{(2, 3)}
"""
name = 'general_sum_of_even_powers'
def matches(self):
if not self.total_degree > 3:
return False
if self.total_degree % 2 != 0:
return False
if not all(k.is_Pow and k.exp == self.total_degree for k in self.coeff if k != 1):
return False
return all(self.coeff[k] == 1 for k in self.coeff if k != 1)
def solve(self, parameters=None, limit=1):
self.pre_solve(parameters)
var = self.free_symbols
coeff = self.coeff
p = None
for q in coeff.keys():
if q.is_Pow and coeff[q]:
p = q.exp
k = len(var)
n = -coeff[1]
result = DiophantineSolutionSet(var, parameters=self.parameters)
if n < 0 or limit < 1:
return result
sign = [-1 if x.is_nonpositive else 1 for x in var]
negs = sign.count(-1) != 0
took = 0
for t in power_representation(n, p, k):
if negs:
result.add([sign[i]*j for i, j in enumerate(t)])
else:
result.add(t)
took += 1
if took == limit:
break
return result
# these types are known (but not necessarily handled)
# note that order is important here (in the current solver state)
all_diop_classes = [
Linear,
Univariate,
BinaryQuadratic,
InhomogeneousTernaryQuadratic,
HomogeneousTernaryQuadraticNormal,
HomogeneousTernaryQuadratic,
InhomogeneousGeneralQuadratic,
HomogeneousGeneralQuadratic,
GeneralSumOfSquares,
GeneralPythagorean,
CubicThue,
GeneralSumOfEvenPowers,
]
diop_known = {diop_class.name for diop_class in all_diop_classes}
def _is_int(i):
try:
as_int(i)
return True
except ValueError:
pass
def _sorted_tuple(*i):
return tuple(sorted(i))
def _remove_gcd(*x):
try:
g = igcd(*x)
except ValueError:
fx = list(filter(None, x))
if len(fx) < 2:
return x
g = igcd(*[i.as_content_primitive()[0] for i in fx])
except TypeError:
raise TypeError('_remove_gcd(a,b,c) or _remove_gcd(*container)')
if g == 1:
return x
return tuple([i//g for i in x])
def _rational_pq(a, b):
# return `(numer, denom)` for a/b; sign in numer and gcd removed
return _remove_gcd(sign(b)*a, abs(b))
def _nint_or_floor(p, q):
# return nearest int to p/q; in case of tie return floor(p/q)
w, r = divmod(p, q)
if abs(r) <= abs(q)//2:
return w
return w + 1
def _odd(i):
return i % 2 != 0
def _even(i):
return i % 2 == 0
def diophantine(eq, param=symbols("t", integer=True), syms=None,
permute=False):
"""
Simplify the solution procedure of diophantine equation ``eq`` by
converting it into a product of terms which should equal zero.
Explanation
===========
For example, when solving, `x^2 - y^2 = 0` this is treated as
`(x + y)(x - y) = 0` and `x + y = 0` and `x - y = 0` are solved
independently and combined. Each term is solved by calling
``diop_solve()``. (Although it is possible to call ``diop_solve()``
directly, one must be careful to pass an equation in the correct
form and to interpret the output correctly; ``diophantine()`` is
the public-facing function to use in general.)
Output of ``diophantine()`` is a set of tuples. The elements of the
tuple are the solutions for each variable in the equation and
are arranged according to the alphabetic ordering of the variables.
e.g. For an equation with two variables, `a` and `b`, the first
element of the tuple is the solution for `a` and the second for `b`.
Usage
=====
``diophantine(eq, t, syms)``: Solve the diophantine
equation ``eq``.
``t`` is the optional parameter to be used by ``diop_solve()``.
``syms`` is an optional list of symbols which determines the
order of the elements in the returned tuple.
By default, only the base solution is returned. If ``permute`` is set to
True then permutations of the base solution and/or permutations of the
signs of the values will be returned when applicable.
Examples
========
>>> from sympy import diophantine
>>> from sympy.abc import a, b
>>> eq = a**4 + b**4 - (2**4 + 3**4)
>>> diophantine(eq)
{(2, 3)}
>>> diophantine(eq, permute=True)
{(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)}
Details
=======
``eq`` should be an expression which is assumed to be zero.
``t`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, z
>>> diophantine(x**2 - y**2)
{(t_0, -t_0), (t_0, t_0)}
>>> diophantine(x*(2*x + 3*y - z))
{(0, n1, n2), (t_0, t_1, 2*t_0 + 3*t_1)}
>>> diophantine(x**2 + 3*x*y + 4*x)
{(0, n1), (3*t_0 - 4, -t_0)}
See Also
========
diop_solve()
sympy.utilities.iterables.permute_signs
sympy.utilities.iterables.signed_permutations
"""
eq = _sympify(eq)
if isinstance(eq, Eq):
eq = eq.lhs - eq.rhs
try:
var = list(eq.expand(force=True).free_symbols)
var.sort(key=default_sort_key)
if syms:
if not is_sequence(syms):
raise TypeError(
'syms should be given as a sequence, e.g. a list')
syms = [i for i in syms if i in var]
if syms != var:
dict_sym_index = dict(zip(syms, range(len(syms))))
return {tuple([t[dict_sym_index[i]] for i in var])
for t in diophantine(eq, param, permute=permute)}
n, d = eq.as_numer_denom()
if n.is_number:
return set()
if not d.is_number:
dsol = diophantine(d)
good = diophantine(n) - dsol
return {s for s in good if _mexpand(d.subs(zip(var, s)))}
else:
eq = n
eq = factor_terms(eq)
assert not eq.is_number
eq = eq.as_independent(*var, as_Add=False)[1]
p = Poly(eq)
assert not any(g.is_number for g in p.gens)
eq = p.as_expr()
assert eq.is_polynomial()
except (GeneratorsNeeded, AssertionError):
raise TypeError(filldedent('''
Equation should be a polynomial with Rational coefficients.'''))
# permute only sign
do_permute_signs = False
# permute sign and values
do_permute_signs_var = False
# permute few signs
permute_few_signs = False
try:
# if we know that factoring should not be attempted, skip
# the factoring step
v, c, t = classify_diop(eq)
# check for permute sign
if permute:
len_var = len(v)
permute_signs_for = [
GeneralSumOfSquares.name,
GeneralSumOfEvenPowers.name]
permute_signs_check = [
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name,
BinaryQuadratic.name]
if t in permute_signs_for:
do_permute_signs_var = True
elif t in permute_signs_check:
# if all the variables in eq have even powers
# then do_permute_sign = True
if len_var == 3:
var_mul = list(subsets(v, 2))
# here var_mul is like [(x, y), (x, z), (y, z)]
xy_coeff = True
x_coeff = True
var1_mul_var2 = map(lambda a: a[0]*a[1], var_mul)
# if coeff(y*z), coeff(y*x), coeff(x*z) is not 0 then
# `xy_coeff` => True and do_permute_sign => False.
# Means no permuted solution.
for v1_mul_v2 in var1_mul_var2:
try:
coeff = c[v1_mul_v2]
except KeyError:
coeff = 0
xy_coeff = bool(xy_coeff) and bool(coeff)
var_mul = list(subsets(v, 1))
# here var_mul is like [(x,), (y, )]
for v1 in var_mul:
try:
coeff = c[v1[0]]
except KeyError:
coeff = 0
x_coeff = bool(x_coeff) and bool(coeff)
if not any((xy_coeff, x_coeff)):
# means only x**2, y**2, z**2, const is present
do_permute_signs = True
elif not x_coeff:
permute_few_signs = True
elif len_var == 2:
var_mul = list(subsets(v, 2))
# here var_mul is like [(x, y)]
xy_coeff = True
x_coeff = True
var1_mul_var2 = map(lambda x: x[0]*x[1], var_mul)
for v1_mul_v2 in var1_mul_var2:
try:
coeff = c[v1_mul_v2]
except KeyError:
coeff = 0
xy_coeff = bool(xy_coeff) and bool(coeff)
var_mul = list(subsets(v, 1))
# here var_mul is like [(x,), (y, )]
for v1 in var_mul:
try:
coeff = c[v1[0]]
except KeyError:
coeff = 0
x_coeff = bool(x_coeff) and bool(coeff)
if not any((xy_coeff, x_coeff)):
# means only x**2, y**2 and const is present
# so we can get more soln by permuting this soln.
do_permute_signs = True
elif not x_coeff:
# when coeff(x), coeff(y) is not present then signs of
# x, y can be permuted such that their sign are same
# as sign of x*y.
# e.g 1. (x_val,y_val)=> (x_val,y_val), (-x_val,-y_val)
# 2. (-x_vall, y_val)=> (-x_val,y_val), (x_val,-y_val)
permute_few_signs = True
if t == 'general_sum_of_squares':
# trying to factor such expressions will sometimes hang
terms = [(eq, 1)]
else:
raise TypeError
except (TypeError, NotImplementedError):
fl = factor_list(eq)
if fl[0].is_Rational and fl[0] != 1:
return diophantine(eq/fl[0], param=param, syms=syms, permute=permute)
terms = fl[1]
sols = set()
for term in terms:
base, _ = term
var_t, _, eq_type = classify_diop(base, _dict=False)
_, base = signsimp(base, evaluate=False).as_coeff_Mul()
solution = diop_solve(base, param)
if eq_type in [
Linear.name,
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name,
GeneralPythagorean.name]:
sols.add(merge_solution(var, var_t, solution))
elif eq_type in [
BinaryQuadratic.name,
GeneralSumOfSquares.name,
GeneralSumOfEvenPowers.name,
Univariate.name]:
for sol in solution:
sols.add(merge_solution(var, var_t, sol))
else:
raise NotImplementedError('unhandled type: %s' % eq_type)
# remove null merge results
if () in sols:
sols.remove(())
null = tuple([0]*len(var))
# if there is no solution, return trivial solution
if not sols and eq.subs(zip(var, null)).is_zero:
sols.add(null)
final_soln = set()
for sol in sols:
if all(_is_int(s) for s in sol):
if do_permute_signs:
permuted_sign = set(permute_signs(sol))
final_soln.update(permuted_sign)
elif permute_few_signs:
lst = list(permute_signs(sol))
lst = list(filter(lambda x: x[0]*x[1] == sol[1]*sol[0], lst))
permuted_sign = set(lst)
final_soln.update(permuted_sign)
elif do_permute_signs_var:
permuted_sign_var = set(signed_permutations(sol))
final_soln.update(permuted_sign_var)
else:
final_soln.add(sol)
else:
final_soln.add(sol)
return final_soln
def merge_solution(var, var_t, solution):
"""
This is used to construct the full solution from the solutions of sub
equations.
Explanation
===========
For example when solving the equation `(x - y)(x^2 + y^2 - z^2) = 0`,
solutions for each of the equations `x - y = 0` and `x^2 + y^2 - z^2` are
found independently. Solutions for `x - y = 0` are `(x, y) = (t, t)`. But
we should introduce a value for z when we output the solution for the
original equation. This function converts `(t, t)` into `(t, t, n_{1})`
where `n_{1}` is an integer parameter.
"""
sol = []
if None in solution:
return ()
solution = iter(solution)
params = numbered_symbols("n", integer=True, start=1)
for v in var:
if v in var_t:
sol.append(next(solution))
else:
sol.append(next(params))
for val, symb in zip(sol, var):
if check_assumptions(val, **symb.assumptions0) is False:
return tuple()
return tuple(sol)
def _diop_solve(eq, params=None):
for diop_type in all_diop_classes:
if diop_type(eq).matches():
return diop_type(eq).solve(parameters=params)
def diop_solve(eq, param=symbols("t", integer=True)):
"""
Solves the diophantine equation ``eq``.
Explanation
===========
Unlike ``diophantine()``, factoring of ``eq`` is not attempted. Uses
``classify_diop()`` to determine the type of the equation and calls
the appropriate solver function.
Use of ``diophantine()`` is recommended over other helper functions.
``diop_solve()`` can return either a set or a tuple depending on the
nature of the equation.
Usage
=====
``diop_solve(eq, t)``: Solve diophantine equation, ``eq`` using ``t``
as a parameter if needed.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``t`` is a parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine import diop_solve
>>> from sympy.abc import x, y, z, w
>>> diop_solve(2*x + 3*y - 5)
(3*t_0 - 5, 5 - 2*t_0)
>>> diop_solve(4*x + 3*y - 4*z + 5)
(t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5)
>>> diop_solve(x + 3*y - 4*z + w - 6)
(t_0, t_0 + t_1, 6*t_0 + 5*t_1 + 4*t_2 - 6, 5*t_0 + 4*t_1 + 3*t_2 - 6)
>>> diop_solve(x**2 + y**2 - 5)
{(-2, -1), (-2, 1), (-1, -2), (-1, 2), (1, -2), (1, 2), (2, -1), (2, 1)}
See Also
========
diophantine()
"""
var, coeff, eq_type = classify_diop(eq, _dict=False)
if eq_type == Linear.name:
return diop_linear(eq, param)
elif eq_type == BinaryQuadratic.name:
return diop_quadratic(eq, param)
elif eq_type == HomogeneousTernaryQuadratic.name:
return diop_ternary_quadratic(eq, parameterize=True)
elif eq_type == HomogeneousTernaryQuadraticNormal.name:
return diop_ternary_quadratic_normal(eq, parameterize=True)
elif eq_type == GeneralPythagorean.name:
return diop_general_pythagorean(eq, param)
elif eq_type == Univariate.name:
return diop_univariate(eq)
elif eq_type == GeneralSumOfSquares.name:
return diop_general_sum_of_squares(eq, limit=S.Infinity)
elif eq_type == GeneralSumOfEvenPowers.name:
return diop_general_sum_of_even_powers(eq, limit=S.Infinity)
if eq_type is not None and eq_type not in diop_known:
raise ValueError(filldedent('''
Alhough this type of equation was identified, it is not yet
handled. It should, however, be listed in `diop_known` at the
top of this file. Developers should see comments at the end of
`classify_diop`.
''')) # pragma: no cover
else:
raise NotImplementedError(
'No solver has been written for %s.' % eq_type)
def classify_diop(eq, _dict=True):
# docstring supplied externally
matched = False
diop_type = None
for diop_class in all_diop_classes:
diop_type = diop_class(eq)
if diop_type.matches():
matched = True
break
if matched:
return diop_type.free_symbols, dict(diop_type.coeff) if _dict else diop_type.coeff, diop_type.name
# new diop type instructions
# --------------------------
# if this error raises and the equation *can* be classified,
# * it should be identified in the if-block above
# * the type should be added to the diop_known
# if a solver can be written for it,
# * a dedicated handler should be written (e.g. diop_linear)
# * it should be passed to that handler in diop_solve
raise NotImplementedError(filldedent('''
This equation is not yet recognized or else has not been
simplified sufficiently to put it in a form recognized by
diop_classify().'''))
classify_diop.func_doc = ( # type: ignore
'''
Helper routine used by diop_solve() to find information about ``eq``.
Explanation
===========
Returns a tuple containing the type of the diophantine equation
along with the variables (free symbols) and their coefficients.
Variables are returned as a list and coefficients are returned
as a dict with the key being the respective term and the constant
term is keyed to 1. The type is one of the following:
* %s
Usage
=====
``classify_diop(eq)``: Return variables, coefficients and type of the
``eq``.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``_dict`` is for internal use: when True (default) a dict is returned,
otherwise a defaultdict which supplies 0 for missing keys is returned.
Examples
========
>>> from sympy.solvers.diophantine import classify_diop
>>> from sympy.abc import x, y, z, w, t
>>> classify_diop(4*x + 6*y - 4)
([x, y], {1: -4, x: 4, y: 6}, 'linear')
>>> classify_diop(x + 3*y -4*z + 5)
([x, y, z], {1: 5, x: 1, y: 3, z: -4}, 'linear')
>>> classify_diop(x**2 + y**2 - x*y + x + 5)
([x, y], {1: 5, x: 1, x**2: 1, y**2: 1, x*y: -1}, 'binary_quadratic')
''' % ('\n * '.join(sorted(diop_known))))
def diop_linear(eq, param=symbols("t", integer=True)):
"""
Solves linear diophantine equations.
A linear diophantine equation is an equation of the form `a_{1}x_{1} +
a_{2}x_{2} + .. + a_{n}x_{n} = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x_{1}, x_{2}, ..x_{n}` are integer variables.
Usage
=====
``diop_linear(eq)``: Returns a tuple containing solutions to the
diophantine equation ``eq``. Values in the tuple is arranged in the same
order as the sorted variables.
Details
=======
``eq`` is a linear diophantine equation which is assumed to be zero.
``param`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_linear
>>> from sympy.abc import x, y, z
>>> diop_linear(2*x - 3*y - 5) # solves equation 2*x - 3*y - 5 == 0
(3*t_0 - 5, 2*t_0 - 5)
Here x = -3*t_0 - 5 and y = -2*t_0 - 5
>>> diop_linear(2*x - 3*y - 4*z -3)
(t_0, 2*t_0 + 4*t_1 + 3, -t_0 - 3*t_1 - 3)
See Also
========
diop_quadratic(), diop_ternary_quadratic(), diop_general_pythagorean(),
diop_general_sum_of_squares()
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == Linear.name:
parameters = None
if param is not None:
parameters = symbols('%s_0:%i' % (param, len(var)), integer=True)
result = Linear(eq).solve(parameters=parameters)
if param is None:
result = result(*[0]*len(result.parameters))
if len(result) > 0:
return list(result)[0]
else:
return tuple([None]*len(result.parameters))
def base_solution_linear(c, a, b, t=None):
"""
Return the base solution for the linear equation, `ax + by = c`.
Explanation
===========
Used by ``diop_linear()`` to find the base solution of a linear
Diophantine equation. If ``t`` is given then the parametrized solution is
returned.
Usage
=====
``base_solution_linear(c, a, b, t)``: ``a``, ``b``, ``c`` are coefficients
in `ax + by = c` and ``t`` is the parameter to be used in the solution.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import base_solution_linear
>>> from sympy.abc import t
>>> base_solution_linear(5, 2, 3) # equation 2*x + 3*y = 5
(-5, 5)
>>> base_solution_linear(0, 5, 7) # equation 5*x + 7*y = 0
(0, 0)
>>> base_solution_linear(5, 2, 3, t) # equation 2*x + 3*y = 5
(3*t - 5, 5 - 2*t)
>>> base_solution_linear(0, 5, 7, t) # equation 5*x + 7*y = 0
(7*t, -5*t)
"""
a, b, c = _remove_gcd(a, b, c)
if c == 0:
if t is not None:
if b < 0:
t = -t
return (b*t, -a*t)
else:
return (0, 0)
else:
x0, y0, d = igcdex(abs(a), abs(b))
x0 *= sign(a)
y0 *= sign(b)
if divisible(c, d):
if t is not None:
if b < 0:
t = -t
return (c*x0 + b*t, c*y0 - a*t)
else:
return (c*x0, c*y0)
else:
return (None, None)
def diop_univariate(eq):
"""
Solves a univariate diophantine equations.
Explanation
===========
A univariate diophantine equation is an equation of the form
`a_{0} + a_{1}x + a_{2}x^2 + .. + a_{n}x^n = 0` where `a_{1}, a_{2}, ..a_{n}` are
integer constants and `x` is an integer variable.
Usage
=====
``diop_univariate(eq)``: Returns a set containing solutions to the
diophantine equation ``eq``.
Details
=======
``eq`` is a univariate diophantine equation which is assumed to be zero.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_univariate
>>> from sympy.abc import x
>>> diop_univariate((x - 2)*(x - 3)**2) # solves equation (x - 2)*(x - 3)**2 == 0
{(2,), (3,)}
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == Univariate.name:
return {(int(i),) for i in solveset_real(
eq, var[0]).intersect(S.Integers)}
def divisible(a, b):
"""
Returns `True` if ``a`` is divisible by ``b`` and `False` otherwise.
"""
return not a % b
def diop_quadratic(eq, param=symbols("t", integer=True)):
"""
Solves quadratic diophantine equations.
i.e. equations of the form `Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`. Returns a
set containing the tuples `(x, y)` which contains the solutions. If there
are no solutions then `(None, None)` is returned.
Usage
=====
``diop_quadratic(eq, param)``: ``eq`` is a quadratic binary diophantine
equation. ``param`` is used to indicate the parameter to be used in the
solution.
Details
=======
``eq`` should be an expression which is assumed to be zero.
``param`` is a parameter to be used in the solution.
Examples
========
>>> from sympy.abc import x, y, t
>>> from sympy.solvers.diophantine.diophantine import diop_quadratic
>>> diop_quadratic(x**2 + y**2 + 2*x + 2*y + 2, t)
{(-1, -1)}
References
==========
.. [1] Methods to solve Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, [online],
Available: http://www.alpertron.com.ar/METHODS.HTM
.. [2] Solving the equation ax^2+ bxy + cy^2 + dx + ey + f= 0, [online],
Available: https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
See Also
========
diop_linear(), diop_ternary_quadratic(), diop_general_sum_of_squares(),
diop_general_pythagorean()
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
if param is not None:
parameters = [param, Symbol("u", integer=True)]
else:
parameters = None
return set(BinaryQuadratic(eq).solve(parameters=parameters))
def is_solution_quad(var, coeff, u, v):
"""
Check whether `(u, v)` is solution to the quadratic binary diophantine
equation with the variable list ``var`` and coefficient dictionary
``coeff``.
Not intended for use by normal users.
"""
reps = dict(zip(var, (u, v)))
eq = Add(*[j*i.xreplace(reps) for i, j in coeff.items()])
return _mexpand(eq) == 0
def diop_DN(D, N, t=symbols("t", integer=True)):
"""
Solves the equation `x^2 - Dy^2 = N`.
Explanation
===========
Mainly concerned with the case `D > 0, D` is not a perfect square,
which is the same as the generalized Pell equation. The LMM
algorithm [1]_ is used to solve this equation.
Returns one solution tuple, (`x, y)` for each class of the solutions.
Other solutions of the class can be constructed according to the
values of ``D`` and ``N``.
Usage
=====
``diop_DN(D, N, t)``: D and N are integers as in `x^2 - Dy^2 = N` and
``t`` is the parameter to be used in the solutions.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
``t`` is the parameter to be used in the solutions.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_DN
>>> diop_DN(13, -4) # Solves equation x**2 - 13*y**2 = -4
[(3, 1), (393, 109), (36, 10)]
The output can be interpreted as follows: There are three fundamental
solutions to the equation `x^2 - 13y^2 = -4` given by (3, 1), (393, 109)
and (36, 10). Each tuple is in the form (x, y), i.e. solution (3, 1) means
that `x = 3` and `y = 1`.
>>> diop_DN(986, 1) # Solves equation x**2 - 986*y**2 = 1
[(49299, 1570)]
See Also
========
find_DN(), diop_bf_DN()
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Pages 16 - 17. [online], Available:
https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
if D < 0:
if N == 0:
return [(0, 0)]
elif N < 0:
return []
elif N > 0:
sol = []
for d in divisors(square_factor(N)):
sols = cornacchia(1, -D, N // d**2)
if sols:
for x, y in sols:
sol.append((d*x, d*y))
if D == -1:
sol.append((d*y, d*x))
return sol
elif D == 0:
if N < 0:
return []
if N == 0:
return [(0, t)]
sN, _exact = integer_nthroot(N, 2)
if _exact:
return [(sN, t)]
else:
return []
else: # D > 0
sD, _exact = integer_nthroot(D, 2)
if _exact:
if N == 0:
return [(sD*t, t)]
else:
sol = []
for y in range(floor(sign(N)*(N - 1)/(2*sD)) + 1):
try:
sq, _exact = integer_nthroot(D*y**2 + N, 2)
except ValueError:
_exact = False
if _exact:
sol.append((sq, y))
return sol
elif 1 < N**2 < D:
# It is much faster to call `_special_diop_DN`.
return _special_diop_DN(D, N)
else:
if N == 0:
return [(0, 0)]
elif abs(N) == 1:
pqa = PQa(0, 1, D)
j = 0
G = []
B = []
for i in pqa:
a = i[2]
G.append(i[5])
B.append(i[4])
if j != 0 and a == 2*sD:
break
j = j + 1
if _odd(j):
if N == -1:
x = G[j - 1]
y = B[j - 1]
else:
count = j
while count < 2*j - 1:
i = next(pqa)
G.append(i[5])
B.append(i[4])
count += 1
x = G[count]
y = B[count]
else:
if N == 1:
x = G[j - 1]
y = B[j - 1]
else:
return []
return [(x, y)]
else:
fs = []
sol = []
div = divisors(N)
for d in div:
if divisible(N, d**2):
fs.append(d)
for f in fs:
m = N // f**2
zs = sqrt_mod(D, abs(m), all_roots=True)
zs = [i for i in zs if i <= abs(m) // 2 ]
if abs(m) != 2:
zs = zs + [-i for i in zs if i] # omit dupl 0
for z in zs:
pqa = PQa(z, abs(m), D)
j = 0
G = []
B = []
for i in pqa:
G.append(i[5])
B.append(i[4])
if j != 0 and abs(i[1]) == 1:
r = G[j-1]
s = B[j-1]
if r**2 - D*s**2 == m:
sol.append((f*r, f*s))
elif diop_DN(D, -1) != []:
a = diop_DN(D, -1)
sol.append((f*(r*a[0][0] + a[0][1]*s*D), f*(r*a[0][1] + s*a[0][0])))
break
j = j + 1
if j == length(z, abs(m), D):
break
return sol
def _special_diop_DN(D, N):
"""
Solves the equation `x^2 - Dy^2 = N` for the special case where
`1 < N**2 < D` and `D` is not a perfect square.
It is better to call `diop_DN` rather than this function, as
the former checks the condition `1 < N**2 < D`, and calls the latter only
if appropriate.
Usage
=====
WARNING: Internal method. Do not call directly!
``_special_diop_DN(D, N)``: D and N are integers as in `x^2 - Dy^2 = N`.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import _special_diop_DN
>>> _special_diop_DN(13, -3) # Solves equation x**2 - 13*y**2 = -3
[(7, 2), (137, 38)]
The output can be interpreted as follows: There are two fundamental
solutions to the equation `x^2 - 13y^2 = -3` given by (7, 2) and
(137, 38). Each tuple is in the form (x, y), i.e. solution (7, 2) means
that `x = 7` and `y = 2`.
>>> _special_diop_DN(2445, -20) # Solves equation x**2 - 2445*y**2 = -20
[(445, 9), (17625560, 356454), (698095554475, 14118073569)]
See Also
========
diop_DN()
References
==========
.. [1] Section 4.4.4 of the following book:
Quadratic Diophantine Equations, T. Andreescu and D. Andrica,
Springer, 2015.
"""
# The following assertion was removed for efficiency, with the understanding
# that this method is not called directly. The parent method, `diop_DN`
# is responsible for performing the appropriate checks.
#
# assert (1 < N**2 < D) and (not integer_nthroot(D, 2)[1])
sqrt_D = sqrt(D)
F = [(N, 1)]
f = 2
while True:
f2 = f**2
if f2 > abs(N):
break
n, r = divmod(N, f2)
if r == 0:
F.append((n, f))
f += 1
P = 0
Q = 1
G0, G1 = 0, 1
B0, B1 = 1, 0
solutions = []
i = 0
while True:
a = floor((P + sqrt_D) / Q)
P = a*Q - P
Q = (D - P**2) // Q
G2 = a*G1 + G0
B2 = a*B1 + B0
for n, f in F:
if G2**2 - D*B2**2 == n:
solutions.append((f*G2, f*B2))
i += 1
if Q == 1 and i % 2 == 0:
break
G0, G1 = G1, G2
B0, B1 = B1, B2
return solutions
def cornacchia(a, b, m):
r"""
Solves `ax^2 + by^2 = m` where `\gcd(a, b) = 1 = gcd(a, m)` and `a, b > 0`.
Explanation
===========
Uses the algorithm due to Cornacchia. The method only finds primitive
solutions, i.e. ones with `\gcd(x, y) = 1`. So this method cannot be used to
find the solutions of `x^2 + y^2 = 20` since the only solution to former is
`(x, y) = (4, 2)` and it is not primitive. When `a = b`, only the
solutions with `x \leq y` are found. For more details, see the References.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import cornacchia
>>> cornacchia(2, 3, 35) # equation 2x**2 + 3y**2 = 35
{(2, 3), (4, 1)}
>>> cornacchia(1, 1, 25) # equation x**2 + y**2 = 25
{(4, 3)}
References
===========
.. [1] A. Nitaj, "L'algorithme de Cornacchia"
.. [2] Solving the diophantine equation ax**2 + by**2 = m by Cornacchia's
method, [online], Available:
http://www.numbertheory.org/php/cornacchia.html
See Also
========
sympy.utilities.iterables.signed_permutations
"""
sols = set()
a1 = igcdex(a, m)[0]
v = sqrt_mod(-b*a1, m, all_roots=True)
if not v:
return None
for t in v:
if t < m // 2:
continue
u, r = t, m
while True:
u, r = r, u % r
if a*r**2 < m:
break
m1 = m - a*r**2
if m1 % b == 0:
m1 = m1 // b
s, _exact = integer_nthroot(m1, 2)
if _exact:
if a == b and r < s:
r, s = s, r
sols.add((int(r), int(s)))
return sols
def PQa(P_0, Q_0, D):
r"""
Returns useful information needed to solve the Pell equation.
Explanation
===========
There are six sequences of integers defined related to the continued
fraction representation of `\\frac{P + \sqrt{D}}{Q}`, namely {`P_{i}`},
{`Q_{i}`}, {`a_{i}`},{`A_{i}`}, {`B_{i}`}, {`G_{i}`}. ``PQa()`` Returns
these values as a 6-tuple in the same order as mentioned above. Refer [1]_
for more detailed information.
Usage
=====
``PQa(P_0, Q_0, D)``: ``P_0``, ``Q_0`` and ``D`` are integers corresponding
to `P_{0}`, `Q_{0}` and `D` in the continued fraction
`\\frac{P_{0} + \sqrt{D}}{Q_{0}}`.
Also it's assumed that `P_{0}^2 == D mod(|Q_{0}|)` and `D` is square free.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import PQa
>>> pqa = PQa(13, 4, 5) # (13 + sqrt(5))/4
>>> next(pqa) # (P_0, Q_0, a_0, A_0, B_0, G_0)
(13, 4, 3, 3, 1, -1)
>>> next(pqa) # (P_1, Q_1, a_1, A_1, B_1, G_1)
(-1, 1, 1, 4, 1, 3)
References
==========
.. [1] Solving the generalized Pell equation x^2 - Dy^2 = N, John P.
Robertson, July 31, 2004, Pages 4 - 8. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
A_i_2 = B_i_1 = 0
A_i_1 = B_i_2 = 1
G_i_2 = -P_0
G_i_1 = Q_0
P_i = P_0
Q_i = Q_0
while True:
a_i = floor((P_i + sqrt(D))/Q_i)
A_i = a_i*A_i_1 + A_i_2
B_i = a_i*B_i_1 + B_i_2
G_i = a_i*G_i_1 + G_i_2
yield P_i, Q_i, a_i, A_i, B_i, G_i
A_i_1, A_i_2 = A_i, A_i_1
B_i_1, B_i_2 = B_i, B_i_1
G_i_1, G_i_2 = G_i, G_i_1
P_i = a_i*Q_i - P_i
Q_i = (D - P_i**2)/Q_i
def diop_bf_DN(D, N, t=symbols("t", integer=True)):
r"""
Uses brute force to solve the equation, `x^2 - Dy^2 = N`.
Explanation
===========
Mainly concerned with the generalized Pell equation which is the case when
`D > 0, D` is not a perfect square. For more information on the case refer
[1]_. Let `(t, u)` be the minimal positive solution of the equation
`x^2 - Dy^2 = 1`. Then this method requires
`\sqrt{\\frac{\mid N \mid (t \pm 1)}{2D}}` to be small.
Usage
=====
``diop_bf_DN(D, N, t)``: ``D`` and ``N`` are coefficients in
`x^2 - Dy^2 = N` and ``t`` is the parameter to be used in the solutions.
Details
=======
``D`` and ``N`` correspond to D and N in the equation.
``t`` is the parameter to be used in the solutions.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_bf_DN
>>> diop_bf_DN(13, -4)
[(3, 1), (-3, 1), (36, 10)]
>>> diop_bf_DN(986, 1)
[(49299, 1570)]
See Also
========
diop_DN()
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Page 15. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
D = as_int(D)
N = as_int(N)
sol = []
a = diop_DN(D, 1)
u = a[0][0]
if abs(N) == 1:
return diop_DN(D, N)
elif N > 1:
L1 = 0
L2 = integer_nthroot(int(N*(u - 1)/(2*D)), 2)[0] + 1
elif N < -1:
L1, _exact = integer_nthroot(-int(N/D), 2)
if not _exact:
L1 += 1
L2 = integer_nthroot(-int(N*(u + 1)/(2*D)), 2)[0] + 1
else: # N = 0
if D < 0:
return [(0, 0)]
elif D == 0:
return [(0, t)]
else:
sD, _exact = integer_nthroot(D, 2)
if _exact:
return [(sD*t, t), (-sD*t, t)]
else:
return [(0, 0)]
for y in range(L1, L2):
try:
x, _exact = integer_nthroot(N + D*y**2, 2)
except ValueError:
_exact = False
if _exact:
sol.append((x, y))
if not equivalent(x, y, -x, y, D, N):
sol.append((-x, y))
return sol
def equivalent(u, v, r, s, D, N):
"""
Returns True if two solutions `(u, v)` and `(r, s)` of `x^2 - Dy^2 = N`
belongs to the same equivalence class and False otherwise.
Explanation
===========
Two solutions `(u, v)` and `(r, s)` to the above equation fall to the same
equivalence class iff both `(ur - Dvs)` and `(us - vr)` are divisible by
`N`. See reference [1]_. No test is performed to test whether `(u, v)` and
`(r, s)` are actually solutions to the equation. User should take care of
this.
Usage
=====
``equivalent(u, v, r, s, D, N)``: `(u, v)` and `(r, s)` are two solutions
of the equation `x^2 - Dy^2 = N` and all parameters involved are integers.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import equivalent
>>> equivalent(18, 5, -18, -5, 13, -1)
True
>>> equivalent(3, 1, -18, 393, 109, -4)
False
References
==========
.. [1] Solving the generalized Pell equation x**2 - D*y**2 = N, John P.
Robertson, July 31, 2004, Page 12. https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
"""
return divisible(u*r - D*v*s, N) and divisible(u*s - v*r, N)
def length(P, Q, D):
r"""
Returns the (length of aperiodic part + length of periodic part) of
continued fraction representation of `\\frac{P + \sqrt{D}}{Q}`.
It is important to remember that this does NOT return the length of the
periodic part but the sum of the lengths of the two parts as mentioned
above.
Usage
=====
``length(P, Q, D)``: ``P``, ``Q`` and ``D`` are integers corresponding to
the continued fraction `\\frac{P + \sqrt{D}}{Q}`.
Details
=======
``P``, ``D`` and ``Q`` corresponds to P, D and Q in the continued fraction,
`\\frac{P + \sqrt{D}}{Q}`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import length
>>> length(-2, 4, 5) # (-2 + sqrt(5))/4
3
>>> length(-5, 4, 17) # (-5 + sqrt(17))/4
4
See Also
========
sympy.ntheory.continued_fraction.continued_fraction_periodic
"""
from sympy.ntheory.continued_fraction import continued_fraction_periodic
v = continued_fraction_periodic(P, Q, D)
if isinstance(v[-1], list):
rpt = len(v[-1])
nonrpt = len(v) - 1
else:
rpt = 0
nonrpt = len(v)
return rpt + nonrpt
def transformation_to_DN(eq):
"""
This function transforms general quadratic,
`ax^2 + bxy + cy^2 + dx + ey + f = 0`
to more easy to deal with `X^2 - DY^2 = N` form.
Explanation
===========
This is used to solve the general quadratic equation by transforming it to
the latter form. Refer to [1]_ for more detailed information on the
transformation. This function returns a tuple (A, B) where A is a 2 X 2
matrix and B is a 2 X 1 matrix such that,
Transpose([x y]) = A * Transpose([X Y]) + B
Usage
=====
``transformation_to_DN(eq)``: where ``eq`` is the quadratic to be
transformed.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import transformation_to_DN
>>> A, B = transformation_to_DN(x**2 - 3*x*y - y**2 - 2*y + 1)
>>> A
Matrix([
[1/26, 3/26],
[ 0, 1/13]])
>>> B
Matrix([
[-6/13],
[-4/13]])
A, B returned are such that Transpose((x y)) = A * Transpose((X Y)) + B.
Substituting these values for `x` and `y` and a bit of simplifying work
will give an equation of the form `x^2 - Dy^2 = N`.
>>> from sympy.abc import X, Y
>>> from sympy import Matrix, simplify
>>> u = (A*Matrix([X, Y]) + B)[0] # Transformation for x
>>> u
X/26 + 3*Y/26 - 6/13
>>> v = (A*Matrix([X, Y]) + B)[1] # Transformation for y
>>> v
Y/13 - 4/13
Next we will substitute these formulas for `x` and `y` and do
``simplify()``.
>>> eq = simplify((x**2 - 3*x*y - y**2 - 2*y + 1).subs(zip((x, y), (u, v))))
>>> eq
X**2/676 - Y**2/52 + 17/13
By multiplying the denominator appropriately, we can get a Pell equation
in the standard form.
>>> eq * 676
X**2 - 13*Y**2 + 884
If only the final equation is needed, ``find_DN()`` can be used.
See Also
========
find_DN()
References
==========
.. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0,
John P.Robertson, May 8, 2003, Page 7 - 11.
https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
return _transformation_to_DN(var, coeff)
def _transformation_to_DN(var, coeff):
x, y = var
a = coeff[x**2]
b = coeff[x*y]
c = coeff[y**2]
d = coeff[x]
e = coeff[y]
f = coeff[1]
a, b, c, d, e, f = [as_int(i) for i in _remove_gcd(a, b, c, d, e, f)]
X, Y = symbols("X, Y", integer=True)
if b:
B, C = _rational_pq(2*a, b)
A, T = _rational_pq(a, B**2)
# eq_1 = A*B*X**2 + B*(c*T - A*C**2)*Y**2 + d*T*X + (B*e*T - d*T*C)*Y + f*T*B
coeff = {X**2: A*B, X*Y: 0, Y**2: B*(c*T - A*C**2), X: d*T, Y: B*e*T - d*T*C, 1: f*T*B}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*A_0, Matrix(2, 2, [S.One/B, -S(C)/B, 0, 1])*B_0
else:
if d:
B, C = _rational_pq(2*a, d)
A, T = _rational_pq(a, B**2)
# eq_2 = A*X**2 + c*T*Y**2 + e*T*Y + f*T - A*C**2
coeff = {X**2: A, X*Y: 0, Y**2: c*T, X: 0, Y: e*T, 1: f*T - A*C**2}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [S.One/B, 0, 0, 1])*A_0, Matrix(2, 2, [S.One/B, 0, 0, 1])*B_0 + Matrix([-S(C)/B, 0])
else:
if e:
B, C = _rational_pq(2*c, e)
A, T = _rational_pq(c, B**2)
# eq_3 = a*T*X**2 + A*Y**2 + f*T - A*C**2
coeff = {X**2: a*T, X*Y: 0, Y**2: A, X: 0, Y: 0, 1: f*T - A*C**2}
A_0, B_0 = _transformation_to_DN([X, Y], coeff)
return Matrix(2, 2, [1, 0, 0, S.One/B])*A_0, Matrix(2, 2, [1, 0, 0, S.One/B])*B_0 + Matrix([0, -S(C)/B])
else:
# TODO: pre-simplification: Not necessary but may simplify
# the equation.
return Matrix(2, 2, [S.One/a, 0, 0, 1]), Matrix([0, 0])
def find_DN(eq):
"""
This function returns a tuple, `(D, N)` of the simplified form,
`x^2 - Dy^2 = N`, corresponding to the general quadratic,
`ax^2 + bxy + cy^2 + dx + ey + f = 0`.
Solving the general quadratic is then equivalent to solving the equation
`X^2 - DY^2 = N` and transforming the solutions by using the transformation
matrices returned by ``transformation_to_DN()``.
Usage
=====
``find_DN(eq)``: where ``eq`` is the quadratic to be transformed.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy.solvers.diophantine.diophantine import find_DN
>>> find_DN(x**2 - 3*x*y - y**2 - 2*y + 1)
(13, -884)
Interpretation of the output is that we get `X^2 -13Y^2 = -884` after
transforming `x^2 - 3xy - y^2 - 2y + 1` using the transformation returned
by ``transformation_to_DN()``.
See Also
========
transformation_to_DN()
References
==========
.. [1] Solving the equation ax^2 + bxy + cy^2 + dx + ey + f = 0,
John P.Robertson, May 8, 2003, Page 7 - 11.
https://web.archive.org/web/20160323033111/http://www.jpr2718.org/ax2p.pdf
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == BinaryQuadratic.name:
return _find_DN(var, coeff)
def _find_DN(var, coeff):
x, y = var
X, Y = symbols("X, Y", integer=True)
A, B = _transformation_to_DN(var, coeff)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
eq = x**2*coeff[x**2] + x*y*coeff[x*y] + y**2*coeff[y**2] + x*coeff[x] + y*coeff[y] + coeff[1]
simplified = _mexpand(eq.subs(zip((x, y), (u, v))))
coeff = simplified.as_coefficients_dict()
return -coeff[Y**2]/coeff[X**2], -coeff[1]/coeff[X**2]
def check_param(x, y, a, params):
"""
If there is a number modulo ``a`` such that ``x`` and ``y`` are both
integers, then return a parametric representation for ``x`` and ``y``
else return (None, None).
Here ``x`` and ``y`` are functions of ``t``.
"""
from sympy.simplify.simplify import clear_coefficients
if x.is_number and not x.is_Integer:
return DiophantineSolutionSet([x, y], parameters=params)
if y.is_number and not y.is_Integer:
return DiophantineSolutionSet([x, y], parameters=params)
m, n = symbols("m, n", integer=True)
c, p = (m*x + n*y).as_content_primitive()
if a % c.q:
return DiophantineSolutionSet([x, y], parameters=params)
# clear_coefficients(mx + b, R)[1] -> (R - b)/m
eq = clear_coefficients(x, m)[1] - clear_coefficients(y, n)[1]
junk, eq = eq.as_content_primitive()
return _diop_solve(eq, params=params)
def diop_ternary_quadratic(eq, parameterize=False):
"""
Solves the general quadratic ternary form,
`ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`.
Returns a tuple `(x, y, z)` which is a base solution for the above
equation. If there are no solutions, `(None, None, None)` is returned.
Usage
=====
``diop_ternary_quadratic(eq)``: Return a tuple containing a basic solution
to ``eq``.
Details
=======
``eq`` should be an homogeneous expression of degree two in three variables
and it is assumed to be zero.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic
>>> diop_ternary_quadratic(x**2 + 3*y**2 - z**2)
(1, 0, 1)
>>> diop_ternary_quadratic(4*x**2 + 5*y**2 - z**2)
(1, 0, 2)
>>> diop_ternary_quadratic(45*x**2 - 7*y**2 - 8*x*y - z**2)
(28, 45, 105)
>>> diop_ternary_quadratic(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y)
(9, 1, 5)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
HomogeneousTernaryQuadratic.name,
HomogeneousTernaryQuadraticNormal.name):
sol = _diop_ternary_quadratic(var, coeff)
if len(sol) > 0:
x_0, y_0, z_0 = list(sol)[0]
else:
x_0, y_0, z_0 = None, None, None
if parameterize:
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
return x_0, y_0, z_0
def _diop_ternary_quadratic(_var, coeff):
eq = sum([i*coeff[i] for i in coeff])
if HomogeneousTernaryQuadratic(eq).matches():
return HomogeneousTernaryQuadratic(eq, free_symbols=_var).solve()
elif HomogeneousTernaryQuadraticNormal(eq).matches():
return HomogeneousTernaryQuadraticNormal(eq, free_symbols=_var).solve()
def transformation_to_normal(eq):
"""
Returns the transformation Matrix that converts a general ternary
quadratic equation ``eq`` (`ax^2 + by^2 + cz^2 + dxy + eyz + fxz`)
to a form without cross terms: `ax^2 + by^2 + cz^2 = 0`. This is
not used in solving ternary quadratics; it is only implemented for
the sake of completeness.
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
"homogeneous_ternary_quadratic",
"homogeneous_ternary_quadratic_normal"):
return _transformation_to_normal(var, coeff)
def _transformation_to_normal(var, coeff):
_var = list(var) # copy
x, y, z = var
if not any(coeff[i**2] for i in var):
# https://math.stackexchange.com/questions/448051/transform-quadratic-ternary-form-to-normal-form/448065#448065
a = coeff[x*y]
b = coeff[y*z]
c = coeff[x*z]
swap = False
if not a: # b can't be 0 or else there aren't 3 vars
swap = True
a, b = b, a
T = Matrix(((1, 1, -b/a), (1, -1, -c/a), (0, 0, 1)))
if swap:
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
if coeff[x**2] == 0:
# If the coefficient of x is zero change the variables
if coeff[y**2] == 0:
_var[0], _var[2] = var[2], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 2)
T.col_swap(0, 2)
return T
else:
_var[0], _var[1] = var[1], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
# Apply the transformation x --> X - (B*Y + C*Z)/(2*A)
if coeff[x*y] != 0 or coeff[x*z] != 0:
A = coeff[x**2]
B = coeff[x*y]
C = coeff[x*z]
D = coeff[y**2]
E = coeff[y*z]
F = coeff[z**2]
_coeff = {}
_coeff[x**2] = 4*A**2
_coeff[y**2] = 4*A*D - B**2
_coeff[z**2] = 4*A*F - C**2
_coeff[y*z] = 4*A*E - 2*B*C
_coeff[x*y] = 0
_coeff[x*z] = 0
T_0 = _transformation_to_normal(_var, _coeff)
return Matrix(3, 3, [1, S(-B)/(2*A), S(-C)/(2*A), 0, 1, 0, 0, 0, 1])*T_0
elif coeff[y*z] != 0:
if coeff[y**2] == 0:
if coeff[z**2] == 0:
# Equations of the form A*x**2 + E*yz = 0.
# Apply transformation y -> Y + Z ans z -> Y - Z
return Matrix(3, 3, [1, 0, 0, 0, 1, 1, 0, 1, -1])
else:
# Ax**2 + E*y*z + F*z**2 = 0
_var[0], _var[2] = var[2], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 2)
T.col_swap(0, 2)
return T
else:
# A*x**2 + D*y**2 + E*y*z + F*z**2 = 0, F may be zero
_var[0], _var[1] = var[1], var[0]
T = _transformation_to_normal(_var, coeff)
T.row_swap(0, 1)
T.col_swap(0, 1)
return T
else:
return Matrix.eye(3)
def parametrize_ternary_quadratic(eq):
"""
Returns the parametrized general solution for the ternary quadratic
equation ``eq`` which has the form
`ax^2 + by^2 + cz^2 + fxy + gyz + hxz = 0`.
Examples
========
>>> from sympy import Tuple, ordered
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import parametrize_ternary_quadratic
The parametrized solution may be returned with three parameters:
>>> parametrize_ternary_quadratic(2*x**2 + y**2 - 2*z**2)
(p**2 - 2*q**2, -2*p**2 + 4*p*q - 4*p*r - 4*q**2, p**2 - 4*p*q + 2*q**2 - 4*q*r)
There might also be only two parameters:
>>> parametrize_ternary_quadratic(4*x**2 + 2*y**2 - 3*z**2)
(2*p**2 - 3*q**2, -4*p**2 + 12*p*q - 6*q**2, 4*p**2 - 8*p*q + 6*q**2)
Notes
=====
Consider ``p`` and ``q`` in the previous 2-parameter
solution and observe that more than one solution can be represented
by a given pair of parameters. If `p` and ``q`` are not coprime, this is
trivially true since the common factor will also be a common factor of the
solution values. But it may also be true even when ``p`` and
``q`` are coprime:
>>> sol = Tuple(*_)
>>> p, q = ordered(sol.free_symbols)
>>> sol.subs([(p, 3), (q, 2)])
(6, 12, 12)
>>> sol.subs([(q, 1), (p, 1)])
(-1, 2, 2)
>>> sol.subs([(q, 0), (p, 1)])
(2, -4, 4)
>>> sol.subs([(q, 1), (p, 0)])
(-3, -6, 6)
Except for sign and a common factor, these are equivalent to
the solution of (1, 2, 2).
References
==========
.. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart,
London Mathematical Society Student Texts 41, Cambridge University
Press, Cambridge, 1998.
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type in (
"homogeneous_ternary_quadratic",
"homogeneous_ternary_quadratic_normal"):
x_0, y_0, z_0 = list(_diop_ternary_quadratic(var, coeff))[0]
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
def _parametrize_ternary_quadratic(solution, _var, coeff):
# called for a*x**2 + b*y**2 + c*z**2 + d*x*y + e*y*z + f*x*z = 0
assert 1 not in coeff
x_0, y_0, z_0 = solution
v = list(_var) # copy
if x_0 is None:
return (None, None, None)
if solution.count(0) >= 2:
# if there are 2 zeros the equation reduces
# to k*X**2 == 0 where X is x, y, or z so X must
# be zero, too. So there is only the trivial
# solution.
return (None, None, None)
if x_0 == 0:
v[0], v[1] = v[1], v[0]
y_p, x_p, z_p = _parametrize_ternary_quadratic(
(y_0, x_0, z_0), v, coeff)
return x_p, y_p, z_p
x, y, z = v
r, p, q = symbols("r, p, q", integer=True)
eq = sum(k*v for k, v in coeff.items())
eq_1 = _mexpand(eq.subs(zip(
(x, y, z), (r*x_0, r*y_0 + p, r*z_0 + q))))
A, B = eq_1.as_independent(r, as_Add=True)
x = A*x_0
y = (A*y_0 - _mexpand(B/r*p))
z = (A*z_0 - _mexpand(B/r*q))
return _remove_gcd(x, y, z)
def diop_ternary_quadratic_normal(eq, parameterize=False):
"""
Solves the quadratic ternary diophantine equation,
`ax^2 + by^2 + cz^2 = 0`.
Explanation
===========
Here the coefficients `a`, `b`, and `c` should be non zero. Otherwise the
equation will be a quadratic binary or univariate equation. If solvable,
returns a tuple `(x, y, z)` that satisfies the given equation. If the
equation does not have integer solutions, `(None, None, None)` is returned.
Usage
=====
``diop_ternary_quadratic_normal(eq)``: where ``eq`` is an equation of the form
`ax^2 + by^2 + cz^2 = 0`.
Examples
========
>>> from sympy.abc import x, y, z
>>> from sympy.solvers.diophantine.diophantine import diop_ternary_quadratic_normal
>>> diop_ternary_quadratic_normal(x**2 + 3*y**2 - z**2)
(1, 0, 1)
>>> diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2)
(1, 0, 2)
>>> diop_ternary_quadratic_normal(34*x**2 - 3*y**2 - 301*z**2)
(4, 9, 1)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == HomogeneousTernaryQuadraticNormal.name:
sol = _diop_ternary_quadratic_normal(var, coeff)
if len(sol) > 0:
x_0, y_0, z_0 = list(sol)[0]
else:
x_0, y_0, z_0 = None, None, None
if parameterize:
return _parametrize_ternary_quadratic(
(x_0, y_0, z_0), var, coeff)
return x_0, y_0, z_0
def _diop_ternary_quadratic_normal(var, coeff):
eq = sum([i * coeff[i] for i in coeff])
return HomogeneousTernaryQuadraticNormal(eq, free_symbols=var).solve()
def sqf_normal(a, b, c, steps=False):
"""
Return `a', b', c'`, the coefficients of the square-free normal
form of `ax^2 + by^2 + cz^2 = 0`, where `a', b', c'` are pairwise
prime. If `steps` is True then also return three tuples:
`sq`, `sqf`, and `(a', b', c')` where `sq` contains the square
factors of `a`, `b` and `c` after removing the `gcd(a, b, c)`;
`sqf` contains the values of `a`, `b` and `c` after removing
both the `gcd(a, b, c)` and the square factors.
The solutions for `ax^2 + by^2 + cz^2 = 0` can be
recovered from the solutions of `a'x^2 + b'y^2 + c'z^2 = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sqf_normal
>>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11)
(11, 1, 5)
>>> sqf_normal(2 * 3**2 * 5, 2 * 5 * 11, 2 * 7**2 * 11, True)
((3, 1, 7), (5, 55, 11), (11, 1, 5))
References
==========
.. [1] Legendre's Theorem, Legrange's Descent,
http://public.csusm.edu/aitken_html/notes/legendre.pdf
See Also
========
reconstruct()
"""
ABC = _remove_gcd(a, b, c)
sq = tuple(square_factor(i) for i in ABC)
sqf = A, B, C = tuple([i//j**2 for i,j in zip(ABC, sq)])
pc = igcd(A, B)
A /= pc
B /= pc
pa = igcd(B, C)
B /= pa
C /= pa
pb = igcd(A, C)
A /= pb
B /= pb
A *= pa
B *= pb
C *= pc
if steps:
return (sq, sqf, (A, B, C))
else:
return A, B, C
def square_factor(a):
r"""
Returns an integer `c` s.t. `a = c^2k, \ c,k \in Z`. Here `k` is square
free. `a` can be given as an integer or a dictionary of factors.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import square_factor
>>> square_factor(24)
2
>>> square_factor(-36*3)
6
>>> square_factor(1)
1
>>> square_factor({3: 2, 2: 1, -1: 1}) # -18
3
See Also
========
sympy.ntheory.factor_.core
"""
f = a if isinstance(a, dict) else factorint(a)
return Mul(*[p**(e//2) for p, e in f.items()])
def reconstruct(A, B, z):
"""
Reconstruct the `z` value of an equivalent solution of `ax^2 + by^2 + cz^2`
from the `z` value of a solution of the square-free normal form of the
equation, `a'*x^2 + b'*y^2 + c'*z^2`, where `a'`, `b'` and `c'` are square
free and `gcd(a', b', c') == 1`.
"""
f = factorint(igcd(A, B))
for p, e in f.items():
if e != 1:
raise ValueError('a and b should be square-free')
z *= p
return z
def ldescent(A, B):
"""
Return a non-trivial solution to `w^2 = Ax^2 + By^2` using
Lagrange's method; return None if there is no such solution.
.
Here, `A \\neq 0` and `B \\neq 0` and `A` and `B` are square free. Output a
tuple `(w_0, x_0, y_0)` which is a solution to the above equation.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import ldescent
>>> ldescent(1, 1) # w^2 = x^2 + y^2
(1, 1, 0)
>>> ldescent(4, -7) # w^2 = 4x^2 - 7y^2
(2, -1, 0)
This means that `x = -1, y = 0` and `w = 2` is a solution to the equation
`w^2 = 4x^2 - 7y^2`
>>> ldescent(5, -1) # w^2 = 5x^2 - y^2
(2, 1, -1)
References
==========
.. [1] The algorithmic resolution of Diophantine equations, Nigel P. Smart,
London Mathematical Society Student Texts 41, Cambridge University
Press, Cambridge, 1998.
.. [2] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
[online], Available:
http://eprints.nottingham.ac.uk/60/1/kvxefz87.pdf
"""
if abs(A) > abs(B):
w, y, x = ldescent(B, A)
return w, x, y
if A == 1:
return (1, 1, 0)
if B == 1:
return (1, 0, 1)
if B == -1: # and A == -1
return
r = sqrt_mod(A, B)
Q = (r**2 - A) // B
if Q == 0:
B_0 = 1
d = 0
else:
div = divisors(Q)
B_0 = None
for i in div:
sQ, _exact = integer_nthroot(abs(Q) // i, 2)
if _exact:
B_0, d = sign(Q)*i, sQ
break
if B_0 is not None:
W, X, Y = ldescent(A, B_0)
return _remove_gcd((-A*X + r*W), (r*X - W), Y*(B_0*d))
def descent(A, B):
"""
Returns a non-trivial solution, (x, y, z), to `x^2 = Ay^2 + Bz^2`
using Lagrange's descent method with lattice-reduction. `A` and `B`
are assumed to be valid for such a solution to exist.
This is faster than the normal Lagrange's descent algorithm because
the Gaussian reduction is used.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import descent
>>> descent(3, 1) # x**2 = 3*y**2 + z**2
(1, 0, 1)
`(x, y, z) = (1, 0, 1)` is a solution to the above equation.
>>> descent(41, -113)
(-16, -3, 1)
References
==========
.. [1] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
"""
if abs(A) > abs(B):
x, y, z = descent(B, A)
return x, z, y
if B == 1:
return (1, 0, 1)
if A == 1:
return (1, 1, 0)
if B == -A:
return (0, 1, 1)
if B == A:
x, z, y = descent(-1, A)
return (A*y, z, x)
w = sqrt_mod(A, B)
x_0, z_0 = gaussian_reduce(w, A, B)
t = (x_0**2 - A*z_0**2) // B
t_2 = square_factor(t)
t_1 = t // t_2**2
x_1, z_1, y_1 = descent(A, t_1)
return _remove_gcd(x_0*x_1 + A*z_0*z_1, z_0*x_1 + x_0*z_1, t_1*t_2*y_1)
def gaussian_reduce(w, a, b):
r"""
Returns a reduced solution `(x, z)` to the congruence
`X^2 - aZ^2 \equiv 0 \ (mod \ b)` so that `x^2 + |a|z^2` is minimal.
Details
=======
Here ``w`` is a solution of the congruence `x^2 \equiv a \ (mod \ b)`
References
==========
.. [1] Gaussian lattice Reduction [online]. Available:
http://home.ie.cuhk.edu.hk/~wkshum/wordpress/?p=404
.. [2] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
"""
u = (0, 1)
v = (1, 0)
if dot(u, v, w, a, b) < 0:
v = (-v[0], -v[1])
if norm(u, w, a, b) < norm(v, w, a, b):
u, v = v, u
while norm(u, w, a, b) > norm(v, w, a, b):
k = dot(u, v, w, a, b) // dot(v, v, w, a, b)
u, v = v, (u[0]- k*v[0], u[1]- k*v[1])
u, v = v, u
if dot(u, v, w, a, b) < dot(v, v, w, a, b)/2 or norm((u[0]-v[0], u[1]-v[1]), w, a, b) > norm(v, w, a, b):
c = v
else:
c = (u[0] - v[0], u[1] - v[1])
return c[0]*w + b*c[1], c[0]
def dot(u, v, w, a, b):
r"""
Returns a special dot product of the vectors `u = (u_{1}, u_{2})` and
`v = (v_{1}, v_{2})` which is defined in order to reduce solution of
the congruence equation `X^2 - aZ^2 \equiv 0 \ (mod \ b)`.
"""
u_1, u_2 = u
v_1, v_2 = v
return (w*u_1 + b*u_2)*(w*v_1 + b*v_2) + abs(a)*u_1*v_1
def norm(u, w, a, b):
r"""
Returns the norm of the vector `u = (u_{1}, u_{2})` under the dot product
defined by `u \cdot v = (wu_{1} + bu_{2})(w*v_{1} + bv_{2}) + |a|*u_{1}*v_{1}`
where `u = (u_{1}, u_{2})` and `v = (v_{1}, v_{2})`.
"""
u_1, u_2 = u
return sqrt(dot((u_1, u_2), (u_1, u_2), w, a, b))
def holzer(x, y, z, a, b, c):
r"""
Simplify the solution `(x, y, z)` of the equation
`ax^2 + by^2 = cz^2` with `a, b, c > 0` and `z^2 \geq \mid ab \mid` to
a new reduced solution `(x', y', z')` such that `z'^2 \leq \mid ab \mid`.
The algorithm is an interpretation of Mordell's reduction as described
on page 8 of Cremona and Rusin's paper [1]_ and the work of Mordell in
reference [2]_.
References
==========
.. [1] Efficient Solution of Rational Conices, J. E. Cremona and D. Rusin,
Mathematics of Computation, Volume 00, Number 0.
.. [2] Diophantine Equations, L. J. Mordell, page 48.
"""
if _odd(c):
k = 2*c
else:
k = c//2
small = a*b*c
step = 0
while True:
t1, t2, t3 = a*x**2, b*y**2, c*z**2
# check that it's a solution
if t1 + t2 != t3:
if step == 0:
raise ValueError('bad starting solution')
break
x_0, y_0, z_0 = x, y, z
if max(t1, t2, t3) <= small:
# Holzer condition
break
uv = u, v = base_solution_linear(k, y_0, -x_0)
if None in uv:
break
p, q = -(a*u*x_0 + b*v*y_0), c*z_0
r = Rational(p, q)
if _even(c):
w = _nint_or_floor(p, q)
assert abs(w - r) <= S.Half
else:
w = p//q # floor
if _odd(a*u + b*v + c*w):
w += 1
assert abs(w - r) <= S.One
A = (a*u**2 + b*v**2 + c*w**2)
B = (a*u*x_0 + b*v*y_0 + c*w*z_0)
x = Rational(x_0*A - 2*u*B, k)
y = Rational(y_0*A - 2*v*B, k)
z = Rational(z_0*A - 2*w*B, k)
assert all(i.is_Integer for i in (x, y, z))
step += 1
return tuple([int(i) for i in (x_0, y_0, z_0)])
def diop_general_pythagorean(eq, param=symbols("m", integer=True)):
"""
Solves the general pythagorean equation,
`a_{1}^2x_{1}^2 + a_{2}^2x_{2}^2 + . . . + a_{n}^2x_{n}^2 - a_{n + 1}^2x_{n + 1}^2 = 0`.
Returns a tuple which contains a parametrized solution to the equation,
sorted in the same order as the input variables.
Usage
=====
``diop_general_pythagorean(eq, param)``: where ``eq`` is a general
pythagorean equation which is assumed to be zero and ``param`` is the base
parameter used to construct other parameters by subscripting.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_pythagorean
>>> from sympy.abc import a, b, c, d, e
>>> diop_general_pythagorean(a**2 + b**2 + c**2 - d**2)
(m1**2 + m2**2 - m3**2, 2*m1*m3, 2*m2*m3, m1**2 + m2**2 + m3**2)
>>> diop_general_pythagorean(9*a**2 - 4*b**2 + 16*c**2 + 25*d**2 + e**2)
(10*m1**2 + 10*m2**2 + 10*m3**2 - 10*m4**2, 15*m1**2 + 15*m2**2 + 15*m3**2 + 15*m4**2, 15*m1*m4, 12*m2*m4, 60*m3*m4)
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralPythagorean.name:
if param is None:
params = None
else:
params = symbols('%s1:%i' % (param, len(var)), integer=True)
return list(GeneralPythagorean(eq).solve(parameters=params))[0]
def diop_general_sum_of_squares(eq, limit=1):
r"""
Solves the equation `x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Returns at most ``limit`` number of solutions.
Usage
=====
``general_sum_of_squares(eq, limit)`` : Here ``eq`` is an expression which
is assumed to be zero. Also, ``eq`` should be in the form,
`x_{1}^2 + x_{2}^2 + . . . + x_{n}^2 - k = 0`.
Details
=======
When `n = 3` if `k = 4^a(8m + 7)` for some `a, m \in Z` then there will be
no solutions. Refer to [1]_ for more details.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_squares
>>> from sympy.abc import a, b, c, d, e
>>> diop_general_sum_of_squares(a**2 + b**2 + c**2 + d**2 + e**2 - 2345)
{(15, 22, 22, 24, 24)}
Reference
=========
.. [1] Representing an integer as a sum of three squares, [online],
Available:
http://www.proofwiki.org/wiki/Integer_as_Sum_of_Three_Squares
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralSumOfSquares.name:
return set(GeneralSumOfSquares(eq).solve(limit=limit))
def diop_general_sum_of_even_powers(eq, limit=1):
"""
Solves the equation `x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`
where `e` is an even, integer power.
Returns at most ``limit`` number of solutions.
Usage
=====
``general_sum_of_even_powers(eq, limit)`` : Here ``eq`` is an expression which
is assumed to be zero. Also, ``eq`` should be in the form,
`x_{1}^e + x_{2}^e + . . . + x_{n}^e - k = 0`.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import diop_general_sum_of_even_powers
>>> from sympy.abc import a, b
>>> diop_general_sum_of_even_powers(a**4 + b**4 - (2**4 + 3**4))
{(2, 3)}
See Also
========
power_representation
"""
var, coeff, diop_type = classify_diop(eq, _dict=False)
if diop_type == GeneralSumOfEvenPowers.name:
return set(GeneralSumOfEvenPowers(eq).solve(limit=limit))
## Functions below this comment can be more suitably grouped under
## an Additive number theory module rather than the Diophantine
## equation module.
def partition(n, k=None, zeros=False):
"""
Returns a generator that can be used to generate partitions of an integer
`n`.
Explanation
===========
A partition of `n` is a set of positive integers which add up to `n`. For
example, partitions of 3 are 3, 1 + 2, 1 + 1 + 1. A partition is returned
as a tuple. If ``k`` equals None, then all possible partitions are returned
irrespective of their size, otherwise only the partitions of size ``k`` are
returned. If the ``zero`` parameter is set to True then a suitable
number of zeros are added at the end of every partition of size less than
``k``.
``zero`` parameter is considered only if ``k`` is not None. When the
partitions are over, the last `next()` call throws the ``StopIteration``
exception, so this function should always be used inside a try - except
block.
Details
=======
``partition(n, k)``: Here ``n`` is a positive integer and ``k`` is the size
of the partition which is also positive integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import partition
>>> f = partition(5)
>>> next(f)
(1, 1, 1, 1, 1)
>>> next(f)
(1, 1, 1, 2)
>>> g = partition(5, 3)
>>> next(g)
(1, 1, 3)
>>> next(g)
(1, 2, 2)
>>> g = partition(5, 3, zeros=True)
>>> next(g)
(0, 0, 5)
"""
if not zeros or k is None:
for i in ordered_partitions(n, k):
yield tuple(i)
else:
for m in range(1, k + 1):
for i in ordered_partitions(n, m):
i = tuple(i)
yield (0,)*(k - len(i)) + i
def prime_as_sum_of_two_squares(p):
"""
Represent a prime `p` as a unique sum of two squares; this can
only be done if the prime is congruent to 1 mod 4.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import prime_as_sum_of_two_squares
>>> prime_as_sum_of_two_squares(7) # can't be done
>>> prime_as_sum_of_two_squares(5)
(1, 2)
Reference
=========
.. [1] Representing a number as a sum of four squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
if not p % 4 == 1:
return
if p % 8 == 5:
b = 2
else:
b = 3
while pow(b, (p - 1) // 2, p) == 1:
b = nextprime(b)
b = pow(b, (p - 1) // 4, p)
a = p
while b**2 > p:
a, b = b, a % b
return (int(a % b), int(b)) # convert from long
def sum_of_three_squares(n):
r"""
Returns a 3-tuple $(a, b, c)$ such that $a^2 + b^2 + c^2 = n$ and
$a, b, c \geq 0$.
Returns None if $n = 4^a(8m + 7)$ for some `a, m \in \mathbb{Z}`. See
[1]_ for more details.
Usage
=====
``sum_of_three_squares(n)``: Here ``n`` is a non-negative integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_three_squares
>>> sum_of_three_squares(44542)
(18, 37, 207)
References
==========
.. [1] Representing a number as a sum of three squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
special = {1:(1, 0, 0), 2:(1, 1, 0), 3:(1, 1, 1), 10: (1, 3, 0), 34: (3, 3, 4), 58:(3, 7, 0),
85:(6, 7, 0), 130:(3, 11, 0), 214:(3, 6, 13), 226:(8, 9, 9), 370:(8, 9, 15),
526:(6, 7, 21), 706:(15, 15, 16), 730:(1, 27, 0), 1414:(6, 17, 33), 1906:(13, 21, 36),
2986: (21, 32, 39), 9634: (56, 57, 57)}
v = 0
if n == 0:
return (0, 0, 0)
v = multiplicity(4, n)
n //= 4**v
if n % 8 == 7:
return
if n in special.keys():
x, y, z = special[n]
return _sorted_tuple(2**v*x, 2**v*y, 2**v*z)
s, _exact = integer_nthroot(n, 2)
if _exact:
return (2**v*s, 0, 0)
x = None
if n % 8 == 3:
s = s if _odd(s) else s - 1
for x in range(s, -1, -2):
N = (n - x**2) // 2
if isprime(N):
y, z = prime_as_sum_of_two_squares(N)
return _sorted_tuple(2**v*x, 2**v*(y + z), 2**v*abs(y - z))
return
if n % 8 in (2, 6):
s = s if _odd(s) else s - 1
else:
s = s - 1 if _odd(s) else s
for x in range(s, -1, -2):
N = n - x**2
if isprime(N):
y, z = prime_as_sum_of_two_squares(N)
return _sorted_tuple(2**v*x, 2**v*y, 2**v*z)
def sum_of_four_squares(n):
r"""
Returns a 4-tuple `(a, b, c, d)` such that `a^2 + b^2 + c^2 + d^2 = n`.
Here `a, b, c, d \geq 0`.
Usage
=====
``sum_of_four_squares(n)``: Here ``n`` is a non-negative integer.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_four_squares
>>> sum_of_four_squares(3456)
(8, 8, 32, 48)
>>> sum_of_four_squares(1294585930293)
(0, 1234, 2161, 1137796)
References
==========
.. [1] Representing a number as a sum of four squares, [online],
Available: http://schorn.ch/lagrange.html
See Also
========
sum_of_squares()
"""
if n == 0:
return (0, 0, 0, 0)
v = multiplicity(4, n)
n //= 4**v
if n % 8 == 7:
d = 2
n = n - 4
elif n % 8 in (2, 6):
d = 1
n = n - 1
else:
d = 0
x, y, z = sum_of_three_squares(n)
return _sorted_tuple(2**v*d, 2**v*x, 2**v*y, 2**v*z)
def power_representation(n, p, k, zeros=False):
r"""
Returns a generator for finding k-tuples of integers,
`(n_{1}, n_{2}, . . . n_{k})`, such that
`n = n_{1}^p + n_{2}^p + . . . n_{k}^p`.
Usage
=====
``power_representation(n, p, k, zeros)``: Represent non-negative number
``n`` as a sum of ``k`` ``p``\ th powers. If ``zeros`` is true, then the
solutions is allowed to contain zeros.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import power_representation
Represent 1729 as a sum of two cubes:
>>> f = power_representation(1729, 3, 2)
>>> next(f)
(9, 10)
>>> next(f)
(1, 12)
If the flag `zeros` is True, the solution may contain tuples with
zeros; any such solutions will be generated after the solutions
without zeros:
>>> list(power_representation(125, 2, 3, zeros=True))
[(5, 6, 8), (3, 4, 10), (0, 5, 10), (0, 2, 11)]
For even `p` the `permute_sign` function can be used to get all
signed values:
>>> from sympy.utilities.iterables import permute_signs
>>> list(permute_signs((1, 12)))
[(1, 12), (-1, 12), (1, -12), (-1, -12)]
All possible signed permutations can also be obtained:
>>> from sympy.utilities.iterables import signed_permutations
>>> list(signed_permutations((1, 12)))
[(1, 12), (-1, 12), (1, -12), (-1, -12), (12, 1), (-12, 1), (12, -1), (-12, -1)]
"""
n, p, k = [as_int(i) for i in (n, p, k)]
if n < 0:
if p % 2:
for t in power_representation(-n, p, k, zeros):
yield tuple(-i for i in t)
return
if p < 1 or k < 1:
raise ValueError(filldedent('''
Expecting positive integers for `(p, k)`, but got `(%s, %s)`'''
% (p, k)))
if n == 0:
if zeros:
yield (0,)*k
return
if k == 1:
if p == 1:
yield (n,)
else:
be = perfect_power(n)
if be:
b, e = be
d, r = divmod(e, p)
if not r:
yield (b**d,)
return
if p == 1:
for t in partition(n, k, zeros=zeros):
yield t
return
if p == 2:
feasible = _can_do_sum_of_squares(n, k)
if not feasible:
return
if not zeros and n > 33 and k >= 5 and k <= n and n - k in (
13, 10, 7, 5, 4, 2, 1):
'''Todd G. Will, "When Is n^2 a Sum of k Squares?", [online].
Available: https://www.maa.org/sites/default/files/Will-MMz-201037918.pdf'''
return
if feasible is not True: # it's prime and k == 2
yield prime_as_sum_of_two_squares(n)
return
if k == 2 and p > 2:
be = perfect_power(n)
if be and be[1] % p == 0:
return # Fermat: a**n + b**n = c**n has no solution for n > 2
if n >= k:
a = integer_nthroot(n - (k - 1), p)[0]
for t in pow_rep_recursive(a, k, n, [], p):
yield tuple(reversed(t))
if zeros:
a = integer_nthroot(n, p)[0]
for i in range(1, k):
for t in pow_rep_recursive(a, i, n, [], p):
yield tuple(reversed(t + (0,)*(k - i)))
sum_of_powers = power_representation
def pow_rep_recursive(n_i, k, n_remaining, terms, p):
# Invalid arguments
if n_i <= 0 or k <= 0:
return
# No solutions may exist
if n_remaining < k:
return
if k * pow(n_i, p) < n_remaining:
return
if k == 0 and n_remaining == 0:
yield tuple(terms)
elif k == 1:
# next_term^p must equal to n_remaining
next_term, exact = integer_nthroot(n_remaining, p)
if exact and next_term <= n_i:
yield tuple(terms + [next_term])
return
else:
# TODO: Fall back to diop_DN when k = 2
if n_i >= 1 and k > 0:
for next_term in range(1, n_i + 1):
residual = n_remaining - pow(next_term, p)
if residual < 0:
break
yield from pow_rep_recursive(next_term, k - 1, residual, terms + [next_term], p)
def sum_of_squares(n, k, zeros=False):
"""Return a generator that yields the k-tuples of nonnegative
values, the squares of which sum to n. If zeros is False (default)
then the solution will not contain zeros. The nonnegative
elements of a tuple are sorted.
* If k == 1 and n is square, (n,) is returned.
* If k == 2 then n can only be written as a sum of squares if
every prime in the factorization of n that has the form
4*k + 3 has an even multiplicity. If n is prime then
it can only be written as a sum of two squares if it is
in the form 4*k + 1.
* if k == 3 then n can be written as a sum of squares if it does
not have the form 4**m*(8*k + 7).
* all integers can be written as the sum of 4 squares.
* if k > 4 then n can be partitioned and each partition can
be written as a sum of 4 squares; if n is not evenly divisible
by 4 then n can be written as a sum of squares only if the
an additional partition can be written as sum of squares.
For example, if k = 6 then n is partitioned into two parts,
the first being written as a sum of 4 squares and the second
being written as a sum of 2 squares -- which can only be
done if the condition above for k = 2 can be met, so this will
automatically reject certain partitions of n.
Examples
========
>>> from sympy.solvers.diophantine.diophantine import sum_of_squares
>>> list(sum_of_squares(25, 2))
[(3, 4)]
>>> list(sum_of_squares(25, 2, True))
[(3, 4), (0, 5)]
>>> list(sum_of_squares(25, 4))
[(1, 2, 2, 4)]
See Also
========
sympy.utilities.iterables.signed_permutations
"""
yield from power_representation(n, 2, k, zeros)
def _can_do_sum_of_squares(n, k):
"""Return True if n can be written as the sum of k squares,
False if it cannot, or 1 if ``k == 2`` and ``n`` is prime (in which
case it *can* be written as a sum of two squares). A False
is returned only if it cannot be written as ``k``-squares, even
if 0s are allowed.
"""
if k < 1:
return False
if n < 0:
return False
if n == 0:
return True
if k == 1:
return is_square(n)
if k == 2:
if n in (1, 2):
return True
if isprime(n):
if n % 4 == 1:
return 1 # signal that it was prime
return False
else:
f = factorint(n)
for p, m in f.items():
# we can proceed iff no prime factor in the form 4*k + 3
# has an odd multiplicity
if (p % 4 == 3) and m % 2:
return False
return True
if k == 3:
if (n//4**multiplicity(4, n)) % 8 == 7:
return False
# every number can be written as a sum of 4 squares; for k > 4 partitions
# can be 0
return True
|
41938b5fd06423904b5c9116efb89f3f773129d4cf78bf98188e7a9a6d517b42 | from sympy.core import Add, Mul, S
from sympy.core.containers import Tuple
from sympy.core.exprtools import factor_terms
from sympy.core.numbers import I
from sympy.core.relational import Eq, Equality
from sympy.core.sorting import default_sort_key, ordered
from sympy.core.symbol import Dummy, Symbol
from sympy.core.function import (expand_mul, expand, Derivative,
AppliedUndef, Function, Subs)
from sympy.functions import (exp, im, cos, sin, re, Piecewise,
piecewise_fold, sqrt, log)
from sympy.functions.combinatorial.factorials import factorial
from sympy.matrices import zeros, Matrix, NonSquareMatrixError, MatrixBase, eye
from sympy.polys import Poly, together
from sympy.simplify import collect, radsimp, signsimp # type: ignore
from sympy.simplify.powsimp import powdenest, powsimp
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import simplify
from sympy.sets.sets import FiniteSet
from sympy.solvers.deutils import ode_order
from sympy.solvers.solveset import NonlinearError, solveset
from sympy.utilities.iterables import (connected_components, iterable,
strongly_connected_components)
from sympy.utilities.misc import filldedent
from sympy.integrals.integrals import Integral, integrate
def _get_func_order(eqs, funcs):
return {func: max(ode_order(eq, func) for eq in eqs) for func in funcs}
class ODEOrderError(ValueError):
"""Raised by linear_ode_to_matrix if the system has the wrong order"""
pass
class ODENonlinearError(NonlinearError):
"""Raised by linear_ode_to_matrix if the system is nonlinear"""
pass
def _simpsol(soleq):
lhs = soleq.lhs
sol = soleq.rhs
sol = powsimp(sol)
gens = list(sol.atoms(exp))
p = Poly(sol, *gens, expand=False)
gens = [factor_terms(g) for g in gens]
if not gens:
gens = p.gens
syms = [Symbol('C1'), Symbol('C2')]
terms = []
for coeff, monom in zip(p.coeffs(), p.monoms()):
coeff = piecewise_fold(coeff)
if isinstance(coeff, Piecewise):
coeff = Piecewise(*((ratsimp(coef).collect(syms), cond) for coef, cond in coeff.args))
else:
coeff = ratsimp(coeff).collect(syms)
monom = Mul(*(g ** i for g, i in zip(gens, monom)))
terms.append(coeff * monom)
return Eq(lhs, Add(*terms))
def _solsimp(e, t):
no_t, has_t = powsimp(expand_mul(e)).as_independent(t)
no_t = ratsimp(no_t)
has_t = has_t.replace(exp, lambda a: exp(factor_terms(a)))
return no_t + has_t
def simpsol(sol, wrt1, wrt2, doit=True):
"""Simplify solutions from dsolve_system."""
# The parameter sol is the solution as returned by dsolve (list of Eq).
#
# The parameters wrt1 and wrt2 are lists of symbols to be collected for
# with those in wrt1 being collected for first. This allows for collecting
# on any factors involving the independent variable before collecting on
# the integration constants or vice versa using e.g.:
#
# sol = simpsol(sol, [t], [C1, C2]) # t first, constants after
# sol = simpsol(sol, [C1, C2], [t]) # constants first, t after
#
# If doit=True (default) then simpsol will begin by evaluating any
# unevaluated integrals. Since many integrals will appear multiple times
# in the solutions this is done intelligently by computing each integral
# only once.
#
# The strategy is to first perform simple cancellation with factor_terms
# and then multiply out all brackets with expand_mul. This gives an Add
# with many terms.
#
# We split each term into two multiplicative factors dep and coeff where
# all factors that involve wrt1 are in dep and any constant factors are in
# coeff e.g.
# sqrt(2)*C1*exp(t) -> ( exp(t), sqrt(2)*C1 )
#
# The dep factors are simplified using powsimp to combine expanded
# exponential factors e.g.
# exp(a*t)*exp(b*t) -> exp(t*(a+b))
#
# We then collect coefficients for all terms having the same (simplified)
# dep. The coefficients are then simplified using together and ratsimp and
# lastly by recursively applying the same transformation to the
# coefficients to collect on wrt2.
#
# Finally the result is recombined into an Add and signsimp is used to
# normalise any minus signs.
def simprhs(rhs, rep, wrt1, wrt2):
"""Simplify the rhs of an ODE solution"""
if rep:
rhs = rhs.subs(rep)
rhs = factor_terms(rhs)
rhs = simp_coeff_dep(rhs, wrt1, wrt2)
rhs = signsimp(rhs)
return rhs
def simp_coeff_dep(expr, wrt1, wrt2=None):
"""Split rhs into terms, split terms into dep and coeff and collect on dep"""
add_dep_terms = lambda e: e.is_Add and e.has(*wrt1)
expandable = lambda e: e.is_Mul and any(map(add_dep_terms, e.args))
expand_func = lambda e: expand_mul(e, deep=False)
expand_mul_mod = lambda e: e.replace(expandable, expand_func)
terms = Add.make_args(expand_mul_mod(expr))
dc = {}
for term in terms:
coeff, dep = term.as_independent(*wrt1, as_Add=False)
# Collect together the coefficients for terms that have the same
# dependence on wrt1 (after dep is normalised using simpdep).
dep = simpdep(dep, wrt1)
# See if the dependence on t cancels out...
if dep is not S.One:
dep2 = factor_terms(dep)
if not dep2.has(*wrt1):
coeff *= dep2
dep = S.One
if dep not in dc:
dc[dep] = coeff
else:
dc[dep] += coeff
# Apply the method recursively to the coefficients but this time
# collecting on wrt2 rather than wrt2.
termpairs = ((simpcoeff(c, wrt2), d) for d, c in dc.items())
if wrt2 is not None:
termpairs = ((simp_coeff_dep(c, wrt2), d) for c, d in termpairs)
return Add(*(c * d for c, d in termpairs))
def simpdep(term, wrt1):
"""Normalise factors involving t with powsimp and recombine exp"""
def canonicalise(a):
# Using factor_terms here isn't quite right because it leads to things
# like exp(t*(1+t)) that we don't want. We do want to cancel factors
# and pull out a common denominator but ideally the numerator would be
# expressed as a standard form polynomial in t so we expand_mul
# and collect afterwards.
a = factor_terms(a)
num, den = a.as_numer_denom()
num = expand_mul(num)
num = collect(num, wrt1)
return num / den
term = powsimp(term)
rep = {e: exp(canonicalise(e.args[0])) for e in term.atoms(exp)}
term = term.subs(rep)
return term
def simpcoeff(coeff, wrt2):
"""Bring to a common fraction and cancel with ratsimp"""
coeff = together(coeff)
if coeff.is_polynomial():
# Calling ratsimp can be expensive. The main reason is to simplify
# sums of terms with irrational denominators so we limit ourselves
# to the case where the expression is polynomial in any symbols.
# Maybe there's a better approach...
coeff = ratsimp(radsimp(coeff))
# collect on secondary variables first and any remaining symbols after
if wrt2 is not None:
syms = list(wrt2) + list(ordered(coeff.free_symbols - set(wrt2)))
else:
syms = list(ordered(coeff.free_symbols))
coeff = collect(coeff, syms)
coeff = together(coeff)
return coeff
# There are often repeated integrals. Collect unique integrals and
# evaluate each once and then substitute into the final result to replace
# all occurrences in each of the solution equations.
if doit:
integrals = set().union(*(s.atoms(Integral) for s in sol))
rep = {i: factor_terms(i).doit() for i in integrals}
else:
rep = {}
sol = [Eq(s.lhs, simprhs(s.rhs, rep, wrt1, wrt2)) for s in sol]
return sol
def linodesolve_type(A, t, b=None):
r"""
Helper function that determines the type of the system of ODEs for solving with :obj:`sympy.solvers.ode.systems.linodesolve()`
Explanation
===========
This function takes in the coefficient matrix and/or the non-homogeneous term
and returns the type of the equation that can be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`.
If the system is constant coefficient homogeneous, then "type1" is returned
If the system is constant coefficient non-homogeneous, then "type2" is returned
If the system is non-constant coefficient homogeneous, then "type3" is returned
If the system is non-constant coefficient non-homogeneous, then "type4" is returned
If the system has a non-constant coefficient matrix which can be factorized into constant
coefficient matrix, then "type5" or "type6" is returned for when the system is homogeneous or
non-homogeneous respectively.
Note that, if the system of ODEs is of "type3" or "type4", then along with the type,
the commutative antiderivative of the coefficient matrix is also returned.
If the system cannot be solved by :obj:`sympy.solvers.ode.systems.linodesolve()`, then
NotImplementedError is raised.
Parameters
==========
A : Matrix
Coefficient matrix of the system of ODEs
b : Matrix or None
Non-homogeneous term of the system. The default value is None.
If this argument is None, then the system is assumed to be homogeneous.
Examples
========
>>> from sympy import symbols, Matrix
>>> from sympy.solvers.ode.systems import linodesolve_type
>>> t = symbols("t")
>>> A = Matrix([[1, 1], [2, 3]])
>>> b = Matrix([t, 1])
>>> linodesolve_type(A, t)
{'antiderivative': None, 'type_of_equation': 'type1'}
>>> linodesolve_type(A, t, b=b)
{'antiderivative': None, 'type_of_equation': 'type2'}
>>> A_t = Matrix([[1, t], [-t, 1]])
>>> linodesolve_type(A_t, t)
{'antiderivative': Matrix([
[ t, t**2/2],
[-t**2/2, t]]), 'type_of_equation': 'type3'}
>>> linodesolve_type(A_t, t, b=b)
{'antiderivative': Matrix([
[ t, t**2/2],
[-t**2/2, t]]), 'type_of_equation': 'type4'}
>>> A_non_commutative = Matrix([[1, t], [t, -1]])
>>> linodesolve_type(A_non_commutative, t)
Traceback (most recent call last):
...
NotImplementedError:
The system does not have a commutative antiderivative, it cannot be
solved by linodesolve.
Returns
=======
Dict
Raises
======
NotImplementedError
When the coefficient matrix does not have a commutative antiderivative
See Also
========
linodesolve: Function for which linodesolve_type gets the information
"""
match = {}
is_non_constant = not _matrix_is_constant(A, t)
is_non_homogeneous = not (b is None or b.is_zero_matrix)
type = "type{}".format(int("{}{}".format(int(is_non_constant), int(is_non_homogeneous)), 2) + 1)
B = None
match.update({"type_of_equation": type, "antiderivative": B})
if is_non_constant:
B, is_commuting = _is_commutative_anti_derivative(A, t)
if not is_commuting:
raise NotImplementedError(filldedent('''
The system does not have a commutative antiderivative, it cannot be solved
by linodesolve.
'''))
match['antiderivative'] = B
match.update(_first_order_type5_6_subs(A, t, b=b))
return match
def _first_order_type5_6_subs(A, t, b=None):
match = {}
factor_terms = _factor_matrix(A, t)
is_homogeneous = b is None or b.is_zero_matrix
if factor_terms is not None:
t_ = Symbol("{}_".format(t))
F_t = integrate(factor_terms[0], t)
inverse = solveset(Eq(t_, F_t), t)
# Note: A simple way to check if a function is invertible
# or not.
if isinstance(inverse, FiniteSet) and not inverse.has(Piecewise)\
and len(inverse) == 1:
A = factor_terms[1]
if not is_homogeneous:
b = b / factor_terms[0]
b = b.subs(t, list(inverse)[0])
type = "type{}".format(5 + (not is_homogeneous))
match.update({'func_coeff': A, 'tau': F_t,
't_': t_, 'type_of_equation': type, 'rhs': b})
return match
def linear_ode_to_matrix(eqs, funcs, t, order):
r"""
Convert a linear system of ODEs to matrix form
Explanation
===========
Express a system of linear ordinary differential equations as a single
matrix differential equation [1]. For example the system $x' = x + y + 1$
and $y' = x - y$ can be represented as
.. math:: A_1 X' = A_0 X + b
where $A_1$ and $A_0$ are $2 \times 2$ matrices and $b$, $X$ and $X'$ are
$2 \times 1$ matrices with $X = [x, y]^T$.
Higher-order systems are represented with additional matrices e.g. a
second-order system would look like
.. math:: A_2 X'' = A_1 X' + A_0 X + b
Examples
========
>>> from sympy import Function, Symbol, Matrix, Eq
>>> from sympy.solvers.ode.systems import linear_ode_to_matrix
>>> t = Symbol('t')
>>> x = Function('x')
>>> y = Function('y')
We can create a system of linear ODEs like
>>> eqs = [
... Eq(x(t).diff(t), x(t) + y(t) + 1),
... Eq(y(t).diff(t), x(t) - y(t)),
... ]
>>> funcs = [x(t), y(t)]
>>> order = 1 # 1st order system
Now ``linear_ode_to_matrix`` can represent this as a matrix
differential equation.
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, order)
>>> A1
Matrix([
[1, 0],
[0, 1]])
>>> A0
Matrix([
[1, 1],
[1, -1]])
>>> b
Matrix([
[1],
[0]])
The original equations can be recovered from these matrices:
>>> eqs_mat = Matrix([eq.lhs - eq.rhs for eq in eqs])
>>> X = Matrix(funcs)
>>> A1 * X.diff(t) - A0 * X - b == eqs_mat
True
If the system of equations has a maximum order greater than the
order of the system specified, a ODEOrderError exception is raised.
>>> eqs = [Eq(x(t).diff(t, 2), x(t).diff(t) + x(t)), Eq(y(t).diff(t), y(t) + x(t))]
>>> linear_ode_to_matrix(eqs, funcs, t, 1)
Traceback (most recent call last):
...
ODEOrderError: Cannot represent system in 1-order form
If the system of equations is nonlinear, then ODENonlinearError is
raised.
>>> eqs = [Eq(x(t).diff(t), x(t) + y(t)), Eq(y(t).diff(t), y(t)**2 + x(t))]
>>> linear_ode_to_matrix(eqs, funcs, t, 1)
Traceback (most recent call last):
...
ODENonlinearError: The system of ODEs is nonlinear.
Parameters
==========
eqs : list of SymPy expressions or equalities
The equations as expressions (assumed equal to zero).
funcs : list of applied functions
The dependent variables of the system of ODEs.
t : symbol
The independent variable.
order : int
The order of the system of ODEs.
Returns
=======
The tuple ``(As, b)`` where ``As`` is a tuple of matrices and ``b`` is the
the matrix representing the rhs of the matrix equation.
Raises
======
ODEOrderError
When the system of ODEs have an order greater than what was specified
ODENonlinearError
When the system of ODEs is nonlinear
See Also
========
linear_eq_to_matrix: for systems of linear algebraic equations.
References
==========
.. [1] https://en.wikipedia.org/wiki/Matrix_differential_equation
"""
from sympy.solvers.solveset import linear_eq_to_matrix
if any(ode_order(eq, func) > order for eq in eqs for func in funcs):
msg = "Cannot represent system in {}-order form"
raise ODEOrderError(msg.format(order))
As = []
for o in range(order, -1, -1):
# Work from the highest derivative down
syms = [func.diff(t, o) for func in funcs]
# Ai is the matrix for X(t).diff(t, o)
# eqs is minus the remainder of the equations.
try:
Ai, b = linear_eq_to_matrix(eqs, syms)
except NonlinearError:
raise ODENonlinearError("The system of ODEs is nonlinear.")
Ai = Ai.applyfunc(expand_mul)
As.append(Ai if o == order else -Ai)
if o:
eqs = [-eq for eq in b]
else:
rhs = b
return As, rhs
def matrix_exp(A, t):
r"""
Matrix exponential $\exp(A*t)$ for the matrix ``A`` and scalar ``t``.
Explanation
===========
This functions returns the $\exp(A*t)$ by doing a simple
matrix multiplication:
.. math:: \exp(A*t) = P * expJ * P^{-1}
where $expJ$ is $\exp(J*t)$. $J$ is the Jordan normal
form of $A$ and $P$ is matrix such that:
.. math:: A = P * J * P^{-1}
The matrix exponential $\exp(A*t)$ appears in the solution of linear
differential equations. For example if $x$ is a vector and $A$ is a matrix
then the initial value problem
.. math:: \frac{dx(t)}{dt} = A \times x(t), x(0) = x0
has the unique solution
.. math:: x(t) = \exp(A t) x0
Examples
========
>>> from sympy import Symbol, Matrix, pprint
>>> from sympy.solvers.ode.systems import matrix_exp
>>> t = Symbol('t')
We will consider a 2x2 matrix for comupting the exponential
>>> A = Matrix([[2, -5], [2, -4]])
>>> pprint(A)
[2 -5]
[ ]
[2 -4]
Now, exp(A*t) is given as follows:
>>> pprint(matrix_exp(A, t))
[ -t -t -t ]
[3*e *sin(t) + e *cos(t) -5*e *sin(t) ]
[ ]
[ -t -t -t ]
[ 2*e *sin(t) - 3*e *sin(t) + e *cos(t)]
Parameters
==========
A : Matrix
The matrix $A$ in the expression $\exp(A*t)$
t : Symbol
The independent variable
See Also
========
matrix_exp_jordan_form: For exponential of Jordan normal form
References
==========
.. [1] https://en.wikipedia.org/wiki/Jordan_normal_form
.. [2] https://en.wikipedia.org/wiki/Matrix_exponential
"""
P, expJ = matrix_exp_jordan_form(A, t)
return P * expJ * P.inv()
def matrix_exp_jordan_form(A, t):
r"""
Matrix exponential $\exp(A*t)$ for the matrix *A* and scalar *t*.
Explanation
===========
Returns the Jordan form of the $\exp(A*t)$ along with the matrix $P$ such that:
.. math::
\exp(A*t) = P * expJ * P^{-1}
Examples
========
>>> from sympy import Matrix, Symbol
>>> from sympy.solvers.ode.systems import matrix_exp, matrix_exp_jordan_form
>>> t = Symbol('t')
We will consider a 2x2 defective matrix. This shows that our method
works even for defective matrices.
>>> A = Matrix([[1, 1], [0, 1]])
It can be observed that this function gives us the Jordan normal form
and the required invertible matrix P.
>>> P, expJ = matrix_exp_jordan_form(A, t)
Here, it is shown that P and expJ returned by this function is correct
as they satisfy the formula: P * expJ * P_inverse = exp(A*t).
>>> P * expJ * P.inv() == matrix_exp(A, t)
True
Parameters
==========
A : Matrix
The matrix $A$ in the expression $\exp(A*t)$
t : Symbol
The independent variable
References
==========
.. [1] https://en.wikipedia.org/wiki/Defective_matrix
.. [2] https://en.wikipedia.org/wiki/Jordan_matrix
.. [3] https://en.wikipedia.org/wiki/Jordan_normal_form
"""
N, M = A.shape
if N != M:
raise ValueError('Needed square matrix but got shape (%s, %s)' % (N, M))
elif A.has(t):
raise ValueError('Matrix A should not depend on t')
def jordan_chains(A):
'''Chains from Jordan normal form analogous to M.eigenvects().
Returns a dict with eignevalues as keys like:
{e1: [[v111,v112,...], [v121, v122,...]], e2:...}
where vijk is the kth vector in the jth chain for eigenvalue i.
'''
P, blocks = A.jordan_cells()
basis = [P[:,i] for i in range(P.shape[1])]
n = 0
chains = {}
for b in blocks:
eigval = b[0, 0]
size = b.shape[0]
if eigval not in chains:
chains[eigval] = []
chains[eigval].append(basis[n:n+size])
n += size
return chains
eigenchains = jordan_chains(A)
# Needed for consistency across Python versions
eigenchains_iter = sorted(eigenchains.items(), key=default_sort_key)
isreal = not A.has(I)
blocks = []
vectors = []
seen_conjugate = set()
for e, chains in eigenchains_iter:
for chain in chains:
n = len(chain)
if isreal and e != e.conjugate() and e.conjugate() in eigenchains:
if e in seen_conjugate:
continue
seen_conjugate.add(e.conjugate())
exprt = exp(re(e) * t)
imrt = im(e) * t
imblock = Matrix([[cos(imrt), sin(imrt)],
[-sin(imrt), cos(imrt)]])
expJblock2 = Matrix(n, n, lambda i,j:
imblock * t**(j-i) / factorial(j-i) if j >= i
else zeros(2, 2))
expJblock = Matrix(2*n, 2*n, lambda i,j: expJblock2[i//2,j//2][i%2,j%2])
blocks.append(exprt * expJblock)
for i in range(n):
vectors.append(re(chain[i]))
vectors.append(im(chain[i]))
else:
vectors.extend(chain)
fun = lambda i,j: t**(j-i)/factorial(j-i) if j >= i else 0
expJblock = Matrix(n, n, fun)
blocks.append(exp(e * t) * expJblock)
expJ = Matrix.diag(*blocks)
P = Matrix(N, N, lambda i,j: vectors[j][i])
return P, expJ
# Note: To add a docstring example with tau
def linodesolve(A, t, b=None, B=None, type="auto", doit=False,
tau=None):
r"""
System of n equations linear first-order differential equations
Explanation
===========
This solver solves the system of ODEs of the follwing form:
.. math::
X'(t) = A(t) X(t) + b(t)
Here, $A(t)$ is the coefficient matrix, $X(t)$ is the vector of n independent variables,
$b(t)$ is the non-homogeneous term and $X'(t)$ is the derivative of $X(t)$
Depending on the properties of $A(t)$ and $b(t)$, this solver evaluates the solution
differently.
When $A(t)$ is constant coefficient matrix and $b(t)$ is zero vector i.e. system is homogeneous,
the system is "type1". The solution is:
.. math::
X(t) = \exp(A t) C
Here, $C$ is a vector of constants and $A$ is the constant coefficient matrix.
When $A(t)$ is constant coefficient matrix and $b(t)$ is non-zero i.e. system is non-homogeneous,
the system is "type2". The solution is:
.. math::
X(t) = e^{A t} ( \int e^{- A t} b \,dt + C)
When $A(t)$ is coefficient matrix such that its commutative with its antiderivative $B(t)$ and
$b(t)$ is a zero vector i.e. system is homogeneous, the system is "type3". The solution is:
.. math::
X(t) = \exp(B(t)) C
When $A(t)$ is commutative with its antiderivative $B(t)$ and $b(t)$ is non-zero i.e. system is
non-homogeneous, the system is "type4". The solution is:
.. math::
X(t) = e^{B(t)} ( \int e^{-B(t)} b(t) \,dt + C)
When $A(t)$ is a coefficient matrix such that it can be factorized into a scalar and a constant
coefficient matrix:
.. math::
A(t) = f(t) * A
Where $f(t)$ is a scalar expression in the independent variable $t$ and $A$ is a constant matrix,
then we can do the following substitutions:
.. math::
tau = \int f(t) dt, X(t) = Y(tau), b(t) = b(f^{-1}(tau))
Here, the substitution for the non-homogeneous term is done only when its non-zero.
Using these substitutions, our original system becomes:
.. math::
Y'(tau) = A * Y(tau) + b(tau)/f(tau)
The above system can be easily solved using the solution for "type1" or "type2" depending
on the homogeneity of the system. After we get the solution for $Y(tau)$, we substitute the
solution for $tau$ as $t$ to get back $X(t)$
.. math::
X(t) = Y(tau)
Systems of "type5" and "type6" have a commutative antiderivative but we use this solution
because its faster to compute.
The final solution is the general solution for all the four equations since a constant coefficient
matrix is always commutative with its antidervative.
An additional feature of this function is, if someone wants to substitute for value of the independent
variable, they can pass the substitution `tau` and the solution will have the independent variable
substituted with the passed expression(`tau`).
Parameters
==========
A : Matrix
Coefficient matrix of the system of linear first order ODEs.
t : Symbol
Independent variable in the system of ODEs.
b : Matrix or None
Non-homogeneous term in the system of ODEs. If None is passed,
a homogeneous system of ODEs is assumed.
B : Matrix or None
Antiderivative of the coefficient matrix. If the antiderivative
is not passed and the solution requires the term, then the solver
would compute it internally.
type : String
Type of the system of ODEs passed. Depending on the type, the
solution is evaluated. The type values allowed and the corresponding
system it solves are: "type1" for constant coefficient homogeneous
"type2" for constant coefficient non-homogeneous, "type3" for non-constant
coefficient homogeneous, "type4" for non-constant coefficient non-homogeneous,
"type5" and "type6" for non-constant coefficient homogeneous and non-homogeneous
systems respectively where the coefficient matrix can be factorized to a constant
coefficient matrix.
The default value is "auto" which will let the solver decide the correct type of
the system passed.
doit : Boolean
Evaluate the solution if True, default value is False
tau: Expression
Used to substitute for the value of `t` after we get the solution of the system.
Examples
========
To solve the system of ODEs using this function directly, several things must be
done in the right order. Wrong inputs to the function will lead to incorrect results.
>>> from sympy import symbols, Function, Eq
>>> from sympy.solvers.ode.systems import canonical_odes, linear_ode_to_matrix, linodesolve, linodesolve_type
>>> from sympy.solvers.ode.subscheck import checkodesol
>>> f, g = symbols("f, g", cls=Function)
>>> x, a = symbols("x, a")
>>> funcs = [f(x), g(x)]
>>> eqs = [Eq(f(x).diff(x) - f(x), a*g(x) + 1), Eq(g(x).diff(x) + g(x), a*f(x))]
Here, it is important to note that before we derive the coefficient matrix, it is
important to get the system of ODEs into the desired form. For that we will use
:obj:`sympy.solvers.ode.systems.canonical_odes()`.
>>> eqs = canonical_odes(eqs, funcs, x)
>>> eqs
[[Eq(Derivative(f(x), x), a*g(x) + f(x) + 1), Eq(Derivative(g(x), x), a*f(x) - g(x))]]
Now, we will use :obj:`sympy.solvers.ode.systems.linear_ode_to_matrix()` to get the coefficient matrix and the
non-homogeneous term if it is there.
>>> eqs = eqs[0]
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1)
>>> A = A0
We have the coefficient matrices and the non-homogeneous term ready. Now, we can use
:obj:`sympy.solvers.ode.systems.linodesolve_type()` to get the information for the system of ODEs
to finally pass it to the solver.
>>> system_info = linodesolve_type(A, x, b=b)
>>> sol_vector = linodesolve(A, x, b=b, B=system_info['antiderivative'], type=system_info['type_of_equation'])
Now, we can prove if the solution is correct or not by using :obj:`sympy.solvers.ode.checkodesol()`
>>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
>>> checkodesol(eqs, sol)
(True, [0, 0])
We can also use the doit method to evaluate the solutions passed by the function.
>>> sol_vector_evaluated = linodesolve(A, x, b=b, type="type2", doit=True)
Now, we will look at a system of ODEs which is non-constant.
>>> eqs = [Eq(f(x).diff(x), f(x) + x*g(x)), Eq(g(x).diff(x), -x*f(x) + g(x))]
The system defined above is already in the desired form, so we do not have to convert it.
>>> (A1, A0), b = linear_ode_to_matrix(eqs, funcs, x, 1)
>>> A = A0
A user can also pass the commutative antiderivative required for type3 and type4 system of ODEs.
Passing an incorrect one will lead to incorrect results. If the coefficient matrix is not commutative
with its antiderivative, then :obj:`sympy.solvers.ode.systems.linodesolve_type()` raises a NotImplementedError.
If it does have a commutative antiderivative, then the function just returns the information about the system.
>>> system_info = linodesolve_type(A, x, b=b)
Now, we can pass the antiderivative as an argument to get the solution. If the system information is not
passed, then the solver will compute the required arguments internally.
>>> sol_vector = linodesolve(A, x, b=b)
Once again, we can verify the solution obtained.
>>> sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
>>> checkodesol(eqs, sol)
(True, [0, 0])
Returns
=======
List
Raises
======
ValueError
This error is raised when the coefficient matrix, non-homogeneous term
or the antiderivative, if passed, are not a matrix or
do not have correct dimensions
NonSquareMatrixError
When the coefficient matrix or its antiderivative, if passed is not a
square matrix
NotImplementedError
If the coefficient matrix does not have a commutative antiderivative
See Also
========
linear_ode_to_matrix: Coefficient matrix computation function
canonical_odes: System of ODEs representation change
linodesolve_type: Getting information about systems of ODEs to pass in this solver
"""
if not isinstance(A, MatrixBase):
raise ValueError(filldedent('''\
The coefficients of the system of ODEs should be of type Matrix
'''))
if not A.is_square:
raise NonSquareMatrixError(filldedent('''\
The coefficient matrix must be a square
'''))
if b is not None:
if not isinstance(b, MatrixBase):
raise ValueError(filldedent('''\
The non-homogeneous terms of the system of ODEs should be of type Matrix
'''))
if A.rows != b.rows:
raise ValueError(filldedent('''\
The system of ODEs should have the same number of non-homogeneous terms and the number of
equations
'''))
if B is not None:
if not isinstance(B, MatrixBase):
raise ValueError(filldedent('''\
The antiderivative of coefficients of the system of ODEs should be of type Matrix
'''))
if not B.is_square:
raise NonSquareMatrixError(filldedent('''\
The antiderivative of the coefficient matrix must be a square
'''))
if A.rows != B.rows:
raise ValueError(filldedent('''\
The coefficient matrix and its antiderivative should have same dimensions
'''))
if not any(type == "type{}".format(i) for i in range(1, 7)) and not type == "auto":
raise ValueError(filldedent('''\
The input type should be a valid one
'''))
n = A.rows
# constants = numbered_symbols(prefix='C', cls=Dummy, start=const_idx+1)
Cvect = Matrix(list(Dummy() for _ in range(n)))
if b is None and any(type == typ for typ in ["type2", "type4", "type6"]):
b = zeros(n, 1)
is_transformed = tau is not None
passed_type = type
if type == "auto":
system_info = linodesolve_type(A, t, b=b)
type = system_info["type_of_equation"]
B = system_info["antiderivative"]
if type in ("type5", "type6"):
is_transformed = True
if passed_type != "auto":
if tau is None:
system_info = _first_order_type5_6_subs(A, t, b=b)
if not system_info:
raise ValueError(filldedent('''
The system passed isn't {}.
'''.format(type)))
tau = system_info['tau']
t = system_info['t_']
A = system_info['A']
b = system_info['b']
intx_wrtt = lambda x: Integral(x, t) if x else 0
if type in ("type1", "type2", "type5", "type6"):
P, J = matrix_exp_jordan_form(A, t)
P = simplify(P)
if type in ("type1", "type5"):
sol_vector = P * (J * Cvect)
else:
Jinv = J.subs(t, -t)
sol_vector = P * J * ((Jinv * P.inv() * b).applyfunc(intx_wrtt) + Cvect)
else:
if B is None:
B, _ = _is_commutative_anti_derivative(A, t)
if type == "type3":
sol_vector = B.exp() * Cvect
else:
sol_vector = B.exp() * (((-B).exp() * b).applyfunc(intx_wrtt) + Cvect)
if is_transformed:
sol_vector = sol_vector.subs(t, tau)
gens = sol_vector.atoms(exp)
if type != "type1":
sol_vector = [expand_mul(s) for s in sol_vector]
sol_vector = [collect(s, ordered(gens), exact=True) for s in sol_vector]
if doit:
sol_vector = [s.doit() for s in sol_vector]
return sol_vector
def _matrix_is_constant(M, t):
"""Checks if the matrix M is independent of t or not."""
return all(coef.as_independent(t, as_Add=True)[1] == 0 for coef in M)
def canonical_odes(eqs, funcs, t):
r"""
Function that solves for highest order derivatives in a system
Explanation
===========
This function inputs a system of ODEs and based on the system,
the dependent variables and their highest order, returns the system
in the following form:
.. math::
X'(t) = A(t) X(t) + b(t)
Here, $X(t)$ is the vector of dependent variables of lower order, $A(t)$ is
the coefficient matrix, $b(t)$ is the non-homogeneous term and $X'(t)$ is the
vector of dependent variables in their respective highest order. We use the term
canonical form to imply the system of ODEs which is of the above form.
If the system passed has a non-linear term with multiple solutions, then a list of
systems is returned in its canonical form.
Parameters
==========
eqs : List
List of the ODEs
funcs : List
List of dependent variables
t : Symbol
Independent variable
Examples
========
>>> from sympy import symbols, Function, Eq, Derivative
>>> from sympy.solvers.ode.systems import canonical_odes
>>> f, g = symbols("f g", cls=Function)
>>> x, y = symbols("x y")
>>> funcs = [f(x), g(x)]
>>> eqs = [Eq(f(x).diff(x) - 7*f(x), 12*g(x)), Eq(g(x).diff(x) + g(x), 20*f(x))]
>>> canonical_eqs = canonical_odes(eqs, funcs, x)
>>> canonical_eqs
[[Eq(Derivative(f(x), x), 7*f(x) + 12*g(x)), Eq(Derivative(g(x), x), 20*f(x) - g(x))]]
>>> system = [Eq(Derivative(f(x), x)**2 - 2*Derivative(f(x), x) + 1, 4), Eq(-y*f(x) + Derivative(g(x), x), 0)]
>>> canonical_system = canonical_odes(system, funcs, x)
>>> canonical_system
[[Eq(Derivative(f(x), x), -1), Eq(Derivative(g(x), x), y*f(x))], [Eq(Derivative(f(x), x), 3), Eq(Derivative(g(x), x), y*f(x))]]
Returns
=======
List
"""
from sympy.solvers.solvers import solve
order = _get_func_order(eqs, funcs)
canon_eqs = solve(eqs, *[func.diff(t, order[func]) for func in funcs], dict=True)
systems = []
for eq in canon_eqs:
system = [Eq(func.diff(t, order[func]), eq[func.diff(t, order[func])]) for func in funcs]
systems.append(system)
return systems
def _is_commutative_anti_derivative(A, t):
r"""
Helper function for determining if the Matrix passed is commutative with its antiderivative
Explanation
===========
This function checks if the Matrix $A$ passed is commutative with its antiderivative with respect
to the independent variable $t$.
.. math::
B(t) = \int A(t) dt
The function outputs two values, first one being the antiderivative $B(t)$, second one being a
boolean value, if True, then the matrix $A(t)$ passed is commutative with $B(t)$, else the matrix
passed isn't commutative with $B(t)$.
Parameters
==========
A : Matrix
The matrix which has to be checked
t : Symbol
Independent variable
Examples
========
>>> from sympy import symbols, Matrix
>>> from sympy.solvers.ode.systems import _is_commutative_anti_derivative
>>> t = symbols("t")
>>> A = Matrix([[1, t], [-t, 1]])
>>> B, is_commuting = _is_commutative_anti_derivative(A, t)
>>> is_commuting
True
Returns
=======
Matrix, Boolean
"""
B = integrate(A, t)
is_commuting = (B*A - A*B).applyfunc(expand).applyfunc(factor_terms).is_zero_matrix
is_commuting = False if is_commuting is None else is_commuting
return B, is_commuting
def _factor_matrix(A, t):
term = None
for element in A:
temp_term = element.as_independent(t)[1]
if temp_term.has(t):
term = temp_term
break
if term is not None:
A_factored = (A/term).applyfunc(ratsimp)
can_factor = _matrix_is_constant(A_factored, t)
term = (term, A_factored) if can_factor else None
return term
def _is_second_order_type2(A, t):
term = _factor_matrix(A, t)
is_type2 = False
if term is not None:
term = 1/term[0]
is_type2 = term.is_polynomial()
if is_type2:
poly = Poly(term.expand(), t)
monoms = poly.monoms()
if monoms[0][0] in (2, 4):
cs = _get_poly_coeffs(poly, 4)
a, b, c, d, e = cs
a1 = powdenest(sqrt(a), force=True)
c1 = powdenest(sqrt(e), force=True)
b1 = powdenest(sqrt(c - 2*a1*c1), force=True)
is_type2 = (b == 2*a1*b1) and (d == 2*b1*c1)
term = a1*t**2 + b1*t + c1
else:
is_type2 = False
return is_type2, term
def _get_poly_coeffs(poly, order):
cs = [0 for _ in range(order+1)]
for c, m in zip(poly.coeffs(), poly.monoms()):
cs[-1-m[0]] = c
return cs
def _match_second_order_type(A1, A0, t, b=None):
r"""
Works only for second order system in its canonical form.
Type 0: Constant coefficient matrix, can be simply solved by
introducing dummy variables.
Type 1: When the substitution: $U = t*X' - X$ works for reducing
the second order system to first order system.
Type 2: When the system is of the form: $poly * X'' = A*X$ where
$poly$ is square of a quadratic polynomial with respect to
*t* and $A$ is a constant coefficient matrix.
"""
match = {"type_of_equation": "type0"}
n = A1.shape[0]
if _matrix_is_constant(A1, t) and _matrix_is_constant(A0, t):
return match
if (A1 + A0*t).applyfunc(expand_mul).is_zero_matrix:
match.update({"type_of_equation": "type1", "A1": A1})
elif A1.is_zero_matrix and (b is None or b.is_zero_matrix):
is_type2, term = _is_second_order_type2(A0, t)
if is_type2:
a, b, c = _get_poly_coeffs(Poly(term, t), 2)
A = (A0*(term**2).expand()).applyfunc(ratsimp) + (b**2/4 - a*c)*eye(n, n)
tau = integrate(1/term, t)
t_ = Symbol("{}_".format(t))
match.update({"type_of_equation": "type2", "A0": A,
"g(t)": sqrt(term), "tau": tau, "is_transformed": True,
"t_": t_})
return match
def _second_order_subs_type1(A, b, funcs, t):
r"""
For a linear, second order system of ODEs, a particular substitution.
A system of the below form can be reduced to a linear first order system of
ODEs:
.. math::
X'' = A(t) * (t*X' - X) + b(t)
By substituting:
.. math:: U = t*X' - X
To get the system:
.. math:: U' = t*(A(t)*U + b(t))
Where $U$ is the vector of dependent variables, $X$ is the vector of dependent
variables in `funcs` and $X'$ is the first order derivative of $X$ with respect to
$t$. It may or may not reduce the system into linear first order system of ODEs.
Then a check is made to determine if the system passed can be reduced or not, if
this substitution works, then the system is reduced and its solved for the new
substitution. After we get the solution for $U$:
.. math:: U = a(t)
We substitute and return the reduced system:
.. math::
a(t) = t*X' - X
Parameters
==========
A: Matrix
Coefficient matrix($A(t)*t$) of the second order system of this form.
b: Matrix
Non-homogeneous term($b(t)$) of the system of ODEs.
funcs: List
List of dependent variables
t: Symbol
Independent variable of the system of ODEs.
Returns
=======
List
"""
U = Matrix([t*func.diff(t) - func for func in funcs])
sol = linodesolve(A, t, t*b)
reduced_eqs = [Eq(u, s) for s, u in zip(sol, U)]
reduced_eqs = canonical_odes(reduced_eqs, funcs, t)[0]
return reduced_eqs
def _second_order_subs_type2(A, funcs, t_):
r"""
Returns a second order system based on the coefficient matrix passed.
Explanation
===========
This function returns a system of second order ODE of the following form:
.. math::
X'' = A * X
Here, $X$ is the vector of dependent variables, but a bit modified, $A$ is the
coefficient matrix passed.
Along with returning the second order system, this function also returns the new
dependent variables with the new independent variable `t_` passed.
Parameters
==========
A: Matrix
Coefficient matrix of the system
funcs: List
List of old dependent variables
t_: Symbol
New independent variable
Returns
=======
List, List
"""
func_names = [func.func.__name__ for func in funcs]
new_funcs = [Function(Dummy("{}_".format(name)))(t_) for name in func_names]
rhss = A * Matrix(new_funcs)
new_eqs = [Eq(func.diff(t_, 2), rhs) for func, rhs in zip(new_funcs, rhss)]
return new_eqs, new_funcs
def _is_euler_system(As, t):
return all(_matrix_is_constant((A*t**i).applyfunc(ratsimp), t) for i, A in enumerate(As))
def _classify_linear_system(eqs, funcs, t, is_canon=False):
r"""
Returns a dictionary with details of the eqs if the system passed is linear
and can be classified by this function else returns None
Explanation
===========
This function takes the eqs, converts it into a form Ax = b where x is a vector of terms
containing dependent variables and their derivatives till their maximum order. If it is
possible to convert eqs into Ax = b, then all the equations in eqs are linear otherwise
they are non-linear.
To check if the equations are constant coefficient, we need to check if all the terms in
A obtained above are constant or not.
To check if the equations are homogeneous or not, we need to check if b is a zero matrix
or not.
Parameters
==========
eqs: List
List of ODEs
funcs: List
List of dependent variables
t: Symbol
Independent variable of the equations in eqs
is_canon: Boolean
If True, then this function will not try to get the
system in canonical form. Default value is False
Returns
=======
match = {
'no_of_equation': len(eqs),
'eq': eqs,
'func': funcs,
'order': order,
'is_linear': is_linear,
'is_constant': is_constant,
'is_homogeneous': is_homogeneous,
}
Dict or list of Dicts or None
Dict with values for keys:
1. no_of_equation: Number of equations
2. eq: The set of equations
3. func: List of dependent variables
4. order: A dictionary that gives the order of the
dependent variable in eqs
5. is_linear: Boolean value indicating if the set of
equations are linear or not.
6. is_constant: Boolean value indicating if the set of
equations have constant coefficients or not.
7. is_homogeneous: Boolean value indicating if the set of
equations are homogeneous or not.
8. commutative_antiderivative: Antiderivative of the coefficient
matrix if the coefficient matrix is non-constant
and commutative with its antiderivative. This key
may or may not exist.
9. is_general: Boolean value indicating if the system of ODEs is
solvable using one of the general case solvers or not.
10. rhs: rhs of the non-homogeneous system of ODEs in Matrix form. This
key may or may not exist.
11. is_higher_order: True if the system passed has an order greater than 1.
This key may or may not exist.
12. is_second_order: True if the system passed is a second order ODE. This
key may or may not exist.
This Dict is the answer returned if the eqs are linear and constant
coefficient. Otherwise, None is returned.
"""
# Error for i == 0 can be added but isn't for now
# Check for len(funcs) == len(eqs)
if len(funcs) != len(eqs):
raise ValueError("Number of functions given is not equal to the number of equations %s" % funcs)
# ValueError when functions have more than one arguments
for func in funcs:
if len(func.args) != 1:
raise ValueError("dsolve() and classify_sysode() work with "
"functions of one variable only, not %s" % func)
# Getting the func_dict and order using the helper
# function
order = _get_func_order(eqs, funcs)
system_order = max(order[func] for func in funcs)
is_higher_order = system_order > 1
is_second_order = system_order == 2 and all(order[func] == 2 for func in funcs)
# Not adding the check if the len(func.args) for
# every func in funcs is 1
# Linearity check
try:
canon_eqs = canonical_odes(eqs, funcs, t) if not is_canon else [eqs]
if len(canon_eqs) == 1:
As, b = linear_ode_to_matrix(canon_eqs[0], funcs, t, system_order)
else:
match = {
'is_implicit': True,
'canon_eqs': canon_eqs
}
return match
# When the system of ODEs is non-linear, an ODENonlinearError is raised.
# This function catches the error and None is returned.
except ODENonlinearError:
return None
is_linear = True
# Homogeneous check
is_homogeneous = True if b.is_zero_matrix else False
# Is general key is used to identify if the system of ODEs can be solved by
# one of the general case solvers or not.
match = {
'no_of_equation': len(eqs),
'eq': eqs,
'func': funcs,
'order': order,
'is_linear': is_linear,
'is_homogeneous': is_homogeneous,
'is_general': True
}
if not is_homogeneous:
match['rhs'] = b
is_constant = all(_matrix_is_constant(A_, t) for A_ in As)
# The match['is_linear'] check will be added in the future when this
# function becomes ready to deal with non-linear systems of ODEs
if not is_higher_order:
A = As[1]
match['func_coeff'] = A
# Constant coefficient check
is_constant = _matrix_is_constant(A, t)
match['is_constant'] = is_constant
try:
system_info = linodesolve_type(A, t, b=b)
except NotImplementedError:
return None
match.update(system_info)
antiderivative = match.pop("antiderivative")
if not is_constant:
match['commutative_antiderivative'] = antiderivative
return match
else:
match['type_of_equation'] = "type0"
if is_second_order:
A1, A0 = As[1:]
match_second_order = _match_second_order_type(A1, A0, t)
match.update(match_second_order)
match['is_second_order'] = True
# If system is constant, then no need to check if its in euler
# form or not. It will be easier and faster to directly proceed
# to solve it.
if match['type_of_equation'] == "type0" and not is_constant:
is_euler = _is_euler_system(As, t)
if is_euler:
t_ = Symbol('{}_'.format(t))
match.update({'is_transformed': True, 'type_of_equation': 'type1',
't_': t_})
else:
is_jordan = lambda M: M == Matrix.jordan_block(M.shape[0], M[0, 0])
terms = _factor_matrix(As[-1], t)
if all(A.is_zero_matrix for A in As[1:-1]) and terms is not None and not is_jordan(terms[1]):
P, J = terms[1].jordan_form()
match.update({'type_of_equation': 'type2', 'J': J,
'f(t)': terms[0], 'P': P, 'is_transformed': True})
if match['type_of_equation'] != 'type0' and is_second_order:
match.pop('is_second_order', None)
match['is_higher_order'] = is_higher_order
return match
def _preprocess_eqs(eqs):
processed_eqs = []
for eq in eqs:
processed_eqs.append(eq if isinstance(eq, Equality) else Eq(eq, 0))
return processed_eqs
def _eqs2dict(eqs, funcs):
eqsorig = {}
eqsmap = {}
funcset = set(funcs)
for eq in eqs:
f1, = eq.lhs.atoms(AppliedUndef)
f2s = (eq.rhs.atoms(AppliedUndef) - {f1}) & funcset
eqsmap[f1] = f2s
eqsorig[f1] = eq
return eqsmap, eqsorig
def _dict2graph(d):
nodes = list(d)
edges = [(f1, f2) for f1, f2s in d.items() for f2 in f2s]
G = (nodes, edges)
return G
def _is_type1(scc, t):
eqs, funcs = scc
try:
(A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, 1)
except (ODENonlinearError, ODEOrderError):
return False
if _matrix_is_constant(A0, t) and b.is_zero_matrix:
return True
return False
def _combine_type1_subsystems(subsystem, funcs, t):
indices = [i for i, sys in enumerate(zip(subsystem, funcs)) if _is_type1(sys, t)]
remove = set()
for ip, i in enumerate(indices):
for j in indices[ip+1:]:
if any(eq2.has(funcs[i]) for eq2 in subsystem[j]):
subsystem[j] = subsystem[i] + subsystem[j]
remove.add(i)
subsystem = [sys for i, sys in enumerate(subsystem) if i not in remove]
return subsystem
def _component_division(eqs, funcs, t):
# Assuming that each eq in eqs is in canonical form,
# that is, [f(x).diff(x) = .., g(x).diff(x) = .., etc]
# and that the system passed is in its first order
eqsmap, eqsorig = _eqs2dict(eqs, funcs)
subsystems = []
for cc in connected_components(_dict2graph(eqsmap)):
eqsmap_c = {f: eqsmap[f] for f in cc}
sccs = strongly_connected_components(_dict2graph(eqsmap_c))
subsystem = [[eqsorig[f] for f in scc] for scc in sccs]
subsystem = _combine_type1_subsystems(subsystem, sccs, t)
subsystems.append(subsystem)
return subsystems
# Returns: List of equations
def _linear_ode_solver(match):
t = match['t']
funcs = match['func']
rhs = match.get('rhs', None)
tau = match.get('tau', None)
t = match['t_'] if 't_' in match else t
A = match['func_coeff']
# Note: To make B None when the matrix has constant
# coefficient
B = match.get('commutative_antiderivative', None)
type = match['type_of_equation']
sol_vector = linodesolve(A, t, b=rhs, B=B,
type=type, tau=tau)
sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
return sol
def _select_equations(eqs, funcs, key=lambda x: x):
eq_dict = {e.lhs: e.rhs for e in eqs}
return [Eq(f, eq_dict[key(f)]) for f in funcs]
def _higher_order_ode_solver(match):
eqs = match["eq"]
funcs = match["func"]
t = match["t"]
sysorder = match['order']
type = match.get('type_of_equation', "type0")
is_second_order = match.get('is_second_order', False)
is_transformed = match.get('is_transformed', False)
is_euler = is_transformed and type == "type1"
is_higher_order_type2 = is_transformed and type == "type2" and 'P' in match
if is_second_order:
new_eqs, new_funcs = _second_order_to_first_order(eqs, funcs, t,
A1=match.get("A1", None), A0=match.get("A0", None),
b=match.get("rhs", None), type=type,
t_=match.get("t_", None))
else:
new_eqs, new_funcs = _higher_order_to_first_order(eqs, sysorder, t, funcs=funcs,
type=type, J=match.get('J', None),
f_t=match.get('f(t)', None),
P=match.get('P', None), b=match.get('rhs', None))
if is_transformed:
t = match.get('t_', t)
if not is_higher_order_type2:
new_eqs = _select_equations(new_eqs, [f.diff(t) for f in new_funcs])
sol = None
# NotImplementedError may be raised when the system may be actually
# solvable if it can be just divided into sub-systems
try:
if not is_higher_order_type2:
sol = _strong_component_solver(new_eqs, new_funcs, t)
except NotImplementedError:
sol = None
# Dividing the system only when it becomes essential
if sol is None:
try:
sol = _component_solver(new_eqs, new_funcs, t)
except NotImplementedError:
sol = None
if sol is None:
return sol
is_second_order_type2 = is_second_order and type == "type2"
underscores = '__' if is_transformed else '_'
sol = _select_equations(sol, funcs,
key=lambda x: Function(Dummy('{}{}0'.format(x.func.__name__, underscores)))(t))
if match.get("is_transformed", False):
if is_second_order_type2:
g_t = match["g(t)"]
tau = match["tau"]
sol = [Eq(s.lhs, s.rhs.subs(t, tau) * g_t) for s in sol]
elif is_euler:
t = match['t']
tau = match['t_']
sol = [s.subs(tau, log(t)) for s in sol]
elif is_higher_order_type2:
P = match['P']
sol_vector = P * Matrix([s.rhs for s in sol])
sol = [Eq(f, s) for f, s in zip(funcs, sol_vector)]
return sol
# Returns: List of equations or None
# If None is returned by this solver, then the system
# of ODEs cannot be solved directly by dsolve_system.
def _strong_component_solver(eqs, funcs, t):
from sympy.solvers.ode.ode import dsolve, constant_renumber
match = _classify_linear_system(eqs, funcs, t, is_canon=True)
sol = None
# Assuming that we can't get an implicit system
# since we are already canonical equations from
# dsolve_system
if match:
match['t'] = t
if match.get('is_higher_order', False):
sol = _higher_order_ode_solver(match)
elif match.get('is_linear', False):
sol = _linear_ode_solver(match)
# Note: For now, only linear systems are handled by this function
# hence, the match condition is added. This can be removed later.
if sol is None and len(eqs) == 1:
sol = dsolve(eqs[0], func=funcs[0])
variables = Tuple(eqs[0]).free_symbols
new_constants = [Dummy() for _ in range(ode_order(eqs[0], funcs[0]))]
sol = constant_renumber(sol, variables=variables, newconstants=new_constants)
sol = [sol]
# To add non-linear case here in future
return sol
def _get_funcs_from_canon(eqs):
return [eq.lhs.args[0] for eq in eqs]
# Returns: List of Equations(a solution)
def _weak_component_solver(wcc, t):
# We will divide the systems into sccs
# only when the wcc cannot be solved as
# a whole
eqs = []
for scc in wcc:
eqs += scc
funcs = _get_funcs_from_canon(eqs)
sol = _strong_component_solver(eqs, funcs, t)
if sol:
return sol
sol = []
for j, scc in enumerate(wcc):
eqs = scc
funcs = _get_funcs_from_canon(eqs)
# Substituting solutions for the dependent
# variables solved in previous SCC, if any solved.
comp_eqs = [eq.subs({s.lhs: s.rhs for s in sol}) for eq in eqs]
scc_sol = _strong_component_solver(comp_eqs, funcs, t)
if scc_sol is None:
raise NotImplementedError(filldedent('''
The system of ODEs passed cannot be solved by dsolve_system.
'''))
# scc_sol: List of equations
# scc_sol is a solution
sol += scc_sol
return sol
# Returns: List of Equations(a solution)
def _component_solver(eqs, funcs, t):
components = _component_division(eqs, funcs, t)
sol = []
for wcc in components:
# wcc_sol: List of Equations
sol += _weak_component_solver(wcc, t)
# sol: List of Equations
return sol
def _second_order_to_first_order(eqs, funcs, t, type="auto", A1=None,
A0=None, b=None, t_=None):
r"""
Expects the system to be in second order and in canonical form
Explanation
===========
Reduces a second order system into a first order one depending on the type of second
order system.
1. "type0": If this is passed, then the system will be reduced to first order by
introducing dummy variables.
2. "type1": If this is passed, then a particular substitution will be used to reduce the
the system into first order.
3. "type2": If this is passed, then the system will be transformed with new dependent
variables and independent variables. This transformation is a part of solving
the corresponding system of ODEs.
`A1` and `A0` are the coefficient matrices from the system and it is assumed that the
second order system has the form given below:
.. math::
A2 * X'' = A1 * X' + A0 * X + b
Here, $A2$ is the coefficient matrix for the vector $X''$ and $b$ is the non-homogeneous
term.
Default value for `b` is None but if `A1` and `A0` are passed and `b` is not passed, then the
system will be assumed homogeneous.
"""
is_a1 = A1 is None
is_a0 = A0 is None
if (type == "type1" and is_a1) or (type == "type2" and is_a0)\
or (type == "auto" and (is_a1 or is_a0)):
(A2, A1, A0), b = linear_ode_to_matrix(eqs, funcs, t, 2)
if not A2.is_Identity:
raise ValueError(filldedent('''
The system must be in its canonical form.
'''))
if type == "auto":
match = _match_second_order_type(A1, A0, t)
type = match["type_of_equation"]
A1 = match.get("A1", None)
A0 = match.get("A0", None)
sys_order = {func: 2 for func in funcs}
if type == "type1":
if b is None:
b = zeros(len(eqs))
eqs = _second_order_subs_type1(A1, b, funcs, t)
sys_order = {func: 1 for func in funcs}
if type == "type2":
if t_ is None:
t_ = Symbol("{}_".format(t))
t = t_
eqs, funcs = _second_order_subs_type2(A0, funcs, t_)
sys_order = {func: 2 for func in funcs}
return _higher_order_to_first_order(eqs, sys_order, t, funcs=funcs)
def _higher_order_type2_to_sub_systems(J, f_t, funcs, t, max_order, b=None, P=None):
# Note: To add a test for this ValueError
if J is None or f_t is None or not _matrix_is_constant(J, t):
raise ValueError(filldedent('''
Correctly input for args 'A' and 'f_t' for Linear, Higher Order,
Type 2
'''))
if P is None and b is not None and not b.is_zero_matrix:
raise ValueError(filldedent('''
Provide the keyword 'P' for matrix P in A = P * J * P-1.
'''))
new_funcs = Matrix([Function(Dummy('{}__0'.format(f.func.__name__)))(t) for f in funcs])
new_eqs = new_funcs.diff(t, max_order) - f_t * J * new_funcs
if b is not None and not b.is_zero_matrix:
new_eqs -= P.inv() * b
new_eqs = canonical_odes(new_eqs, new_funcs, t)[0]
return new_eqs, new_funcs
def _higher_order_to_first_order(eqs, sys_order, t, funcs=None, type="type0", **kwargs):
if funcs is None:
funcs = sys_order.keys()
# Standard Cauchy Euler system
if type == "type1":
t_ = Symbol('{}_'.format(t))
new_funcs = [Function(Dummy('{}_'.format(f.func.__name__)))(t_) for f in funcs]
max_order = max(sys_order[func] for func in funcs)
subs_dict = {func: new_func for func, new_func in zip(funcs, new_funcs)}
subs_dict[t] = exp(t_)
free_function = Function(Dummy())
def _get_coeffs_from_subs_expression(expr):
if isinstance(expr, Subs):
free_symbol = expr.args[1][0]
term = expr.args[0]
return {ode_order(term, free_symbol): 1}
if isinstance(expr, Mul):
coeff = expr.args[0]
order = list(_get_coeffs_from_subs_expression(expr.args[1]).keys())[0]
return {order: coeff}
if isinstance(expr, Add):
coeffs = {}
for arg in expr.args:
if isinstance(arg, Mul):
coeffs.update(_get_coeffs_from_subs_expression(arg))
else:
order = list(_get_coeffs_from_subs_expression(arg).keys())[0]
coeffs[order] = 1
return coeffs
for o in range(1, max_order + 1):
expr = free_function(log(t_)).diff(t_, o)*t_**o
coeff_dict = _get_coeffs_from_subs_expression(expr)
coeffs = [coeff_dict[order] if order in coeff_dict else 0 for order in range(o + 1)]
expr_to_subs = sum(free_function(t_).diff(t_, i) * c for i, c in
enumerate(coeffs)) / t**o
subs_dict.update({f.diff(t, o): expr_to_subs.subs(free_function(t_), nf)
for f, nf in zip(funcs, new_funcs)})
new_eqs = [eq.subs(subs_dict) for eq in eqs]
new_sys_order = {nf: sys_order[f] for f, nf in zip(funcs, new_funcs)}
new_eqs = canonical_odes(new_eqs, new_funcs, t_)[0]
return _higher_order_to_first_order(new_eqs, new_sys_order, t_, funcs=new_funcs)
# Systems of the form: X(n)(t) = f(t)*A*X + b
# where X(n)(t) is the nth derivative of the vector of dependent variables
# with respect to the independent variable and A is a constant matrix.
if type == "type2":
J = kwargs.get('J', None)
f_t = kwargs.get('f_t', None)
b = kwargs.get('b', None)
P = kwargs.get('P', None)
max_order = max(sys_order[func] for func in funcs)
return _higher_order_type2_to_sub_systems(J, f_t, funcs, t, max_order, P=P, b=b)
# Note: To be changed to this after doit option is disabled for default cases
# new_sysorder = _get_func_order(new_eqs, new_funcs)
#
# return _higher_order_to_first_order(new_eqs, new_sysorder, t, funcs=new_funcs)
new_funcs = []
for prev_func in funcs:
func_name = prev_func.func.__name__
func = Function(Dummy('{}_0'.format(func_name)))(t)
new_funcs.append(func)
subs_dict = {prev_func: func}
new_eqs = []
for i in range(1, sys_order[prev_func]):
new_func = Function(Dummy('{}_{}'.format(func_name, i)))(t)
subs_dict[prev_func.diff(t, i)] = new_func
new_funcs.append(new_func)
prev_f = subs_dict[prev_func.diff(t, i-1)]
new_eq = Eq(prev_f.diff(t), new_func)
new_eqs.append(new_eq)
eqs = [eq.subs(subs_dict) for eq in eqs] + new_eqs
return eqs, new_funcs
def dsolve_system(eqs, funcs=None, t=None, ics=None, doit=False, simplify=True):
r"""
Solves any(supported) system of Ordinary Differential Equations
Explanation
===========
This function takes a system of ODEs as an input, determines if the
it is solvable by this function, and returns the solution if found any.
This function can handle:
1. Linear, First Order, Constant coefficient homogeneous system of ODEs
2. Linear, First Order, Constant coefficient non-homogeneous system of ODEs
3. Linear, First Order, non-constant coefficient homogeneous system of ODEs
4. Linear, First Order, non-constant coefficient non-homogeneous system of ODEs
5. Any implicit system which can be divided into system of ODEs which is of the above 4 forms
6. Any higher order linear system of ODEs that can be reduced to one of the 5 forms of systems described above.
The types of systems described above are not limited by the number of equations, i.e. this
function can solve the above types irrespective of the number of equations in the system passed.
But, the bigger the system, the more time it will take to solve the system.
This function returns a list of solutions. Each solution is a list of equations where LHS is
the dependent variable and RHS is an expression in terms of the independent variable.
Among the non constant coefficient types, not all the systems are solvable by this function. Only
those which have either a coefficient matrix with a commutative antiderivative or those systems which
may be divided further so that the divided systems may have coefficient matrix with commutative antiderivative.
Parameters
==========
eqs : List
system of ODEs to be solved
funcs : List or None
List of dependent variables that make up the system of ODEs
t : Symbol or None
Independent variable in the system of ODEs
ics : Dict or None
Set of initial boundary/conditions for the system of ODEs
doit : Boolean
Evaluate the solutions if True. Default value is True. Can be
set to false if the integral evaluation takes too much time and/or
is not required.
simplify: Boolean
Simplify the solutions for the systems. Default value is True.
Can be set to false if simplification takes too much time and/or
is not required.
Examples
========
>>> from sympy import symbols, Eq, Function
>>> from sympy.solvers.ode.systems import dsolve_system
>>> f, g = symbols("f g", cls=Function)
>>> x = symbols("x")
>>> eqs = [Eq(f(x).diff(x), g(x)), Eq(g(x).diff(x), f(x))]
>>> dsolve_system(eqs)
[[Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), C1*exp(-x) + C2*exp(x))]]
You can also pass the initial conditions for the system of ODEs:
>>> dsolve_system(eqs, ics={f(0): 1, g(0): 0})
[[Eq(f(x), exp(x)/2 + exp(-x)/2), Eq(g(x), exp(x)/2 - exp(-x)/2)]]
Optionally, you can pass the dependent variables and the independent
variable for which the system is to be solved:
>>> funcs = [f(x), g(x)]
>>> dsolve_system(eqs, funcs=funcs, t=x)
[[Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), C1*exp(-x) + C2*exp(x))]]
Lets look at an implicit system of ODEs:
>>> eqs = [Eq(f(x).diff(x)**2, g(x)**2), Eq(g(x).diff(x), g(x))]
>>> dsolve_system(eqs)
[[Eq(f(x), C1 - C2*exp(x)), Eq(g(x), C2*exp(x))], [Eq(f(x), C1 + C2*exp(x)), Eq(g(x), C2*exp(x))]]
Returns
=======
List of List of Equations
Raises
======
NotImplementedError
When the system of ODEs is not solvable by this function.
ValueError
When the parameters passed are not in the required form.
"""
from sympy.solvers.ode.ode import solve_ics, _extract_funcs, constant_renumber
if not iterable(eqs):
raise ValueError(filldedent('''
List of equations should be passed. The input is not valid.
'''))
eqs = _preprocess_eqs(eqs)
if funcs is not None and not isinstance(funcs, list):
raise ValueError(filldedent('''
Input to the funcs should be a list of functions.
'''))
if funcs is None:
funcs = _extract_funcs(eqs)
if any(len(func.args) != 1 for func in funcs):
raise ValueError(filldedent('''
dsolve_system can solve a system of ODEs with only one independent
variable.
'''))
if len(eqs) != len(funcs):
raise ValueError(filldedent('''
Number of equations and number of functions do not match
'''))
if t is not None and not isinstance(t, Symbol):
raise ValueError(filldedent('''
The indepedent variable must be of type Symbol
'''))
if t is None:
t = list(list(eqs[0].atoms(Derivative))[0].atoms(Symbol))[0]
sols = []
canon_eqs = canonical_odes(eqs, funcs, t)
for canon_eq in canon_eqs:
try:
sol = _strong_component_solver(canon_eq, funcs, t)
except NotImplementedError:
sol = None
if sol is None:
sol = _component_solver(canon_eq, funcs, t)
sols.append(sol)
if sols:
final_sols = []
variables = Tuple(*eqs).free_symbols
for sol in sols:
sol = _select_equations(sol, funcs)
sol = constant_renumber(sol, variables=variables)
if ics:
constants = Tuple(*sol).free_symbols - variables
solved_constants = solve_ics(sol, funcs, constants, ics)
sol = [s.subs(solved_constants) for s in sol]
if simplify:
constants = Tuple(*sol).free_symbols - variables
sol = simpsol(sol, [t], constants, doit=doit)
final_sols.append(sol)
sols = final_sols
return sols
|
14d4dd53e32f6946b4e2438ba3aa6994884cd3445db9b33539f2958d005324ea | from math import isclose
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import (Function, Lambda, nfloat, diff)
from sympy.core.mod import Mod
from sympy.core.numbers import (E, I, Rational, oo, pi, Integer)
from sympy.core.relational import (Eq, Gt, Ne, Ge)
from sympy.core.singleton import S
from sympy.core.sorting import ordered
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.complexes import (Abs, arg, im, re, sign, conjugate)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (HyperbolicFunction,
sinh, tanh, cosh, sech, coth)
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (
TrigonometricFunction, acos, acot, acsc, asec, asin, atan, atan2,
cos, cot, csc, sec, sin, tan)
from sympy.functions.special.error_functions import (erf, erfc,
erfcinv, erfinv)
from sympy.logic.boolalg import And
from sympy.matrices.dense import MutableDenseMatrix as Matrix
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.polys.polytools import Poly
from sympy.polys.rootoftools import CRootOf
from sympy.sets.contains import Contains
from sympy.sets.conditionset import ConditionSet
from sympy.sets.fancysets import ImageSet, Range
from sympy.sets.sets import (Complement, FiniteSet,
Intersection, Interval, Union, imageset, ProductSet)
from sympy.simplify import simplify
from sympy.tensor.indexed import Indexed
from sympy.utilities.iterables import numbered_symbols
from sympy.testing.pytest import (XFAIL, raises, skip, slow, SKIP, _both_exp_pow)
from sympy.core.random import verify_numerically as tn
from sympy.physics.units import cm
from sympy.solvers import solve
from sympy.solvers.solveset import (
solveset_real, domain_check, solveset_complex, linear_eq_to_matrix,
linsolve, _is_function_class_equation, invert_real, invert_complex,
solveset, solve_decomposition, substitution, nonlinsolve, solvify,
_is_finite_with_finite_vars, _transolve, _is_exponential,
_solve_exponential, _is_logarithmic, _is_lambert,
_solve_logarithm, _term_factors, _is_modular, NonlinearError)
from sympy.abc import (a, b, c, d, e, f, g, h, i, j, k, l, m, n, q, r,
t, w, x, y, z)
def dumeq(i, j):
if type(i) in (list, tuple):
return all(dumeq(i, j) for i, j in zip(i, j))
return i == j or i.dummy_eq(j)
def assert_close_ss(sol1, sol2):
"""Test solutions with floats from solveset are close"""
sol1 = sympify(sol1)
sol2 = sympify(sol2)
assert isinstance(sol1, FiniteSet)
assert isinstance(sol2, FiniteSet)
assert len(sol1) == len(sol2)
assert all(isclose(v1, v2) for v1, v2 in zip(sol1, sol2))
def assert_close_nl(sol1, sol2):
"""Test solutions with floats from nonlinsolve are close"""
sol1 = sympify(sol1)
sol2 = sympify(sol2)
assert isinstance(sol1, FiniteSet)
assert isinstance(sol2, FiniteSet)
assert len(sol1) == len(sol2)
for s1, s2 in zip(sol1, sol2):
assert len(s1) == len(s2)
assert all(isclose(v1, v2) for v1, v2 in zip(s1, s2))
@_both_exp_pow
def test_invert_real():
x = Symbol('x', real=True)
def ireal(x, s=S.Reals):
return Intersection(s, x)
assert invert_real(exp(x), z, x) == (x, ireal(FiniteSet(log(z))))
y = Symbol('y', positive=True)
n = Symbol('n', real=True)
assert invert_real(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_real(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_real(exp(x), y, x) == (x, FiniteSet(log(y)))
assert invert_real(exp(3*x), y, x) == (x, FiniteSet(log(y) / 3))
assert invert_real(exp(x + 3), y, x) == (x, FiniteSet(log(y) - 3))
assert invert_real(exp(x) + 3, y, x) == (x, ireal(FiniteSet(log(y - 3))))
assert invert_real(exp(x)*3, y, x) == (x, FiniteSet(log(y / 3)))
assert invert_real(log(x), y, x) == (x, FiniteSet(exp(y)))
assert invert_real(log(3*x), y, x) == (x, FiniteSet(exp(y) / 3))
assert invert_real(log(x + 3), y, x) == (x, FiniteSet(exp(y) - 3))
assert invert_real(Abs(x), y, x) == (x, FiniteSet(y, -y))
assert invert_real(2**x, y, x) == (x, FiniteSet(log(y)/log(2)))
assert invert_real(2**exp(x), y, x) == (x, ireal(FiniteSet(log(log(y)/log(2)))))
assert invert_real(x**2, y, x) == (x, FiniteSet(sqrt(y), -sqrt(y)))
assert invert_real(x**S.Half, y, x) == (x, FiniteSet(y**2))
raises(ValueError, lambda: invert_real(x, x, x))
# issue 21236
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
assert invert_real(x**pi, -E, x) == (x, S.EmptySet)
assert invert_real(x**Rational(3/2), 1000, x) == (x, FiniteSet(100))
assert invert_real(x**1.0, 1, x) == (x**1.0, FiniteSet(1))
raises(ValueError, lambda: invert_real(S.One, y, x))
assert invert_real(x**31 + x, y, x) == (x**31 + x, FiniteSet(y))
lhs = x**31 + x
base_values = FiniteSet(y - 1, -y - 1)
assert invert_real(Abs(x**31 + x + 1), y, x) == (lhs, base_values)
assert dumeq(invert_real(sin(x), y, x),
(x, imageset(Lambda(n, n*pi + (-1)**n*asin(y)), S.Integers)))
assert dumeq(invert_real(sin(exp(x)), y, x),
(x, imageset(Lambda(n, log((-1)**n*asin(y) + n*pi)), S.Integers)))
assert dumeq(invert_real(csc(x), y, x),
(x, imageset(Lambda(n, n*pi + (-1)**n*acsc(y)), S.Integers)))
assert dumeq(invert_real(csc(exp(x)), y, x),
(x, imageset(Lambda(n, log((-1)**n*acsc(y) + n*pi)), S.Integers)))
assert dumeq(invert_real(cos(x), y, x),
(x, Union(imageset(Lambda(n, 2*n*pi + acos(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - acos(y)), S.Integers))))
assert dumeq(invert_real(cos(exp(x)), y, x),
(x, Union(imageset(Lambda(n, log(2*n*pi + acos(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - acos(y))), S.Integers))))
assert dumeq(invert_real(sec(x), y, x),
(x, Union(imageset(Lambda(n, 2*n*pi + asec(y)), S.Integers), \
imageset(Lambda(n, 2*n*pi - asec(y)), S.Integers))))
assert dumeq(invert_real(sec(exp(x)), y, x),
(x, Union(imageset(Lambda(n, log(2*n*pi + asec(y))), S.Integers), \
imageset(Lambda(n, log(2*n*pi - asec(y))), S.Integers))))
assert dumeq(invert_real(tan(x), y, x),
(x, imageset(Lambda(n, n*pi + atan(y)), S.Integers)))
assert dumeq(invert_real(tan(exp(x)), y, x),
(x, imageset(Lambda(n, log(n*pi + atan(y))), S.Integers)))
assert dumeq(invert_real(cot(x), y, x),
(x, imageset(Lambda(n, n*pi + acot(y)), S.Integers)))
assert dumeq(invert_real(cot(exp(x)), y, x),
(x, imageset(Lambda(n, log(n*pi + acot(y))), S.Integers)))
assert dumeq(invert_real(tan(tan(x)), y, x),
(tan(x), imageset(Lambda(n, n*pi + atan(y)), S.Integers)))
x = Symbol('x', positive=True)
assert invert_real(x**pi, y, x) == (x, FiniteSet(y**(1/pi)))
def test_invert_complex():
assert invert_complex(x + 3, y, x) == (x, FiniteSet(y - 3))
assert invert_complex(x*3, y, x) == (x, FiniteSet(y / 3))
assert invert_complex((x - 1)**3, 0, x) == (x, FiniteSet(1))
assert dumeq(invert_complex(exp(x), y, x),
(x, imageset(Lambda(n, I*(2*pi*n + arg(y)) + log(Abs(y))), S.Integers)))
assert invert_complex(log(x), y, x) == (x, FiniteSet(exp(y)))
raises(ValueError, lambda: invert_real(1, y, x))
raises(ValueError, lambda: invert_complex(x, x, x))
raises(ValueError, lambda: invert_complex(x, x, 1))
# https://github.com/skirpichev/omg/issues/16
assert invert_complex(sinh(x), 0, x) != (x, FiniteSet(0))
def test_domain_check():
assert domain_check(1/(1 + (1/(x+1))**2), x, -1) is False
assert domain_check(x**2, x, 0) is True
assert domain_check(x, x, oo) is False
assert domain_check(0, x, oo) is False
def test_issue_11536():
assert solveset(0**x - 100, x, S.Reals) == S.EmptySet
assert solveset(0**x - 1, x, S.Reals) == FiniteSet(0)
def test_issue_17479():
f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2)
fx = f.diff(x)
fy = f.diff(y)
fz = f.diff(z)
sol = nonlinsolve([fx, fy, fz], [x, y, z])
assert len(sol) >= 4 and len(sol) <= 20
# nonlinsolve has been giving a varying number of solutions
# (originally 18, then 20, now 19) due to various internal changes.
# Unfortunately not all the solutions are actually valid and some are
# redundant. Since the original issue was that an exception was raised,
# this first test only checks that nonlinsolve returns a "plausible"
# solution set. The next test checks the result for correctness.
@XFAIL
def test_issue_18449():
x, y, z = symbols("x, y, z")
f = (x**2 + y**2)**2 + (x**2 + z**2)**2 - 2*(2*x**2 + y**2 + z**2)
fx = diff(f, x)
fy = diff(f, y)
fz = diff(f, z)
sol = nonlinsolve([fx, fy, fz], [x, y, z])
for (xs, ys, zs) in sol:
d = {x: xs, y: ys, z: zs}
assert tuple(_.subs(d).simplify() for _ in (fx, fy, fz)) == (0, 0, 0)
# After simplification and removal of duplicate elements, there should
# only be 4 parametric solutions left:
# simplifiedsolutions = FiniteSet((sqrt(1 - z**2), z, z),
# (-sqrt(1 - z**2), z, z),
# (sqrt(1 - z**2), -z, z),
# (-sqrt(1 - z**2), -z, z))
# TODO: Is the above solution set definitely complete?
def test_issue_21047():
f = (2 - x)**2 + (sqrt(x - 1) - 1)**6
assert solveset(f, x, S.Reals) == FiniteSet(2)
f = (sqrt(x)-1)**2 + (sqrt(x)+1)**2 -2*x**2 + sqrt(2)
assert solveset(f, x, S.Reals) == FiniteSet(
S.Half - sqrt(2*sqrt(2) + 5)/2, S.Half + sqrt(2*sqrt(2) + 5)/2)
def test_is_function_class_equation():
assert _is_function_class_equation(TrigonometricFunction,
tan(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + sin(x) - a, x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x + a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
sin(x)*tan(x*a) + sin(x), x) is True
assert _is_function_class_equation(TrigonometricFunction,
a*tan(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**2 + sin(x) - 1, x) is True
assert _is_function_class_equation(TrigonometricFunction,
tan(x) + x, x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x**2) + sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(x)**sin(x), x) is False
assert _is_function_class_equation(TrigonometricFunction,
tan(sin(x)) + sin(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + sinh(x) - a, x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x + a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
sinh(x)*tanh(x*a) + sinh(x), x) is True
assert _is_function_class_equation(HyperbolicFunction,
a*tanh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**2 + sinh(x) - 1, x) is True
assert _is_function_class_equation(HyperbolicFunction,
tanh(x) + x, x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x**2) + sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(x)**sinh(x), x) is False
assert _is_function_class_equation(HyperbolicFunction,
tanh(sinh(x)) + sinh(x), x) is False
def test_garbage_input():
raises(ValueError, lambda: solveset_real([y], y))
x = Symbol('x', real=True)
assert solveset_real(x, 1) == S.EmptySet
assert solveset_real(x - 1, 1) == FiniteSet(x)
assert solveset_real(x, pi) == S.EmptySet
assert solveset_real(x, x**2) == S.EmptySet
raises(ValueError, lambda: solveset_complex([x], x))
assert solveset_complex(x, pi) == S.EmptySet
raises(ValueError, lambda: solveset((x, y), x))
raises(ValueError, lambda: solveset(x + 1, S.Reals))
raises(ValueError, lambda: solveset(x + 1, x, 2))
def test_solve_mul():
assert solveset_real((a*x + b)*(exp(x) - 3), x) == \
Union({log(3)}, Intersection({-b/a}, S.Reals))
anz = Symbol('anz', nonzero=True)
bb = Symbol('bb', real=True)
assert solveset_real((anz*x + bb)*(exp(x) - 3), x) == \
FiniteSet(-bb/anz, log(3))
assert solveset_real((2*x + 8)*(8 + exp(x)), x) == FiniteSet(S(-4))
assert solveset_real(x/log(x), x) is S.EmptySet
def test_solve_invert():
assert solveset_real(exp(x) - 3, x) == FiniteSet(log(3))
assert solveset_real(log(x) - 3, x) == FiniteSet(exp(3))
assert solveset_real(3**(x + 2), x) == FiniteSet()
assert solveset_real(3**(2 - x), x) == FiniteSet()
assert solveset_real(y - b*exp(a/x), x) == Intersection(
S.Reals, FiniteSet(a/log(y/b)))
# issue 4504
assert solveset_real(2**x - 10, x) == FiniteSet(1 + log(5)/log(2))
def test_errorinverses():
assert solveset_real(erf(x) - S.Half, x) == \
FiniteSet(erfinv(S.Half))
assert solveset_real(erfinv(x) - 2, x) == \
FiniteSet(erf(2))
assert solveset_real(erfc(x) - S.One, x) == \
FiniteSet(erfcinv(S.One))
assert solveset_real(erfcinv(x) - 2, x) == FiniteSet(erfc(2))
def test_solve_polynomial():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset_real(3*x - 2, x) == FiniteSet(Rational(2, 3))
assert solveset_real(x**2 - 1, x) == FiniteSet(-S.One, S.One)
assert solveset_real(x - y**3, x) == FiniteSet(y ** 3)
assert solveset_real(x**3 - 15*x - 4, x) == FiniteSet(
-2 + 3 ** S.Half,
S(4),
-2 - 3 ** S.Half)
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert len(solveset_real(x**5 + x**3 + 1, x)) == 1
assert len(solveset_real(-2*x**3 + 4*x**2 - 2*x + 6, x)) > 0
assert solveset_real(x**6 + x**4 + I, x) is S.EmptySet
def test_return_root_of():
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get CRootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(list(solveset_complex(x**5 + 3*x**3 + 7, x))[0],
exponent=False) == CRootOf(x**5 + 3*x**3 + 7, 0).n()
sol = list(solveset_complex(x**6 - 2*x + 2, x))
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = list(solveset_complex(f, x))
for root in s:
assert root.func == CRootOf
s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4)
assert solveset_complex(s, x) == \
FiniteSet(*Poly(s*4, domain='ZZ').all_roots())
# Refer issue #7876
eq = x*(x - 1)**2*(x + 1)*(x**6 - x + 1)
assert solveset_complex(eq, x) == \
FiniteSet(-1, 0, 1, CRootOf(x**6 - x + 1, 0),
CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2),
CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4),
CRootOf(x**6 - x + 1, 5))
def test_solveset_sqrt_1():
assert solveset_real(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S.One, S(2))
assert solveset_real(sqrt(x - 1) - x + 7, x) == FiniteSet(10)
assert solveset_real(sqrt(x - 2) - 5, x) == FiniteSet(27)
assert solveset_real(sqrt(x) - 2 - 5, x) == FiniteSet(49)
assert solveset_real(sqrt(x**3), x) == FiniteSet(0)
assert solveset_real(sqrt(x - 1), x) == FiniteSet(1)
assert solveset_real(sqrt((x-3)/x), x) == FiniteSet(3)
assert solveset_real(sqrt((x-3)/x)-Rational(1, 2), x) == \
FiniteSet(4)
def test_solveset_sqrt_2():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
# http://tutorial.math.lamar.edu/Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solveset_real(sqrt(2*x - 1) - sqrt(x - 4) - 2, x) == \
FiniteSet(S(5), S(13))
assert solveset_real(sqrt(x + 7) + 2 - sqrt(3 - x), x) == \
FiniteSet(-6)
# http://www.purplemath.com/modules/solverad.htm
assert solveset_real(sqrt(17*x - sqrt(x**2 - 5)) - 7, x) == \
FiniteSet(3)
eq = x + 1 - (x**4 + 4*x**3 - x)**Rational(1, 4)
assert solveset_real(eq, x) == FiniteSet(Rational(-1, 2), Rational(-1, 3))
eq = sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)
assert solveset_real(eq, x) == FiniteSet(0)
eq = sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)
assert solveset_real(eq, x) == FiniteSet(5)
eq = sqrt(x)*sqrt(x - 7) - 12
assert solveset_real(eq, x) == FiniteSet(16)
eq = sqrt(x - 3) + sqrt(x) - 3
assert solveset_real(eq, x) == FiniteSet(4)
eq = sqrt(2*x**2 - 7) - (3 - x)
assert solveset_real(eq, x) == FiniteSet(-S(8), S(2))
# others
eq = sqrt(9*x**2 + 4) - (3*x + 2)
assert solveset_real(eq, x) == FiniteSet(0)
assert solveset_real(sqrt(x - 3) - sqrt(x) - 3, x) == FiniteSet()
eq = (2*x - 5)**Rational(1, 3) - 3
assert solveset_real(eq, x) == FiniteSet(16)
assert solveset_real(sqrt(x) + sqrt(sqrt(x)) - 4, x) == \
FiniteSet((Rational(-1, 2) + sqrt(17)/2)**4)
eq = sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))
assert solveset_real(eq, x) == FiniteSet()
eq = (x - 4)**2 + (sqrt(x) - 2)**4
assert solveset_real(eq, x) == FiniteSet(-4, 4)
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
ans = solveset_real(eq, x)
ra = S('''-1484/375 - 4*(-S(1)/2 + sqrt(3)*I/2)*(-12459439/52734375 +
114*sqrt(12657)/78125)**(S(1)/3) - 172564/(140625*(-S(1)/2 +
sqrt(3)*I/2)*(-12459439/52734375 + 114*sqrt(12657)/78125)**(S(1)/3))''')
rb = Rational(4, 5)
assert all(abs(eq.subs(x, i).n()) < 1e-10 for i in (ra, rb)) and \
len(ans) == 2 and \
{i.n(chop=True) for i in ans} == \
{i.n(chop=True) for i in (ra, rb)}
assert solveset_real(sqrt(x) + x**Rational(1, 3) +
x**Rational(1, 4), x) == FiniteSet(0)
assert solveset_real(x/sqrt(x**2 + 1), x) == FiniteSet(0)
eq = (x - y**3)/((y**2)*sqrt(1 - y**2))
assert solveset_real(eq, x) == FiniteSet(y**3)
# issue 4497
assert solveset_real(1/(5 + x)**Rational(1, 5) - 9, x) == \
FiniteSet(Rational(-295244, 59049))
@XFAIL
def test_solve_sqrt_fail():
# this only works if we check real_root(eq.subs(x, Rational(1, 3)))
# but checksol doesn't work like that
eq = (x**3 - 3*x**2)**Rational(1, 3) + 1 - x
assert solveset_real(eq, x) == FiniteSet(Rational(1, 3))
@slow
def test_solve_sqrt_3():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solveset_complex(eq, R)
fset = [Rational(5, 3) + 4*sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3,
-sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 +
40*re(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 +
sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 + Rational(5, 3) +
I*(-sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
40*im(1/((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9)]
cset = [40*re(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 -
sqrt(10)*cos(atan(3*sqrt(111)/251)/3)/3 - sqrt(30)*sin(atan(3*sqrt(111)/251)/3)/3 +
Rational(5, 3) +
I*(40*im(1/((Rational(-1, 2) + sqrt(3)*I/2)*(Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3)))/9 -
sqrt(10)*sin(atan(3*sqrt(111)/251)/3)/3 +
sqrt(30)*cos(atan(3*sqrt(111)/251)/3)/3)]
assert sol._args[0] == FiniteSet(*fset)
assert sol._args[1] == ConditionSet(
R,
Eq(sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1), 0),
FiniteSet(*cset))
# the number of real roots will depend on the value of m: for m=1 there are 4
# and for m=-1 there are none.
eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2)
unsolved_object = ConditionSet(q, Eq(sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) -
sqrt((-m**2/2 - sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m -
sqrt(4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2), 0), S.Reals)
assert solveset_real(eq, q) == unsolved_object
def test_solve_polynomial_symbolic_param():
assert solveset_complex((x**2 - 1)**2 - a, x) == \
FiniteSet(sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a)))
# issue 4507
assert solveset_complex(y - b/(1 + a*x), x) == \
FiniteSet((b/y - 1)/a) - FiniteSet(-1/a)
# issue 4508
assert solveset_complex(y - b*x/(a + x), x) == \
FiniteSet(-a*y/(y - b)) - FiniteSet(-a)
def test_solve_rational():
assert solveset_real(1/x + 1, x) == FiniteSet(-S.One)
assert solveset_real(1/exp(x) - 1, x) == FiniteSet(0)
assert solveset_real(x*(1 - 5/x), x) == FiniteSet(5)
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
assert solveset_real((x**2/(7 - x)).diff(x), x) == \
FiniteSet(S.Zero, S(14))
def test_solveset_real_gen_is_pow():
assert solveset_real(sqrt(1) + 1, x) is S.EmptySet
def test_no_sol():
assert solveset(1 - oo*x) is S.EmptySet
assert solveset(oo*x, x) is S.EmptySet
assert solveset(oo*x - oo, x) is S.EmptySet
assert solveset_real(4, x) is S.EmptySet
assert solveset_real(exp(x), x) is S.EmptySet
assert solveset_real(x**2 + 1, x) is S.EmptySet
assert solveset_real(-3*a/sqrt(x), x) is S.EmptySet
assert solveset_real(1/x, x) is S.EmptySet
assert solveset_real(-(1 + x)/(2 + x)**2 + 1/(2 + x), x
) is S.EmptySet
def test_sol_zero_real():
assert solveset_real(0, x) == S.Reals
assert solveset(0, x, Interval(1, 2)) == Interval(1, 2)
assert solveset_real(-x**2 - 2*x + (x + 1)**2 - 1, x) == S.Reals
def test_no_sol_rational_extragenous():
assert solveset_real((x/(x + 1) + 3)**(-2), x) is S.EmptySet
assert solveset_real((x - 1)/(1 + 1/(x - 1)), x) is S.EmptySet
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to
a polynomial equation using the change of variable y -> x**Rational(p, q)
"""
assert solveset_real(sqrt(x) - 1, x) == FiniteSet(1)
assert solveset_real(sqrt(x) - 2, x) == FiniteSet(4)
assert solveset_real(x**Rational(1, 4) - 2, x) == FiniteSet(16)
assert solveset_real(x**Rational(1, 3) - 3, x) == FiniteSet(27)
assert solveset_real(x*(x**(S.One / 3) - 3), x) == \
FiniteSet(S.Zero, S(27))
def test_solveset_real_rational():
"""Test solveset_real for rational functions"""
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset_real((x - y**3) / ((y**2)*sqrt(1 - y**2)), x) \
== FiniteSet(y**3)
# issue 4486
assert solveset_real(2*x/(x + 2) - 1, x) == FiniteSet(2)
def test_solveset_real_log():
assert solveset_real(log((x-1)*(x+1)), x) == \
FiniteSet(sqrt(2), -sqrt(2))
def test_poly_gens():
assert solveset_real(4**(2*(x**2) + 2*x) - 8, x) == \
FiniteSet(Rational(-3, 2), S.Half)
def test_solve_abs():
n = Dummy('n')
raises(ValueError, lambda: solveset(Abs(x) - 1, x))
assert solveset(Abs(x) - n, x, S.Reals).dummy_eq(
ConditionSet(x, Contains(n, Interval(0, oo)), {-n, n}))
assert solveset_real(Abs(x) - 2, x) == FiniteSet(-2, 2)
assert solveset_real(Abs(x) + 2, x) is S.EmptySet
assert solveset_real(Abs(x + 3) - 2*Abs(x - 3), x) == \
FiniteSet(1, 9)
assert solveset_real(2*Abs(x) - Abs(x - 1), x) == \
FiniteSet(-1, Rational(1, 3))
sol = ConditionSet(
x,
And(
Contains(b, Interval(0, oo)),
Contains(a + b, Interval(0, oo)),
Contains(a - b, Interval(0, oo))),
FiniteSet(-a - b - 3, -a + b - 3, a - b - 3, a + b - 3))
eq = Abs(Abs(x + 3) - a) - b
assert invert_real(eq, 0, x)[1] == sol
reps = {a: 3, b: 1}
eqab = eq.subs(reps)
for si in sol.subs(reps):
assert not eqab.subs(x, si)
assert dumeq(solveset(Eq(sin(Abs(x)), 1), x, domain=S.Reals), Union(
Intersection(Interval(0, oo),
ImageSet(Lambda(n, (-1)**n*pi/2 + n*pi), S.Integers)),
Intersection(Interval(-oo, 0),
ImageSet(Lambda(n, n*pi - (-1)**(-n)*pi/2), S.Integers))))
def test_issue_9824():
assert dumeq(solveset(sin(x)**2 - 2*sin(x) + 1, x), ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers))
assert dumeq(solveset(cos(x)**2 - 2*cos(x) + 1, x), ImageSet(Lambda(n, 2*n*pi), S.Integers))
def test_issue_9565():
assert solveset_real(Abs((x - 1)/(x - 5)) <= Rational(1, 3), x) == Interval(-1, 2)
def test_issue_10069():
eq = abs(1/(x - 1)) - 1 > 0
assert solveset_real(eq, x) == Union(
Interval.open(0, 1), Interval.open(1, 2))
def test_real_imag_splitting():
a, b = symbols('a b', real=True)
assert solveset_real(sqrt(a**2 - b**2) - 3, a) == \
FiniteSet(-sqrt(b**2 + 9), sqrt(b**2 + 9))
assert solveset_real(sqrt(a**2 + b**2) - 3, a) != \
S.EmptySet
def test_units():
assert solveset_real(1/x - 1/(2*cm), x) == FiniteSet(2*cm)
def test_solve_only_exp_1():
y = Symbol('y', positive=True)
assert solveset_real(exp(x) - y, x) == FiniteSet(log(y))
assert solveset_real(exp(x) + exp(-x) - 4, x) == \
FiniteSet(log(-sqrt(3) + 2), log(sqrt(3) + 2))
assert solveset_real(exp(x) + exp(-x) - y, x) != S.EmptySet
def test_atan2():
# The .inverse() method on atan2 works only if x.is_real is True and the
# second argument is a real constant
assert solveset_real(atan2(x, 2) - pi/3, x) == FiniteSet(2*sqrt(3))
def test_piecewise_solveset():
eq = Piecewise((x - 2, Gt(x, 2)), (2 - x, True)) - 3
assert set(solveset_real(eq, x)) == set(FiniteSet(-1, 5))
absxm3 = Piecewise(
(x - 3, 0 <= x - 3),
(3 - x, 0 > x - 3))
y = Symbol('y', positive=True)
assert solveset_real(absxm3 - y, x) == FiniteSet(-y + 3, y + 3)
f = Piecewise(((x - 2)**2, x >= 0), (0, True))
assert solveset(f, x, domain=S.Reals) == Union(FiniteSet(2), Interval(-oo, 0, True, True))
assert solveset(
Piecewise((x + 1, x > 0), (I, True)) - I, x, S.Reals
) == Interval(-oo, 0)
assert solveset(Piecewise((x - 1, Ne(x, I)), (x, True)), x) == FiniteSet(1)
# issue 19718
g = Piecewise((1, x > 10), (0, True))
assert solveset(g > 0, x, S.Reals) == Interval.open(10, oo)
from sympy.logic.boolalg import BooleanTrue
f = BooleanTrue()
assert solveset(f, x, domain=Interval(-3, 10)) == Interval(-3, 10)
# issue 20552
f = Piecewise((0, Eq(x, 0)), (x**2/Abs(x), True))
g = Piecewise((0, Eq(x, pi)), ((x - pi)/sin(x), True))
assert solveset(f, x, domain=S.Reals) == FiniteSet(0)
assert solveset(g) == FiniteSet(pi)
def test_solveset_complex_polynomial():
assert solveset_complex(a*x**2 + b*x + c, x) == \
FiniteSet(-b/(2*a) - sqrt(-4*a*c + b**2)/(2*a),
-b/(2*a) + sqrt(-4*a*c + b**2)/(2*a))
assert solveset_complex(x - y**3, y) == FiniteSet(
(-x**Rational(1, 3))/2 + I*sqrt(3)*x**Rational(1, 3)/2,
x**Rational(1, 3),
(-x**Rational(1, 3))/2 - I*sqrt(3)*x**Rational(1, 3)/2)
assert solveset_complex(x + 1/x - 1, x) == \
FiniteSet(S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2)
def test_sol_zero_complex():
assert solveset_complex(0, x) is S.Complexes
def test_solveset_complex_rational():
assert solveset_complex((x - 1)*(x - I)/(x - 3), x) == \
FiniteSet(1, I)
assert solveset_complex((x - y**3)/((y**2)*sqrt(1 - y**2)), x) == \
FiniteSet(y**3)
assert solveset_complex(-x**2 - I, x) == \
FiniteSet(-sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2)
def test_solve_quintics():
skip("This test is too slow")
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
f = x**5 + 15*x + 12
s = solveset_complex(f, x)
for root in s:
res = f.subs(x, root.n()).n()
assert tn(res, 0)
def test_solveset_complex_exp():
assert dumeq(solveset_complex(exp(x) - 1, x),
imageset(Lambda(n, I*2*n*pi), S.Integers))
assert dumeq(solveset_complex(exp(x) - I, x),
imageset(Lambda(n, I*(2*n*pi + pi/2)), S.Integers))
assert solveset_complex(1/exp(x), x) == S.EmptySet
assert dumeq(solveset_complex(sinh(x).rewrite(exp), x),
imageset(Lambda(n, n*pi*I), S.Integers))
def test_solveset_real_exp():
assert solveset(Eq((-2)**x, 4), x, S.Reals) == FiniteSet(2)
assert solveset(Eq(-2**x, 4), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**x, 27), x, S.Reals) == S.EmptySet
assert solveset(Eq((-5)**(x+1), 625), x, S.Reals) == FiniteSet(3)
assert solveset(Eq(2**(x-3), -16), x, S.Reals) == S.EmptySet
assert solveset(Eq((-3)**(x - 3), -3**39), x, S.Reals) == FiniteSet(42)
assert solveset(Eq(2**x, y), x, S.Reals) == Intersection(S.Reals, FiniteSet(log(y)/log(2)))
assert invert_real((-2)**(2*x) - 16, 0, x) == (x, FiniteSet(2))
def test_solve_complex_log():
assert solveset_complex(log(x), x) == FiniteSet(1)
assert solveset_complex(1 - log(a + 4*x**2), x) == \
FiniteSet(-sqrt(-a + E)/2, sqrt(-a + E)/2)
def test_solve_complex_sqrt():
assert solveset_complex(sqrt(5*x + 6) - 2 - x, x) == \
FiniteSet(-S.One, S(2))
assert solveset_complex(sqrt(5*x + 6) - (2 + 2*I) - x, x) == \
FiniteSet(-S(2), 3 - 4*I)
assert solveset_complex(4*x*(1 - a * sqrt(x)), x) == \
FiniteSet(S.Zero, 1 / a ** 2)
def test_solveset_complex_tan():
s = solveset_complex(tan(x).rewrite(exp), x)
assert dumeq(s, imageset(Lambda(n, pi*n), S.Integers) - \
imageset(Lambda(n, pi*n + pi/2), S.Integers))
@_both_exp_pow
def test_solve_trig():
assert dumeq(solveset_real(sin(x), x),
Union(imageset(Lambda(n, 2*pi*n), S.Integers),
imageset(Lambda(n, 2*pi*n + pi), S.Integers)))
assert dumeq(solveset_real(sin(x) - 1, x),
imageset(Lambda(n, 2*pi*n + pi/2), S.Integers))
assert dumeq(solveset_real(cos(x), x),
Union(imageset(Lambda(n, 2*pi*n + pi/2), S.Integers),
imageset(Lambda(n, 2*pi*n + pi*Rational(3, 2)), S.Integers)))
assert dumeq(solveset_real(sin(x) + cos(x), x),
Union(imageset(Lambda(n, 2*n*pi + pi*Rational(3, 4)), S.Integers),
imageset(Lambda(n, 2*n*pi + pi*Rational(7, 4)), S.Integers)))
assert solveset_real(sin(x)**2 + cos(x)**2, x) == S.EmptySet
assert dumeq(solveset_complex(cos(x) - S.Half, x),
Union(imageset(Lambda(n, 2*n*pi + pi*Rational(5, 3)), S.Integers),
imageset(Lambda(n, 2*n*pi + pi/3), S.Integers)))
assert dumeq(solveset(sin(y + a) - sin(y), a, domain=S.Reals),
Union(ImageSet(Lambda(n, 2*n*pi), S.Integers),
Intersection(ImageSet(Lambda(n, -I*(I*(
2*n*pi + arg(-exp(-2*I*y))) +
2*im(y))), S.Integers), S.Reals)))
assert dumeq(solveset_real(sin(2*x)*cos(x) + cos(2*x)*sin(x)-1, x),
ImageSet(Lambda(n, n*pi*Rational(2, 3) + pi/6), S.Integers))
assert dumeq(solveset_real(2*tan(x)*sin(x) + 1, x), Union(
ImageSet(Lambda(n, 2*n*pi + atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(1 - sqrt(17))) + pi), S.Integers),
ImageSet(Lambda(n, 2*n*pi - atan(sqrt(2)*sqrt(-1 + sqrt(17))/
(1 - sqrt(17))) + pi), S.Integers)))
assert dumeq(solveset_real(cos(2*x)*cos(4*x) - 1, x),
ImageSet(Lambda(n, n*pi), S.Integers))
assert dumeq(solveset(sin(x/10) + Rational(3, 4)), Union(
ImageSet(Lambda(n, 20*n*pi + 10*atan(3*sqrt(7)/7) + 10*pi), S.Integers),
ImageSet(Lambda(n, 20*n*pi - 10*atan(3*sqrt(7)/7) + 20*pi), S.Integers)))
assert dumeq(solveset(cos(x/15) + cos(x/5)), Union(
ImageSet(Lambda(n, 30*n*pi + 15*pi/2), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 45*pi/2), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 75*pi/4), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 45*pi/4), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 105*pi/4), S.Integers),
ImageSet(Lambda(n, 30*n*pi + 15*pi/4), S.Integers)))
assert dumeq(solveset(sec(sqrt(2)*x/3) + 5), Union(
ImageSet(Lambda(n, 3*sqrt(2)*(2*n*pi - pi + atan(2*sqrt(6)))/2), S.Integers),
ImageSet(Lambda(n, 3*sqrt(2)*(2*n*pi - atan(2*sqrt(6)) + pi)/2), S.Integers)))
assert dumeq(simplify(solveset(tan(pi*x) - cot(pi/2*x))), Union(
ImageSet(Lambda(n, 4*n + 1), S.Integers),
ImageSet(Lambda(n, 4*n + 3), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(7, 3)), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(5, 3)), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(11, 3)), S.Integers),
ImageSet(Lambda(n, 4*n + Rational(1, 3)), S.Integers)))
assert dumeq(solveset(cos(9*x)), Union(
ImageSet(Lambda(n, 2*n*pi/9 + pi/18), S.Integers),
ImageSet(Lambda(n, 2*n*pi/9 + pi/6), S.Integers)))
assert dumeq(solveset(sin(8*x) + cot(12*x), x, S.Reals), Union(
ImageSet(Lambda(n, n*pi/2 + pi/8), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 3*pi/8), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 5*pi/16), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 3*pi/16), S.Integers),
ImageSet(Lambda(n, n*pi/2 + 7*pi/16), S.Integers),
ImageSet(Lambda(n, n*pi/2 + pi/16), S.Integers)))
# This is the only remaining solveset test that actually ends up being solved
# by _solve_trig2(). All others are handled by the improved _solve_trig1.
assert dumeq(solveset_real(2*cos(x)*cos(2*x) - 1, x),
Union(ImageSet(Lambda(n, 2*n*pi + 2*atan(sqrt(-2*2**Rational(1, 3)*(67 +
9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 +
9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6)))), S.Integers),
ImageSet(Lambda(n, 2*n*pi - 2*atan(sqrt(-2*2**Rational(1, 3)*(67 +
9*sqrt(57))**Rational(2, 3) + 8*2**Rational(2, 3) + 11*(67 +
9*sqrt(57))**Rational(1, 3))/(3*(67 + 9*sqrt(57))**Rational(1, 6))) +
2*pi), S.Integers)))
# issue #16870
assert dumeq(simplify(solveset(sin(x/180*pi) - S.Half, x, S.Reals)), Union(
ImageSet(Lambda(n, 360*n + 150), S.Integers),
ImageSet(Lambda(n, 360*n + 30), S.Integers)))
def test_solve_hyperbolic():
# actual solver: _solve_trig1
n = Dummy('n')
assert solveset(sinh(x) + cosh(x), x) == S.EmptySet
assert solveset(sinh(x) + cos(x), x) == ConditionSet(x,
Eq(cos(x) + sinh(x), 0), S.Complexes)
assert solveset_real(sinh(x) + sech(x), x) == FiniteSet(
log(sqrt(sqrt(5) - 2)))
assert solveset_real(3*cosh(2*x) - 5, x) == FiniteSet(
-log(3)/2, log(3)/2)
assert solveset_real(sinh(x - 3) - 2, x) == FiniteSet(
log((2 + sqrt(5))*exp(3)))
assert solveset_real(cosh(2*x) + 2*sinh(x) - 5, x) == FiniteSet(
log(-2 + sqrt(5)), log(1 + sqrt(2)))
assert solveset_real((coth(x) + sinh(2*x))/cosh(x) - 3, x) == FiniteSet(
log(S.Half + sqrt(5)/2), log(1 + sqrt(2)))
assert solveset_real(cosh(x)*sinh(x) - 2, x) == FiniteSet(
log(4 + sqrt(17))/2)
assert solveset_real(sinh(x) + tanh(x) - 1, x) == FiniteSet(
log(sqrt(2)/2 + sqrt(-S(1)/2 + sqrt(2))))
assert dumeq(solveset_complex(sinh(x) - I/2, x), Union(
ImageSet(Lambda(n, I*(2*n*pi + 5*pi/6)), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi/6)), S.Integers)))
assert dumeq(solveset_complex(sinh(x) + sech(x), x), Union(
ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(-2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sqrt(-2 + sqrt(5)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - pi/2) + log(sqrt(2 + sqrt(5)))), S.Integers)))
assert dumeq(solveset(sinh(x/10) + Rational(3, 4)), Union(
ImageSet(Lambda(n, 10*I*(2*n*pi + pi) + 10*log(2)), S.Integers),
ImageSet(Lambda(n, 20*n*I*pi - 10*log(2)), S.Integers)))
assert dumeq(solveset(cosh(x/15) + cosh(x/5)), Union(
ImageSet(Lambda(n, 15*I*(2*n*pi + pi/2)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi - pi/2)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi - 3*pi/4)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi + 3*pi/4)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi - pi/4)), S.Integers),
ImageSet(Lambda(n, 15*I*(2*n*pi + pi/4)), S.Integers)))
assert dumeq(solveset(sech(sqrt(2)*x/3) + 5), Union(
ImageSet(Lambda(n, 3*sqrt(2)*I*(2*n*pi - pi + atan(2*sqrt(6)))/2), S.Integers),
ImageSet(Lambda(n, 3*sqrt(2)*I*(2*n*pi - atan(2*sqrt(6)) + pi)/2), S.Integers)))
assert dumeq(solveset(tanh(pi*x) - coth(pi/2*x)), Union(
ImageSet(Lambda(n, 2*I*(2*n*pi + pi/2)/pi), S.Integers),
ImageSet(Lambda(n, 2*I*(2*n*pi - pi/2)/pi), S.Integers)))
assert dumeq(solveset(cosh(9*x)), Union(
ImageSet(Lambda(n, I*(2*n*pi + pi/2)/9), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - pi/2)/9), S.Integers)))
# issues #9606 / #9531:
assert solveset(sinh(x), x, S.Reals) == FiniteSet(0)
assert dumeq(solveset(sinh(x), x, S.Complexes), Union(
ImageSet(Lambda(n, I*(2*n*pi + pi)), S.Integers),
ImageSet(Lambda(n, 2*n*I*pi), S.Integers)))
# issues #11218 / #18427
assert dumeq(solveset(sin(pi*x), x, S.Reals), Union(
ImageSet(Lambda(n, (2*n*pi + pi)/pi), S.Integers),
ImageSet(Lambda(n, 2*n), S.Integers)))
assert dumeq(solveset(sin(pi*x), x), Union(
ImageSet(Lambda(n, (2*n*pi + pi)/pi), S.Integers),
ImageSet(Lambda(n, 2*n), S.Integers)))
# issue #17543
assert dumeq(simplify(solveset(I*cot(8*x - 8*E), x)), Union(
ImageSet(Lambda(n, n*pi/4 - 13*pi/16 + E), S.Integers),
ImageSet(Lambda(n, n*pi/4 - 11*pi/16 + E), S.Integers)))
# issues #18490 / #19489
assert solveset(cosh(x) + cosh(3*x) - cosh(5*x), x, S.Reals
).dummy_eq(ConditionSet(x,
Eq(cosh(x) + cosh(3*x) - cosh(5*x), 0), S.Reals))
assert solveset(sinh(8*x) + coth(12*x)).dummy_eq(
ConditionSet(x, Eq(sinh(8*x) + coth(12*x), 0), S.Complexes))
def test_solve_trig_hyp_symbolic():
# actual solver: _solve_trig1
assert dumeq(solveset(sin(a*x), x), ConditionSet(x, Ne(a, 0), Union(
ImageSet(Lambda(n, (2*n*pi + pi)/a), S.Integers),
ImageSet(Lambda(n, 2*n*pi/a), S.Integers))))
assert dumeq(solveset(cosh(x/a), x), ConditionSet(x, Ne(a, 0), Union(
ImageSet(Lambda(n, I*a*(2*n*pi + pi/2)), S.Integers),
ImageSet(Lambda(n, I*a*(2*n*pi - pi/2)), S.Integers))))
assert dumeq(solveset(sin(2*sqrt(3)/3*a**2/(b*pi)*x)
+ cos(4*sqrt(3)/3*a**2/(b*pi)*x), x),
ConditionSet(x, Ne(b, 0) & Ne(a**2, 0), Union(
ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi + pi/2)/(2*a**2)), S.Integers),
ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi - 5*pi/6)/(2*a**2)), S.Integers),
ImageSet(Lambda(n, sqrt(3)*pi*b*(2*n*pi - pi/6)/(2*a**2)), S.Integers))))
assert dumeq(simplify(solveset(cot((1 + I)*x) - cot((3 + 3*I)*x), x)), Union(
ImageSet(Lambda(n, pi*(1 - I)*(4*n + 1)/4), S.Integers),
ImageSet(Lambda(n, pi*(1 - I)*(4*n - 1)/4), S.Integers)))
assert dumeq(solveset(cosh((a**2 + 1)*x) - 3, x),
ConditionSet(x, Ne(a**2 + 1, 0), Union(
ImageSet(Lambda(n, (2*n*I*pi + log(3 - 2*sqrt(2)))/(a**2 + 1)), S.Integers),
ImageSet(Lambda(n, (2*n*I*pi + log(2*sqrt(2) + 3))/(a**2 + 1)), S.Integers))))
ar = Symbol('ar', real=True)
assert solveset(cosh((ar**2 + 1)*x) - 2, x, S.Reals) == FiniteSet(
log(sqrt(3) + 2)/(ar**2 + 1), log(2 - sqrt(3))/(ar**2 + 1))
def test_issue_9616():
assert dumeq(solveset(sinh(x) + tanh(x) - 1, x), Union(
ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - atan(sqrt(2)*sqrt(S.Half + sqrt(2))) + pi)
+ log(sqrt(1 + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi) + log(-sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi - pi + atan(sqrt(2)*sqrt(S.Half + sqrt(2))))
+ log(sqrt(1 + sqrt(2)))), S.Integers)))
f1 = (sinh(x)).rewrite(exp)
f2 = (tanh(x)).rewrite(exp)
assert dumeq(solveset(f1 + f2 - 1, x), Union(
Complement(ImageSet(
Lambda(n, I*(2*n*pi + pi) + log(-sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)),
Complement(ImageSet(Lambda(n, I*(2*n*pi - pi + atan(sqrt(2)*sqrt(S.Half + sqrt(2))))
+ log(sqrt(1 + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)),
Complement(ImageSet(Lambda(n, I*(2*n*pi - atan(sqrt(2)*sqrt(S.Half + sqrt(2))) + pi)
+ log(sqrt(1 + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers)),
Complement(
ImageSet(Lambda(n, 2*n*I*pi + log(sqrt(2)/2 + sqrt(-S.Half + sqrt(2)))), S.Integers),
ImageSet(Lambda(n, I*(2*n*pi + pi)/2), S.Integers))))
def test_solve_invalid_sol():
assert 0 not in solveset_real(sin(x)/x, x)
assert 0 not in solveset_complex((exp(x) - 1)/x, x)
@XFAIL
def test_solve_trig_simplified():
n = Dummy('n')
assert dumeq(solveset_real(sin(x), x),
imageset(Lambda(n, n*pi), S.Integers))
assert dumeq(solveset_real(cos(x), x),
imageset(Lambda(n, n*pi + pi/2), S.Integers))
assert dumeq(solveset_real(cos(x) + sin(x), x),
imageset(Lambda(n, n*pi - pi/4), S.Integers))
@XFAIL
def test_solve_lambert():
assert solveset_real(x*exp(x) - 1, x) == FiniteSet(LambertW(1))
assert solveset_real(exp(x) + x, x) == FiniteSet(-LambertW(1))
assert solveset_real(x + 2**x, x) == \
FiniteSet(-LambertW(log(2))/log(2))
# issue 4739
ans = solveset_real(3*x + 5 + 2**(-5*x + 3), x)
assert ans == FiniteSet(Rational(-5, 3) +
LambertW(-10240*2**Rational(1, 3)*log(2)/3)/(5*log(2)))
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solveset_real(eq, x)
ans = FiniteSet((log(2401) +
5*LambertW(-log(7**(7*3**Rational(1, 5)/5))))/(3*log(7))/-1)
assert result == ans
assert solveset_real(eq.expand(), x) == result
assert solveset_real(5*x - 1 + 3*exp(2 - 7*x), x) == \
FiniteSet(Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7)
assert solveset_real(2*x + 5 + log(3*x - 2), x) == \
FiniteSet(Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2)
assert solveset_real(3*x + log(4*x), x) == \
FiniteSet(LambertW(Rational(3, 4))/3)
assert solveset_real(x**x - 2) == FiniteSet(exp(LambertW(log(2))))
a = Symbol('a')
assert solveset_real(-a*x + 2*x*log(x), x) == FiniteSet(exp(a/2))
a = Symbol('a', real=True)
assert solveset_real(a/x + exp(x/2), x) == \
FiniteSet(2*LambertW(-a/2))
assert solveset_real((a/x + exp(x/2)).diff(x), x) == \
FiniteSet(4*LambertW(sqrt(2)*sqrt(a)/4))
# coverage test
assert solveset_real(tanh(x + 3)*tanh(x - 3) - 1, x) is S.EmptySet
assert solveset_real((x**2 - 2*x + 1).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*S.Exp1)/3)
assert solveset_real((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) == \
FiniteSet(LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3)
assert solveset_real((x**2 - 2*x - 2).subs(x, log(x) + 3*x), x) == \
FiniteSet(LambertW(3*exp(1 + sqrt(3)))/3, LambertW(3*exp(-sqrt(3) + 1))/3)
assert solveset_real(x*log(x) + 3*x + 1, x) == \
FiniteSet(exp(-3 + LambertW(-exp(3))))
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solveset_real(eq, x) == \
FiniteSet(LambertW(3*exp(-LambertW(3))))
assert solveset_real(3*log(a**(3*x + 5)) + a**(3*x + 5), x) == \
FiniteSet(-((log(a**5) + LambertW(Rational(1, 3)))/(3*log(a))))
p = symbols('p', positive=True)
assert solveset_real(3*log(p**(3*x + 5)) + p**(3*x + 5), x) == \
FiniteSet(
log((-3**Rational(1, 3) - 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p),
log((-3**Rational(1, 3) + 3**Rational(5, 6)*I)*LambertW(Rational(1, 3))**Rational(1, 3)/(2*p**Rational(5, 3)))/log(p),
log((3*LambertW(Rational(1, 3))/p**5)**(1/(3*log(p)))),) # checked numerically
# check collection
b = Symbol('b')
eq = 3*log(a**(3*x + 5)) + b*log(a**(3*x + 5)) + a**(3*x + 5)
assert solveset_real(eq, x) == FiniteSet(
-((log(a**5) + LambertW(1/(b + 3)))/(3*log(a))))
# issue 4271
assert solveset_real((a/x + exp(x/2)).diff(x, 2), x) == FiniteSet(
6*LambertW((-1)**Rational(1, 3)*a**Rational(1, 3)/3))
assert solveset_real(x**3 - 3**x, x) == \
FiniteSet(-3/log(3)*LambertW(-log(3)/3))
assert solveset_real(3**cos(x) - cos(x)**3) == FiniteSet(
acos(-3*LambertW(-log(3)/3)/log(3)))
assert solveset_real(x**2 - 2**x, x) == \
solveset_real(-x**2 + 2**x, x)
assert solveset_real(3*log(x) - x*log(3)) == FiniteSet(
-3*LambertW(-log(3)/3)/log(3),
-3*LambertW(-log(3)/3, -1)/log(3))
assert solveset_real(LambertW(2*x) - y) == FiniteSet(
y*exp(y)/2)
@XFAIL
def test_other_lambert():
a = Rational(6, 5)
assert solveset_real(x**a - a**x, x) == FiniteSet(
a, -a*LambertW(-log(a)/a)/log(a))
@_both_exp_pow
def test_solveset():
f = Function('f')
raises(ValueError, lambda: solveset(x + y))
assert solveset(x, 1) == S.EmptySet
assert solveset(f(1)**2 + y + 1, f(1)
) == FiniteSet(-sqrt(-y - 1), sqrt(-y - 1))
assert solveset(f(1)**2 - 1, f(1), S.Reals) == FiniteSet(-1, 1)
assert solveset(f(1)**2 + 1, f(1)) == FiniteSet(-I, I)
assert solveset(x - 1, 1) == FiniteSet(x)
assert solveset(sin(x) - cos(x), sin(x)) == FiniteSet(cos(x))
assert solveset(0, domain=S.Reals) == S.Reals
assert solveset(1) == S.EmptySet
assert solveset(True, domain=S.Reals) == S.Reals # issue 10197
assert solveset(False, domain=S.Reals) == S.EmptySet
assert solveset(exp(x) - 1, domain=S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, x, S.Reals) == FiniteSet(0)
assert solveset(Eq(exp(x), 1), x, S.Reals) == FiniteSet(0)
assert solveset(exp(x) - 1, exp(x), S.Reals) == FiniteSet(1)
A = Indexed('A', x)
assert solveset(A - 1, A, S.Reals) == FiniteSet(1)
assert solveset(x - 1 >= 0, x, S.Reals) == Interval(1, oo)
assert solveset(exp(x) - 1 >= 0, x, S.Reals) == Interval(0, oo)
assert dumeq(solveset(exp(x) - 1, x), imageset(Lambda(n, 2*I*pi*n), S.Integers))
assert dumeq(solveset(Eq(exp(x), 1), x), imageset(Lambda(n, 2*I*pi*n),
S.Integers))
# issue 13825
assert solveset(x**2 + f(0) + 1, x) == {-sqrt(-f(0) - 1), sqrt(-f(0) - 1)}
# issue 19977
assert solveset(atan(log(x)) > 0, x, domain=Interval.open(0, oo)) == Interval.open(1, oo)
@_both_exp_pow
def test_multi_exp():
k1, k2, k3 = symbols('k1, k2, k3')
assert dumeq(solveset(exp(exp(x)) - 5, x),\
imageset(Lambda(((k1, n),), I*(2*k1*pi + arg(2*n*I*pi + log(5))) + log(Abs(2*n*I*pi + log(5)))),\
ProductSet(S.Integers, S.Integers)))
assert dumeq(solveset((d*exp(exp(a*x + b)) + c), x),\
imageset(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k1, n),), \
I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))), \
ProductSet(S.Integers, S.Integers))))
assert dumeq(solveset((d*exp(exp(exp(a*x + b))) + c), x),\
imageset(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k2, k1, n),), \
I*(2*k2*pi + arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + \
log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + \
log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))))), \
ProductSet(S.Integers, S.Integers, S.Integers))))
assert dumeq(solveset((d*exp(exp(exp(exp(a*x + b)))) + c), x),\
ImageSet(Lambda(x, (-b + x)/a), ImageSet(Lambda(((k3, k2, k1, n),), \
I*(2*k3*pi + arg(I*(2*k2*pi + arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + \
log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + \
log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))))) + log(Abs(I*(2*k2*pi + \
arg(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))))) + \
log(Abs(I*(2*k1*pi + arg(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d)))) + log(Abs(I*(2*n*pi + arg(-c/d)) + log(Abs(c/d))))))))), \
ProductSet(S.Integers, S.Integers, S.Integers, S.Integers))))
def test__solveset_multi():
from sympy.solvers.solveset import _solveset_multi
from sympy.sets import Reals
# Basic univariate case:
assert _solveset_multi([x**2-1], [x], [S.Reals]) == FiniteSet((1,), (-1,))
# Linear systems of two equations
assert _solveset_multi([x+y, x+1], [x, y], [Reals, Reals]) == FiniteSet((-1, 1))
assert _solveset_multi([x+y, x+1], [y, x], [Reals, Reals]) == FiniteSet((1, -1))
assert _solveset_multi([x+y, x-y-1], [x, y], [Reals, Reals]) == FiniteSet((S(1)/2, -S(1)/2))
assert _solveset_multi([x-1, y-2], [x, y], [Reals, Reals]) == FiniteSet((1, 2))
# assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), ImageSet(Lambda(x, (x, -x)), Reals))
assert dumeq(_solveset_multi([x+y], [x, y], [Reals, Reals]), Union(
ImageSet(Lambda(((x,),), (x, -x)), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (-y, y)), ProductSet(Reals))))
assert _solveset_multi([x+y, x+y+1], [x, y], [Reals, Reals]) == S.EmptySet
assert _solveset_multi([x+y, x-y, x-1], [x, y], [Reals, Reals]) == S.EmptySet
assert _solveset_multi([x+y, x-y, x-1], [y, x], [Reals, Reals]) == S.EmptySet
# Systems of three equations:
assert _solveset_multi([x+y+z-1, x+y-z-2, x-y-z-3], [x, y, z], [Reals,
Reals, Reals]) == FiniteSet((2, -S.Half, -S.Half))
# Nonlinear systems:
from sympy.abc import theta
assert _solveset_multi([x**2+y**2-2, x+y], [x, y], [Reals, Reals]) == FiniteSet((-1, 1), (1, -1))
assert _solveset_multi([x**2-1, y], [x, y], [Reals, Reals]) == FiniteSet((1, 0), (-1, 0))
#assert _solveset_multi([x**2-y**2], [x, y], [Reals, Reals]) == Union(
# ImageSet(Lambda(x, (x, -x)), Reals), ImageSet(Lambda(x, (x, x)), Reals))
assert dumeq(_solveset_multi([x**2-y**2], [x, y], [Reals, Reals]), Union(
ImageSet(Lambda(((x,),), (x, -Abs(x))), ProductSet(Reals)),
ImageSet(Lambda(((x,),), (x, Abs(x))), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (-Abs(y), y)), ProductSet(Reals)),
ImageSet(Lambda(((y,),), (Abs(y), y)), ProductSet(Reals))))
assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [theta, r],
[Interval(0, pi), Interval(-1, 1)]) == FiniteSet((0, 1), (pi, -1))
assert _solveset_multi([r*cos(theta)-1, r*sin(theta)], [r, theta],
[Interval(0, 1), Interval(0, pi)]) == FiniteSet((1, 0))
#assert _solveset_multi([r*cos(theta)-r, r*sin(theta)], [r, theta],
# [Interval(0, 1), Interval(0, pi)]) == ?
assert dumeq(_solveset_multi([r*cos(theta)-r, r*sin(theta)], [r, theta],
[Interval(0, 1), Interval(0, pi)]), Union(
ImageSet(Lambda(((r,),), (r, 0)), ImageSet(Lambda(r, (r,)), Interval(0, 1))),
ImageSet(Lambda(((theta,),), (0, theta)), ImageSet(Lambda(theta, (theta,)), Interval(0, pi)))))
def test_conditionset():
assert solveset(Eq(sin(x)**2 + cos(x)**2, 1), x, domain=S.Reals
) is S.Reals
assert solveset(Eq(x**2 + x*sin(x), 1), x, domain=S.Reals
).dummy_eq(ConditionSet(x, Eq(x**2 + x*sin(x) - 1, 0), S.Reals))
assert dumeq(solveset(Eq(-I*(exp(I*x) - exp(-I*x))/2, 1), x
), imageset(Lambda(n, 2*n*pi + pi/2), S.Integers))
assert solveset(x + sin(x) > 1, x, domain=S.Reals
).dummy_eq(ConditionSet(x, x + sin(x) > 1, S.Reals))
assert solveset(Eq(sin(Abs(x)), x), x, domain=S.Reals
).dummy_eq(ConditionSet(x, Eq(-x + sin(Abs(x)), 0), S.Reals))
assert solveset(y**x-z, x, S.Reals
).dummy_eq(ConditionSet(x, Eq(y**x - z, 0), S.Reals))
@XFAIL
def test_conditionset_equality():
''' Checking equality of different representations of ConditionSet'''
assert solveset(Eq(tan(x), y), x) == ConditionSet(x, Eq(tan(x), y), S.Complexes)
def test_solveset_domain():
assert solveset(x**2 - x - 6, x, Interval(0, oo)) == FiniteSet(3)
assert solveset(x**2 - 1, x, Interval(0, oo)) == FiniteSet(1)
assert solveset(x**4 - 16, x, Interval(0, 10)) == FiniteSet(2)
def test_improve_coverage():
solution = solveset(exp(x) + sin(x), x, S.Reals)
unsolved_object = ConditionSet(x, Eq(exp(x) + sin(x), 0), S.Reals)
assert solution.dummy_eq(unsolved_object)
def test_issue_9522():
expr1 = Eq(1/(x**2 - 4) + x, 1/(x**2 - 4) + 2)
expr2 = Eq(1/x + x, 1/x)
assert solveset(expr1, x, S.Reals) is S.EmptySet
assert solveset(expr2, x, S.Reals) is S.EmptySet
def test_solvify():
assert solvify(x**2 + 10, x, S.Reals) == []
assert solvify(x**3 + 1, x, S.Complexes) == [-1, S.Half - sqrt(3)*I/2,
S.Half + sqrt(3)*I/2]
assert solvify(log(x), x, S.Reals) == [1]
assert solvify(cos(x), x, S.Reals) == [pi/2, pi*Rational(3, 2)]
assert solvify(sin(x) + 1, x, S.Reals) == [pi*Rational(3, 2)]
raises(NotImplementedError, lambda: solvify(sin(exp(x)), x, S.Complexes))
def test_solvify_piecewise():
p1 = Piecewise((0, x < -1), (x**2, x <= 1), (log(x), True))
p2 = Piecewise((0, x < -10), (x**2 + 5*x - 6, x >= -9))
p3 = Piecewise((0, Eq(x, 0)), (x**2/Abs(x), True))
p4 = Piecewise((0, Eq(x, pi)), ((x - pi)/sin(x), True))
# issue 21079
assert solvify(p1, x, S.Reals) == [0]
assert solvify(p2, x, S.Reals) == [-6, 1]
assert solvify(p3, x, S.Reals) == [0]
assert solvify(p4, x, S.Reals) == [pi]
def test_abs_invert_solvify():
x = Symbol('x',positive=True)
assert solvify(sin(Abs(x)), x, S.Reals) == [0, pi]
x = Symbol('x')
assert solvify(sin(Abs(x)), x, S.Reals) is None
def test_linear_eq_to_matrix():
assert linear_eq_to_matrix(0, x) == (Matrix([[0]]), Matrix([[0]]))
assert linear_eq_to_matrix(1, x) == (Matrix([[0]]), Matrix([[-1]]))
# integer coefficients
eqns1 = [2*x + y - 2*z - 3, x - y - z, x + y + 3*z - 12]
eqns2 = [Eq(3*x + 2*y - z, 1), Eq(2*x - 2*y + 4*z, -2), -2*x + y - 2*z]
A, B = linear_eq_to_matrix(eqns1, x, y, z)
assert A == Matrix([[2, 1, -2], [1, -1, -1], [1, 1, 3]])
assert B == Matrix([[3], [0], [12]])
A, B = linear_eq_to_matrix(eqns2, x, y, z)
assert A == Matrix([[3, 2, -1], [2, -2, 4], [-2, 1, -2]])
assert B == Matrix([[1], [-2], [0]])
# Pure symbolic coefficients
eqns3 = [a*b*x + b*y + c*z - d, e*x + d*x + f*y + g*z - h, i*x + j*y + k*z - l]
A, B = linear_eq_to_matrix(eqns3, x, y, z)
assert A == Matrix([[a*b, b, c], [d + e, f, g], [i, j, k]])
assert B == Matrix([[d], [h], [l]])
# raise Errors if
# 1) no symbols are given
raises(ValueError, lambda: linear_eq_to_matrix(eqns3))
# 2) there are duplicates
raises(ValueError, lambda: linear_eq_to_matrix(eqns3, [x, x, y]))
# 3) a nonlinear term is detected in the original expression
raises(NonlinearError, lambda: linear_eq_to_matrix(Eq(1/x + x, 1/x), [x]))
raises(NonlinearError, lambda: linear_eq_to_matrix([x**2], [x]))
raises(NonlinearError, lambda: linear_eq_to_matrix([x*y], [x, y]))
# 4) Eq being used to represent equations autoevaluates
# (use unevaluated Eq instead)
raises(ValueError, lambda: linear_eq_to_matrix(Eq(x, x), x))
raises(ValueError, lambda: linear_eq_to_matrix(Eq(x, x + 1), x))
# if non-symbols are passed, the user is responsible for interpreting
assert linear_eq_to_matrix([x], [1/x]) == (Matrix([[0]]), Matrix([[-x]]))
# issue 15195
assert linear_eq_to_matrix(x + y*(z*(3*x + 2) + 3), x) == (
Matrix([[3*y*z + 1]]), Matrix([[-y*(2*z + 3)]]))
assert linear_eq_to_matrix(Matrix(
[[a*x + b*y - 7], [5*x + 6*y - c]]), x, y) == (
Matrix([[a, b], [5, 6]]), Matrix([[7], [c]]))
# issue 15312
assert linear_eq_to_matrix(Eq(x + 2, 1), x) == (
Matrix([[1]]), Matrix([[-1]]))
def test_issue_16577():
assert linear_eq_to_matrix(Eq(a*(2*x + 3*y) + 4*y, 5), x, y) == (
Matrix([[2*a, 3*a + 4]]), Matrix([[5]]))
def test_issue_10085():
assert invert_real(exp(x),0,x) == (x, S.EmptySet)
def test_linsolve():
x1, x2, x3, x4 = symbols('x1, x2, x3, x4')
# Test for different input forms
M = Matrix([[1, 2, 1, 1, 7], [1, 2, 2, -1, 12], [2, 4, 0, 6, 4]])
system1 = A, B = M[:, :-1], M[:, -1]
Eqns = [x1 + 2*x2 + x3 + x4 - 7, x1 + 2*x2 + 2*x3 - x4 - 12,
2*x1 + 4*x2 + 6*x4 - 4]
sol = FiniteSet((-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
assert linsolve(Eqns, (x1, x2, x3, x4)) == sol
assert linsolve(Eqns, *(x1, x2, x3, x4)) == sol
assert linsolve(system1, (x1, x2, x3, x4)) == sol
assert linsolve(system1, *(x1, x2, x3, x4)) == sol
# issue 9667 - symbols can be Dummy symbols
x1, x2, x3, x4 = symbols('x:4', cls=Dummy)
assert linsolve(system1, x1, x2, x3, x4) == FiniteSet(
(-2*x2 - 3*x4 + 2, x2, 2*x4 + 5, x4))
# raise ValueError for garbage value
raises(ValueError, lambda: linsolve(Eqns))
raises(ValueError, lambda: linsolve(x1))
raises(ValueError, lambda: linsolve(x1, x2))
raises(ValueError, lambda: linsolve((A,), x1, x2))
raises(ValueError, lambda: linsolve(A, B, x1, x2))
raises(ValueError, lambda: linsolve([x1], x1, x1))
raises(ValueError, lambda: linsolve([x1], (i for i in (x1, x1))))
#raise ValueError if equations are non-linear in given variables
raises(NonlinearError, lambda: linsolve([x + y - 1, x ** 2 + y - 3], [x, y]))
raises(NonlinearError, lambda: linsolve([cos(x) + y, x + y], [x, y]))
assert linsolve([x + z - 1, x ** 2 + y - 3], [z, y]) == {(-x + 1, -x**2 + 3)}
# Fully symbolic test
A = Matrix([[a, b], [c, d]])
B = Matrix([[e], [g]])
system2 = (A, B)
sol = FiniteSet(((-b*g + d*e)/(a*d - b*c), (a*g - c*e)/(a*d - b*c)))
assert linsolve(system2, [x, y]) == sol
# No solution
A = Matrix([[1, 2, 3], [2, 4, 6], [3, 6, 9]])
B = Matrix([0, 0, 1])
assert linsolve((A, B), (x, y, z)) is S.EmptySet
# Issue #10056
A, B, J1, J2 = symbols('A B J1 J2')
Augmatrix = Matrix([
[2*I*J1, 2*I*J2, -2/J1],
[-2*I*J2, -2*I*J1, 2/J2],
[0, 2, 2*I/(J1*J2)],
[2, 0, 0],
])
assert linsolve(Augmatrix, A, B) == FiniteSet((0, I/(J1*J2)))
# Issue #10121 - Assignment of free variables
Augmatrix = Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0]])
assert linsolve(Augmatrix, a, b, c, d, e) == FiniteSet((a, 0, c, 0, e))
#raises(IndexError, lambda: linsolve(Augmatrix, a, b, c))
x0, x1, x2, _x0 = symbols('tau0 tau1 tau2 _tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('tau00 tau01 tau02 tau0')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
x0, x1, x2, _x0 = symbols('tau00 tau01 tau02 tau1')
assert linsolve(Matrix([[0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, _x0]])
) == FiniteSet((x0, 0, x1, _x0, x2))
# symbols can be given as generators
x0, x2, x4 = symbols('x0, x2, x4')
assert linsolve(Augmatrix, numbered_symbols('x')
) == FiniteSet((x0, 0, x2, 0, x4))
Augmatrix[-1, -1] = x0
# use Dummy to avoid clash; the names may clash but the symbols
# will not
Augmatrix[-1, -1] = symbols('_x0')
assert len(linsolve(
Augmatrix, numbered_symbols('x', cls=Dummy)).free_symbols) == 4
# Issue #12604
f = Function('f')
assert linsolve([f(x) - 5], f(x)) == FiniteSet((5,))
# Issue #14860
from sympy.physics.units import meter, newton, kilo
kN = kilo*newton
Eqns = [8*kN + x + y, 28*kN*meter + 3*x*meter]
assert linsolve(Eqns, x, y) == {
(kilo*newton*Rational(-28, 3), kN*Rational(4, 3))}
# linsolve does not allow expansion (real or implemented)
# to remove singularities, but it will cancel linear terms
assert linsolve([Eq(x, x + y)], [x, y]) == {(x, 0)}
assert linsolve([Eq(x + x*y, 1 + y)], [x]) == {(1,)}
assert linsolve([Eq(1 + y, x + x*y)], [x]) == {(1,)}
raises(NonlinearError, lambda:
linsolve([Eq(x**2, x**2 + y)], [x, y]))
# corner cases
#
# XXX: The case below should give the same as for [0]
# assert linsolve([], [x]) == {(x,)}
assert linsolve([], [x]) is S.EmptySet
assert linsolve([0], [x]) == {(x,)}
assert linsolve([x], [x, y]) == {(0, y)}
assert linsolve([x, 0], [x, y]) == {(0, y)}
def test_linsolve_large_sparse():
#
# This is mainly a performance test
#
def _mk_eqs_sol(n):
xs = symbols('x:{}'.format(n))
ys = symbols('y:{}'.format(n))
syms = xs + ys
eqs = []
sol = (-S.Half,) * n + (S.Half,) * n
for xi, yi in zip(xs, ys):
eqs.extend([xi + yi, xi - yi + 1])
return eqs, syms, FiniteSet(sol)
n = 500
eqs, syms, sol = _mk_eqs_sol(n)
assert linsolve(eqs, syms) == sol
def test_linsolve_immutable():
A = ImmutableDenseMatrix([[1, 1, 2], [0, 1, 2], [0, 0, 1]])
B = ImmutableDenseMatrix([2, 1, -1])
assert linsolve([A, B], (x, y, z)) == FiniteSet((1, 3, -1))
A = ImmutableDenseMatrix([[1, 1, 7], [1, -1, 3]])
assert linsolve(A) == FiniteSet((5, 2))
def test_solve_decomposition():
n = Dummy('n')
f1 = exp(3*x) - 6*exp(2*x) + 11*exp(x) - 6
f2 = sin(x)**2 - 2*sin(x) + 1
f3 = sin(x)**2 - sin(x)
f4 = sin(x + 1)
f5 = exp(x + 2) - 1
f6 = 1/log(x)
f7 = 1/x
s1 = ImageSet(Lambda(n, 2*n*pi), S.Integers)
s2 = ImageSet(Lambda(n, 2*n*pi + pi), S.Integers)
s3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
s4 = ImageSet(Lambda(n, 2*n*pi - 1), S.Integers)
s5 = ImageSet(Lambda(n, 2*n*pi - 1 + pi), S.Integers)
assert solve_decomposition(f1, x, S.Reals) == FiniteSet(0, log(2), log(3))
assert dumeq(solve_decomposition(f2, x, S.Reals), s3)
assert dumeq(solve_decomposition(f3, x, S.Reals), Union(s1, s2, s3))
assert dumeq(solve_decomposition(f4, x, S.Reals), Union(s4, s5))
assert solve_decomposition(f5, x, S.Reals) == FiniteSet(-2)
assert solve_decomposition(f6, x, S.Reals) == S.EmptySet
assert solve_decomposition(f7, x, S.Reals) == S.EmptySet
assert solve_decomposition(x, x, Interval(1, 2)) == S.EmptySet
# nonlinsolve testcases
def test_nonlinsolve_basic():
assert nonlinsolve([],[]) == S.EmptySet
assert nonlinsolve([],[x, y]) == S.EmptySet
system = [x, y - x - 5]
assert nonlinsolve([x],[x, y]) == FiniteSet((0, y))
assert nonlinsolve(system, [y]) == S.EmptySet
soln = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
assert dumeq(nonlinsolve([sin(x) - 1], [x]), FiniteSet(tuple(soln)))
soln = ((ImageSet(Lambda(n, 2*n*pi + pi), S.Integers), FiniteSet(1)),
(ImageSet(Lambda(n, 2*n*pi), S.Integers), FiniteSet(1,)))
assert dumeq(nonlinsolve([sin(x), y - 1], [x, y]), FiniteSet(*soln))
assert nonlinsolve([x**2 - 1], [x]) == FiniteSet((-1,), (1,))
soln = FiniteSet((y, y))
assert nonlinsolve([x - y, 0], x, y) == soln
assert nonlinsolve([0, x - y], x, y) == soln
assert nonlinsolve([x - y, x - y], x, y) == soln
assert nonlinsolve([x, 0], x, y) == FiniteSet((0, y))
f = Function('f')
assert nonlinsolve([f(x), 0], f(x), y) == FiniteSet((0, y))
assert nonlinsolve([f(x), 0], f(x), f(y)) == FiniteSet((0, f(y)))
A = Indexed('A', x)
assert nonlinsolve([A, 0], A, y) == FiniteSet((0, y))
assert nonlinsolve([x**2 -1], [sin(x)]) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], sin(x)) == FiniteSet((S.EmptySet,))
assert nonlinsolve([x**2 -1], 1) == FiniteSet((x**2,))
assert nonlinsolve([x**2 -1], x + y) == FiniteSet((S.EmptySet,))
assert nonlinsolve([Eq(1, x + y), Eq(1, -x + y - 1), Eq(1, -x + y - 1)], x, y) == FiniteSet(
(-S.Half, 3*S.Half))
def test_nonlinsolve_abs():
soln = FiniteSet((y, y), (-y, y))
assert nonlinsolve([Abs(x) - y], x, y) == soln
def test_raise_exception_nonlinsolve():
raises(IndexError, lambda: nonlinsolve([x**2 -1], []))
raises(ValueError, lambda: nonlinsolve([x**2 -1]))
def test_trig_system():
# TODO: add more simple testcases when solveset returns
# simplified soln for Trig eq
assert nonlinsolve([sin(x) - 1, cos(x) -1 ], x) == S.EmptySet
soln1 = (ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers),)
soln = FiniteSet(soln1)
assert dumeq(nonlinsolve([sin(x) - 1, cos(x)], x), soln)
@XFAIL
def test_trig_system_fail():
# fails because solveset trig solver is not much smart.
sys = [x + y - pi/2, sin(x) + sin(y) - 1]
# solveset returns conditionset for sin(x) + sin(y) - 1
soln_1 = (ImageSet(Lambda(n, n*pi + pi/2), S.Integers),
ImageSet(Lambda(n, n*pi), S.Integers))
soln_1 = FiniteSet(soln_1)
soln_2 = (ImageSet(Lambda(n, n*pi), S.Integers),
ImageSet(Lambda(n, n*pi+ pi/2), S.Integers))
soln_2 = FiniteSet(soln_2)
soln = soln_1 + soln_2
assert dumeq(nonlinsolve(sys, [x, y]), soln)
# Add more cases from here
# http://www.vitutor.com/geometry/trigonometry/equations_systems.html#uno
sys = [sin(x) + sin(y) - (sqrt(3)+1)/2, sin(x) - sin(y) - (sqrt(3) - 1)/2]
soln_x = Union(ImageSet(Lambda(n, 2*n*pi + pi/3), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*Rational(2, 3)), S.Integers))
soln_y = Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*Rational(5, 6)), S.Integers))
assert dumeq(nonlinsolve(sys, [x, y]), FiniteSet((soln_x, soln_y)))
def test_nonlinsolve_positive_dimensional():
x, y, a, b, c, d = symbols('x, y, a, b, c, d', extended_real=True)
assert nonlinsolve([x*y, x*y - x], [x, y]) == FiniteSet((0, y))
system = [a**2 + a*c, a - b]
assert nonlinsolve(system, [a, b]) == FiniteSet((0, 0), (-c, -c))
# here (a= 0, b = 0) is independent soln so both is printed.
# if symbols = [a, b, c] then only {a : -c ,b : -c}
eq1 = a + b + c + d
eq2 = a*b + b*c + c*d + d*a
eq3 = a*b*c + b*c*d + c*d*a + d*a*b
eq4 = a*b*c*d - 1
system = [eq1, eq2, eq3, eq4]
sol1 = (-1/d, -d, 1/d, FiniteSet(d) - FiniteSet(0))
sol2 = (1/d, -d, -1/d, FiniteSet(d) - FiniteSet(0))
soln = FiniteSet(sol1, sol2)
assert nonlinsolve(system, [a, b, c, d]) == soln
assert nonlinsolve([x**4 - 3*x**2 + y*x, x*z**2, y*z - 1], [x, y, z]) == \
{(0, 1/z, z)}
def test_nonlinsolve_polysys():
x, y, z = symbols('x, y, z', real=True)
assert nonlinsolve([x**2 + y - 2, x**2 + y], [x, y]) == S.EmptySet
s = (-y + 2, y)
assert nonlinsolve([(x + y)**2 - 4, x + y - 2], [x, y]) == FiniteSet(s)
system = [x**2 - y**2]
soln_real = FiniteSet((-y, y), (y, y))
soln_complex = FiniteSet((-Abs(y), y), (Abs(y), y))
soln =soln_real + soln_complex
assert nonlinsolve(system, [x, y]) == soln
system = [x**2 - y**2]
soln_real= FiniteSet((y, -y), (y, y))
soln_complex = FiniteSet((y, -Abs(y)), (y, Abs(y)))
soln = soln_real + soln_complex
assert nonlinsolve(system, [y, x]) == soln
system = [x**2 + y - 3, x - y - 4]
assert nonlinsolve(system, (x, y)) != nonlinsolve(system, (y, x))
assert nonlinsolve([-x**2 - y**2 + z, -2*x, -2*y, S.One], [x, y, z]) == S.EmptySet
assert nonlinsolve([x + y + z, S.One, S.One, S.One], [x, y, z]) == S.EmptySet
system = [-x**2*z**2 + x*y*z + y**4, -2*x*z**2 + y*z, x*z + 4*y**3, -2*x**2*z + x*y]
assert nonlinsolve(system, [x, y, z]) == FiniteSet((0, 0, z), (x, 0, 0))
def test_nonlinsolve_using_substitution():
x, y, z, n = symbols('x, y, z, n', real = True)
system = [(x + y)*n - y**2 + 2]
s_x = (n*y - y**2 + 2)/n
soln = (-s_x, y)
assert nonlinsolve(system, [x, y]) == FiniteSet(soln)
system = [z**2*x**2 - z**2*y**2/exp(x)]
soln_real_1 = (y, x, 0)
soln_real_2 = (-exp(x/2)*Abs(x), x, z)
soln_real_3 = (exp(x/2)*Abs(x), x, z)
soln_complex_1 = (-x*exp(x/2), x, z)
soln_complex_2 = (x*exp(x/2), x, z)
syms = [y, x, z]
soln = FiniteSet(soln_real_1, soln_complex_1, soln_complex_2,\
soln_real_2, soln_real_3)
assert nonlinsolve(system,syms) == soln
def test_nonlinsolve_complex():
n = Dummy('n')
assert dumeq(nonlinsolve([exp(x) - sin(y), 1/y - 3], [x, y]), {
(ImageSet(Lambda(n, 2*n*I*pi + log(sin(Rational(1, 3)))), S.Integers), Rational(1, 3))})
system = [exp(x) - sin(y), 1/exp(y) - 3]
assert dumeq(nonlinsolve(system, [x, y]), {
(ImageSet(Lambda(n, I*(2*n*pi + pi)
+ log(sin(log(3)))), S.Integers), -log(3)),
(ImageSet(Lambda(n, I*(2*n*pi + arg(sin(2*n*I*pi - log(3))))
+ log(Abs(sin(2*n*I*pi - log(3))))), S.Integers),
ImageSet(Lambda(n, 2*n*I*pi - log(3)), S.Integers))})
system = [exp(x) - sin(y), y**2 - 4]
assert dumeq(nonlinsolve(system, [x, y]), {
(ImageSet(Lambda(n, I*(2*n*pi + pi) + log(sin(2))), S.Integers), -2),
(ImageSet(Lambda(n, 2*n*I*pi + log(sin(2))), S.Integers), 2)})
system = [exp(x) - 2, y ** 2 - 2]
assert dumeq(nonlinsolve(system, [x, y]), {
(log(2), -sqrt(2)), (log(2), sqrt(2)),
(ImageSet(Lambda(n, 2*n*I*pi + log(2)), S.Integers), FiniteSet(-sqrt(2))),
(ImageSet(Lambda(n, 2 * n * I * pi + log(2)), S.Integers), FiniteSet(sqrt(2)))})
def test_nonlinsolve_radical():
assert nonlinsolve([sqrt(y) - x - z, y - 1], [x, y, z]) == {(1 - z, 1, z)}
def test_nonlinsolve_inexact():
sol = [(-1.625, -1.375), (1.625, 1.375)]
res = nonlinsolve([(x + y)**2 - 9, x**2 - y**2 - 0.75], [x, y])
assert all(abs(res.args[i][j]-sol[i][j]) < 1e-9
for i in range(2) for j in range(2))
assert nonlinsolve([(x + y)**2 - 9, (x + y)**2 - 0.75], [x, y]) == S.EmptySet
assert nonlinsolve([y**2 + (x - 0.5)**2 - 0.0625, 2*x - 1.0, 2*y], [x, y]) == \
S.EmptySet
res = nonlinsolve([x**2 + y - 0.5, (x + y)**2, log(z)], [x, y, z])
sol = [(-0.366025403784439, 0.366025403784439, 1),
(-0.366025403784439, 0.366025403784439, 1),
(1.36602540378444, -1.36602540378444, 1)]
assert all(abs(res.args[i][j]-sol[i][j]) < 1e-9
for i in range(3) for j in range(3))
res = nonlinsolve([y - x**2, x**5 - x + 1.0], [x, y])
sol = [(-1.16730397826142, 1.36259857766493),
(-0.181232444469876 - 1.08395410131771*I,
-1.14211129483496 + 0.392895302949911*I),
(-0.181232444469876 + 1.08395410131771*I,
-1.14211129483496 - 0.392895302949911*I),
(0.764884433600585 - 0.352471546031726*I,
0.460812006002492 - 0.539199997693599*I),
(0.764884433600585 + 0.352471546031726*I,
0.460812006002492 + 0.539199997693599*I)]
assert all(abs(res.args[i][j] - sol[i][j]) < 1e-9
for i in range(5) for j in range(2))
@XFAIL
def test_solve_nonlinear_trans():
# After the transcendental equation solver these will work
x, y = symbols('x, y', real=True)
soln1 = FiniteSet((2*LambertW(y/2), y))
soln2 = FiniteSet((-x*sqrt(exp(x)), y), (x*sqrt(exp(x)), y))
soln3 = FiniteSet((x*exp(x/2), x))
soln4 = FiniteSet(2*LambertW(y/2), y)
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln1
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln2
assert nonlinsolve([x**2 - y**2/exp(x)], [y, x]) == soln3
assert nonlinsolve([x**2 - y**2/exp(x)], [x, y]) == soln4
def test_issue_14642():
x = Symbol('x')
n1 = 0.5*x**3+x**2+0.5+I #add I in the Polynomials
solution = solveset(n1, x)
assert abs(solution.args[0] - (-2.28267560928153 - 0.312325580497716*I)) <= 1e-9
assert abs(solution.args[1] - (-0.297354141679308 + 1.01904778618762*I)) <= 1e-9
assert abs(solution.args[2] - (0.580029750960839 - 0.706722205689907*I)) <= 1e-9
# Symbolic
n1 = S.Half*x**3+x**2+S.Half+I
res = FiniteSet(-((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)
/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2))/3)/3 - S(2)/3 - 4*cos(atan((27 +
3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*
31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2))/3)/(3*((3*
sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 +
(27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)**2)**(S(1)/
6)) + I*(-((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/
2)/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(
atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)
/2)/2 + S(43)/2))/3)/3 + 4*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)
/49)/2)/2 + S(43)/2))/3)/(3*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6))), -S(2)/3 - sqrt(3)*((3*
sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 +
(27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)**2)**(S(1)
/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 + S(43)/2))
/3)/6 - 4*re(1/((-S(1)/2 - sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 +
(43 + 54*I)**2)/2)**(S(1)/3)))/3 + ((3*sqrt(3)*31985**(S(1)/4)*sin(
atan(S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*
31985**(S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*
sin(atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 + I*(-4*im(1/((-S(1)/2 -
sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)/
3)))/3 + ((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2))/3)/6 + sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/
49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/
4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2))/3)/6), -S(2)/3 - 4*re(1/((-S(1)/2 +
sqrt(3)*I/2)*(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)
/3)))/3 + sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2))/3)/6 + ((3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(S(172)/49)/2)
/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(S(1)/4)*cos(atan(
S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(atan(S(172)/49)/2)/2 +
S(43)/2))/3)/6 + I*(-sqrt(3)*((3*sqrt(3)*31985**(S(1)/4)*sin(atan(
S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*cos(
atan(S(172)/49)/2)/2)**2)**(S(1)/6)*cos(atan((27 + 3*sqrt(3)*31985**(
S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(
atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 + ((3*sqrt(3)*31985**(S(1)/4)*
sin(atan(S(172)/49)/2)/2 + S(43)/2)**2 + (27 + 3*sqrt(3)*31985**(S(1)/4)*
cos(atan(S(172)/49)/2)/2)**2)**(S(1)/6)*sin(atan((27 + 3*sqrt(3)*31985**(
S(1)/4)*cos(atan(S(172)/49)/2)/2)/(3*sqrt(3)*31985**(S(1)/4)*sin(
atan(S(172)/49)/2)/2 + S(43)/2))/3)/6 - 4*im(1/((-S(1)/2 + sqrt(3)*I/2)*
(S(43)/2 + 27*I + sqrt(-256 + (43 + 54*I)**2)/2)**(S(1)/3)))/3))
assert solveset(n1, x) == res
def test_issue_13961():
V = (ax, bx, cx, gx, jx, lx, mx, nx, q) = symbols('ax bx cx gx jx lx mx nx q')
S = (ax*q - lx*q - mx, ax - gx*q - lx, bx*q**2 + cx*q - jx*q - nx, q*(-ax*q + lx*q + mx), q*(-ax + gx*q + lx))
sol = FiniteSet((lx + mx/q, (-cx*q + jx*q + nx)/q**2, cx, mx/q**2, jx, lx, mx, nx, Complement({q}, {0})),
(lx + mx/q, (cx*q - jx*q - nx)/q**2*-1, cx, mx/q**2, jx, lx, mx, nx, Complement({q}, {0})))
assert nonlinsolve(S, *V) == sol
# The two solutions are in fact identical, so even better if only one is returned
def test_issue_14541():
solutions = solveset(sqrt(-x**2 - 2.0), x)
assert abs(solutions.args[0]+1.4142135623731*I) <= 1e-9
assert abs(solutions.args[1]-1.4142135623731*I) <= 1e-9
def test_issue_13396():
expr = -2*y*exp(-x**2 - y**2)*Abs(x)
sol = FiniteSet(0)
assert solveset(expr, y, domain=S.Reals) == sol
# Related type of equation also solved here
assert solveset(atan(x**2 - y**2)-pi/2, y, S.Reals) is S.EmptySet
def test_issue_12032():
sol = FiniteSet(-sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 +
sqrt(Abs(-2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2,
-sqrt(Abs(-2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2 -
sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2,
sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 -
I*sqrt(Abs(-2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) -
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))))/2,
sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)))/2 +
I*sqrt(Abs(-2/sqrt(-2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) +
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3))) -
2*(Rational(1, 16) + sqrt(849)/144)**(Rational(1, 3)) +
2/(3*(Rational(1, 16) + sqrt(849)/144)**(Rational(1,3)))))/2)
assert solveset(x**4 + x - 1, x) == sol
def test_issue_10876():
assert solveset(1/sqrt(x), x) == S.EmptySet
def test_issue_19050():
# test_issue_19050 --> TypeError removed
assert dumeq(nonlinsolve([x + y, sin(y)], [x, y]),
FiniteSet((ImageSet(Lambda(n, -2*n*pi), S.Integers), ImageSet(Lambda(n, 2*n*pi), S.Integers)),\
(ImageSet(Lambda(n, -2*n*pi - pi), S.Integers), ImageSet(Lambda(n, 2*n*pi + pi), S.Integers))))
assert dumeq(nonlinsolve([x + y, sin(y) + cos(y)], [x, y]),
FiniteSet((ImageSet(Lambda(n, -2*n*pi - 3*pi/4), S.Integers), ImageSet(Lambda(n, 2*n*pi + 3*pi/4), S.Integers)), \
(ImageSet(Lambda(n, -2*n*pi - 7*pi/4), S.Integers), ImageSet(Lambda(n, 2*n*pi + 7*pi/4), S.Integers))))
def test_issue_16618():
# AttributeError is removed !
eqn = [sin(x)*sin(y), cos(x)*cos(y) - 1]
ans = FiniteSet((x, 2*n*pi), (2*n*pi, y), (x, 2*n*pi + pi), (2*n*pi + pi, y))
sol = nonlinsolve(eqn, [x, y])
for i0, j0 in zip(ordered(sol), ordered(ans)):
assert len(i0) == len(j0) == 2
assert all(a.dummy_eq(b) for a, b in zip(i0, j0))
assert len(sol) == len(ans)
def test_issue_17566():
assert nonlinsolve([32*(2**x)/2**(-y) - 4**y, 27*(3**x) - S(1)/3**y], x, y) ==\
FiniteSet((-log(81)/log(3), 1))
def test_issue_16643():
n = Dummy('n')
assert solveset(x**2*sin(x), x).dummy_eq(Union(ImageSet(Lambda(n, 2*n*pi + pi), S.Integers),
ImageSet(Lambda(n, 2*n*pi), S.Integers)))
def test_issue_19587():
n,m = symbols('n m')
assert nonlinsolve([32*2**m*2**n - 4**n, 27*3**m - 3**(-n)], m, n) ==\
FiniteSet((-log(81)/log(3), 1))
def test_issue_5132_1():
system = [sqrt(x**2 + y**2) - sqrt(10), x + y - 4]
assert nonlinsolve(system, [x, y]) == FiniteSet((1, 3), (3, 1))
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + -log(3))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2)
)
soln = soln_real + soln_complex
assert dumeq(nonlinsolve(eqs, [y, z]), soln)
def test_issue_5132_2():
x, y = symbols('x, y', real=True)
eqs = [exp(x)**2 - sin(y) + z**2]
n = Dummy('n')
soln_real = (log(-z**2 + sin(y))/2, z)
lam = Lambda( n, I*(2*n*pi + arg(-z**2 + sin(y)))/2 + log(Abs(z**2 - sin(y)))/2)
img = ImageSet(lam, S.Integers)
# not sure about the complex soln. But it looks correct.
soln_complex = (img, z)
soln = FiniteSet(soln_real, soln_complex)
assert dumeq(nonlinsolve(eqs, [x, z]), soln)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x = sqrt(r/(tan(t)**2 + 1))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x, s_y), (-s_x, -s_y))
assert nonlinsolve(system, [x, y]) == soln
def test_issue_6752():
a, b = symbols('a, b', real=True)
assert nonlinsolve([a**2 + a, a - b], [a, b]) == {(-1, -1), (0, 0)}
@SKIP("slow")
def test_issue_5114_solveset():
# slow testcase
from sympy.abc import o, p
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = [a, b, c, f, h, k, n]
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(nonlinsolve(eqs, syms)) == 1
@SKIP("Hangs")
def _test_issue_5335():
# Not able to check zero dimensional system.
# is_zero_dimensional Hangs
lam, a0, conc = symbols('lam a0 conc')
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions but only two are valid
assert len(nonlinsolve(eqs, sym)) == 2
# float
eqs = [lam + 2*y - a0*(1 - x/2)*x - 0.005*x/2*x,
a0*(1 - x/2)*x - 1*y - 0.743436700916726*y,
x + y - conc]
sym = [x, y, a0]
assert len(nonlinsolve(eqs, sym)) == 2
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = Rational(191, 20), 3*sqrt(391)/20
ans = {(a, -b), (a, b)}
assert nonlinsolve((e1, e2), (x, y)) == ans
assert nonlinsolve((e1, e2/(x - a)), (x, y)) == S.EmptySet
# make the 2nd circle's radius be -3
e2 += 6
assert nonlinsolve((e1, e2), (x, y)) == S.EmptySet
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = [x, y, z]
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x2 - x)**2 + (y2 - y)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = [f1, f2, f3]
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = [g1, g2, g3]
# both soln same
A = nonlinsolve(F, v)
B = nonlinsolve(G, v)
assert A == B
def test_nonlinsolve_conditionset():
# when solveset failed to solve all the eq
# return conditionset
f = Function('f')
f1 = f(x) - pi/2
f2 = f(y) - pi*Rational(3, 2)
intermediate_system = Eq(2*f(x) - pi, 0) & Eq(2*f(y) - 3*pi, 0)
syms = Tuple(x, y)
soln = ConditionSet(
syms,
intermediate_system,
S.Complexes**2)
assert nonlinsolve([f1, f2], [x, y]) == soln
def test_substitution_basic():
assert substitution([], [x, y]) == S.EmptySet
assert substitution([], []) == S.EmptySet
system = [2*x**2 + 3*y**2 - 30, 3*x**2 - 2*y**2 - 19]
soln = FiniteSet((-3, -2), (-3, 2), (3, -2), (3, 2))
assert substitution(system, [x, y]) == soln
soln = FiniteSet((-1, 1))
assert substitution([x + y], [x], [{y: 1}], [y], set(), [x, y]) == soln
assert substitution(
[x + y], [x], [{y: 1}], [y],
{x + 1}, [y, x]) == S.EmptySet
def test_substitution_incorrect():
# the solutions in the following two tests are incorrect. The
# correct result is EmptySet in both cases.
assert substitution([h - 1, k - 1, f - 2, f - 4, -2 * k],
[h, k, f]) == {(1, 1, f)}
assert substitution([x + y + z, S.One, S.One, S.One], [x, y, z]) == \
{(-y - z, y, z)}
# the correct result in the test below is {(-I, I, I, -I),
# (I, -I, -I, I)}
assert substitution([a - d, b + d, c + d, d**2 + 1], [a, b, c, d]) == \
{(d, -d, -d, d)}
# the result in the test below is incomplete. The complete result
# is {(0, b), (log(2), 2)}
assert substitution([a*(a - log(b)), a*(b - 2)], [a, b]) == \
{(0, b)}
# The system in the test below is zero-dimensional, so the result
# should have no free symbols
assert substitution([-k*y + 6*x - 4*y, -81*k + 49*y**2 - 270,
-3*k*z + k + z**3, k**2 - 2*k + 4],
[x, y, z, k]).free_symbols == {z}
def test_substitution_redundant():
# the third and fourth solutions are redundant in the test below
assert substitution([x**2 - y**2, z - 1], [x, z]) == \
{(-y, 1), (y, 1), (-sqrt(y**2), 1), (sqrt(y**2), 1)}
# the system below has three solutions. Two of the solutions
# returned by substitution are redundant.
res = substitution([x - y, y**3 - 3*y**2 + 1], [x, y])
assert len(res) == 5
def test_issue_5132_substitution():
x, y, z, r, t = symbols('x, y, z, r, t', real=True)
system = [r - x**2 - y**2, tan(t) - y/x]
s_x_1 = Complement(FiniteSet(-sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_x_2 = Complement(FiniteSet(sqrt(r/(tan(t)**2 + 1))), FiniteSet(0))
s_y = sqrt(r/(tan(t)**2 + 1))*tan(t)
soln = FiniteSet((s_x_2, s_y)) + FiniteSet((s_x_1, -s_y))
assert substitution(system, [x, y]) == soln
n = Dummy('n')
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
s_real_y = -log(3)
s_real_z = sqrt(-exp(2*x) - sin(log(3)))
soln_real = FiniteSet((s_real_y, s_real_z), (s_real_y, -s_real_z))
lam = Lambda(n, 2*n*I*pi + -log(3))
s_complex_y = ImageSet(lam, S.Integers)
lam = Lambda(n, sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_1 = ImageSet(lam, S.Integers)
lam = Lambda(n, -sqrt(-exp(2*x) + sin(2*n*I*pi + -log(3))))
s_complex_z_2 = ImageSet(lam, S.Integers)
soln_complex = FiniteSet(
(s_complex_y, s_complex_z_1),
(s_complex_y, s_complex_z_2))
soln = soln_real + soln_complex
assert dumeq(substitution(eqs, [y, z]), soln)
def test_raises_substitution():
raises(ValueError, lambda: substitution([x**2 -1], []))
raises(TypeError, lambda: substitution([x**2 -1]))
raises(ValueError, lambda: substitution([x**2 -1], [sin(x)]))
raises(TypeError, lambda: substitution([x**2 -1], x))
raises(TypeError, lambda: substitution([x**2 -1], 1))
def test_issue_21022():
from sympy.core.sympify import sympify
eqs = [
'k-16',
'p-8',
'y*y+z*z-x*x',
'd - x + p',
'd*d+k*k-y*y',
'z*z-p*p-k*k',
'abc-efg',
]
efg = Symbol('efg')
eqs = [sympify(x) for x in eqs]
syb = list(ordered(set.union(*[x.free_symbols for x in eqs])))
res = nonlinsolve(eqs, syb)
ans = FiniteSet(
(efg, 32, efg, 16, 8, 40, -16*sqrt(5), -8*sqrt(5)),
(efg, 32, efg, 16, 8, 40, -16*sqrt(5), 8*sqrt(5)),
(efg, 32, efg, 16, 8, 40, 16*sqrt(5), -8*sqrt(5)),
(efg, 32, efg, 16, 8, 40, 16*sqrt(5), 8*sqrt(5)),
)
assert len(res) == len(ans) == 4
assert res == ans
for result in res.args:
assert len(result) == 8
def test_issue_17940():
n = Dummy('n')
k1 = Dummy('k1')
sol = ImageSet(Lambda(((k1, n),), I*(2*k1*pi + arg(2*n*I*pi + log(5)))
+ log(Abs(2*n*I*pi + log(5)))),
ProductSet(S.Integers, S.Integers))
assert solveset(exp(exp(x)) - 5, x).dummy_eq(sol)
def test_issue_17906():
assert solveset(7**(x**2 - 80) - 49**x, x) == FiniteSet(-8, 10)
def test_issue_17933():
eq1 = x*sin(45) - y*cos(q)
eq2 = x*cos(45) - y*sin(q)
eq3 = 9*x*sin(45)/10 + y*cos(q)
eq4 = 9*x*cos(45)/10 + y*sin(z) - z
assert nonlinsolve([eq1, eq2, eq3, eq4], x, y, z, q) ==\
FiniteSet((0, 0, 0, q))
def test_issue_14565():
# removed redundancy
assert dumeq(nonlinsolve([k + m, k + m*exp(-2*pi*k)], [k, m]) ,
FiniteSet((-n*I, ImageSet(Lambda(n, n*I), S.Integers))))
# end of tests for nonlinsolve
def test_issue_9556():
b = Symbol('b', positive=True)
assert solveset(Abs(x) + 1, x, S.Reals) is S.EmptySet
assert solveset(Abs(x) + b, x, S.Reals) is S.EmptySet
assert solveset(Eq(b, -1), b, S.Reals) is S.EmptySet
def test_issue_9611():
assert solveset(Eq(x - x + a, a), x, S.Reals) == S.Reals
assert solveset(Eq(y - y + a, a), y) == S.Complexes
def test_issue_9557():
assert solveset(x**2 + a, x, S.Reals) == Intersection(S.Reals,
FiniteSet(-sqrt(-a), sqrt(-a)))
def test_issue_9778():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert solveset(x**3 + 1, x, S.Reals) == FiniteSet(-1)
assert solveset(x**Rational(3, 5) + 1, x, S.Reals) == S.EmptySet
assert solveset(x**3 + y, x, S.Reals) == \
FiniteSet(-Abs(y)**Rational(1, 3)*sign(y))
def test_issue_10214():
assert solveset(x**Rational(3, 2) + 4, x, S.Reals) == S.EmptySet
assert solveset(x**(Rational(-3, 2)) + 4, x, S.Reals) == S.EmptySet
ans = FiniteSet(-2**Rational(2, 3))
assert solveset(x**(S(3)) + 4, x, S.Reals) == ans
assert (x**(S(3)) + 4).subs(x,list(ans)[0]) == 0 # substituting ans and verifying the result.
assert (x**(S(3)) + 4).subs(x,-(-2)**Rational(2, 3)) == 0
def test_issue_9849():
assert solveset(Abs(sin(x)) + 1, x, S.Reals) == S.EmptySet
def test_issue_9953():
assert linsolve([ ], x) == S.EmptySet
def test_issue_9913():
assert solveset(2*x + 1/(x - 10)**2, x, S.Reals) == \
FiniteSet(-(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)/3 - 100/
(3*(3*sqrt(24081)/4 + Rational(4027, 4))**Rational(1, 3)) + Rational(20, 3))
def test_issue_10397():
assert solveset(sqrt(x), x, S.Complexes) == FiniteSet(0)
def test_issue_14987():
raises(ValueError, lambda: linear_eq_to_matrix(
[x**2], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(-3/x + 1) + 2*y - a], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x**2 - 3*x)/(x - 3) - 3], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)**3 - x**3 - 3*x**2 + 7], x))
raises(ValueError, lambda: linear_eq_to_matrix(
[x*(1/x + 1) + y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[(x + 1)*y], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(1/x, 1/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(y/x, y/x + y)], [x, y]))
raises(ValueError, lambda: linear_eq_to_matrix(
[Eq(x*(x + 1), x**2 + y)], [x, y]))
def test_simplification():
eq = x + (a - b)/(-2*a + 2*b)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == Intersection({-((a - b)/(-2*a + 2*b))}, S.Reals)
# So that ap - bn is not zero:
ap = Symbol('ap', positive=True)
bn = Symbol('bn', negative=True)
eq = x + (ap - bn)/(-2*ap + 2*bn)
assert solveset(eq, x) == FiniteSet(S.Half)
assert solveset(eq, x, S.Reals) == FiniteSet(S.Half)
def test_integer_domain_relational():
eq1 = 2*x + 3 > 0
eq2 = x**2 + 3*x - 2 >= 0
eq3 = x + 1/x > -2 + 1/x
eq4 = x + sqrt(x**2 - 5) > 0
eq = x + 1/x > -2 + 1/x
eq5 = eq.subs(x,log(x))
eq6 = log(x)/x <= 0
eq7 = log(x)/x < 0
eq8 = x/(x-3) < 3
eq9 = x/(x**2-3) < 3
assert solveset(eq1, x, S.Integers) == Range(-1, oo, 1)
assert solveset(eq2, x, S.Integers) == Union(Range(-oo, -3, 1), Range(1, oo, 1))
assert solveset(eq3, x, S.Integers) == Union(Range(-1, 0, 1), Range(1, oo, 1))
assert solveset(eq4, x, S.Integers) == Range(3, oo, 1)
assert solveset(eq5, x, S.Integers) == Range(2, oo, 1)
assert solveset(eq6, x, S.Integers) == Range(1, 2, 1)
assert solveset(eq7, x, S.Integers) == S.EmptySet
assert solveset(eq8, x, domain=Range(0,5)) == Range(0, 3, 1)
assert solveset(eq9, x, domain=Range(0,5)) == Union(Range(0, 2, 1), Range(2, 5, 1))
# test_issue_19794
assert solveset(x + 2 < 0, x, S.Integers) == Range(-oo, -2, 1)
def test_issue_10555():
f = Function('f')
g = Function('g')
assert solveset(f(x) - pi/2, x, S.Reals).dummy_eq(
ConditionSet(x, Eq(f(x) - pi/2, 0), S.Reals))
assert solveset(f(g(x)) - pi/2, g(x), S.Reals).dummy_eq(
ConditionSet(g(x), Eq(f(g(x)) - pi/2, 0), S.Reals))
def test_issue_8715():
eq = x + 1/x > -2 + 1/x
assert solveset(eq, x, S.Reals) == \
(Interval.open(-2, oo) - FiniteSet(0))
assert solveset(eq.subs(x,log(x)), x, S.Reals) == \
Interval.open(exp(-2), oo) - FiniteSet(1)
def test_issue_11174():
eq = z**2 + exp(2*x) - sin(y)
soln = Intersection(S.Reals, FiniteSet(log(-z**2 + sin(y))/2))
assert solveset(eq, x, S.Reals) == soln
eq = sqrt(r)*Abs(tan(t))/sqrt(tan(t)**2 + 1) + x*tan(t)
s = -sqrt(r)*Abs(tan(t))/(sqrt(tan(t)**2 + 1)*tan(t))
soln = Intersection(S.Reals, FiniteSet(s))
assert solveset(eq, x, S.Reals) == soln
def test_issue_11534():
# eq and eq2 should give the same solution as a Complement
x = Symbol('x', real=True)
y = Symbol('y', real=True)
eq = -y + x/sqrt(-x**2 + 1)
eq2 = -y**2 + x**2/(-x**2 + 1)
soln = Complement(FiniteSet(-y/sqrt(y**2 + 1), y/sqrt(y**2 + 1)), FiniteSet(-1, 1))
assert solveset(eq, x, S.Reals) == soln
assert solveset(eq2, x, S.Reals) == soln
def test_issue_10477():
assert solveset((x**2 + 4*x - 3)/x < 2, x, S.Reals) == \
Union(Interval.open(-oo, -3), Interval.open(0, 1))
def test_issue_10671():
assert solveset(sin(y), y, Interval(0, pi)) == FiniteSet(0, pi)
i = Interval(1, 10)
assert solveset((1/x).diff(x) < 0, x, i) == i
def test_issue_11064():
eq = x + sqrt(x**2 - 5)
assert solveset(eq > 0, x, S.Reals) == \
Interval(sqrt(5), oo)
assert solveset(eq < 0, x, S.Reals) == \
Interval(-oo, -sqrt(5))
assert solveset(eq > sqrt(5), x, S.Reals) == \
Interval.Lopen(sqrt(5), oo)
def test_issue_12478():
eq = sqrt(x - 2) + 2
soln = solveset_real(eq, x)
assert soln is S.EmptySet
assert solveset(eq < 0, x, S.Reals) is S.EmptySet
assert solveset(eq > 0, x, S.Reals) == Interval(2, oo)
def test_issue_12429():
eq = solveset(log(x)/x <= 0, x, S.Reals)
sol = Interval.Lopen(0, 1)
assert eq == sol
def test_issue_19506():
eq = arg(x + I)
C = Dummy('C')
assert solveset(eq).dummy_eq(Intersection(ConditionSet(C, Eq(im(C) + 1, 0), S.Complexes),
ConditionSet(C, re(C) > 0, S.Complexes)))
def test_solveset_arg():
assert solveset(arg(x), x, S.Reals) == Interval.open(0, oo)
assert solveset(arg(4*x -3), x, S.Reals) == Interval.open(Rational(3, 4), oo)
def test__is_finite_with_finite_vars():
f = _is_finite_with_finite_vars
# issue 12482
assert all(f(1/x) is None for x in (
Dummy(), Dummy(real=True), Dummy(complex=True)))
assert f(1/Dummy(real=False)) is True # b/c it's finite but not 0
def test_issue_13550():
assert solveset(x**2 - 2*x - 15, symbol = x, domain = Interval(-oo, 0)) == FiniteSet(-3)
def test_issue_13849():
assert nonlinsolve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) is S.EmptySet
def test_issue_14223():
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
S.Reals) == FiniteSet(-1, 1)
assert solveset((Abs(x + Min(x, 2)) - 2).rewrite(Piecewise), x,
Interval(0, 2)) == FiniteSet(1)
assert solveset(x, x, FiniteSet(1, 2)) is S.EmptySet
def test_issue_10158():
dom = S.Reals
assert solveset(x*Max(x, 15) - 10, x, dom) == FiniteSet(Rational(2, 3))
assert solveset(x*Min(x, 15) - 10, x, dom) == FiniteSet(-sqrt(10), sqrt(10))
assert solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom) == FiniteSet(-1, 1)
assert solveset(Abs(x - 1) - Abs(y), x, dom) == FiniteSet(-Abs(y) + 1, Abs(y) + 1)
assert solveset(Abs(x + 4*Abs(x + 1)), x, dom) == FiniteSet(Rational(-4, 3), Rational(-4, 5))
assert solveset(2*Abs(x + Abs(x + Max(3, x))) - 2, x, S.Reals) == FiniteSet(-1, -2)
dom = S.Complexes
raises(ValueError, lambda: solveset(x*Max(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(x*Min(x, 15) - 10, x, dom))
raises(ValueError, lambda: solveset(Max(Abs(x - 3) - 1, x + 2) - 3, x, dom))
raises(ValueError, lambda: solveset(Abs(x - 1) - Abs(y), x, dom))
raises(ValueError, lambda: solveset(Abs(x + 4*Abs(x + 1)), x, dom))
def test_issue_14300():
f = 1 - exp(-18000000*x) - y
a1 = FiniteSet(-log(-y + 1)/18000000)
assert solveset(f, x, S.Reals) == \
Intersection(S.Reals, a1)
assert dumeq(solveset(f, x),
ImageSet(Lambda(n, -I*(2*n*pi + arg(-y + 1))/18000000 -
log(Abs(y - 1))/18000000), S.Integers))
def test_issue_14454():
number = CRootOf(x**4 + x - 1, 2)
raises(ValueError, lambda: invert_real(number, 0, x))
assert invert_real(x**2, number, x) # no error
def test_issue_17882():
assert solveset(-8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3)), x, S.Complexes) == \
FiniteSet(sqrt(3), -sqrt(3))
def test_term_factors():
assert list(_term_factors(3**x - 2)) == [-2, 3**x]
expr = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
assert set(_term_factors(expr)) == {
3**(x + 2), 4**(x + 2), 3**(x + 3), 4**(x - 1), -1, 4**(x + 1)}
#################### tests for transolve and its helpers ###############
def test_transolve():
assert _transolve(3**x, x, S.Reals) == S.EmptySet
assert _transolve(3**x - 9**(x + 5), x, S.Reals) == FiniteSet(-10)
def test_issue_21276():
eq = (2*x*(y - z) - y*erf(y - z) - y + z*erf(y - z) + z)**2
assert solveset(eq.expand(), y) == FiniteSet(z, z + erfinv(2*x - 1))
# exponential tests
def test_exponential_real():
from sympy.abc import y
e1 = 3**(2*x) - 2**(x + 3)
e2 = 4**(5 - 9*x) - 8**(2 - x)
e3 = 2**x + 4**x
e4 = exp(log(5)*x) - 2**x
e5 = exp(x/y)*exp(-z/y) - 2
e6 = 5**(x/2) - 2**(x/3)
e7 = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
e8 = -9*exp(-2*x + 5) + 4*exp(3*x + 1)
e9 = 2**x + 4**x + 8**x - 84
e10 = 29*2**(x + 1)*615**(x) - 123*2726**(x)
assert solveset(e1, x, S.Reals) == FiniteSet(
-3*log(2)/(-2*log(3) + log(2)))
assert solveset(e2, x, S.Reals) == FiniteSet(Rational(4, 15))
assert solveset(e3, x, S.Reals) == S.EmptySet
assert solveset(e4, x, S.Reals) == FiniteSet(0)
assert solveset(e5, x, S.Reals) == Intersection(
S.Reals, FiniteSet(y*log(2*exp(z/y))))
assert solveset(e6, x, S.Reals) == FiniteSet(0)
assert solveset(e7, x, S.Reals) == FiniteSet(2)
assert solveset(e8, x, S.Reals) == FiniteSet(-2*log(2)/5 + 2*log(3)/5 + Rational(4, 5))
assert solveset(e9, x, S.Reals) == FiniteSet(2)
assert solveset(e10,x, S.Reals) == FiniteSet((-log(29) - log(2) + log(123))/(-log(2726) + log(2) + log(615)))
assert solveset_real(-9*exp(-2*x + 5) + 2**(x + 1), x) == FiniteSet(
-((-5 - 2*log(3) + log(2))/(log(2) + 2)))
assert solveset_real(4**(x/2) - 2**(x/3), x) == FiniteSet(0)
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solveset_real(5**(x/2) - 2**(3/x), x) == FiniteSet(-b, b)
# coverage test
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solveset_real(C1 + C2/x**2 - exp(-f(x)), f(x)) == Intersection(
S.Reals, FiniteSet(-log(C1 + C2/x**2)))
y = symbols('y', positive=True)
assert solveset_real(x**2 - y**2/exp(x), y) == Intersection(
S.Reals, FiniteSet(-sqrt(x**2*exp(x)), sqrt(x**2*exp(x))))
p = Symbol('p', positive=True)
assert solveset_real((1/p + 1)**(p + 1), p).dummy_eq(
ConditionSet(x, Eq((1 + 1/x)**(x + 1), 0), S.Reals))
@XFAIL
def test_exponential_complex():
n = Dummy('n')
assert dumeq(solveset_complex(2**x + 4**x, x),imageset(
Lambda(n, I*(2*n*pi + pi)/log(2)), S.Integers))
assert solveset_complex(x**z*y**z - 2, z) == FiniteSet(
log(2)/(log(x) + log(y)))
assert dumeq(solveset_complex(4**(x/2) - 2**(x/3), x), imageset(
Lambda(n, 3*n*I*pi/log(2)), S.Integers))
assert dumeq(solveset(2**x + 32, x), imageset(
Lambda(n, (I*(2*n*pi + pi) + 5*log(2))/log(2)), S.Integers))
eq = (2**exp(y**2/x) + 2)/(x**2 + 15)
a = sqrt(x)*sqrt(-log(log(2)) + log(log(2) + 2*n*I*pi))
assert solveset_complex(eq, y) == FiniteSet(-a, a)
union1 = imageset(Lambda(n, I*(2*n*pi - pi*Rational(2, 3))/log(2)), S.Integers)
union2 = imageset(Lambda(n, I*(2*n*pi + pi*Rational(2, 3))/log(2)), S.Integers)
assert dumeq(solveset(2**x + 4**x + 8**x, x), Union(union1, union2))
eq = 4**(x + 1) + 4**(x + 2) + 4**(x - 1) - 3**(x + 2) - 3**(x + 3)
res = solveset(eq, x)
num = 2*n*I*pi - 4*log(2) + 2*log(3)
den = -2*log(2) + log(3)
ans = imageset(Lambda(n, num/den), S.Integers)
assert dumeq(res, ans)
def test_expo_conditionset():
f1 = (exp(x) + 1)**x - 2
f2 = (x + 2)**y*x - 3
f3 = 2**x - exp(x) - 3
f4 = log(x) - exp(x)
f5 = 2**x + 3**x - 5**x
assert solveset(f1, x, S.Reals).dummy_eq(ConditionSet(
x, Eq((exp(x) + 1)**x - 2, 0), S.Reals))
assert solveset(f2, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(x*(x + 2)**y - 3, 0), S.Reals))
assert solveset(f3, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(2**x - exp(x) - 3, 0), S.Reals))
assert solveset(f4, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(-exp(x) + log(x), 0), S.Reals))
assert solveset(f5, x, S.Reals).dummy_eq(ConditionSet(
x, Eq(2**x + 3**x - 5**x, 0), S.Reals))
def test_exponential_symbols():
x, y, z = symbols('x y z', positive=True)
xr, zr = symbols('xr, zr', real=True)
assert solveset(z**x - y, x, S.Reals) == Intersection(
S.Reals, FiniteSet(log(y)/log(z)))
f1 = 2*x**w - 4*y**w
f2 = (x/y)**w - 2
sol1 = Intersection({log(2)/(log(x) - log(y))}, S.Reals)
sol2 = Intersection({log(2)/log(x/y)}, S.Reals)
assert solveset(f1, w, S.Reals) == sol1, solveset(f1, w, S.Reals)
assert solveset(f2, w, S.Reals) == sol2, solveset(f2, w, S.Reals)
assert solveset(x**x, x, Interval.Lopen(0,oo)).dummy_eq(
ConditionSet(w, Eq(w**w, 0), Interval.open(0, oo)))
assert solveset(x**y - 1, y, S.Reals) == FiniteSet(0)
assert solveset(exp(x/y)*exp(-z/y) - 2, y, S.Reals) == \
Complement(ConditionSet(y, Eq(im(x)/y, 0) & Eq(im(z)/y, 0), \
Complement(Intersection(FiniteSet((x - z)/log(2)), S.Reals), FiniteSet(0))), FiniteSet(0))
assert solveset(exp(xr/y)*exp(-zr/y) - 2, y, S.Reals) == \
Complement(FiniteSet((xr - zr)/log(2)), FiniteSet(0))
assert solveset(a**x - b**x, x).dummy_eq(ConditionSet(
w, Ne(a, 0) & Ne(b, 0), FiniteSet(0)))
def test_ignore_assumptions():
# make sure assumptions are ignored
xpos = symbols('x', positive=True)
x = symbols('x')
assert solveset_complex(xpos**2 - 4, xpos
) == solveset_complex(x**2 - 4, x)
@XFAIL
def test_issue_10864():
assert solveset(x**(y*z) - x, x, S.Reals) == FiniteSet(1)
@XFAIL
def test_solve_only_exp_2():
assert solveset_real(sqrt(exp(x)) + sqrt(exp(-x)) - 4, x) == \
FiniteSet(2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2))
def test_is_exponential():
assert _is_exponential(y, x) is False
assert _is_exponential(3**x - 2, x) is True
assert _is_exponential(5**x - 7**(2 - x), x) is True
assert _is_exponential(sin(2**x) - 4*x, x) is False
assert _is_exponential(x**y - z, y) is True
assert _is_exponential(x**y - z, x) is False
assert _is_exponential(2**x + 4**x - 1, x) is True
assert _is_exponential(x**(y*z) - x, x) is False
assert _is_exponential(x**(2*x) - 3**x, x) is False
assert _is_exponential(x**y - y*z, y) is False
assert _is_exponential(x**y - x*z, y) is True
def test_solve_exponential():
assert _solve_exponential(3**(2*x) - 2**(x + 3), 0, x, S.Reals) == \
FiniteSet(-3*log(2)/(-2*log(3) + log(2)))
assert _solve_exponential(2**y + 4**y, 1, y, S.Reals) == \
FiniteSet(log(Rational(-1, 2) + sqrt(5)/2)/log(2))
assert _solve_exponential(2**y + 4**y, 0, y, S.Reals) == \
S.EmptySet
assert _solve_exponential(2**x + 3**x - 5**x, 0, x, S.Reals) == \
ConditionSet(x, Eq(2**x + 3**x - 5**x, 0), S.Reals)
# end of exponential tests
# logarithmic tests
def test_logarithmic():
assert solveset_real(log(x - 3) + log(x + 3), x) == FiniteSet(
-sqrt(10), sqrt(10))
assert solveset_real(log(x + 1) - log(2*x - 1), x) == FiniteSet(2)
assert solveset_real(log(x + 3) + log(1 + 3/x) - 3, x) == FiniteSet(
-3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2)
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solveset_real(eq, x) == \
Intersection(S.Reals, FiniteSet(-sqrt(y**2 - y*exp(z)),
sqrt(y**2 - y*exp(z)))) - \
Intersection(S.Reals, FiniteSet(-sqrt(y**2), sqrt(y**2)))
assert solveset_real(
log(3*x) - log(-x + 1) - log(4*x + 1), x) == FiniteSet(Rational(-1, 2), S.Half)
assert solveset(log(x**y) - y*log(x), x, S.Reals) == S.Reals
@XFAIL
def test_uselogcombine_2():
eq = log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)
assert solveset_real(eq, x) is S.EmptySet
eq = log(8*x) - log(sqrt(x) + 1) - 2
assert solveset_real(eq, x) is S.EmptySet
def test_is_logarithmic():
assert _is_logarithmic(y, x) is False
assert _is_logarithmic(log(x), x) is True
assert _is_logarithmic(log(x) - 3, x) is True
assert _is_logarithmic(log(x)*log(y), x) is True
assert _is_logarithmic(log(x)**2, x) is False
assert _is_logarithmic(log(x - 3) + log(x + 3), x) is True
assert _is_logarithmic(log(x**y) - y*log(x), x) is True
assert _is_logarithmic(sin(log(x)), x) is False
assert _is_logarithmic(x + y, x) is False
assert _is_logarithmic(log(3*x) - log(1 - x) + 4, x) is True
assert _is_logarithmic(log(x) + log(y) + x, x) is False
assert _is_logarithmic(log(log(x - 3)) + log(x - 3), x) is True
assert _is_logarithmic(log(log(3) + x) + log(x), x) is True
assert _is_logarithmic(log(x)*(y + 3) + log(x), y) is False
def test_solve_logarithm():
y = Symbol('y')
assert _solve_logarithm(log(x**y) - y*log(x), 0, x, S.Reals) == S.Reals
y = Symbol('y', positive=True)
assert _solve_logarithm(log(x)*log(y), 0, x, S.Reals) == FiniteSet(1)
# end of logarithmic tests
# lambert tests
def test_is_lambert():
a, b, c = symbols('a,b,c')
assert _is_lambert(x**2, x) is False
assert _is_lambert(a**x**2+b*x+c, x) is True
assert _is_lambert(E**2, x) is False
assert _is_lambert(x*E**2, x) is False
assert _is_lambert(3*log(x) - x*log(3), x) is True
assert _is_lambert(log(log(x - 3)) + log(x-3), x) is True
assert _is_lambert(5*x - 1 + 3*exp(2 - 7*x), x) is True
assert _is_lambert((a/x + exp(x/2)).diff(x, 2), x) is True
assert _is_lambert((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1), x) is True
assert _is_lambert(x*sinh(x) - 1, x) is True
assert _is_lambert(x*cos(x) - 5, x) is True
assert _is_lambert(tanh(x) - 5*x, x) is True
assert _is_lambert(cosh(x) - sinh(x), x) is False
# end of lambert tests
def test_linear_coeffs():
from sympy.solvers.solveset import linear_coeffs
assert linear_coeffs(0, x) == [0, 0]
assert all(i is S.Zero for i in linear_coeffs(0, x))
assert linear_coeffs(x + 2*y + 3, x, y) == [1, 2, 3]
assert linear_coeffs(x + 2*y + 3, y, x) == [2, 1, 3]
assert linear_coeffs(x + 2*x**2 + 3, x, x**2) == [1, 2, 3]
raises(ValueError, lambda:
linear_coeffs(x + 2*x**2 + x**3, x, x**2))
raises(ValueError, lambda:
linear_coeffs(1/x*(x - 1) + 1/x, x))
raises(ValueError, lambda:
linear_coeffs(x, x, x))
assert linear_coeffs(a*(x + y), x, y) == [a, a, 0]
assert linear_coeffs(1.0, x, y) == [0, 0, 1.0]
# don't include coefficients of 0
assert linear_coeffs(Eq(x, x + y), x, y, dict=True) == {y: -1}
assert linear_coeffs(0, x, y, dict=True) == {}
def test_is_modular():
assert _is_modular(y, x) is False
assert _is_modular(Mod(x, 3) - 1, x) is True
assert _is_modular(Mod(x**3 - 3*x**2 - x + 1, 3) - 1, x) is True
assert _is_modular(Mod(exp(x + y), 3) - 2, x) is True
assert _is_modular(Mod(exp(x + y), 3) - log(x), x) is True
assert _is_modular(Mod(x, 3) - 1, y) is False
assert _is_modular(Mod(x, 3)**2 - 5, x) is False
assert _is_modular(Mod(x, 3)**2 - y, x) is False
assert _is_modular(exp(Mod(x, 3)) - 1, x) is False
assert _is_modular(Mod(3, y) - 1, y) is False
def test_invert_modular():
n = Dummy('n', integer=True)
from sympy.solvers.solveset import _invert_modular as invert_modular
# non invertible cases
assert invert_modular(Mod(sin(x), 7), S(5), n, x) == (Mod(sin(x), 7), 5)
assert invert_modular(Mod(exp(x), 7), S(5), n, x) == (Mod(exp(x), 7), 5)
assert invert_modular(Mod(log(x), 7), S(5), n, x) == (Mod(log(x), 7), 5)
# a is symbol
assert dumeq(invert_modular(Mod(x, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 5), S.Integers)))
# a.is_Add
assert dumeq(invert_modular(Mod(x + 8, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 4), S.Integers)))
assert invert_modular(Mod(x**2 + x, 7), S(5), n, x) == \
(Mod(x**2 + x, 7), 5)
# a.is_Mul
assert dumeq(invert_modular(Mod(3*x, 7), S(5), n, x),
(x, ImageSet(Lambda(n, 7*n + 4), S.Integers)))
assert invert_modular(Mod((x + 1)*(x + 2), 7), S(5), n, x) == \
(Mod((x + 1)*(x + 2), 7), 5)
# a.is_Pow
assert invert_modular(Mod(x**4, 7), S(5), n, x) == \
(x, S.EmptySet)
assert dumeq(invert_modular(Mod(3**x, 4), S(3), n, x),
(x, ImageSet(Lambda(n, 2*n + 1), S.Naturals0)))
assert dumeq(invert_modular(Mod(2**(x**2 + x + 1), 7), S(2), n, x),
(x**2 + x + 1, ImageSet(Lambda(n, 3*n + 1), S.Naturals0)))
assert invert_modular(Mod(sin(x)**4, 7), S(5), n, x) == (x, S.EmptySet)
def test_solve_modular():
n = Dummy('n', integer=True)
# if rhs has symbol (need to be implemented in future).
assert solveset(Mod(x, 4) - x, x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(-x + Mod(x, 4), 0),
S.Integers))
# when _invert_modular fails to invert
assert solveset(3 - Mod(sin(x), 7), x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(sin(x), 7) - 3, 0), S.Integers))
assert solveset(3 - Mod(log(x), 7), x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(log(x), 7) - 3, 0), S.Integers))
assert solveset(3 - Mod(exp(x), 7), x, S.Integers
).dummy_eq(ConditionSet(x, Eq(Mod(exp(x), 7) - 3, 0),
S.Integers))
# EmptySet solution definitely
assert solveset(7 - Mod(x, 5), x, S.Integers) is S.EmptySet
assert solveset(5 - Mod(x, 5), x, S.Integers) is S.EmptySet
# Negative m
assert dumeq(solveset(2 + Mod(x, -3), x, S.Integers),
ImageSet(Lambda(n, -3*n - 2), S.Integers))
assert solveset(4 + Mod(x, -3), x, S.Integers) is S.EmptySet
# linear expression in Mod
assert dumeq(solveset(3 - Mod(x, 5), x, S.Integers),
ImageSet(Lambda(n, 5*n + 3), S.Integers))
assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Integers),
ImageSet(Lambda(n, 7*n + 5), S.Integers))
assert dumeq(solveset(3 - Mod(5*x, 7), x, S.Integers),
ImageSet(Lambda(n, 7*n + 2), S.Integers))
# higher degree expression in Mod
assert dumeq(solveset(Mod(x**2, 160) - 9, x, S.Integers),
Union(ImageSet(Lambda(n, 160*n + 3), S.Integers),
ImageSet(Lambda(n, 160*n + 13), S.Integers),
ImageSet(Lambda(n, 160*n + 67), S.Integers),
ImageSet(Lambda(n, 160*n + 77), S.Integers),
ImageSet(Lambda(n, 160*n + 83), S.Integers),
ImageSet(Lambda(n, 160*n + 93), S.Integers),
ImageSet(Lambda(n, 160*n + 147), S.Integers),
ImageSet(Lambda(n, 160*n + 157), S.Integers)))
assert solveset(3 - Mod(x**4, 7), x, S.Integers) is S.EmptySet
assert dumeq(solveset(Mod(x**4, 17) - 13, x, S.Integers),
Union(ImageSet(Lambda(n, 17*n + 3), S.Integers),
ImageSet(Lambda(n, 17*n + 5), S.Integers),
ImageSet(Lambda(n, 17*n + 12), S.Integers),
ImageSet(Lambda(n, 17*n + 14), S.Integers)))
# a.is_Pow tests
assert dumeq(solveset(Mod(7**x, 41) - 15, x, S.Integers),
ImageSet(Lambda(n, 40*n + 3), S.Naturals0))
assert dumeq(solveset(Mod(12**x, 21) - 18, x, S.Integers),
ImageSet(Lambda(n, 6*n + 2), S.Naturals0))
assert dumeq(solveset(Mod(3**x, 4) - 3, x, S.Integers),
ImageSet(Lambda(n, 2*n + 1), S.Naturals0))
assert dumeq(solveset(Mod(2**x, 7) - 2 , x, S.Integers),
ImageSet(Lambda(n, 3*n + 1), S.Naturals0))
assert dumeq(solveset(Mod(3**(3**x), 4) - 3, x, S.Integers),
Intersection(ImageSet(Lambda(n, Intersection({log(2*n + 1)/log(3)},
S.Integers)), S.Naturals0), S.Integers))
# Implemented for m without primitive root
assert solveset(Mod(x**3, 7) - 2, x, S.Integers) is S.EmptySet
assert dumeq(solveset(Mod(x**3, 8) - 1, x, S.Integers),
ImageSet(Lambda(n, 8*n + 1), S.Integers))
assert dumeq(solveset(Mod(x**4, 9) - 4, x, S.Integers),
Union(ImageSet(Lambda(n, 9*n + 4), S.Integers),
ImageSet(Lambda(n, 9*n + 5), S.Integers)))
# domain intersection
assert dumeq(solveset(3 - Mod(5*x - 8, 7), x, S.Naturals0),
Intersection(ImageSet(Lambda(n, 7*n + 5), S.Integers), S.Naturals0))
# Complex args
assert solveset(Mod(x, 3) - I, x, S.Integers) == \
S.EmptySet
assert solveset(Mod(I*x, 3) - 2, x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(I*x, 3) - 2, 0), S.Integers))
assert solveset(Mod(I + x, 3) - 2, x, S.Integers
).dummy_eq(
ConditionSet(x, Eq(Mod(x + I, 3) - 2, 0), S.Integers))
# issue 17373 (https://github.com/sympy/sympy/issues/17373)
assert dumeq(solveset(Mod(x**4, 14) - 11, x, S.Integers),
Union(ImageSet(Lambda(n, 14*n + 3), S.Integers),
ImageSet(Lambda(n, 14*n + 11), S.Integers)))
assert dumeq(solveset(Mod(x**31, 74) - 43, x, S.Integers),
ImageSet(Lambda(n, 74*n + 31), S.Integers))
# issue 13178
n = symbols('n', integer=True)
a = 742938285
b = 1898888478
m = 2**31 - 1
c = 20170816
assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Integers),
ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0))
assert dumeq(solveset(c - Mod(a**n*b, m), n, S.Naturals0),
Intersection(ImageSet(Lambda(n, 2147483646*n + 100), S.Naturals0),
S.Naturals0))
assert dumeq(solveset(c - Mod(a**(2*n)*b, m), n, S.Integers),
Intersection(ImageSet(Lambda(n, 1073741823*n + 50), S.Naturals0),
S.Integers))
assert solveset(c - Mod(a**(2*n + 7)*b, m), n, S.Integers) is S.EmptySet
assert dumeq(solveset(c - Mod(a**(n - 4)*b, m), n, S.Integers),
Intersection(ImageSet(Lambda(n, 2147483646*n + 104), S.Naturals0),
S.Integers))
# end of modular tests
def test_issue_17276():
assert nonlinsolve([Eq(x, 5**(S(1)/5)), Eq(x*y, 25*sqrt(5))], x, y) == \
FiniteSet((5**(S(1)/5), 25*5**(S(3)/10)))
def test_issue_10426():
x = Dummy('x')
a = Symbol('a')
n = Dummy('n')
assert (solveset(sin(x + a) - sin(x), a)).dummy_eq(Dummy('x')) == (Union(
ImageSet(Lambda(n, 2*n*pi), S.Integers),
Intersection(S.Complexes, ImageSet(Lambda(n, -I*(I*(2*n*pi + arg(-exp(-2*I*x))) + 2*im(x))),
S.Integers)))).dummy_eq(Dummy('x,n'))
def test_solveset_conjugate():
"""Test solveset for simple conjugate functions"""
assert solveset(conjugate(x) -3 + I) == FiniteSet(3 + I)
def test_issue_18208():
variables = symbols('x0:16') + symbols('y0:12')
x0, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15,\
y0, y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11 = variables
eqs = [x0 + x1 + x2 + x3 - 51,
x0 + x1 + x4 + x5 - 46,
x2 + x3 + x6 + x7 - 39,
x0 + x3 + x4 + x7 - 50,
x1 + x2 + x5 + x6 - 35,
x4 + x5 + x6 + x7 - 34,
x4 + x5 + x8 + x9 - 46,
x10 + x11 + x6 + x7 - 23,
x11 + x4 + x7 + x8 - 25,
x10 + x5 + x6 + x9 - 44,
x10 + x11 + x8 + x9 - 35,
x12 + x13 + x8 + x9 - 35,
x10 + x11 + x14 + x15 - 29,
x11 + x12 + x15 + x8 - 35,
x10 + x13 + x14 + x9 - 29,
x12 + x13 + x14 + x15 - 29,
y0 + y1 + y2 + y3 - 55,
y0 + y1 + y4 + y5 - 53,
y2 + y3 + y6 + y7 - 56,
y0 + y3 + y4 + y7 - 57,
y1 + y2 + y5 + y6 - 52,
y4 + y5 + y6 + y7 - 54,
y4 + y5 + y8 + y9 - 48,
y10 + y11 + y6 + y7 - 60,
y11 + y4 + y7 + y8 - 51,
y10 + y5 + y6 + y9 - 57,
y10 + y11 + y8 + y9 - 54,
x10 - 2,
x11 - 5,
x12 - 1,
x13 - 6,
x14 - 1,
x15 - 21,
y0 - 12,
y1 - 20]
expected = [38 - x3, x3 - 10, 23 - x3, x3, 12 - x7, x7 + 6, 16 - x7, x7,
8, 20, 2, 5, 1, 6, 1, 21, 12, 20, -y11 + y9 + 2, y11 - y9 + 21,
-y11 - y7 + y9 + 24, y11 + y7 - y9 - 3, 33 - y7, y7, 27 - y9, y9,
27 - y11, y11]
A, b = linear_eq_to_matrix(eqs, variables)
# solve
solve_expected = {v:eq for v, eq in zip(variables, expected) if v != eq}
assert solve(eqs, variables) == solve_expected
# linsolve
linsolve_expected = FiniteSet(Tuple(*expected))
assert linsolve(eqs, variables) == linsolve_expected
assert linsolve((A, b), variables) == linsolve_expected
# gauss_jordan_solve
gj_solve, new_vars = A.gauss_jordan_solve(b)
gj_solve = [i for i in gj_solve]
gj_expected = linsolve_expected.subs(zip([x3, x7, y7, y9, y11], new_vars))
assert FiniteSet(Tuple(*gj_solve)) == gj_expected
# nonlinsolve
# The solution set of nonlinsolve is currently equivalent to linsolve and is
# also correct. However, we would prefer to use the same symbols as parameters
# for the solution to the underdetermined system in all cases if possible.
# We want a solution that is not just equivalent but also given in the same form.
# This test may be changed should nonlinsolve be modified in this way.
nonlinsolve_expected = FiniteSet((38 - x3, x3 - 10, 23 - x3, x3, 12 - x7, x7 + 6,
16 - x7, x7, 8, 20, 2, 5, 1, 6, 1, 21, 12, 20,
-y5 + y7 - 1, y5 - y7 + 24, 21 - y5, y5, 33 - y7,
y7, 27 - y9, y9, -y5 + y7 - y9 + 24, y5 - y7 + y9 + 3))
assert nonlinsolve(eqs, variables) == nonlinsolve_expected
def test_substitution_with_infeasible_solution():
a00, a01, a10, a11, l0, l1, l2, l3, m0, m1, m2, m3, m4, m5, m6, m7, c00, c01, c10, c11, p00, p01, p10, p11 = symbols(
'a00, a01, a10, a11, l0, l1, l2, l3, m0, m1, m2, m3, m4, m5, m6, m7, c00, c01, c10, c11, p00, p01, p10, p11'
)
solvefor = [p00, p01, p10, p11, c00, c01, c10, c11, m0, m1, m3, l0, l1, l2, l3]
system = [
-l0 * c00 - l1 * c01 + m0 + c00 + c01,
-l0 * c10 - l1 * c11 + m1,
-l2 * c00 - l3 * c01 + c00 + c01,
-l2 * c10 - l3 * c11 + m3,
-l0 * p00 - l2 * p10 + p00 + p10,
-l1 * p00 - l3 * p10 + p00 + p10,
-l0 * p01 - l2 * p11,
-l1 * p01 - l3 * p11,
-a00 + c00 * p00 + c10 * p01,
-a01 + c01 * p00 + c11 * p01,
-a10 + c00 * p10 + c10 * p11,
-a11 + c01 * p10 + c11 * p11,
-m0 * p00,
-m1 * p01,
-m2 * p10,
-m3 * p11,
-m4 * c00,
-m5 * c01,
-m6 * c10,
-m7 * c11,
m2,
m4,
m5,
m6,
m7
]
sol = FiniteSet(
(0, Complement(FiniteSet(p01), FiniteSet(0)), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, l2, l3),
(p00, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, 1, 1, -p01/p11, -p01/p11),
(0, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, 1, -l3*p11/p01, -p01/p11, l3),
(0, Complement(FiniteSet(p01), FiniteSet(0)), 0, p11, 0, 0, 0, 0, 0, 0, 0, -l2*p11/p01, -l3*p11/p01, l2, l3),
)
assert sol != nonlinsolve(system, solvefor)
def test_issue_20097():
assert solveset(1/sqrt(x)) is S.EmptySet
def test_issue_15350():
assert solveset(diff(sqrt(1/x+x))) == FiniteSet(-1, 1)
def test_issue_18359():
c1 = Piecewise((0, x < 0), (Min(1, x)/2 - Min(2, x)/2 + Min(3, x)/2, True))
c2 = Piecewise((Piecewise((0, x < 0), (Min(1, x)/2 - Min(2, x)/2 + Min(3, x)/2, True)), x >= 0), (0, True))
correct_result = Interval(1, 2)
result1 = solveset(c1 - Rational(1, 2), x, Interval(0, 3))
result2 = solveset(c2 - Rational(1, 2), x, Interval(0, 3))
assert result1 == correct_result
assert result2 == correct_result
def test_issue_17604():
lhs = -2**(3*x/11)*exp(x/11) + pi**(x/11)
assert _is_exponential(lhs, x)
assert _solve_exponential(lhs, 0, x, S.Complexes) == FiniteSet(0)
def test_issue_17580():
assert solveset(1/(1 - x**3)**2, x, S.Reals) is S.EmptySet
def test_issue_17566_actual():
sys = [2**x + 2**y - 3, 4**x + 9**y - 5]
# Not clear this is the correct result, but at least no recursion error
assert nonlinsolve(sys, x, y) == FiniteSet((log(3 - 2**y)/log(2), y))
def test_issue_17565():
eq = Ge(2*(x - 2)**2/(3*(x + 1)**(Integer(1)/3)) + 2*(x - 2)*(x + 1)**(Integer(2)/3), 0)
res = Union(Interval.Lopen(-1, -Rational(1, 4)), Interval(2, oo))
assert solveset(eq, x, S.Reals) == res
def test_issue_15024():
function = (x + 5)/sqrt(-x**2 - 10*x)
assert solveset(function, x, S.Reals) == FiniteSet(Integer(-5))
def test_issue_16877():
assert dumeq(nonlinsolve([x - 1, sin(y)], x, y),
FiniteSet((FiniteSet(1), ImageSet(Lambda(n, 2*n*pi), S.Integers)),
(FiniteSet(1), ImageSet(Lambda(n, 2*n*pi + pi), S.Integers))))
# Even better if (FiniteSet(1), ImageSet(Lambda(n, n*pi), S.Integers)) is obtained
def test_issue_16876():
assert dumeq(nonlinsolve([sin(x), 2*x - 4*y], x, y),
FiniteSet((ImageSet(Lambda(n, 2*n*pi), S.Integers),
ImageSet(Lambda(n, n*pi), S.Integers)),
(ImageSet(Lambda(n, 2*n*pi + pi), S.Integers),
ImageSet(Lambda(n, n*pi + pi/2), S.Integers))))
# Even better if (ImageSet(Lambda(n, n*pi), S.Integers),
# ImageSet(Lambda(n, n*pi/2), S.Integers)) is obtained
def test_issue_21236():
x, z = symbols("x z")
y = symbols('y', rational=True)
assert solveset(x**y - z, x, S.Reals) == ConditionSet(x, Eq(x**y - z, 0), S.Reals)
e1, e2 = symbols('e1 e2', even=True)
y = e1/e2 # don't know if num or den will be odd and the other even
assert solveset(x**y - z, x, S.Reals) == ConditionSet(x, Eq(x**y - z, 0), S.Reals)
def test_issue_21908():
assert nonlinsolve([(x**2 + 2*x - y**2)*exp(x), -2*y*exp(x)], x, y
) == {(-2, 0), (0, 0)}
def test_issue_19144():
# test case 1
expr1 = [x + y - 1, y**2 + 1]
eq1 = [Eq(i, 0) for i in expr1]
soln1 = {(1 - I, I), (1 + I, -I)}
soln_expr1 = nonlinsolve(expr1, [x, y])
soln_eq1 = nonlinsolve(eq1, [x, y])
assert soln_eq1 == soln_expr1 == soln1
# test case 2 - with denoms
expr2 = [x/y - 1, y**2 + 1]
eq2 = [Eq(i, 0) for i in expr2]
soln2 = {(-I, -I), (I, I)}
soln_expr2 = nonlinsolve(expr2, [x, y])
soln_eq2 = nonlinsolve(eq2, [x, y])
assert soln_eq2 == soln_expr2 == soln2
# denominators that cancel in expression
assert nonlinsolve([Eq(x + 1/x, 1/x)], [x]) == FiniteSet((S.EmptySet,))
def test_issue_22413():
res = nonlinsolve((4*y*(2*x + 2*exp(y) + 1)*exp(2*x),
4*x*exp(2*x) + 4*y*exp(2*x + y) + 4*exp(2*x + y) + 1),
x, y)
# First solution is not correct, but the issue was an exception
sols = FiniteSet((x, S.Zero), (-exp(y) - S.Half, y))
assert res == sols
def test_issue_23318():
eqs_eq = [
Eq(53.5780461486929, x * log(y / (5.0 - y) + 1) / y),
Eq(x, 0.0015 * z),
Eq(0.0015, 7845.32 * y / z),
]
eqs_expr = [eq.rewrite(Add) for eq in eqs_eq]
sol = {(266.97755814852, 0.0340301680681629, 177985.03876568)}
assert_close_nl(nonlinsolve(eqs_eq, [x, y, z]), sol)
assert_close_nl(nonlinsolve(eqs_expr, [x, y, z]), sol)
logterm = log(1.91196789933362e-7*z/(5.0 - 1.91196789933362e-7*z) + 1)
eq = -0.0015*z*logterm + 1.02439504345316e-5*z
assert_close_ss(solveset(eq, z), {0, 177985.038765679})
def test_issue_19814():
assert nonlinsolve([ 2**m - 2**(2*n), 4*2**m - 2**(4*n)], m, n
) == FiniteSet((log(2**(2*n))/log(2), S.Complexes))
def test_issue_22058():
sol = solveset(-sqrt(t)*x**2 + 2*x + sqrt(t), x, S.Reals)
# doesn't fail (and following numerical check)
assert sol.xreplace({t: 1}) == {1 - sqrt(2), 1 + sqrt(2)}, sol.xreplace({t: 1})
def test_issue_11184():
assert solveset(20*sqrt(y**2 + (sqrt(-(y - 10)*(y + 10)) + 10)**2) - 60, y, S.Reals) is S.EmptySet
def test_issue_21890():
e = S(2)/3
assert nonlinsolve([4*x**3*y**4 - 2*y, 4*x**4*y**3 - 2*x], x, y) == {
(2**e/(2*y), y), ((-2**e/4 - 2**e*sqrt(3)*I/4)/y, y),
((-2**e/4 + 2**e*sqrt(3)*I/4)/y, y)}
assert nonlinsolve([(1 - 4*x**2)*exp(-2*x**2 - 2*y**2),
-4*x*y*exp(-2*x**2)*exp(-2*y**2)], x, y) == {(-S(1)/2, 0), (S(1)/2, 0)}
rx, ry = symbols('x y', real=True)
sol = nonlinsolve([4*rx**3*ry**4 - 2*ry, 4*rx**4*ry**3 - 2*rx], rx, ry)
ans = {(2**(S(2)/3)/(2*ry), ry),
((-2**(S(2)/3)/4 - 2**(S(2)/3)*sqrt(3)*I/4)/ry, ry),
((-2**(S(2)/3)/4 + 2**(S(2)/3)*sqrt(3)*I/4)/ry, ry)}
assert sol == ans
def test_issue_22628():
assert nonlinsolve([h - 1, k - 1, f - 2, f - 4, -2*k], h, k, f) == S.EmptySet
assert nonlinsolve([x**3 - 1, x + y, x**2 - 4], [x, y]) == S.EmptySet
|
53a24d171b6994feaa1233270ab67e12c39bff9ebe8fa3b71fa638db18402b17 | from sympy.assumptions.ask import (Q, ask)
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import (Derivative, Function, diff)
from sympy.core.mul import Mul
from sympy.core import (GoldenRatio, TribonacciConstant)
from sympy.core.numbers import (E, Float, I, Rational, oo, pi)
from sympy.core.relational import (Eq, Gt, Lt, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, Wild, symbols)
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import binomial
from sympy.functions.elementary.complexes import (Abs, arg, conjugate, im, re)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (atanh, cosh, sinh, tanh)
from sympy.functions.elementary.miscellaneous import (cbrt, root, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, asin, atan, atan2, cos, sec, sin, tan)
from sympy.functions.special.error_functions import (erf, erfc, erfcinv, erfinv)
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import (And, Or)
from sympy.matrices.dense import Matrix
from sympy.matrices import SparseMatrix
from sympy.polys.polytools import Poly
from sympy.printing.str import sstr
from sympy.simplify.radsimp import denom
from sympy.solvers.solvers import (nsolve, solve, solve_linear)
from sympy.core.function import nfloat
from sympy.solvers import solve_linear_system, solve_linear_system_LU, \
solve_undetermined_coeffs
from sympy.solvers.bivariate import _filtered_gens, _solve_lambert, _lambert
from sympy.solvers.solvers import _invert, unrad, checksol, posify, _ispow, \
det_quick, det_perm, det_minor, _simple_dens, denoms
from sympy.physics.units import cm
from sympy.polys.rootoftools import CRootOf
from sympy.testing.pytest import slow, XFAIL, SKIP, raises
from sympy.core.random import verify_numerically as tn
from sympy.abc import a, b, c, d, e, k, h, p, x, y, z, t, q, m, R
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_swap_back():
f, g = map(Function, 'fg')
fx, gx = f(x), g(x)
assert solve([fx + y - 2, fx - gx - 5], fx, y, gx) == \
{fx: gx + 5, y: -gx - 3}
assert solve(fx + gx*x - 2, [fx, gx], dict=True) == [{fx: 2, gx: 0}]
assert solve(fx + gx**2*x - y, [fx, gx], dict=True) == [{fx: y, gx: 0}]
assert solve([f(1) - 2, x + 2], dict=True) == [{x: -2, f(1): 2}]
def guess_solve_strategy(eq, symbol):
try:
solve(eq, symbol)
return True
except (TypeError, NotImplementedError):
return False
def test_guess_poly():
# polynomial equations
assert guess_solve_strategy( S(4), x ) # == GS_POLY
assert guess_solve_strategy( x, x ) # == GS_POLY
assert guess_solve_strategy( x + a, x ) # == GS_POLY
assert guess_solve_strategy( 2*x, x ) # == GS_POLY
assert guess_solve_strategy( x + sqrt(2), x) # == GS_POLY
assert guess_solve_strategy( x + 2**Rational(1, 4), x) # == GS_POLY
assert guess_solve_strategy( x**2 + 1, x ) # == GS_POLY
assert guess_solve_strategy( x**2 - 1, x ) # == GS_POLY
assert guess_solve_strategy( x*y + y, x ) # == GS_POLY
assert guess_solve_strategy( x*exp(y) + y, x) # == GS_POLY
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), x) # == GS_POLY
def test_guess_poly_cv():
# polynomial equations via a change of variable
assert guess_solve_strategy( sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy(
x**Rational(1, 3) + sqrt(x) + 1, x ) # == GS_POLY_CV_1
assert guess_solve_strategy( 4*x*(1 - sqrt(x)), x ) # == GS_POLY_CV_1
# polynomial equation multiplying both sides by x**n
assert guess_solve_strategy( x + 1/x + y, x ) # == GS_POLY_CV_2
def test_guess_rational_cv():
# rational functions
assert guess_solve_strategy( (x + 1)/(x**2 + 2), x) # == GS_RATIONAL
assert guess_solve_strategy(
(x - y**3)/(y**2*sqrt(1 - y**2)), y) # == GS_RATIONAL_CV_1
# rational functions via the change of variable y -> x**n
assert guess_solve_strategy( (sqrt(x) + 1)/(x**Rational(1, 3) + sqrt(x) + 1), x ) \
#== GS_RATIONAL_CV_1
def test_guess_transcendental():
#transcendental functions
assert guess_solve_strategy( exp(x) + 1, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy( 2*cos(x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(
exp(x) + exp(-x) - y, x ) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(3**x - 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(-3**x + 10, x) # == GS_TRANSCENDENTAL
assert guess_solve_strategy(a*x**b - y, x) # == GS_TRANSCENDENTAL
def test_solve_args():
# equation container, issue 5113
ans = {x: -3, y: 1}
eqs = (x + 5*y - 2, -3*x + 6*y - 15)
assert all(solve(container(eqs), x, y) == ans for container in
(tuple, list, set, frozenset))
assert solve(Tuple(*eqs), x, y) == ans
# implicit symbol to solve for
assert set(solve(x**2 - 4)) == {S(2), -S(2)}
assert solve([x + y - 3, x - y - 5]) == {x: 4, y: -1}
assert solve(x - exp(x), x, implicit=True) == [exp(x)]
# no symbol to solve for
assert solve(42) == solve(42, x) == []
assert solve([1, 2]) == []
assert solve([sqrt(2)],[x]) == []
# duplicate symbols raises
raises(ValueError, lambda: solve((x - 3, y + 2), x, y, x))
raises(ValueError, lambda: solve(x, x, x))
# no error in exclude
assert solve(x, x, exclude=[y, y]) == [0]
# duplicate symbols raises
raises(ValueError, lambda: solve((x - 3, y + 2), x, y, x))
raises(ValueError, lambda: solve(x, x, x))
# no error in exclude
assert solve(x, x, exclude=[y, y]) == [0]
# unordered symbols
# only 1
assert solve(y - 3, {y}) == [3]
# more than 1
assert solve(y - 3, {x, y}) == [{y: 3}]
# multiple symbols: take the first linear solution+
# - return as tuple with values for all requested symbols
assert solve(x + y - 3, [x, y]) == [(3 - y, y)]
# - unless dict is True
assert solve(x + y - 3, [x, y], dict=True) == [{x: 3 - y}]
# - or no symbols are given
assert solve(x + y - 3) == [{x: 3 - y}]
# multiple symbols might represent an undetermined coefficients system
assert solve(a + b*x - 2, [a, b]) == {a: 2, b: 0}
assert solve((a + b)*x + b - c, [a, b]) == {a: -c, b: c}
eq = a*x**2 + b*x + c - ((x - h)**2 + 4*p*k)/4/p
# - check that flags are obeyed
sol = solve(eq, [h, p, k], exclude=[a, b, c])
assert sol == {h: -b/(2*a), k: (4*a*c - b**2)/(4*a), p: 1/(4*a)}
assert solve(eq, [h, p, k], dict=True) == [sol]
assert solve(eq, [h, p, k], set=True) == \
([h, p, k], {(-b/(2*a), 1/(4*a), (4*a*c - b**2)/(4*a))})
# issue 23889 - polysys not simplified
assert solve(eq, [h, p, k], exclude=[a, b, c], simplify=False) == \
{h: -b/(2*a), k: (4*a*c - b**2)/(4*a), p: 1/(4*a)}
# but this only happens when system has a single solution
args = (a + b)*x - b**2 + 2, a, b
assert solve(*args) == [((b**2 - b*x - 2)/x, b)]
# and if the system has a solution; the following doesn't so
# an algebraic solution is returned
assert solve(a*x + b**2/(x + 4) - 3*x - 4/x, a, b, dict=True) == \
[{a: (-b**2*x + 3*x**3 + 12*x**2 + 4*x + 16)/(x**2*(x + 4))}]
# failed single equation
assert solve(1/(1/x - y + exp(y))) == []
raises(
NotImplementedError, lambda: solve(exp(x) + sin(x) + exp(y) + sin(y)))
# failed system
# -- when no symbols given, 1 fails
assert solve([y, exp(x) + x]) == [{x: -LambertW(1), y: 0}]
# both fail
assert solve(
(exp(x) - x, exp(y) - y)) == [{x: -LambertW(-1), y: -LambertW(-1)}]
# -- when symbols given
assert solve([y, exp(x) + x], x, y) == [(-LambertW(1), 0)]
# symbol is a number
assert solve(x**2 - pi, pi) == [x**2]
# no equations
assert solve([], [x]) == []
# nonlinear systen
assert solve((x**2 - 4, y - 2), x, y) == [(-2, 2), (2, 2)]
assert solve((x**2 - 4, y - 2), y, x) == [(2, -2), (2, 2)]
assert solve((x**2 - 4 + z, y - 2 - z), a, z, y, x, set=True
) == ([a, z, y, x], {
(a, z, z + 2, -sqrt(4 - z)),
(a, z, z + 2, sqrt(4 - z))})
# overdetermined system
# - nonlinear
assert solve([(x + y)**2 - 4, x + y - 2]) == [{x: -y + 2}]
# - linear
assert solve((x + y - 2, 2*x + 2*y - 4)) == {x: -y + 2}
# When one or more args are Boolean
assert solve(Eq(x**2, 0.0)) == [0] # issue 19048
assert solve([True, Eq(x, 0)], [x], dict=True) == [{x: 0}]
assert solve([Eq(x, x), Eq(x, 0), Eq(x, x+1)], [x], dict=True) == []
assert not solve([Eq(x, x+1), x < 2], x)
assert solve([Eq(x, 0), x+1<2]) == Eq(x, 0)
assert solve([Eq(x, x), Eq(x, x+1)], x) == []
assert solve(True, x) == []
assert solve([x - 1, False], [x], set=True) == ([], set())
assert solve([-y*(x + y - 1)/2, (y - 1)/x/y + 1/y],
set=True, check=False) == ([x, y], {(1 - y, y), (x, 0)})
# ordering should be canonical, fastest to order by keys instead
# of by size
assert list(solve((y - 1, x - sqrt(3)*z)).keys()) == [x, y]
# as set always returns as symbols, set even if no solution
assert solve([x - 1, x], (y, x), set=True) == ([y, x], set())
assert solve([x - 1, x], {y, x}, set=True) == ([x, y], set())
def test_solve_polynomial1():
assert solve(3*x - 2, x) == [Rational(2, 3)]
assert solve(Eq(3*x, 2), x) == [Rational(2, 3)]
assert set(solve(x**2 - 1, x)) == {-S.One, S.One}
assert set(solve(Eq(x**2, 1), x)) == {-S.One, S.One}
assert solve(x - y**3, x) == [y**3]
rx = root(x, 3)
assert solve(x - y**3, y) == [
rx, -rx/2 - sqrt(3)*I*rx/2, -rx/2 + sqrt(3)*I*rx/2]
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
assert solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y) == \
{
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
solution = {x: S.Zero, y: S.Zero}
assert solve((x - y, x + y), x, y ) == solution
assert solve((x - y, x + y), (x, y)) == solution
assert solve((x - y, x + y), [x, y]) == solution
assert set(solve(x**3 - 15*x - 4, x)) == {
-2 + 3**S.Half,
S(4),
-2 - 3**S.Half
}
assert set(solve((x**2 - 1)**2 - a, x)) == \
{sqrt(1 + sqrt(a)), -sqrt(1 + sqrt(a)),
sqrt(1 - sqrt(a)), -sqrt(1 - sqrt(a))}
def test_solve_polynomial2():
assert solve(4, x) == []
def test_solve_polynomial_cv_1a():
"""
Test for solving on equations that can be converted to a polynomial equation
using the change of variable y -> x**Rational(p, q)
"""
assert solve( sqrt(x) - 1, x) == [1]
assert solve( sqrt(x) - 2, x) == [4]
assert solve( x**Rational(1, 4) - 2, x) == [16]
assert solve( x**Rational(1, 3) - 3, x) == [27]
assert solve(sqrt(x) + x**Rational(1, 3) + x**Rational(1, 4), x) == [0]
def test_solve_polynomial_cv_1b():
assert set(solve(4*x*(1 - a*sqrt(x)), x)) == {S.Zero, 1/a**2}
assert set(solve(x*(root(x, 3) - 3), x)) == {S.Zero, S(27)}
def test_solve_polynomial_cv_2():
"""
Test for solving on equations that can be converted to a polynomial equation
multiplying both sides of the equation by x**m
"""
assert solve(x + 1/x - 1, x) in \
[[ S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2],
[ S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]]
def test_quintics_1():
f = x**5 - 110*x**3 - 55*x**2 + 2310*x + 979
s = solve(f, check=False)
for r in s:
res = f.subs(x, r.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for r in s:
assert r.func == CRootOf
# if one uses solve to get the roots of a polynomial that has a CRootOf
# solution, make sure that the use of nfloat during the solve process
# doesn't fail. Note: if you want numerical solutions to a polynomial
# it is *much* faster to use nroots to get them than to solve the
# equation only to get RootOf solutions which are then numerically
# evaluated. So for eq = x**5 + 3*x + 7 do Poly(eq).nroots() rather
# than [i.n() for i in solve(eq)] to get the numerical roots of eq.
assert nfloat(solve(x**5 + 3*x**3 + 7)[0], exponent=False) == \
CRootOf(x**5 + 3*x**3 + 7, 0).n()
def test_quintics_2():
f = x**5 + 15*x + 12
s = solve(f, check=False)
for r in s:
res = f.subs(x, r.n()).n()
assert tn(res, 0)
f = x**5 - 15*x**3 - 5*x**2 + 10*x + 20
s = solve(f)
for r in s:
assert r.func == CRootOf
assert solve(x**5 - 6*x**3 - 6*x**2 + x - 6) == [
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 0),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 1),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 2),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 3),
CRootOf(x**5 - 6*x**3 - 6*x**2 + x - 6, 4)]
def test_quintics_3():
y = x**5 + x**3 - 2**Rational(1, 3)
assert solve(y) == solve(-y) == []
def test_highorder_poly():
# just testing that the uniq generator is unpacked
sol = solve(x**6 - 2*x + 2)
assert all(isinstance(i, CRootOf) for i in sol) and len(sol) == 6
def test_solve_rational():
"""Test solve for rational functions"""
assert solve( ( x - y**3 )/( (y**2)*sqrt(1 - y**2) ), x) == [y**3]
def test_solve_conjugate():
"""Test solve for simple conjugate functions"""
assert solve(conjugate(x) -3 + I) == [3 + I]
def test_solve_nonlinear():
assert solve(x**2 - y**2, x, y, dict=True) == [{x: -y}, {x: y}]
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: -x*sqrt(exp(x))},
{y: x*sqrt(exp(x))}]
def test_issue_8666():
x = symbols('x')
assert solve(Eq(x**2 - 1/(x**2 - 4), 4 - 1/(x**2 - 4)), x) == []
assert solve(Eq(x + 1/x, 1/x), x) == []
def test_issue_7228():
assert solve(4**(2*(x**2) + 2*x) - 8, x) == [Rational(-3, 2), S.Half]
def test_issue_7190():
assert solve(log(x-3) + log(x+3), x) == [sqrt(10)]
def test_issue_21004():
x = symbols('x')
f = x/sqrt(x**2+1)
f_diff = f.diff(x)
assert solve(f_diff, x) == []
def test_linear_system():
x, y, z, t, n = symbols('x, y, z, t, n')
assert solve([x - 1, x - y, x - 2*y, y - 1], [x, y]) == []
assert solve([x - 1, x - y, x - 2*y, x - 1], [x, y]) == []
assert solve([x - 1, x - 1, x - y, x - 2*y], [x, y]) == []
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == {x: -3, y: 1}
M = Matrix([[0, 0, n*(n + 1), (n + 1)**2, 0],
[n + 1, n + 1, -2*n - 1, -(n + 1), 0],
[-1, 0, 1, 0, 0]])
assert solve_linear_system(M, x, y, z, t) == \
{x: t*(-n-1)/n, y: 0, z: t*(-n-1)/n}
assert solve([x + y + z + t, -z - t], x, y, z, t) == {x: -y, z: -t}
@XFAIL
def test_linear_system_xfail():
# https://github.com/sympy/sympy/issues/6420
M = Matrix([[0, 15.0, 10.0, 700.0],
[1, 1, 1, 100.0],
[0, 10.0, 5.0, 200.0],
[-5.0, 0, 0, 0 ]])
assert solve_linear_system(M, x, y, z) == {x: 0, y: -60.0, z: 160.0}
def test_linear_system_function():
a = Function('a')
assert solve([a(0, 0) + a(0, 1) + a(1, 0) + a(1, 1), -a(1, 0) - a(1, 1)],
a(0, 0), a(0, 1), a(1, 0), a(1, 1)) == {a(1, 0): -a(1, 1), a(0, 0): -a(0, 1)}
def test_linear_system_symbols_doesnt_hang_1():
def _mk_eqs(wy):
# Equations for fitting a wy*2 - 1 degree polynomial between two points,
# at end points derivatives are known up to order: wy - 1
order = 2*wy - 1
x, x0, x1 = symbols('x, x0, x1', real=True)
y0s = symbols('y0_:{}'.format(wy), real=True)
y1s = symbols('y1_:{}'.format(wy), real=True)
c = symbols('c_:{}'.format(order+1), real=True)
expr = sum([coeff*x**o for o, coeff in enumerate(c)])
eqs = []
for i in range(wy):
eqs.append(expr.diff(x, i).subs({x: x0}) - y0s[i])
eqs.append(expr.diff(x, i).subs({x: x1}) - y1s[i])
return eqs, c
#
# The purpose of this test is just to see that these calls don't hang. The
# expressions returned are complicated so are not included here. Testing
# their correctness takes longer than solving the system.
#
for n in range(1, 7+1):
eqs, c = _mk_eqs(n)
solve(eqs, c)
def test_linear_system_symbols_doesnt_hang_2():
M = Matrix([
[66, 24, 39, 50, 88, 40, 37, 96, 16, 65, 31, 11, 37, 72, 16, 19, 55, 37, 28, 76],
[10, 93, 34, 98, 59, 44, 67, 74, 74, 94, 71, 61, 60, 23, 6, 2, 57, 8, 29, 78],
[19, 91, 57, 13, 64, 65, 24, 53, 77, 34, 85, 58, 87, 39, 39, 7, 36, 67, 91, 3],
[74, 70, 15, 53, 68, 43, 86, 83, 81, 72, 25, 46, 67, 17, 59, 25, 78, 39, 63, 6],
[69, 40, 67, 21, 67, 40, 17, 13, 93, 44, 46, 89, 62, 31, 30, 38, 18, 20, 12, 81],
[50, 22, 74, 76, 34, 45, 19, 76, 28, 28, 11, 99, 97, 82, 8, 46, 99, 57, 68, 35],
[58, 18, 45, 88, 10, 64, 9, 34, 90, 82, 17, 41, 43, 81, 45, 83, 22, 88, 24, 39],
[42, 21, 70, 68, 6, 33, 64, 81, 83, 15, 86, 75, 86, 17, 77, 34, 62, 72, 20, 24],
[ 7, 8, 2, 72, 71, 52, 96, 5, 32, 51, 31, 36, 79, 88, 25, 77, 29, 26, 33, 13],
[19, 31, 30, 85, 81, 39, 63, 28, 19, 12, 16, 49, 37, 66, 38, 13, 3, 71, 61, 51],
[29, 82, 80, 49, 26, 85, 1, 37, 2, 74, 54, 82, 26, 47, 54, 9, 35, 0, 99, 40],
[15, 49, 82, 91, 93, 57, 45, 25, 45, 97, 15, 98, 48, 52, 66, 24, 62, 54, 97, 37],
[62, 23, 73, 53, 52, 86, 28, 38, 0, 74, 92, 38, 97, 70, 71, 29, 26, 90, 67, 45],
[ 2, 32, 23, 24, 71, 37, 25, 71, 5, 41, 97, 65, 93, 13, 65, 45, 25, 88, 69, 50],
[40, 56, 1, 29, 79, 98, 79, 62, 37, 28, 45, 47, 3, 1, 32, 74, 98, 35, 84, 32],
[33, 15, 87, 79, 65, 9, 14, 63, 24, 19, 46, 28, 74, 20, 29, 96, 84, 91, 93, 1],
[97, 18, 12, 52, 1, 2, 50, 14, 52, 76, 19, 82, 41, 73, 51, 79, 13, 3, 82, 96],
[40, 28, 52, 10, 10, 71, 56, 78, 82, 5, 29, 48, 1, 26, 16, 18, 50, 76, 86, 52],
[38, 89, 83, 43, 29, 52, 90, 77, 57, 0, 67, 20, 81, 88, 48, 96, 88, 58, 14, 3]])
syms = x0,x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x11,x12,x13,x14,x15,x16,x17,x18 = symbols('x:19')
sol = {
x0: -S(1967374186044955317099186851240896179)/3166636564687820453598895768302256588,
x1: -S(84268280268757263347292368432053826)/791659141171955113399723942075564147,
x2: -S(229962957341664730974463872411844965)/1583318282343910226799447884151128294,
x3: S(990156781744251750886760432229180537)/6333273129375640907197791536604513176,
x4: -S(2169830351210066092046760299593096265)/18999819388126922721593374609813539528,
x5: S(4680868883477577389628494526618745355)/9499909694063461360796687304906769764,
x6: -S(1590820774344371990683178396480879213)/3166636564687820453598895768302256588,
x7: -S(54104723404825537735226491634383072)/339282489073695048599881689460956063,
x8: S(3182076494196560075964847771774733847)/6333273129375640907197791536604513176,
x9: -S(10870817431029210431989147852497539675)/18999819388126922721593374609813539528,
x10: -S(13118019242576506476316318268573312603)/18999819388126922721593374609813539528,
x11: -S(5173852969886775824855781403820641259)/4749954847031730680398343652453384882,
x12: S(4261112042731942783763341580651820563)/4749954847031730680398343652453384882,
x13: -S(821833082694661608993818117038209051)/6333273129375640907197791536604513176,
x14: S(906881575107250690508618713632090559)/904753304196520129599684505229216168,
x15: -S(732162528717458388995329317371283987)/6333273129375640907197791536604513176,
x16: S(4524215476705983545537087360959896817)/9499909694063461360796687304906769764,
x17: -S(3898571347562055611881270844646055217)/6333273129375640907197791536604513176,
x18: S(7513502486176995632751685137907442269)/18999819388126922721593374609813539528
}
eqs = list(M * Matrix(syms + (1,)))
assert solve(eqs, syms) == sol
y = Symbol('y')
eqs = list(y * M * Matrix(syms + (1,)))
assert solve(eqs, syms) == sol
def test_linear_systemLU():
n = Symbol('n')
M = Matrix([[1, 2, 0, 1], [1, 3, 2*n, 1], [4, -1, n**2, 1]])
assert solve_linear_system_LU(M, [x, y, z]) == {z: -3/(n**2 + 18*n),
x: 1 - 12*n/(n**2 + 18*n),
y: 6*n/(n**2 + 18*n)}
# Note: multiple solutions exist for some of these equations, so the tests
# should be expected to break if the implementation of the solver changes
# in such a way that a different branch is chosen
@slow
def test_solve_transcendental():
from sympy.abc import a, b
assert solve(exp(x) - 3, x) == [log(3)]
assert set(solve((a*x + b)*(exp(x) - 3), x)) == {-b/a, log(3)}
assert solve(cos(x) - y, x) == [-acos(y) + 2*pi, acos(y)]
assert solve(2*cos(x) - y, x) == [-acos(y/2) + 2*pi, acos(y/2)]
assert solve(Eq(cos(x), sin(x)), x) == [pi/4]
assert set(solve(exp(x) + exp(-x) - y, x)) in [{
log(y/2 - sqrt(y**2 - 4)/2),
log(y/2 + sqrt(y**2 - 4)/2),
}, {
log(y - sqrt(y**2 - 4)) - log(2),
log(y + sqrt(y**2 - 4)) - log(2)},
{
log(y/2 - sqrt((y - 2)*(y + 2))/2),
log(y/2 + sqrt((y - 2)*(y + 2))/2)}]
assert solve(exp(x) - 3, x) == [log(3)]
assert solve(Eq(exp(x), 3), x) == [log(3)]
assert solve(log(x) - 3, x) == [exp(3)]
assert solve(sqrt(3*x) - 4, x) == [Rational(16, 3)]
assert solve(3**(x + 2), x) == []
assert solve(3**(2 - x), x) == []
assert solve(x + 2**x, x) == [-LambertW(log(2))/log(2)]
assert solve(2*x + 5 + log(3*x - 2), x) == \
[Rational(2, 3) + LambertW(2*exp(Rational(-19, 3))/3)/2]
assert solve(3*x + log(4*x), x) == [LambertW(Rational(3, 4))/3]
assert set(solve((2*x + 8)*(8 + exp(x)), x)) == {S(-4), log(8) + pi*I}
eq = 2*exp(3*x + 4) - 3
ans = solve(eq, x) # this generated a failure in flatten
assert len(ans) == 3 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(2*log(3*x + 4) - 3, x) == [(exp(Rational(3, 2)) - 4)/3]
assert solve(exp(x) + 1, x) == [pi*I]
eq = 2*(3*x + 4)**5 - 6*7**(3*x + 9)
result = solve(eq, x)
x0 = -log(2401)
x1 = 3**Rational(1, 5)
x2 = log(7**(7*x1/20))
x3 = sqrt(2)
x4 = sqrt(5)
x5 = x3*sqrt(x4 - 5)
x6 = x4 + 1
x7 = 1/(3*log(7))
x8 = -x4
x9 = x3*sqrt(x8 - 5)
x10 = x8 + 1
ans = [x7*(x0 - 5*LambertW(x2*(-x5 + x6))),
x7*(x0 - 5*LambertW(x2*(x5 + x6))),
x7*(x0 - 5*LambertW(x2*(x10 - x9))),
x7*(x0 - 5*LambertW(x2*(x10 + x9))),
x7*(x0 - 5*LambertW(-log(7**(7*x1/5))))]
assert result == ans, result
# it works if expanded, too
assert solve(eq.expand(), x) == result
assert solve(z*cos(x) - y, x) == [-acos(y/z) + 2*pi, acos(y/z)]
assert solve(z*cos(2*x) - y, x) == [-acos(y/z)/2 + pi, acos(y/z)/2]
assert solve(z*cos(sin(x)) - y, x) == [
pi - asin(acos(y/z)), asin(acos(y/z) - 2*pi) + pi,
-asin(acos(y/z) - 2*pi), asin(acos(y/z))]
assert solve(z*cos(x), x) == [pi/2, pi*Rational(3, 2)]
# issue 4508
assert solve(y - b*x/(a + x), x) in [[-a*y/(y - b)], [a*y/(b - y)]]
assert solve(y - b*exp(a/x), x) == [a/log(y/b)]
# issue 4507
assert solve(y - b/(1 + a*x), x) in [[(b - y)/(a*y)], [-((y - b)/(a*y))]]
# issue 4506
assert solve(y - a*x**b, x) == [(y/a)**(1/b)]
# issue 4505
assert solve(z**x - y, x) == [log(y)/log(z)]
# issue 4504
assert solve(2**x - 10, x) == [1 + log(5)/log(2)]
# issue 6744
assert solve(x*y) == [{x: 0}, {y: 0}]
assert solve([x*y]) == [{x: 0}, {y: 0}]
assert solve(x**y - 1) == [{x: 1}, {y: 0}]
assert solve([x**y - 1]) == [{x: 1}, {y: 0}]
assert solve(x*y*(x**2 - y**2)) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
assert solve([x*y*(x**2 - y**2)]) == [{x: 0}, {x: -y}, {x: y}, {y: 0}]
# issue 4739
assert solve(exp(log(5)*x) - 2**x, x) == [0]
# issue 14791
assert solve(exp(log(5)*x) - exp(log(2)*x), x) == [0]
f = Function('f')
assert solve(y*f(log(5)*x) - y*f(log(2)*x), x) == [0]
assert solve(f(x) - f(0), x) == [0]
assert solve(f(x) - f(2 - x), x) == [1]
raises(NotImplementedError, lambda: solve(f(x, y) - f(1, 2), x))
raises(NotImplementedError, lambda: solve(f(x, y) - f(2 - x, 2), x))
raises(ValueError, lambda: solve(f(x, y) - f(1 - x), x))
raises(ValueError, lambda: solve(f(x, y) - f(1), x))
# misc
# make sure that the right variables is picked up in tsolve
# shouldn't generate a GeneratorsNeeded error in _tsolve when the NaN is generated
# for eq_down. Actual answers, as determined numerically are approx. +/- 0.83
raises(NotImplementedError, lambda:
solve(sinh(x)*sinh(sinh(x)) + cosh(x)*cosh(sinh(x)) - 3))
# watch out for recursive loop in tsolve
raises(NotImplementedError, lambda: solve((x + 2)**y*x - 3, x))
# issue 7245
assert solve(sin(sqrt(x))) == [0, pi**2]
# issue 7602
a, b = symbols('a, b', real=True, negative=False)
assert str(solve(Eq(a, 0.5 - cos(pi*b)/2), b)) == \
'[2.0 - 0.318309886183791*acos(1.0 - 2.0*a), 0.318309886183791*acos(1.0 - 2.0*a)]'
# issue 15325
assert solve(y**(1/x) - z, x) == [log(y)/log(z)]
def test_solve_for_functions_derivatives():
t = Symbol('t')
x = Function('x')(t)
y = Function('y')(t)
a11, a12, a21, a22, b1, b2 = symbols('a11,a12,a21,a22,b1,b2')
soln = solve([a11*x + a12*y - b1, a21*x + a22*y - b2], x, y)
assert soln == {
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21),
y: (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
}
assert solve(x - 1, x) == [1]
assert solve(3*x - 2, x) == [Rational(2, 3)]
soln = solve([a11*x.diff(t) + a12*y.diff(t) - b1, a21*x.diff(t) +
a22*y.diff(t) - b2], x.diff(t), y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x.diff(t): (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
assert solve(x.diff(t) - 1, x.diff(t)) == [1]
assert solve(3*x.diff(t) - 2, x.diff(t)) == [Rational(2, 3)]
eqns = {3*x - 1, 2*y - 4}
assert solve(eqns, {x, y}) == { x: Rational(1, 3), y: 2 }
x = Symbol('x')
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
assert solve(F.diff(x), diff(f(x), x)) == [(-x + 2)/f(x)]
# Mixed cased with a Symbol and a Function
x = Symbol('x')
y = Function('y')(t)
soln = solve([a11*x + a12*y.diff(t) - b1, a21*x +
a22*y.diff(t) - b2], x, y.diff(t))
assert soln == { y.diff(t): (a11*b2 - a21*b1)/(a11*a22 - a12*a21),
x: (a22*b1 - a12*b2)/(a11*a22 - a12*a21) }
# issue 13263
x = Symbol('x')
f = Function('f')
soln = solve([f(x).diff(x) + f(x).diff(x, 2) - 1, f(x).diff(x) - f(x).diff(x, 2)],
f(x).diff(x), f(x).diff(x, 2))
assert soln == { f(x).diff(x, 2): 1/2, f(x).diff(x): 1/2 }
soln = solve([f(x).diff(x, 2) + f(x).diff(x, 3) - 1, 1 - f(x).diff(x, 2) -
f(x).diff(x, 3), 1 - f(x).diff(x,3)], f(x).diff(x, 2), f(x).diff(x, 3))
assert soln == { f(x).diff(x, 2): 0, f(x).diff(x, 3): 1 }
def test_issue_3725():
f = Function('f')
F = x**2 + f(x)**2 - 4*x - 1
e = F.diff(x)
assert solve(e, f(x).diff(x)) in [[(2 - x)/f(x)], [-((x - 2)/f(x))]]
def test_issue_3870():
a, b, c, d = symbols('a b c d')
A = Matrix(2, 2, [a, b, c, d])
B = Matrix(2, 2, [0, 2, -3, 0])
C = Matrix(2, 2, [1, 2, 3, 4])
assert solve(A*B - C, [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve([A*B - C], [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve(Eq(A*B, C), [a, b, c, d]) == {a: 1, b: Rational(-1, 3), c: 2, d: -1}
assert solve([A*B - B*A], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c}
assert solve([A*C - C*A], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c}
assert solve([A*B - B*A, A*C - C*A], [a, b, c, d]) == {a: d, b: 0, c: 0}
assert solve([Eq(A*B, B*A)], [a, b, c, d]) == {a: d, b: Rational(-2, 3)*c}
assert solve([Eq(A*C, C*A)], [a, b, c, d]) == {a: d - c, b: Rational(2, 3)*c}
assert solve([Eq(A*B, B*A), Eq(A*C, C*A)], [a, b, c, d]) == {a: d, b: 0, c: 0}
def test_solve_linear():
w = Wild('w')
assert solve_linear(x, x) == (0, 1)
assert solve_linear(x, exclude=[x]) == (0, 1)
assert solve_linear(x, symbols=[w]) == (0, 1)
assert solve_linear(x, y - 2*x) in [(x, y/3), (y, 3*x)]
assert solve_linear(x, y - 2*x, exclude=[x]) == (y, 3*x)
assert solve_linear(3*x - y, 0) in [(x, y/3), (y, 3*x)]
assert solve_linear(3*x - y, 0, [x]) == (x, y/3)
assert solve_linear(3*x - y, 0, [y]) == (y, 3*x)
assert solve_linear(x**2/y, 1) == (y, x**2)
assert solve_linear(w, x) in [(w, x), (x, w)]
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y) == \
(y, -2 - cos(x)**2 - sin(x)**2)
assert solve_linear(cos(x)**2 + sin(x)**2 + 2 + y, symbols=[x]) == (0, 1)
assert solve_linear(Eq(x, 3)) == (x, 3)
assert solve_linear(1/(1/x - 2)) == (0, 0)
assert solve_linear((x + 1)*exp(-x), symbols=[x]) == (x, -1)
assert solve_linear((x + 1)*exp(x), symbols=[x]) == ((x + 1)*exp(x), 1)
assert solve_linear(x*exp(-x**2), symbols=[x]) == (x, 0)
assert solve_linear(0**x - 1) == (0**x - 1, 1)
assert solve_linear(1 + 1/(x - 1)) == (x, 0)
eq = y*cos(x)**2 + y*sin(x)**2 - y # = y*(1 - 1) = 0
assert solve_linear(eq) == (0, 1)
eq = cos(x)**2 + sin(x)**2 # = 1
assert solve_linear(eq) == (0, 1)
raises(ValueError, lambda: solve_linear(Eq(x, 3), 3))
def test_solve_undetermined_coeffs():
assert solve_undetermined_coeffs(a*x**2 + b*x**2 + b*x + 2*c*x + c + 1, [a, b, c], x) == \
{a: -2, b: 2, c: -1}
# Test that rational functions work
assert solve_undetermined_coeffs(a/x + b/(x + 1) - (2*x + 1)/(x**2 + x), [a, b], x) == \
{a: 1, b: 1}
# Test cancellation in rational functions
assert solve_undetermined_coeffs(((c + 1)*a*x**2 + (c + 1)*b*x**2 +
(c + 1)*b*x + (c + 1)*2*c*x + (c + 1)**2)/(c + 1), [a, b, c], x) == \
{a: -2, b: 2, c: -1}
def test_solve_inequalities():
x = Symbol('x')
sol = And(S.Zero < x, x < oo)
assert solve(x + 1 > 1) == sol
assert solve([x + 1 > 1]) == sol
assert solve([x + 1 > 1], x) == sol
assert solve([x + 1 > 1], [x]) == sol
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
And(Or(And(Lt(-sqrt(2), x), Lt(x, -1)),
And(Lt(1, x), Lt(x, sqrt(2)))), Eq(0, 0))
x = Symbol('x', real=True)
system = [Lt(x**2 - 2, 0), Gt(x**2 - 1, 0)]
assert solve(system) == \
Or(And(Lt(-sqrt(2), x), Lt(x, -1)), And(Lt(1, x), Lt(x, sqrt(2))))
# issues 6627, 3448
assert solve((x - 3)/(x - 2) < 0, x) == And(Lt(2, x), Lt(x, 3))
assert solve(x/(x + 1) > 1, x) == And(Lt(-oo, x), Lt(x, -1))
assert solve(sin(x) > S.Half) == And(pi/6 < x, x < pi*Rational(5, 6))
assert solve(Eq(False, x < 1)) == (S.One <= x) & (x < oo)
assert solve(Eq(True, x < 1)) == (-oo < x) & (x < 1)
assert solve(Eq(x < 1, False)) == (S.One <= x) & (x < oo)
assert solve(Eq(x < 1, True)) == (-oo < x) & (x < 1)
assert solve(Eq(False, x)) == False
assert solve(Eq(0, x)) == [0]
assert solve(Eq(True, x)) == True
assert solve(Eq(1, x)) == [1]
assert solve(Eq(False, ~x)) == True
assert solve(Eq(True, ~x)) == False
assert solve(Ne(True, x)) == False
assert solve(Ne(1, x)) == (x > -oo) & (x < oo) & Ne(x, 1)
def test_issue_4793():
assert solve(1/x) == []
assert solve(x*(1 - 5/x)) == [5]
assert solve(x + sqrt(x) - 2) == [1]
assert solve(-(1 + x)/(2 + x)**2 + 1/(2 + x)) == []
assert solve(-x**2 - 2*x + (x + 1)**2 - 1) == []
assert solve((x/(x + 1) + 3)**(-2)) == []
assert solve(x/sqrt(x**2 + 1), x) == [0]
assert solve(exp(x) - y, x) == [log(y)]
assert solve(exp(x)) == []
assert solve(x**2 + x + sin(y)**2 + cos(y)**2 - 1, x) in [[0, -1], [-1, 0]]
eq = 4*3**(5*x + 2) - 7
ans = solve(eq, x)
assert len(ans) == 5 and all(eq.subs(x, a).n(chop=True) == 0 for a in ans)
assert solve(log(x**2) - y**2/exp(x), x, y, set=True) == (
[x, y],
{(x, sqrt(exp(x) * log(x ** 2))), (x, -sqrt(exp(x) * log(x ** 2)))})
assert solve(x**2*z**2 - z**2*y**2) == [{x: -y}, {x: y}, {z: 0}]
assert solve((x - 1)/(1 + 1/(x - 1))) == []
assert solve(x**(y*z) - x, x) == [1]
raises(NotImplementedError, lambda: solve(log(x) - exp(x), x))
raises(NotImplementedError, lambda: solve(2**x - exp(x) - 3))
def test_PR1964():
# issue 5171
assert solve(sqrt(x)) == solve(sqrt(x**3)) == [0]
assert solve(sqrt(x - 1)) == [1]
# issue 4462
a = Symbol('a')
assert solve(-3*a/sqrt(x), x) == []
# issue 4486
assert solve(2*x/(x + 2) - 1, x) == [2]
# issue 4496
assert set(solve((x**2/(7 - x)).diff(x))) == {S.Zero, S(14)}
# issue 4695
f = Function('f')
assert solve((3 - 5*x/f(x))*f(x), f(x)) == [x*Rational(5, 3)]
# issue 4497
assert solve(1/root(5 + x, 5) - 9, x) == [Rational(-295244, 59049)]
assert solve(sqrt(x) + sqrt(sqrt(x)) - 4) == [(Rational(-1, 2) + sqrt(17)/2)**4]
assert set(solve(Poly(sqrt(exp(x)) + sqrt(exp(-x)) - 4))) in \
[
{log((-sqrt(3) + 2)**2), log((sqrt(3) + 2)**2)},
{2*log(-sqrt(3) + 2), 2*log(sqrt(3) + 2)},
{log(-4*sqrt(3) + 7), log(4*sqrt(3) + 7)},
]
assert set(solve(Poly(exp(x) + exp(-x) - 4))) == \
{log(-sqrt(3) + 2), log(sqrt(3) + 2)}
assert set(solve(x**y + x**(2*y) - 1, x)) == \
{(Rational(-1, 2) + sqrt(5)/2)**(1/y), (Rational(-1, 2) - sqrt(5)/2)**(1/y)}
assert solve(exp(x/y)*exp(-z/y) - 2, y) == [(x - z)/log(2)]
assert solve(
x**z*y**z - 2, z) in [[log(2)/(log(x) + log(y))], [log(2)/(log(x*y))]]
# if you do inversion too soon then multiple roots (as for the following)
# will be missed, e.g. if exp(3*x) = exp(3) -> 3*x = 3
E = S.Exp1
assert solve(exp(3*x) - exp(3), x) in [
[1, log(E*(Rational(-1, 2) - sqrt(3)*I/2)), log(E*(Rational(-1, 2) + sqrt(3)*I/2))],
[1, log(-E/2 - sqrt(3)*E*I/2), log(-E/2 + sqrt(3)*E*I/2)],
]
# coverage test
p = Symbol('p', positive=True)
assert solve((1/p + 1)**(p + 1)) == []
def test_issue_5197():
x = Symbol('x', real=True)
assert solve(x**2 + 1, x) == []
n = Symbol('n', integer=True, positive=True)
assert solve((n - 1)*(n + 2)*(2*n - 1), n) == [1]
x = Symbol('x', positive=True)
y = Symbol('y')
assert solve([x + 5*y - 2, -3*x + 6*y - 15], x, y) == []
# not {x: -3, y: 1} b/c x is positive
# The solution following should not contain (-sqrt(2), sqrt(2))
assert solve([(x + y), 2 - y**2], x, y) == [(sqrt(2), -sqrt(2))]
y = Symbol('y', positive=True)
# The solution following should not contain {y: -x*exp(x/2)}
assert solve(x**2 - y**2/exp(x), y, x, dict=True) == [{y: x*exp(x/2)}]
x, y, z = symbols('x y z', positive=True)
assert solve(z**2*x**2 - z**2*y**2/exp(x), y, x, z, dict=True) == [{y: x*exp(x/2)}]
def test_checking():
assert set(
solve(x*(x - y/x), x, check=False)) == {sqrt(y), S.Zero, -sqrt(y)}
assert set(solve(x*(x - y/x), x, check=True)) == {sqrt(y), -sqrt(y)}
# {x: 0, y: 4} sets denominator to 0 in the following so system should return None
assert solve((1/(1/x + 2), 1/(y - 3) - 1)) == []
# 0 sets denominator of 1/x to zero so None is returned
assert solve(1/(1/x + 2)) == []
def test_issue_4671_4463_4467():
assert solve(sqrt(x**2 - 1) - 2) in ([sqrt(5), -sqrt(5)],
[-sqrt(5), sqrt(5)])
assert solve((2**exp(y**2/x) + 2)/(x**2 + 15), y) == [
-sqrt(x*log(1 + I*pi/log(2))), sqrt(x*log(1 + I*pi/log(2)))]
C1, C2 = symbols('C1 C2')
f = Function('f')
assert solve(C1 + C2/x**2 - exp(-f(x)), f(x)) == [log(x**2/(C1*x**2 + C2))]
a = Symbol('a')
E = S.Exp1
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2]
)
assert solve(log(a**(-3) - x**2)/a, x) in (
[-sqrt(-1 + a**(-3)), sqrt(-1 + a**(-3))],
[sqrt(-1 + a**(-3)), -sqrt(-1 + a**(-3))],)
assert solve(1 - log(a + 4*x**2), x) in (
[-sqrt(-a + E)/2, sqrt(-a + E)/2],
[sqrt(-a + E)/2, -sqrt(-a + E)/2],)
assert solve((a**2 + 1)*(sin(a*x) + cos(a*x)), x) == [-pi/(4*a)]
assert solve(3 - (sinh(a*x) + cosh(a*x)), x) == [log(3)/a]
assert set(solve(3 - (sinh(a*x) + cosh(a*x)**2), x)) == \
{log(-2 + sqrt(5))/a, log(-sqrt(2) + 1)/a,
log(-sqrt(5) - 2)/a, log(1 + sqrt(2))/a}
assert solve(atan(x) - 1) == [tan(1)]
def test_issue_5132():
r, t = symbols('r,t')
assert set(solve([r - x**2 - y**2, tan(t) - y/x], [x, y])) == \
{(
-sqrt(r*cos(t)**2), -1*sqrt(r*cos(t)**2)*tan(t)),
(sqrt(r*cos(t)**2), sqrt(r*cos(t)**2)*tan(t))}
assert solve([exp(x) - sin(y), 1/y - 3], [x, y]) == \
[(log(sin(Rational(1, 3))), Rational(1, 3))]
assert solve([exp(x) - sin(y), 1/exp(y) - 3], [x, y]) == \
[(log(-sin(log(3))), -log(3))]
assert set(solve([exp(x) - sin(y), y**2 - 4], [x, y])) == \
{(log(-sin(2)), -S(2)), (log(sin(2)), S(2))}
eqs = [exp(x)**2 - sin(y) + z**2, 1/exp(y) - 3]
assert solve(eqs, set=True) == \
([y, z], {
(-log(3), sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), -sqrt(-exp(2*x) - sin(log(3))))})
assert solve(eqs, x, z, set=True) == (
[x, z],
{(x, sqrt(-exp(2*x) + sin(y))), (x, -sqrt(-exp(2*x) + sin(y)))})
assert set(solve(eqs, x, y)) == \
{
(log(-sqrt(-z**2 - sin(log(3)))), -log(3)),
(log(-z**2 - sin(log(3)))/2, -log(3))}
assert set(solve(eqs, y, z)) == \
{
(-log(3), -sqrt(-exp(2*x) - sin(log(3)))),
(-log(3), sqrt(-exp(2*x) - sin(log(3))))}
eqs = [exp(x)**2 - sin(y) + z, 1/exp(y) - 3]
assert solve(eqs, set=True) == ([y, z], {
(-log(3), -exp(2*x) - sin(log(3)))})
assert solve(eqs, x, z, set=True) == (
[x, z], {(x, -exp(2*x) + sin(y))})
assert set(solve(eqs, x, y)) == {
(log(-sqrt(-z - sin(log(3)))), -log(3)),
(log(-z - sin(log(3)))/2, -log(3))}
assert solve(eqs, z, y) == \
[(-exp(2*x) - sin(log(3)), -log(3))]
assert solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), set=True) == (
[x, y], {(S.One, S(3)), (S(3), S.One)})
assert set(solve((sqrt(x**2 + y**2) - sqrt(10), x + y - 4), x, y)) == \
{(S.One, S(3)), (S(3), S.One)}
def test_issue_5335():
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
# there are 4 solutions obtained manually but only two are valid
assert len(solve(eqs, sym, manual=True, minimal=True)) == 2
assert len(solve(eqs, sym)) == 2 # cf below with rational=False
@SKIP("Hangs")
def _test_issue_5335_float():
# gives ZeroDivisionError: polynomial division
lam, a0, conc = symbols('lam a0 conc')
a = 0.005
b = 0.743436700916726
eqs = [lam + 2*y - a0*(1 - x/2)*x - a*x/2*x,
a0*(1 - x/2)*x - 1*y - b*y,
x + y - conc]
sym = [x, y, a0]
assert len(solve(eqs, sym, rational=False)) == 2
def test_issue_5767():
assert set(solve([x**2 + y + 4], [x])) == \
{(-sqrt(-y - 4),), (sqrt(-y - 4),)}
def test_polysys():
assert set(solve([x**2 + 2/y - 2, x + y - 3], [x, y])) == \
{(S.One, S(2)), (1 + sqrt(5), 2 - sqrt(5)),
(1 - sqrt(5), 2 + sqrt(5))}
assert solve([x**2 + y - 2, x**2 + y]) == []
# the ordering should be whatever the user requested
assert solve([x**2 + y - 3, x - y - 4], (x, y)) != solve([x**2 +
y - 3, x - y - 4], (y, x))
@slow
def test_unrad1():
raises(NotImplementedError, lambda:
unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x)) + 3))
raises(NotImplementedError, lambda:
unrad(sqrt(x) + (x + 1)**Rational(1, 3) + 2*sqrt(y)))
s = symbols('s', cls=Dummy)
# checkers to deal with possibility of answer coming
# back with a sign change (cf issue 5203)
def check(rv, ans):
assert bool(rv[1]) == bool(ans[1])
if ans[1]:
return s_check(rv, ans)
e = rv[0].expand()
a = ans[0].expand()
return e in [a, -a] and rv[1] == ans[1]
def s_check(rv, ans):
# get the dummy
rv = list(rv)
d = rv[0].atoms(Dummy)
reps = list(zip(d, [s]*len(d)))
# replace s with this dummy
rv = (rv[0].subs(reps).expand(), [rv[1][0].subs(reps), rv[1][1].subs(reps)])
ans = (ans[0].subs(reps).expand(), [ans[1][0].subs(reps), ans[1][1].subs(reps)])
return str(rv[0]) in [str(ans[0]), str(-ans[0])] and \
str(rv[1]) == str(ans[1])
assert unrad(1) is None
assert check(unrad(sqrt(x)),
(x, []))
assert check(unrad(sqrt(x) + 1),
(x - 1, []))
assert check(unrad(sqrt(x) + root(x, 3) + 2),
(s**3 + s**2 + 2, [s, s**6 - x]))
assert check(unrad(sqrt(x)*root(x, 3) + 2),
(x**5 - 64, []))
assert check(unrad(sqrt(x) + (x + 1)**Rational(1, 3)),
(x**3 - (x + 1)**2, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(2*x)),
(-2*sqrt(2)*x - 2*x + 1, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + 2),
(16*x - 9, []))
assert check(unrad(sqrt(x) + sqrt(x + 1) + sqrt(1 - x)),
(5*x**2 - 4*x, []))
assert check(unrad(a*sqrt(x) + b*sqrt(x) + c*sqrt(y) + d*sqrt(y)),
((a*sqrt(x) + b*sqrt(x))**2 - (c*sqrt(y) + d*sqrt(y))**2, []))
assert check(unrad(sqrt(x) + sqrt(1 - x)),
(2*x - 1, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) - 3),
(x**2 - x + 16, []))
assert check(unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x)),
(5*x**2 - 2*x + 1, []))
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - 3) in [
(25*x**4 + 376*x**3 + 1256*x**2 - 2272*x + 784, []),
(25*x**8 - 476*x**6 + 2534*x**4 - 1468*x**2 + 169, [])]
assert unrad(sqrt(x) + sqrt(1 - x) + sqrt(2 + x) - sqrt(1 - 2*x)) == \
(41*x**4 + 40*x**3 + 232*x**2 - 160*x + 16, []) # orig root at 0.487
assert check(unrad(sqrt(x) + sqrt(x + 1)), (S.One, []))
eq = sqrt(x) + sqrt(x + 1) + sqrt(1 - sqrt(x))
assert check(unrad(eq),
(16*x**2 - 9*x, []))
assert set(solve(eq, check=False)) == {S.Zero, Rational(9, 16)}
assert solve(eq) == []
# but this one really does have those solutions
assert set(solve(sqrt(x) - sqrt(x + 1) + sqrt(1 - sqrt(x)))) == \
{S.Zero, Rational(9, 16)}
assert check(unrad(sqrt(x) + root(x + 1, 3) + 2*sqrt(y), y),
(S('2*sqrt(x)*(x + 1)**(1/3) + x - 4*y + (x + 1)**(2/3)'), []))
assert check(unrad(sqrt(x/(1 - x)) + (x + 1)**Rational(1, 3)),
(x**5 - x**4 - x**3 + 2*x**2 + x - 1, []))
assert check(unrad(sqrt(x/(1 - x)) + 2*sqrt(y), y),
(4*x*y + x - 4*y, []))
assert check(unrad(sqrt(x)*sqrt(1 - x) + 2, x),
(x**2 - x + 4, []))
# http://tutorial.math.lamar.edu/
# Classes/Alg/SolveRadicalEqns.aspx#Solve_Rad_Ex2_a
assert solve(Eq(x, sqrt(x + 6))) == [3]
assert solve(Eq(x + sqrt(x - 4), 4)) == [4]
assert solve(Eq(1, x + sqrt(2*x - 3))) == []
assert set(solve(Eq(sqrt(5*x + 6) - 2, x))) == {-S.One, S(2)}
assert set(solve(Eq(sqrt(2*x - 1) - sqrt(x - 4), 2))) == {S(5), S(13)}
assert solve(Eq(sqrt(x + 7) + 2, sqrt(3 - x))) == [-6]
# http://www.purplemath.com/modules/solverad.htm
assert solve((2*x - 5)**Rational(1, 3) - 3) == [16]
assert set(solve(x + 1 - root(x**4 + 4*x**3 - x, 4))) == \
{Rational(-1, 2), Rational(-1, 3)}
assert set(solve(sqrt(2*x**2 - 7) - (3 - x))) == {-S(8), S(2)}
assert solve(sqrt(2*x + 9) - sqrt(x + 1) - sqrt(x + 4)) == [0]
assert solve(sqrt(x + 4) + sqrt(2*x - 1) - 3*sqrt(x - 1)) == [5]
assert solve(sqrt(x)*sqrt(x - 7) - 12) == [16]
assert solve(sqrt(x - 3) + sqrt(x) - 3) == [4]
assert solve(sqrt(9*x**2 + 4) - (3*x + 2)) == [0]
assert solve(sqrt(x) - 2 - 5) == [49]
assert solve(sqrt(x - 3) - sqrt(x) - 3) == []
assert solve(sqrt(x - 1) - x + 7) == [10]
assert solve(sqrt(x - 2) - 5) == [27]
assert solve(sqrt(17*x - sqrt(x**2 - 5)) - 7) == [3]
assert solve(sqrt(x) - sqrt(x - 1) + sqrt(sqrt(x))) == []
# don't posify the expression in unrad and do use _mexpand
z = sqrt(2*x + 1)/sqrt(x) - sqrt(2 + 1/x)
p = posify(z)[0]
assert solve(p) == []
assert solve(z) == []
assert solve(z + 6*I) == [Rational(-1, 11)]
assert solve(p + 6*I) == []
# issue 8622
assert unrad(root(x + 1, 5) - root(x, 3)) == (
-(x**5 - x**3 - 3*x**2 - 3*x - 1), [])
# issue #8679
assert check(unrad(x + root(x, 3) + root(x, 3)**2 + sqrt(y), x),
(s**3 + s**2 + s + sqrt(y), [s, s**3 - x]))
# for coverage
assert check(unrad(sqrt(x) + root(x, 3) + y),
(s**3 + s**2 + y, [s, s**6 - x]))
assert solve(sqrt(x) + root(x, 3) - 2) == [1]
raises(NotImplementedError, lambda:
solve(sqrt(x) + root(x, 3) + root(x + 1, 5) - 2))
# fails through a different code path
raises(NotImplementedError, lambda: solve(-sqrt(2) + cosh(x)/x))
# unrad some
assert solve(sqrt(x + root(x, 3))+root(x - y, 5), y) == [
x + (x**Rational(1, 3) + x)**Rational(5, 2)]
assert check(unrad(sqrt(x) - root(x + 1, 3)*sqrt(x + 2) + 2),
(s**10 + 8*s**8 + 24*s**6 - 12*s**5 - 22*s**4 - 160*s**3 - 212*s**2 -
192*s - 56, [s, s**2 - x]))
e = root(x + 1, 3) + root(x, 3)
assert unrad(e) == (2*x + 1, [])
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
(15625*x**4 + 173000*x**3 + 355600*x**2 - 817920*x + 331776, []))
assert check(unrad(root(x, 4) + root(x, 4)**3 - 1),
(s**3 + s - 1, [s, s**4 - x]))
assert check(unrad(root(x, 2) + root(x, 2)**3 - 1),
(x**3 + 2*x**2 + x - 1, []))
assert unrad(x**0.5) is None
assert check(unrad(t + root(x + y, 5) + root(x + y, 5)**3),
(s**3 + s + t, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, y),
(s**3 + s + x, [s, s**5 - x - y]))
assert check(unrad(x + root(x + y, 5) + root(x + y, 5)**3, x),
(s**5 + s**3 + s - y, [s, s**5 - x - y]))
assert check(unrad(root(x - 1, 3) + root(x + 1, 5) + root(2, 5)),
(s**5 + 5*2**Rational(1, 5)*s**4 + s**3 + 10*2**Rational(2, 5)*s**3 +
10*2**Rational(3, 5)*s**2 + 5*2**Rational(4, 5)*s + 4, [s, s**3 - x + 1]))
raises(NotImplementedError, lambda:
unrad((root(x, 2) + root(x, 3) + root(x, 4)).subs(x, x**5 - x + 1)))
# the simplify flag should be reset to False for unrad results;
# if it's not then this next test will take a long time
assert solve(root(x, 3) + root(x, 5) - 2) == [1]
eq = (sqrt(x) + sqrt(x + 1) + sqrt(1 - x) - 6*sqrt(5)/5)
assert check(unrad(eq),
((5*x - 4)*(3125*x**3 + 37100*x**2 + 100800*x - 82944), []))
ans = S('''
[4/5, -1484/375 + 172564/(140625*(114*sqrt(12657)/78125 +
12459439/52734375)**(1/3)) +
4*(114*sqrt(12657)/78125 + 12459439/52734375)**(1/3)]''')
assert solve(eq) == ans
# duplicate radical handling
assert check(unrad(sqrt(x + root(x + 1, 3)) - root(x + 1, 3) - 2),
(s**3 - s**2 - 3*s - 5, [s, s**3 - x - 1]))
# cov post-processing
e = root(x**2 + 1, 3) - root(x**2 - 1, 5) - 2
assert check(unrad(e),
(s**5 - 10*s**4 + 39*s**3 - 80*s**2 + 80*s - 30,
[s, s**3 - x**2 - 1]))
e = sqrt(x + root(x + 1, 2)) - root(x + 1, 3) - 2
assert check(unrad(e),
(s**6 - 2*s**5 - 7*s**4 - 3*s**3 + 26*s**2 + 40*s + 25,
[s, s**3 - x - 1]))
assert check(unrad(e, _reverse=True),
(s**6 - 14*s**5 + 73*s**4 - 187*s**3 + 276*s**2 - 228*s + 89,
[s, s**2 - x - sqrt(x + 1)]))
# this one needs r0, r1 reversal to work
assert check(unrad(sqrt(x + sqrt(root(x, 3) - 1)) - root(x, 6) - 2),
(s**12 - 2*s**8 - 8*s**7 - 8*s**6 + s**4 + 8*s**3 + 23*s**2 +
32*s + 17, [s, s**6 - x]))
# why does this pass
assert unrad(root(cosh(x), 3)/x*root(x + 1, 5) - 1) == (
-(x**15 - x**3*cosh(x)**5 - 3*x**2*cosh(x)**5 - 3*x*cosh(x)**5
- cosh(x)**5), [])
# and this fail?
#assert unrad(sqrt(cosh(x)/x) + root(x + 1, 3)*sqrt(x) - 1) == (
# -s**6 + 6*s**5 - 15*s**4 + 20*s**3 - 15*s**2 + 6*s + x**5 +
# 2*x**4 + x**3 - 1, [s, s**2 - cosh(x)/x])
# watch for symbols in exponents
assert unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1')) is None
assert check(unrad(S('(x+y)**(2*y/3) + (x+y)**(1/3) + 1'), x),
(s**(2*y) + s + 1, [s, s**3 - x - y]))
# should _Q be so lenient?
assert unrad(x**(S.Half/y) + y, x) == (x**(1/y) - y**2, [])
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests that the use of
# composite
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 -
1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3
# watch out for when the cov doesn't involve the symbol of interest
eq = S('-x + (7*y/8 - (27*x/2 + 27*sqrt(x**2)/2)**(1/3)/3)**3 - 1')
assert solve(eq, y) == [
2**(S(2)/3)*(27*x + 27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 +
S(512)/343)**(S(1)/3)*(-S(1)/2 - sqrt(3)*I/2), 2**(S(2)/3)*(27*x +
27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 +
S(512)/343)**(S(1)/3)*(-S(1)/2 + sqrt(3)*I/2), 2**(S(2)/3)*(27*x +
27*sqrt(x**2))**(S(1)/3)*S(4)/21 + (512*x/343 + S(512)/343)**(S(1)/3)]
eq = root(x + 1, 3) - (root(x, 3) + root(x, 5))
assert check(unrad(eq),
(3*s**13 + 3*s**11 + s**9 - 1, [s, s**15 - x]))
assert check(unrad(eq - 2),
(3*s**13 + 3*s**11 + 6*s**10 + s**9 + 12*s**8 + 6*s**6 + 12*s**5 +
12*s**3 + 7, [s, s**15 - x]))
assert check(unrad(root(x, 3) - root(x + 1, 4)/2 + root(x + 2, 3)),
(s*(4096*s**9 + 960*s**8 + 48*s**7 - s**6 - 1728),
[s, s**4 - x - 1])) # orig expr has two real roots: -1, -.389
assert check(unrad(root(x, 3) + root(x + 1, 4) - root(x + 2, 3)/2),
(343*s**13 + 2904*s**12 + 1344*s**11 + 512*s**10 - 1323*s**9 -
3024*s**8 - 1728*s**7 + 1701*s**5 + 216*s**4 - 729*s, [s, s**4 - x -
1])) # orig expr has one real root: -0.048
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3)),
(729*s**13 - 216*s**12 + 1728*s**11 - 512*s**10 + 1701*s**9 -
3024*s**8 + 1344*s**7 + 1323*s**5 - 2904*s**4 + 343*s, [s, s**4 - x -
1])) # orig expr has 2 real roots: -0.91, -0.15
assert check(unrad(root(x, 3)/2 - root(x + 1, 4) + root(x + 2, 3) - 2),
(729*s**13 + 1242*s**12 + 18496*s**10 + 129701*s**9 + 388602*s**8 +
453312*s**7 - 612864*s**6 - 3337173*s**5 - 6332418*s**4 - 7134912*s**3
- 5064768*s**2 - 2111913*s - 398034, [s, s**4 - x - 1]))
# orig expr has 1 real root: 19.53
ans = solve(sqrt(x) + sqrt(x + 1) -
sqrt(1 - x) - sqrt(2 + x))
assert len(ans) == 1 and NS(ans[0])[:4] == '0.73'
# the fence optimization problem
# https://github.com/sympy/sympy/issues/4793#issuecomment-36994519
F = Symbol('F')
eq = F - (2*x + 2*y + sqrt(x**2 + y**2))
ans = F*Rational(2, 7) - sqrt(2)*F/14
X = solve(eq, x, check=False)
for xi in reversed(X): # reverse since currently, ans is the 2nd one
Y = solve((x*y).subs(x, xi).diff(y), y, simplify=False, check=False)
if any((a - ans).expand().is_zero for a in Y):
break
else:
assert None # no answer was found
assert solve(sqrt(x + 1) + root(x, 3) - 2) == S('''
[(-11/(9*(47/54 + sqrt(93)/6)**(1/3)) + 1/3 + (47/54 +
sqrt(93)/6)**(1/3))**3]''')
assert solve(sqrt(sqrt(x + 1)) + x**Rational(1, 3) - 2) == S('''
[(-sqrt(-2*(-1/16 + sqrt(6913)/16)**(1/3) + 6/(-1/16 +
sqrt(6913)/16)**(1/3) + 17/2 + 121/(4*sqrt(-6/(-1/16 +
sqrt(6913)/16)**(1/3) + 2*(-1/16 + sqrt(6913)/16)**(1/3) + 17/4)))/2 +
sqrt(-6/(-1/16 + sqrt(6913)/16)**(1/3) + 2*(-1/16 +
sqrt(6913)/16)**(1/3) + 17/4)/2 + 9/4)**3]''')
assert solve(sqrt(x) + root(sqrt(x) + 1, 3) - 2) == S('''
[(-(81/2 + 3*sqrt(741)/2)**(1/3)/3 + (81/2 + 3*sqrt(741)/2)**(-1/3) +
2)**2]''')
eq = S('''
-x + (1/2 - sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3
+ x*(3*x**2 - 34) + 90)**2/4 - 39304/27) - 45)**(1/3) + 34/(3*(1/2 -
sqrt(3)*I/2)*(3*x**3/2 - x*(3*x**2 - 34)/2 + sqrt((-3*x**3 + x*(3*x**2
- 34) + 90)**2/4 - 39304/27) - 45)**(1/3))''')
assert check(unrad(eq),
(s*-(-s**6 + sqrt(3)*s**6*I - 153*2**Rational(2, 3)*3**Rational(1, 3)*s**4 +
51*12**Rational(1, 3)*s**4 - 102*2**Rational(2, 3)*3**Rational(5, 6)*s**4*I - 1620*s**3 +
1620*sqrt(3)*s**3*I + 13872*18**Rational(1, 3)*s**2 - 471648 +
471648*sqrt(3)*I), [s, s**3 - 306*x - sqrt(3)*sqrt(31212*x**2 -
165240*x + 61484) + 810]))
assert solve(eq) == [] # not other code errors
eq = root(x, 3) - root(y, 3) + root(x, 5)
assert check(unrad(eq),
(s**15 + 3*s**13 + 3*s**11 + s**9 - y, [s, s**15 - x]))
eq = root(x, 3) + root(y, 3) + root(x*y, 4)
assert check(unrad(eq),
(s*y*(-s**12 - 3*s**11*y - 3*s**10*y**2 - s**9*y**3 -
3*s**8*y**2 + 21*s**7*y**3 - 3*s**6*y**4 - 3*s**4*y**4 -
3*s**3*y**5 - y**6), [s, s**4 - x*y]))
raises(NotImplementedError,
lambda: unrad(root(x, 3) + root(y, 3) + root(x*y, 5)))
# Test unrad with an Equality
eq = Eq(-x**(S(1)/5) + x**(S(1)/3), -3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5))
assert check(unrad(eq),
(-s**5 + s**3 - 3**(S(1)/3) - (-1)**(S(3)/5)*3**(S(1)/5), [s, s**15 - x]))
# make sure buried radicals are exposed
s = sqrt(x) - 1
assert unrad(s**2 - s**3) == (x**3 - 6*x**2 + 9*x - 4, [])
# make sure numerators which are already polynomial are rejected
assert unrad((x/(x + 1) + 3)**(-2), x) is None
# https://github.com/sympy/sympy/issues/23707
eq = sqrt(x - y)*exp(t*sqrt(x - y)) - exp(t*sqrt(x - y))
assert solve(eq, y) == [x - 1]
assert unrad(eq) is None
@slow
def test_unrad_slow():
# this has roots with multiplicity > 1; there should be no
# repeats in roots obtained, however
eq = (sqrt(1 + sqrt(1 - 4*x**2)) - x*(1 + sqrt(1 + 2*sqrt(1 - 4*x**2))))
assert solve(eq) == [S.Half]
@XFAIL
def test_unrad_fail():
# this only works if we check real_root(eq.subs(x, Rational(1, 3)))
# but checksol doesn't work like that
assert solve(root(x**3 - 3*x**2, 3) + 1 - x) == [Rational(1, 3)]
assert solve(root(x + 1, 3) + root(x**2 - 2, 5) + 1) == [
-1, -1 + CRootOf(x**5 + x**4 + 5*x**3 + 8*x**2 + 10*x + 5, 0)**3]
def test_checksol():
x, y, r, t = symbols('x, y, r, t')
eq = r - x**2 - y**2
dict_var_soln = {y: - sqrt(r) / sqrt(tan(t)**2 + 1),
x: -sqrt(r)*tan(t)/sqrt(tan(t)**2 + 1)}
assert checksol(eq, dict_var_soln) == True
assert checksol(Eq(x, False), {x: False}) is True
assert checksol(Ne(x, False), {x: False}) is False
assert checksol(Eq(x < 1, True), {x: 0}) is True
assert checksol(Eq(x < 1, True), {x: 1}) is False
assert checksol(Eq(x < 1, False), {x: 1}) is True
assert checksol(Eq(x < 1, False), {x: 0}) is False
assert checksol(Eq(x + 1, x**2 + 1), {x: 1}) is True
assert checksol([x - 1, x**2 - 1], x, 1) is True
assert checksol([x - 1, x**2 - 2], x, 1) is False
assert checksol(Poly(x**2 - 1), x, 1) is True
assert checksol(0, {}) is True
assert checksol([1e-10, x - 2], x, 2) is False
assert checksol([0.5, 0, x], x, 0) is False
assert checksol(y, x, 2) is False
assert checksol(x+1e-10, x, 0, numerical=True) is True
assert checksol(x+1e-10, x, 0, numerical=False) is False
raises(ValueError, lambda: checksol(x, 1))
raises(ValueError, lambda: checksol([], x, 1))
def test__invert():
assert _invert(x - 2) == (2, x)
assert _invert(2) == (2, 0)
assert _invert(exp(1/x) - 3, x) == (1/log(3), x)
assert _invert(exp(1/x + a/x) - 3, x) == ((a + 1)/log(3), x)
assert _invert(a, x) == (a, 0)
def test_issue_4463():
assert solve(-a*x + 2*x*log(x), x) == [exp(a/2)]
assert solve(x**x) == []
assert solve(x**x - 2) == [exp(LambertW(log(2)))]
assert solve(((x - 3)*(x - 2))**((x - 3)*(x - 4))) == [2]
@slow
def test_issue_5114_solvers():
a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('a:r')
# there is no 'a' in the equation set but this is how the
# problem was originally posed
syms = a, b, c, f, h, k, n
eqs = [b + r/d - c/d,
c*(1/d + 1/e + 1/g) - f/g - r/d,
f*(1/g + 1/i + 1/j) - c/g - h/i,
h*(1/i + 1/l + 1/m) - f/i - k/m,
k*(1/m + 1/o + 1/p) - h/m - n/p,
n*(1/p + 1/q) - k/p]
assert len(solve(eqs, syms, manual=True, check=False, simplify=False)) == 1
def test_issue_5849():
#
# XXX: This system does not have a solution for most values of the
# parameters. Generally solve returns the empty set for systems that are
# generically inconsistent.
#
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
ans = [{
I1: I2 + I3,
dI1: -4*I2 - 8*I3 - 4*I5 - 6*I6 + 24,
I4: I3 - I5,
dQ4: I3 - I5,
Q4: -I3/2 + 3*I5/2 - dI4/2,
dQ2: I2,
Q2: 2*I3 + 2*I5 + 3*I6}]
v = I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4
assert solve(e, *v, manual=True, check=False, dict=True) == ans
assert solve(e, *v, manual=True, check=False) == [
tuple([a.get(i, i) for i in v]) for a in ans]
assert solve(e, *v, manual=True) == []
assert solve(e, *v) == []
# the matrix solver (tested below) doesn't like this because it produces
# a zero row in the matrix. Is this related to issue 4551?
assert [ei.subs(
ans[0]) for ei in e] == [0, 0, I3 - I6, -I3 + I6, 0, 0, 0, 0, 0]
def test_issue_5849_matrix():
'''Same as test_issue_5849 but solved with the matrix solver.
A solution only exists if I3 == I6 which is not generically true,
but `solve` does not return conditions under which the solution is
valid, only a solution that is canonical and consistent with the input.
'''
# a simple example with the same issue
# assert solve([x+y+z, x+y], [x, y]) == {x: y}
# the longer example
I1, I2, I3, I4, I5, I6 = symbols('I1:7')
dI1, dI4, dQ2, dQ4, Q2, Q4 = symbols('dI1,dI4,dQ2,dQ4,Q2,Q4')
e = (
I1 - I2 - I3,
I3 - I4 - I5,
I4 + I5 - I6,
-I1 + I2 + I6,
-2*I1 - 2*I3 - 2*I5 - 3*I6 - dI1/2 + 12,
-I4 + dQ4,
-I2 + dQ2,
2*I3 + 2*I5 + 3*I6 - Q2,
I4 - 2*I5 + 2*Q4 + dI4
)
assert solve(e, I1, I4, Q2, Q4, dI1, dI4, dQ2, dQ4) == []
def test_issue_21882():
a, b, c, d, f, g, k = unknowns = symbols('a, b, c, d, f, g, k')
equations = [
-k*a + b + 5*f/6 + 2*c/9 + 5*d/6 + 4*a/3,
-k*f + 4*f/3 + d/2,
-k*d + f/6 + d,
13*b/18 + 13*c/18 + 13*a/18,
-k*c + b/2 + 20*c/9 + a,
-k*b + b + c/18 + a/6,
5*b/3 + c/3 + a,
2*b/3 + 2*c + 4*a/3,
-g,
]
answer = [
{a: 0, f: 0, b: 0, d: 0, c: 0, g: 0},
{a: 0, f: -d, b: 0, k: S(5)/6, c: 0, g: 0},
{a: -2*c, f: 0, b: c, d: 0, k: S(13)/18, g: 0}]
# but not {a: 0, f: 0, b: 0, k: S(3)/2, c: 0, d: 0, g: 0}
# since this is already covered by the first solution
got = solve(equations, unknowns, dict=True)
assert got == answer, (got,answer)
def test_issue_5901():
f, g, h = map(Function, 'fgh')
a = Symbol('a')
D = Derivative(f(x), x)
G = Derivative(g(a), a)
assert solve(f(x) + f(x).diff(x), f(x)) == \
[-D]
assert solve(f(x) - 3, f(x)) == \
[3]
assert solve(f(x) - 3*f(x).diff(x), f(x)) == \
[3*D]
assert solve([f(x) - 3*f(x).diff(x)], f(x)) == \
{f(x): 3*D}
assert solve([f(x) - 3*f(x).diff(x), f(x)**2 - y + 4], f(x), y) == \
[(3*D, 9*D**2 + 4)]
assert solve(-f(a)**2*g(a)**2 + f(a)**2*h(a)**2 + g(a).diff(a),
h(a), g(a), set=True) == \
([h(a), g(a)], {
(-sqrt(f(a)**2*g(a)**2 - G)/f(a), g(a)),
(sqrt(f(a)**2*g(a)**2 - G)/f(a), g(a))}), solve(-f(a)**2*g(a)**2 + f(a)**2*h(a)**2 + g(a).diff(a),
h(a), g(a), set=True)
args = [[f(x).diff(x, 2)*(f(x) + g(x)), 2 - g(x)**2], f(x), g(x)]
assert solve(*args, set=True)[1] == \
{(-sqrt(2), sqrt(2)), (sqrt(2), -sqrt(2))}
eqs = [f(x)**2 + g(x) - 2*f(x).diff(x), g(x)**2 - 4]
assert solve(eqs, f(x), g(x), set=True) == \
([f(x), g(x)], {
(-sqrt(2*D - 2), S(2)),
(sqrt(2*D - 2), S(2)),
(-sqrt(2*D + 2), -S(2)),
(sqrt(2*D + 2), -S(2))})
# the underlying problem was in solve_linear that was not masking off
# anything but a Mul or Add; it now raises an error if it gets anything
# but a symbol and solve handles the substitutions necessary so solve_linear
# won't make this error
raises(
ValueError, lambda: solve_linear(f(x) + f(x).diff(x), symbols=[f(x)]))
assert solve_linear(f(x) + f(x).diff(x), symbols=[x]) == \
(f(x) + Derivative(f(x), x), 1)
assert solve_linear(f(x) + Integral(x, (x, y)), symbols=[x]) == \
(f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(x) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x + f(x) + Integral(x, (x, y)), 1)
assert solve_linear(f(y) + Integral(x, (x, y)) + x, symbols=[x]) == \
(x, -f(y) - Integral(x, (x, y)))
assert solve_linear(x - f(x)/a + (f(x) - 1)/a, symbols=[x]) == \
(x, 1/a)
assert solve_linear(x + Derivative(2*x, x)) == \
(x, -2)
assert solve_linear(x + Integral(x, y), symbols=[x]) == \
(x, 0)
assert solve_linear(x + Integral(x, y) - 2, symbols=[x]) == \
(x, 2/(y + 1))
assert set(solve(x + exp(x)**2, exp(x))) == \
{-sqrt(-x), sqrt(-x)}
assert solve(x + exp(x), x, implicit=True) == \
[-exp(x)]
assert solve(cos(x) - sin(x), x, implicit=True) == []
assert solve(x - sin(x), x, implicit=True) == \
[sin(x)]
assert solve(x**2 + x - 3, x, implicit=True) == \
[-x**2 + 3]
assert solve(x**2 + x - 3, x**2, implicit=True) == \
[-x + 3]
def test_issue_5912():
assert set(solve(x**2 - x - 0.1, rational=True)) == \
{S.Half + sqrt(35)/10, -sqrt(35)/10 + S.Half}
ans = solve(x**2 - x - 0.1, rational=False)
assert len(ans) == 2 and all(a.is_Number for a in ans)
ans = solve(x**2 - x - 0.1)
assert len(ans) == 2 and all(a.is_Number for a in ans)
def test_float_handling():
def test(e1, e2):
return len(e1.atoms(Float)) == len(e2.atoms(Float))
assert solve(x - 0.5, rational=True)[0].is_Rational
assert solve(x - 0.5, rational=False)[0].is_Float
assert solve(x - S.Half, rational=False)[0].is_Rational
assert solve(x - 0.5, rational=None)[0].is_Float
assert solve(x - S.Half, rational=None)[0].is_Rational
assert test(nfloat(1 + 2*x), 1.0 + 2.0*x)
for contain in [list, tuple, set]:
ans = nfloat(contain([1 + 2*x]))
assert type(ans) is contain and test(list(ans)[0], 1.0 + 2.0*x)
k, v = list(nfloat({2*x: [1 + 2*x]}).items())[0]
assert test(k, 2*x) and test(v[0], 1.0 + 2.0*x)
assert test(nfloat(cos(2*x)), cos(2.0*x))
assert test(nfloat(3*x**2), 3.0*x**2)
assert test(nfloat(3*x**2, exponent=True), 3.0*x**2.0)
assert test(nfloat(exp(2*x)), exp(2.0*x))
assert test(nfloat(x/3), x/3.0)
assert test(nfloat(x**4 + 2*x + cos(Rational(1, 3)) + 1),
x**4 + 2.0*x + 1.94495694631474)
# don't call nfloat if there is no solution
tot = 100 + c + z + t
assert solve(((.7 + c)/tot - .6, (.2 + z)/tot - .3, t/tot - .1)) == []
def test_check_assumptions():
x = symbols('x', positive=True)
assert solve(x**2 - 1) == [1]
def test_issue_6056():
assert solve(tanh(x + 3)*tanh(x - 3) - 1) == []
assert solve(tanh(x - 1)*tanh(x + 1) + 1) == \
[I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)]
assert solve((tanh(x + 3)*tanh(x - 3) + 1)**2) == \
[I*pi*Rational(-3, 4), -I*pi/4, I*pi/4, I*pi*Rational(3, 4)]
def test_issue_5673():
eq = -x + exp(exp(LambertW(log(x)))*LambertW(log(x)))
assert checksol(eq, x, 2) is True
assert checksol(eq, x, 2, numerical=False) is None
def test_exclude():
R, C, Ri, Vout, V1, Vminus, Vplus, s = \
symbols('R, C, Ri, Vout, V1, Vminus, Vplus, s')
Rf = symbols('Rf', positive=True) # to eliminate Rf = 0 soln
eqs = [C*V1*s + Vplus*(-2*C*s - 1/R),
Vminus*(-1/Ri - 1/Rf) + Vout/Rf,
C*Vplus*s + V1*(-C*s - 1/R) + Vout/R,
-Vminus + Vplus]
assert solve(eqs, exclude=s*C*R) == [
{
Rf: Ri*(C*R*s + 1)**2/(C*R*s),
Vminus: Vplus,
V1: 2*Vplus + Vplus/(C*R*s),
Vout: C*R*Vplus*s + 3*Vplus + Vplus/(C*R*s)},
{
Vplus: 0,
Vminus: 0,
V1: 0,
Vout: 0},
]
# TODO: Investigate why currently solution [0] is preferred over [1].
assert solve(eqs, exclude=[Vplus, s, C]) in [[{
Vminus: Vplus,
V1: Vout/2 + Vplus/2 + sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus - sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}, {
Vminus: Vplus,
V1: Vout/2 + Vplus/2 - sqrt((Vout - 5*Vplus)*(Vout - Vplus))/2,
R: (Vout - 3*Vplus + sqrt(Vout**2 - 6*Vout*Vplus + 5*Vplus**2))/(2*C*Vplus*s),
Rf: Ri*(Vout - Vplus)/Vplus,
}], [{
Vminus: Vplus,
Vout: (V1**2 - V1*Vplus - Vplus**2)/(V1 - 2*Vplus),
Rf: Ri*(V1 - Vplus)**2/(Vplus*(V1 - 2*Vplus)),
R: Vplus/(C*s*(V1 - 2*Vplus)),
}]]
def test_high_order_roots():
s = x**5 + 4*x**3 + 3*x**2 + Rational(7, 4)
assert set(solve(s)) == set(Poly(s*4, domain='ZZ').all_roots())
def test_minsolve_linear_system():
pqt = dict(quick=True, particular=True)
pqf = dict(quick=False, particular=True)
assert solve([x + y - 5, 2*x - y - 1], **pqt) == {x: 2, y: 3}
assert solve([x + y - 5, 2*x - y - 1], **pqf) == {x: 2, y: 3}
def count(dic):
return len([x for x in dic.values() if x == 0])
assert count(solve([x + y + z, y + z + a + t], **pqt)) == 3
assert count(solve([x + y + z, y + z + a + t], **pqf)) == 3
assert count(solve([x + y + z, y + z + a], **pqt)) == 1
assert count(solve([x + y + z, y + z + a], **pqf)) == 2
# issue 22718
A = Matrix([
[ 1, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0],
[ 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, -1, -1, 0, 0],
[-1, -1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 1, 0, 1],
[ 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, -1, 0, -1, 0],
[-1, 0, -1, 0, 0, -1, 0, 0, 0, 0, 1, 0, 1, 1],
[-1, 0, 0, -1, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1],
[ 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, -1, -1, 0],
[ 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1],
[ 0, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, -1, 0, -1],
[ 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, -1],
[ 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0],
[ 0, 0, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 0, 0]])
v = Matrix(symbols("v:14", integer=True))
B = Matrix([[2], [-2], [0], [0], [0], [0], [0], [0], [0],
[0], [0], [0]])
eqs = A@v-B
assert solve(eqs) == []
assert solve(eqs, particular=True) == [] # assumption violated
assert all(v for v in solve([x + y + z, y + z + a]).values())
for _q in (True, False):
assert not all(v for v in solve(
[x + y + z, y + z + a], quick=_q,
particular=True).values())
# raise error if quick used w/o particular=True
raises(ValueError, lambda: solve([x + 1], quick=_q))
raises(ValueError, lambda: solve([x + 1], quick=_q, particular=False))
# and give a good error message if someone tries to use
# particular with a single equation
raises(ValueError, lambda: solve(x + 1, particular=True))
def test_real_roots():
# cf. issue 6650
x = Symbol('x', real=True)
assert len(solve(x**5 + x**3 + 1)) == 1
def test_issue_6528():
eqs = [
327600995*x**2 - 37869137*x + 1809975124*y**2 - 9998905626,
895613949*x**2 - 273830224*x*y + 530506983*y**2 - 10000000000]
# two expressions encountered are > 1400 ops long so if this hangs
# it is likely because simplification is being done
assert len(solve(eqs, y, x, check=False)) == 4
def test_overdetermined():
x = symbols('x', real=True)
eqs = [Abs(4*x - 7) - 5, Abs(3 - 8*x) - 1]
assert solve(eqs, x) == [(S.Half,)]
assert solve(eqs, x, manual=True) == [(S.Half,)]
assert solve(eqs, x, manual=True, check=False) == [(S.Half,), (S(3),)]
def test_issue_6605():
x = symbols('x')
assert solve(4**(x/2) - 2**(x/3)) == [0, 3*I*pi/log(2)]
# while the first one passed, this one failed
x = symbols('x', real=True)
assert solve(5**(x/2) - 2**(x/3)) == [0]
b = sqrt(6)*sqrt(log(2))/sqrt(log(5))
assert solve(5**(x/2) - 2**(3/x)) == [-b, b]
def test__ispow():
assert _ispow(x**2)
assert not _ispow(x)
assert not _ispow(True)
def test_issue_6644():
eq = -sqrt((m - q)**2 + (-m/(2*q) + S.Half)**2) + sqrt((-m**2/2 - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2 + (m**2/2 - m - sqrt(
4*m**4 - 4*m**2 + 8*m + 1)/4 - Rational(1, 4))**2)
sol = solve(eq, q, simplify=False, check=False)
assert len(sol) == 5
def test_issue_6752():
assert solve([a**2 + a, a - b], [a, b]) == [(-1, -1), (0, 0)]
assert solve([a**2 + a*c, a - b], [a, b]) == [(0, 0), (-c, -c)]
def test_issue_6792():
assert solve(x*(x - 1)**2*(x + 1)*(x**6 - x + 1)) == [
-1, 0, 1, CRootOf(x**6 - x + 1, 0), CRootOf(x**6 - x + 1, 1),
CRootOf(x**6 - x + 1, 2), CRootOf(x**6 - x + 1, 3),
CRootOf(x**6 - x + 1, 4), CRootOf(x**6 - x + 1, 5)]
def test_issues_6819_6820_6821_6248_8692():
# issue 6821
x, y = symbols('x y', real=True)
assert solve(abs(x + 3) - 2*abs(x - 3)) == [1, 9]
assert solve([abs(x) - 2, arg(x) - pi], x) == [(-2,)]
assert set(solve(abs(x - 7) - 8)) == {-S.One, S(15)}
# issue 8692
assert solve(Eq(Abs(x + 1) + Abs(x**2 - 7), 9), x) == [
Rational(-1, 2) + sqrt(61)/2, -sqrt(69)/2 + S.Half]
# issue 7145
assert solve(2*abs(x) - abs(x - 1)) == [-1, Rational(1, 3)]
x = symbols('x')
assert solve([re(x) - 1, im(x) - 2], x) == [
{re(x): 1, x: 1 + 2*I, im(x): 2}]
# check for 'dict' handling of solution
eq = sqrt(re(x)**2 + im(x)**2) - 3
assert solve(eq) == solve(eq, x)
i = symbols('i', imaginary=True)
assert solve(abs(i) - 3) == [-3*I, 3*I]
raises(NotImplementedError, lambda: solve(abs(x) - 3))
w = symbols('w', integer=True)
assert solve(2*x**w - 4*y**w, w) == solve((x/y)**w - 2, w)
x, y = symbols('x y', real=True)
assert solve(x + y*I + 3) == {y: 0, x: -3}
# issue 2642
assert solve(x*(1 + I)) == [0]
x, y = symbols('x y', imaginary=True)
assert solve(x + y*I + 3 + 2*I) == {x: -2*I, y: 3*I}
x = symbols('x', real=True)
assert solve(x + y + 3 + 2*I) == {x: -3, y: -2*I}
# issue 6248
f = Function('f')
assert solve(f(x + 1) - f(2*x - 1)) == [2]
assert solve(log(x + 1) - log(2*x - 1)) == [2]
x = symbols('x')
assert solve(2**x + 4**x) == [I*pi/log(2)]
def test_issue_14607():
# issue 14607
s, tau_c, tau_1, tau_2, phi, K = symbols(
's, tau_c, tau_1, tau_2, phi, K')
target = (s**2*tau_1*tau_2 + s*tau_1 + s*tau_2 + 1)/(K*s*(-phi + tau_c))
K_C, tau_I, tau_D = symbols('K_C, tau_I, tau_D',
positive=True, nonzero=True)
PID = K_C*(1 + 1/(tau_I*s) + tau_D*s)
eq = (target - PID).together()
eq *= denom(eq).simplify()
eq = Poly(eq, s)
c = eq.coeffs()
vars = [K_C, tau_I, tau_D]
s = solve(c, vars, dict=True)
assert len(s) == 1
knownsolution = {K_C: -(tau_1 + tau_2)/(K*(phi - tau_c)),
tau_I: tau_1 + tau_2,
tau_D: tau_1*tau_2/(tau_1 + tau_2)}
for var in vars:
assert s[0][var].simplify() == knownsolution[var].simplify()
def test_lambert_multivariate():
from sympy.abc import x, y
assert _filtered_gens(Poly(x + 1/x + exp(x) + y), x) == {x, exp(x)}
assert _lambert(x, x) == []
assert solve((x**2 - 2*x + 1).subs(x, log(x) + 3*x)) == [LambertW(3*S.Exp1)/3]
assert solve((x**2 - 2*x + 1).subs(x, (log(x) + 3*x)**2 - 1)) == \
[LambertW(3*exp(-sqrt(2)))/3, LambertW(3*exp(sqrt(2)))/3]
assert solve((x**2 - 2*x - 2).subs(x, log(x) + 3*x)) == \
[LambertW(3*exp(1 - sqrt(3)))/3, LambertW(3*exp(1 + sqrt(3)))/3]
eq = (x*exp(x) - 3).subs(x, x*exp(x))
assert solve(eq) == [LambertW(3*exp(-LambertW(3)))]
# coverage test
raises(NotImplementedError, lambda: solve(x - sin(x)*log(y - x), x))
ans = [3, -3*LambertW(-log(3)/3)/log(3)] # 3 and 2.478...
assert solve(x**3 - 3**x, x) == ans
assert set(solve(3*log(x) - x*log(3))) == set(ans)
assert solve(LambertW(2*x) - y, x) == [y*exp(y)/2]
@XFAIL
def test_other_lambert():
assert solve(3*sin(x) - x*sin(3), x) == [3]
assert set(solve(x**a - a**x), x) == {
a, -a*LambertW(-log(a)/a)/log(a)}
@slow
def test_lambert_bivariate():
# tests passing current implementation
assert solve((x**2 + x)*exp(x**2 + x) - 1) == [
Rational(-1, 2) + sqrt(1 + 4*LambertW(1))/2,
Rational(-1, 2) - sqrt(1 + 4*LambertW(1))/2]
assert solve((x**2 + x)*exp((x**2 + x)*2) - 1) == [
Rational(-1, 2) + sqrt(1 + 2*LambertW(2))/2,
Rational(-1, 2) - sqrt(1 + 2*LambertW(2))/2]
assert solve(a/x + exp(x/2), x) == [2*LambertW(-a/2)]
assert solve((a/x + exp(x/2)).diff(x), x) == \
[4*LambertW(-sqrt(2)*sqrt(a)/4), 4*LambertW(sqrt(2)*sqrt(a)/4)]
assert solve((1/x + exp(x/2)).diff(x), x) == \
[4*LambertW(-sqrt(2)/4),
4*LambertW(sqrt(2)/4), # nsimplifies as 2*2**(141/299)*3**(206/299)*5**(205/299)*7**(37/299)/21
4*LambertW(-sqrt(2)/4, -1)]
assert solve(x*log(x) + 3*x + 1, x) == \
[exp(-3 + LambertW(-exp(3)))]
assert solve(-x**2 + 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)]
assert solve(x**2 - 2**x, x) == [2, 4, -2*LambertW(log(2)/2)/log(2)]
ans = solve(3*x + 5 + 2**(-5*x + 3), x)
assert len(ans) == 1 and ans[0].expand() == \
Rational(-5, 3) + LambertW(-10240*root(2, 3)*log(2)/3)/(5*log(2))
assert solve(5*x - 1 + 3*exp(2 - 7*x), x) == \
[Rational(1, 5) + LambertW(-21*exp(Rational(3, 5))/5)/7]
assert solve((log(x) + x).subs(x, x**2 + 1)) == [
-I*sqrt(-LambertW(1) + 1), sqrt(-1 + LambertW(1))]
# check collection
ax = a**(3*x + 5)
ans = solve(3*log(ax) + b*log(ax) + ax, x)
x0 = 1/log(a)
x1 = sqrt(3)*I
x2 = b + 3
x3 = x2*LambertW(1/x2)/a**5
x4 = x3**Rational(1, 3)/2
assert ans == [
x0*log(x4*(-x1 - 1)),
x0*log(x4*(x1 - 1)),
x0*log(x3)/3]
x1 = LambertW(Rational(1, 3))
x2 = a**(-5)
x3 = -3**Rational(1, 3)
x4 = 3**Rational(5, 6)*I
x5 = x1**Rational(1, 3)*x2**Rational(1, 3)/2
ans = solve(3*log(ax) + ax, x)
assert ans == [
x0*log(3*x1*x2)/3,
x0*log(x5*(x3 - x4)),
x0*log(x5*(x3 + x4))]
# coverage
p = symbols('p', positive=True)
eq = 4*2**(2*p + 3) - 2*p - 3
assert _solve_lambert(eq, p, _filtered_gens(Poly(eq), p)) == [
Rational(-3, 2) - LambertW(-4*log(2))/(2*log(2))]
assert set(solve(3**cos(x) - cos(x)**3)) == {
acos(3), acos(-3*LambertW(-log(3)/3)/log(3))}
# should give only one solution after using `uniq`
assert solve(2*log(x) - 2*log(z) + log(z + log(x) + log(z)), x) == [
exp(-z + LambertW(2*z**4*exp(2*z))/2)/z]
# cases when p != S.One
# issue 4271
ans = solve((a/x + exp(x/2)).diff(x, 2), x)
x0 = (-a)**Rational(1, 3)
x1 = sqrt(3)*I
x2 = x0/6
assert ans == [
6*LambertW(x0/3),
6*LambertW(x2*(-x1 - 1)),
6*LambertW(x2*(x1 - 1))]
assert solve((1/x + exp(x/2)).diff(x, 2), x) == \
[6*LambertW(Rational(-1, 3)), 6*LambertW(Rational(1, 6) - sqrt(3)*I/6), \
6*LambertW(Rational(1, 6) + sqrt(3)*I/6), 6*LambertW(Rational(-1, 3), -1)]
assert solve(x**2 - y**2/exp(x), x, y, dict=True) == \
[{x: 2*LambertW(-y/2)}, {x: 2*LambertW(y/2)}]
# this is slow but not exceedingly slow
assert solve((x**3)**(x/2) + pi/2, x) == [
exp(LambertW(-2*log(2)/3 + 2*log(pi)/3 + I*pi*Rational(2, 3)))]
# issue 23253
assert solve((1/log(sqrt(x) + 2)**2 - 1/x)) == [
(LambertW(-exp(-2), -1) + 2)**2]
assert solve((1/log(1/sqrt(x) + 2)**2 - x)) == [
(LambertW(-exp(-2), -1) + 2)**-2]
assert solve((1/log(x**2 + 2)**2 - x**-4)) == [
-I*sqrt(2 - LambertW(exp(2))),
-I*sqrt(LambertW(-exp(-2)) + 2),
sqrt(-2 - LambertW(-exp(-2))),
sqrt(-2 + LambertW(exp(2))),
-sqrt(-2 - LambertW(-exp(-2), -1)),
sqrt(-2 - LambertW(-exp(-2), -1))]
def test_rewrite_trig():
assert solve(sin(x) + tan(x)) == [0, -pi, pi, 2*pi]
assert solve(sin(x) + sec(x)) == [
-2*atan(Rational(-1, 2) + sqrt(2)*sqrt(1 - sqrt(3)*I)/2 + sqrt(3)*I/2),
2*atan(S.Half - sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half
+ sqrt(2)*sqrt(1 + sqrt(3)*I)/2 + sqrt(3)*I/2), 2*atan(S.Half -
sqrt(3)*I/2 + sqrt(2)*sqrt(1 - sqrt(3)*I)/2)]
assert solve(sinh(x) + tanh(x)) == [0, I*pi]
# issue 6157
assert solve(2*sin(x) - cos(x), x) == [atan(S.Half)]
@XFAIL
def test_rewrite_trigh():
# if this import passes then the test below should also pass
from sympy.functions.elementary.hyperbolic import sech
assert solve(sinh(x) + sech(x)) == [
2*atanh(Rational(-1, 2) + sqrt(5)/2 - sqrt(-2*sqrt(5) + 2)/2),
2*atanh(Rational(-1, 2) + sqrt(5)/2 + sqrt(-2*sqrt(5) + 2)/2),
2*atanh(-sqrt(5)/2 - S.Half + sqrt(2 + 2*sqrt(5))/2),
2*atanh(-sqrt(2 + 2*sqrt(5))/2 - sqrt(5)/2 - S.Half)]
def test_uselogcombine():
eq = z - log(x) + log(y/(x*(-1 + y**2/x**2)))
assert solve(eq, x, force=True) == [-sqrt(y*(y - exp(z))), sqrt(y*(y - exp(z)))]
assert solve(log(x + 3) + log(1 + 3/x) - 3) in [
[-3 + sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 + exp(3)/2,
-sqrt(-12 + exp(3))*exp(Rational(3, 2))/2 - 3 + exp(3)/2],
[-3 + sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2,
-3 - sqrt(-36 + (-exp(3) + 6)**2)/2 + exp(3)/2],
]
assert solve(log(exp(2*x) + 1) + log(-tanh(x) + 1) - log(2)) == []
def test_atan2():
assert solve(atan2(x, 2) - pi/3, x) == [2*sqrt(3)]
def test_errorinverses():
assert solve(erf(x) - y, x) == [erfinv(y)]
assert solve(erfinv(x) - y, x) == [erf(y)]
assert solve(erfc(x) - y, x) == [erfcinv(y)]
assert solve(erfcinv(x) - y, x) == [erfc(y)]
def test_issue_2725():
R = Symbol('R')
eq = sqrt(2)*R*sqrt(1/(R + 1)) + (R + 1)*(sqrt(2)*sqrt(1/(R + 1)) - 1)
sol = solve(eq, R, set=True)[1]
assert sol == {(Rational(5, 3) + (Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3) + 40/(9*((Rational(-1, 2) - sqrt(3)*I/2)*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3))),), (Rational(5, 3) + 40/(9*(Rational(251, 27) +
sqrt(111)*I/9)**Rational(1, 3)) + (Rational(251, 27) + sqrt(111)*I/9)**Rational(1, 3),)}
def test_issue_5114_6611():
# See that it doesn't hang; this solves in about 2 seconds.
# Also check that the solution is relatively small.
# Note: the system in issue 6611 solves in about 5 seconds and has
# an op-count of 138336 (with simplify=False).
b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r = symbols('b:r')
eqs = Matrix([
[b - c/d + r/d], [c*(1/g + 1/e + 1/d) - f/g - r/d],
[-c/g + f*(1/j + 1/i + 1/g) - h/i], [-f/i + h*(1/m + 1/l + 1/i) - k/m],
[-h/m + k*(1/p + 1/o + 1/m) - n/p], [-k/p + n*(1/q + 1/p)]])
v = Matrix([f, h, k, n, b, c])
ans = solve(list(eqs), list(v), simplify=False)
# If time is taken to simplify then then 2617 below becomes
# 1168 and the time is about 50 seconds instead of 2.
assert sum([s.count_ops() for s in ans.values()]) <= 3270
def test_det_quick():
m = Matrix(3, 3, symbols('a:9'))
assert m.det() == det_quick(m) # calls det_perm
m[0, 0] = 1
assert m.det() == det_quick(m) # calls det_minor
m = Matrix(3, 3, list(range(9)))
assert m.det() == det_quick(m) # defaults to .det()
# make sure they work with Sparse
s = SparseMatrix(2, 2, (1, 2, 1, 4))
assert det_perm(s) == det_minor(s) == s.det()
def test_real_imag_splitting():
a, b = symbols('a b', real=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == \
[-sqrt(-b**2 + 9), sqrt(-b**2 + 9)]
a, b = symbols('a b', imaginary=True)
assert solve(sqrt(a**2 + b**2) - 3, a) == []
def test_issue_7110():
y = -2*x**3 + 4*x**2 - 2*x + 5
assert any(ask(Q.real(i)) for i in solve(y))
def test_units():
assert solve(1/x - 1/(2*cm)) == [2*cm]
def test_issue_7547():
A, B, V = symbols('A,B,V')
eq1 = Eq(630.26*(V - 39.0)*V*(V + 39) - A + B, 0)
eq2 = Eq(B, 1.36*10**8*(V - 39))
eq3 = Eq(A, 5.75*10**5*V*(V + 39.0))
sol = Matrix(nsolve(Tuple(eq1, eq2, eq3), [A, B, V], (0, 0, 0)))
assert str(sol) == str(Matrix(
[['4442890172.68209'],
['4289299466.1432'],
['70.5389666628177']]))
def test_issue_7895():
r = symbols('r', real=True)
assert solve(sqrt(r) - 2) == [4]
def test_issue_2777():
# the equations represent two circles
x, y = symbols('x y', real=True)
e1, e2 = sqrt(x**2 + y**2) - 10, sqrt(y**2 + (-x + 10)**2) - 3
a, b = Rational(191, 20), 3*sqrt(391)/20
ans = [(a, -b), (a, b)]
assert solve((e1, e2), (x, y)) == ans
assert solve((e1, e2/(x - a)), (x, y)) == []
# make the 2nd circle's radius be -3
e2 += 6
assert solve((e1, e2), (x, y)) == []
assert solve((e1, e2), (x, y), check=False) == ans
def test_issue_7322():
number = 5.62527e-35
assert solve(x - number, x)[0] == number
def test_nsolve():
raises(ValueError, lambda: nsolve(x, (-1, 1), method='bisect'))
raises(TypeError, lambda: nsolve((x - y + 3,x + y,z - y),(x,y,z),(-50,50)))
raises(TypeError, lambda: nsolve((x + y, x - y), (0, 1)))
@slow
def test_high_order_multivariate():
assert len(solve(a*x**3 - x + 1, x)) == 3
assert len(solve(a*x**4 - x + 1, x)) == 4
assert solve(a*x**5 - x + 1, x) == [] # incomplete solution allowed
raises(NotImplementedError, lambda:
solve(a*x**5 - x + 1, x, incomplete=False))
# result checking must always consider the denominator and CRootOf
# must be checked, too
d = x**5 - x + 1
assert solve(d*(1 + 1/d)) == [CRootOf(d + 1, i) for i in range(5)]
d = x - 1
assert solve(d*(2 + 1/d)) == [S.Half]
def test_base_0_exp_0():
assert solve(0**x - 1) == [0]
assert solve(0**(x - 2) - 1) == [2]
assert solve(S('x*(1/x**0 - x)', evaluate=False)) == \
[0, 1]
def test__simple_dens():
assert _simple_dens(1/x**0, [x]) == set()
assert _simple_dens(1/x**y, [x]) == {x**y}
assert _simple_dens(1/root(x, 3), [x]) == {x}
def test_issue_8755():
# This tests two things: that if full unrad is attempted and fails
# the solution should still be found; also it tests the use of
# keyword `composite`.
assert len(solve(sqrt(y)*x + x**3 - 1, x)) == 3
assert len(solve(-512*y**3 + 1344*(x + 2)**Rational(1, 3)*y**2 -
1176*(x + 2)**Rational(2, 3)*y - 169*x + 686, y, _unrad=False)) == 3
@slow
def test_issue_8828():
x1 = 0
y1 = -620
r1 = 920
x2 = 126
y2 = 276
x3 = 51
y3 = 205
r3 = 104
v = x, y, z
f1 = (x - x1)**2 + (y - y1)**2 - (r1 - z)**2
f2 = (x - x2)**2 + (y - y2)**2 - z**2
f3 = (x - x3)**2 + (y - y3)**2 - (r3 - z)**2
F = f1,f2,f3
g1 = sqrt((x - x1)**2 + (y - y1)**2) + z - r1
g2 = f2
g3 = sqrt((x - x3)**2 + (y - y3)**2) + z - r3
G = g1,g2,g3
A = solve(F, v)
B = solve(G, v)
C = solve(G, v, manual=True)
p, q, r = [{tuple(i.evalf(2) for i in j) for j in R} for R in [A, B, C]]
assert p == q == r
@slow
def test_issue_2840_8155():
assert solve(sin(3*x) + sin(6*x)) == [
0, pi*Rational(-5, 3), pi*Rational(-4, 3), -pi, pi*Rational(-2, 3),
pi*Rational(-4, 9), -pi/3, pi*Rational(-2, 9), pi*Rational(2, 9),
pi/3, pi*Rational(4, 9), pi*Rational(2, 3), pi, pi*Rational(4, 3),
pi*Rational(14, 9), pi*Rational(5, 3), pi*Rational(16, 9), 2*pi,
-2*I*log(-(-1)**Rational(1, 9)), -2*I*log(-(-1)**Rational(2, 9)),
-2*I*log(-sin(pi/18) - I*cos(pi/18)),
-2*I*log(-sin(pi/18) + I*cos(pi/18)),
-2*I*log(sin(pi/18) - I*cos(pi/18)),
-2*I*log(sin(pi/18) + I*cos(pi/18))]
assert solve(2*sin(x) - 2*sin(2*x)) == [
0, pi*Rational(-5, 3), -pi, -pi/3, pi/3, pi, pi*Rational(5, 3)]
def test_issue_9567():
assert solve(1 + 1/(x - 1)) == [0]
def test_issue_11538():
assert solve(x + E) == [-E]
assert solve(x**2 + E) == [-I*sqrt(E), I*sqrt(E)]
assert solve(x**3 + 2*E) == [
-cbrt(2 * E),
cbrt(2)*cbrt(E)/2 - cbrt(2)*sqrt(3)*I*cbrt(E)/2,
cbrt(2)*cbrt(E)/2 + cbrt(2)*sqrt(3)*I*cbrt(E)/2]
assert solve([x + 4, y + E], x, y) == {x: -4, y: -E}
assert solve([x**2 + 4, y + E], x, y) == [
(-2*I, -E), (2*I, -E)]
e1 = x - y**3 + 4
e2 = x + y + 4 + 4 * E
assert len(solve([e1, e2], x, y)) == 3
@slow
def test_issue_12114():
a, b, c, d, e, f, g = symbols('a,b,c,d,e,f,g')
terms = [1 + a*b + d*e, 1 + a*c + d*f, 1 + b*c + e*f,
g - a**2 - d**2, g - b**2 - e**2, g - c**2 - f**2]
sol = solve(terms, [a, b, c, d, e, f, g], dict=True)
s = sqrt(-f**2 - 1)
s2 = sqrt(2 - f**2)
s3 = sqrt(6 - 3*f**2)
s4 = sqrt(3)*f
s5 = sqrt(3)*s2
assert sol == [
{a: -s, b: -s, c: -s, d: f, e: f, g: -1},
{a: s, b: s, c: s, d: f, e: f, g: -1},
{a: -s4/2 - s2/2, b: s4/2 - s2/2, c: s2,
d: -f/2 + s3/2, e: -f/2 - s5/2, g: 2},
{a: -s4/2 + s2/2, b: s4/2 + s2/2, c: -s2,
d: -f/2 - s3/2, e: -f/2 + s5/2, g: 2},
{a: s4/2 - s2/2, b: -s4/2 - s2/2, c: s2,
d: -f/2 - s3/2, e: -f/2 + s5/2, g: 2},
{a: s4/2 + s2/2, b: -s4/2 + s2/2, c: -s2,
d: -f/2 + s3/2, e: -f/2 - s5/2, g: 2}]
def test_inf():
assert solve(1 - oo*x) == []
assert solve(oo*x, x) == []
assert solve(oo*x - oo, x) == []
def test_issue_12448():
f = Function('f')
fun = [f(i) for i in range(15)]
sym = symbols('x:15')
reps = dict(zip(fun, sym))
(x, y, z), c = sym[:3], sym[3:]
ssym = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
(x, y, z), c = fun[:3], fun[3:]
sfun = solve([c[4*i]*x + c[4*i + 1]*y + c[4*i + 2]*z + c[4*i + 3]
for i in range(3)], (x, y, z))
assert sfun[fun[0]].xreplace(reps).count_ops() == \
ssym[sym[0]].count_ops()
def test_denoms():
assert denoms(x/2 + 1/y) == {2, y}
assert denoms(x/2 + 1/y, y) == {y}
assert denoms(x/2 + 1/y, [y]) == {y}
assert denoms(1/x + 1/y + 1/z, [x, y]) == {x, y}
assert denoms(1/x + 1/y + 1/z, x, y) == {x, y}
assert denoms(1/x + 1/y + 1/z, {x, y}) == {x, y}
def test_issue_12476():
x0, x1, x2, x3, x4, x5 = symbols('x0 x1 x2 x3 x4 x5')
eqns = [x0**2 - x0, x0*x1 - x1, x0*x2 - x2, x0*x3 - x3, x0*x4 - x4, x0*x5 - x5,
x0*x1 - x1, -x0/3 + x1**2 - 2*x2/3, x1*x2 - x1/3 - x2/3 - x3/3,
x1*x3 - x2/3 - x3/3 - x4/3, x1*x4 - 2*x3/3 - x5/3, x1*x5 - x4, x0*x2 - x2,
x1*x2 - x1/3 - x2/3 - x3/3, -x0/6 - x1/6 + x2**2 - x2/6 - x3/3 - x4/6,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6, x2*x4 - x2/3 - x3/3 - x4/3,
x2*x5 - x3, x0*x3 - x3, x1*x3 - x2/3 - x3/3 - x4/3,
-x1/6 + x2*x3 - x2/3 - x3/6 - x4/6 - x5/6,
-x0/6 - x1/6 - x2/6 + x3**2 - x3/3 - x4/6, -x1/3 - x2/3 + x3*x4 - x3/3,
-x2 + x3*x5, x0*x4 - x4, x1*x4 - 2*x3/3 - x5/3, x2*x4 - x2/3 - x3/3 - x4/3,
-x1/3 - x2/3 + x3*x4 - x3/3, -x0/3 - 2*x2/3 + x4**2, -x1 + x4*x5, x0*x5 - x5,
x1*x5 - x4, x2*x5 - x3, -x2 + x3*x5, -x1 + x4*x5, -x0 + x5**2, x0 - 1]
sols = [{x0: 1, x3: Rational(1, 6), x2: Rational(1, 6), x4: Rational(-2, 3), x1: Rational(-2, 3), x5: 1},
{x0: 1, x3: S.Half, x2: Rational(-1, 2), x4: 0, x1: 0, x5: -1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(-1, 3), x4: Rational(1, 3), x1: Rational(1, 3), x5: 1},
{x0: 1, x3: 1, x2: 1, x4: 1, x1: 1, x5: 1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: sqrt(5)/3, x1: -sqrt(5)/3, x5: -1},
{x0: 1, x3: Rational(-1, 3), x2: Rational(1, 3), x4: -sqrt(5)/3, x1: sqrt(5)/3, x5: -1}]
assert solve(eqns) == sols
def test_issue_13849():
t = symbols('t')
assert solve((t*(sqrt(5) + sqrt(2)) - sqrt(2), t), t) == []
def test_issue_14860():
from sympy.physics.units import newton, kilo
assert solve(8*kilo*newton + x + y, x) == [-8000*newton - y]
def test_issue_14721():
k, h, a, b = symbols(':4')
assert solve([
-1 + (-k + 1)**2/b**2 + (-h - 1)**2/a**2,
-1 + (-k + 1)**2/b**2 + (-h + 1)**2/a**2,
h, k + 2], h, k, a, b) == [
(0, -2, -b*sqrt(1/(b**2 - 9)), b),
(0, -2, b*sqrt(1/(b**2 - 9)), b)]
assert solve([
h, h/a + 1/b**2 - 2, -h/2 + 1/b**2 - 2], a, h, b) == [
(a, 0, -sqrt(2)/2), (a, 0, sqrt(2)/2)]
assert solve((a + b**2 - 1, a + b**2 - 2)) == []
def test_issue_14779():
x = symbols('x', real=True)
assert solve(sqrt(x**4 - 130*x**2 + 1089) + sqrt(x**4 - 130*x**2
+ 3969) - 96*Abs(x)/x,x) == [sqrt(130)]
def test_issue_15307():
assert solve((y - 2, Mul(x + 3,x - 2, evaluate=False))) == \
[{x: -3, y: 2}, {x: 2, y: 2}]
assert solve((y - 2, Mul(3, x - 2, evaluate=False))) == \
{x: 2, y: 2}
assert solve((y - 2, Add(x + 4, x - 2, evaluate=False))) == \
{x: -1, y: 2}
eq1 = Eq(12513*x + 2*y - 219093, -5726*x - y)
eq2 = Eq(-2*x + 8, 2*x - 40)
assert solve([eq1, eq2]) == {x:12, y:75}
def test_issue_15415():
assert solve(x - 3, x) == [3]
assert solve([x - 3], x) == {x:3}
assert solve(Eq(y + 3*x**2/2, y + 3*x), y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x)], y) == []
assert solve([Eq(y + 3*x**2/2, y + 3*x), Eq(x, 1)], y) == []
@slow
def test_issue_15731():
# f(x)**g(x)=c
assert solve(Eq((x**2 - 7*x + 11)**(x**2 - 13*x + 42), 1)) == [2, 3, 4, 5, 6, 7]
assert solve((x)**(x + 4) - 4) == [-2]
assert solve((-x)**(-x + 4) - 4) == [2]
assert solve((x**2 - 6)**(x**2 - 2) - 4) == [-2, 2]
assert solve((x**2 - 2*x - 1)**(x**2 - 3) - 1/(1 - 2*sqrt(2))) == [sqrt(2)]
assert solve(x**(x + S.Half) - 4*sqrt(2)) == [S(2)]
assert solve((x**2 + 1)**x - 25) == [2]
assert solve(x**(2/x) - 2) == [2, 4]
assert solve((x/2)**(2/x) - sqrt(2)) == [4, 8]
assert solve(x**(x + S.Half) - Rational(9, 4)) == [Rational(3, 2)]
# a**g(x)=c
assert solve((-sqrt(sqrt(2)))**x - 2) == [4, log(2)/(log(2**Rational(1, 4)) + I*pi)]
assert solve((sqrt(2))**x - sqrt(sqrt(2))) == [S.Half]
assert solve((-sqrt(2))**x + 2*(sqrt(2))) == [3,
(3*log(2)**2 + 4*pi**2 - 4*I*pi*log(2))/(log(2)**2 + 4*pi**2)]
assert solve((sqrt(2))**x - 2*(sqrt(2))) == [3]
assert solve(I**x + 1) == [2]
assert solve((1 + I)**x - 2*I) == [2]
assert solve((sqrt(2) + sqrt(3))**x - (2*sqrt(6) + 5)**Rational(1, 3)) == [Rational(2, 3)]
# bases of both sides are equal
b = Symbol('b')
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
assert solve(b**x - b, x) == [1]
b = Symbol('b', positive=True)
assert solve(b**x - b**2, x) == [2]
assert solve(b**x - 1/b, x) == [-1]
def test_issue_10933():
assert solve(x**4 + y*(x + 0.1), x) # doesn't fail
assert solve(I*x**4 + x**3 + x**2 + 1.) # doesn't fail
def test_Abs_handling():
x = symbols('x', real=True)
assert solve(abs(x/y), x) == [0]
def test_issue_7982():
x = Symbol('x')
# Test that no exception happens
assert solve([2*x**2 + 5*x + 20 <= 0, x >= 1.5], x) is S.false
# From #8040
assert solve([x**3 - 8.08*x**2 - 56.48*x/5 - 106 >= 0, x - 1 <= 0], [x]) is S.false
def test_issue_14645():
x, y = symbols('x y')
assert solve([x*y - x - y, x*y - x - y], [x, y]) == [(y/(y - 1), y)]
def test_issue_12024():
x, y = symbols('x y')
assert solve(Piecewise((0.0, x < 0.1), (x, x >= 0.1)) - y) == \
[{y: Piecewise((0.0, x < 0.1), (x, True))}]
def test_issue_17452():
assert solve((7**x)**x + pi, x) == [-sqrt(log(pi) + I*pi)/sqrt(log(7)),
sqrt(log(pi) + I*pi)/sqrt(log(7))]
assert solve(x**(x/11) + pi/11, x) == [exp(LambertW(-11*log(11) + 11*log(pi) + 11*I*pi))]
def test_issue_17799():
assert solve(-erf(x**(S(1)/3))**pi + I, x) == []
def test_issue_17650():
x = Symbol('x', real=True)
assert solve(abs(abs(x**2 - 1) - x) - x) == [1, -1 + sqrt(2), 1 + sqrt(2)]
def test_issue_17882():
eq = -8*x**2/(9*(x**2 - 1)**(S(4)/3)) + 4/(3*(x**2 - 1)**(S(1)/3))
assert unrad(eq) is None
def test_issue_17949():
assert solve(exp(+x+x**2), x) == []
assert solve(exp(-x+x**2), x) == []
assert solve(exp(+x-x**2), x) == []
assert solve(exp(-x-x**2), x) == []
def test_issue_10993():
assert solve(Eq(binomial(x, 2), 3)) == [-2, 3]
assert solve(Eq(pow(x, 2) + binomial(x, 3), x)) == [-4, 0, 1]
assert solve(Eq(binomial(x, 2), 0)) == [0, 1]
assert solve(a+binomial(x, 3), a) == [-binomial(x, 3)]
assert solve(x-binomial(a, 3) + binomial(y, 2) + sin(a), x) == [-sin(a) + binomial(a, 3) - binomial(y, 2)]
assert solve((x+1)-binomial(x+1, 3), x) == [-2, -1, 3]
def test_issue_11553():
eq1 = x + y + 1
eq2 = x + GoldenRatio
assert solve([eq1, eq2], x, y) == {x: -GoldenRatio, y: -1 + GoldenRatio}
eq3 = x + 2 + TribonacciConstant
assert solve([eq1, eq3], x, y) == {x: -2 - TribonacciConstant, y: 1 + TribonacciConstant}
def test_issue_19113_19102():
t = S(1)/3
solve(cos(x)**5-sin(x)**5)
assert solve(4*cos(x)**3 - 2*sin(x)**3) == [
atan(2**(t)), -atan(2**(t)*(1 - sqrt(3)*I)/2),
-atan(2**(t)*(1 + sqrt(3)*I)/2)]
h = S.Half
assert solve(cos(x)**2 + sin(x)) == [
2*atan(-h + sqrt(5)/2 + sqrt(2)*sqrt(1 - sqrt(5))/2),
-2*atan(h + sqrt(5)/2 + sqrt(2)*sqrt(1 + sqrt(5))/2),
-2*atan(-sqrt(5)/2 + h + sqrt(2)*sqrt(1 - sqrt(5))/2),
-2*atan(-sqrt(2)*sqrt(1 + sqrt(5))/2 + h + sqrt(5)/2)]
assert solve(3*cos(x) - sin(x)) == [atan(3)]
def test_issue_19509():
a = S(3)/4
b = S(5)/8
c = sqrt(5)/8
d = sqrt(5)/4
assert solve(1/(x -1)**5 - 1) == [2,
-d + a - sqrt(-b + c),
-d + a + sqrt(-b + c),
d + a - sqrt(-b - c),
d + a + sqrt(-b - c)]
def test_issue_20747():
THT, HT, DBH, dib, c0, c1, c2, c3, c4 = symbols('THT HT DBH dib c0 c1 c2 c3 c4')
f = DBH*c3 + THT*c4 + c2
rhs = 1 - ((HT - 1)/(THT - 1))**c1*(1 - exp(c0/f))
eq = dib - DBH*(c0 - f*log(rhs))
term = ((1 - exp((DBH*c0 - dib)/(DBH*(DBH*c3 + THT*c4 + c2))))
/ (1 - exp(c0/(DBH*c3 + THT*c4 + c2))))
sol = [THT*term**(1/c1) - term**(1/c1) + 1]
assert solve(eq, HT) == sol
def test_issue_20902():
f = (t / ((1 + t) ** 2))
assert solve(f.subs({t: 3 * x + 2}).diff(x) > 0, x) == (S(-1) < x) & (x < S(-1)/3)
assert solve(f.subs({t: 3 * x + 3}).diff(x) > 0, x) == (S(-4)/3 < x) & (x < S(-2)/3)
assert solve(f.subs({t: 3 * x + 4}).diff(x) > 0, x) == (S(-5)/3 < x) & (x < S(-1))
assert solve(f.subs({t: 3 * x + 2}).diff(x) > 0, x) == (S(-1) < x) & (x < S(-1)/3)
def test_issue_21034():
a = symbols('a', real=True)
system = [x - cosh(cos(4)), y - sinh(cos(a)), z - tanh(x)]
# constants inside hyperbolic functions should not be rewritten in terms of exp
assert solve(system, x, y, z) == [(cosh(cos(4)), sinh(cos(a)), tanh(cosh(cos(4))))]
# but if the variable of interest is present in a hyperbolic function,
# then it should be rewritten in terms of exp and solved further
newsystem = [(exp(x) - exp(-x)) - tanh(x)*(exp(x) + exp(-x)) + x - 5]
assert solve(newsystem, x) == {x: 5}
def test_issue_4886():
z = a*sqrt(R**2*a**2 + R**2*b**2 - c**2)/(a**2 + b**2)
t = b*c/(a**2 + b**2)
sol = [((b*(t - z) - c)/(-a), t - z), ((b*(t + z) - c)/(-a), t + z)]
assert solve([x**2 + y**2 - R**2, a*x + b*y - c], x, y) == sol
def test_issue_6819():
a, b, c, d = symbols('a b c d', positive=True)
assert solve(a*b**x - c*d**x, x) == [log(c/a)/log(b/d)]
def test_issue_17454():
x = Symbol('x')
assert solve((1 - x - I)**4, x) == [1 - I]
def test_issue_21852():
solution = [21 - 21*sqrt(2)/2]
assert solve(2*x + sqrt(2*x**2) - 21) == solution
def test_issue_21942():
eq = -d + (a*c**(1 - e) + b**(1 - e)*(1 - a))**(1/(1 - e))
sol = solve(eq, c, simplify=False, check=False)
assert sol == [(b/b**e - b/(a*b**e) + d**(1 - e)/a)**(1/(1 - e))]
def test_solver_flags():
root = solve(x**5 + x**2 - x - 1, cubics=False)
rad = solve(x**5 + x**2 - x - 1, cubics=True)
assert root != rad
def test_issue_22768():
eq = 2*x**3 - 16*(y - 1)**6*z**3
assert solve(eq.expand(), x, simplify=False
) == [2*z*(y - 1)**2, z*(-1 + sqrt(3)*I)*(y - 1)**2,
-z*(1 + sqrt(3)*I)*(y - 1)**2]
def test_issue_22717():
assert solve((-y**2 + log(y**2/x) + 2, -2*x*y + 2*x/y)) == [
{y: -1, x: E}, {y: 1, x: E}]
def test_issue_10169():
eq = S(-8*a - x**5*(a + b + c + e) - x**4*(4*a - 2**Rational(3,4)*c + 4*c +
d + 2**Rational(3,4)*e + 4*e + k) - x**3*(-4*2**Rational(3,4)*c + sqrt(2)*c -
2**Rational(3,4)*d + 4*d + sqrt(2)*e + 4*2**Rational(3,4)*e + 2**Rational(3,4)*k + 4*k) -
x**2*(4*sqrt(2)*c - 4*2**Rational(3,4)*d + sqrt(2)*d + 4*sqrt(2)*e +
sqrt(2)*k + 4*2**Rational(3,4)*k) - x*(2*a + 2*b + 4*sqrt(2)*d +
4*sqrt(2)*k) + 5)
assert solve_undetermined_coeffs(eq, [a, b, c, d, e, k], x) == {
a: Rational(5,8),
b: Rational(-5,1032),
c: Rational(-40,129) - 5*2**Rational(3,4)/129 + 5*2**Rational(1,4)/1032,
d: -20*2**Rational(3,4)/129 - 10*sqrt(2)/129 - 5*2**Rational(1,4)/258,
e: Rational(-40,129) - 5*2**Rational(1,4)/1032 + 5*2**Rational(3,4)/129,
k: -10*sqrt(2)/129 + 5*2**Rational(1,4)/258 + 20*2**Rational(3,4)/129
}
|
557c51b22720b7222b66d2638af6211e7fd31f6c2cfd97cf96b72a59454e92e7 | from sympy.core.numbers import (Float, I, Rational, pi)
from sympy.core.relational import Eq
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import sin
from sympy.integrals.integrals import Integral
from sympy.matrices.dense import Matrix
from mpmath import mnorm, mpf
from sympy.solvers import nsolve
from sympy.utilities.lambdify import lambdify
from sympy.testing.pytest import raises, XFAIL
from sympy.utilities.decorator import conserve_mpmath_dps
@XFAIL
def test_nsolve_fail():
x = symbols('x')
# Sometimes it is better to use the numerator (issue 4829)
# but sometimes it is not (issue 11768) so leave this to
# the discretion of the user
ans = nsolve(x**2/(1 - x)/(1 - 2*x)**2 - 100, x, 0)
assert ans > 0.46 and ans < 0.47
def test_nsolve_denominator():
x = symbols('x')
# Test that nsolve uses the full expression (numerator and denominator).
ans = nsolve((x**2 + 3*x + 2)/(x + 2), -2.1)
# The root -2 was divided out, so make sure we don't find it.
assert ans == -1.0
def test_nsolve():
# onedimensional
x = Symbol('x')
assert nsolve(sin(x), 2) - pi.evalf() < 1e-15
assert nsolve(Eq(2*x, 2), x, -10) == nsolve(2*x - 2, -10)
# Testing checks on number of inputs
raises(TypeError, lambda: nsolve(Eq(2*x, 2)))
raises(TypeError, lambda: nsolve(Eq(2*x, 2), x, 1, 2))
# multidimensional
x1 = Symbol('x1')
x2 = Symbol('x2')
f1 = 3 * x1**2 - 2 * x2**2 - 1
f2 = x1**2 - 2 * x1 + x2**2 + 2 * x2 - 8
f = Matrix((f1, f2)).T
F = lambdify((x1, x2), f.T, modules='mpmath')
for x0 in [(-1, 1), (1, -2), (4, 4), (-4, -4)]:
x = nsolve(f, (x1, x2), x0, tol=1.e-8)
assert mnorm(F(*x), 1) <= 1.e-10
# The Chinese mathematician Zhu Shijie was the very first to solve this
# nonlinear system 700 years ago (z was added to make it 3-dimensional)
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
f1 = -x + 2*y
f2 = (x**2 + x*(y**2 - 2) - 4*y) / (x + 4)
f3 = sqrt(x**2 + y**2)*z
f = Matrix((f1, f2, f3)).T
F = lambdify((x, y, z), f.T, modules='mpmath')
def getroot(x0):
root = nsolve(f, (x, y, z), x0)
assert mnorm(F(*root), 1) <= 1.e-8
return root
assert list(map(round, getroot((1, 1, 1)))) == [2.0, 1.0, 0.0]
assert nsolve([Eq(
f1, 0), Eq(f2, 0), Eq(f3, 0)], [x, y, z], (1, 1, 1)) # just see that it works
a = Symbol('a')
assert abs(nsolve(1/(0.001 + a)**3 - 6/(0.9 - a)**3, a, 0.3) -
mpf('0.31883011387318591')) < 1e-15
def test_issue_6408():
x = Symbol('x')
assert nsolve(Piecewise((x, x < 1), (x**2, True)), x, 2) == 0.0
def test_issue_6408_integral():
x, y = symbols('x y')
assert nsolve(Integral(x*y, (x, 0, 5)), y, 2) == 0.0
@conserve_mpmath_dps
def test_increased_dps():
# Issue 8564
import mpmath
mpmath.mp.dps = 128
x = Symbol('x')
e1 = x**2 - pi
q = nsolve(e1, x, 3.0)
assert abs(sqrt(pi).evalf(128) - q) < 1e-128
def test_nsolve_precision():
x, y = symbols('x y')
sol = nsolve(x**2 - pi, x, 3, prec=128)
assert abs(sqrt(pi).evalf(128) - sol) < 1e-128
assert isinstance(sol, Float)
sols = nsolve((y**2 - x, x**2 - pi), (x, y), (3, 3), prec=128)
assert isinstance(sols, Matrix)
assert sols.shape == (2, 1)
assert abs(sqrt(pi).evalf(128) - sols[0]) < 1e-128
assert abs(sqrt(sqrt(pi)).evalf(128) - sols[1]) < 1e-128
assert all(isinstance(i, Float) for i in sols)
def test_nsolve_complex():
x, y = symbols('x y')
assert nsolve(x**2 + 2, 1j) == sqrt(2.)*I
assert nsolve(x**2 + 2, I) == sqrt(2.)*I
assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I])
assert nsolve([x**2 + 2, y**2 + 2], [x, y], [I, I]) == Matrix([sqrt(2.)*I, sqrt(2.)*I])
def test_nsolve_dict_kwarg():
x, y = symbols('x y')
# one variable
assert nsolve(x**2 - 2, 1, dict = True) == \
[{x: sqrt(2.)}]
# one variable with complex solution
assert nsolve(x**2 + 2, I, dict = True) == \
[{x: sqrt(2.)*I}]
# two variables
assert nsolve([x**2 + y**2 - 5, x**2 - y**2 + 1], [x, y], [1, 1], dict = True) == \
[{x: sqrt(2.), y: sqrt(3.)}]
def test_nsolve_rational():
x = symbols('x')
assert nsolve(x - Rational(1, 3), 0, prec=100) == Rational(1, 3).evalf(100)
def test_issue_14950():
x = Matrix(symbols('t s'))
x0 = Matrix([17, 23])
eqn = x + x0
assert nsolve(eqn, x, x0) == -x0
assert nsolve(eqn.T, x.T, x0.T) == -x0
|
1dab77b2cd4a1a01c07e9bfe831b929b3509837d962614f95b80f1c87f0436a1 | from sympy.core.function import (Function, Lambda, expand)
from sympy.core.numbers import (I, Rational)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.combinatorial.factorials import (rf, binomial, factorial)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.polys.polytools import factor
from sympy.solvers.recurr import rsolve, rsolve_hyper, rsolve_poly, rsolve_ratio
from sympy.testing.pytest import raises, slow, XFAIL
from sympy.abc import a, b
y = Function('y')
n, k = symbols('n,k', integer=True)
C0, C1, C2 = symbols('C0,C1,C2')
def test_rsolve_poly():
assert rsolve_poly([-1, -1, 1], 0, n) == 0
assert rsolve_poly([-1, -1, 1], 1, n) == -1
assert rsolve_poly([-1, n + 1], n, n) == 1
assert rsolve_poly([-1, 1], n, n) == C0 + (n**2 - n)/2
assert rsolve_poly([-n - 1, n], 1, n) == C0*n - 1
assert rsolve_poly([-4*n - 2, 1], 4*n + 1, n) == -1
assert rsolve_poly([-1, 1], n**5 + n**3, n) == \
C0 - n**3 / 2 - n**5 / 2 + n**2 / 6 + n**6 / 6 + 2*n**4 / 3
def test_rsolve_ratio():
solution = rsolve_ratio([-2*n**3 + n**2 + 2*n - 1, 2*n**3 + n**2 - 6*n,
-2*n**3 - 11*n**2 - 18*n - 9, 2*n**3 + 13*n**2 + 22*n + 8], 0, n)
assert solution == C0*(2*n - 3)/(n**2 - 1)/2
def test_rsolve_hyper():
assert rsolve_hyper([-1, -1, 1], 0, n) in [
C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
]
assert rsolve_hyper([n**2 - 2, -2*n - 1, 1], 0, n) in [
C0*rf(sqrt(2), n) + C1*rf(-sqrt(2), n),
C1*rf(sqrt(2), n) + C0*rf(-sqrt(2), n),
]
assert rsolve_hyper([n**2 - k, -2*n - 1, 1], 0, n) in [
C0*rf(sqrt(k), n) + C1*rf(-sqrt(k), n),
C1*rf(sqrt(k), n) + C0*rf(-sqrt(k), n),
]
assert rsolve_hyper(
[2*n*(n + 1), -n**2 - 3*n + 2, n - 1], 0, n) == C1*factorial(n) + C0*2**n
assert rsolve_hyper(
[n + 2, -(2*n + 3)*(17*n**2 + 51*n + 39), n + 1], 0, n) == 0
assert rsolve_hyper([-n - 1, -1, 1], 0, n) == 0
assert rsolve_hyper([-1, 1], n, n).expand() == C0 + n**2/2 - n/2
assert rsolve_hyper([-1, 1], 1 + n, n).expand() == C0 + n**2/2 + n/2
assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n
assert rsolve_hyper([-a, 1],0,n).expand() == C0*a**n
assert rsolve_hyper([-a, 0, 1], 0, n).expand() == (-1)**n*C1*a**(n/2) + C0*a**(n/2)
assert rsolve_hyper([1, 1, 1], 0, n).expand() == \
C0*(Rational(-1, 2) - sqrt(3)*I/2)**n + C1*(Rational(-1, 2) + sqrt(3)*I/2)**n
assert rsolve_hyper([1, -2*n/a - 2/a, 1], 0, n) == 0
@XFAIL
def test_rsolve_ratio_missed():
# this arises during computation
# assert rsolve_hyper([-1, 1], 3*(n + n**2), n).expand() == C0 + n**3 - n
assert rsolve_ratio([-n, n + 2], n, n) is not None
def recurrence_term(c, f):
"""Compute RHS of recurrence in f(n) with coefficients in c."""
return sum(c[i]*f.subs(n, n + i) for i in range(len(c)))
def test_rsolve_bulk():
"""Some bulk-generated tests."""
funcs = [ n, n + 1, n**2, n**3, n**4, n + n**2, 27*n + 52*n**2 - 3*
n**3 + 12*n**4 - 52*n**5 ]
coeffs = [ [-2, 1], [-2, -1, 1], [-1, 1, 1, -1, 1], [-n, 1], [n**2 -
n + 12, 1] ]
for p in funcs:
# compute difference
for c in coeffs:
q = recurrence_term(c, p)
if p.is_polynomial(n):
assert rsolve_poly(c, q, n) == p
# See issue 3956:
if p.is_hypergeometric(n) and len(c) <= 3:
assert rsolve_hyper(c, q, n).subs(zip(symbols('C:3'), [0, 0, 0])).expand() == p
def test_rsolve_0_sol_homogeneous():
# fixed by cherry-pick from
# https://github.com/diofant/diofant/commit/e1d2e52125199eb3df59f12e8944f8a5f24b00a5
assert rsolve_hyper([n**2 - n + 12, 1], n*(n**2 - n + 12) + n + 1, n) == n
def test_rsolve():
f = y(n + 2) - y(n + 1) - y(n)
h = sqrt(5)*(S.Half + S.Half*sqrt(5))**n \
- sqrt(5)*(S.Half - S.Half*sqrt(5))**n
assert rsolve(f, y(n)) in [
C0*(S.Half - S.Half*sqrt(5))**n + C1*(S.Half + S.Half*sqrt(5))**n,
C1*(S.Half - S.Half*sqrt(5))**n + C0*(S.Half + S.Half*sqrt(5))**n,
]
assert rsolve(f, y(n), [0, 5]) == h
assert rsolve(f, y(n), {0: 0, 1: 5}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 5}) == h
assert rsolve(y(n) - y(n - 1) - y(n - 2), y(n), [0, 5]) == h
assert rsolve(Eq(y(n), y(n - 1) + y(n - 2)), y(n), [0, 5]) == h
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = (n - 1)*y(n + 2) - (n**2 + 3*n - 2)*y(n + 1) + 2*n*(n + 1)*y(n)
g = C1*factorial(n) + C0*2**n
h = -3*factorial(n) + 3*2**n
assert rsolve(f, y(n)) == g
assert rsolve(f, y(n), []) == g
assert rsolve(f, y(n), {}) == g
assert rsolve(f, y(n), [0, 3]) == h
assert rsolve(f, y(n), {0: 0, 1: 3}) == h
assert rsolve(f, y(n), {y(0): 0, y(1): 3}) == h
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - y(n - 1) - 2
assert rsolve(f, y(n), {y(0): 0}) == 2*n
assert rsolve(f, y(n), {y(0): 1}) == 2*n + 1
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) is None
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = 3*y(n - 1) - y(n) - 1
assert rsolve(f, y(n), {y(0): 0}) == -3**n/2 + S.Half
assert rsolve(f, y(n), {y(0): 1}) == 3**n/2 + S.Half
assert rsolve(f, y(n), {y(0): 2}) == 3*3**n/2 + S.Half
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - 1/n*y(n - 1)
assert rsolve(f, y(n)) == C0/factorial(n)
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = y(n) - 1/n*y(n - 1) - 1
assert rsolve(f, y(n)) is None
f = 2*y(n - 1) + (1 - n)*y(n)/n
assert rsolve(f, y(n), {y(1): 1}) == 2**(n - 1)*n
assert rsolve(f, y(n), {y(1): 2}) == 2**(n - 1)*n*2
assert rsolve(f, y(n), {y(1): 3}) == 2**(n - 1)*n*3
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
f = (n - 1)*(n - 2)*y(n + 2) - (n + 1)*(n + 2)*y(n)
assert rsolve(f, y(n), {y(3): 6, y(4): 24}) == n*(n - 1)*(n - 2)
assert rsolve(
f, y(n), {y(3): 6, y(4): -24}) == -n*(n - 1)*(n - 2)*(-1)**(n)
assert f.subs(y, Lambda(k, rsolve(f, y(n)).subs(n, k))).simplify() == 0
assert rsolve(Eq(y(n + 1), a*y(n)), y(n), {y(1): a}).simplify() == a**n
assert rsolve(y(n) - a*y(n-2),y(n), \
{y(1): sqrt(a)*(a + b), y(2): a*(a - b)}).simplify() == \
a**(n/2)*(-(-1)**n*b + a)
f = (-16*n**2 + 32*n - 12)*y(n - 1) + (4*n**2 - 12*n + 9)*y(n)
yn = rsolve(f, y(n), {y(1): binomial(2*n + 1, 3)})
sol = 2**(2*n)*n*(2*n - 1)**2*(2*n + 1)/12
assert factor(expand(yn, func=True)) == sol
sol = rsolve(y(n) + a*(y(n + 1) + y(n - 1))/2, y(n))
Y = lambda i: sol.subs(n, i)
assert (Y(n) + a*(Y(n + 1) + Y(n - 1))/2).expand().cancel() == 0
assert rsolve((k + 1)*y(k), y(k)) is None
assert (rsolve((k + 1)*y(k) + (k + 3)*y(k + 1) + (k + 5)*y(k + 2), y(k))
is None)
assert rsolve(y(n) + y(n + 1) + 2**n + 3**n, y(n)) == (-1)**n*C0 - 2**n/3 - 3**n/4
def test_rsolve_raises():
x = Function('x')
raises(ValueError, lambda: rsolve(y(n) - y(k + 1), y(n)))
raises(ValueError, lambda: rsolve(y(n) - y(n + 1), x(n)))
raises(ValueError, lambda: rsolve(y(n) - x(n + 1), y(n)))
raises(ValueError, lambda: rsolve(y(n) - sqrt(n)*y(n + 1), y(n)))
raises(ValueError, lambda: rsolve(y(n) - y(n + 1), y(n), {x(0): 0}))
raises(ValueError, lambda: rsolve(y(n) + y(n + 1) + 2**n + cos(n), y(n)))
def test_issue_6844():
f = y(n + 2) - y(n + 1) + y(n)/4
assert rsolve(f, y(n)) == 2**(-n + 1)*C1*n + 2**(-n)*C0
assert rsolve(f, y(n), {y(0): 0, y(1): 1}) == 2**(1 - n)*n
def test_issue_18751():
r = Symbol('r', positive=True)
theta = Symbol('theta', real=True)
f = y(n) - 2 * r * cos(theta) * y(n - 1) + r**2 * y(n - 2)
assert rsolve(f, y(n)) == \
C0*(r*(cos(theta) - I*Abs(sin(theta))))**n + C1*(r*(cos(theta) + I*Abs(sin(theta))))**n
def test_constant_naming():
#issue 8697
assert rsolve(y(n+3) - y(n+2) - y(n+1) + y(n), y(n)) == (-1)**n*C1 + C0 + C2*n
assert rsolve(y(n+3)+3*y(n+2)+3*y(n+1)+y(n), y(n)).expand() == (-1)**n*C0 - (-1)**n*C1*n - (-1)**n*C2*n**2
assert rsolve(y(n) - 2*y(n - 3) + 5*y(n - 2) - 4*y(n - 1),y(n),[1,3,8]) == 3*2**n - n - 2
#issue 19630
assert rsolve(y(n+3) - 3*y(n+1) + 2*y(n), y(n), {y(1):0, y(2):8, y(3):-2}) == (-2)**n + 2*n
@slow
def test_issue_15751():
f = y(n) + 21*y(n + 1) - 273*y(n + 2) - 1092*y(n + 3) + 1820*y(n + 4) + 1092*y(n + 5) - 273*y(n + 6) - 21*y(n + 7) + y(n + 8)
assert rsolve(f, y(n)) is not None
def test_issue_17990():
f = -10*y(n) + 4*y(n + 1) + 6*y(n + 2) + 46*y(n + 3)
sol = rsolve(f, y(n))
expected = C0*((86*18**(S(1)/3)/69 + (-12 + (-1 + sqrt(3)*I)*(290412 +
3036*sqrt(9165))**(S(1)/3))*(1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))**
(S(1)/3)/276)/((1 - sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3))
)**n + C1*((86*18**(S(1)/3)/69 + (-12 + (-1 - sqrt(3)*I)*(290412 + 3036
*sqrt(9165))**(S(1)/3))*(1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))**
(S(1)/3)/276)/((1 + sqrt(3)*I)*(24201 + 253*sqrt(9165))**(S(1)/3))
)**n + C2*(-43*18**(S(1)/3)/(69*(24201 + 253*sqrt(9165))**(S(1)/3)) -
S(1)/23 + (290412 + 3036*sqrt(9165))**(S(1)/3)/138)**n
assert sol == expected
e = sol.subs({C0: 1, C1: 1, C2: 1, n: 1}).evalf()
assert abs(e + 0.130434782608696) < 1e-13
def test_issue_8697():
a = Function('a')
eq = a(n + 3) - a(n + 2) - a(n + 1) + a(n)
assert rsolve(eq, a(n)) == (-1)**n*C1 + C0 + C2*n
eq2 = a(n + 3) + 3*a(n + 2) + 3*a(n + 1) + a(n)
assert (rsolve(eq2, a(n)) ==
(-1)**n*C0 + (-1)**(n + 1)*C1*n + (-1)**(n + 1)*C2*n**2)
assert rsolve(a(n) - 2*a(n - 3) + 5*a(n - 2) - 4*a(n - 1),
a(n), {a(0): 1, a(1): 3, a(2): 8}) == 3*2**n - n - 2
# From issue thread (but fixed by https://github.com/diofant/diofant/commit/da9789c6cd7d0c2ceeea19fbf59645987125b289):
assert rsolve(a(n) - 2*a(n - 1) - n, a(n), {a(0): 1}) == 3*2**n - n - 2
def test_diofantissue_294():
f = y(n) - y(n - 1) - 2*y(n - 2) - 2*n
assert rsolve(f, y(n)) == (-1)**n*C0 + 2**n*C1 - n - Rational(5, 2)
# issue sympy/sympy#11261
assert rsolve(f, y(n), {y(0): -1, y(1): 1}) == (-(-1)**n/2 + 2*2**n -
n - Rational(5, 2))
# issue sympy/sympy#7055
assert rsolve(-2*y(n) + y(n + 1) + n - 1, y(n)) == 2**n*C0 + n
def test_issue_15553():
f = Function("f")
assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n)) == 2**n*C0 - n - 2
assert rsolve(Eq(f(n + 1), 2*f(n) + n**2 + 1), f(n)) == 2**n*C0 - n**2 - 2*n - 4
assert rsolve(Eq(f(n + 1), 2*f(n) + n**2 + 1), f(n), {f(1): 0}) == 7*2**n/2 - n**2 - 2*n - 4
assert rsolve(Eq(f(n), 2*f(n - 1) + 3*n**2), f(n)) == 2**n*C0 - 3*n**2 - 12*n - 18
assert rsolve(Eq(f(n), 2*f(n - 1) + n**2), f(n)) == 2**n*C0 - n**2 - 4*n - 6
assert rsolve(Eq(f(n), 2*f(n - 1) + n), f(n), {f(0): 1}) == 3*2**n - n - 2
|
4ebf4aad81219f61be977a81ab1e531a172e54d5787faa249e418b915f9956e4 | from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.numbers import (Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.matrices.dense import Matrix
from sympy.ntheory.factor_ import factorint
from sympy.simplify.powsimp import powsimp
from sympy.core.function import _mexpand
from sympy.core.sorting import default_sort_key, ordered
from sympy.functions.elementary.trigonometric import sin
from sympy.solvers.diophantine import diophantine
from sympy.solvers.diophantine.diophantine import (diop_DN,
diop_solve, diop_ternary_quadratic_normal,
diop_general_pythagorean, diop_ternary_quadratic, diop_linear,
diop_quadratic, diop_general_sum_of_squares, diop_general_sum_of_even_powers,
descent, diop_bf_DN, divisible, equivalent, find_DN, ldescent, length,
reconstruct, partition, power_representation,
prime_as_sum_of_two_squares, square_factor, sum_of_four_squares,
sum_of_three_squares, transformation_to_DN, transformation_to_normal,
classify_diop, base_solution_linear, cornacchia, sqf_normal, gaussian_reduce, holzer,
check_param, parametrize_ternary_quadratic, sum_of_powers, sum_of_squares,
_diop_ternary_quadratic_normal, _nint_or_floor,
_odd, _even, _remove_gcd, _can_do_sum_of_squares, DiophantineSolutionSet, GeneralPythagorean,
BinaryQuadratic)
from sympy.testing.pytest import slow, raises, XFAIL
from sympy.utilities.iterables import (
signed_permutations)
a, b, c, d, p, q, x, y, z, w, t, u, v, X, Y, Z = symbols(
"a, b, c, d, p, q, x, y, z, w, t, u, v, X, Y, Z", integer=True)
t_0, t_1, t_2, t_3, t_4, t_5, t_6 = symbols("t_:7", integer=True)
m1, m2, m3 = symbols('m1:4', integer=True)
n1 = symbols('n1', integer=True)
def diop_simplify(eq):
return _mexpand(powsimp(_mexpand(eq)))
def test_input_format():
raises(TypeError, lambda: diophantine(sin(x)))
raises(TypeError, lambda: diophantine(x/pi - 3))
def test_nosols():
# diophantine should sympify eq so that these are equivalent
assert diophantine(3) == set()
assert diophantine(S(3)) == set()
def test_univariate():
assert diop_solve((x - 1)*(x - 2)**2) == {(1,), (2,)}
assert diop_solve((x - 1)*(x - 2)) == {(1,), (2,)}
def test_classify_diop():
raises(TypeError, lambda: classify_diop(x**2/3 - 1))
raises(ValueError, lambda: classify_diop(1))
raises(NotImplementedError, lambda: classify_diop(w*x*y*z - 1))
raises(NotImplementedError, lambda: classify_diop(x**3 + y**3 + z**4 - 90))
assert classify_diop(14*x**2 + 15*x - 42) == (
[x], {1: -42, x: 15, x**2: 14}, 'univariate')
assert classify_diop(x*y + z) == (
[x, y, z], {x*y: 1, z: 1}, 'inhomogeneous_ternary_quadratic')
assert classify_diop(x*y + z + w + x**2) == (
[w, x, y, z], {x*y: 1, w: 1, x**2: 1, z: 1}, 'inhomogeneous_general_quadratic')
assert classify_diop(x*y + x*z + x**2 + 1) == (
[x, y, z], {x*y: 1, x*z: 1, x**2: 1, 1: 1}, 'inhomogeneous_general_quadratic')
assert classify_diop(x*y + z + w + 42) == (
[w, x, y, z], {x*y: 1, w: 1, 1: 42, z: 1}, 'inhomogeneous_general_quadratic')
assert classify_diop(x*y + z*w) == (
[w, x, y, z], {x*y: 1, w*z: 1}, 'homogeneous_general_quadratic')
assert classify_diop(x*y**2 + 1) == (
[x, y], {x*y**2: 1, 1: 1}, 'cubic_thue')
assert classify_diop(x**4 + y**4 + z**4 - (1 + 16 + 81)) == (
[x, y, z], {1: -98, x**4: 1, z**4: 1, y**4: 1}, 'general_sum_of_even_powers')
assert classify_diop(x**2 + y**2 + z**2) == (
[x, y, z], {x**2: 1, y**2: 1, z**2: 1}, 'homogeneous_ternary_quadratic_normal')
def test_linear():
assert diop_solve(x) == (0,)
assert diop_solve(1*x) == (0,)
assert diop_solve(3*x) == (0,)
assert diop_solve(x + 1) == (-1,)
assert diop_solve(2*x + 1) == (None,)
assert diop_solve(2*x + 4) == (-2,)
assert diop_solve(y + x) == (t_0, -t_0)
assert diop_solve(y + x + 0) == (t_0, -t_0)
assert diop_solve(y + x - 0) == (t_0, -t_0)
assert diop_solve(0*x - y - 5) == (-5,)
assert diop_solve(3*y + 2*x - 5) == (3*t_0 - 5, -2*t_0 + 5)
assert diop_solve(2*x - 3*y - 5) == (3*t_0 - 5, 2*t_0 - 5)
assert diop_solve(-2*x - 3*y - 5) == (3*t_0 + 5, -2*t_0 - 5)
assert diop_solve(7*x + 5*y) == (5*t_0, -7*t_0)
assert diop_solve(2*x + 4*y) == (2*t_0, -t_0)
assert diop_solve(4*x + 6*y - 4) == (3*t_0 - 2, -2*t_0 + 2)
assert diop_solve(4*x + 6*y - 3) == (None, None)
assert diop_solve(0*x + 3*y - 4*z + 5) == (4*t_0 + 5, 3*t_0 + 5)
assert diop_solve(4*x + 3*y - 4*z + 5) == (t_0, 8*t_0 + 4*t_1 + 5, 7*t_0 + 3*t_1 + 5)
assert diop_solve(4*x + 3*y - 4*z + 5, None) == (0, 5, 5)
assert diop_solve(4*x + 2*y + 8*z - 5) == (None, None, None)
assert diop_solve(5*x + 7*y - 2*z - 6) == (t_0, -3*t_0 + 2*t_1 + 6, -8*t_0 + 7*t_1 + 18)
assert diop_solve(3*x - 6*y + 12*z - 9) == (2*t_0 + 3, t_0 + 2*t_1, t_1)
assert diop_solve(6*w + 9*x + 20*y - z) == (t_0, t_1, t_1 + t_2, 6*t_0 + 29*t_1 + 20*t_2)
# to ignore constant factors, use diophantine
raises(TypeError, lambda: diop_solve(x/2))
def test_quadratic_simple_hyperbolic_case():
# Simple Hyperbolic case: A = C = 0 and B != 0
assert diop_solve(3*x*y + 34*x - 12*y + 1) == \
{(-133, -11), (5, -57)}
assert diop_solve(6*x*y + 2*x + 3*y + 1) == set()
assert diop_solve(-13*x*y + 2*x - 4*y - 54) == {(27, 0)}
assert diop_solve(-27*x*y - 30*x - 12*y - 54) == {(-14, -1)}
assert diop_solve(2*x*y + 5*x + 56*y + 7) == {(-161, -3), (-47, -6), (-35, -12),
(-29, -69), (-27, 64), (-21, 7),
(-9, 1), (105, -2)}
assert diop_solve(6*x*y + 9*x + 2*y + 3) == set()
assert diop_solve(x*y + x + y + 1) == {(-1, t), (t, -1)}
assert diophantine(48*x*y)
def test_quadratic_elliptical_case():
# Elliptical case: B**2 - 4AC < 0
assert diop_solve(42*x**2 + 8*x*y + 15*y**2 + 23*x + 17*y - 4915) == {(-11, -1)}
assert diop_solve(4*x**2 + 3*y**2 + 5*x - 11*y + 12) == set()
assert diop_solve(x**2 + y**2 + 2*x + 2*y + 2) == {(-1, -1)}
assert diop_solve(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950) == {(-15, 6)}
assert diop_solve(10*x**2 + 12*x*y + 12*y**2 - 34) == \
{(-1, -1), (-1, 2), (1, -2), (1, 1)}
def test_quadratic_parabolic_case():
# Parabolic case: B**2 - 4AC = 0
assert check_solutions(8*x**2 - 24*x*y + 18*y**2 + 5*x + 7*y + 16)
assert check_solutions(8*x**2 - 24*x*y + 18*y**2 + 6*x + 12*y - 6)
assert check_solutions(8*x**2 + 24*x*y + 18*y**2 + 4*x + 6*y - 7)
assert check_solutions(-4*x**2 + 4*x*y - y**2 + 2*x - 3)
assert check_solutions(x**2 + 2*x*y + y**2 + 2*x + 2*y + 1)
assert check_solutions(x**2 - 2*x*y + y**2 + 2*x + 2*y + 1)
assert check_solutions(y**2 - 41*x + 40)
def test_quadratic_perfect_square():
# B**2 - 4*A*C > 0
# B**2 - 4*A*C is a perfect square
assert check_solutions(48*x*y)
assert check_solutions(4*x**2 - 5*x*y + y**2 + 2)
assert check_solutions(-2*x**2 - 3*x*y + 2*y**2 -2*x - 17*y + 25)
assert check_solutions(12*x**2 + 13*x*y + 3*y**2 - 2*x + 3*y - 12)
assert check_solutions(8*x**2 + 10*x*y + 2*y**2 - 32*x - 13*y - 23)
assert check_solutions(4*x**2 - 4*x*y - 3*y- 8*x - 3)
assert check_solutions(- 4*x*y - 4*y**2 - 3*y- 5*x - 10)
assert check_solutions(x**2 - y**2 - 2*x - 2*y)
assert check_solutions(x**2 - 9*y**2 - 2*x - 6*y)
assert check_solutions(4*x**2 - 9*y**2 - 4*x - 12*y - 3)
def test_quadratic_non_perfect_square():
# B**2 - 4*A*C is not a perfect square
# Used check_solutions() since the solutions are complex expressions involving
# square roots and exponents
assert check_solutions(x**2 - 2*x - 5*y**2)
assert check_solutions(3*x**2 - 2*y**2 - 2*x - 2*y)
assert check_solutions(x**2 - x*y - y**2 - 3*y)
assert check_solutions(x**2 - 9*y**2 - 2*x - 6*y)
assert BinaryQuadratic(x**2 + y**2 + 2*x + 2*y + 2).solve() == {(-1, -1)}
def test_issue_9106():
eq = -48 - 2*x*(3*x - 1) + y*(3*y - 1)
v = (x, y)
for sol in diophantine(eq):
assert not diop_simplify(eq.xreplace(dict(zip(v, sol))))
def test_issue_18138():
eq = x**2 - x - y**2
v = (x, y)
for sol in diophantine(eq):
assert not diop_simplify(eq.xreplace(dict(zip(v, sol))))
@slow
def test_quadratic_non_perfect_slow():
assert check_solutions(8*x**2 + 10*x*y - 2*y**2 - 32*x - 13*y - 23)
# This leads to very large numbers.
# assert check_solutions(5*x**2 - 13*x*y + y**2 - 4*x - 4*y - 15)
assert check_solutions(-3*x**2 - 2*x*y + 7*y**2 - 5*x - 7)
assert check_solutions(-4 - x + 4*x**2 - y - 3*x*y - 4*y**2)
assert check_solutions(1 + 2*x + 2*x**2 + 2*y + x*y - 2*y**2)
def test_DN():
# Most of the test cases were adapted from,
# Solving the generalized Pell equation x**2 - D*y**2 = N, John P. Robertson, July 31, 2004.
# https://web.archive.org/web/20160323033128/http://www.jpr2718.org/pell.pdf
# others are verified using Wolfram Alpha.
# Covers cases where D <= 0 or D > 0 and D is a square or N = 0
# Solutions are straightforward in these cases.
assert diop_DN(3, 0) == [(0, 0)]
assert diop_DN(-17, -5) == []
assert diop_DN(-19, 23) == [(2, 1)]
assert diop_DN(-13, 17) == [(2, 1)]
assert diop_DN(-15, 13) == []
assert diop_DN(0, 5) == []
assert diop_DN(0, 9) == [(3, t)]
assert diop_DN(9, 0) == [(3*t, t)]
assert diop_DN(16, 24) == []
assert diop_DN(9, 180) == [(18, 4)]
assert diop_DN(9, -180) == [(12, 6)]
assert diop_DN(7, 0) == [(0, 0)]
# When equation is x**2 + y**2 = N
# Solutions are interchangeable
assert diop_DN(-1, 5) == [(2, 1), (1, 2)]
assert diop_DN(-1, 169) == [(12, 5), (5, 12), (13, 0), (0, 13)]
# D > 0 and D is not a square
# N = 1
assert diop_DN(13, 1) == [(649, 180)]
assert diop_DN(980, 1) == [(51841, 1656)]
assert diop_DN(981, 1) == [(158070671986249, 5046808151700)]
assert diop_DN(986, 1) == [(49299, 1570)]
assert diop_DN(991, 1) == [(379516400906811930638014896080, 12055735790331359447442538767)]
assert diop_DN(17, 1) == [(33, 8)]
assert diop_DN(19, 1) == [(170, 39)]
# N = -1
assert diop_DN(13, -1) == [(18, 5)]
assert diop_DN(991, -1) == []
assert diop_DN(41, -1) == [(32, 5)]
assert diop_DN(290, -1) == [(17, 1)]
assert diop_DN(21257, -1) == [(13913102721304, 95427381109)]
assert diop_DN(32, -1) == []
# |N| > 1
# Some tests were created using calculator at
# http://www.numbertheory.org/php/patz.html
assert diop_DN(13, -4) == [(3, 1), (393, 109), (36, 10)]
# Source I referred returned (3, 1), (393, 109) and (-3, 1) as fundamental solutions
# So (-3, 1) and (393, 109) should be in the same equivalent class
assert equivalent(-3, 1, 393, 109, 13, -4) == True
assert diop_DN(13, 27) == [(220, 61), (40, 11), (768, 213), (12, 3)]
assert set(diop_DN(157, 12)) == {(13, 1), (10663, 851), (579160, 46222),
(483790960, 38610722), (26277068347, 2097138361),
(21950079635497, 1751807067011)}
assert diop_DN(13, 25) == [(3245, 900)]
assert diop_DN(192, 18) == []
assert diop_DN(23, 13) == [(-6, 1), (6, 1)]
assert diop_DN(167, 2) == [(13, 1)]
assert diop_DN(167, -2) == []
assert diop_DN(123, -2) == [(11, 1)]
# One calculator returned [(11, 1), (-11, 1)] but both of these are in
# the same equivalence class
assert equivalent(11, 1, -11, 1, 123, -2)
assert diop_DN(123, -23) == [(-10, 1), (10, 1)]
assert diop_DN(0, 0, t) == [(0, t)]
assert diop_DN(0, -1, t) == []
def test_bf_pell():
assert diop_bf_DN(13, -4) == [(3, 1), (-3, 1), (36, 10)]
assert diop_bf_DN(13, 27) == [(12, 3), (-12, 3), (40, 11), (-40, 11)]
assert diop_bf_DN(167, -2) == []
assert diop_bf_DN(1729, 1) == [(44611924489705, 1072885712316)]
assert diop_bf_DN(89, -8) == [(9, 1), (-9, 1)]
assert diop_bf_DN(21257, -1) == [(13913102721304, 95427381109)]
assert diop_bf_DN(340, -4) == [(756, 41)]
assert diop_bf_DN(-1, 0, t) == [(0, 0)]
assert diop_bf_DN(0, 0, t) == [(0, t)]
assert diop_bf_DN(4, 0, t) == [(2*t, t), (-2*t, t)]
assert diop_bf_DN(3, 0, t) == [(0, 0)]
assert diop_bf_DN(1, -2, t) == []
def test_length():
assert length(2, 1, 0) == 1
assert length(-2, 4, 5) == 3
assert length(-5, 4, 17) == 4
assert length(0, 4, 13) == 6
assert length(7, 13, 11) == 23
assert length(1, 6, 4) == 2
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
simplified = diop_simplify(eq.subs(zip((x, y), (u, v))))
coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X*Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, 1]:
if term not in coeff.keys():
coeff[term] = 0
if coeff[X**2] != 0:
return divisible(coeff[Y**2], coeff[X**2]) and \
divisible(coeff[1], coeff[X**2])
return True
def test_transformation_to_pell():
assert is_pell_transformation_ok(-13*x**2 - 7*x*y + y**2 + 2*x - 2*y - 14)
assert is_pell_transformation_ok(-17*x**2 + 19*x*y - 7*y**2 - 5*x - 13*y - 23)
assert is_pell_transformation_ok(x**2 - y**2 + 17)
assert is_pell_transformation_ok(-x**2 + 7*y**2 - 23)
assert is_pell_transformation_ok(25*x**2 - 45*x*y + 5*y**2 - 5*x - 10*y + 5)
assert is_pell_transformation_ok(190*x**2 + 30*x*y + y**2 - 3*y - 170*x - 130)
assert is_pell_transformation_ok(x**2 - 2*x*y -190*y**2 - 7*y - 23*x - 89)
assert is_pell_transformation_ok(15*x**2 - 9*x*y + 14*y**2 - 23*x - 14*y - 4950)
def test_find_DN():
assert find_DN(x**2 - 2*x - y**2) == (1, 1)
assert find_DN(x**2 - 3*y**2 - 5) == (3, 5)
assert find_DN(x**2 - 2*x*y - 4*y**2 - 7) == (5, 7)
assert find_DN(4*x**2 - 8*x*y - y**2 - 9) == (20, 36)
assert find_DN(7*x**2 - 2*x*y - y**2 - 12) == (8, 84)
assert find_DN(-3*x**2 + 4*x*y -y**2) == (1, 0)
assert find_DN(-13*x**2 - 7*x*y + y**2 + 2*x - 2*y -14) == (101, -7825480)
def test_ldescent():
# Equations which have solutions
u = ([(13, 23), (3, -11), (41, -113), (4, -7), (-7, 4), (91, -3), (1, 1), (1, -1),
(4, 32), (17, 13), (123689, 1), (19, -570)])
for a, b in u:
w, x, y = ldescent(a, b)
assert a*x**2 + b*y**2 == w**2
assert ldescent(-1, -1) is None
def test_diop_ternary_quadratic_normal():
assert check_solutions(234*x**2 - 65601*y**2 - z**2)
assert check_solutions(23*x**2 + 616*y**2 - z**2)
assert check_solutions(5*x**2 + 4*y**2 - z**2)
assert check_solutions(3*x**2 + 6*y**2 - 3*z**2)
assert check_solutions(x**2 + 3*y**2 - z**2)
assert check_solutions(4*x**2 + 5*y**2 - z**2)
assert check_solutions(x**2 + y**2 - z**2)
assert check_solutions(16*x**2 + y**2 - 25*z**2)
assert check_solutions(6*x**2 - y**2 + 10*z**2)
assert check_solutions(213*x**2 + 12*y**2 - 9*z**2)
assert check_solutions(34*x**2 - 3*y**2 - 301*z**2)
assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2)
def is_normal_transformation_ok(eq):
A = transformation_to_normal(eq)
X, Y, Z = A*Matrix([x, y, z])
simplified = diop_simplify(eq.subs(zip((x, y, z), (X, Y, Z))))
coeff = dict([reversed(t.as_independent(*[X, Y, Z])) for t in simplified.args])
for term in [X*Y, Y*Z, X*Z]:
if term in coeff.keys():
return False
return True
def test_transformation_to_normal():
assert is_normal_transformation_ok(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z)
assert is_normal_transformation_ok(x**2 + 3*y**2 - 100*z**2)
assert is_normal_transformation_ok(x**2 + 23*y*z)
assert is_normal_transformation_ok(3*y**2 - 100*z**2 - 12*x*y)
assert is_normal_transformation_ok(x**2 + 23*x*y - 34*y*z + 12*x*z)
assert is_normal_transformation_ok(z**2 + 34*x*y - 23*y*z + x*z)
assert is_normal_transformation_ok(x**2 + y**2 + z**2 - x*y - y*z - x*z)
assert is_normal_transformation_ok(x**2 + 2*y*z + 3*z**2)
assert is_normal_transformation_ok(x*y + 2*x*z + 3*y*z)
assert is_normal_transformation_ok(2*x*z + 3*y*z)
def test_diop_ternary_quadratic():
assert check_solutions(2*x**2 + z**2 + y**2 - 4*x*y)
assert check_solutions(x**2 - y**2 - z**2 - x*y - y*z)
assert check_solutions(3*x**2 - x*y - y*z - x*z)
assert check_solutions(x**2 - y*z - x*z)
assert check_solutions(5*x**2 - 3*x*y - x*z)
assert check_solutions(4*x**2 - 5*y**2 - x*z)
assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z)
assert check_solutions(8*x**2 - 12*y*z)
assert check_solutions(45*x**2 - 7*y**2 - 8*x*y - z**2)
assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y -8*x*y)
assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 17*y*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - x*y - 16*y*z + 12*x*z)
assert check_solutions(x**2 + 3*y**2 + z**2 - 13*x*y - 16*y*z + 12*x*z)
assert check_solutions(x*y - 7*y*z + 13*x*z)
assert diop_ternary_quadratic_normal(x**2 + y**2 + z**2) == (None, None, None)
assert diop_ternary_quadratic_normal(x**2 + y**2) is None
raises(ValueError, lambda:
_diop_ternary_quadratic_normal((x, y, z),
{x*y: 1, x**2: 2, y**2: 3, z**2: 0}))
eq = -2*x*y - 6*x*z + 7*y**2 - 3*y*z + 4*z**2
assert diop_ternary_quadratic(eq) == (7, 2, 0)
assert diop_ternary_quadratic_normal(4*x**2 + 5*y**2 - z**2) == \
(1, 0, 2)
assert diop_ternary_quadratic(x*y + 2*y*z) == \
(-2, 0, n1)
eq = -5*x*y - 8*x*z - 3*y*z + 8*z**2
assert parametrize_ternary_quadratic(eq) == \
(8*p**2 - 3*p*q, -8*p*q + 8*q**2, 5*p*q)
# this cannot be tested with diophantine because it will
# factor into a product
assert diop_solve(x*y + 2*y*z) == (-2*p*q, -n1*p**2 + p**2, p*q)
def test_square_factor():
assert square_factor(1) == square_factor(-1) == 1
assert square_factor(0) == 1
assert square_factor(5) == square_factor(-5) == 1
assert square_factor(4) == square_factor(-4) == 2
assert square_factor(12) == square_factor(-12) == 2
assert square_factor(6) == 1
assert square_factor(18) == 3
assert square_factor(52) == 2
assert square_factor(49) == 7
assert square_factor(392) == 14
assert square_factor(factorint(-12)) == 2
def test_parametrize_ternary_quadratic():
assert check_solutions(x**2 + y**2 - z**2)
assert check_solutions(x**2 + 2*x*y + z**2)
assert check_solutions(234*x**2 - 65601*y**2 - z**2)
assert check_solutions(3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z)
assert check_solutions(x**2 - y**2 - z**2)
assert check_solutions(x**2 - 49*y**2 - z**2 + 13*z*y - 8*x*y)
assert check_solutions(8*x*y + z**2)
assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2)
assert check_solutions(236*x**2 - 225*y**2 - 11*x*y - 13*y*z - 17*x*z)
assert check_solutions(90*x**2 + 3*y**2 + 5*x*y + 2*z*y + 5*x*z)
assert check_solutions(124*x**2 - 30*y**2 - 7729*z**2)
def test_no_square_ternary_quadratic():
assert check_solutions(2*x*y + y*z - 3*x*z)
assert check_solutions(189*x*y - 345*y*z - 12*x*z)
assert check_solutions(23*x*y + 34*y*z)
assert check_solutions(x*y + y*z + z*x)
assert check_solutions(23*x*y + 23*y*z + 23*x*z)
def test_descent():
u = ([(13, 23), (3, -11), (41, -113), (91, -3), (1, 1), (1, -1), (17, 13), (123689, 1), (19, -570)])
for a, b in u:
w, x, y = descent(a, b)
assert a*x**2 + b*y**2 == w**2
# the docstring warns against bad input, so these are expected results
# - can't both be negative
raises(TypeError, lambda: descent(-1, -3))
# A can't be zero unless B != 1
raises(ZeroDivisionError, lambda: descent(0, 3))
# supposed to be square-free
raises(TypeError, lambda: descent(4, 3))
def test_diophantine():
assert check_solutions((x - y)*(y - z)*(z - x))
assert check_solutions((x - y)*(x**2 + y**2 - z**2))
assert check_solutions((x - 3*y + 7*z)*(x**2 + y**2 - z**2))
assert check_solutions(x**2 - 3*y**2 - 1)
assert check_solutions(y**2 + 7*x*y)
assert check_solutions(x**2 - 3*x*y + y**2)
assert check_solutions(z*(x**2 - y**2 - 15))
assert check_solutions(x*(2*y - 2*z + 5))
assert check_solutions((x**2 - 3*y**2 - 1)*(x**2 - y**2 - 15))
assert check_solutions((x**2 - 3*y**2 - 1)*(y - 7*z))
assert check_solutions((x**2 + y**2 - z**2)*(x - 7*y - 3*z + 4*w))
# Following test case caused problems in parametric representation
# But this can be solved by factoring out y.
# No need to use methods for ternary quadratic equations.
assert check_solutions(y**2 - 7*x*y + 4*y*z)
assert check_solutions(x**2 - 2*x + 1)
assert diophantine(x - y) == diophantine(Eq(x, y))
# 18196
eq = x**4 + y**4 - 97
assert diophantine(eq, permute=True) == diophantine(-eq, permute=True)
assert diophantine(3*x*pi - 2*y*pi) == {(2*t_0, 3*t_0)}
eq = x**2 + y**2 + z**2 - 14
base_sol = {(1, 2, 3)}
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
assert diophantine(x**2 + x*Rational(15, 14) - 3) == set()
# test issue 11049
eq = 92*x**2 - 99*y**2 - z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
{(9, 7, 51)}
assert diophantine(eq) == {(
891*p**2 + 9*q**2, -693*p**2 - 102*p*q + 7*q**2,
5049*p**2 - 1386*p*q - 51*q**2)}
eq = 2*x**2 + 2*y**2 - z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
{(1, 1, 2)}
assert diophantine(eq) == {(
2*p**2 - q**2, -2*p**2 + 4*p*q - q**2,
4*p**2 - 4*p*q + 2*q**2)}
eq = 411*x**2+57*y**2-221*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
{(2021, 2645, 3066)}
assert diophantine(eq) == \
{(115197*p**2 - 446641*q**2, -150765*p**2 + 1355172*p*q -
584545*q**2, 174762*p**2 - 301530*p*q + 677586*q**2)}
eq = 573*x**2+267*y**2-984*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
{(49, 233, 127)}
assert diophantine(eq) == \
{(4361*p**2 - 16072*q**2, -20737*p**2 + 83312*p*q - 76424*q**2,
11303*p**2 - 41474*p*q + 41656*q**2)}
# this produces factors during reconstruction
eq = x**2 + 3*y**2 - 12*z**2
coeff = eq.as_coefficients_dict()
assert _diop_ternary_quadratic_normal((x, y, z), coeff) == \
{(0, 2, 1)}
assert diophantine(eq) == \
{(24*p*q, 2*p**2 - 24*q**2, p**2 + 12*q**2)}
# solvers have not been written for every type
raises(NotImplementedError, lambda: diophantine(x*y**2 + 1))
# rational expressions
assert diophantine(1/x) == set()
assert diophantine(1/x + 1/y - S.Half) == {(6, 3), (-2, 1), (4, 4), (1, -2), (3, 6)}
assert diophantine(x**2 + y**2 +3*x- 5, permute=True) == \
{(-1, 1), (-4, -1), (1, -1), (1, 1), (-4, 1), (-1, -1), (4, 1), (4, -1)}
#test issue 18186
assert diophantine(y**4 + x**4 - 2**4 - 3**4, syms=(x, y), permute=True) == \
{(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)}
assert diophantine(y**4 + x**4 - 2**4 - 3**4, syms=(y, x), permute=True) == \
{(-3, -2), (-3, 2), (-2, -3), (-2, 3), (2, -3), (2, 3), (3, -2), (3, 2)}
# issue 18122
assert check_solutions(x**2-y)
assert check_solutions(y**2-x)
assert diophantine((x**2-y), t) == {(t, t**2)}
assert diophantine((y**2-x), t) == {(t**2, -t)}
def test_general_pythagorean():
from sympy.abc import a, b, c, d, e
assert check_solutions(a**2 + b**2 + c**2 - d**2)
assert check_solutions(a**2 + 4*b**2 + 4*c**2 - d**2)
assert check_solutions(9*a**2 + 4*b**2 + 4*c**2 - d**2)
assert check_solutions(9*a**2 + 4*b**2 - 25*d**2 + 4*c**2 )
assert check_solutions(9*a**2 - 16*d**2 + 4*b**2 + 4*c**2)
assert check_solutions(-e**2 + 9*a**2 + 4*b**2 + 4*c**2 + 25*d**2)
assert check_solutions(16*a**2 - b**2 + 9*c**2 + d**2 + 25*e**2)
assert GeneralPythagorean(a**2 + b**2 + c**2 - d**2).solve(parameters=[x, y, z]) == \
{(x**2 + y**2 - z**2, 2*x*z, 2*y*z, x**2 + y**2 + z**2)}
def test_diop_general_sum_of_squares_quick():
for i in range(3, 10):
assert check_solutions(sum(i**2 for i in symbols(':%i' % i)) - i)
assert diop_general_sum_of_squares(x**2 + y**2 - 2) is None
assert diop_general_sum_of_squares(x**2 + y**2 + z**2 + 2) == set()
eq = x**2 + y**2 + z**2 - (1 + 4 + 9)
assert diop_general_sum_of_squares(eq) == \
{(1, 2, 3)}
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 1313
assert len(diop_general_sum_of_squares(eq, 3)) == 3
# issue 11016
var = symbols(':5') + (symbols('6', negative=True),)
eq = Add(*[i**2 for i in var]) - 112
base_soln = {(0, 1, 1, 5, 6, -7), (1, 1, 1, 3, 6, -8), (2, 3, 3, 4, 5, -7), (0, 1, 1, 1, 3, -10),
(0, 0, 4, 4, 4, -8), (1, 2, 3, 3, 5, -8), (0, 1, 2, 3, 7, -7), (2, 2, 4, 4, 6, -6),
(1, 1, 3, 4, 6, -7), (0, 2, 3, 3, 3, -9), (0, 0, 2, 2, 2, -10), (1, 1, 2, 3, 4, -9),
(0, 1, 1, 2, 5, -9), (0, 0, 2, 6, 6, -6), (1, 3, 4, 5, 5, -6), (0, 2, 2, 2, 6, -8),
(0, 3, 3, 3, 6, -7), (0, 2, 3, 5, 5, -7), (0, 1, 5, 5, 5, -6)}
assert diophantine(eq) == base_soln
assert len(diophantine(eq, permute=True)) == 196800
# handle negated squares with signsimp
assert diophantine(12 - x**2 - y**2 - z**2) == {(2, 2, 2)}
# diophantine handles simplification, so classify_diop should
# not have to look for additional patterns that are removed
# by diophantine
eq = a**2 + b**2 + c**2 + d**2 - 4
raises(NotImplementedError, lambda: classify_diop(-eq))
def test_issue_23807():
# fixes recursion error
eq = x**2 + y**2 + z**2 - 1000000
base_soln = {(0, 0, 1000), (0, 352, 936), (480, 600, 640), (24, 640, 768), (192, 640, 744),
(192, 480, 856), (168, 224, 960), (0, 600, 800), (280, 576, 768), (152, 480, 864),
(0, 280, 960), (352, 360, 864), (424, 480, 768), (360, 480, 800), (224, 600, 768),
(96, 360, 928), (168, 576, 800), (96, 480, 872)}
assert diophantine(eq) == base_soln
def test_diop_partition():
for n in [8, 10]:
for k in range(1, 8):
for p in partition(n, k):
assert len(p) == k
assert [p for p in partition(3, 5)] == []
assert [list(p) for p in partition(3, 5, 1)] == [
[0, 0, 0, 0, 3], [0, 0, 0, 1, 2], [0, 0, 1, 1, 1]]
assert list(partition(0)) == [()]
assert list(partition(1, 0)) == [()]
assert [list(i) for i in partition(3)] == [[1, 1, 1], [1, 2], [3]]
def test_prime_as_sum_of_two_squares():
for i in [5, 13, 17, 29, 37, 41, 2341, 3557, 34841, 64601]:
a, b = prime_as_sum_of_two_squares(i)
assert a**2 + b**2 == i
assert prime_as_sum_of_two_squares(7) is None
ans = prime_as_sum_of_two_squares(800029)
assert ans == (450, 773) and type(ans[0]) is int
def test_sum_of_three_squares():
for i in [0, 1, 2, 34, 123, 34304595905, 34304595905394941, 343045959052344,
800, 801, 802, 803, 804, 805, 806]:
a, b, c = sum_of_three_squares(i)
assert a**2 + b**2 + c**2 == i
assert sum_of_three_squares(7) is None
assert sum_of_three_squares((4**5)*15) is None
assert sum_of_three_squares(25) == (5, 0, 0)
assert sum_of_three_squares(4) == (0, 0, 2)
def test_sum_of_four_squares():
from sympy.core.random import randint
# this should never fail
n = randint(1, 100000000000000)
assert sum(i**2 for i in sum_of_four_squares(n)) == n
assert sum_of_four_squares(0) == (0, 0, 0, 0)
assert sum_of_four_squares(14) == (0, 1, 2, 3)
assert sum_of_four_squares(15) == (1, 1, 2, 3)
assert sum_of_four_squares(18) == (1, 2, 2, 3)
assert sum_of_four_squares(19) == (0, 1, 3, 3)
assert sum_of_four_squares(48) == (0, 4, 4, 4)
def test_power_representation():
tests = [(1729, 3, 2), (234, 2, 4), (2, 1, 2), (3, 1, 3), (5, 2, 2), (12352, 2, 4),
(32760, 2, 3)]
for test in tests:
n, p, k = test
f = power_representation(n, p, k)
while True:
try:
l = next(f)
assert len(l) == k
chk_sum = 0
for l_i in l:
chk_sum = chk_sum + l_i**p
assert chk_sum == n
except StopIteration:
break
assert list(power_representation(20, 2, 4, True)) == \
[(1, 1, 3, 3), (0, 0, 2, 4)]
raises(ValueError, lambda: list(power_representation(1.2, 2, 2)))
raises(ValueError, lambda: list(power_representation(2, 0, 2)))
raises(ValueError, lambda: list(power_representation(2, 2, 0)))
assert list(power_representation(-1, 2, 2)) == []
assert list(power_representation(1, 1, 1)) == [(1,)]
assert list(power_representation(3, 2, 1)) == []
assert list(power_representation(4, 2, 1)) == [(2,)]
assert list(power_representation(3**4, 4, 6, zeros=True)) == \
[(1, 2, 2, 2, 2, 2), (0, 0, 0, 0, 0, 3)]
assert list(power_representation(3**4, 4, 5, zeros=False)) == []
assert list(power_representation(-2, 3, 2)) == [(-1, -1)]
assert list(power_representation(-2, 4, 2)) == []
assert list(power_representation(0, 3, 2, True)) == [(0, 0)]
assert list(power_representation(0, 3, 2, False)) == []
# when we are dealing with squares, do feasibility checks
assert len(list(power_representation(4**10*(8*10 + 7), 2, 3))) == 0
# there will be a recursion error if these aren't recognized
big = 2**30
for i in [13, 10, 7, 5, 4, 2, 1]:
assert list(sum_of_powers(big, 2, big - i)) == []
def test_assumptions():
"""
Test whether diophantine respects the assumptions.
"""
#Test case taken from the below so question regarding assumptions in diophantine module
#https://stackoverflow.com/questions/23301941/how-can-i-declare-natural-symbols-with-sympy
m, n = symbols('m n', integer=True, positive=True)
diof = diophantine(n**2 + m*n - 500)
assert diof == {(5, 20), (40, 10), (95, 5), (121, 4), (248, 2), (499, 1)}
a, b = symbols('a b', integer=True, positive=False)
diof = diophantine(a*b + 2*a + 3*b - 6)
assert diof == {(-15, -3), (-9, -4), (-7, -5), (-6, -6), (-5, -8), (-4, -14)}
def check_solutions(eq):
"""
Determines whether solutions returned by diophantine() satisfy the original
equation. Hope to generalize this so we can remove functions like check_ternay_quadratic,
check_solutions_normal, check_solutions()
"""
s = diophantine(eq)
factors = Mul.make_args(eq)
var = list(eq.free_symbols)
var.sort(key=default_sort_key)
while s:
solution = s.pop()
for f in factors:
if diop_simplify(f.subs(zip(var, solution))) == 0:
break
else:
return False
return True
def test_diopcoverage():
eq = (2*x + y + 1)**2
assert diop_solve(eq) == {(t_0, -2*t_0 - 1)}
eq = 2*x**2 + 6*x*y + 12*x + 4*y**2 + 18*y + 18
assert diop_solve(eq) == {(t, -t - 3), (2*t - 3, -t)}
assert diop_quadratic(x + y**2 - 3) == {(-t**2 + 3, -t)}
assert diop_linear(x + y - 3) == (t_0, 3 - t_0)
assert base_solution_linear(0, 1, 2, t=None) == (0, 0)
ans = (3*t - 1, -2*t + 1)
assert base_solution_linear(4, 8, 12, t) == ans
assert base_solution_linear(4, 8, 12, t=None) == tuple(_.subs(t, 0) for _ in ans)
assert cornacchia(1, 1, 20) is None
assert cornacchia(1, 1, 5) == {(2, 1)}
assert cornacchia(1, 2, 17) == {(3, 2)}
raises(ValueError, lambda: reconstruct(4, 20, 1))
assert gaussian_reduce(4, 1, 3) == (1, 1)
eq = -w**2 - x**2 - y**2 + z**2
assert diop_general_pythagorean(eq) == \
diop_general_pythagorean(-eq) == \
(m1**2 + m2**2 - m3**2, 2*m1*m3,
2*m2*m3, m1**2 + m2**2 + m3**2)
assert len(check_param(S(3) + x/3, S(4) + x/2, S(2), [x])) == 0
assert len(check_param(Rational(3, 2), S(4) + x, S(2), [x])) == 0
assert len(check_param(S(4) + x, Rational(3, 2), S(2), [x])) == 0
assert _nint_or_floor(16, 10) == 2
assert _odd(1) == (not _even(1)) == True
assert _odd(0) == (not _even(0)) == False
assert _remove_gcd(2, 4, 6) == (1, 2, 3)
raises(TypeError, lambda: _remove_gcd((2, 4, 6)))
assert sqf_normal(2*3**2*5, 2*5*11, 2*7**2*11) == \
(11, 1, 5)
# it's ok if these pass some day when the solvers are implemented
raises(NotImplementedError, lambda: diophantine(x**2 + y**2 + x*y + 2*y*z - 12))
raises(NotImplementedError, lambda: diophantine(x**3 + y**2))
assert diop_quadratic(x**2 + y**2 - 1**2 - 3**4) == \
{(-9, -1), (-9, 1), (-1, -9), (-1, 9), (1, -9), (1, 9), (9, -1), (9, 1)}
def test_holzer():
# if the input is good, don't let it diverge in holzer()
# (but see test_fail_holzer below)
assert holzer(2, 7, 13, 4, 79, 23) == (2, 7, 13)
# None in uv condition met; solution is not Holzer reduced
# so this will hopefully change but is here for coverage
assert holzer(2, 6, 2, 1, 1, 10) == (2, 6, 2)
raises(ValueError, lambda: holzer(2, 7, 14, 4, 79, 23))
@XFAIL
def test_fail_holzer():
eq = lambda x, y, z: a*x**2 + b*y**2 - c*z**2
a, b, c = 4, 79, 23
x, y, z = xyz = 26, 1, 11
X, Y, Z = ans = 2, 7, 13
assert eq(*xyz) == 0
assert eq(*ans) == 0
assert max(a*x**2, b*y**2, c*z**2) <= a*b*c
assert max(a*X**2, b*Y**2, c*Z**2) <= a*b*c
h = holzer(x, y, z, a, b, c)
assert h == ans # it would be nice to get the smaller soln
def test_issue_9539():
assert diophantine(6*w + 9*y + 20*x - z) == \
{(t_0, t_1, t_1 + t_2, 6*t_0 + 29*t_1 + 9*t_2)}
def test_issue_8943():
assert diophantine(
3*(x**2 + y**2 + z**2) - 14*(x*y + y*z + z*x)) == \
{(0, 0, 0)}
def test_diop_sum_of_even_powers():
eq = x**4 + y**4 + z**4 - 2673
assert diop_solve(eq) == {(3, 6, 6), (2, 4, 7)}
assert diop_general_sum_of_even_powers(eq, 2) == {(3, 6, 6), (2, 4, 7)}
raises(NotImplementedError, lambda: diop_general_sum_of_even_powers(-eq, 2))
neg = symbols('neg', negative=True)
eq = x**4 + y**4 + neg**4 - 2673
assert diop_general_sum_of_even_powers(eq) == {(-3, 6, 6)}
assert diophantine(x**4 + y**4 + 2) == set()
assert diop_general_sum_of_even_powers(x**4 + y**4 - 2, limit=0) == set()
def test_sum_of_squares_powers():
tru = {(0, 0, 1, 1, 11), (0, 0, 5, 7, 7), (0, 1, 3, 7, 8), (0, 1, 4, 5, 9), (0, 3, 4, 7, 7), (0, 3, 5, 5, 8),
(1, 1, 2, 6, 9), (1, 1, 6, 6, 7), (1, 2, 3, 3, 10), (1, 3, 4, 4, 9), (1, 5, 5, 6, 6), (2, 2, 3, 5, 9),
(2, 3, 5, 6, 7), (3, 3, 4, 5, 8)}
eq = u**2 + v**2 + x**2 + y**2 + z**2 - 123
ans = diop_general_sum_of_squares(eq, oo) # allow oo to be used
assert len(ans) == 14
assert ans == tru
raises(ValueError, lambda: list(sum_of_squares(10, -1)))
assert list(sum_of_squares(-10, 2)) == []
assert list(sum_of_squares(2, 3)) == []
assert list(sum_of_squares(0, 3, True)) == [(0, 0, 0)]
assert list(sum_of_squares(0, 3)) == []
assert list(sum_of_squares(4, 1)) == [(2,)]
assert list(sum_of_squares(5, 1)) == []
assert list(sum_of_squares(50, 2)) == [(5, 5), (1, 7)]
assert list(sum_of_squares(11, 5, True)) == [
(1, 1, 1, 2, 2), (0, 0, 1, 1, 3)]
assert list(sum_of_squares(8, 8)) == [(1, 1, 1, 1, 1, 1, 1, 1)]
assert [len(list(sum_of_squares(i, 5, True))) for i in range(30)] == [
1, 1, 1, 1, 2,
2, 1, 1, 2, 2,
2, 2, 2, 3, 2,
1, 3, 3, 3, 3,
4, 3, 3, 2, 2,
4, 4, 4, 4, 5]
assert [len(list(sum_of_squares(i, 5))) for i in range(30)] == [
0, 0, 0, 0, 0,
1, 0, 0, 1, 0,
0, 1, 0, 1, 1,
0, 1, 1, 0, 1,
2, 1, 1, 1, 1,
1, 1, 1, 1, 3]
for i in range(30):
s1 = set(sum_of_squares(i, 5, True))
assert not s1 or all(sum(j**2 for j in t) == i for t in s1)
s2 = set(sum_of_squares(i, 5))
assert all(sum(j**2 for j in t) == i for t in s2)
raises(ValueError, lambda: list(sum_of_powers(2, -1, 1)))
raises(ValueError, lambda: list(sum_of_powers(2, 1, -1)))
assert list(sum_of_powers(-2, 3, 2)) == [(-1, -1)]
assert list(sum_of_powers(-2, 4, 2)) == []
assert list(sum_of_powers(2, 1, 1)) == [(2,)]
assert list(sum_of_powers(2, 1, 3, True)) == [(0, 0, 2), (0, 1, 1)]
assert list(sum_of_powers(5, 1, 2, True)) == [(0, 5), (1, 4), (2, 3)]
assert list(sum_of_powers(6, 2, 2)) == []
assert list(sum_of_powers(3**5, 3, 1)) == []
assert list(sum_of_powers(3**6, 3, 1)) == [(9,)] and (9**3 == 3**6)
assert list(sum_of_powers(2**1000, 5, 2)) == []
def test__can_do_sum_of_squares():
assert _can_do_sum_of_squares(3, -1) is False
assert _can_do_sum_of_squares(-3, 1) is False
assert _can_do_sum_of_squares(0, 1)
assert _can_do_sum_of_squares(4, 1)
assert _can_do_sum_of_squares(1, 2)
assert _can_do_sum_of_squares(2, 2)
assert _can_do_sum_of_squares(3, 2) is False
def test_diophantine_permute_sign():
from sympy.abc import a, b, c, d, e
eq = a**4 + b**4 - (2**4 + 3**4)
base_sol = {(2, 3)}
assert diophantine(eq) == base_sol
complete_soln = set(signed_permutations(base_sol.pop()))
assert diophantine(eq, permute=True) == complete_soln
eq = a**2 + b**2 + c**2 + d**2 + e**2 - 234
assert len(diophantine(eq)) == 35
assert len(diophantine(eq, permute=True)) == 62000
soln = {(-1, -1), (-1, 2), (1, -2), (1, 1)}
assert diophantine(10*x**2 + 12*x*y + 12*y**2 - 34, permute=True) == soln
@XFAIL
def test_not_implemented():
eq = x**2 + y**4 - 1**2 - 3**4
assert diophantine(eq, syms=[x, y]) == {(9, 1), (1, 3)}
def test_issue_9538():
eq = x - 3*y + 2
assert diophantine(eq, syms=[y,x]) == {(t_0, 3*t_0 - 2)}
raises(TypeError, lambda: diophantine(eq, syms={y, x}))
def test_ternary_quadratic():
# solution with 3 parameters
s = diophantine(2*x**2 + y**2 - 2*z**2)
p, q, r = ordered(S(s).free_symbols)
assert s == {(
p**2 - 2*q**2,
-2*p**2 + 4*p*q - 4*p*r - 4*q**2,
p**2 - 4*p*q + 2*q**2 - 4*q*r)}
# solution with Mul in solution
s = diophantine(x**2 + 2*y**2 - 2*z**2)
assert s == {(4*p*q, p**2 - 2*q**2, p**2 + 2*q**2)}
# solution with no Mul in solution
s = diophantine(2*x**2 + 2*y**2 - z**2)
assert s == {(2*p**2 - q**2, -2*p**2 + 4*p*q - q**2,
4*p**2 - 4*p*q + 2*q**2)}
# reduced form when parametrized
s = diophantine(3*x**2 + 72*y**2 - 27*z**2)
assert s == {(24*p**2 - 9*q**2, 6*p*q, 8*p**2 + 3*q**2)}
assert parametrize_ternary_quadratic(
3*x**2 + 2*y**2 - z**2 - 2*x*y + 5*y*z - 7*y*z) == (
2*p**2 - 2*p*q - q**2, 2*p**2 + 2*p*q - q**2, 2*p**2 -
2*p*q + 3*q**2)
assert parametrize_ternary_quadratic(
124*x**2 - 30*y**2 - 7729*z**2) == (
-1410*p**2 - 363263*q**2, 2700*p**2 + 30916*p*q -
695610*q**2, -60*p**2 + 5400*p*q + 15458*q**2)
def test_diophantine_solution_set():
s1 = DiophantineSolutionSet([], [])
assert set(s1) == set()
assert s1.symbols == ()
assert s1.parameters == ()
raises(ValueError, lambda: s1.add((x,)))
assert list(s1.dict_iterator()) == []
s2 = DiophantineSolutionSet([x, y], [t, u])
assert s2.symbols == (x, y)
assert s2.parameters == (t, u)
raises(ValueError, lambda: s2.add((1,)))
s2.add((3, 4))
assert set(s2) == {(3, 4)}
s2.update((3, 4), (-1, u))
assert set(s2) == {(3, 4), (-1, u)}
raises(ValueError, lambda: s1.update(s2))
assert list(s2.dict_iterator()) == [{x: -1, y: u}, {x: 3, y: 4}]
s3 = DiophantineSolutionSet([x, y, z], [t, u])
assert len(s3.parameters) == 2
s3.add((t**2 + u, t - u, 1))
assert set(s3) == {(t**2 + u, t - u, 1)}
assert s3.subs(t, 2) == {(u + 4, 2 - u, 1)}
assert s3(2) == {(u + 4, 2 - u, 1)}
assert s3.subs({t: 7, u: 8}) == {(57, -1, 1)}
assert s3(7, 8) == {(57, -1, 1)}
assert s3.subs({t: 5}) == {(u + 25, 5 - u, 1)}
assert s3(5) == {(u + 25, 5 - u, 1)}
assert s3.subs(u, -3) == {(t**2 - 3, t + 3, 1)}
assert s3(None, -3) == {(t**2 - 3, t + 3, 1)}
assert s3.subs({t: 2, u: 8}) == {(12, -6, 1)}
assert s3(2, 8) == {(12, -6, 1)}
assert s3.subs({t: 5, u: -3}) == {(22, 8, 1)}
assert s3(5, -3) == {(22, 8, 1)}
raises(ValueError, lambda: s3.subs(x=1))
raises(ValueError, lambda: s3.subs(1, 2, 3))
raises(ValueError, lambda: s3.add(()))
raises(ValueError, lambda: s3.add((1, 2, 3, 4)))
raises(ValueError, lambda: s3.add((1, 2)))
raises(ValueError, lambda: s3(1, 2, 3))
raises(TypeError, lambda: s3(t=1))
s4 = DiophantineSolutionSet([x, y], [t, u])
s4.add((t, 11*t))
s4.add((-t, 22*t))
assert s4(0, 0) == {(0, 0)}
def test_quadratic_parameter_passing():
eq = -33*x*y + 3*y**2
solution = BinaryQuadratic(eq).solve(parameters=[t, u])
# test that parameters are passed all the way to the final solution
assert solution == {(t, 11*t), (-t, 22*t)}
assert solution(0, 0) == {(0, 0)}
|
0fdf11c6b5199e024bc56ff23e4a886e78785a8899f03903f9ed7e4fc82bc08c | from sympy.core.function import (Derivative, Function, diff)
from sympy.core.mul import Mul
from sympy.core.numbers import (I, Rational, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.hyperbolic import sinh
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.matrices.dense import Matrix
from sympy.core.containers import Tuple
from sympy.functions import exp, cos, sin, log, Ci, Si, erf, erfi
from sympy.matrices import dotprodsimp, NonSquareMatrixError
from sympy.solvers.ode import dsolve
from sympy.solvers.ode.ode import constant_renumber
from sympy.solvers.ode.subscheck import checksysodesol
from sympy.solvers.ode.systems import (_classify_linear_system, linear_ode_to_matrix,
ODEOrderError, ODENonlinearError, _simpsol,
_is_commutative_anti_derivative, linodesolve,
canonical_odes, dsolve_system, _component_division,
_eqs2dict, _dict2graph)
from sympy.functions import airyai, airybi
from sympy.integrals.integrals import Integral
from sympy.simplify.ratsimp import ratsimp
from sympy.testing.pytest import ON_TRAVIS, raises, slow, skip, XFAIL
C0, C1, C2, C3, C4, C5, C6, C7, C8, C9, C10 = symbols('C0:11')
x = symbols('x')
f = Function('f')
g = Function('g')
h = Function('h')
def test_linear_ode_to_matrix():
f, g, h = symbols("f, g, h", cls=Function)
t = Symbol("t")
funcs = [f(t), g(t), h(t)]
f1 = f(t).diff(t)
g1 = g(t).diff(t)
h1 = h(t).diff(t)
f2 = f(t).diff(t, 2)
g2 = g(t).diff(t, 2)
h2 = h(t).diff(t, 2)
eqs_1 = [Eq(f1, g(t)), Eq(g1, f(t))]
sol_1 = ([Matrix([[1, 0], [0, 1]]), Matrix([[ 0, 1], [1, 0]])], Matrix([[0],[0]]))
assert linear_ode_to_matrix(eqs_1, funcs[:-1], t, 1) == sol_1
eqs_2 = [Eq(f1, f(t) + 2*g(t)), Eq(g1, h(t)), Eq(h1, g(t) + h(t) + f(t))]
sol_2 = ([Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]), Matrix([[1, 2, 0], [ 0, 0, 1], [1, 1, 1]])],
Matrix([[0], [0], [0]]))
assert linear_ode_to_matrix(eqs_2, funcs, t, 1) == sol_2
eqs_3 = [Eq(2*f1 + 3*h1, f(t) + g(t)), Eq(4*h1 + 5*g1, f(t) + h(t)), Eq(5*f1 + 4*g1, g(t) + h(t))]
sol_3 = ([Matrix([[2, 0, 3], [0, 5, 4], [5, 4, 0]]), Matrix([[1, 1, 0], [1, 0, 1], [0, 1, 1]])],
Matrix([[0], [0], [0]]))
assert linear_ode_to_matrix(eqs_3, funcs, t, 1) == sol_3
eqs_4 = [Eq(f2 + h(t), f1 + g(t)), Eq(2*h2 + g2 + g1 + g(t), 0), Eq(3*h1, 4)]
sol_4 = ([Matrix([[1, 0, 0], [0, 1, 2], [0, 0, 0]]), Matrix([[1, 0, 0], [0, -1, 0], [0, 0, -3]]),
Matrix([[0, 1, -1], [0, -1, 0], [0, 0, 0]])], Matrix([[0], [0], [4]]))
assert linear_ode_to_matrix(eqs_4, funcs, t, 2) == sol_4
eqs_5 = [Eq(f2, g(t)), Eq(f1 + g1, f(t))]
raises(ODEOrderError, lambda: linear_ode_to_matrix(eqs_5, funcs[:-1], t, 1))
eqs_6 = [Eq(f1, f(t)**2), Eq(g1, f(t) + g(t))]
raises(ODENonlinearError, lambda: linear_ode_to_matrix(eqs_6, funcs[:-1], t, 1))
def test__classify_linear_system():
x, y, z, w = symbols('x, y, z, w', cls=Function)
t, k, l = symbols('t k l')
x1 = diff(x(t), t)
y1 = diff(y(t), t)
z1 = diff(z(t), t)
w1 = diff(w(t), t)
x2 = diff(x(t), t, t)
y2 = diff(y(t), t, t)
funcs = [x(t), y(t)]
funcs_2 = funcs + [z(t), w(t)]
eqs_1 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * t * x(t) + 3 * y(t) + t))
assert _classify_linear_system(eqs_1, funcs, t) is None
eqs_2 = (5 * (x1**2) + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * t * x(t) + 3 * y(t) + t))
sol2 = {'is_implicit': True,
'canon_eqs': [[Eq(Derivative(x(t), t), -sqrt(-12*x(t)/5 + 6*y(t)/5)),
Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)],
[Eq(Derivative(x(t), t), sqrt(-12*x(t)/5 + 6*y(t)/5)),
Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)]]}
assert _classify_linear_system(eqs_2, funcs, t) == sol2
eqs_2_1 = [Eq(Derivative(x(t), t), -sqrt(-12*x(t)/5 + 6*y(t)/5)),
Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)]
assert _classify_linear_system(eqs_2_1, funcs, t) is None
eqs_2_2 = [Eq(Derivative(x(t), t), sqrt(-12*x(t)/5 + 6*y(t)/5)),
Eq(Derivative(y(t), t), 11*t*x(t)/2 - t/2 - 3*y(t)/2)]
assert _classify_linear_system(eqs_2_2, funcs, t) is None
eqs_3 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * x(t) + 3 * y(t)), (5 * w1 + z(t)), (z1 + w(t)))
answer_3 = {'no_of_equation': 4,
'eq': (12*x(t) - 6*y(t) + 5*Derivative(x(t), t),
-11*x(t) + 3*y(t) + 2*Derivative(y(t), t),
z(t) + 5*Derivative(w(t), t),
w(t) + Derivative(z(t), t)),
'func': [x(t), y(t), z(t), w(t)],
'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1},
'is_linear': True,
'is_constant': True,
'is_homogeneous': True,
'func_coeff': -Matrix([
[Rational(12, 5), Rational(-6, 5), 0, 0],
[Rational(-11, 2), Rational(3, 2), 0, 0],
[0, 0, 0, 1],
[0, 0, Rational(1, 5), 0]]),
'type_of_equation': 'type1',
'is_general': True}
assert _classify_linear_system(eqs_3, funcs_2, t) == answer_3
eqs_4 = (5 * x1 + 12 * x(t) - 6 * (y(t)), (2 * y1 - 11 * x(t) + 3 * y(t)), (z1 - w(t)), (w1 - z(t)))
answer_4 = {'no_of_equation': 4,
'eq': (12 * x(t) - 6 * y(t) + 5 * Derivative(x(t), t),
-11 * x(t) + 3 * y(t) + 2 * Derivative(y(t), t),
-w(t) + Derivative(z(t), t),
-z(t) + Derivative(w(t), t)),
'func': [x(t), y(t), z(t), w(t)],
'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1},
'is_linear': True,
'is_constant': True,
'is_homogeneous': True,
'func_coeff': -Matrix([
[Rational(12, 5), Rational(-6, 5), 0, 0],
[Rational(-11, 2), Rational(3, 2), 0, 0],
[0, 0, 0, -1],
[0, 0, -1, 0]]),
'type_of_equation': 'type1',
'is_general': True}
assert _classify_linear_system(eqs_4, funcs_2, t) == answer_4
eqs_5 = (5*x1 + 12*x(t) - 6*(y(t)) + x2, (2*y1 - 11*x(t) + 3*y(t)), (z1 - w(t)), (w1 - z(t)))
answer_5 = {'no_of_equation': 4, 'eq': (12*x(t) - 6*y(t) + 5*Derivative(x(t), t) + Derivative(x(t), (t, 2)),
-11*x(t) + 3*y(t) + 2*Derivative(y(t), t), -w(t) + Derivative(z(t), t), -z(t) + Derivative(w(t),
t)), 'func': [x(t), y(t), z(t), w(t)], 'order': {x(t): 2, y(t): 1, z(t): 1, w(t): 1}, 'is_linear':
True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type0', 'is_higher_order': True}
assert _classify_linear_system(eqs_5, funcs_2, t) == answer_5
eqs_6 = (Eq(x1, 3*y(t) - 11*z(t)), Eq(y1, 7*z(t) - 3*x(t)), Eq(z1, 11*x(t) - 7*y(t)))
answer_6 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)),
Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1},
'is_linear': True, 'is_constant': True, 'is_homogeneous': True,
'func_coeff': -Matrix([
[ 0, -3, 11],
[ 3, 0, -7],
[-11, 7, 0]]),
'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eqs_6, funcs_2[:-1], t) == answer_6
eqs_7 = (Eq(x1, y(t)), Eq(y1, x(t)))
answer_7 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))),
'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True,
'is_homogeneous': True, 'func_coeff': -Matrix([
[ 0, -1],
[-1, 0]]),
'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eqs_7, funcs, t) == answer_7
eqs_8 = (Eq(x1, 21*x(t)), Eq(y1, 17*x(t) + 3*y(t)), Eq(z1, 5*x(t) + 7*y(t) + 9*z(t)))
answer_8 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 21*x(t)), Eq(Derivative(y(t), t), 17*x(t) + 3*y(t)),
Eq(Derivative(z(t), t), 5*x(t) + 7*y(t) + 9*z(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1},
'is_linear': True, 'is_constant': True, 'is_homogeneous': True,
'func_coeff': -Matrix([
[-21, 0, 0],
[-17, -3, 0],
[ -5, -7, -9]]),
'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eqs_8, funcs_2[:-1], t) == answer_8
eqs_9 = (Eq(x1, 4*x(t) + 5*y(t) + 2*z(t)), Eq(y1, x(t) + 13*y(t) + 9*z(t)), Eq(z1, 32*x(t) + 41*y(t) + 11*z(t)))
answer_9 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 4*x(t) + 5*y(t) + 2*z(t)),
Eq(Derivative(y(t), t), x(t) + 13*y(t) + 9*z(t)), Eq(Derivative(z(t), t), 32*x(t) + 41*y(t) + 11*z(t))),
'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True,
'is_constant': True, 'is_homogeneous': True,
'func_coeff': -Matrix([
[ -4, -5, -2],
[ -1, -13, -9],
[-32, -41, -11]]),
'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eqs_9, funcs_2[:-1], t) == answer_9
eqs_10 = (Eq(3*x1, 4*5*(y(t) - z(t))), Eq(4*y1, 3*5*(z(t) - x(t))), Eq(5*z1, 3*4*(x(t) - y(t))))
answer_10 = {'no_of_equation': 3, 'eq': (Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)),
Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)), Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))),
'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True,
'is_constant': True, 'is_homogeneous': True,
'func_coeff': -Matrix([
[ 0, Rational(-20, 3), Rational(20, 3)],
[Rational(15, 4), 0, Rational(-15, 4)],
[Rational(-12, 5), Rational(12, 5), 0]]),
'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eqs_10, funcs_2[:-1], t) == answer_10
eq11 = (Eq(x1, 3*y(t) - 11*z(t)), Eq(y1, 7*z(t) - 3*x(t)), Eq(z1, 11*x(t) - 7*y(t)))
sol11 = {'no_of_equation': 3, 'eq': (Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)), Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)),
Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))), 'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1},
'is_linear': True, 'is_constant': True, 'is_homogeneous': True, 'func_coeff': -Matrix([
[ 0, -3, 11], [ 3, 0, -7], [-11, 7, 0]]), 'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eq11, funcs_2[:-1], t) == sol11
eq12 = (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t)))
sol12 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), y(t)), Eq(Derivative(y(t), t), x(t))),
'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True,
'is_homogeneous': True, 'func_coeff': -Matrix([
[0, -1],
[-1, 0]]), 'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eq12, [x(t), y(t)], t) == sol12
eq13 = (Eq(Derivative(x(t), t), 21*x(t)), Eq(Derivative(y(t), t), 17*x(t) + 3*y(t)),
Eq(Derivative(z(t), t), 5*x(t) + 7*y(t) + 9*z(t)))
sol13 = {'no_of_equation': 3, 'eq': (
Eq(Derivative(x(t), t), 21 * x(t)), Eq(Derivative(y(t), t), 17 * x(t) + 3 * y(t)),
Eq(Derivative(z(t), t), 5 * x(t) + 7 * y(t) + 9 * z(t))), 'func': [x(t), y(t), z(t)],
'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True,
'func_coeff': -Matrix([
[-21, 0, 0],
[-17, -3, 0],
[-5, -7, -9]]), 'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eq13, [x(t), y(t), z(t)], t) == sol13
eq14 = (
Eq(Derivative(x(t), t), 4*x(t) + 5*y(t) + 2*z(t)), Eq(Derivative(y(t), t), x(t) + 13*y(t) + 9*z(t)),
Eq(Derivative(z(t), t), 32*x(t) + 41*y(t) + 11*z(t)))
sol14 = {'no_of_equation': 3, 'eq': (
Eq(Derivative(x(t), t), 4 * x(t) + 5 * y(t) + 2 * z(t)), Eq(Derivative(y(t), t), x(t) + 13 * y(t) + 9 * z(t)),
Eq(Derivative(z(t), t), 32 * x(t) + 41 * y(t) + 11 * z(t))), 'func': [x(t), y(t), z(t)],
'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True,
'func_coeff': -Matrix([
[-4, -5, -2],
[-1, -13, -9],
[-32, -41, -11]]), 'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eq14, [x(t), y(t), z(t)], t) == sol14
eq15 = (Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)), Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)),
Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t)))
sol15 = {'no_of_equation': 3, 'eq': (
Eq(3 * Derivative(x(t), t), 20 * y(t) - 20 * z(t)), Eq(4 * Derivative(y(t), t), -15 * x(t) + 15 * z(t)),
Eq(5 * Derivative(z(t), t), 12 * x(t) - 12 * y(t))), 'func': [x(t), y(t), z(t)],
'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': True,
'func_coeff': -Matrix([
[0, Rational(-20, 3), Rational(20, 3)],
[Rational(15, 4), 0, Rational(-15, 4)],
[Rational(-12, 5), Rational(12, 5), 0]]), 'type_of_equation': 'type1', 'is_general': True}
assert _classify_linear_system(eq15, [x(t), y(t), z(t)], t) == sol15
# Constant coefficient homogeneous ODEs
eq1 = (Eq(diff(x(t), t), x(t) + y(t) + 9), Eq(diff(y(t), t), 2*x(t) + 5*y(t) + 23))
sol1 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), x(t) + y(t) + 9),
Eq(Derivative(y(t), t), 2*x(t) + 5*y(t) + 23)), 'func': [x(t), y(t)],
'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': True, 'is_homogeneous': False, 'is_general': True,
'func_coeff': -Matrix([[-1, -1], [-2, -5]]), 'rhs': Matrix([[ 9], [23]]), 'type_of_equation': 'type2'}
assert _classify_linear_system(eq1, funcs, t) == sol1
# Non constant coefficient homogeneous ODEs
eq1 = (Eq(diff(x(t), t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t), t), 2*x(t) + 5*t*y(t)))
sol1 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), t), 5*t*x(t) + 2*y(t)), Eq(Derivative(y(t), t), 5*t*y(t) + 2*x(t))),
'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True, 'is_constant': False,
'is_homogeneous': True, 'func_coeff': -Matrix([ [-5*t, -2], [ -2, -5*t]]), 'commutative_antiderivative': Matrix([
[5*t**2/2, 2*t], [ 2*t, 5*t**2/2]]), 'type_of_equation': 'type3', 'is_general': True}
assert _classify_linear_system(eq1, funcs, t) == sol1
# Non constant coefficient non-homogeneous ODEs
eq1 = [Eq(x1, x(t) + t*y(t) + t), Eq(y1, t*x(t) + y(t))]
sol1 = {'no_of_equation': 2, 'eq': [Eq(Derivative(x(t), t), t*y(t) + t + x(t)), Eq(Derivative(y(t), t),
t*x(t) + y(t))], 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True,
'is_constant': False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-1, -t],
[-t, -1]]), 'commutative_antiderivative': Matrix([ [ t, t**2/2], [t**2/2, t]]), 'rhs':
Matrix([ [t], [0]]), 'type_of_equation': 'type4'}
assert _classify_linear_system(eq1, funcs, t) == sol1
eq2 = [Eq(x1, t*x(t) + t*y(t) + t), Eq(y1, t*x(t) + t*y(t) + cos(t))]
sol2 = {'no_of_equation': 2, 'eq': [Eq(Derivative(x(t), t), t*x(t) + t*y(t) + t), Eq(Derivative(y(t), t),
t*x(t) + t*y(t) + cos(t))], 'func': [x(t), y(t)], 'order': {x(t): 1, y(t): 1}, 'is_linear': True,
'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ [ t], [cos(t)]]), 'func_coeff':
Matrix([ [t, t], [t, t]]), 'is_constant': False, 'type_of_equation': 'type4',
'commutative_antiderivative': Matrix([ [t**2/2, t**2/2], [t**2/2, t**2/2]])}
assert _classify_linear_system(eq2, funcs, t) == sol2
eq3 = [Eq(x1, t*(x(t) + y(t) + z(t) + 1)), Eq(y1, t*(x(t) + y(t) + z(t))), Eq(z1, t*(x(t) + y(t) + z(t)))]
sol3 = {'no_of_equation': 3, 'eq': [Eq(Derivative(x(t), t), t*(x(t) + y(t) + z(t) + 1)),
Eq(Derivative(y(t), t), t*(x(t) + y(t) + z(t))), Eq(Derivative(z(t), t), t*(x(t) + y(t) + z(t)))],
'func': [x(t), y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant':
False, 'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-t, -t, -t], [-t, -t,
-t], [-t, -t, -t]]), 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2, t**2/2], [t**2/2,
t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2]]), 'rhs': Matrix([ [t], [0], [0]]), 'type_of_equation':
'type4'}
assert _classify_linear_system(eq3, funcs_2[:-1], t) == sol3
eq4 = [Eq(x1, x(t) + y(t) + t*z(t) + 1), Eq(y1, x(t) + t*y(t) + z(t) + 10), Eq(z1, t*x(t) + y(t) + z(t) + t)]
sol4 = {'no_of_equation': 3, 'eq': [Eq(Derivative(x(t), t), t*z(t) + x(t) + y(t) + 1), Eq(Derivative(y(t),
t), t*y(t) + x(t) + z(t) + 10), Eq(Derivative(z(t), t), t*x(t) + t + y(t) + z(t))], 'func': [x(t),
y(t), z(t)], 'order': {x(t): 1, y(t): 1, z(t): 1}, 'is_linear': True, 'is_constant': False,
'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-1, -1, -t], [-1, -t, -1], [-t,
-1, -1]]), 'commutative_antiderivative': Matrix([ [ t, t, t**2/2], [ t, t**2/2,
t], [t**2/2, t, t]]), 'rhs': Matrix([ [ 1], [10], [ t]]), 'type_of_equation': 'type4'}
assert _classify_linear_system(eq4, funcs_2[:-1], t) == sol4
sum_terms = t*(x(t) + y(t) + z(t) + w(t))
eq5 = [Eq(x1, sum_terms), Eq(y1, sum_terms), Eq(z1, sum_terms + 1), Eq(w1, sum_terms)]
sol5 = {'no_of_equation': 4, 'eq': [Eq(Derivative(x(t), t), t*(w(t) + x(t) + y(t) + z(t))),
Eq(Derivative(y(t), t), t*(w(t) + x(t) + y(t) + z(t))), Eq(Derivative(z(t), t), t*(w(t) + x(t) +
y(t) + z(t)) + 1), Eq(Derivative(w(t), t), t*(w(t) + x(t) + y(t) + z(t)))], 'func': [x(t), y(t),
z(t), w(t)], 'order': {x(t): 1, y(t): 1, z(t): 1, w(t): 1}, 'is_linear': True, 'is_constant': False,
'is_homogeneous': False, 'is_general': True, 'func_coeff': -Matrix([ [-t, -t, -t, -t], [-t, -t, -t,
-t], [-t, -t, -t, -t], [-t, -t, -t, -t]]), 'commutative_antiderivative': Matrix([ [t**2/2, t**2/2,
t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2, t**2/2], [t**2/2, t**2/2, t**2/2, t**2/2], [t**2/2,
t**2/2, t**2/2, t**2/2]]), 'rhs': Matrix([ [0], [0], [1], [0]]), 'type_of_equation': 'type4'}
assert _classify_linear_system(eq5, funcs_2, t) == sol5
# Second Order
t_ = symbols("t_")
eq1 = (Eq(9*x(t) + 7*y(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) + 3*Derivative(y(t), t), 11*exp(I*t)),
Eq(3*x(t) + 12*y(t) + 5*Derivative(x(t), t) + 8*Derivative(y(t), t) + Derivative(y(t), (t, 2)), 2*exp(I*t)))
sol1 = {'no_of_equation': 2, 'eq': (Eq(9*x(t) + 7*y(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)) +
3*Derivative(y(t), t), 11*exp(I*t)), Eq(3*x(t) + 12*y(t) + 5*Derivative(x(t), t) +
8*Derivative(y(t), t) + Derivative(y(t), (t, 2)), 2*exp(I*t))), 'func': [x(t), y(t)], 'order':
{x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([
[11*exp(I*t)], [ 2*exp(I*t)]]), 'type_of_equation': 'type0', 'is_second_order': True,
'is_higher_order': True}
assert _classify_linear_system(eq1, funcs, t) == sol1
eq2 = (Eq((4*t**2 + 7*t + 1)**2*Derivative(x(t), (t, 2)), 5*x(t) + 35*y(t)),
Eq((4*t**2 + 7*t + 1)**2*Derivative(y(t), (t, 2)), x(t) + 9*y(t)))
sol2 = {'no_of_equation': 2, 'eq': (Eq((4*t**2 + 7*t + 1)**2*Derivative(x(t), (t, 2)), 5*x(t) + 35*y(t)),
Eq((4*t**2 + 7*t + 1)**2*Derivative(y(t), (t, 2)), x(t) + 9*y(t))), 'func': [x(t), y(t)], 'order':
{x(t): 2, y(t): 2}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True,
'type_of_equation': 'type2', 'A0': Matrix([ [Rational(53, 4), 35], [ 1, Rational(69, 4)]]), 'g(t)': sqrt(4*t**2 + 7*t
+ 1), 'tau': sqrt(33)*log(t - sqrt(33)/8 + Rational(7, 8))/33 - sqrt(33)*log(t + sqrt(33)/8 + Rational(7, 8))/33,
'is_transformed': True, 't_': t_, 'is_second_order': True, 'is_higher_order': True}
assert _classify_linear_system(eq2, funcs, t) == sol2
eq3 = ((t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) - y(t))*exp(t) + Derivative(x(t), (t, 2)),
t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t), t) - y(t)) + Derivative(y(t), (t, 2)))
sol3 = {'no_of_equation': 2, 'eq': ((t*Derivative(x(t), t) - x(t))*log(t) + (t*Derivative(y(t), t) -
y(t))*exp(t) + Derivative(x(t), (t, 2)), t**2*(t*Derivative(x(t), t) - x(t)) + t*(t*Derivative(y(t),
t) - y(t)) + Derivative(y(t), (t, 2))), 'func': [x(t), y(t)], 'order': {x(t): 2, y(t): 2},
'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation': 'type1', 'A1':
Matrix([ [-t*log(t), -t*exp(t)], [ -t**3, -t**2]]), 'is_second_order': True,
'is_higher_order': True}
assert _classify_linear_system(eq3, funcs, t) == sol3
eq4 = (Eq(x2, k*x(t) - l*y1), Eq(y2, l*x1 + k*y(t)))
sol4 = {'no_of_equation': 2, 'eq': (Eq(Derivative(x(t), (t, 2)), k*x(t) - l*Derivative(y(t), t)),
Eq(Derivative(y(t), (t, 2)), k*y(t) + l*Derivative(x(t), t))), 'func': [x(t), y(t)], 'order': {x(t):
2, y(t): 2}, 'is_linear': True, 'is_homogeneous': True, 'is_general': True, 'type_of_equation':
'type0', 'is_second_order': True, 'is_higher_order': True}
assert _classify_linear_system(eq4, funcs, t) == sol4
# Multiple matchs
f, g = symbols("f g", cls=Function)
y, t_ = symbols("y t_")
funcs = [f(t), g(t)]
eq1 = [Eq(Derivative(f(t), t)**2 - 2*Derivative(f(t), t) + 1, 4),
Eq(-y*f(t) + Derivative(g(t), t), 0)]
sol1 = {'is_implicit': True,
'canon_eqs': [[Eq(Derivative(f(t), t), -1), Eq(Derivative(g(t), t), y*f(t))],
[Eq(Derivative(f(t), t), 3), Eq(Derivative(g(t), t), y*f(t))]]}
assert _classify_linear_system(eq1, funcs, t) == sol1
raises(ValueError, lambda: _classify_linear_system(eq1, funcs[:1], t))
eq2 = [Eq(Derivative(f(t), t), (2*f(t) + g(t) + 1)/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)]
sol2 = {'no_of_equation': 2, 'eq': [Eq(Derivative(f(t), t), (2*f(t) + g(t) + 1)/t), Eq(Derivative(g(t), t),
(f(t) + 2*g(t))/t)], 'func': [f(t), g(t)], 'order': {f(t): 1, g(t): 1}, 'is_linear': True,
'is_homogeneous': False, 'is_general': True, 'rhs': Matrix([ [1], [0]]), 'func_coeff': Matrix([ [2,
1], [1, 2]]), 'is_constant': False, 'type_of_equation': 'type6', 't_': t_, 'tau': log(t),
'commutative_antiderivative': Matrix([ [2*log(t), log(t)], [ log(t), 2*log(t)]])}
assert _classify_linear_system(eq2, funcs, t) == sol2
eq3 = [Eq(Derivative(f(t), t), (2*f(t) + g(t))/t), Eq(Derivative(g(t), t), (f(t) + 2*g(t))/t)]
sol3 = {'no_of_equation': 2, 'eq': [Eq(Derivative(f(t), t), (2*f(t) + g(t))/t), Eq(Derivative(g(t), t),
(f(t) + 2*g(t))/t)], 'func': [f(t), g(t)], 'order': {f(t): 1, g(t): 1}, 'is_linear': True,
'is_homogeneous': True, 'is_general': True, 'func_coeff': Matrix([ [2, 1], [1, 2]]), 'is_constant':
False, 'type_of_equation': 'type5', 't_': t_, 'rhs': Matrix([ [0], [0]]), 'tau': log(t),
'commutative_antiderivative': Matrix([ [2*log(t), log(t)], [ log(t), 2*log(t)]])}
assert _classify_linear_system(eq3, funcs, t) == sol3
def test_matrix_exp():
from sympy.matrices.dense import Matrix, eye, zeros
from sympy.solvers.ode.systems import matrix_exp
t = Symbol('t')
for n in range(1, 6+1):
assert matrix_exp(zeros(n), t) == eye(n)
for n in range(1, 6+1):
A = eye(n)
expAt = exp(t) * eye(n)
assert matrix_exp(A, t) == expAt
for n in range(1, 6+1):
A = Matrix(n, n, lambda i,j: i+1 if i==j else 0)
expAt = Matrix(n, n, lambda i,j: exp((i+1)*t) if i==j else 0)
assert matrix_exp(A, t) == expAt
A = Matrix([[0, 1], [-1, 0]])
expAt = Matrix([[cos(t), sin(t)], [-sin(t), cos(t)]])
assert matrix_exp(A, t) == expAt
A = Matrix([[2, -5], [2, -4]])
expAt = Matrix([
[3*exp(-t)*sin(t) + exp(-t)*cos(t), -5*exp(-t)*sin(t)],
[2*exp(-t)*sin(t), -3*exp(-t)*sin(t) + exp(-t)*cos(t)]
])
assert matrix_exp(A, t) == expAt
A = Matrix([[21, 17, 6], [-5, -1, -6], [4, 4, 16]])
# TO update this.
# expAt = Matrix([
# [(8*t*exp(12*t) + 5*exp(12*t) - 1)*exp(4*t)/4,
# (8*t*exp(12*t) + 5*exp(12*t) - 5)*exp(4*t)/4,
# (exp(12*t) - 1)*exp(4*t)/2],
# [(-8*t*exp(12*t) - exp(12*t) + 1)*exp(4*t)/4,
# (-8*t*exp(12*t) - exp(12*t) + 5)*exp(4*t)/4,
# (-exp(12*t) + 1)*exp(4*t)/2],
# [4*t*exp(16*t), 4*t*exp(16*t), exp(16*t)]])
expAt = Matrix([
[2*t*exp(16*t) + 5*exp(16*t)/4 - exp(4*t)/4, 2*t*exp(16*t) + 5*exp(16*t)/4 - 5*exp(4*t)/4, exp(16*t)/2 - exp(4*t)/2],
[ -2*t*exp(16*t) - exp(16*t)/4 + exp(4*t)/4, -2*t*exp(16*t) - exp(16*t)/4 + 5*exp(4*t)/4, -exp(16*t)/2 + exp(4*t)/2],
[ 4*t*exp(16*t), 4*t*exp(16*t), exp(16*t)]
])
assert matrix_exp(A, t) == expAt
A = Matrix([[1, 1, 0, 0],
[0, 1, 1, 0],
[0, 0, 1, -S(1)/8],
[0, 0, S(1)/2, S(1)/2]])
expAt = Matrix([
[exp(t), t*exp(t), 4*t*exp(3*t/4) + 8*t*exp(t) + 48*exp(3*t/4) - 48*exp(t),
-2*t*exp(3*t/4) - 2*t*exp(t) - 16*exp(3*t/4) + 16*exp(t)],
[0, exp(t), -t*exp(3*t/4) - 8*exp(3*t/4) + 8*exp(t), t*exp(3*t/4)/2 + 2*exp(3*t/4) - 2*exp(t)],
[0, 0, t*exp(3*t/4)/4 + exp(3*t/4), -t*exp(3*t/4)/8],
[0, 0, t*exp(3*t/4)/2, -t*exp(3*t/4)/4 + exp(3*t/4)]
])
assert matrix_exp(A, t) == expAt
A = Matrix([
[ 0, 1, 0, 0],
[-1, 0, 0, 0],
[ 0, 0, 0, 1],
[ 0, 0, -1, 0]])
expAt = Matrix([
[ cos(t), sin(t), 0, 0],
[-sin(t), cos(t), 0, 0],
[ 0, 0, cos(t), sin(t)],
[ 0, 0, -sin(t), cos(t)]])
assert matrix_exp(A, t) == expAt
A = Matrix([
[ 0, 1, 1, 0],
[-1, 0, 0, 1],
[ 0, 0, 0, 1],
[ 0, 0, -1, 0]])
expAt = Matrix([
[ cos(t), sin(t), t*cos(t), t*sin(t)],
[-sin(t), cos(t), -t*sin(t), t*cos(t)],
[ 0, 0, cos(t), sin(t)],
[ 0, 0, -sin(t), cos(t)]])
assert matrix_exp(A, t) == expAt
# This case is unacceptably slow right now but should be solvable...
#a, b, c, d, e, f = symbols('a b c d e f')
#A = Matrix([
#[-a, b, c, d],
#[ a, -b, e, 0],
#[ 0, 0, -c - e - f, 0],
#[ 0, 0, f, -d]])
A = Matrix([[0, I], [I, 0]])
expAt = Matrix([
[exp(I*t)/2 + exp(-I*t)/2, exp(I*t)/2 - exp(-I*t)/2],
[exp(I*t)/2 - exp(-I*t)/2, exp(I*t)/2 + exp(-I*t)/2]])
assert matrix_exp(A, t) == expAt
# Testing Errors
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 7, 7]])
M1 = Matrix([[t, 1], [1, 1]])
raises(ValueError, lambda: matrix_exp(M[:, :2], t))
raises(ValueError, lambda: matrix_exp(M[:2, :], t))
raises(ValueError, lambda: matrix_exp(M1, t))
raises(ValueError, lambda: matrix_exp(M1[:1, :1], t))
def test_canonical_odes():
f, g, h = symbols('f g h', cls=Function)
x = symbols('x')
funcs = [f(x), g(x), h(x)]
eqs1 = [Eq(f(x).diff(x, x), f(x) + 2*g(x)), Eq(g(x) + 1, g(x).diff(x) + f(x))]
sol1 = [[Eq(Derivative(f(x), (x, 2)), f(x) + 2*g(x)), Eq(Derivative(g(x), x), -f(x) + g(x) + 1)]]
assert canonical_odes(eqs1, funcs[:2], x) == sol1
eqs2 = [Eq(f(x).diff(x), h(x).diff(x) + f(x)), Eq(g(x).diff(x)**2, f(x) + h(x)), Eq(h(x).diff(x), f(x))]
sol2 = [[Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), -sqrt(f(x) + h(x))), Eq(Derivative(h(x), x), f(x))],
[Eq(Derivative(f(x), x), 2*f(x)), Eq(Derivative(g(x), x), sqrt(f(x) + h(x))), Eq(Derivative(h(x), x), f(x))]]
assert canonical_odes(eqs2, funcs, x) == sol2
def test_sysode_linear_neq_order1_type1():
f, g, x, y, h = symbols('f g x y h', cls=Function)
a, b, c, t = symbols('a b c t')
eqs1 = [Eq(Derivative(x(t), t), x(t)),
Eq(Derivative(y(t), t), y(t))]
sol1 = [Eq(x(t), C1*exp(t)),
Eq(y(t), C2*exp(t))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
eqs2 = [Eq(Derivative(x(t), t), 2*x(t)),
Eq(Derivative(y(t), t), 3*y(t))]
sol2 = [Eq(x(t), C1*exp(2*t)),
Eq(y(t), C2*exp(3*t))]
assert dsolve(eqs2) == sol2
assert checksysodesol(eqs2, sol2) == (True, [0, 0])
eqs3 = [Eq(Derivative(x(t), t), a*x(t)),
Eq(Derivative(y(t), t), a*y(t))]
sol3 = [Eq(x(t), C1*exp(a*t)),
Eq(y(t), C2*exp(a*t))]
assert dsolve(eqs3) == sol3
assert checksysodesol(eqs3, sol3) == (True, [0, 0])
# Regression test case for issue #15474
# https://github.com/sympy/sympy/issues/15474
eqs4 = [Eq(Derivative(x(t), t), a*x(t)),
Eq(Derivative(y(t), t), b*y(t))]
sol4 = [Eq(x(t), C1*exp(a*t)),
Eq(y(t), C2*exp(b*t))]
assert dsolve(eqs4) == sol4
assert checksysodesol(eqs4, sol4) == (True, [0, 0])
eqs5 = [Eq(Derivative(x(t), t), -y(t)),
Eq(Derivative(y(t), t), x(t))]
sol5 = [Eq(x(t), -C1*sin(t) - C2*cos(t)),
Eq(y(t), C1*cos(t) - C2*sin(t))]
assert dsolve(eqs5) == sol5
assert checksysodesol(eqs5, sol5) == (True, [0, 0])
eqs6 = [Eq(Derivative(x(t), t), -2*y(t)),
Eq(Derivative(y(t), t), 2*x(t))]
sol6 = [Eq(x(t), -C1*sin(2*t) - C2*cos(2*t)),
Eq(y(t), C1*cos(2*t) - C2*sin(2*t))]
assert dsolve(eqs6) == sol6
assert checksysodesol(eqs6, sol6) == (True, [0, 0])
eqs7 = [Eq(Derivative(x(t), t), I*y(t)),
Eq(Derivative(y(t), t), I*x(t))]
sol7 = [Eq(x(t), -C1*exp(-I*t) + C2*exp(I*t)),
Eq(y(t), C1*exp(-I*t) + C2*exp(I*t))]
assert dsolve(eqs7) == sol7
assert checksysodesol(eqs7, sol7) == (True, [0, 0])
eqs8 = [Eq(Derivative(x(t), t), -a*y(t)),
Eq(Derivative(y(t), t), a*x(t))]
sol8 = [Eq(x(t), -I*C1*exp(-I*a*t) + I*C2*exp(I*a*t)),
Eq(y(t), C1*exp(-I*a*t) + C2*exp(I*a*t))]
assert dsolve(eqs8) == sol8
assert checksysodesol(eqs8, sol8) == (True, [0, 0])
eqs9 = [Eq(Derivative(x(t), t), x(t) + y(t)),
Eq(Derivative(y(t), t), x(t) - y(t))]
sol9 = [Eq(x(t), C1*(1 - sqrt(2))*exp(-sqrt(2)*t) + C2*(1 + sqrt(2))*exp(sqrt(2)*t)),
Eq(y(t), C1*exp(-sqrt(2)*t) + C2*exp(sqrt(2)*t))]
assert dsolve(eqs9) == sol9
assert checksysodesol(eqs9, sol9) == (True, [0, 0])
eqs10 = [Eq(Derivative(x(t), t), x(t) + y(t)),
Eq(Derivative(y(t), t), x(t) + y(t))]
sol10 = [Eq(x(t), -C1 + C2*exp(2*t)),
Eq(y(t), C1 + C2*exp(2*t))]
assert dsolve(eqs10) == sol10
assert checksysodesol(eqs10, sol10) == (True, [0, 0])
eqs11 = [Eq(Derivative(x(t), t), 2*x(t) + y(t)),
Eq(Derivative(y(t), t), -x(t) + 2*y(t))]
sol11 = [Eq(x(t), C1*exp(2*t)*sin(t) + C2*exp(2*t)*cos(t)),
Eq(y(t), C1*exp(2*t)*cos(t) - C2*exp(2*t)*sin(t))]
assert dsolve(eqs11) == sol11
assert checksysodesol(eqs11, sol11) == (True, [0, 0])
eqs12 = [Eq(Derivative(x(t), t), x(t) + 2*y(t)),
Eq(Derivative(y(t), t), 2*x(t) + y(t))]
sol12 = [Eq(x(t), -C1*exp(-t) + C2*exp(3*t)),
Eq(y(t), C1*exp(-t) + C2*exp(3*t))]
assert dsolve(eqs12) == sol12
assert checksysodesol(eqs12, sol12) == (True, [0, 0])
eqs13 = [Eq(Derivative(x(t), t), 4*x(t) + y(t)),
Eq(Derivative(y(t), t), -x(t) + 2*y(t))]
sol13 = [Eq(x(t), C2*t*exp(3*t) + (C1 + C2)*exp(3*t)),
Eq(y(t), -C1*exp(3*t) - C2*t*exp(3*t))]
assert dsolve(eqs13) == sol13
assert checksysodesol(eqs13, sol13) == (True, [0, 0])
eqs14 = [Eq(Derivative(x(t), t), a*y(t)),
Eq(Derivative(y(t), t), a*x(t))]
sol14 = [Eq(x(t), -C1*exp(-a*t) + C2*exp(a*t)),
Eq(y(t), C1*exp(-a*t) + C2*exp(a*t))]
assert dsolve(eqs14) == sol14
assert checksysodesol(eqs14, sol14) == (True, [0, 0])
eqs15 = [Eq(Derivative(x(t), t), a*y(t)),
Eq(Derivative(y(t), t), b*x(t))]
sol15 = [Eq(x(t), -C1*a*exp(-t*sqrt(a*b))/sqrt(a*b) + C2*a*exp(t*sqrt(a*b))/sqrt(a*b)),
Eq(y(t), C1*exp(-t*sqrt(a*b)) + C2*exp(t*sqrt(a*b)))]
assert dsolve(eqs15) == sol15
assert checksysodesol(eqs15, sol15) == (True, [0, 0])
eqs16 = [Eq(Derivative(x(t), t), a*x(t) + b*y(t)),
Eq(Derivative(y(t), t), c*x(t))]
sol16 = [Eq(x(t), -2*C1*b*exp(t*(a + sqrt(a**2 + 4*b*c))/2)/(a - sqrt(a**2 + 4*b*c)) - 2*C2*b*exp(t*(a -
sqrt(a**2 + 4*b*c))/2)/(a + sqrt(a**2 + 4*b*c))),
Eq(y(t), C1*exp(t*(a + sqrt(a**2 + 4*b*c))/2) + C2*exp(t*(a - sqrt(a**2 + 4*b*c))/2))]
assert dsolve(eqs16) == sol16
assert checksysodesol(eqs16, sol16) == (True, [0, 0])
# Regression test case for issue #18562
# https://github.com/sympy/sympy/issues/18562
eqs17 = [Eq(Derivative(x(t), t), a*y(t) + x(t)),
Eq(Derivative(y(t), t), a*x(t) - y(t))]
sol17 = [Eq(x(t), C1*a*exp(t*sqrt(a**2 + 1))/(sqrt(a**2 + 1) - 1) - C2*a*exp(-t*sqrt(a**2 + 1))/(sqrt(a**2 +
1) + 1)),
Eq(y(t), C1*exp(t*sqrt(a**2 + 1)) + C2*exp(-t*sqrt(a**2 + 1)))]
assert dsolve(eqs17) == sol17
assert checksysodesol(eqs17, sol17) == (True, [0, 0])
eqs18 = [Eq(Derivative(x(t), t), 0),
Eq(Derivative(y(t), t), 0)]
sol18 = [Eq(x(t), C1),
Eq(y(t), C2)]
assert dsolve(eqs18) == sol18
assert checksysodesol(eqs18, sol18) == (True, [0, 0])
eqs19 = [Eq(Derivative(x(t), t), 2*x(t) - y(t)),
Eq(Derivative(y(t), t), x(t))]
sol19 = [Eq(x(t), C2*t*exp(t) + (C1 + C2)*exp(t)),
Eq(y(t), C1*exp(t) + C2*t*exp(t))]
assert dsolve(eqs19) == sol19
assert checksysodesol(eqs19, sol19) == (True, [0, 0])
eqs20 = [Eq(Derivative(x(t), t), x(t)),
Eq(Derivative(y(t), t), x(t) + y(t))]
sol20 = [Eq(x(t), C1*exp(t)),
Eq(y(t), C1*t*exp(t) + C2*exp(t))]
assert dsolve(eqs20) == sol20
assert checksysodesol(eqs20, sol20) == (True, [0, 0])
eqs21 = [Eq(Derivative(x(t), t), 3*x(t)),
Eq(Derivative(y(t), t), x(t) + y(t))]
sol21 = [Eq(x(t), 2*C1*exp(3*t)),
Eq(y(t), C1*exp(3*t) + C2*exp(t))]
assert dsolve(eqs21) == sol21
assert checksysodesol(eqs21, sol21) == (True, [0, 0])
eqs22 = [Eq(Derivative(x(t), t), 3*x(t)),
Eq(Derivative(y(t), t), y(t))]
sol22 = [Eq(x(t), C1*exp(3*t)),
Eq(y(t), C2*exp(t))]
assert dsolve(eqs22) == sol22
assert checksysodesol(eqs22, sol22) == (True, [0, 0])
@slow
def test_sysode_linear_neq_order1_type1_slow():
t = Symbol('t')
Z0 = Function('Z0')
Z1 = Function('Z1')
Z2 = Function('Z2')
Z3 = Function('Z3')
k01, k10, k20, k21, k23, k30 = symbols('k01 k10 k20 k21 k23 k30')
eqs1 = [Eq(Derivative(Z0(t), t), -k01*Z0(t) + k10*Z1(t) + k20*Z2(t) + k30*Z3(t)),
Eq(Derivative(Z1(t), t), k01*Z0(t) - k10*Z1(t) + k21*Z2(t)),
Eq(Derivative(Z2(t), t), (-k20 - k21 - k23)*Z2(t)),
Eq(Derivative(Z3(t), t), k23*Z2(t) - k30*Z3(t))]
sol1 = [Eq(Z0(t), C1*k10/k01 - C2*(k10 - k30)*exp(-k30*t)/(k01 + k10 - k30) - C3*(k10*(k20 + k21 - k30) -
k20**2 - k20*(k21 + k23 - k30) + k23*k30)*exp(-t*(k20 + k21 + k23))/(k23*(-k01 - k10 + k20 + k21 +
k23)) - C4*exp(-t*(k01 + k10))),
Eq(Z1(t), C1 - C2*k01*exp(-k30*t)/(k01 + k10 - k30) + C3*(-k01*(k20 + k21 - k30) + k20*k21 + k21**2
+ k21*(k23 - k30))*exp(-t*(k20 + k21 + k23))/(k23*(-k01 - k10 + k20 + k21 + k23)) + C4*exp(-t*(k01 +
k10))),
Eq(Z2(t), -C3*(k20 + k21 + k23 - k30)*exp(-t*(k20 + k21 + k23))/k23),
Eq(Z3(t), C2*exp(-k30*t) + C3*exp(-t*(k20 + k21 + k23)))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0, 0, 0])
x, y, z, u, v, w = symbols('x y z u v w', cls=Function)
k2, k3 = symbols('k2 k3')
a_b, a_c = symbols('a_b a_c', real=True)
eqs2 = [Eq(Derivative(z(t), t), k2*y(t)),
Eq(Derivative(x(t), t), k3*y(t)),
Eq(Derivative(y(t), t), (-k2 - k3)*y(t))]
sol2 = [Eq(z(t), C1 - C2*k2*exp(-t*(k2 + k3))/(k2 + k3)),
Eq(x(t), -C2*k3*exp(-t*(k2 + k3))/(k2 + k3) + C3),
Eq(y(t), C2*exp(-t*(k2 + k3)))]
assert dsolve(eqs2) == sol2
assert checksysodesol(eqs2, sol2) == (True, [0, 0, 0])
eqs3 = [4*u(t) - v(t) - 2*w(t) + Derivative(u(t), t),
2*u(t) + v(t) - 2*w(t) + Derivative(v(t), t),
5*u(t) + v(t) - 3*w(t) + Derivative(w(t), t)]
sol3 = [Eq(u(t), C3*exp(-2*t) + (C1/2 + sqrt(3)*C2/6)*cos(sqrt(3)*t) + sin(sqrt(3)*t)*(sqrt(3)*C1/6 +
C2*Rational(-1, 2))),
Eq(v(t), (C1/2 + sqrt(3)*C2/6)*cos(sqrt(3)*t) + sin(sqrt(3)*t)*(sqrt(3)*C1/6 + C2*Rational(-1, 2))),
Eq(w(t), C1*cos(sqrt(3)*t) - C2*sin(sqrt(3)*t) + C3*exp(-2*t))]
assert dsolve(eqs3) == sol3
assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0])
eqs4 = [Eq(Derivative(x(t), t), w(t)*Rational(-2, 9) + 2*x(t) + y(t) + z(t)*Rational(-8, 9)),
Eq(Derivative(y(t), t), w(t)*Rational(4, 9) + 2*y(t) + z(t)*Rational(16, 9)),
Eq(Derivative(z(t), t), w(t)*Rational(-2, 9) + z(t)*Rational(37, 9)),
Eq(Derivative(w(t), t), w(t)*Rational(44, 9) + z(t)*Rational(-4, 9))]
sol4 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t)),
Eq(y(t), C2*exp(2*t) + 2*C3*exp(4*t)),
Eq(z(t), 2*C3*exp(4*t) + C4*exp(5*t)*Rational(-1, 4)),
Eq(w(t), C3*exp(4*t) + C4*exp(5*t))]
assert dsolve(eqs4) == sol4
assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0, 0])
# Regression test case for issue #15574
# https://github.com/sympy/sympy/issues/15574
eq5 = [Eq(x(t).diff(t), x(t)), Eq(y(t).diff(t), y(t)), Eq(z(t).diff(t), z(t)), Eq(w(t).diff(t), w(t))]
sol5 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t)), Eq(w(t), C4*exp(t))]
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0, 0, 0])
eqs6 = [Eq(Derivative(x(t), t), x(t) + y(t)),
Eq(Derivative(y(t), t), y(t) + z(t)),
Eq(Derivative(z(t), t), w(t)*Rational(-1, 8) + z(t)),
Eq(Derivative(w(t), t), w(t)/2 + z(t)/2)]
sol6 = [Eq(x(t), C1*exp(t) + C2*t*exp(t) + 4*C4*t*exp(t*Rational(3, 4)) + (4*C3 + 48*C4)*exp(t*Rational(3,
4))),
Eq(y(t), C2*exp(t) - C4*t*exp(t*Rational(3, 4)) - (C3 + 8*C4)*exp(t*Rational(3, 4))),
Eq(z(t), C4*t*exp(t*Rational(3, 4))/4 + (C3/4 + C4)*exp(t*Rational(3, 4))),
Eq(w(t), C3*exp(t*Rational(3, 4))/2 + C4*t*exp(t*Rational(3, 4))/2)]
assert dsolve(eqs6) == sol6
assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0])
# Regression test case for issue #15574
# https://github.com/sympy/sympy/issues/15574
eq7 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), y(t)), Eq(Derivative(z(t), t), z(t)),
Eq(Derivative(w(t), t), w(t)), Eq(Derivative(u(t), t), u(t))]
sol7 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t)), Eq(w(t), C4*exp(t)),
Eq(u(t), C5*exp(t))]
assert dsolve(eq7) == sol7
assert checksysodesol(eq7, sol7) == (True, [0, 0, 0, 0, 0])
eqs8 = [Eq(Derivative(x(t), t), 2*x(t) + y(t)),
Eq(Derivative(y(t), t), 2*y(t)),
Eq(Derivative(z(t), t), 4*z(t)),
Eq(Derivative(w(t), t), u(t) + 5*w(t)),
Eq(Derivative(u(t), t), 5*u(t))]
sol8 = [Eq(x(t), C1*exp(2*t) + C2*t*exp(2*t)),
Eq(y(t), C2*exp(2*t)),
Eq(z(t), C3*exp(4*t)),
Eq(w(t), C4*exp(5*t) + C5*t*exp(5*t)),
Eq(u(t), C5*exp(5*t))]
assert dsolve(eqs8) == sol8
assert checksysodesol(eqs8, sol8) == (True, [0, 0, 0, 0, 0])
# Regression test case for issue #15574
# https://github.com/sympy/sympy/issues/15574
eq9 = [Eq(Derivative(x(t), t), x(t)), Eq(Derivative(y(t), t), y(t)), Eq(Derivative(z(t), t), z(t))]
sol9 = [Eq(x(t), C1*exp(t)), Eq(y(t), C2*exp(t)), Eq(z(t), C3*exp(t))]
assert dsolve(eq9) == sol9
assert checksysodesol(eq9, sol9) == (True, [0, 0, 0])
# Regression test case for issue #15407
# https://github.com/sympy/sympy/issues/15407
eqs10 = [Eq(Derivative(x(t), t), (-a_b - a_c)*x(t)),
Eq(Derivative(y(t), t), a_b*y(t)),
Eq(Derivative(z(t), t), a_c*x(t))]
sol10 = [Eq(x(t), -C1*(a_b + a_c)*exp(-t*(a_b + a_c))/a_c),
Eq(y(t), C2*exp(a_b*t)),
Eq(z(t), C1*exp(-t*(a_b + a_c)) + C3)]
assert dsolve(eqs10) == sol10
assert checksysodesol(eqs10, sol10) == (True, [0, 0, 0])
# Regression test case for issue #14312
# https://github.com/sympy/sympy/issues/14312
eqs11 = [Eq(Derivative(x(t), t), k3*y(t)),
Eq(Derivative(y(t), t), (-k2 - k3)*y(t)),
Eq(Derivative(z(t), t), k2*y(t))]
sol11 = [Eq(x(t), C1 + C2*k3*exp(-t*(k2 + k3))/k2),
Eq(y(t), -C2*(k2 + k3)*exp(-t*(k2 + k3))/k2),
Eq(z(t), C2*exp(-t*(k2 + k3)) + C3)]
assert dsolve(eqs11) == sol11
assert checksysodesol(eqs11, sol11) == (True, [0, 0, 0])
# Regression test case for issue #14312
# https://github.com/sympy/sympy/issues/14312
eqs12 = [Eq(Derivative(z(t), t), k2*y(t)),
Eq(Derivative(x(t), t), k3*y(t)),
Eq(Derivative(y(t), t), (-k2 - k3)*y(t))]
sol12 = [Eq(z(t), C1 - C2*k2*exp(-t*(k2 + k3))/(k2 + k3)),
Eq(x(t), -C2*k3*exp(-t*(k2 + k3))/(k2 + k3) + C3),
Eq(y(t), C2*exp(-t*(k2 + k3)))]
assert dsolve(eqs12) == sol12
assert checksysodesol(eqs12, sol12) == (True, [0, 0, 0])
f, g, h = symbols('f, g, h', cls=Function)
a, b, c = symbols('a, b, c')
# Regression test case for issue #15474
# https://github.com/sympy/sympy/issues/15474
eqs13 = [Eq(Derivative(f(t), t), 2*f(t) + g(t)),
Eq(Derivative(g(t), t), a*f(t))]
sol13 = [Eq(f(t), C1*exp(t*(sqrt(a + 1) + 1))/(sqrt(a + 1) - 1) - C2*exp(-t*(sqrt(a + 1) - 1))/(sqrt(a + 1) +
1)),
Eq(g(t), C1*exp(t*(sqrt(a + 1) + 1)) + C2*exp(-t*(sqrt(a + 1) - 1)))]
assert dsolve(eqs13) == sol13
assert checksysodesol(eqs13, sol13) == (True, [0, 0])
eqs14 = [Eq(Derivative(f(t), t), 2*g(t) - 3*h(t)),
Eq(Derivative(g(t), t), -2*f(t) + 4*h(t)),
Eq(Derivative(h(t), t), 3*f(t) - 4*g(t))]
sol14 = [Eq(f(t), 2*C1 - sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(3, 25) + C3*Rational(-8, 25)) -
cos(sqrt(29)*t)*(C2*Rational(8, 25) + sqrt(29)*C3*Rational(3, 25))),
Eq(g(t), C1*Rational(3, 2) + sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(4, 25) + C3*Rational(6, 25)) -
cos(sqrt(29)*t)*(C2*Rational(6, 25) + sqrt(29)*C3*Rational(-4, 25))),
Eq(h(t), C1 + C2*cos(sqrt(29)*t) - C3*sin(sqrt(29)*t))]
assert dsolve(eqs14) == sol14
assert checksysodesol(eqs14, sol14) == (True, [0, 0, 0])
eqs15 = [Eq(2*Derivative(f(t), t), 12*g(t) - 12*h(t)),
Eq(3*Derivative(g(t), t), -8*f(t) + 8*h(t)),
Eq(4*Derivative(h(t), t), 6*f(t) - 6*g(t))]
sol15 = [Eq(f(t), C1 - sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(6, 13) + C3*Rational(-16, 13)) -
cos(sqrt(29)*t)*(C2*Rational(16, 13) + sqrt(29)*C3*Rational(6, 13))),
Eq(g(t), C1 + sin(sqrt(29)*t)*(sqrt(29)*C2*Rational(8, 39) + C3*Rational(16, 13)) -
cos(sqrt(29)*t)*(C2*Rational(16, 13) + sqrt(29)*C3*Rational(-8, 39))),
Eq(h(t), C1 + C2*cos(sqrt(29)*t) - C3*sin(sqrt(29)*t))]
assert dsolve(eqs15) == sol15
assert checksysodesol(eqs15, sol15) == (True, [0, 0, 0])
eq16 = (Eq(diff(x(t), t), 21*x(t)), Eq(diff(y(t), t), 17*x(t) + 3*y(t)),
Eq(diff(z(t), t), 5*x(t) + 7*y(t) + 9*z(t)))
sol16 = [Eq(x(t), 216*C1*exp(21*t)/209),
Eq(y(t), 204*C1*exp(21*t)/209 - 6*C2*exp(3*t)/7),
Eq(z(t), C1*exp(21*t) + C2*exp(3*t) + C3*exp(9*t))]
assert dsolve(eq16) == sol16
assert checksysodesol(eq16, sol16) == (True, [0, 0, 0])
eqs17 = [Eq(Derivative(x(t), t), 3*y(t) - 11*z(t)),
Eq(Derivative(y(t), t), -3*x(t) + 7*z(t)),
Eq(Derivative(z(t), t), 11*x(t) - 7*y(t))]
sol17 = [Eq(x(t), C1*Rational(7, 3) - sin(sqrt(179)*t)*(sqrt(179)*C2*Rational(11, 170) + C3*Rational(-21,
170)) - cos(sqrt(179)*t)*(C2*Rational(21, 170) + sqrt(179)*C3*Rational(11, 170))),
Eq(y(t), C1*Rational(11, 3) + sin(sqrt(179)*t)*(sqrt(179)*C2*Rational(7, 170) + C3*Rational(33,
170)) - cos(sqrt(179)*t)*(C2*Rational(33, 170) + sqrt(179)*C3*Rational(-7, 170))),
Eq(z(t), C1 + C2*cos(sqrt(179)*t) - C3*sin(sqrt(179)*t))]
assert dsolve(eqs17) == sol17
assert checksysodesol(eqs17, sol17) == (True, [0, 0, 0])
eqs18 = [Eq(3*Derivative(x(t), t), 20*y(t) - 20*z(t)),
Eq(4*Derivative(y(t), t), -15*x(t) + 15*z(t)),
Eq(5*Derivative(z(t), t), 12*x(t) - 12*y(t))]
sol18 = [Eq(x(t), C1 - sin(5*sqrt(2)*t)*(sqrt(2)*C2*Rational(4, 3) - C3) - cos(5*sqrt(2)*t)*(C2 +
sqrt(2)*C3*Rational(4, 3))),
Eq(y(t), C1 + sin(5*sqrt(2)*t)*(sqrt(2)*C2*Rational(3, 4) + C3) - cos(5*sqrt(2)*t)*(C2 +
sqrt(2)*C3*Rational(-3, 4))),
Eq(z(t), C1 + C2*cos(5*sqrt(2)*t) - C3*sin(5*sqrt(2)*t))]
assert dsolve(eqs18) == sol18
assert checksysodesol(eqs18, sol18) == (True, [0, 0, 0])
eqs19 = [Eq(Derivative(x(t), t), 4*x(t) - z(t)),
Eq(Derivative(y(t), t), 2*x(t) + 2*y(t) - z(t)),
Eq(Derivative(z(t), t), 3*x(t) + y(t))]
sol19 = [Eq(x(t), C2*t**2*exp(2*t)/2 + t*(2*C2 + C3)*exp(2*t) + (C1 + C2 + 2*C3)*exp(2*t)),
Eq(y(t), C2*t**2*exp(2*t)/2 + t*(2*C2 + C3)*exp(2*t) + (C1 + 2*C3)*exp(2*t)),
Eq(z(t), C2*t**2*exp(2*t) + t*(3*C2 + 2*C3)*exp(2*t) + (2*C1 + 3*C3)*exp(2*t))]
assert dsolve(eqs19) == sol19
assert checksysodesol(eqs19, sol19) == (True, [0, 0, 0])
eqs20 = [Eq(Derivative(x(t), t), 4*x(t) - y(t) - 2*z(t)),
Eq(Derivative(y(t), t), 2*x(t) + y(t) - 2*z(t)),
Eq(Derivative(z(t), t), 5*x(t) - 3*z(t))]
sol20 = [Eq(x(t), C1*exp(2*t) - sin(t)*(C2*Rational(3, 5) + C3/5) - cos(t)*(C2/5 + C3*Rational(-3, 5))),
Eq(y(t), -sin(t)*(C2*Rational(3, 5) + C3/5) - cos(t)*(C2/5 + C3*Rational(-3, 5))),
Eq(z(t), C1*exp(2*t) - C2*sin(t) + C3*cos(t))]
assert dsolve(eqs20) == sol20
assert checksysodesol(eqs20, sol20) == (True, [0, 0, 0])
eq21 = (Eq(diff(x(t), t), 9*y(t)), Eq(diff(y(t), t), 12*x(t)))
sol21 = [Eq(x(t), -sqrt(3)*C1*exp(-6*sqrt(3)*t)/2 + sqrt(3)*C2*exp(6*sqrt(3)*t)/2),
Eq(y(t), C1*exp(-6*sqrt(3)*t) + C2*exp(6*sqrt(3)*t))]
assert dsolve(eq21) == sol21
assert checksysodesol(eq21, sol21) == (True, [0, 0])
eqs22 = [Eq(Derivative(x(t), t), 2*x(t) + 4*y(t)),
Eq(Derivative(y(t), t), 12*x(t) + 41*y(t))]
sol22 = [Eq(x(t), C1*(39 - sqrt(1713))*exp(t*(sqrt(1713) + 43)/2)*Rational(-1, 24) + C2*(39 +
sqrt(1713))*exp(t*(43 - sqrt(1713))/2)*Rational(-1, 24)),
Eq(y(t), C1*exp(t*(sqrt(1713) + 43)/2) + C2*exp(t*(43 - sqrt(1713))/2))]
assert dsolve(eqs22) == sol22
assert checksysodesol(eqs22, sol22) == (True, [0, 0])
eqs23 = [Eq(Derivative(x(t), t), x(t) + y(t)),
Eq(Derivative(y(t), t), -2*x(t) + 2*y(t))]
sol23 = [Eq(x(t), (C1/4 + sqrt(7)*C2/4)*cos(sqrt(7)*t/2)*exp(t*Rational(3, 2)) +
sin(sqrt(7)*t/2)*(sqrt(7)*C1/4 + C2*Rational(-1, 4))*exp(t*Rational(3, 2))),
Eq(y(t), C1*cos(sqrt(7)*t/2)*exp(t*Rational(3, 2)) - C2*sin(sqrt(7)*t/2)*exp(t*Rational(3, 2)))]
assert dsolve(eqs23) == sol23
assert checksysodesol(eqs23, sol23) == (True, [0, 0])
# Regression test case for issue #15474
# https://github.com/sympy/sympy/issues/15474
a = Symbol("a", real=True)
eq24 = [x(t).diff(t) - a*y(t), y(t).diff(t) + a*x(t)]
sol24 = [Eq(x(t), C1*sin(a*t) + C2*cos(a*t)), Eq(y(t), C1*cos(a*t) - C2*sin(a*t))]
assert dsolve(eq24) == sol24
assert checksysodesol(eq24, sol24) == (True, [0, 0])
# Regression test case for issue #19150
# https://github.com/sympy/sympy/issues/19150
eqs25 = [Eq(Derivative(f(t), t), 0),
Eq(Derivative(g(t), t), (f(t) - 2*g(t) + x(t))/(b*c)),
Eq(Derivative(x(t), t), (g(t) - 2*x(t) + y(t))/(b*c)),
Eq(Derivative(y(t), t), (h(t) + x(t) - 2*y(t))/(b*c)),
Eq(Derivative(h(t), t), 0)]
sol25 = [Eq(f(t), -3*C1 + 4*C2),
Eq(g(t), -2*C1 + 3*C2 - C3*exp(-2*t/(b*c)) + C4*exp(-t*(sqrt(2) + 2)/(b*c)) + C5*exp(-t*(2 -
sqrt(2))/(b*c))),
Eq(x(t), -C1 + 2*C2 - sqrt(2)*C4*exp(-t*(sqrt(2) + 2)/(b*c)) + sqrt(2)*C5*exp(-t*(2 -
sqrt(2))/(b*c))),
Eq(y(t), C2 + C3*exp(-2*t/(b*c)) + C4*exp(-t*(sqrt(2) + 2)/(b*c)) + C5*exp(-t*(2 - sqrt(2))/(b*c))),
Eq(h(t), C1)]
assert dsolve(eqs25) == sol25
assert checksysodesol(eqs25, sol25) == (True, [0, 0, 0, 0, 0])
eq26 = [Eq(Derivative(f(t), t), 2*f(t)), Eq(Derivative(g(t), t), 3*f(t) + 7*g(t))]
sol26 = [Eq(f(t), -5*C1*exp(2*t)/3), Eq(g(t), C1*exp(2*t) + C2*exp(7*t))]
assert dsolve(eq26) == sol26
assert checksysodesol(eq26, sol26) == (True, [0, 0])
eq27 = [Eq(Derivative(f(t), t), -9*I*f(t) - 4*g(t)), Eq(Derivative(g(t), t), -4*I*g(t))]
sol27 = [Eq(f(t), 4*I*C1*exp(-4*I*t)/5 + C2*exp(-9*I*t)), Eq(g(t), C1*exp(-4*I*t))]
assert dsolve(eq27) == sol27
assert checksysodesol(eq27, sol27) == (True, [0, 0])
eq28 = [Eq(Derivative(f(t), t), -9*I*f(t)), Eq(Derivative(g(t), t), -4*I*g(t))]
sol28 = [Eq(f(t), C1*exp(-9*I*t)), Eq(g(t), C2*exp(-4*I*t))]
assert dsolve(eq28) == sol28
assert checksysodesol(eq28, sol28) == (True, [0, 0])
eq29 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), 0)]
sol29 = [Eq(f(t), C1), Eq(g(t), C2)]
assert dsolve(eq29) == sol29
assert checksysodesol(eq29, sol29) == (True, [0, 0])
eq30 = [Eq(Derivative(f(t), t), f(t)), Eq(Derivative(g(t), t), 0)]
sol30 = [Eq(f(t), C1*exp(t)), Eq(g(t), C2)]
assert dsolve(eq30) == sol30
assert checksysodesol(eq30, sol30) == (True, [0, 0])
eq31 = [Eq(Derivative(f(t), t), g(t)), Eq(Derivative(g(t), t), 0)]
sol31 = [Eq(f(t), C1 + C2*t), Eq(g(t), C2)]
assert dsolve(eq31) == sol31
assert checksysodesol(eq31, sol31) == (True, [0, 0])
eq32 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), f(t))]
sol32 = [Eq(f(t), C1), Eq(g(t), C1*t + C2)]
assert dsolve(eq32) == sol32
assert checksysodesol(eq32, sol32) == (True, [0, 0])
eq33 = [Eq(Derivative(f(t), t), 0), Eq(Derivative(g(t), t), g(t))]
sol33 = [Eq(f(t), C1), Eq(g(t), C2*exp(t))]
assert dsolve(eq33) == sol33
assert checksysodesol(eq33, sol33) == (True, [0, 0])
eq34 = [Eq(Derivative(f(t), t), f(t)), Eq(Derivative(g(t), t), I*g(t))]
sol34 = [Eq(f(t), C1*exp(t)), Eq(g(t), C2*exp(I*t))]
assert dsolve(eq34) == sol34
assert checksysodesol(eq34, sol34) == (True, [0, 0])
eq35 = [Eq(Derivative(f(t), t), I*f(t)), Eq(Derivative(g(t), t), -I*g(t))]
sol35 = [Eq(f(t), C1*exp(I*t)), Eq(g(t), C2*exp(-I*t))]
assert dsolve(eq35) == sol35
assert checksysodesol(eq35, sol35) == (True, [0, 0])
eq36 = [Eq(Derivative(f(t), t), I*g(t)), Eq(Derivative(g(t), t), 0)]
sol36 = [Eq(f(t), I*C1 + I*C2*t), Eq(g(t), C2)]
assert dsolve(eq36) == sol36
assert checksysodesol(eq36, sol36) == (True, [0, 0])
eq37 = [Eq(Derivative(f(t), t), I*g(t)), Eq(Derivative(g(t), t), I*f(t))]
sol37 = [Eq(f(t), -C1*exp(-I*t) + C2*exp(I*t)), Eq(g(t), C1*exp(-I*t) + C2*exp(I*t))]
assert dsolve(eq37) == sol37
assert checksysodesol(eq37, sol37) == (True, [0, 0])
# Multiple systems
eq1 = [Eq(Derivative(f(t), t)**2, g(t)**2), Eq(-f(t) + Derivative(g(t), t), 0)]
sol1 = [[Eq(f(t), -C1*sin(t) - C2*cos(t)),
Eq(g(t), C1*cos(t) - C2*sin(t))],
[Eq(f(t), -C1*exp(-t) + C2*exp(t)),
Eq(g(t), C1*exp(-t) + C2*exp(t))]]
assert dsolve(eq1) == sol1
for sol in sol1:
assert checksysodesol(eq1, sol) == (True, [0, 0])
def test_sysode_linear_neq_order1_type2():
f, g, h, k = symbols('f g h k', cls=Function)
x, t, a, b, c, d, y = symbols('x t a b c d y')
k1, k2 = symbols('k1 k2')
eqs1 = [Eq(Derivative(f(x), x), f(x) + g(x) + 5),
Eq(Derivative(g(x), x), -f(x) - g(x) + 7)]
sol1 = [Eq(f(x), C1 + C2 + 6*x**2 + x*(C2 + 5)),
Eq(g(x), -C1 - 6*x**2 - x*(C2 - 7))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
eqs2 = [Eq(Derivative(f(x), x), f(x) + g(x) + 5),
Eq(Derivative(g(x), x), f(x) + g(x) + 7)]
sol2 = [Eq(f(x), -C1 + C2*exp(2*x) - x - 3),
Eq(g(x), C1 + C2*exp(2*x) + x - 3)]
assert dsolve(eqs2) == sol2
assert checksysodesol(eqs2, sol2) == (True, [0, 0])
eqs3 = [Eq(Derivative(f(x), x), f(x) + 5),
Eq(Derivative(g(x), x), f(x) + 7)]
sol3 = [Eq(f(x), C1*exp(x) - 5),
Eq(g(x), C1*exp(x) + C2 + 2*x - 5)]
assert dsolve(eqs3) == sol3
assert checksysodesol(eqs3, sol3) == (True, [0, 0])
eqs4 = [Eq(Derivative(f(x), x), f(x) + exp(x)),
Eq(Derivative(g(x), x), x*exp(x) + f(x) + g(x))]
sol4 = [Eq(f(x), C1*exp(x) + x*exp(x)),
Eq(g(x), C1*x*exp(x) + C2*exp(x) + x**2*exp(x))]
assert dsolve(eqs4) == sol4
assert checksysodesol(eqs4, sol4) == (True, [0, 0])
eqs5 = [Eq(Derivative(f(x), x), 5*x + f(x) + g(x)),
Eq(Derivative(g(x), x), f(x) - g(x))]
sol5 = [Eq(f(x), C1*(1 + sqrt(2))*exp(sqrt(2)*x) + C2*(1 - sqrt(2))*exp(-sqrt(2)*x) + x*Rational(-5, 2) +
Rational(-5, 2)),
Eq(g(x), C1*exp(sqrt(2)*x) + C2*exp(-sqrt(2)*x) + x*Rational(-5, 2))]
assert dsolve(eqs5) == sol5
assert checksysodesol(eqs5, sol5) == (True, [0, 0])
eqs6 = [Eq(Derivative(f(x), x), -9*f(x) - 4*g(x)),
Eq(Derivative(g(x), x), -4*g(x)),
Eq(Derivative(h(x), x), h(x) + exp(x))]
sol6 = [Eq(f(x), C2*exp(-4*x)*Rational(-4, 5) + C1*exp(-9*x)),
Eq(g(x), C2*exp(-4*x)),
Eq(h(x), C3*exp(x) + x*exp(x))]
assert dsolve(eqs6) == sol6
assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0])
# Regression test case for issue #8859
# https://github.com/sympy/sympy/issues/8859
eqs7 = [Eq(Derivative(f(t), t), 3*t + f(t)),
Eq(Derivative(g(t), t), g(t))]
sol7 = [Eq(f(t), C1*exp(t) - 3*t - 3),
Eq(g(t), C2*exp(t))]
assert dsolve(eqs7) == sol7
assert checksysodesol(eqs7, sol7) == (True, [0, 0])
# Regression test case for issue #8567
# https://github.com/sympy/sympy/issues/8567
eqs8 = [Eq(Derivative(f(t), t), f(t) + 2*g(t)),
Eq(Derivative(g(t), t), -2*f(t) + g(t) + 2*exp(t))]
sol8 = [Eq(f(t), C1*exp(t)*sin(2*t) + C2*exp(t)*cos(2*t)
+ exp(t)*sin(2*t)**2 + exp(t)*cos(2*t)**2),
Eq(g(t), C1*exp(t)*cos(2*t) - C2*exp(t)*sin(2*t))]
assert dsolve(eqs8) == sol8
assert checksysodesol(eqs8, sol8) == (True, [0, 0])
# Regression test case for issue #19150
# https://github.com/sympy/sympy/issues/19150
eqs9 = [Eq(Derivative(f(t), t), (c - 2*f(t) + g(t))/(a*b)),
Eq(Derivative(g(t), t), (f(t) - 2*g(t) + h(t))/(a*b)),
Eq(Derivative(h(t), t), (d + g(t) - 2*h(t))/(a*b))]
sol9 = [Eq(f(t), -C1*exp(-2*t/(a*b)) + C2*exp(-t*(sqrt(2) + 2)/(a*b)) + C3*exp(-t*(2 - sqrt(2))/(a*b)) +
Mul(Rational(1, 4), 3*c + d, evaluate=False)),
Eq(g(t), -sqrt(2)*C2*exp(-t*(sqrt(2) + 2)/(a*b)) + sqrt(2)*C3*exp(-t*(2 - sqrt(2))/(a*b)) +
Mul(Rational(1, 2), c + d, evaluate=False)),
Eq(h(t), C1*exp(-2*t/(a*b)) + C2*exp(-t*(sqrt(2) + 2)/(a*b)) + C3*exp(-t*(2 - sqrt(2))/(a*b)) +
Mul(Rational(1, 4), c + 3*d, evaluate=False))]
assert dsolve(eqs9) == sol9
assert checksysodesol(eqs9, sol9) == (True, [0, 0, 0])
# Regression test case for issue #16635
# https://github.com/sympy/sympy/issues/16635
eqs10 = [Eq(Derivative(f(t), t), 15*t + f(t) - g(t) - 10),
Eq(Derivative(g(t), t), -15*t + f(t) - g(t) - 5)]
sol10 = [Eq(f(t), C1 + C2 + 5*t**3 + 5*t**2 + t*(C2 - 10)),
Eq(g(t), C1 + 5*t**3 - 10*t**2 + t*(C2 - 5))]
assert dsolve(eqs10) == sol10
assert checksysodesol(eqs10, sol10) == (True, [0, 0])
# Multiple solutions
eqs11 = [Eq(Derivative(f(t), t)**2 - 2*Derivative(f(t), t) + 1, 4),
Eq(-y*f(t) + Derivative(g(t), t), 0)]
sol11 = [[Eq(f(t), C1 - t), Eq(g(t), C1*t*y + C2*y + t**2*y*Rational(-1, 2))],
[Eq(f(t), C1 + 3*t), Eq(g(t), C1*t*y + C2*y + t**2*y*Rational(3, 2))]]
assert dsolve(eqs11) == sol11
for s11 in sol11:
assert checksysodesol(eqs11, s11) == (True, [0, 0])
# test case for issue #19831
# https://github.com/sympy/sympy/issues/19831
n = symbols('n', positive=True)
x0 = symbols('x_0')
t0 = symbols('t_0')
x_0 = symbols('x_0')
t_0 = symbols('t_0')
t = symbols('t')
x = Function('x')
y = Function('y')
T = symbols('T')
eqs12 = [Eq(Derivative(y(t), t), x(t)),
Eq(Derivative(x(t), t), n*(y(t) + 1))]
sol12 = [Eq(y(t), C1*exp(sqrt(n)*t)*n**Rational(-1, 2) - C2*exp(-sqrt(n)*t)*n**Rational(-1, 2) - 1),
Eq(x(t), C1*exp(sqrt(n)*t) + C2*exp(-sqrt(n)*t))]
assert dsolve(eqs12) == sol12
assert checksysodesol(eqs12, sol12) == (True, [0, 0])
sol12b = [
Eq(y(t), (T*exp(-sqrt(n)*t_0)/2 + exp(-sqrt(n)*t_0)/2 +
x_0*exp(-sqrt(n)*t_0)/(2*sqrt(n)))*exp(sqrt(n)*t) +
(T*exp(sqrt(n)*t_0)/2 + exp(sqrt(n)*t_0)/2 -
x_0*exp(sqrt(n)*t_0)/(2*sqrt(n)))*exp(-sqrt(n)*t) - 1),
Eq(x(t), (T*sqrt(n)*exp(-sqrt(n)*t_0)/2 + sqrt(n)*exp(-sqrt(n)*t_0)/2
+ x_0*exp(-sqrt(n)*t_0)/2)*exp(sqrt(n)*t)
- (T*sqrt(n)*exp(sqrt(n)*t_0)/2 + sqrt(n)*exp(sqrt(n)*t_0)/2 -
x_0*exp(sqrt(n)*t_0)/2)*exp(-sqrt(n)*t))
]
assert dsolve(eqs12, ics={y(t0): T, x(t0): x0}) == sol12b
assert checksysodesol(eqs12, sol12b) == (True, [0, 0])
#Test cases added for the issue 19763
#https://github.com/sympy/sympy/issues/19763
eq13 = [Eq(Derivative(f(t), t), f(t) + g(t) + 9),
Eq(Derivative(g(t), t), 2*f(t) + 5*g(t) + 23)]
sol13 = [Eq(f(t), -C1*(2 + sqrt(6))*exp(t*(3 - sqrt(6)))/2 - C2*(2 - sqrt(6))*exp(t*(sqrt(6) + 3))/2 -
Rational(22,3)),
Eq(g(t), C1*exp(t*(3 - sqrt(6))) + C2*exp(t*(sqrt(6) + 3)) - Rational(5,3))]
assert dsolve(eq13) == sol13
assert checksysodesol(eq13, sol13) == (True, [0, 0])
eq14 = [Eq(Derivative(f(t), t), f(t) + g(t) + 81),
Eq(Derivative(g(t), t), -2*f(t) + g(t) + 23)]
sol14 = [Eq(f(t), sqrt(2)*C1*exp(t)*sin(sqrt(2)*t)/2
+ sqrt(2)*C2*exp(t)*cos(sqrt(2)*t)/2
- 58*sin(sqrt(2)*t)**2/3 - 58*cos(sqrt(2)*t)**2/3),
Eq(g(t), C1*exp(t)*cos(sqrt(2)*t) - C2*exp(t)*sin(sqrt(2)*t)
- 185*sin(sqrt(2)*t)**2/3 - 185*cos(sqrt(2)*t)**2/3)]
assert dsolve(eq14) == sol14
assert checksysodesol(eq14, sol14) == (True, [0,0])
eq15 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1),
Eq(Derivative(g(t), t), 3*f(t) + 4*g(t) + k2)]
sol15 = [Eq(f(t), -C1*(3 - sqrt(33))*exp(t*(5 + sqrt(33))/2)/6 -
C2*(3 + sqrt(33))*exp(t*(5 - sqrt(33))/2)/6 + 2*k1 - k2),
Eq(g(t), C1*exp(t*(5 + sqrt(33))/2) + C2*exp(t*(5 - sqrt(33))/2) -
Mul(Rational(1,2), 3*k1 - k2, evaluate = False))]
assert dsolve(eq15) == sol15
assert checksysodesol(eq15, sol15) == (True, [0,0])
eq16 = [Eq(Derivative(f(t), t), k1),
Eq(Derivative(g(t), t), k2)]
sol16 = [Eq(f(t), C1 + k1*t),
Eq(g(t), C2 + k2*t)]
assert dsolve(eq16) == sol16
assert checksysodesol(eq16, sol16) == (True, [0,0])
eq17 = [Eq(Derivative(f(t), t), 0),
Eq(Derivative(g(t), t), c*f(t) + k2)]
sol17 = [Eq(f(t), C1),
Eq(g(t), C2*c + t*(C1*c + k2))]
assert dsolve(eq17) == sol17
assert checksysodesol(eq17 , sol17) == (True , [0,0])
eq18 = [Eq(Derivative(f(t), t), k1),
Eq(Derivative(g(t), t), f(t) + k2)]
sol18 = [Eq(f(t), C1 + k1*t),
Eq(g(t), C2 + k1*t**2/2 + t*(C1 + k2))]
assert dsolve(eq18) == sol18
assert checksysodesol(eq18 , sol18) == (True , [0,0])
eq19 = [Eq(Derivative(f(t), t), k1),
Eq(Derivative(g(t), t), f(t) + 2*g(t) + k2)]
sol19 = [Eq(f(t), -2*C1 + k1*t),
Eq(g(t), C1 + C2*exp(2*t) - k1*t/2 - Mul(Rational(1,4), k1 + 2*k2 , evaluate = False))]
assert dsolve(eq19) == sol19
assert checksysodesol(eq19 , sol19) == (True , [0,0])
eq20 = [Eq(diff(f(t), t), f(t) + k1),
Eq(diff(g(t), t), k2)]
sol20 = [Eq(f(t), C1*exp(t) - k1),
Eq(g(t), C2 + k2*t)]
assert dsolve(eq20) == sol20
assert checksysodesol(eq20 , sol20) == (True , [0,0])
eq21 = [Eq(diff(f(t), t), g(t) + k1),
Eq(diff(g(t), t), 0)]
sol21 = [Eq(f(t), C1 + t*(C2 + k1)),
Eq(g(t), C2)]
assert dsolve(eq21) == sol21
assert checksysodesol(eq21 , sol21) == (True , [0,0])
eq22 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1),
Eq(Derivative(g(t), t), k2)]
sol22 = [Eq(f(t), -2*C1 + C2*exp(t) - k1 - 2*k2*t - 2*k2),
Eq(g(t), C1 + k2*t)]
assert dsolve(eq22) == sol22
assert checksysodesol(eq22 , sol22) == (True , [0,0])
eq23 = [Eq(Derivative(f(t), t), g(t) + k1),
Eq(Derivative(g(t), t), 2*g(t) + k2)]
sol23 = [Eq(f(t), C1 + C2*exp(2*t)/2 - k2/4 + t*(2*k1 - k2)/2),
Eq(g(t), C2*exp(2*t) - k2/2)]
assert dsolve(eq23) == sol23
assert checksysodesol(eq23 , sol23) == (True , [0,0])
eq24 = [Eq(Derivative(f(t), t), f(t) + k1),
Eq(Derivative(g(t), t), 2*f(t) + k2)]
sol24 = [Eq(f(t), C1*exp(t)/2 - k1),
Eq(g(t), C1*exp(t) + C2 - 2*k1 - t*(2*k1 - k2))]
assert dsolve(eq24) == sol24
assert checksysodesol(eq24 , sol24) == (True , [0,0])
eq25 = [Eq(Derivative(f(t), t), f(t) + 2*g(t) + k1),
Eq(Derivative(g(t), t), 3*f(t) + 6*g(t) + k2)]
sol25 = [Eq(f(t), -2*C1 + C2*exp(7*t)/3 + 2*t*(3*k1 - k2)/7 -
Mul(Rational(1,49), k1 + 2*k2 , evaluate = False)),
Eq(g(t), C1 + C2*exp(7*t) - t*(3*k1 - k2)/7 -
Mul(Rational(3,49), k1 + 2*k2 , evaluate = False))]
assert dsolve(eq25) == sol25
assert checksysodesol(eq25 , sol25) == (True , [0,0])
eq26 = [Eq(Derivative(f(t), t), 2*f(t) - g(t) + k1),
Eq(Derivative(g(t), t), 4*f(t) - 2*g(t) + 2*k1)]
sol26 = [Eq(f(t), C1 + 2*C2 + t*(2*C1 + k1)),
Eq(g(t), 4*C2 + t*(4*C1 + 2*k1))]
assert dsolve(eq26) == sol26
assert checksysodesol(eq26 , sol26) == (True , [0,0])
# Test Case added for issue #22715
# https://github.com/sympy/sympy/issues/22715
eq27 = [Eq(diff(x(t),t),-1*y(t)+10), Eq(diff(y(t),t),5*x(t)-2*y(t)+3)]
sol27 = [Eq(x(t), (C1/5 - 2*C2/5)*exp(-t)*cos(2*t)
- (2*C1/5 + C2/5)*exp(-t)*sin(2*t)
+ 17*sin(2*t)**2/5 + 17*cos(2*t)**2/5),
Eq(y(t), C1*exp(-t)*cos(2*t) - C2*exp(-t)*sin(2*t)
+ 10*sin(2*t)**2 + 10*cos(2*t)**2)]
assert dsolve(eq27) == sol27
assert checksysodesol(eq27 , sol27) == (True , [0,0])
def test_sysode_linear_neq_order1_type3():
f, g, h, k, x0 , y0 = symbols('f g h k x0 y0', cls=Function)
x, t, a = symbols('x t a')
r = symbols('r', real=True)
eqs1 = [Eq(Derivative(f(r), r), r*g(r) + f(r)),
Eq(Derivative(g(r), r), -r*f(r) + g(r))]
sol1 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2)),
Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
eqs2 = [Eq(Derivative(f(x), x), x**2*g(x) + x*f(x)),
Eq(Derivative(g(x), x), 2*x**2*f(x) + (3*x**2 + x)*g(x))]
sol2 = [Eq(f(x), (sqrt(17)*C1/17 + C2*(17 - 3*sqrt(17))/34)*exp(x**3*(3 + sqrt(17))/6 + x**2/2) -
exp(x**3*(3 - sqrt(17))/6 + x**2/2)*(sqrt(17)*C1/17 + C2*(3*sqrt(17) + 17)*Rational(-1, 34))),
Eq(g(x), exp(x**3*(3 - sqrt(17))/6 + x**2/2)*(C1*(17 - 3*sqrt(17))/34 + sqrt(17)*C2*Rational(-2,
17)) + exp(x**3*(3 + sqrt(17))/6 + x**2/2)*(C1*(3*sqrt(17) + 17)/34 + sqrt(17)*C2*Rational(2, 17)))]
assert dsolve(eqs2) == sol2
assert checksysodesol(eqs2, sol2) == (True, [0, 0])
eqs3 = [Eq(f(x).diff(x), x*f(x) + g(x)),
Eq(g(x).diff(x), -f(x) + x*g(x))]
sol3 = [Eq(f(x), (C1/2 + I*C2/2)*exp(x**2/2 - I*x) + exp(x**2/2 + I*x)*(C1/2 + I*C2*Rational(-1, 2))),
Eq(g(x), (I*C1/2 + C2/2)*exp(x**2/2 + I*x) - exp(x**2/2 - I*x)*(I*C1/2 + C2*Rational(-1, 2)))]
assert dsolve(eqs3) == sol3
assert checksysodesol(eqs3, sol3) == (True, [0, 0])
eqs4 = [Eq(f(x).diff(x), x*(f(x) + g(x) + h(x))), Eq(g(x).diff(x), x*(f(x) + g(x) + h(x))),
Eq(h(x).diff(x), x*(f(x) + g(x) + h(x)))]
sol4 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)),
Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)),
Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2))]
assert dsolve(eqs4) == sol4
assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0])
eqs5 = [Eq(f(x).diff(x), x**2*(f(x) + g(x) + h(x))), Eq(g(x).diff(x), x**2*(f(x) + g(x) + h(x))),
Eq(h(x).diff(x), x**2*(f(x) + g(x) + h(x)))]
sol5 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3)),
Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3)),
Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(x**3))]
assert dsolve(eqs5) == sol5
assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0])
eqs6 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x) + k(x))),
Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x) + k(x))),
Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x) + k(x))),
Eq(Derivative(k(x), x), x*(f(x) + g(x) + h(x) + k(x)))]
sol6 = [Eq(f(x), -C1/4 - C2/4 - C3/4 + 3*C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)),
Eq(g(x), 3*C1/4 - C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)),
Eq(h(x), -C1/4 + 3*C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2)),
Eq(k(x), -C1/4 - C2/4 + 3*C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 + C4/4)*exp(2*x**2))]
assert dsolve(eqs6) == sol6
assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0])
y = symbols("y", real=True)
eqs7 = [Eq(Derivative(f(y), y), y*f(y) + g(y)),
Eq(Derivative(g(y), y), y*g(y) - f(y))]
sol7 = [Eq(f(y), C1*exp(y**2/2)*sin(y) + C2*exp(y**2/2)*cos(y)),
Eq(g(y), C1*exp(y**2/2)*cos(y) - C2*exp(y**2/2)*sin(y))]
assert dsolve(eqs7) == sol7
assert checksysodesol(eqs7, sol7) == (True, [0, 0])
#Test cases added for the issue 19763
#https://github.com/sympy/sympy/issues/19763
eqs8 = [Eq(Derivative(f(t), t), 5*t*f(t) + 2*h(t)),
Eq(Derivative(h(t), t), 2*f(t) + 5*t*h(t))]
sol8 = [Eq(f(t), Mul(-1, (C1/2 - C2/2), evaluate = False)*exp(5*t**2/2 - 2*t) + (C1/2 + C2/2)*exp(5*t**2/2 + 2*t)),
Eq(h(t), (C1/2 - C2/2)*exp(5*t**2/2 - 2*t) + (C1/2 + C2/2)*exp(5*t**2/2 + 2*t))]
assert dsolve(eqs8) == sol8
assert checksysodesol(eqs8, sol8) == (True, [0, 0])
eqs9 = [Eq(diff(f(t), t), 5*t*f(t) + t**2*g(t)),
Eq(diff(g(t), t), -t**2*f(t) + 5*t*g(t))]
sol9 = [Eq(f(t), (C1/2 - I*C2/2)*exp(I*t**3/3 + 5*t**2/2) + (C1/2 + I*C2/2)*exp(-I*t**3/3 + 5*t**2/2)),
Eq(g(t), Mul(-1, (I*C1/2 - C2/2) , evaluate = False)*exp(-I*t**3/3 + 5*t**2/2) + (I*C1/2 + C2/2)*exp(I*t**3/3 + 5*t**2/2))]
assert dsolve(eqs9) == sol9
assert checksysodesol(eqs9 , sol9) == (True , [0,0])
eqs10 = [Eq(diff(f(t), t), t**2*g(t) + 5*t*f(t)),
Eq(diff(g(t), t), -t**2*f(t) + (9*t**2 + 5*t)*g(t))]
sol10 = [Eq(f(t), (C1*(77 - 9*sqrt(77))/154 + sqrt(77)*C2/77)*exp(t**3*(sqrt(77) + 9)/6 + 5*t**2/2) + (C1*(77 + 9*sqrt(77))/154 - sqrt(77)*C2/77)*exp(t**3*(9 - sqrt(77))/6 + 5*t**2/2)),
Eq(g(t), (sqrt(77)*C1/77 + C2*(77 - 9*sqrt(77))/154)*exp(t**3*(9 - sqrt(77))/6 + 5*t**2/2) - (sqrt(77)*C1/77 - C2*(77 + 9*sqrt(77))/154)*exp(t**3*(sqrt(77) + 9)/6 + 5*t**2/2))]
assert dsolve(eqs10) == sol10
assert checksysodesol(eqs10 , sol10) == (True , [0,0])
eqs11 = [Eq(diff(f(t), t), 5*t*f(t) + t**2*g(t)),
Eq(diff(g(t), t), (1-t**2)*f(t) + (5*t + 9*t**2)*g(t))]
sol11 = [Eq(f(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)),
Eq(g(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))]
assert dsolve(eqs11) == sol11
@slow
def test_sysode_linear_neq_order1_type4():
f, g, h, k = symbols('f g h k', cls=Function)
x, t, a = symbols('x t a')
r = symbols('r', real=True)
eqs1 = [Eq(diff(f(r), r), f(r) + r*g(r) + r**2), Eq(diff(g(r), r), -r*f(r) + g(r) + r)]
sol1 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2) + exp(r)*sin(r**2/2)*Integral(r**2*exp(-r)*sin(r**2/2) +
r*exp(-r)*cos(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r**2*exp(-r)*cos(r**2/2) - r*exp(-r)*sin(r**2/2), r)),
Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2) - exp(r)*sin(r**2/2)*Integral(r**2*exp(-r)*cos(r**2/2) -
r*exp(-r)*sin(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r**2*exp(-r)*sin(r**2/2) + r*exp(-r)*cos(r**2/2), r))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
eqs2 = [Eq(diff(f(r), r), f(r) + r*g(r) + r), Eq(diff(g(r), r), -r*f(r) + g(r) + log(r))]
sol2 = [Eq(f(r), C1*exp(r)*sin(r**2/2) + C2*exp(r)*cos(r**2/2) + exp(r)*sin(r**2/2)*Integral(r*exp(-r)*sin(r**2/2) +
exp(-r)*log(r)*cos(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r*exp(-r)*cos(r**2/2) - exp(-r)*log(r)*sin(
r**2/2), r)),
Eq(g(r), C1*exp(r)*cos(r**2/2) - C2*exp(r)*sin(r**2/2) - exp(r)*sin(r**2/2)*Integral(r*exp(-r)*cos(r**2/2) -
exp(-r)*log(r)*sin(r**2/2), r) + exp(r)*cos(r**2/2)*Integral(r*exp(-r)*sin(r**2/2) + exp(-r)*log(r)*cos(
r**2/2), r))]
# XXX: dsolve hangs for this in integration
assert dsolve_system(eqs2, simplify=False, doit=False) == [sol2]
assert checksysodesol(eqs2, sol2) == (True, [0, 0])
eqs3 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x)) + x),
Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x)) + x),
Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x)) + 1)]
sol3 = [Eq(f(x), C1*Rational(-1, 3) + C2*Rational(-1, 3) + C3*Rational(2, 3) + x**2/6 + x*Rational(-1, 3) +
(C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) +
sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9)),
Eq(g(x), C1*Rational(2, 3) + C2*Rational(-1, 3) + C3*Rational(-1, 3) + x**2/6 + x*Rational(-1, 3) +
(C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) +
sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9)),
Eq(h(x), C1*Rational(-1, 3) + C2*Rational(2, 3) + C3*Rational(-1, 3) + x**2*Rational(-1, 3) +
x*Rational(2, 3) + (C1/3 + C2/3 + C3/3)*exp(x**2*Rational(3, 2)) +
sqrt(6)*sqrt(pi)*erf(sqrt(6)*x/2)*exp(x**2*Rational(3, 2))/18 + Rational(-2, 9))]
assert dsolve(eqs3) == sol3
assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0])
eqs4 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x)) + sin(x)),
Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x)) + sin(x)),
Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x)) + sin(x))]
sol4 = [Eq(f(x), C1*Rational(-1, 3) + C2*Rational(-1, 3) + C3*Rational(2, 3) + (C1/3 + C2/3 +
C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3,
2))),
Eq(g(x), C1*Rational(2, 3) + C2*Rational(-1, 3) + C3*Rational(-1, 3) + (C1/3 + C2/3 +
C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3,
2))),
Eq(h(x), C1*Rational(-1, 3) + C2*Rational(2, 3) + C3*Rational(-1, 3) + (C1/3 + C2/3 +
C3/3)*exp(x**2*Rational(3, 2)) + Integral(sin(x)*exp(x**2*Rational(-3, 2)), x)*exp(x**2*Rational(3,
2)))]
assert dsolve(eqs4) == sol4
assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0])
eqs5 = [Eq(Derivative(f(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)),
Eq(Derivative(g(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)),
Eq(Derivative(h(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1)),
Eq(Derivative(k(x), x), x*(f(x) + g(x) + h(x) + k(x) + 1))]
sol5 = [Eq(f(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(3, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)),
Eq(g(x), C1*Rational(3, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)),
Eq(h(x), C1*Rational(-1, 4) + C2*Rational(3, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4)),
Eq(k(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(3, 4) + C4*Rational(-1, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(2*x**2) + Rational(-1, 4))]
assert dsolve(eqs5) == sol5
assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0, 0])
eqs6 = [Eq(Derivative(f(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)),
Eq(Derivative(g(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)),
Eq(Derivative(h(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1)),
Eq(Derivative(k(x), x), x**2*(f(x) + g(x) + h(x) + k(x) + 1))]
sol6 = [Eq(f(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(3, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)),
Eq(g(x), C1*Rational(3, 4) + C2*Rational(-1, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)),
Eq(h(x), C1*Rational(-1, 4) + C2*Rational(3, 4) + C3*Rational(-1, 4) + C4*Rational(-1, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4)),
Eq(k(x), C1*Rational(-1, 4) + C2*Rational(-1, 4) + C3*Rational(3, 4) + C4*Rational(-1, 4) + (C1/4 +
C2/4 + C3/4 + C4/4)*exp(x**3*Rational(4, 3)) + Rational(-1, 4))]
assert dsolve(eqs6) == sol6
assert checksysodesol(eqs6, sol6) == (True, [0, 0, 0, 0])
eqs7 = [Eq(Derivative(f(x), x), (f(x) + g(x) + h(x))*log(x) + sin(x)), Eq(Derivative(g(x), x), (f(x) + g(x)
+ h(x))*log(x) + sin(x)), Eq(Derivative(h(x), x), (f(x) + g(x) + h(x))*log(x) + sin(x))]
sol7 = [Eq(f(x), -C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 +
C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) -
3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x)),
Eq(g(x), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 +
C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) -
3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x)),
Eq(h(x), -C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 +
C3/3)*exp(x*(3*log(x) - 3)) + exp(x*(3*log(x) -
3))*Integral(exp(3*x)*exp(-3*x*log(x))*sin(x), x))]
with dotprodsimp(True):
assert dsolve(eqs7, simplify=False, doit=False) == sol7
assert checksysodesol(eqs7, sol7) == (True, [0, 0, 0])
eqs8 = [Eq(Derivative(f(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x)), Eq(Derivative(g(x), x), (f(x)
+ g(x) + h(x) + k(x))*log(x) + sin(x)), Eq(Derivative(h(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) +
sin(x)), Eq(Derivative(k(x), x), (f(x) + g(x) + h(x) + k(x))*log(x) + sin(x))]
sol8 = [Eq(f(x), -C1/4 - C2/4 - C3/4 + 3*C4/4 + (C1/4 + C2/4 + C3/4 +
C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) -
4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)),
Eq(g(x), 3*C1/4 - C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 +
C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) -
4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)),
Eq(h(x), -C1/4 + 3*C2/4 - C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 +
C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) -
4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x)),
Eq(k(x), -C1/4 - C2/4 + 3*C3/4 - C4/4 + (C1/4 + C2/4 + C3/4 +
C4/4)*exp(x*(4*log(x) - 4)) + exp(x*(4*log(x) -
4))*Integral(exp(4*x)*exp(-4*x*log(x))*sin(x), x))]
with dotprodsimp(True):
assert dsolve(eqs8) == sol8
assert checksysodesol(eqs8, sol8) == (True, [0, 0, 0, 0])
def test_sysode_linear_neq_order1_type5_type6():
f, g = symbols("f g", cls=Function)
x, x_ = symbols("x x_")
# Type 5
eqs1 = [Eq(Derivative(f(x), x), (2*f(x) + g(x))/x), Eq(Derivative(g(x), x), (f(x) + 2*g(x))/x)]
sol1 = [Eq(f(x), -C1*x + C2*x**3), Eq(g(x), C1*x + C2*x**3)]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
# Type 6
eqs2 = [Eq(Derivative(f(x), x), (2*f(x) + g(x) + 1)/x),
Eq(Derivative(g(x), x), (x + f(x) + 2*g(x))/x)]
sol2 = [Eq(f(x), C2*x**3 - x*(C1 + Rational(1, 4)) + x*log(x)*Rational(-1, 2) + Rational(-2, 3)),
Eq(g(x), C2*x**3 + x*log(x)/2 + x*(C1 + Rational(-1, 4)) + Rational(1, 3))]
assert dsolve(eqs2) == sol2
assert checksysodesol(eqs2, sol2) == (True, [0, 0])
def test_higher_order_to_first_order():
f, g = symbols('f g', cls=Function)
x = symbols('x')
eqs1 = [Eq(Derivative(f(x), (x, 2)), 2*f(x) + g(x)),
Eq(Derivative(g(x), (x, 2)), -f(x))]
sol1 = [Eq(f(x), -C2*x*exp(-x) + C3*x*exp(x) - (C1 - C2)*exp(-x) + (C3 + C4)*exp(x)),
Eq(g(x), C2*x*exp(-x) - C3*x*exp(x) + (C1 + C2)*exp(-x) + (C3 - C4)*exp(x))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
eqs2 = [Eq(f(x).diff(x, 2), 0), Eq(g(x).diff(x, 2), f(x))]
sol2 = [Eq(f(x), C1 + C2*x), Eq(g(x), C1*x**2/2 + C2*x**3/6 + C3 + C4*x)]
assert dsolve(eqs2) == sol2
assert checksysodesol(eqs2, sol2) == (True, [0, 0])
eqs3 = [Eq(Derivative(f(x), (x, 2)), 2*f(x)),
Eq(Derivative(g(x), (x, 2)), -f(x) + 2*g(x))]
sol3 = [Eq(f(x), 4*C1*exp(-sqrt(2)*x) + 4*C2*exp(sqrt(2)*x)),
Eq(g(x), sqrt(2)*C1*x*exp(-sqrt(2)*x) - sqrt(2)*C2*x*exp(sqrt(2)*x) + (C1 +
sqrt(2)*C4)*exp(-sqrt(2)*x) + (C2 - sqrt(2)*C3)*exp(sqrt(2)*x))]
assert dsolve(eqs3) == sol3
assert checksysodesol(eqs3, sol3) == (True, [0, 0])
eqs4 = [Eq(Derivative(f(x), (x, 2)), 2*f(x) + g(x)),
Eq(Derivative(g(x), (x, 2)), 2*g(x))]
sol4 = [Eq(f(x), C1*x*exp(sqrt(2)*x)/4 + C3*x*exp(-sqrt(2)*x)/4 + (C2/4 + sqrt(2)*C3/8)*exp(-sqrt(2)*x) -
exp(sqrt(2)*x)*(sqrt(2)*C1/8 + C4*Rational(-1, 4))),
Eq(g(x), sqrt(2)*C1*exp(sqrt(2)*x)/2 + sqrt(2)*C3*exp(-sqrt(2)*x)*Rational(-1, 2))]
assert dsolve(eqs4) == sol4
assert checksysodesol(eqs4, sol4) == (True, [0, 0])
eqs5 = [Eq(f(x).diff(x, 2), f(x)), Eq(g(x).diff(x, 2), f(x))]
sol5 = [Eq(f(x), -C1*exp(-x) + C2*exp(x)), Eq(g(x), -C1*exp(-x) + C2*exp(x) + C3 + C4*x)]
assert dsolve(eqs5) == sol5
assert checksysodesol(eqs5, sol5) == (True, [0, 0])
eqs6 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x)),
Eq(Derivative(g(x), (x, 2)), -f(x) - g(x))]
sol6 = [Eq(f(x), C1 + C2*x**2/2 + C2 + C4*x**3/6 + x*(C3 + C4)),
Eq(g(x), -C1 + C2*x**2*Rational(-1, 2) - C3*x + C4*x**3*Rational(-1, 6))]
assert dsolve(eqs6) == sol6
assert checksysodesol(eqs6, sol6) == (True, [0, 0])
eqs7 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x) + 1),
Eq(Derivative(g(x), (x, 2)), f(x) + g(x) + 1)]
sol7 = [Eq(f(x), -C1 - C2*x + sqrt(2)*C3*exp(sqrt(2)*x)/2 + sqrt(2)*C4*exp(-sqrt(2)*x)*Rational(-1, 2) +
Rational(-1, 2)),
Eq(g(x), C1 + C2*x + sqrt(2)*C3*exp(sqrt(2)*x)/2 + sqrt(2)*C4*exp(-sqrt(2)*x)*Rational(-1, 2) +
Rational(-1, 2))]
assert dsolve(eqs7) == sol7
assert checksysodesol(eqs7, sol7) == (True, [0, 0])
eqs8 = [Eq(Derivative(f(x), (x, 2)), f(x) + g(x) + 1),
Eq(Derivative(g(x), (x, 2)), -f(x) - g(x) + 1)]
sol8 = [Eq(f(x), C1 + C2 + C4*x**3/6 + x**4/12 + x**2*(C2/2 + Rational(1, 2)) + x*(C3 + C4)),
Eq(g(x), -C1 - C3*x + C4*x**3*Rational(-1, 6) + x**4*Rational(-1, 12) - x**2*(C2/2 + Rational(-1,
2)))]
assert dsolve(eqs8) == sol8
assert checksysodesol(eqs8, sol8) == (True, [0, 0])
x, y = symbols('x, y', cls=Function)
t, l = symbols('t, l')
eqs10 = [Eq(Derivative(x(t), (t, 2)), 5*x(t) + 43*y(t)),
Eq(Derivative(y(t), (t, 2)), x(t) + 9*y(t))]
sol10 = [Eq(x(t), C1*(61 - 9*sqrt(47))*sqrt(sqrt(47) + 7)*exp(-t*sqrt(sqrt(47) + 7))/2 + C2*sqrt(7 -
sqrt(47))*(61 + 9*sqrt(47))*exp(-t*sqrt(7 - sqrt(47)))/2 + C3*(61 - 9*sqrt(47))*sqrt(sqrt(47) +
7)*exp(t*sqrt(sqrt(47) + 7))*Rational(-1, 2) + C4*sqrt(7 - sqrt(47))*(61 + 9*sqrt(47))*exp(t*sqrt(7
- sqrt(47)))*Rational(-1, 2)),
Eq(y(t), C1*(7 - sqrt(47))*sqrt(sqrt(47) + 7)*exp(-t*sqrt(sqrt(47) + 7))*Rational(-1, 2) + C2*sqrt(7
- sqrt(47))*(sqrt(47) + 7)*exp(-t*sqrt(7 - sqrt(47)))*Rational(-1, 2) + C3*(7 -
sqrt(47))*sqrt(sqrt(47) + 7)*exp(t*sqrt(sqrt(47) + 7))/2 + C4*sqrt(7 - sqrt(47))*(sqrt(47) +
7)*exp(t*sqrt(7 - sqrt(47)))/2)]
assert dsolve(eqs10) == sol10
assert checksysodesol(eqs10, sol10) == (True, [0, 0])
eqs11 = [Eq(7*x(t) + Derivative(x(t), (t, 2)) - 9*Derivative(y(t), t), 0),
Eq(7*y(t) + 9*Derivative(x(t), t) + Derivative(y(t), (t, 2)), 0)]
sol11 = [Eq(y(t), C1*(9 - sqrt(109))*sin(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)/14 + C2*(9 -
sqrt(109))*cos(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C3*(9 +
sqrt(109))*sin(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)/14 + C4*(9 + sqrt(109))*cos(sqrt(2)*t*sqrt(95 -
9*sqrt(109))/2)*Rational(-1, 14)),
Eq(x(t), C1*(9 - sqrt(109))*cos(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C2*(9 -
sqrt(109))*sin(sqrt(2)*t*sqrt(9*sqrt(109) + 95)/2)*Rational(-1, 14) + C3*(9 +
sqrt(109))*cos(sqrt(2)*t*sqrt(95 - 9*sqrt(109))/2)/14 + C4*(9 + sqrt(109))*sin(sqrt(2)*t*sqrt(95 -
9*sqrt(109))/2)/14)]
assert dsolve(eqs11) == sol11
assert checksysodesol(eqs11, sol11) == (True, [0, 0])
# Euler Systems
# Note: To add examples of euler systems solver with non-homogeneous term.
eqs13 = [Eq(Derivative(f(t), (t, 2)), Derivative(f(t), t)/t + f(t)/t**2 + g(t)/t**2),
Eq(Derivative(g(t), (t, 2)), g(t)/t**2)]
sol13 = [Eq(f(t), C1*(sqrt(5) + 3)*Rational(-1, 2)*t**(Rational(1, 2) +
sqrt(5)*Rational(-1, 2)) + C2*t**(Rational(1, 2) +
sqrt(5)/2)*(3 - sqrt(5))*Rational(-1, 2) - C3*t**(1 -
sqrt(2))*(1 + sqrt(2)) - C4*t**(1 + sqrt(2))*(1 - sqrt(2))),
Eq(g(t), C1*(1 + sqrt(5))*Rational(-1, 2)*t**(Rational(1, 2) +
sqrt(5)*Rational(-1, 2)) + C2*t**(Rational(1, 2) +
sqrt(5)/2)*(1 - sqrt(5))*Rational(-1, 2))]
assert dsolve(eqs13) == sol13
assert checksysodesol(eqs13, sol13) == (True, [0, 0])
# Solving systems using dsolve separately
eqs14 = [Eq(Derivative(f(t), (t, 2)), t*f(t)),
Eq(Derivative(g(t), (t, 2)), t*g(t))]
sol14 = [Eq(f(t), C1*airyai(t) + C2*airybi(t)),
Eq(g(t), C3*airyai(t) + C4*airybi(t))]
assert dsolve(eqs14) == sol14
assert checksysodesol(eqs14, sol14) == (True, [0, 0])
eqs15 = [Eq(Derivative(x(t), (t, 2)), t*(4*Derivative(x(t), t) + 8*Derivative(y(t), t))),
Eq(Derivative(y(t), (t, 2)), t*(12*Derivative(x(t), t) - 6*Derivative(y(t), t)))]
sol15 = [Eq(x(t), C1 - erf(sqrt(6)*t)*(sqrt(6)*sqrt(pi)*C2/33 + sqrt(6)*sqrt(pi)*C3*Rational(-1, 44)) +
erfi(sqrt(5)*t)*(sqrt(5)*sqrt(pi)*C2*Rational(2, 55) + sqrt(5)*sqrt(pi)*C3*Rational(4, 55))),
Eq(y(t), C4 + erf(sqrt(6)*t)*(sqrt(6)*sqrt(pi)*C2*Rational(2, 33) + sqrt(6)*sqrt(pi)*C3*Rational(-1,
22)) + erfi(sqrt(5)*t)*(sqrt(5)*sqrt(pi)*C2*Rational(3, 110) + sqrt(5)*sqrt(pi)*C3*Rational(3, 55)))]
assert dsolve(eqs15) == sol15
assert checksysodesol(eqs15, sol15) == (True, [0, 0])
@slow
def test_higher_order_to_first_order_9():
f, g = symbols('f g', cls=Function)
x = symbols('x')
eqs9 = [f(x) + g(x) - 2*exp(I*x) + 2*Derivative(f(x), x) + Derivative(f(x), (x, 2)),
f(x) + g(x) - 2*exp(I*x) + 2*Derivative(g(x), x) + Derivative(g(x), (x, 2))]
sol9 = [Eq(f(x), -C1 + C4*exp(-2*x)/2 - (C2/2 - C3/2)*exp(-x)*cos(x)
+ (C2/2 + C3/2)*exp(-x)*sin(x) + 2*((1 - 2*I)*exp(I*x)*sin(x)**2/5)
+ 2*((1 - 2*I)*exp(I*x)*cos(x)**2/5)),
Eq(g(x), C1 - C4*exp(-2*x)/2 - (C2/2 - C3/2)*exp(-x)*cos(x)
+ (C2/2 + C3/2)*exp(-x)*sin(x) + 2*((1 - 2*I)*exp(I*x)*sin(x)**2/5)
+ 2*((1 - 2*I)*exp(I*x)*cos(x)**2/5))]
assert dsolve(eqs9) == sol9
assert checksysodesol(eqs9, sol9) == (True, [0, 0])
def test_higher_order_to_first_order_12():
f, g = symbols('f g', cls=Function)
x = symbols('x')
x, y = symbols('x, y', cls=Function)
t, l = symbols('t, l')
eqs12 = [Eq(4*x(t) + Derivative(x(t), (t, 2)) + 8*Derivative(y(t), t), 0),
Eq(4*y(t) - 8*Derivative(x(t), t) + Derivative(y(t), (t, 2)), 0)]
sol12 = [Eq(y(t), C1*(2 - sqrt(5))*sin(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C2*(2 -
sqrt(5))*cos(2*t*sqrt(4*sqrt(5) + 9))/2 + C3*(2 + sqrt(5))*sin(2*t*sqrt(9 - 4*sqrt(5)))*Rational(-1,
2) + C4*(2 + sqrt(5))*cos(2*t*sqrt(9 - 4*sqrt(5)))/2),
Eq(x(t), C1*(2 - sqrt(5))*cos(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C2*(2 -
sqrt(5))*sin(2*t*sqrt(4*sqrt(5) + 9))*Rational(-1, 2) + C3*(2 + sqrt(5))*cos(2*t*sqrt(9 -
4*sqrt(5)))/2 + C4*(2 + sqrt(5))*sin(2*t*sqrt(9 - 4*sqrt(5)))/2)]
assert dsolve(eqs12) == sol12
assert checksysodesol(eqs12, sol12) == (True, [0, 0])
def test_second_order_to_first_order_2():
f, g = symbols("f g", cls=Function)
x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m")
eqs2 = [Eq(f(x).diff(x, 2), 2*(x*g(x).diff(x) - g(x))),
Eq(g(x).diff(x, 2),-2*(x*f(x).diff(x) - f(x)))]
sol2 = [Eq(f(x), C1*x + x*Integral(C2*exp(-x_)*exp(I*exp(2*x_))/2 + C2*exp(-x_)*exp(-I*exp(2*x_))/2 -
I*C3*exp(-x_)*exp(I*exp(2*x_))/2 + I*C3*exp(-x_)*exp(-I*exp(2*x_))/2, (x_, log(x)))),
Eq(g(x), C4*x + x*Integral(I*C2*exp(-x_)*exp(I*exp(2*x_))/2 - I*C2*exp(-x_)*exp(-I*exp(2*x_))/2 +
C3*exp(-x_)*exp(I*exp(2*x_))/2 + C3*exp(-x_)*exp(-I*exp(2*x_))/2, (x_, log(x))))]
# XXX: dsolve hangs for this in integration
assert dsolve_system(eqs2, simplify=False, doit=False) == [sol2]
assert checksysodesol(eqs2, sol2) == (True, [0, 0])
eqs3 = (Eq(diff(f(t),t,t), 9*t*diff(g(t),t)-9*g(t)), Eq(diff(g(t),t,t),7*t*diff(f(t),t)-7*f(t)))
sol3 = [Eq(f(t), C1*t + t*Integral(C2*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/2 + C2*exp(-t_)*
exp(-3*sqrt(7)*exp(2*t_)/2)/2 + 3*sqrt(7)*C3*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/14 -
3*sqrt(7)*C3*exp(-t_)*exp(-3*sqrt(7)*exp(2*t_)/2)/14, (t_, log(t)))),
Eq(g(t), C4*t + t*Integral(sqrt(7)*C2*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/6 - sqrt(7)*C2*exp(-t_)*
exp(-3*sqrt(7)*exp(2*t_)/2)/6 + C3*exp(-t_)*exp(3*sqrt(7)*exp(2*t_)/2)/2 + C3*exp(-t_)*exp(-3*sqrt(7)*
exp(2*t_)/2)/2, (t_, log(t))))]
# XXX: dsolve hangs for this in integration
assert dsolve_system(eqs3, simplify=False, doit=False) == [sol3]
assert checksysodesol(eqs3, sol3) == (True, [0, 0])
# Regression Test case for sympy#19238
# https://github.com/sympy/sympy/issues/19238
# Note: When the doit method is removed, these particular types of systems
# can be divided first so that we have lesser number of big matrices.
eqs5 = [Eq(Derivative(g(t), (t, 2)), a*m),
Eq(Derivative(f(t), (t, 2)), 0)]
sol5 = [Eq(g(t), C1 + C2*t + a*m*t**2/2),
Eq(f(t), C3 + C4*t)]
assert dsolve(eqs5) == sol5
assert checksysodesol(eqs5, sol5) == (True, [0, 0])
# Type 2
eqs6 = [Eq(Derivative(f(t), (t, 2)), f(t)/t**4),
Eq(Derivative(g(t), (t, 2)), d*g(t)/t**4)]
sol6 = [Eq(f(t), C1*sqrt(t**2)*exp(-1/t) - C2*sqrt(t**2)*exp(1/t)),
Eq(g(t), C3*sqrt(t**2)*exp(-sqrt(d)/t)*d**Rational(-1, 2) -
C4*sqrt(t**2)*exp(sqrt(d)/t)*d**Rational(-1, 2))]
assert dsolve(eqs6) == sol6
assert checksysodesol(eqs6, sol6) == (True, [0, 0])
@slow
def test_second_order_to_first_order_slow1():
f, g = symbols("f g", cls=Function)
x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m")
# Type 1
eqs1 = [Eq(f(x).diff(x, 2), 2/x *(x*g(x).diff(x) - g(x))),
Eq(g(x).diff(x, 2),-2/x *(x*f(x).diff(x) - f(x)))]
sol1 = [Eq(f(x), C1*x + 2*C2*x*Ci(2*x) - C2*sin(2*x) - 2*C3*x*Si(2*x) - C3*cos(2*x)),
Eq(g(x), -2*C2*x*Si(2*x) - C2*cos(2*x) - 2*C3*x*Ci(2*x) + C3*sin(2*x) + C4*x)]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
def test_second_order_to_first_order_slow4():
f, g = symbols("f g", cls=Function)
x, t, x_, t_, d, a, m = symbols("x t x_ t_ d a m")
eqs4 = [Eq(Derivative(f(t), (t, 2)), t*sin(t)*Derivative(g(t), t) - g(t)*sin(t)),
Eq(Derivative(g(t), (t, 2)), t*sin(t)*Derivative(f(t), t) - f(t)*sin(t))]
sol4 = [Eq(f(t), C1*t + t*Integral(C2*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*exp(-sin(exp(t_)))/2 +
C2*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2 - C3*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*
exp(-sin(exp(t_)))/2 +
C3*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2, (t_, log(t)))),
Eq(g(t), C4*t + t*Integral(-C2*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*exp(-sin(exp(t_)))/2 +
C2*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2 + C3*exp(-t_)*exp(exp(t_)*cos(exp(t_)))*
exp(-sin(exp(t_)))/2 + C3*exp(-t_)*exp(-exp(t_)*cos(exp(t_)))*exp(sin(exp(t_)))/2, (t_, log(t))))]
# XXX: dsolve hangs for this in integration
assert dsolve_system(eqs4, simplify=False, doit=False) == [sol4]
assert checksysodesol(eqs4, sol4) == (True, [0, 0])
def test_component_division():
f, g, h, k = symbols('f g h k', cls=Function)
x = symbols("x")
funcs = [f(x), g(x), h(x), k(x)]
eqs1 = [Eq(Derivative(f(x), x), 2*f(x)),
Eq(Derivative(g(x), x), f(x)),
Eq(Derivative(h(x), x), h(x)),
Eq(Derivative(k(x), x), h(x)**4 + k(x))]
sol1 = [Eq(f(x), 2*C1*exp(2*x)),
Eq(g(x), C1*exp(2*x) + C2),
Eq(h(x), C3*exp(x)),
Eq(k(x), C3**4*exp(4*x)/3 + C4*exp(x))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0, 0, 0])
components1 = {((Eq(Derivative(f(x), x), 2*f(x)),), (Eq(Derivative(g(x), x), f(x)),)),
((Eq(Derivative(h(x), x), h(x)),), (Eq(Derivative(k(x), x), h(x)**4 + k(x)),))}
eqsdict1 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {h(x)}},
{f(x): Eq(Derivative(f(x), x), 2*f(x)),
g(x): Eq(Derivative(g(x), x), f(x)),
h(x): Eq(Derivative(h(x), x), h(x)),
k(x): Eq(Derivative(k(x), x), h(x)**4 + k(x))})
graph1 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), h(x))}]
assert {tuple(tuple(scc) for scc in wcc) for wcc in _component_division(eqs1, funcs, x)} == components1
assert _eqs2dict(eqs1, funcs) == eqsdict1
assert [set(element) for element in _dict2graph(eqsdict1[0])] == graph1
eqs2 = [Eq(Derivative(f(x), x), 2*f(x)),
Eq(Derivative(g(x), x), f(x)),
Eq(Derivative(h(x), x), h(x)),
Eq(Derivative(k(x), x), f(x)**4 + k(x))]
sol2 = [Eq(f(x), C1*exp(2*x)),
Eq(g(x), C1*exp(2*x)/2 + C2),
Eq(h(x), C3*exp(x)),
Eq(k(x), C1**4*exp(8*x)/7 + C4*exp(x))]
assert dsolve(eqs2) == sol2
assert checksysodesol(eqs2, sol2) == (True, [0, 0, 0, 0])
components2 = {frozenset([(Eq(Derivative(f(x), x), 2*f(x)),),
(Eq(Derivative(g(x), x), f(x)),),
(Eq(Derivative(k(x), x), f(x)**4 + k(x)),)]),
frozenset([(Eq(Derivative(h(x), x), h(x)),)])}
eqsdict2 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {f(x)}},
{f(x): Eq(Derivative(f(x), x), 2*f(x)),
g(x): Eq(Derivative(g(x), x), f(x)),
h(x): Eq(Derivative(h(x), x), h(x)),
k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))})
graph2 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), f(x))}]
assert {frozenset(tuple(scc) for scc in wcc) for wcc in _component_division(eqs2, funcs, x)} == components2
assert _eqs2dict(eqs2, funcs) == eqsdict2
assert [set(element) for element in _dict2graph(eqsdict2[0])] == graph2
eqs3 = [Eq(Derivative(f(x), x), 2*f(x)),
Eq(Derivative(g(x), x), x + f(x)),
Eq(Derivative(h(x), x), h(x)),
Eq(Derivative(k(x), x), f(x)**4 + k(x))]
sol3 = [Eq(f(x), C1*exp(2*x)),
Eq(g(x), C1*exp(2*x)/2 + C2 + x**2/2),
Eq(h(x), C3*exp(x)),
Eq(k(x), C1**4*exp(8*x)/7 + C4*exp(x))]
assert dsolve(eqs3) == sol3
assert checksysodesol(eqs3, sol3) == (True, [0, 0, 0, 0])
components3 = {frozenset([(Eq(Derivative(f(x), x), 2*f(x)),),
(Eq(Derivative(g(x), x), x + f(x)),),
(Eq(Derivative(k(x), x), f(x)**4 + k(x)),)]),
frozenset([(Eq(Derivative(h(x), x), h(x)),),])}
eqsdict3 = ({f(x): set(), g(x): {f(x)}, h(x): set(), k(x): {f(x)}},
{f(x): Eq(Derivative(f(x), x), 2*f(x)),
g(x): Eq(Derivative(g(x), x), x + f(x)),
h(x): Eq(Derivative(h(x), x), h(x)),
k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))})
graph3 = [{f(x), g(x), h(x), k(x)}, {(g(x), f(x)), (k(x), f(x))}]
assert {frozenset(tuple(scc) for scc in wcc) for wcc in _component_division(eqs3, funcs, x)} == components3
assert _eqs2dict(eqs3, funcs) == eqsdict3
assert [set(l) for l in _dict2graph(eqsdict3[0])] == graph3
# Note: To be uncommented when the default option to call dsolve first for
# single ODE system can be rearranged. This can be done after the doit
# option in dsolve is made False by default.
eqs4 = [Eq(Derivative(f(x), x), x*f(x) + 2*g(x)),
Eq(Derivative(g(x), x), f(x) + x*g(x) + x),
Eq(Derivative(h(x), x), h(x)),
Eq(Derivative(k(x), x), f(x)**4 + k(x))]
sol4 = [Eq(f(x), (C1/2 - sqrt(2)*C2/2 - sqrt(2)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 +\
sqrt(2)*x)/2, x)/2 + Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 +\
sqrt(2)*x)/2, x)/2)*exp(x**2/2 - sqrt(2)*x) + (C1/2 + sqrt(2)*C2/2 + sqrt(2)*Integral(x*exp(-x**2/2
- sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2
- sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2)*exp(x**2/2 + sqrt(2)*x)),
Eq(g(x), (-sqrt(2)*C1/4 + C2/2 + Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 -\
sqrt(2)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2,
x)/4)*exp(x**2/2 - sqrt(2)*x) + (sqrt(2)*C1/4 + C2/2 + Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 +
x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + sqrt(2)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 -
sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/4)*exp(x**2/2 + sqrt(2)*x)),
Eq(h(x), C3*exp(x)),
Eq(k(x), C4*exp(x) + exp(x)*Integral((C1*exp(x**2/2 - sqrt(2)*x)/2 + C1*exp(x**2/2 + sqrt(2)*x)/2 -
sqrt(2)*C2*exp(x**2/2 - sqrt(2)*x)/2 + sqrt(2)*C2*exp(x**2/2 + sqrt(2)*x)/2 - sqrt(2)*exp(x**2/2 -
sqrt(2)*x)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2 + exp(x**2/2 -
sqrt(2)*x)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 - sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2,
x)/2 + sqrt(2)*exp(x**2/2 + sqrt(2)*x)*Integral(x*exp(-x**2/2 - sqrt(2)*x)/2 + x*exp(-x**2/2 +
sqrt(2)*x)/2, x)/2 + exp(x**2/2 + sqrt(2)*x)*Integral(sqrt(2)*x*exp(-x**2/2 - sqrt(2)*x)/2 -
sqrt(2)*x*exp(-x**2/2 + sqrt(2)*x)/2, x)/2)**4*exp(-x), x))]
components4 = {(frozenset([Eq(Derivative(f(x), x), x*f(x) + 2*g(x)),
Eq(Derivative(g(x), x), x*g(x) + x + f(x))]),
frozenset([Eq(Derivative(k(x), x), f(x)**4 + k(x)),])),
(frozenset([Eq(Derivative(h(x), x), h(x)),]),)}
eqsdict4 = ({f(x): {g(x)}, g(x): {f(x)}, h(x): set(), k(x): {f(x)}},
{f(x): Eq(Derivative(f(x), x), x*f(x) + 2*g(x)),
g(x): Eq(Derivative(g(x), x), x*g(x) + x + f(x)),
h(x): Eq(Derivative(h(x), x), h(x)),
k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))})
graph4 = [{f(x), g(x), h(x), k(x)}, {(f(x), g(x)), (g(x), f(x)), (k(x), f(x))}]
assert {tuple(frozenset(scc) for scc in wcc) for wcc in _component_division(eqs4, funcs, x)} == components4
assert _eqs2dict(eqs4, funcs) == eqsdict4
assert [set(element) for element in _dict2graph(eqsdict4[0])] == graph4
# XXX: dsolve hangs in integration here:
assert dsolve_system(eqs4, simplify=False, doit=False) == [sol4]
assert checksysodesol(eqs4, sol4) == (True, [0, 0, 0, 0])
eqs5 = [Eq(Derivative(f(x), x), x*f(x) + 2*g(x)),
Eq(Derivative(g(x), x), x*g(x) + f(x)),
Eq(Derivative(h(x), x), h(x)),
Eq(Derivative(k(x), x), f(x)**4 + k(x))]
sol5 = [Eq(f(x), (C1/2 - sqrt(2)*C2/2)*exp(x**2/2 - sqrt(2)*x) + (C1/2 + sqrt(2)*C2/2)*exp(x**2/2 + sqrt(2)*x)),
Eq(g(x), (-sqrt(2)*C1/4 + C2/2)*exp(x**2/2 - sqrt(2)*x) + (sqrt(2)*C1/4 + C2/2)*exp(x**2/2 + sqrt(2)*x)),
Eq(h(x), C3*exp(x)),
Eq(k(x), C4*exp(x) + exp(x)*Integral((C1*exp(x**2/2 - sqrt(2)*x)/2 + C1*exp(x**2/2 + sqrt(2)*x)/2 -
sqrt(2)*C2*exp(x**2/2 - sqrt(2)*x)/2 + sqrt(2)*C2*exp(x**2/2 + sqrt(2)*x)/2)**4*exp(-x), x))]
components5 = {(frozenset([Eq(Derivative(f(x), x), x*f(x) + 2*g(x)),
Eq(Derivative(g(x), x), x*g(x) + f(x))]),
frozenset([Eq(Derivative(k(x), x), f(x)**4 + k(x)),])),
(frozenset([Eq(Derivative(h(x), x), h(x)),]),)}
eqsdict5 = ({f(x): {g(x)}, g(x): {f(x)}, h(x): set(), k(x): {f(x)}},
{f(x): Eq(Derivative(f(x), x), x*f(x) + 2*g(x)),
g(x): Eq(Derivative(g(x), x), x*g(x) + f(x)),
h(x): Eq(Derivative(h(x), x), h(x)),
k(x): Eq(Derivative(k(x), x), f(x)**4 + k(x))})
graph5 = [{f(x), g(x), h(x), k(x)}, {(f(x), g(x)), (g(x), f(x)), (k(x), f(x))}]
assert {tuple(frozenset(scc) for scc in wcc) for wcc in _component_division(eqs5, funcs, x)} == components5
assert _eqs2dict(eqs5, funcs) == eqsdict5
assert [set(element) for element in _dict2graph(eqsdict5[0])] == graph5
# XXX: dsolve hangs in integration here:
assert dsolve_system(eqs5, simplify=False, doit=False) == [sol5]
assert checksysodesol(eqs5, sol5) == (True, [0, 0, 0, 0])
def test_linodesolve():
t, x, a = symbols("t x a")
f, g, h = symbols("f g h", cls=Function)
# Testing the Errors
raises(ValueError, lambda: linodesolve(1, t))
raises(ValueError, lambda: linodesolve(a, t))
A1 = Matrix([[1, 2], [2, 4], [4, 6]])
raises(NonSquareMatrixError, lambda: linodesolve(A1, t))
A2 = Matrix([[1, 2, 1], [3, 1, 2]])
raises(NonSquareMatrixError, lambda: linodesolve(A2, t))
# Testing auto functionality
func = [f(t), g(t)]
eq = [Eq(f(t).diff(t) + g(t).diff(t), g(t)), Eq(g(t).diff(t), f(t))]
ceq = canonical_odes(eq, func, t)
(A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1)
A = A0
sol = [C1*(-Rational(1, 2) + sqrt(5)/2)*exp(t*(-Rational(1, 2) + sqrt(5)/2)) + C2*(-sqrt(5)/2 - Rational(1, 2))*
exp(t*(-sqrt(5)/2 - Rational(1, 2))),
C1*exp(t*(-Rational(1, 2) + sqrt(5)/2)) + C2*exp(t*(-sqrt(5)/2 - Rational(1, 2)))]
assert constant_renumber(linodesolve(A, t), variables=Tuple(*eq).free_symbols) == sol
# Testing the Errors
raises(ValueError, lambda: linodesolve(1, t, b=Matrix([t+1])))
raises(ValueError, lambda: linodesolve(a, t, b=Matrix([log(t) + sin(t)])))
raises(ValueError, lambda: linodesolve(Matrix([7]), t, b=t**2))
raises(ValueError, lambda: linodesolve(Matrix([a+10]), t, b=log(t)*cos(t)))
raises(ValueError, lambda: linodesolve(7, t, b=t**2))
raises(ValueError, lambda: linodesolve(a, t, b=log(t) + sin(t)))
A1 = Matrix([[1, 2], [2, 4], [4, 6]])
b1 = Matrix([t, 1, t**2])
raises(NonSquareMatrixError, lambda: linodesolve(A1, t, b=b1))
A2 = Matrix([[1, 2, 1], [3, 1, 2]])
b2 = Matrix([t, t**2])
raises(NonSquareMatrixError, lambda: linodesolve(A2, t, b=b2))
raises(ValueError, lambda: linodesolve(A1[:2, :], t, b=b1))
raises(ValueError, lambda: linodesolve(A1[:2, :], t, b=b1[:1]))
# DOIT check
A1 = Matrix([[1, -1], [1, -1]])
b1 = Matrix([15*t - 10, -15*t - 5])
sol1 = [C1 + C2*t + C2 - 10*t**3 + 10*t**2 + t*(15*t**2 - 5*t) - 10*t,
C1 + C2*t - 10*t**3 - 5*t**2 + t*(15*t**2 - 5*t) - 5*t]
assert constant_renumber(linodesolve(A1, t, b=b1, type="type2", doit=True),
variables=[t]) == sol1
# Testing auto functionality
func = [f(t), g(t)]
eq = [Eq(f(t).diff(t) + g(t).diff(t), g(t) + t), Eq(g(t).diff(t), f(t))]
ceq = canonical_odes(eq, func, t)
(A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1)
A = A0
sol = [-C1*exp(-t/2 + sqrt(5)*t/2)/2 + sqrt(5)*C1*exp(-t/2 + sqrt(5)*t/2)/2 - sqrt(5)*C2*exp(-sqrt(5)*t/2 -
t/2)/2 - C2*exp(-sqrt(5)*t/2 - t/2)/2 - exp(-t/2 + sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 +
t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)), t)/2 + sqrt(5)*exp(-t/2 +
sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 +
t/2)/(-5 + sqrt(5)), t)/2 - sqrt(5)*exp(-sqrt(5)*t/2 - t/2)*Integral(-sqrt(5)*t*exp(t/2 +
sqrt(5)*t/2)/5, t)/2 - exp(-sqrt(5)*t/2 - t/2)*Integral(-sqrt(5)*t*exp(t/2 + sqrt(5)*t/2)/5, t)/2,
C1*exp(-t/2 + sqrt(5)*t/2) + C2*exp(-sqrt(5)*t/2 - t/2) + exp(-t/2 +
sqrt(5)*t/2)*Integral(t*exp(-sqrt(5)*t/2 + t/2)/(-5 + sqrt(5)) - sqrt(5)*t*exp(-sqrt(5)*t/2 +
t/2)/(-5 + sqrt(5)), t) + exp(-sqrt(5)*t/2 -
t/2)*Integral(-sqrt(5)*t*exp(t/2 + sqrt(5)*t/2)/5, t)]
assert constant_renumber(linodesolve(A, t, b=b), variables=[t]) == sol
# non-homogeneous term assumed to be 0
sol1 = [-C1*exp(-t/2 + sqrt(5)*t/2)/2 + sqrt(5)*C1*exp(-t/2 + sqrt(5)*t/2)/2 - sqrt(5)*C2*exp(-sqrt(5)*t/2
- t/2)/2 - C2*exp(-sqrt(5)*t/2 - t/2)/2,
C1*exp(-t/2 + sqrt(5)*t/2) + C2*exp(-sqrt(5)*t/2 - t/2)]
assert constant_renumber(linodesolve(A, t, type="type2"), variables=[t]) == sol1
# Testing the Errors
raises(ValueError, lambda: linodesolve(t+10, t))
raises(ValueError, lambda: linodesolve(a*t, t))
A1 = Matrix([[1, t], [-t, 1]])
B1, _ = _is_commutative_anti_derivative(A1, t)
raises(NonSquareMatrixError, lambda: linodesolve(A1[:, :1], t, B=B1))
raises(ValueError, lambda: linodesolve(A1, t, B=1))
A2 = Matrix([[t, t, t], [t, t, t], [t, t, t]])
B2, _ = _is_commutative_anti_derivative(A2, t)
raises(NonSquareMatrixError, lambda: linodesolve(A2, t, B=B2[:2, :]))
raises(ValueError, lambda: linodesolve(A2, t, B=2))
raises(ValueError, lambda: linodesolve(A2, t, B=B2, type="type31"))
raises(ValueError, lambda: linodesolve(A1, t, B=B2))
raises(ValueError, lambda: linodesolve(A2, t, B=B1))
# Testing auto functionality
func = [f(t), g(t)]
eq = [Eq(f(t).diff(t), f(t) + t*g(t)), Eq(g(t).diff(t), -t*f(t) + g(t))]
ceq = canonical_odes(eq, func, t)
(A1, A0), b = linear_ode_to_matrix(ceq[0], func, t, 1)
A = A0
sol = [(C1/2 - I*C2/2)*exp(I*t**2/2 + t) + (C1/2 + I*C2/2)*exp(-I*t**2/2 + t),
(-I*C1/2 + C2/2)*exp(-I*t**2/2 + t) + (I*C1/2 + C2/2)*exp(I*t**2/2 + t)]
assert constant_renumber(linodesolve(A, t), variables=Tuple(*eq).free_symbols) == sol
assert constant_renumber(linodesolve(A, t, type="type3"), variables=Tuple(*eq).free_symbols) == sol
A1 = Matrix([[t, 1], [t, -1]])
raises(NotImplementedError, lambda: linodesolve(A1, t))
# Testing the Errors
raises(ValueError, lambda: linodesolve(t+10, t, b=Matrix([t+1])))
raises(ValueError, lambda: linodesolve(a*t, t, b=Matrix([log(t) + sin(t)])))
raises(ValueError, lambda: linodesolve(Matrix([7*t]), t, b=t**2))
raises(ValueError, lambda: linodesolve(Matrix([a + 10*log(t)]), t, b=log(t)*cos(t)))
raises(ValueError, lambda: linodesolve(7*t, t, b=t**2))
raises(ValueError, lambda: linodesolve(a*t**2, t, b=log(t) + sin(t)))
A1 = Matrix([[1, t], [-t, 1]])
b1 = Matrix([t, t ** 2])
B1, _ = _is_commutative_anti_derivative(A1, t)
raises(NonSquareMatrixError, lambda: linodesolve(A1[:, :1], t, b=b1))
A2 = Matrix([[t, t, t], [t, t, t], [t, t, t]])
b2 = Matrix([t, 1, t**2])
B2, _ = _is_commutative_anti_derivative(A2, t)
raises(NonSquareMatrixError, lambda: linodesolve(A2[:2, :], t, b=b2))
raises(ValueError, lambda: linodesolve(A1, t, b=b2))
raises(ValueError, lambda: linodesolve(A2, t, b=b1))
raises(ValueError, lambda: linodesolve(A1, t, b=b1, B=B2))
raises(ValueError, lambda: linodesolve(A2, t, b=b2, B=B1))
# Testing auto functionality
func = [f(x), g(x), h(x)]
eq = [Eq(f(x).diff(x), x*(f(x) + g(x) + h(x)) + x),
Eq(g(x).diff(x), x*(f(x) + g(x) + h(x)) + x),
Eq(h(x).diff(x), x*(f(x) + g(x) + h(x)) + 1)]
ceq = canonical_odes(eq, func, x)
(A1, A0), b = linear_ode_to_matrix(ceq[0], func, x, 1)
A = A0
_x1 = exp(-3*x**2/2)
_x2 = exp(3*x**2/2)
_x3 = Integral(2*_x1*x/3 + _x1/3 + x/3 - Rational(1, 3), x)
_x4 = 2*_x2*_x3/3
_x5 = Integral(2*_x1*x/3 + _x1/3 - 2*x/3 + Rational(2, 3), x)
sol = [
C1*_x2/3 - C1/3 + C2*_x2/3 - C2/3 + C3*_x2/3 + 2*C3/3 + _x2*_x5/3 + _x3/3 + _x4 - _x5/3,
C1*_x2/3 + 2*C1/3 + C2*_x2/3 - C2/3 + C3*_x2/3 - C3/3 + _x2*_x5/3 + _x3/3 + _x4 - _x5/3,
C1*_x2/3 - C1/3 + C2*_x2/3 + 2*C2/3 + C3*_x2/3 - C3/3 + _x2*_x5/3 - 2*_x3/3 + _x4 + 2*_x5/3,
]
assert constant_renumber(linodesolve(A, x, b=b), variables=Tuple(*eq).free_symbols) == sol
assert constant_renumber(linodesolve(A, x, b=b, type="type4"),
variables=Tuple(*eq).free_symbols) == sol
A1 = Matrix([[t, 1], [t, -1]])
raises(NotImplementedError, lambda: linodesolve(A1, t, b=b1))
# non-homogeneous term not passed
sol1 = [-C1/3 - C2/3 + 2*C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2), 2*C1/3 - C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2),
-C1/3 + 2*C2/3 - C3/3 + (C1/3 + C2/3 + C3/3)*exp(3*x**2/2)]
assert constant_renumber(linodesolve(A, x, type="type4", doit=True), variables=Tuple(*eq).free_symbols) == sol1
@slow
def test_linear_3eq_order1_type4_slow():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
f = t ** 3 + log(t)
g = t ** 2 + sin(t)
eq1 = (Eq(diff(x(t), t), (4 * f + g) * x(t) - f * y(t) - 2 * f * z(t)),
Eq(diff(y(t), t), 2 * f * x(t) + (f + g) * y(t) - 2 * f * z(t)), Eq(diff(z(t), t), 5 * f * x(t) + f * y(
t) + (-3 * f + g) * z(t)))
with dotprodsimp(True):
dsolve(eq1)
@slow
def test_linear_neq_order1_type2_slow1():
i, r1, c1, r2, c2, t = symbols('i, r1, c1, r2, c2, t')
x1 = Function('x1')
x2 = Function('x2')
eq1 = r1*c1*Derivative(x1(t), t) + x1(t) - x2(t) - r1*i
eq2 = r2*c1*Derivative(x1(t), t) + r2*c2*Derivative(x2(t), t) + x2(t) - r2*i
eq = [eq1, eq2]
# XXX: Solution is too complicated
[sol] = dsolve_system(eq, simplify=False, doit=False)
assert checksysodesol(eq, sol) == (True, [0, 0])
# Regression test case for issue #9204
# https://github.com/sympy/sympy/issues/9204
@slow
def test_linear_new_order1_type2_de_lorentz_slow_check():
if ON_TRAVIS:
skip("Too slow for travis.")
m = Symbol("m", real=True)
q = Symbol("q", real=True)
t = Symbol("t", real=True)
e1, e2, e3 = symbols("e1:4", real=True)
b1, b2, b3 = symbols("b1:4", real=True)
v1, v2, v3 = symbols("v1:4", cls=Function, real=True)
eqs = [
-e1*q + m*Derivative(v1(t), t) - q*(-b2*v3(t) + b3*v2(t)),
-e2*q + m*Derivative(v2(t), t) - q*(b1*v3(t) - b3*v1(t)),
-e3*q + m*Derivative(v3(t), t) - q*(-b1*v2(t) + b2*v1(t))
]
sol = dsolve(eqs)
assert checksysodesol(eqs, sol) == (True, [0, 0, 0])
# Regression test case for issue #14001
# https://github.com/sympy/sympy/issues/14001
@slow
def test_linear_neq_order1_type2_slow_check():
RC, t, C, Vs, L, R1, V0, I0 = symbols("RC t C Vs L R1 V0 I0")
V = Function("V")
I = Function("I")
system = [Eq(V(t).diff(t), -1/RC*V(t) + I(t)/C), Eq(I(t).diff(t), -R1/L*I(t) - 1/L*V(t) + Vs/L)]
[sol] = dsolve_system(system, simplify=False, doit=False)
assert checksysodesol(system, sol) == (True, [0, 0])
def _linear_3eq_order1_type4_long():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
f = t ** 3 + log(t)
g = t ** 2 + sin(t)
eq1 = (Eq(diff(x(t), t), (4*f + g)*x(t) - f*y(t) - 2*f*z(t)),
Eq(diff(y(t), t), 2*f*x(t) + (f + g)*y(t) - 2*f*z(t)), Eq(diff(z(t), t), 5*f*x(t) + f*y(
t) + (-3*f + g)*z(t)))
dsolve_sol = dsolve(eq1)
dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol]
x_1 = sqrt(-t**6 - 8*t**3*log(t) + 8*t**3 - 16*log(t)**2 + 32*log(t) - 16)
x_2 = sqrt(3)
x_3 = 8324372644*C1*x_1*x_2 + 4162186322*C2*x_1*x_2 - 8324372644*C3*x_1*x_2
x_4 = 1 / (1903457163*t**3 + 3825881643*x_1*x_2 + 7613828652*log(t) - 7613828652)
x_5 = exp(t**3/3 + t*x_1*x_2/4 - cos(t))
x_6 = exp(t**3/3 - t*x_1*x_2/4 - cos(t))
x_7 = exp(t**4/2 + t**3/3 + 2*t*log(t) - 2*t - cos(t))
x_8 = 91238*C1*x_1*x_2 + 91238*C2*x_1*x_2 - 91238*C3*x_1*x_2
x_9 = 1 / (66049*t**3 - 50629*x_1*x_2 + 264196*log(t) - 264196)
x_10 = 50629 * C1 / 25189 + 37909*C2/25189 - 50629*C3/25189 - x_3*x_4
x_11 = -50629*C1/25189 - 12720*C2/25189 + 50629*C3/25189 + x_3*x_4
sol = [Eq(x(t), x_10*x_5 + x_11*x_6 + x_7*(C1 - C2)), Eq(y(t), x_10*x_5 + x_11*x_6), Eq(z(t), x_5*(
-424*C1/257 - 167*C2/257 + 424*C3/257 - x_8*x_9) + x_6*(167*C1/257 + 424*C2/257 -
167*C3/257 + x_8*x_9) + x_7*(C1 - C2))]
assert dsolve_sol1 == sol
assert checksysodesol(eq1, dsolve_sol1) == (True, [0, 0, 0])
@slow
def test_neq_order1_type4_slow_check1():
f, g = symbols("f g", cls=Function)
x = symbols("x")
eqs = [Eq(diff(f(x), x), x*f(x) + x**2*g(x) + x),
Eq(diff(g(x), x), 2*x**2*f(x) + (x + 3*x**2)*g(x) + 1)]
sol = dsolve(eqs)
assert checksysodesol(eqs, sol) == (True, [0, 0])
@slow
def test_neq_order1_type4_slow_check2():
f, g, h = symbols("f, g, h", cls=Function)
x = Symbol("x")
eqs = [
Eq(Derivative(f(x), x), x*h(x) + f(x) + g(x) + 1),
Eq(Derivative(g(x), x), x*g(x) + f(x) + h(x) + 10),
Eq(Derivative(h(x), x), x*f(x) + x + g(x) + h(x))
]
with dotprodsimp(True):
sol = dsolve(eqs)
assert checksysodesol(eqs, sol) == (True, [0, 0, 0])
def _neq_order1_type4_slow3():
f, g = symbols("f g", cls=Function)
x = symbols("x")
eqs = [
Eq(Derivative(f(x), x), x*f(x) + g(x) + sin(x)),
Eq(Derivative(g(x), x), x**2 + x*g(x) - f(x))
]
sol = [
Eq(f(x), (C1/2 - I*C2/2 - I*Integral(x**2*exp(-x**2/2 - I*x)/2 +
x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 -
I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 + Integral(-I*x**2*exp(-x**2/2
- I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 -
I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 +
I*x) + (C1/2 + I*C2/2 + I*Integral(x**2*exp(-x**2/2 - I*x)/2 +
x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 -
I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 + Integral(-I*x**2*exp(-x**2/2
- I*x)/2 + I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 -
I*x)*sin(x)/2 + exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 -
I*x)),
Eq(g(x), (-I*C1/2 + C2/2 + Integral(x**2*exp(-x**2/2 - I*x)/2 +
x**2*exp(-x**2/2 + I*x)/2 + I*exp(-x**2/2 - I*x)*sin(x)/2 -
I*exp(-x**2/2 + I*x)*sin(x)/2, x)/2 -
I*Integral(-I*x**2*exp(-x**2/2 - I*x)/2 + I*x**2*exp(-x**2/2 +
I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 + exp(-x**2/2 +
I*x)*sin(x)/2, x)/2)*exp(x**2/2 - I*x) + (I*C1/2 + C2/2 +
Integral(x**2*exp(-x**2/2 - I*x)/2 + x**2*exp(-x**2/2 + I*x)/2 +
I*exp(-x**2/2 - I*x)*sin(x)/2 - I*exp(-x**2/2 + I*x)*sin(x)/2,
x)/2 + I*Integral(-I*x**2*exp(-x**2/2 - I*x)/2 +
I*x**2*exp(-x**2/2 + I*x)/2 + exp(-x**2/2 - I*x)*sin(x)/2 +
exp(-x**2/2 + I*x)*sin(x)/2, x)/2)*exp(x**2/2 + I*x))
]
return eqs, sol
def test_neq_order1_type4_slow3():
eqs, sol = _neq_order1_type4_slow3()
assert dsolve_system(eqs, simplify=False, doit=False) == [sol]
# XXX: dsolve gives an error in integration:
# assert dsolve(eqs) == sol
# https://github.com/sympy/sympy/issues/20155
@slow
def test_neq_order1_type4_slow_check3():
eqs, sol = _neq_order1_type4_slow3()
assert checksysodesol(eqs, sol) == (True, [0, 0])
@XFAIL
@slow
def test_linear_3eq_order1_type4_long_dsolve_slow_xfail():
if ON_TRAVIS:
skip("Too slow for travis.")
eq, sol = _linear_3eq_order1_type4_long()
dsolve_sol = dsolve(eq)
dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol]
assert dsolve_sol1 == sol
@slow
def test_linear_3eq_order1_type4_long_dsolve_dotprodsimp():
if ON_TRAVIS:
skip("Too slow for travis.")
eq, sol = _linear_3eq_order1_type4_long()
# XXX: Only works with dotprodsimp see
# test_linear_3eq_order1_type4_long_dsolve_slow_xfail which is too slow
with dotprodsimp(True):
dsolve_sol = dsolve(eq)
dsolve_sol1 = [_simpsol(sol) for sol in dsolve_sol]
assert dsolve_sol1 == sol
@slow
def test_linear_3eq_order1_type4_long_check():
if ON_TRAVIS:
skip("Too slow for travis.")
eq, sol = _linear_3eq_order1_type4_long()
assert checksysodesol(eq, sol) == (True, [0, 0, 0])
def test_dsolve_system():
f, g = symbols("f g", cls=Function)
x = symbols("x")
eqs = [Eq(f(x).diff(x), f(x) + g(x)), Eq(g(x).diff(x), f(x) + g(x))]
funcs = [f(x), g(x)]
sol = [[Eq(f(x), -C1 + C2*exp(2*x)), Eq(g(x), C1 + C2*exp(2*x))]]
assert dsolve_system(eqs, funcs=funcs, t=x, doit=True) == sol
raises(ValueError, lambda: dsolve_system(1))
raises(ValueError, lambda: dsolve_system(eqs, 1))
raises(ValueError, lambda: dsolve_system(eqs, funcs, 1))
raises(ValueError, lambda: dsolve_system(eqs, funcs[:1], x))
eq = (Eq(f(x).diff(x), 12 * f(x) - 6 * g(x)), Eq(g(x).diff(x) ** 2, 11 * f(x) + 3 * g(x)))
raises(NotImplementedError, lambda: dsolve_system(eq) == ([], []))
raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)]) == ([], []))
raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], t=x) == ([], []))
raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], t=x, ics={f(0): 1, g(0): 1}) == ([], []))
raises(NotImplementedError, lambda: dsolve_system(eq, t=x, ics={f(0): 1, g(0): 1}) == ([], []))
raises(NotImplementedError, lambda: dsolve_system(eq, ics={f(0): 1, g(0): 1}) == ([], []))
raises(NotImplementedError, lambda: dsolve_system(eq, funcs=[f(x), g(x)], ics={f(0): 1, g(0): 1}) == ([], []))
def test_dsolve():
f, g = symbols('f g', cls=Function)
x, y = symbols('x y')
eqs = [f(x).diff(x) - x, f(x).diff(x) + x]
with raises(ValueError):
dsolve(eqs)
eqs = [f(x, y).diff(x)]
with raises(ValueError):
dsolve(eqs)
eqs = [f(x, y).diff(x)+g(x).diff(x), g(x).diff(x)]
with raises(ValueError):
dsolve(eqs)
@slow
def test_higher_order1_slow1():
x, y = symbols("x y", cls=Function)
t = symbols("t")
eq = [
Eq(diff(x(t),t,t), (log(t)+t**2)*diff(x(t),t)+(log(t)+t**2)*3*diff(y(t),t)),
Eq(diff(y(t),t,t), (log(t)+t**2)*2*diff(x(t),t)+(log(t)+t**2)*9*diff(y(t),t))
]
sol, = dsolve_system(eq, simplify=False, doit=False)
# The solution is too long to write out explicitly and checkodesol is too
# slow so we test for particular values of t:
for e in eq:
res = (e.lhs - e.rhs).subs({sol[0].lhs:sol[0].rhs, sol[1].lhs:sol[1].rhs})
res = res.subs({d: d.doit(deep=False) for d in res.atoms(Derivative)})
assert ratsimp(res.subs(t, 1)) == 0
def test_second_order_type2_slow1():
x, y, z = symbols('x, y, z', cls=Function)
t, l = symbols('t, l')
eqs1 = [Eq(Derivative(x(t), (t, 2)), t*(2*x(t) + y(t))),
Eq(Derivative(y(t), (t, 2)), t*(-x(t) + 2*y(t)))]
sol1 = [Eq(x(t), I*C1*airyai(t*(2 - I)**(S(1)/3)) + I*C2*airybi(t*(2 - I)**(S(1)/3)) - I*C3*airyai(t*(2 +
I)**(S(1)/3)) - I*C4*airybi(t*(2 + I)**(S(1)/3))),
Eq(y(t), C1*airyai(t*(2 - I)**(S(1)/3)) + C2*airybi(t*(2 - I)**(S(1)/3)) + C3*airyai(t*(2 + I)**(S(1)/3)) +
C4*airybi(t*(2 + I)**(S(1)/3)))]
assert dsolve(eqs1) == sol1
assert checksysodesol(eqs1, sol1) == (True, [0, 0])
@slow
@XFAIL
def test_nonlinear_3eq_order1_type1():
if ON_TRAVIS:
skip("Too slow for travis.")
a, b, c = symbols('a b c')
eqs = [
a * f(x).diff(x) - (b - c) * g(x) * h(x),
b * g(x).diff(x) - (c - a) * h(x) * f(x),
c * h(x).diff(x) - (a - b) * f(x) * g(x),
]
assert dsolve(eqs) # NotImplementedError
@XFAIL
def test_nonlinear_3eq_order1_type4():
eqs = [
Eq(f(x).diff(x), (2*h(x)*g(x) - 3*g(x)*h(x))),
Eq(g(x).diff(x), (4*f(x)*h(x) - 2*h(x)*f(x))),
Eq(h(x).diff(x), (3*g(x)*f(x) - 4*f(x)*g(x))),
]
dsolve(eqs) # KeyError when matching
# sol = ?
# assert dsolve_sol == sol
# assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0])
@slow
@XFAIL
def test_nonlinear_3eq_order1_type3():
if ON_TRAVIS:
skip("Too slow for travis.")
eqs = [
Eq(f(x).diff(x), (2*f(x)**2 - 3 )),
Eq(g(x).diff(x), (4 - 2*h(x) )),
Eq(h(x).diff(x), (3*h(x) - 4*f(x)**2)),
]
dsolve(eqs) # Not sure if this finishes...
# sol = ?
# assert dsolve_sol == sol
# assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0])
@XFAIL
def test_nonlinear_3eq_order1_type5():
eqs = [
Eq(f(x).diff(x), f(x)*(2*f(x) - 3*g(x))),
Eq(g(x).diff(x), g(x)*(4*g(x) - 2*h(x))),
Eq(h(x).diff(x), h(x)*(3*h(x) - 4*f(x))),
]
dsolve(eqs) # KeyError
# sol = ?
# assert dsolve_sol == sol
# assert checksysodesol(eqs, dsolve_sol) == (True, [0, 0, 0])
def test_linear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
k, l, m, n = symbols('k, l, m, n', Integer=True)
t = Symbol('t')
x0, y0 = symbols('x0, y0', cls=Function)
eq1 = (Eq(diff(x(t),t), x(t) + y(t) + 9), Eq(diff(y(t),t), 2*x(t) + 5*y(t) + 23))
sol1 = [Eq(x(t), C1*exp(t*(sqrt(6) + 3)) + C2*exp(t*(-sqrt(6) + 3)) - Rational(22, 3)), \
Eq(y(t), C1*(2 + sqrt(6))*exp(t*(sqrt(6) + 3)) + C2*(-sqrt(6) + 2)*exp(t*(-sqrt(6) + 3)) - Rational(5, 3))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), x(t) + y(t) + 81), Eq(diff(y(t),t), -2*x(t) + y(t) + 23))
sol2 = [Eq(x(t), (C1*cos(sqrt(2)*t) + C2*sin(sqrt(2)*t))*exp(t) - Rational(58, 3)), \
Eq(y(t), (-sqrt(2)*C1*sin(sqrt(2)*t) + sqrt(2)*C2*cos(sqrt(2)*t))*exp(t) - Rational(185, 3))]
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), 5*t*x(t) + 2*y(t)), Eq(diff(y(t),t), 2*x(t) + 5*t*y(t)))
sol3 = [Eq(x(t), (C1*exp(2*t) + C2*exp(-2*t))*exp(Rational(5, 2)*t**2)), \
Eq(y(t), (C1*exp(2*t) - C2*exp(-2*t))*exp(Rational(5, 2)*t**2))]
assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + 5*t*y(t)))
sol4 = [Eq(x(t), (C1*cos((t**3)/3) + C2*sin((t**3)/3))*exp(Rational(5, 2)*t**2)), \
Eq(y(t), (-C1*sin((t**3)/3) + C2*cos((t**3)/3))*exp(Rational(5, 2)*t**2))]
assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), -t**2*x(t) + (5*t+9*t**2)*y(t)))
sol5 = [Eq(x(t), (C1*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \
C2*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2)), \
Eq(y(t), (C1*(sqrt(77)/2 + Rational(9, 2))*exp((sqrt(77)/2 + Rational(9, 2))*(t**3)/3) + \
C2*(-sqrt(77)/2 + Rational(9, 2))*exp((-sqrt(77)/2 + Rational(9, 2))*(t**3)/3))*exp(Rational(5, 2)*t**2))]
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t), 5*t*x(t) + t**2*y(t)), Eq(diff(y(t),t), (1-t**2)*x(t) + (5*t+9*t**2)*y(t)))
sol6 = [Eq(x(t), C1*x0(t) + C2*x0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t)), \
Eq(y(t), C1*y0(t) + C2*(y0(t)*Integral(t**2*exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)**2, t) + \
exp(Integral(5*t, t))*exp(Integral(9*t**2 + 5*t, t))/x0(t)))]
s = dsolve(eq6)
assert s == sol6 # too complicated to test with subs and simplify
# assert checksysodesol(eq10, sol10) == (True, [0, 0]) # this one fails
def test_nonlinear_2eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t = Symbol('t')
eq1 = (Eq(diff(x(t),t),x(t)*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol1 = [
Eq(x(t), C1*exp((-1/(4*C2 + 4*t))**(Rational(-1, 4)))),
Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), C1*exp(-1/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), C1*exp(-I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), C1*exp(I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert dsolve(eq1) == sol1
assert checksysodesol(eq1, sol1) == (True, [0, 0])
eq2 = (Eq(diff(x(t),t), exp(3*x(t))*y(t)**3),Eq(diff(y(t),t), y(t)**5))
sol2 = [
Eq(x(t), -log(C1 - 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), -log(C1 + 3/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), -log(C1 + 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), -log(C1 - 3*I/(-1/(4*C2 + 4*t))**Rational(1, 4))/3),
Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert dsolve(eq2) == sol2
assert checksysodesol(eq2, sol2) == (True, [0, 0])
eq3 = (Eq(diff(x(t),t), y(t)*x(t)), Eq(diff(y(t),t), x(t)**3))
tt = Rational(2, 3)
sol3 = [
Eq(x(t), 6**tt/(6*(-sinh(sqrt(C1)*(C2 + t)/2)/sqrt(C1))**tt)),
Eq(y(t), sqrt(C1 + C1/sinh(sqrt(C1)*(C2 + t)/2)**2)/3)]
assert dsolve(eq3) == sol3
# FIXME: assert checksysodesol(eq3, sol3) == (True, [0, 0])
eq4 = (Eq(diff(x(t),t),x(t)*y(t)*sin(t)**2), Eq(diff(y(t),t),y(t)**2*sin(t)**2))
sol4 = {Eq(x(t), -2*exp(C1)/(C2*exp(C1) + t - sin(2*t)/2)), Eq(y(t), -2/(C1 + t - sin(2*t)/2))}
assert dsolve(eq4) == sol4
# FIXME: assert checksysodesol(eq4, sol4) == (True, [0, 0])
eq5 = (Eq(x(t),t*diff(x(t),t)+diff(x(t),t)*diff(y(t),t)), Eq(y(t),t*diff(y(t),t)+diff(y(t),t)**2))
sol5 = {Eq(x(t), C1*C2 + C1*t), Eq(y(t), C2**2 + C2*t)}
assert dsolve(eq5) == sol5
assert checksysodesol(eq5, sol5) == (True, [0, 0])
eq6 = (Eq(diff(x(t),t),x(t)**2*y(t)**3), Eq(diff(y(t),t),y(t)**5))
sol6 = [
Eq(x(t), 1/(C1 - 1/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), -(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), 1/(C1 + (-1/(4*C2 + 4*t))**(Rational(-1, 4)))),
Eq(y(t), (-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), 1/(C1 + I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), -I*(-1/(4*C2 + 4*t))**Rational(1, 4)),
Eq(x(t), 1/(C1 - I/(-1/(4*C2 + 4*t))**Rational(1, 4))),
Eq(y(t), I*(-1/(4*C2 + 4*t))**Rational(1, 4))]
assert dsolve(eq6) == sol6
assert checksysodesol(eq6, sol6) == (True, [0, 0])
@slow
def test_nonlinear_3eq_order1():
x, y, z = symbols('x, y, z', cls=Function)
t, u = symbols('t u')
eq1 = (4*diff(x(t),t) + 2*y(t)*z(t), 3*diff(y(t),t) - z(t)*x(t), 5*diff(z(t),t) - x(t)*y(t))
sol1 = [Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)), (u, x(t))),
C3 - sqrt(15)*t/15), Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)),
(u, y(t))), C3 + sqrt(5)*t/10), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*t/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq1), sol1)]
# FIXME: assert checksysodesol(eq1, sol1) == (True, [0, 0, 0])
eq2 = (4*diff(x(t),t) + 2*y(t)*z(t)*sin(t), 3*diff(y(t),t) - z(t)*x(t)*sin(t), 5*diff(z(t),t) - x(t)*y(t)*sin(t))
sol2 = [Eq(3*Integral(1/(sqrt(-6*u**2 - C1 + 5*C2)*sqrt(3*u**2 + C1 - 4*C2)), (u, x(t))), C3 +
sqrt(5)*cos(t)/10), Eq(4*Integral(1/(sqrt(-4*u**2 - 3*C1 + C2)*sqrt(-4*u**2 + 5*C1 - C2)),
(u, y(t))), C3 - sqrt(15)*cos(t)/15), Eq(5*Integral(1/(sqrt(-10*u**2 - 3*C1 + C2)*
sqrt(5*u**2 + 4*C1 - C2)), (u, z(t))), C3 + sqrt(3)*cos(t)/6)]
assert [i.dummy_eq(j) for i, j in zip(dsolve(eq2), sol2)]
# FIXME: assert checksysodesol(eq2, sol2) == (True, [0, 0, 0])
def test_C1_function_9239():
t = Symbol('t')
C1 = Function('C1')
C2 = Function('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
eq = (Eq(diff(C1(t), t), 9*C2(t)), Eq(diff(C2(t), t), 12*C1(t)))
sol = [Eq(C1(t), 9*C3*exp(6*sqrt(3)*t) + 9*C4*exp(-6*sqrt(3)*t)),
Eq(C2(t), 6*sqrt(3)*C3*exp(6*sqrt(3)*t) - 6*sqrt(3)*C4*exp(-6*sqrt(3)*t))]
assert checksysodesol(eq, sol) == (True, [0, 0])
def test_dsolve_linsystem_symbol():
eps = Symbol('epsilon', positive=True)
eq1 = (Eq(diff(f(x), x), -eps*g(x)), Eq(diff(g(x), x), eps*f(x)))
sol1 = [Eq(f(x), -C1*eps*cos(eps*x) - C2*eps*sin(eps*x)),
Eq(g(x), -C1*eps*sin(eps*x) + C2*eps*cos(eps*x))]
assert checksysodesol(eq1, sol1) == (True, [0, 0])
|
9ba2eb28230878d7ae89c7ee30c146dfe29efbba328d3b03d5045b29d4d15589 | from sympy.core.numbers import (E, Rational, pi)
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.core import S, symbols, I
from sympy.discrete.convolutions import (
convolution, convolution_fft, convolution_ntt, convolution_fwht,
convolution_subset, covering_product, intersecting_product)
from sympy.testing.pytest import raises
from sympy.abc import x, y
def test_convolution():
# fft
a = [1, Rational(5, 3), sqrt(3), Rational(7, 5)]
b = [9, 5, 5, 4, 3, 2]
c = [3, 5, 3, 7, 8]
d = [1422, 6572, 3213, 5552]
assert convolution(a, b) == convolution_fft(a, b)
assert convolution(a, b, dps=9) == convolution_fft(a, b, dps=9)
assert convolution(a, d, dps=7) == convolution_fft(d, a, dps=7)
assert convolution(a, d[1:], dps=3) == convolution_fft(d[1:], a, dps=3)
# prime moduli of the form (m*2**k + 1), sequence length
# should be a divisor of 2**k
p = 7*17*2**23 + 1
q = 19*2**10 + 1
# ntt
assert convolution(d, b, prime=q) == convolution_ntt(b, d, prime=q)
assert convolution(c, b, prime=p) == convolution_ntt(b, c, prime=p)
assert convolution(d, c, prime=p) == convolution_ntt(c, d, prime=p)
raises(TypeError, lambda: convolution(b, d, dps=5, prime=q))
raises(TypeError, lambda: convolution(b, d, dps=6, prime=q))
# fwht
assert convolution(a, b, dyadic=True) == convolution_fwht(a, b)
assert convolution(a, b, dyadic=False) == convolution(a, b)
raises(TypeError, lambda: convolution(b, d, dps=2, dyadic=True))
raises(TypeError, lambda: convolution(b, d, prime=p, dyadic=True))
raises(TypeError, lambda: convolution(a, b, dps=2, dyadic=True))
raises(TypeError, lambda: convolution(b, c, prime=p, dyadic=True))
# subset
assert convolution(a, b, subset=True) == convolution_subset(a, b) == \
convolution(a, b, subset=True, dyadic=False) == \
convolution(a, b, subset=True)
assert convolution(a, b, subset=False) == convolution(a, b)
raises(TypeError, lambda: convolution(a, b, subset=True, dyadic=True))
raises(TypeError, lambda: convolution(c, d, subset=True, dps=6))
raises(TypeError, lambda: convolution(a, c, subset=True, prime=q))
def test_cyclic_convolution():
# fft
a = [1, Rational(5, 3), sqrt(3), Rational(7, 5)]
b = [9, 5, 5, 4, 3, 2]
assert convolution([1, 2, 3], [4, 5, 6], cycle=0) == \
convolution([1, 2, 3], [4, 5, 6], cycle=5) == \
convolution([1, 2, 3], [4, 5, 6])
assert convolution([1, 2, 3], [4, 5, 6], cycle=3) == [31, 31, 28]
a = [Rational(1, 3), Rational(7, 3), Rational(5, 9), Rational(2, 7), Rational(5, 8)]
b = [Rational(3, 5), Rational(4, 7), Rational(7, 8), Rational(8, 9)]
assert convolution(a, b, cycle=0) == \
convolution(a, b, cycle=len(a) + len(b) - 1)
assert convolution(a, b, cycle=4) == [Rational(87277, 26460), Rational(30521, 11340),
Rational(11125, 4032), Rational(3653, 1080)]
assert convolution(a, b, cycle=6) == [Rational(20177, 20160), Rational(676, 315), Rational(47, 24),
Rational(3053, 1080), Rational(16397, 5292), Rational(2497, 2268)]
assert convolution(a, b, cycle=9) == \
convolution(a, b, cycle=0) + [S.Zero]
# ntt
a = [2313, 5323532, S(3232), 42142, 42242421]
b = [S(33456), 56757, 45754, 432423]
assert convolution(a, b, prime=19*2**10 + 1, cycle=0) == \
convolution(a, b, prime=19*2**10 + 1, cycle=8) == \
convolution(a, b, prime=19*2**10 + 1)
assert convolution(a, b, prime=19*2**10 + 1, cycle=5) == [96, 17146, 2664,
15534, 3517]
assert convolution(a, b, prime=19*2**10 + 1, cycle=7) == [4643, 3458, 1260,
15534, 3517, 16314, 13688]
assert convolution(a, b, prime=19*2**10 + 1, cycle=9) == \
convolution(a, b, prime=19*2**10 + 1) + [0]
# fwht
u, v, w, x, y = symbols('u v w x y')
p, q, r, s, t = symbols('p q r s t')
c = [u, v, w, x, y]
d = [p, q, r, s, t]
assert convolution(a, b, dyadic=True, cycle=3) == \
[2499522285783, 19861417974796, 4702176579021]
assert convolution(a, b, dyadic=True, cycle=5) == [2718149225143,
2114320852171, 20571217906407, 246166418903, 1413262436976]
assert convolution(c, d, dyadic=True, cycle=4) == \
[p*u + p*y + q*v + r*w + s*x + t*u + t*y,
p*v + q*u + q*y + r*x + s*w + t*v,
p*w + q*x + r*u + r*y + s*v + t*w,
p*x + q*w + r*v + s*u + s*y + t*x]
assert convolution(c, d, dyadic=True, cycle=6) == \
[p*u + q*v + r*w + r*y + s*x + t*w + t*y,
p*v + q*u + r*x + s*w + s*y + t*x,
p*w + q*x + r*u + s*v,
p*x + q*w + r*v + s*u,
p*y + t*u,
q*y + t*v]
# subset
assert convolution(a, b, subset=True, cycle=7) == [18266671799811,
178235365533, 213958794, 246166418903, 1413262436976,
2397553088697, 1932759730434]
assert convolution(a[1:], b, subset=True, cycle=4) == \
[178104086592, 302255835516, 244982785880, 3717819845434]
assert convolution(a, b[:-1], subset=True, cycle=6) == [1932837114162,
178235365533, 213958794, 245166224504, 1413262436976, 2397553088697]
assert convolution(c, d, subset=True, cycle=3) == \
[p*u + p*x + q*w + r*v + r*y + s*u + t*w,
p*v + p*y + q*u + s*y + t*u + t*x,
p*w + q*y + r*u + t*v]
assert convolution(c, d, subset=True, cycle=5) == \
[p*u + q*y + t*v,
p*v + q*u + r*y + t*w,
p*w + r*u + s*y + t*x,
p*x + q*w + r*v + s*u,
p*y + t*u]
raises(ValueError, lambda: convolution([1, 2, 3], [4, 5, 6], cycle=-1))
def test_convolution_fft():
assert all(convolution_fft([], x, dps=y) == [] for x in ([], [1]) for y in (None, 3))
assert convolution_fft([1, 2, 3], [4, 5, 6]) == [4, 13, 28, 27, 18]
assert convolution_fft([1], [5, 6, 7]) == [5, 6, 7]
assert convolution_fft([1, 3], [5, 6, 7]) == [5, 21, 25, 21]
assert convolution_fft([1 + 2*I], [2 + 3*I]) == [-4 + 7*I]
assert convolution_fft([1 + 2*I, 3 + 4*I, 5 + 3*I/5], [Rational(2, 5) + 4*I/7]) == \
[Rational(-26, 35) + I*48/35, Rational(-38, 35) + I*116/35, Rational(58, 35) + I*542/175]
assert convolution_fft([Rational(3, 4), Rational(5, 6)], [Rational(7, 8), Rational(1, 3), Rational(2, 5)]) == \
[Rational(21, 32), Rational(47, 48), Rational(26, 45), Rational(1, 3)]
assert convolution_fft([Rational(1, 9), Rational(2, 3), Rational(3, 5)], [Rational(2, 5), Rational(3, 7), Rational(4, 9)]) == \
[Rational(2, 45), Rational(11, 35), Rational(8152, 14175), Rational(523, 945), Rational(4, 15)]
assert convolution_fft([pi, E, sqrt(2)], [sqrt(3), 1/pi, 1/E]) == \
[sqrt(3)*pi, 1 + sqrt(3)*E, E/pi + pi*exp(-1) + sqrt(6),
sqrt(2)/pi + 1, sqrt(2)*exp(-1)]
assert convolution_fft([2321, 33123], [5321, 6321, 71323]) == \
[12350041, 190918524, 374911166, 2362431729]
assert convolution_fft([312313, 31278232], [32139631, 319631]) == \
[10037624576503, 1005370659728895, 9997492572392]
raises(TypeError, lambda: convolution_fft(x, y))
raises(ValueError, lambda: convolution_fft([x, y], [y, x]))
def test_convolution_ntt():
# prime moduli of the form (m*2**k + 1), sequence length
# should be a divisor of 2**k
p = 7*17*2**23 + 1
q = 19*2**10 + 1
r = 2*500000003 + 1 # only for sequences of length 1 or 2
# s = 2*3*5*7 # composite modulus
assert all(convolution_ntt([], x, prime=y) == [] for x in ([], [1]) for y in (p, q, r))
assert convolution_ntt([2], [3], r) == [6]
assert convolution_ntt([2, 3], [4], r) == [8, 12]
assert convolution_ntt([32121, 42144, 4214, 4241], [32132, 3232, 87242], p) == [33867619,
459741727, 79180879, 831885249, 381344700, 369993322]
assert convolution_ntt([121913, 3171831, 31888131, 12], [17882, 21292, 29921, 312], q) == \
[8158, 3065, 3682, 7090, 1239, 2232, 3744]
assert convolution_ntt([12, 19, 21, 98, 67], [2, 6, 7, 8, 9], p) == \
convolution_ntt([12, 19, 21, 98, 67], [2, 6, 7, 8, 9], q)
assert convolution_ntt([12, 19, 21, 98, 67], [21, 76, 17, 78, 69], p) == \
convolution_ntt([12, 19, 21, 98, 67], [21, 76, 17, 78, 69], q)
raises(ValueError, lambda: convolution_ntt([2, 3], [4, 5], r))
raises(ValueError, lambda: convolution_ntt([x, y], [y, x], q))
raises(TypeError, lambda: convolution_ntt(x, y, p))
def test_convolution_fwht():
assert convolution_fwht([], []) == []
assert convolution_fwht([], [1]) == []
assert convolution_fwht([1, 2, 3], [4, 5, 6]) == [32, 13, 18, 27]
assert convolution_fwht([Rational(5, 7), Rational(6, 8), Rational(7, 3)], [2, 4, Rational(6, 7)]) == \
[Rational(45, 7), Rational(61, 14), Rational(776, 147), Rational(419, 42)]
a = [1, Rational(5, 3), sqrt(3), Rational(7, 5), 4 + 5*I]
b = [94, 51, 53, 45, 31, 27, 13]
c = [3 + 4*I, 5 + 7*I, 3, Rational(7, 6), 8]
assert convolution_fwht(a, b) == [53*sqrt(3) + 366 + 155*I,
45*sqrt(3) + Rational(5848, 15) + 135*I,
94*sqrt(3) + Rational(1257, 5) + 65*I,
51*sqrt(3) + Rational(3974, 15),
13*sqrt(3) + 452 + 470*I,
Rational(4513, 15) + 255*I,
31*sqrt(3) + Rational(1314, 5) + 265*I,
27*sqrt(3) + Rational(3676, 15) + 225*I]
assert convolution_fwht(b, c) == [Rational(1993, 2) + 733*I, Rational(6215, 6) + 862*I,
Rational(1659, 2) + 527*I, Rational(1988, 3) + 551*I, 1019 + 313*I, Rational(3955, 6) + 325*I,
Rational(1175, 2) + 52*I, Rational(3253, 6) + 91*I]
assert convolution_fwht(a[3:], c) == [Rational(-54, 5) + I*293/5, -1 + I*204/5,
Rational(133, 15) + I*35/6, Rational(409, 30) + 15*I, Rational(56, 5), 32 + 40*I, 0, 0]
u, v, w, x, y, z = symbols('u v w x y z')
assert convolution_fwht([u, v], [x, y]) == [u*x + v*y, u*y + v*x]
assert convolution_fwht([u, v, w], [x, y]) == \
[u*x + v*y, u*y + v*x, w*x, w*y]
assert convolution_fwht([u, v, w], [x, y, z]) == \
[u*x + v*y + w*z, u*y + v*x, u*z + w*x, v*z + w*y]
raises(TypeError, lambda: convolution_fwht(x, y))
raises(TypeError, lambda: convolution_fwht(x*y, u + v))
def test_convolution_subset():
assert convolution_subset([], []) == []
assert convolution_subset([], [Rational(1, 3)]) == []
assert convolution_subset([6 + I*3/7], [Rational(2, 3)]) == [4 + I*2/7]
a = [1, Rational(5, 3), sqrt(3), 4 + 5*I]
b = [64, 71, 55, 47, 33, 29, 15]
c = [3 + I*2/3, 5 + 7*I, 7, Rational(7, 5), 9]
assert convolution_subset(a, b) == [64, Rational(533, 3), 55 + 64*sqrt(3),
71*sqrt(3) + Rational(1184, 3) + 320*I, 33, 84,
15 + 33*sqrt(3), 29*sqrt(3) + 157 + 165*I]
assert convolution_subset(b, c) == [192 + I*128/3, 533 + I*1486/3,
613 + I*110/3, Rational(5013, 5) + I*1249/3,
675 + 22*I, 891 + I*751/3,
771 + 10*I, Rational(3736, 5) + 105*I]
assert convolution_subset(a, c) == convolution_subset(c, a)
assert convolution_subset(a[:2], b) == \
[64, Rational(533, 3), 55, Rational(416, 3), 33, 84, 15, 25]
assert convolution_subset(a[:2], c) == \
[3 + I*2/3, 10 + I*73/9, 7, Rational(196, 15), 9, 15, 0, 0]
u, v, w, x, y, z = symbols('u v w x y z')
assert convolution_subset([u, v, w], [x, y]) == [u*x, u*y + v*x, w*x, w*y]
assert convolution_subset([u, v, w, x], [y, z]) == \
[u*y, u*z + v*y, w*y, w*z + x*y]
assert convolution_subset([u, v], [x, y, z]) == \
convolution_subset([x, y, z], [u, v])
raises(TypeError, lambda: convolution_subset(x, z))
raises(TypeError, lambda: convolution_subset(Rational(7, 3), u))
def test_covering_product():
assert covering_product([], []) == []
assert covering_product([], [Rational(1, 3)]) == []
assert covering_product([6 + I*3/7], [Rational(2, 3)]) == [4 + I*2/7]
a = [1, Rational(5, 8), sqrt(7), 4 + 9*I]
b = [66, 81, 95, 49, 37, 89, 17]
c = [3 + I*2/3, 51 + 72*I, 7, Rational(7, 15), 91]
assert covering_product(a, b) == [66, Rational(1383, 8), 95 + 161*sqrt(7),
130*sqrt(7) + 1303 + 2619*I, 37,
Rational(671, 4), 17 + 54*sqrt(7),
89*sqrt(7) + Rational(4661, 8) + 1287*I]
assert covering_product(b, c) == [198 + 44*I, 7740 + 10638*I,
1412 + I*190/3, Rational(42684, 5) + I*31202/3,
9484 + I*74/3, 22163 + I*27394/3,
10621 + I*34/3, Rational(90236, 15) + 1224*I]
assert covering_product(a, c) == covering_product(c, a)
assert covering_product(b, c[:-1]) == [198 + 44*I, 7740 + 10638*I,
1412 + I*190/3, Rational(42684, 5) + I*31202/3,
111 + I*74/3, 6693 + I*27394/3,
429 + I*34/3, Rational(23351, 15) + 1224*I]
assert covering_product(a, c[:-1]) == [3 + I*2/3,
Rational(339, 4) + I*1409/12, 7 + 10*sqrt(7) + 2*sqrt(7)*I/3,
-403 + 772*sqrt(7)/15 + 72*sqrt(7)*I + I*12658/15]
u, v, w, x, y, z = symbols('u v w x y z')
assert covering_product([u, v, w], [x, y]) == \
[u*x, u*y + v*x + v*y, w*x, w*y]
assert covering_product([u, v, w, x], [y, z]) == \
[u*y, u*z + v*y + v*z, w*y, w*z + x*y + x*z]
assert covering_product([u, v], [x, y, z]) == \
covering_product([x, y, z], [u, v])
raises(TypeError, lambda: covering_product(x, z))
raises(TypeError, lambda: covering_product(Rational(7, 3), u))
def test_intersecting_product():
assert intersecting_product([], []) == []
assert intersecting_product([], [Rational(1, 3)]) == []
assert intersecting_product([6 + I*3/7], [Rational(2, 3)]) == [4 + I*2/7]
a = [1, sqrt(5), Rational(3, 8) + 5*I, 4 + 7*I]
b = [67, 51, 65, 48, 36, 79, 27]
c = [3 + I*2/5, 5 + 9*I, 7, Rational(7, 19), 13]
assert intersecting_product(a, b) == [195*sqrt(5) + Rational(6979, 8) + 1886*I,
178*sqrt(5) + 520 + 910*I, Rational(841, 2) + 1344*I,
192 + 336*I, 0, 0, 0, 0]
assert intersecting_product(b, c) == [Rational(128553, 19) + I*9521/5,
Rational(17820, 19) + 1602*I, Rational(19264, 19), Rational(336, 19), 1846, 0, 0, 0]
assert intersecting_product(a, c) == intersecting_product(c, a)
assert intersecting_product(b[1:], c[:-1]) == [Rational(64788, 19) + I*8622/5,
Rational(12804, 19) + 1152*I, Rational(11508, 19), Rational(252, 19), 0, 0, 0, 0]
assert intersecting_product(a, c[:-2]) == \
[Rational(-99, 5) + 10*sqrt(5) + 2*sqrt(5)*I/5 + I*3021/40,
-43 + 5*sqrt(5) + 9*sqrt(5)*I + 71*I, Rational(245, 8) + 84*I, 0]
u, v, w, x, y, z = symbols('u v w x y z')
assert intersecting_product([u, v, w], [x, y]) == \
[u*x + u*y + v*x + w*x + w*y, v*y, 0, 0]
assert intersecting_product([u, v, w, x], [y, z]) == \
[u*y + u*z + v*y + w*y + w*z + x*y, v*z + x*z, 0, 0]
assert intersecting_product([u, v], [x, y, z]) == \
intersecting_product([x, y, z], [u, v])
raises(TypeError, lambda: intersecting_product(x, z))
raises(TypeError, lambda: intersecting_product(u, Rational(8, 3)))
|
3a5c9ae3a1fbc3c1ac9c68c960db1143e33c1a46f442de3e9a66eb6b4ce7bdcc | # Tests that require installed backends go into
# sympy/test_external/test_autowrap
import os
import tempfile
import shutil
from io import StringIO
from sympy.core import symbols, Eq
from sympy.utilities.autowrap import (autowrap, binary_function,
CythonCodeWrapper, UfuncifyCodeWrapper, CodeWrapper)
from sympy.utilities.codegen import (
CCodeGen, C99CodeGen, CodeGenArgumentListError, make_routine
)
from sympy.testing.pytest import raises
from sympy.testing.tmpfiles import TmpFileManager
def get_string(dump_fn, routines, prefix="file", **kwargs):
"""Wrapper for dump_fn. dump_fn writes its results to a stream object and
this wrapper returns the contents of that stream as a string. This
auxiliary function is used by many tests below.
The header and the empty lines are not generator to facilitate the
testing of the output.
"""
output = StringIO()
dump_fn(routines, output, prefix, **kwargs)
source = output.getvalue()
output.close()
return source
def test_cython_wrapper_scalar_function():
x, y, z = symbols('x,y,z')
expr = (x + y)*z
routine = make_routine("test", expr)
code_gen = CythonCodeWrapper(CCodeGen())
source = get_string(code_gen.dump_pyx, [routine])
expected = (
"cdef extern from 'file.h':\n"
" double test(double x, double y, double z)\n"
"\n"
"def test_c(double x, double y, double z):\n"
"\n"
" return test(x, y, z)")
assert source == expected
def test_cython_wrapper_outarg():
from sympy.core.relational import Equality
x, y, z = symbols('x,y,z')
code_gen = CythonCodeWrapper(C99CodeGen())
routine = make_routine("test", Equality(z, x + y))
source = get_string(code_gen.dump_pyx, [routine])
expected = (
"cdef extern from 'file.h':\n"
" void test(double x, double y, double *z)\n"
"\n"
"def test_c(double x, double y):\n"
"\n"
" cdef double z = 0\n"
" test(x, y, &z)\n"
" return z")
assert source == expected
def test_cython_wrapper_inoutarg():
from sympy.core.relational import Equality
x, y, z = symbols('x,y,z')
code_gen = CythonCodeWrapper(C99CodeGen())
routine = make_routine("test", Equality(z, x + y + z))
source = get_string(code_gen.dump_pyx, [routine])
expected = (
"cdef extern from 'file.h':\n"
" void test(double x, double y, double *z)\n"
"\n"
"def test_c(double x, double y, double z):\n"
"\n"
" test(x, y, &z)\n"
" return z")
assert source == expected
def test_cython_wrapper_compile_flags():
from sympy.core.relational import Equality
x, y, z = symbols('x,y,z')
routine = make_routine("test", Equality(z, x + y))
code_gen = CythonCodeWrapper(CCodeGen())
expected = """\
try:
from setuptools import setup
from setuptools import Extension
except ImportError:
from distutils.core import setup
from distutils.extension import Extension
from Cython.Build import cythonize
cy_opts = {'compiler_directives': {'language_level': '3'}}
ext_mods = [Extension(
'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'],
include_dirs=[],
library_dirs=[],
libraries=[],
extra_compile_args=['-std=c99'],
extra_link_args=[]
)]
setup(ext_modules=cythonize(ext_mods, **cy_opts))
""" % {'num': CodeWrapper._module_counter}
temp_dir = tempfile.mkdtemp()
TmpFileManager.tmp_folder(temp_dir)
setup_file_path = os.path.join(temp_dir, 'setup.py')
code_gen._prepare_files(routine, build_dir=temp_dir)
with open(setup_file_path) as f:
setup_text = f.read()
assert setup_text == expected
code_gen = CythonCodeWrapper(CCodeGen(),
include_dirs=['/usr/local/include', '/opt/booger/include'],
library_dirs=['/user/local/lib'],
libraries=['thelib', 'nilib'],
extra_compile_args=['-slow-math'],
extra_link_args=['-lswamp', '-ltrident'],
cythonize_options={'compiler_directives': {'boundscheck': False}}
)
expected = """\
try:
from setuptools import setup
from setuptools import Extension
except ImportError:
from distutils.core import setup
from distutils.extension import Extension
from Cython.Build import cythonize
cy_opts = {'compiler_directives': {'boundscheck': False}}
ext_mods = [Extension(
'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'],
include_dirs=['/usr/local/include', '/opt/booger/include'],
library_dirs=['/user/local/lib'],
libraries=['thelib', 'nilib'],
extra_compile_args=['-slow-math', '-std=c99'],
extra_link_args=['-lswamp', '-ltrident']
)]
setup(ext_modules=cythonize(ext_mods, **cy_opts))
""" % {'num': CodeWrapper._module_counter}
code_gen._prepare_files(routine, build_dir=temp_dir)
with open(setup_file_path) as f:
setup_text = f.read()
assert setup_text == expected
expected = """\
try:
from setuptools import setup
from setuptools import Extension
except ImportError:
from distutils.core import setup
from distutils.extension import Extension
from Cython.Build import cythonize
cy_opts = {'compiler_directives': {'boundscheck': False}}
import numpy as np
ext_mods = [Extension(
'wrapper_module_%(num)s', ['wrapper_module_%(num)s.pyx', 'wrapped_code_%(num)s.c'],
include_dirs=['/usr/local/include', '/opt/booger/include', np.get_include()],
library_dirs=['/user/local/lib'],
libraries=['thelib', 'nilib'],
extra_compile_args=['-slow-math', '-std=c99'],
extra_link_args=['-lswamp', '-ltrident']
)]
setup(ext_modules=cythonize(ext_mods, **cy_opts))
""" % {'num': CodeWrapper._module_counter}
code_gen._need_numpy = True
code_gen._prepare_files(routine, build_dir=temp_dir)
with open(setup_file_path) as f:
setup_text = f.read()
assert setup_text == expected
TmpFileManager.cleanup()
def test_cython_wrapper_unique_dummyvars():
from sympy.core.relational import Equality
from sympy.core.symbol import Dummy
x, y, z = Dummy('x'), Dummy('y'), Dummy('z')
x_id, y_id, z_id = [str(d.dummy_index) for d in [x, y, z]]
expr = Equality(z, x + y)
routine = make_routine("test", expr)
code_gen = CythonCodeWrapper(CCodeGen())
source = get_string(code_gen.dump_pyx, [routine])
expected_template = (
"cdef extern from 'file.h':\n"
" void test(double x_{x_id}, double y_{y_id}, double *z_{z_id})\n"
"\n"
"def test_c(double x_{x_id}, double y_{y_id}):\n"
"\n"
" cdef double z_{z_id} = 0\n"
" test(x_{x_id}, y_{y_id}, &z_{z_id})\n"
" return z_{z_id}")
expected = expected_template.format(x_id=x_id, y_id=y_id, z_id=z_id)
assert source == expected
def test_autowrap_dummy():
x, y, z = symbols('x y z')
# Uses DummyWrapper to test that codegen works as expected
f = autowrap(x + y, backend='dummy')
assert f() == str(x + y)
assert f.args == "x, y"
assert f.returns == "nameless"
f = autowrap(Eq(z, x + y), backend='dummy')
assert f() == str(x + y)
assert f.args == "x, y"
assert f.returns == "z"
f = autowrap(Eq(z, x + y + z), backend='dummy')
assert f() == str(x + y + z)
assert f.args == "x, y, z"
assert f.returns == "z"
def test_autowrap_args():
x, y, z = symbols('x y z')
raises(CodeGenArgumentListError, lambda: autowrap(Eq(z, x + y),
backend='dummy', args=[x]))
f = autowrap(Eq(z, x + y), backend='dummy', args=[y, x])
assert f() == str(x + y)
assert f.args == "y, x"
assert f.returns == "z"
raises(CodeGenArgumentListError, lambda: autowrap(Eq(z, x + y + z),
backend='dummy', args=[x, y]))
f = autowrap(Eq(z, x + y + z), backend='dummy', args=[y, x, z])
assert f() == str(x + y + z)
assert f.args == "y, x, z"
assert f.returns == "z"
f = autowrap(Eq(z, x + y + z), backend='dummy', args=(y, x, z))
assert f() == str(x + y + z)
assert f.args == "y, x, z"
assert f.returns == "z"
def test_autowrap_store_files():
x, y = symbols('x y')
tmp = tempfile.mkdtemp()
TmpFileManager.tmp_folder(tmp)
f = autowrap(x + y, backend='dummy', tempdir=tmp)
assert f() == str(x + y)
assert os.access(tmp, os.F_OK)
TmpFileManager.cleanup()
def test_autowrap_store_files_issue_gh12939():
x, y = symbols('x y')
tmp = './tmp'
try:
f = autowrap(x + y, backend='dummy', tempdir=tmp)
assert f() == str(x + y)
assert os.access(tmp, os.F_OK)
finally:
shutil.rmtree(tmp)
def test_binary_function():
x, y = symbols('x y')
f = binary_function('f', x + y, backend='dummy')
assert f._imp_() == str(x + y)
def test_ufuncify_source():
x, y, z = symbols('x,y,z')
code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify"))
routine = make_routine("test", x + y + z)
source = get_string(code_wrapper.dump_c, [routine])
expected = """\
#include "Python.h"
#include "math.h"
#include "numpy/ndarraytypes.h"
#include "numpy/ufuncobject.h"
#include "numpy/halffloat.h"
#include "file.h"
static PyMethodDef wrapper_module_%(num)sMethods[] = {
{NULL, NULL, 0, NULL}
};
static void test_ufunc(char **args, npy_intp *dimensions, npy_intp* steps, void* data)
{
npy_intp i;
npy_intp n = dimensions[0];
char *in0 = args[0];
char *in1 = args[1];
char *in2 = args[2];
char *out0 = args[3];
npy_intp in0_step = steps[0];
npy_intp in1_step = steps[1];
npy_intp in2_step = steps[2];
npy_intp out0_step = steps[3];
for (i = 0; i < n; i++) {
*((double *)out0) = test(*(double *)in0, *(double *)in1, *(double *)in2);
in0 += in0_step;
in1 += in1_step;
in2 += in2_step;
out0 += out0_step;
}
}
PyUFuncGenericFunction test_funcs[1] = {&test_ufunc};
static char test_types[4] = {NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE};
static void *test_data[1] = {NULL};
#if PY_VERSION_HEX >= 0x03000000
static struct PyModuleDef moduledef = {
PyModuleDef_HEAD_INIT,
"wrapper_module_%(num)s",
NULL,
-1,
wrapper_module_%(num)sMethods,
NULL,
NULL,
NULL,
NULL
};
PyMODINIT_FUNC PyInit_wrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = PyModule_Create(&moduledef);
if (!m) {
return NULL;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(test_funcs, test_data, test_types, 1, 3, 1,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "test", ufunc0);
Py_DECREF(ufunc0);
return m;
}
#else
PyMODINIT_FUNC initwrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = Py_InitModule("wrapper_module_%(num)s", wrapper_module_%(num)sMethods);
if (m == NULL) {
return;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(test_funcs, test_data, test_types, 1, 3, 1,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "test", ufunc0);
Py_DECREF(ufunc0);
}
#endif""" % {'num': CodeWrapper._module_counter}
assert source == expected
def test_ufuncify_source_multioutput():
x, y, z = symbols('x,y,z')
var_symbols = (x, y, z)
expr = x + y**3 + 10*z**2
code_wrapper = UfuncifyCodeWrapper(C99CodeGen("ufuncify"))
routines = [make_routine("func{}".format(i), expr.diff(var_symbols[i]), var_symbols) for i in range(len(var_symbols))]
source = get_string(code_wrapper.dump_c, routines, funcname='multitest')
expected = """\
#include "Python.h"
#include "math.h"
#include "numpy/ndarraytypes.h"
#include "numpy/ufuncobject.h"
#include "numpy/halffloat.h"
#include "file.h"
static PyMethodDef wrapper_module_%(num)sMethods[] = {
{NULL, NULL, 0, NULL}
};
static void multitest_ufunc(char **args, npy_intp *dimensions, npy_intp* steps, void* data)
{
npy_intp i;
npy_intp n = dimensions[0];
char *in0 = args[0];
char *in1 = args[1];
char *in2 = args[2];
char *out0 = args[3];
char *out1 = args[4];
char *out2 = args[5];
npy_intp in0_step = steps[0];
npy_intp in1_step = steps[1];
npy_intp in2_step = steps[2];
npy_intp out0_step = steps[3];
npy_intp out1_step = steps[4];
npy_intp out2_step = steps[5];
for (i = 0; i < n; i++) {
*((double *)out0) = func0(*(double *)in0, *(double *)in1, *(double *)in2);
*((double *)out1) = func1(*(double *)in0, *(double *)in1, *(double *)in2);
*((double *)out2) = func2(*(double *)in0, *(double *)in1, *(double *)in2);
in0 += in0_step;
in1 += in1_step;
in2 += in2_step;
out0 += out0_step;
out1 += out1_step;
out2 += out2_step;
}
}
PyUFuncGenericFunction multitest_funcs[1] = {&multitest_ufunc};
static char multitest_types[6] = {NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE, NPY_DOUBLE};
static void *multitest_data[1] = {NULL};
#if PY_VERSION_HEX >= 0x03000000
static struct PyModuleDef moduledef = {
PyModuleDef_HEAD_INIT,
"wrapper_module_%(num)s",
NULL,
-1,
wrapper_module_%(num)sMethods,
NULL,
NULL,
NULL,
NULL
};
PyMODINIT_FUNC PyInit_wrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = PyModule_Create(&moduledef);
if (!m) {
return NULL;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(multitest_funcs, multitest_data, multitest_types, 1, 3, 3,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "multitest", ufunc0);
Py_DECREF(ufunc0);
return m;
}
#else
PyMODINIT_FUNC initwrapper_module_%(num)s(void)
{
PyObject *m, *d;
PyObject *ufunc0;
m = Py_InitModule("wrapper_module_%(num)s", wrapper_module_%(num)sMethods);
if (m == NULL) {
return;
}
import_array();
import_umath();
d = PyModule_GetDict(m);
ufunc0 = PyUFunc_FromFuncAndData(multitest_funcs, multitest_data, multitest_types, 1, 3, 3,
PyUFunc_None, "wrapper_module_%(num)s", "Created in SymPy with Ufuncify", 0);
PyDict_SetItemString(d, "multitest", ufunc0);
Py_DECREF(ufunc0);
}
#endif""" % {'num': CodeWrapper._module_counter}
assert source == expected
|
54647a6c86f3105905617ee6d0c5a84ff1345276c613e455b0847dc0d1678136 | from itertools import product
import math
import inspect
import mpmath
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.concrete.summations import Sum
from sympy.core.function import (Function, Lambda, diff)
from sympy.core.numbers import (E, Float, I, Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions.combinatorial.factorials import (RisingFactorial, factorial)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.hyperbolic import acosh
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, cos, cot, sin,
sinc, tan)
from sympy.functions.special.bessel import (besseli, besselj, besselk, bessely)
from sympy.functions.special.beta_functions import (beta, betainc, betainc_regularized)
from sympy.functions.special.delta_functions import (Heaviside)
from sympy.functions.special.error_functions import (Ei, erf, erfc, fresnelc, fresnels)
from sympy.functions.special.gamma_functions import (digamma, gamma, loggamma)
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import (And, false, ITE, Not, Or, true)
from sympy.matrices.expressions.dotproduct import DotProduct
from sympy.tensor.array import derive_by_array, Array
from sympy.tensor.indexed import IndexedBase
from sympy.utilities.lambdify import lambdify
from sympy.core.expr import UnevaluatedExpr
from sympy.codegen.cfunctions import expm1, log1p, exp2, log2, log10, hypot
from sympy.codegen.numpy_nodes import logaddexp, logaddexp2
from sympy.codegen.scipy_nodes import cosm1
from sympy.functions.elementary.complexes import re, im, arg
from sympy.functions.special.polynomials import \
chebyshevt, chebyshevu, legendre, hermite, laguerre, gegenbauer, \
assoc_legendre, assoc_laguerre, jacobi
from sympy.matrices import Matrix, MatrixSymbol, SparseMatrix
from sympy.printing.lambdarepr import LambdaPrinter
from sympy.printing.numpy import NumPyPrinter
from sympy.utilities.lambdify import implemented_function, lambdastr
from sympy.testing.pytest import skip
from sympy.utilities.decorator import conserve_mpmath_dps
from sympy.utilities.exceptions import ignore_warnings
from sympy.external import import_module
from sympy.functions.special.gamma_functions import uppergamma, lowergamma
import sympy
MutableDenseMatrix = Matrix
numpy = import_module('numpy')
scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']})
numexpr = import_module('numexpr')
tensorflow = import_module('tensorflow')
cupy = import_module('cupy')
jax = import_module('jax')
numba = import_module('numba')
if tensorflow:
# Hide Tensorflow warnings
import os
os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2'
w, x, y, z = symbols('w,x,y,z')
#================== Test different arguments =======================
def test_no_args():
f = lambdify([], 1)
raises(TypeError, lambda: f(-1))
assert f() == 1
def test_single_arg():
f = lambdify(x, 2*x)
assert f(1) == 2
def test_list_args():
f = lambdify([x, y], x + y)
assert f(1, 2) == 3
def test_nested_args():
f1 = lambdify([[w, x]], [w, x])
assert f1([91, 2]) == [91, 2]
raises(TypeError, lambda: f1(1, 2))
f2 = lambdify([(w, x), (y, z)], [w, x, y, z])
assert f2((18, 12), (73, 4)) == [18, 12, 73, 4]
raises(TypeError, lambda: f2(3, 4))
f3 = lambdify([w, [[[x]], y], z], [w, x, y, z])
assert f3(10, [[[52]], 31], 44) == [10, 52, 31, 44]
def test_str_args():
f = lambdify('x,y,z', 'z,y,x')
assert f(3, 2, 1) == (1, 2, 3)
assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0)
# make sure correct number of args required
raises(TypeError, lambda: f(0))
def test_own_namespace_1():
myfunc = lambda x: 1
f = lambdify(x, sin(x), {"sin": myfunc})
assert f(0.1) == 1
assert f(100) == 1
def test_own_namespace_2():
def myfunc(x):
return 1
f = lambdify(x, sin(x), {'sin': myfunc})
assert f(0.1) == 1
assert f(100) == 1
def test_own_module():
f = lambdify(x, sin(x), math)
assert f(0) == 0.0
p, q, r = symbols("p q r", real=True)
ae = abs(exp(p+UnevaluatedExpr(q+r)))
f = lambdify([p, q, r], [ae, ae], modules=math)
results = f(1.0, 1e18, -1e18)
refvals = [math.exp(1.0)]*2
for res, ref in zip(results, refvals):
assert abs((res-ref)/ref) < 1e-15
def test_bad_args():
# no vargs given
raises(TypeError, lambda: lambdify(1))
# same with vector exprs
raises(TypeError, lambda: lambdify([1, 2]))
def test_atoms():
# Non-Symbol atoms should not be pulled out from the expression namespace
f = lambdify(x, pi + x, {"pi": 3.14})
assert f(0) == 3.14
f = lambdify(x, I + x, {"I": 1j})
assert f(1) == 1 + 1j
#================== Test different modules =========================
# high precision output of sin(0.2*pi) is used to detect if precision is lost unwanted
@conserve_mpmath_dps
def test_sympy_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "sympy")
assert f(x) == sin(x)
prec = 1e-15
assert -prec < f(Rational(1, 5)).evalf() - Float(str(sin02)) < prec
# arctan is in numpy module and should not be available
# The arctan below gives NameError. What is this supposed to test?
# raises(NameError, lambda: lambdify(x, arctan(x), "sympy"))
@conserve_mpmath_dps
def test_math_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "math")
prec = 1e-15
assert -prec < f(0.2) - sin02 < prec
raises(TypeError, lambda: f(x))
# if this succeeds, it can't be a Python math function
@conserve_mpmath_dps
def test_mpmath_lambda():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin(x), "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(mpmath.mpf("0.2")) - sin02 < prec
raises(TypeError, lambda: f(x))
# if this succeeds, it can't be a mpmath function
@conserve_mpmath_dps
def test_number_precision():
mpmath.mp.dps = 50
sin02 = mpmath.mpf("0.19866933079506121545941262711838975037020672954020")
f = lambdify(x, sin02, "mpmath")
prec = 1e-49 # mpmath precision is around 50 decimal places
assert -prec < f(0) - sin02 < prec
@conserve_mpmath_dps
def test_mpmath_precision():
mpmath.mp.dps = 100
assert str(lambdify((), pi.evalf(100), 'mpmath')()) == str(pi.evalf(100))
#================== Test Translations ==============================
# We can only check if all translated functions are valid. It has to be checked
# by hand if they are complete.
def test_math_transl():
from sympy.utilities.lambdify import MATH_TRANSLATIONS
for sym, mat in MATH_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert mat in math.__dict__
def test_mpmath_transl():
from sympy.utilities.lambdify import MPMATH_TRANSLATIONS
for sym, mat in MPMATH_TRANSLATIONS.items():
assert sym in sympy.__dict__ or sym == 'Matrix'
assert mat in mpmath.__dict__
def test_numpy_transl():
if not numpy:
skip("numpy not installed.")
from sympy.utilities.lambdify import NUMPY_TRANSLATIONS
for sym, nump in NUMPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert nump in numpy.__dict__
def test_scipy_transl():
if not scipy:
skip("scipy not installed.")
from sympy.utilities.lambdify import SCIPY_TRANSLATIONS
for sym, scip in SCIPY_TRANSLATIONS.items():
assert sym in sympy.__dict__
assert scip in scipy.__dict__ or scip in scipy.special.__dict__
def test_numpy_translation_abs():
if not numpy:
skip("numpy not installed.")
f = lambdify(x, Abs(x), "numpy")
assert f(-1) == 1
assert f(1) == 1
def test_numexpr_printer():
if not numexpr:
skip("numexpr not installed.")
# if translation/printing is done incorrectly then evaluating
# a lambdified numexpr expression will throw an exception
from sympy.printing.lambdarepr import NumExprPrinter
blacklist = ('where', 'complex', 'contains')
arg_tuple = (x, y, z) # some functions take more than one argument
for sym in NumExprPrinter._numexpr_functions.keys():
if sym in blacklist:
continue
ssym = S(sym)
if hasattr(ssym, '_nargs'):
nargs = ssym._nargs[0]
else:
nargs = 1
args = arg_tuple[:nargs]
f = lambdify(args, ssym(*args), modules='numexpr')
assert f(*(1, )*nargs) is not None
def test_issue_9334():
if not numexpr:
skip("numexpr not installed.")
if not numpy:
skip("numpy not installed.")
expr = S('b*a - sqrt(a**2)')
a, b = sorted(expr.free_symbols, key=lambda s: s.name)
func_numexpr = lambdify((a,b), expr, modules=[numexpr], dummify=False)
foo, bar = numpy.random.random((2, 4))
func_numexpr(foo, bar)
def test_issue_12984():
if not numexpr:
skip("numexpr not installed.")
func_numexpr = lambdify((x,y,z), Piecewise((y, x >= 0), (z, x > -1)), numexpr)
with ignore_warnings(RuntimeWarning):
assert func_numexpr(1, 24, 42) == 24
assert str(func_numexpr(-1, 24, 42)) == 'nan'
def test_empty_modules():
x, y = symbols('x y')
expr = -(x % y)
no_modules = lambdify([x, y], expr)
empty_modules = lambdify([x, y], expr, modules=[])
assert no_modules(3, 7) == empty_modules(3, 7)
assert no_modules(3, 7) == -3
def test_exponentiation():
f = lambdify(x, x**2)
assert f(-1) == 1
assert f(0) == 0
assert f(1) == 1
assert f(-2) == 4
assert f(2) == 4
assert f(2.5) == 6.25
def test_sqrt():
f = lambdify(x, sqrt(x))
assert f(0) == 0.0
assert f(1) == 1.0
assert f(4) == 2.0
assert abs(f(2) - 1.414) < 0.001
assert f(6.25) == 2.5
def test_trig():
f = lambdify([x], [cos(x), sin(x)], 'math')
d = f(pi)
prec = 1e-11
assert -prec < d[0] + 1 < prec
assert -prec < d[1] < prec
d = f(3.14159)
prec = 1e-5
assert -prec < d[0] + 1 < prec
assert -prec < d[1] < prec
def test_integral():
if numpy and not scipy:
skip("scipy not installed.")
f = Lambda(x, exp(-x**2))
l = lambdify(y, Integral(f(x), (x, y, oo)))
d = l(-oo)
assert 1.77245385 < d < 1.772453851
def test_double_integral():
if numpy and not scipy:
skip("scipy not installed.")
# example from http://mpmath.org/doc/current/calculus/integration.html
i = Integral(1/(1 - x**2*y**2), (x, 0, 1), (y, 0, z))
l = lambdify([z], i)
d = l(1)
assert 1.23370055 < d < 1.233700551
#================== Test vectors ===================================
def test_vector_simple():
f = lambdify((x, y, z), (z, y, x))
assert f(3, 2, 1) == (1, 2, 3)
assert f(1.0, 2.0, 3.0) == (3.0, 2.0, 1.0)
# make sure correct number of args required
raises(TypeError, lambda: f(0))
def test_vector_discontinuous():
f = lambdify(x, (-1/x, 1/x))
raises(ZeroDivisionError, lambda: f(0))
assert f(1) == (-1.0, 1.0)
assert f(2) == (-0.5, 0.5)
assert f(-2) == (0.5, -0.5)
def test_trig_symbolic():
f = lambdify([x], [cos(x), sin(x)], 'math')
d = f(pi)
assert abs(d[0] + 1) < 0.0001
assert abs(d[1] - 0) < 0.0001
def test_trig_float():
f = lambdify([x], [cos(x), sin(x)])
d = f(3.14159)
assert abs(d[0] + 1) < 0.0001
assert abs(d[1] - 0) < 0.0001
def test_docs():
f = lambdify(x, x**2)
assert f(2) == 4
f = lambdify([x, y, z], [z, y, x])
assert f(1, 2, 3) == [3, 2, 1]
f = lambdify(x, sqrt(x))
assert f(4) == 2.0
f = lambdify((x, y), sin(x*y)**2)
assert f(0, 5) == 0
def test_math():
f = lambdify((x, y), sin(x), modules="math")
assert f(0, 5) == 0
def test_sin():
f = lambdify(x, sin(x)**2)
assert isinstance(f(2), float)
f = lambdify(x, sin(x)**2, modules="math")
assert isinstance(f(2), float)
def test_matrix():
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol = Matrix([[1, 2], [sin(3) + 4, 1]])
f = lambdify((x, y, z), A, modules="sympy")
assert f(1, 2, 3) == sol
f = lambdify((x, y, z), (A, [A]), modules="sympy")
assert f(1, 2, 3) == (sol, [sol])
J = Matrix((x, x + y)).jacobian((x, y))
v = Matrix((x, y))
sol = Matrix([[1, 0], [1, 1]])
assert lambdify(v, J, modules='sympy')(1, 2) == sol
assert lambdify(v.T, J, modules='sympy')(1, 2) == sol
def test_numpy_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
#Lambdify array first, to ensure return to array as default
f = lambdify((x, y, z), A, ['numpy'])
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
#Check that the types are arrays and matrices
assert isinstance(f(1, 2, 3), numpy.ndarray)
# gh-15071
class dot(Function):
pass
x_dot_mtx = dot(x, Matrix([[2], [1], [0]]))
f_dot1 = lambdify(x, x_dot_mtx)
inp = numpy.zeros((17, 3))
assert numpy.all(f_dot1(inp) == 0)
strict_kw = dict(allow_unknown_functions=False, inline=True, fully_qualified_modules=False)
p2 = NumPyPrinter(dict(user_functions={'dot': 'dot'}, **strict_kw))
f_dot2 = lambdify(x, x_dot_mtx, printer=p2)
assert numpy.all(f_dot2(inp) == 0)
p3 = NumPyPrinter(strict_kw)
# The line below should probably fail upon construction (before calling with "(inp)"):
raises(Exception, lambda: lambdify(x, x_dot_mtx, printer=p3)(inp))
def test_numpy_transpose():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A.T, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, 0], [2, 1]]))
def test_numpy_dotproduct():
if not numpy:
skip("numpy not installed")
A = Matrix([x, y, z])
f1 = lambdify([x, y, z], DotProduct(A, A), modules='numpy')
f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy')
f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='numpy')
f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='numpy')
assert f1(1, 2, 3) == \
f2(1, 2, 3) == \
f3(1, 2, 3) == \
f4(1, 2, 3) == \
numpy.array([14])
def test_numpy_inverse():
if not numpy:
skip("numpy not installed.")
A = Matrix([[1, x], [0, 1]])
f = lambdify((x), A**-1, modules="numpy")
numpy.testing.assert_array_equal(f(2), numpy.array([[1, -2], [0, 1]]))
def test_numpy_old_matrix():
if not numpy:
skip("numpy not installed.")
A = Matrix([[x, x*y], [sin(z) + 4, x**z]])
sol_arr = numpy.array([[1, 2], [numpy.sin(3) + 4, 1]])
f = lambdify((x, y, z), A, [{'ImmutableDenseMatrix': numpy.matrix}, 'numpy'])
with ignore_warnings(PendingDeprecationWarning):
numpy.testing.assert_allclose(f(1, 2, 3), sol_arr)
assert isinstance(f(1, 2, 3), numpy.matrix)
def test_scipy_sparse_matrix():
if not scipy:
skip("scipy not installed.")
A = SparseMatrix([[x, 0], [0, y]])
f = lambdify((x, y), A, modules="scipy")
B = f(1, 2)
assert isinstance(B, scipy.sparse.coo_matrix)
def test_python_div_zero_issue_11306():
if not numpy:
skip("numpy not installed.")
p = Piecewise((1 / x, y < -1), (x, y < 1), (1 / x, True))
f = lambdify([x, y], p, modules='numpy')
numpy.seterr(divide='ignore')
assert float(f(numpy.array([0]),numpy.array([0.5]))) == 0
assert str(float(f(numpy.array([0]),numpy.array([1])))) == 'inf'
numpy.seterr(divide='warn')
def test_issue9474():
mods = [None, 'math']
if numpy:
mods.append('numpy')
if mpmath:
mods.append('mpmath')
for mod in mods:
f = lambdify(x, S.One/x, modules=mod)
assert f(2) == 0.5
f = lambdify(x, floor(S.One/x), modules=mod)
assert f(2) == 0
for absfunc, modules in product([Abs, abs], mods):
f = lambdify(x, absfunc(x), modules=modules)
assert f(-1) == 1
assert f(1) == 1
assert f(3+4j) == 5
def test_issue_9871():
if not numexpr:
skip("numexpr not installed.")
if not numpy:
skip("numpy not installed.")
r = sqrt(x**2 + y**2)
expr = diff(1/r, x)
xn = yn = numpy.linspace(1, 10, 16)
# expr(xn, xn) = -xn/(sqrt(2)*xn)^3
fv_exact = -numpy.sqrt(2.)**-3 * xn**-2
fv_numpy = lambdify((x, y), expr, modules='numpy')(xn, yn)
fv_numexpr = lambdify((x, y), expr, modules='numexpr')(xn, yn)
numpy.testing.assert_allclose(fv_numpy, fv_exact, rtol=1e-10)
numpy.testing.assert_allclose(fv_numexpr, fv_exact, rtol=1e-10)
def test_numpy_piecewise():
if not numpy:
skip("numpy not installed.")
pieces = Piecewise((x, x < 3), (x**2, x > 5), (0, True))
f = lambdify(x, pieces, modules="numpy")
numpy.testing.assert_array_equal(f(numpy.arange(10)),
numpy.array([0, 1, 2, 0, 0, 0, 36, 49, 64, 81]))
# If we evaluate somewhere all conditions are False, we should get back NaN
nodef_func = lambdify(x, Piecewise((x, x > 0), (-x, x < 0)))
numpy.testing.assert_array_equal(nodef_func(numpy.array([-1, 0, 1])),
numpy.array([1, numpy.nan, 1]))
def test_numpy_logical_ops():
if not numpy:
skip("numpy not installed.")
and_func = lambdify((x, y), And(x, y), modules="numpy")
and_func_3 = lambdify((x, y, z), And(x, y, z), modules="numpy")
or_func = lambdify((x, y), Or(x, y), modules="numpy")
or_func_3 = lambdify((x, y, z), Or(x, y, z), modules="numpy")
not_func = lambdify((x), Not(x), modules="numpy")
arr1 = numpy.array([True, True])
arr2 = numpy.array([False, True])
arr3 = numpy.array([True, False])
numpy.testing.assert_array_equal(and_func(arr1, arr2), numpy.array([False, True]))
numpy.testing.assert_array_equal(and_func_3(arr1, arr2, arr3), numpy.array([False, False]))
numpy.testing.assert_array_equal(or_func(arr1, arr2), numpy.array([True, True]))
numpy.testing.assert_array_equal(or_func_3(arr1, arr2, arr3), numpy.array([True, True]))
numpy.testing.assert_array_equal(not_func(arr2), numpy.array([True, False]))
def test_numpy_matmul():
if not numpy:
skip("numpy not installed.")
xmat = Matrix([[x, y], [z, 1+z]])
ymat = Matrix([[x**2], [Abs(x)]])
mat_func = lambdify((x, y, z), xmat*ymat, modules="numpy")
numpy.testing.assert_array_equal(mat_func(0.5, 3, 4), numpy.array([[1.625], [3.5]]))
numpy.testing.assert_array_equal(mat_func(-0.5, 3, 4), numpy.array([[1.375], [3.5]]))
# Multiple matrices chained together in multiplication
f = lambdify((x, y, z), xmat*xmat*xmat, modules="numpy")
numpy.testing.assert_array_equal(f(0.5, 3, 4), numpy.array([[72.125, 119.25],
[159, 251]]))
def test_numpy_numexpr():
if not numpy:
skip("numpy not installed.")
if not numexpr:
skip("numexpr not installed.")
a, b, c = numpy.random.randn(3, 128, 128)
# ensure that numpy and numexpr return same value for complicated expression
expr = sin(x) + cos(y) + tan(z)**2 + Abs(z-y)*acos(sin(y*z)) + \
Abs(y-z)*acosh(2+exp(y-x))- sqrt(x**2+I*y**2)
npfunc = lambdify((x, y, z), expr, modules='numpy')
nefunc = lambdify((x, y, z), expr, modules='numexpr')
assert numpy.allclose(npfunc(a, b, c), nefunc(a, b, c))
def test_numexpr_userfunctions():
if not numpy:
skip("numpy not installed.")
if not numexpr:
skip("numexpr not installed.")
a, b = numpy.random.randn(2, 10)
uf = type('uf', (Function, ),
{'eval' : classmethod(lambda x, y : y**2+1)})
func = lambdify(x, 1-uf(x), modules='numexpr')
assert numpy.allclose(func(a), -(a**2))
uf = implemented_function(Function('uf'), lambda x, y : 2*x*y+1)
func = lambdify((x, y), uf(x, y), modules='numexpr')
assert numpy.allclose(func(a, b), 2*a*b+1)
def test_tensorflow_basic_math():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
a = tensorflow.constant(0, dtype=tensorflow.float32)
assert func(a).eval(session=s) == 0.5
def test_tensorflow_placeholders():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
a = tensorflow.compat.v1.placeholder(dtype=tensorflow.float32)
assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5
def test_tensorflow_variables():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(sin(x), Abs(1/(x+2)))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
a = tensorflow.Variable(0, dtype=tensorflow.float32)
s.run(a.initializer)
assert func(a).eval(session=s, feed_dict={a: 0}) == 0.5
def test_tensorflow_logical_operations():
if not tensorflow:
skip("tensorflow not installed.")
expr = Not(And(Or(x, y), y))
func = lambdify([x, y], expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(False, True).eval(session=s) == False
def test_tensorflow_piecewise():
if not tensorflow:
skip("tensorflow not installed.")
expr = Piecewise((0, Eq(x,0)), (-1, x < 0), (1, x > 0))
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(-1).eval(session=s) == -1
assert func(0).eval(session=s) == 0
assert func(1).eval(session=s) == 1
def test_tensorflow_multi_max():
if not tensorflow:
skip("tensorflow not installed.")
expr = Max(x, -x, x**2)
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(-2).eval(session=s) == 4
def test_tensorflow_multi_min():
if not tensorflow:
skip("tensorflow not installed.")
expr = Min(x, -x, x**2)
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(-2).eval(session=s) == -2
def test_tensorflow_relational():
if not tensorflow:
skip("tensorflow not installed.")
expr = x >= 0
func = lambdify(x, expr, modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
assert func(1).eval(session=s) == True
def test_tensorflow_complexes():
if not tensorflow:
skip("tensorflow not installed")
func1 = lambdify(x, re(x), modules="tensorflow")
func2 = lambdify(x, im(x), modules="tensorflow")
func3 = lambdify(x, Abs(x), modules="tensorflow")
func4 = lambdify(x, arg(x), modules="tensorflow")
with tensorflow.compat.v1.Session() as s:
# For versions before
# https://github.com/tensorflow/tensorflow/issues/30029
# resolved, using Python numeric types may not work
a = tensorflow.constant(1+2j)
assert func1(a).eval(session=s) == 1
assert func2(a).eval(session=s) == 2
tensorflow_result = func3(a).eval(session=s)
sympy_result = Abs(1 + 2j).evalf()
assert abs(tensorflow_result-sympy_result) < 10**-6
tensorflow_result = func4(a).eval(session=s)
sympy_result = arg(1 + 2j).evalf()
assert abs(tensorflow_result-sympy_result) < 10**-6
def test_tensorflow_array_arg():
# Test for issue 14655 (tensorflow part)
if not tensorflow:
skip("tensorflow not installed.")
f = lambdify([[x, y]], x*x + y, 'tensorflow')
with tensorflow.compat.v1.Session() as s:
fcall = f(tensorflow.constant([2.0, 1.0]))
assert fcall.eval(session=s) == 5.0
#================== Test symbolic ==================================
def test_sym_single_arg():
f = lambdify(x, x * y)
assert f(z) == z * y
def test_sym_list_args():
f = lambdify([x, y], x + y + z)
assert f(1, 2) == 3 + z
def test_sym_integral():
f = Lambda(x, exp(-x**2))
l = lambdify(x, Integral(f(x), (x, -oo, oo)), modules="sympy")
assert l(y) == Integral(exp(-y**2), (y, -oo, oo))
assert l(y).doit() == sqrt(pi)
def test_namespace_order():
# lambdify had a bug, such that module dictionaries or cached module
# dictionaries would pull earlier namespaces into themselves.
# Because the module dictionaries form the namespace of the
# generated lambda, this meant that the behavior of a previously
# generated lambda function could change as a result of later calls
# to lambdify.
n1 = {'f': lambda x: 'first f'}
n2 = {'f': lambda x: 'second f',
'g': lambda x: 'function g'}
f = sympy.Function('f')
g = sympy.Function('g')
if1 = lambdify(x, f(x), modules=(n1, "sympy"))
assert if1(1) == 'first f'
if2 = lambdify(x, g(x), modules=(n2, "sympy"))
# previously gave 'second f'
assert if1(1) == 'first f'
assert if2(1) == 'function g'
def test_imps():
# Here we check if the default returned functions are anonymous - in
# the sense that we can have more than one function with the same name
f = implemented_function('f', lambda x: 2*x)
g = implemented_function('f', lambda x: math.sqrt(x))
l1 = lambdify(x, f(x))
l2 = lambdify(x, g(x))
assert str(f(x)) == str(g(x))
assert l1(3) == 6
assert l2(3) == math.sqrt(3)
# check that we can pass in a Function as input
func = sympy.Function('myfunc')
assert not hasattr(func, '_imp_')
my_f = implemented_function(func, lambda x: 2*x)
assert hasattr(my_f, '_imp_')
# Error for functions with same name and different implementation
f2 = implemented_function("f", lambda x: x + 101)
raises(ValueError, lambda: lambdify(x, f(f2(x))))
def test_imps_errors():
# Test errors that implemented functions can return, and still be able to
# form expressions.
# See: https://github.com/sympy/sympy/issues/10810
#
# XXX: Removed AttributeError here. This test was added due to issue 10810
# but that issue was about ValueError. It doesn't seem reasonable to
# "support" catching AttributeError in the same context...
for val, error_class in product((0, 0., 2, 2.0), (TypeError, ValueError)):
def myfunc(a):
if a == 0:
raise error_class
return 1
f = implemented_function('f', myfunc)
expr = f(val)
assert expr == f(val)
def test_imps_wrong_args():
raises(ValueError, lambda: implemented_function(sin, lambda x: x))
def test_lambdify_imps():
# Test lambdify with implemented functions
# first test basic (sympy) lambdify
f = sympy.cos
assert lambdify(x, f(x))(0) == 1
assert lambdify(x, 1 + f(x))(0) == 2
assert lambdify((x, y), y + f(x))(0, 1) == 2
# make an implemented function and test
f = implemented_function("f", lambda x: x + 100)
assert lambdify(x, f(x))(0) == 100
assert lambdify(x, 1 + f(x))(0) == 101
assert lambdify((x, y), y + f(x))(0, 1) == 101
# Can also handle tuples, lists, dicts as expressions
lam = lambdify(x, (f(x), x))
assert lam(3) == (103, 3)
lam = lambdify(x, [f(x), x])
assert lam(3) == [103, 3]
lam = lambdify(x, [f(x), (f(x), x)])
assert lam(3) == [103, (103, 3)]
lam = lambdify(x, {f(x): x})
assert lam(3) == {103: 3}
lam = lambdify(x, {f(x): x})
assert lam(3) == {103: 3}
lam = lambdify(x, {x: f(x)})
assert lam(3) == {3: 103}
# Check that imp preferred to other namespaces by default
d = {'f': lambda x: x + 99}
lam = lambdify(x, f(x), d)
assert lam(3) == 103
# Unless flag passed
lam = lambdify(x, f(x), d, use_imps=False)
assert lam(3) == 102
def test_dummification():
t = symbols('t')
F = Function('F')
G = Function('G')
#"\alpha" is not a valid Python variable name
#lambdify should sub in a dummy for it, and return
#without a syntax error
alpha = symbols(r'\alpha')
some_expr = 2 * F(t)**2 / G(t)
lam = lambdify((F(t), G(t)), some_expr)
assert lam(3, 9) == 2
lam = lambdify(sin(t), 2 * sin(t)**2)
assert lam(F(t)) == 2 * F(t)**2
#Test that \alpha was properly dummified
lam = lambdify((alpha, t), 2*alpha + t)
assert lam(2, 1) == 5
raises(SyntaxError, lambda: lambdify(F(t) * G(t), F(t) * G(t) + 5))
raises(SyntaxError, lambda: lambdify(2 * F(t), 2 * F(t) + 5))
raises(SyntaxError, lambda: lambdify(2 * F(t), 4 * F(t) + 5))
def test_curly_matrix_symbol():
# Issue #15009
curlyv = sympy.MatrixSymbol("{v}", 2, 1)
lam = lambdify(curlyv, curlyv)
assert lam(1)==1
lam = lambdify(curlyv, curlyv, dummify=True)
assert lam(1)==1
def test_python_keywords():
# Test for issue 7452. The automatic dummification should ensure use of
# Python reserved keywords as symbol names will create valid lambda
# functions. This is an additional regression test.
python_if = symbols('if')
expr = python_if / 2
f = lambdify(python_if, expr)
assert f(4.0) == 2.0
def test_lambdify_docstring():
func = lambdify((w, x, y, z), w + x + y + z)
ref = (
"Created with lambdify. Signature:\n\n"
"func(w, x, y, z)\n\n"
"Expression:\n\n"
"w + x + y + z"
).splitlines()
assert func.__doc__.splitlines()[:len(ref)] == ref
syms = symbols('a1:26')
func = lambdify(syms, sum(syms))
ref = (
"Created with lambdify. Signature:\n\n"
"func(a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12, a13, a14, a15,\n"
" a16, a17, a18, a19, a20, a21, a22, a23, a24, a25)\n\n"
"Expression:\n\n"
"a1 + a10 + a11 + a12 + a13 + a14 + a15 + a16 + a17 + a18 + a19 + a2 + a20 +..."
).splitlines()
assert func.__doc__.splitlines()[:len(ref)] == ref
#================== Test special printers ==========================
def test_special_printers():
from sympy.printing.lambdarepr import IntervalPrinter
def intervalrepr(expr):
return IntervalPrinter().doprint(expr)
expr = sqrt(sqrt(2) + sqrt(3)) + S.Half
func0 = lambdify((), expr, modules="mpmath", printer=intervalrepr)
func1 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter)
func2 = lambdify((), expr, modules="mpmath", printer=IntervalPrinter())
mpi = type(mpmath.mpi(1, 2))
assert isinstance(func0(), mpi)
assert isinstance(func1(), mpi)
assert isinstance(func2(), mpi)
# To check Is lambdify loggamma works for mpmath or not
exp1 = lambdify(x, loggamma(x), 'mpmath')(5)
exp2 = lambdify(x, loggamma(x), 'mpmath')(1.8)
exp3 = lambdify(x, loggamma(x), 'mpmath')(15)
exp_ls = [exp1, exp2, exp3]
sol1 = mpmath.loggamma(5)
sol2 = mpmath.loggamma(1.8)
sol3 = mpmath.loggamma(15)
sol_ls = [sol1, sol2, sol3]
assert exp_ls == sol_ls
def test_true_false():
# We want exact is comparison here, not just ==
assert lambdify([], true)() is True
assert lambdify([], false)() is False
def test_issue_2790():
assert lambdify((x, (y, z)), x + y)(1, (2, 4)) == 3
assert lambdify((x, (y, (w, z))), w + x + y + z)(1, (2, (3, 4))) == 10
assert lambdify(x, x + 1, dummify=False)(1) == 2
def test_issue_12092():
f = implemented_function('f', lambda x: x**2)
assert f(f(2)).evalf() == Float(16)
def test_issue_14911():
class Variable(sympy.Symbol):
def _sympystr(self, printer):
return printer.doprint(self.name)
_lambdacode = _sympystr
_numpycode = _sympystr
x = Variable('x')
y = 2 * x
code = LambdaPrinter().doprint(y)
assert code.replace(' ', '') == '2*x'
def test_ITE():
assert lambdify((x, y, z), ITE(x, y, z))(True, 5, 3) == 5
assert lambdify((x, y, z), ITE(x, y, z))(False, 5, 3) == 3
def test_Min_Max():
# see gh-10375
assert lambdify((x, y, z), Min(x, y, z))(1, 2, 3) == 1
assert lambdify((x, y, z), Max(x, y, z))(1, 2, 3) == 3
def test_Indexed():
# Issue #10934
if not numpy:
skip("numpy not installed")
a = IndexedBase('a')
i, j = symbols('i j')
b = numpy.array([[1, 2], [3, 4]])
assert lambdify(a, Sum(a[x, y], (x, 0, 1), (y, 0, 1)))(b) == 10
def test_issue_12173():
#test for issue 12173
expr1 = lambdify((x, y), uppergamma(x, y),"mpmath")(1, 2)
expr2 = lambdify((x, y), lowergamma(x, y),"mpmath")(1, 2)
assert expr1 == uppergamma(1, 2).evalf()
assert expr2 == lowergamma(1, 2).evalf()
def test_issue_13642():
if not numpy:
skip("numpy not installed")
f = lambdify(x, sinc(x))
assert Abs(f(1) - sinc(1)).n() < 1e-15
def test_sinc_mpmath():
f = lambdify(x, sinc(x), "mpmath")
assert Abs(f(1) - sinc(1)).n() < 1e-15
def test_lambdify_dummy_arg():
d1 = Dummy()
f1 = lambdify(d1, d1 + 1, dummify=False)
assert f1(2) == 3
f1b = lambdify(d1, d1 + 1)
assert f1b(2) == 3
d2 = Dummy('x')
f2 = lambdify(d2, d2 + 1)
assert f2(2) == 3
f3 = lambdify([[d2]], d2 + 1)
assert f3([2]) == 3
def test_lambdify_mixed_symbol_dummy_args():
d = Dummy()
# Contrived example of name clash
dsym = symbols(str(d))
f = lambdify([d, dsym], d - dsym)
assert f(4, 1) == 3
def test_numpy_array_arg():
# Test for issue 14655 (numpy part)
if not numpy:
skip("numpy not installed")
f = lambdify([[x, y]], x*x + y, 'numpy')
assert f(numpy.array([2.0, 1.0])) == 5
def test_scipy_fns():
if not scipy:
skip("scipy not installed")
single_arg_sympy_fns = [Ei, erf, erfc, factorial, gamma, loggamma, digamma]
single_arg_scipy_fns = [scipy.special.expi, scipy.special.erf, scipy.special.erfc,
scipy.special.factorial, scipy.special.gamma, scipy.special.gammaln,
scipy.special.psi]
numpy.random.seed(0)
for (sympy_fn, scipy_fn) in zip(single_arg_sympy_fns, single_arg_scipy_fns):
f = lambdify(x, sympy_fn(x), modules="scipy")
for i in range(20):
tv = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy thinks that factorial(z) is 0 when re(z) < 0 and
# does not support complex numbers.
# SymPy does not think so.
if sympy_fn == factorial:
tv = numpy.abs(tv)
# SciPy supports gammaln for real arguments only,
# and there is also a branch cut along the negative real axis
if sympy_fn == loggamma:
tv = numpy.abs(tv)
# SymPy's digamma evaluates as polygamma(0, z)
# which SciPy supports for real arguments only
if sympy_fn == digamma:
tv = numpy.real(tv)
sympy_result = sympy_fn(tv).evalf()
assert abs(f(tv) - sympy_result) < 1e-13*(1 + abs(sympy_result))
assert abs(f(tv) - scipy_fn(tv)) < 1e-13*(1 + abs(sympy_result))
double_arg_sympy_fns = [RisingFactorial, besselj, bessely, besseli,
besselk]
double_arg_scipy_fns = [scipy.special.poch, scipy.special.jv,
scipy.special.yv, scipy.special.iv, scipy.special.kv]
for (sympy_fn, scipy_fn) in zip(double_arg_sympy_fns, double_arg_scipy_fns):
f = lambdify((x, y), sympy_fn(x, y), modules="scipy")
for i in range(20):
# SciPy supports only real orders of Bessel functions
tv1 = numpy.random.uniform(-10, 10)
tv2 = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy supports poch for real arguments only
if sympy_fn == RisingFactorial:
tv2 = numpy.real(tv2)
sympy_result = sympy_fn(tv1, tv2).evalf()
assert abs(f(tv1, tv2) - sympy_result) < 1e-13*(1 + abs(sympy_result))
assert abs(f(tv1, tv2) - scipy_fn(tv1, tv2)) < 1e-13*(1 + abs(sympy_result))
def test_scipy_polys():
if not scipy:
skip("scipy not installed")
numpy.random.seed(0)
params = symbols('n k a b')
# list polynomials with the number of parameters
polys = [
(chebyshevt, 1),
(chebyshevu, 1),
(legendre, 1),
(hermite, 1),
(laguerre, 1),
(gegenbauer, 2),
(assoc_legendre, 2),
(assoc_laguerre, 2),
(jacobi, 3)
]
msg = \
"The random test of the function {func} with the arguments " \
"{args} had failed because the SymPy result {sympy_result} " \
"and SciPy result {scipy_result} had failed to converge " \
"within the tolerance {tol} " \
"(Actual absolute difference : {diff})"
for sympy_fn, num_params in polys:
args = params[:num_params] + (x,)
f = lambdify(args, sympy_fn(*args))
for _ in range(10):
tn = numpy.random.randint(3, 10)
tparams = tuple(numpy.random.uniform(0, 5, size=num_params-1))
tv = numpy.random.uniform(-10, 10) + 1j*numpy.random.uniform(-5, 5)
# SciPy supports hermite for real arguments only
if sympy_fn == hermite:
tv = numpy.real(tv)
# assoc_legendre needs x in (-1, 1) and integer param at most n
if sympy_fn == assoc_legendre:
tv = numpy.random.uniform(-1, 1)
tparams = tuple(numpy.random.randint(1, tn, size=1))
vals = (tn,) + tparams + (tv,)
scipy_result = f(*vals)
sympy_result = sympy_fn(*vals).evalf()
atol = 1e-9*(1 + abs(sympy_result))
diff = abs(scipy_result - sympy_result)
try:
assert diff < atol
except TypeError:
raise AssertionError(
msg.format(
func=repr(sympy_fn),
args=repr(vals),
sympy_result=repr(sympy_result),
scipy_result=repr(scipy_result),
diff=diff,
tol=atol)
)
def test_lambdify_inspect():
f = lambdify(x, x**2)
# Test that inspect.getsource works but don't hard-code implementation
# details
assert 'x**2' in inspect.getsource(f)
def test_issue_14941():
x, y = Dummy(), Dummy()
# test dict
f1 = lambdify([x, y], {x: 3, y: 3}, 'sympy')
assert f1(2, 3) == {2: 3, 3: 3}
# test tuple
f2 = lambdify([x, y], (y, x), 'sympy')
assert f2(2, 3) == (3, 2)
f2b = lambdify([], (1,)) # gh-23224
assert f2b() == (1,)
# test list
f3 = lambdify([x, y], [y, x], 'sympy')
assert f3(2, 3) == [3, 2]
def test_lambdify_Derivative_arg_issue_16468():
f = Function('f')(x)
fx = f.diff()
assert lambdify((f, fx), f + fx)(10, 5) == 15
assert eval(lambdastr((f, fx), f/fx))(10, 5) == 2
raises(SyntaxError, lambda:
eval(lambdastr((f, fx), f/fx, dummify=False)))
assert eval(lambdastr((f, fx), f/fx, dummify=True))(10, 5) == 2
assert eval(lambdastr((fx, f), f/fx, dummify=True))(S(10), 5) == S.Half
assert lambdify(fx, 1 + fx)(41) == 42
assert eval(lambdastr(fx, 1 + fx, dummify=True))(41) == 42
def test_imag_real():
f_re = lambdify([z], sympy.re(z))
val = 3+2j
assert f_re(val) == val.real
f_im = lambdify([z], sympy.im(z)) # see #15400
assert f_im(val) == val.imag
def test_MatrixSymbol_issue_15578():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol('A', 2, 2)
A0 = numpy.array([[1, 2], [3, 4]])
f = lambdify(A, A**(-1))
assert numpy.allclose(f(A0), numpy.array([[-2., 1.], [1.5, -0.5]]))
g = lambdify(A, A**3)
assert numpy.allclose(g(A0), numpy.array([[37, 54], [81, 118]]))
def test_issue_15654():
if not scipy:
skip("scipy not installed")
from sympy.abc import n, l, r, Z
from sympy.physics import hydrogen
nv, lv, rv, Zv = 1, 0, 3, 1
sympy_value = hydrogen.R_nl(nv, lv, rv, Zv).evalf()
f = lambdify((n, l, r, Z), hydrogen.R_nl(n, l, r, Z))
scipy_value = f(nv, lv, rv, Zv)
assert abs(sympy_value - scipy_value) < 1e-15
def test_issue_15827():
if not numpy:
skip("numpy not installed")
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 2, 3)
C = MatrixSymbol("C", 3, 4)
D = MatrixSymbol("D", 4, 5)
k=symbols("k")
f = lambdify(A, (2*k)*A)
g = lambdify(A, (2+k)*A)
h = lambdify(A, 2*A)
i = lambdify((B, C, D), 2*B*C*D)
assert numpy.array_equal(f(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2*k, 4*k, 6*k], [2*k, 4*k, 6*k], [2*k, 4*k, 6*k]], dtype=object))
assert numpy.array_equal(g(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[k + 2, 2*k + 4, 3*k + 6], [k + 2, 2*k + 4, 3*k + 6], \
[k + 2, 2*k + 4, 3*k + 6]], dtype=object))
assert numpy.array_equal(h(numpy.array([[1, 2, 3], [1, 2, 3], [1, 2, 3]])), \
numpy.array([[2, 4, 6], [2, 4, 6], [2, 4, 6]]))
assert numpy.array_equal(i(numpy.array([[1, 2, 3], [1, 2, 3]]), numpy.array([[1, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 4]]), \
numpy.array([[1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5]])), numpy.array([[ 120, 240, 360, 480, 600], \
[ 120, 240, 360, 480, 600]]))
def test_issue_16930():
if not scipy:
skip("scipy not installed")
x = symbols("x")
f = lambda x: S.GoldenRatio * x**2
f_ = lambdify(x, f(x), modules='scipy')
assert f_(1) == scipy.constants.golden_ratio
def test_issue_17898():
if not scipy:
skip("scipy not installed")
x = symbols("x")
f_ = lambdify([x], sympy.LambertW(x,-1), modules='scipy')
assert f_(0.1) == mpmath.lambertw(0.1, -1)
def test_issue_13167_21411():
if not numpy:
skip("numpy not installed")
f1 = lambdify(x, sympy.Heaviside(x))
f2 = lambdify(x, sympy.Heaviside(x, 1))
res1 = f1([-1, 0, 1])
res2 = f2([-1, 0, 1])
assert Abs(res1[0]).n() < 1e-15 # First functionality: only one argument passed
assert Abs(res1[1] - 1/2).n() < 1e-15
assert Abs(res1[2] - 1).n() < 1e-15
assert Abs(res2[0]).n() < 1e-15 # Second functionality: two arguments passed
assert Abs(res2[1] - 1).n() < 1e-15
assert Abs(res2[2] - 1).n() < 1e-15
def test_single_e():
f = lambdify(x, E)
assert f(23) == exp(1.0)
def test_issue_16536():
if not scipy:
skip("scipy not installed")
a = symbols('a')
f1 = lowergamma(a, x)
F = lambdify((a, x), f1, modules='scipy')
assert abs(lowergamma(1, 3) - F(1, 3)) <= 1e-10
f2 = uppergamma(a, x)
F = lambdify((a, x), f2, modules='scipy')
assert abs(uppergamma(1, 3) - F(1, 3)) <= 1e-10
def test_issue_22726():
if not numpy:
skip("numpy not installed")
x1, x2 = symbols('x1 x2')
f = Max(S.Zero, Min(x1, x2))
g = derive_by_array(f, (x1, x2))
G = lambdify((x1, x2), g, modules='numpy')
point = {x1: 1, x2: 2}
assert (abs(g.subs(point) - G(*point.values())) <= 1e-10).all()
def test_issue_22739():
if not numpy:
skip("numpy not installed")
x1, x2 = symbols('x1 x2')
f = Heaviside(Min(x1, x2))
F = lambdify((x1, x2), f, modules='numpy')
point = {x1: 1, x2: 2}
assert abs(f.subs(point) - F(*point.values())) <= 1e-10
def test_issue_22992():
if not numpy:
skip("numpy not installed")
a, t = symbols('a t')
expr = a*(log(cot(t/2)) - cos(t))
F = lambdify([a, t], expr, 'numpy')
point = {a: 10, t: 2}
assert abs(expr.subs(point) - F(*point.values())) <= 1e-10
# Standard math
F = lambdify([a, t], expr)
assert abs(expr.subs(point) - F(*point.values())) <= 1e-10
def test_issue_19764():
if not numpy:
skip("numpy not installed")
expr = Array([x, x**2])
f = lambdify(x, expr, 'numpy')
assert f(1).__class__ == numpy.ndarray
def test_issue_20070():
if not numba:
skip("numba not installed")
f = lambdify(x, sin(x), 'numpy')
assert numba.jit(f)(1)==0.8414709848078965
def test_fresnel_integrals_scipy():
if not scipy:
skip("scipy not installed")
f1 = fresnelc(x)
f2 = fresnels(x)
F1 = lambdify(x, f1, modules='scipy')
F2 = lambdify(x, f2, modules='scipy')
assert abs(fresnelc(1.3) - F1(1.3)) <= 1e-10
assert abs(fresnels(1.3) - F2(1.3)) <= 1e-10
def test_beta_scipy():
if not scipy:
skip("scipy not installed")
f = beta(x, y)
F = lambdify((x, y), f, modules='scipy')
assert abs(beta(1.3, 2.3) - F(1.3, 2.3)) <= 1e-10
def test_beta_math():
f = beta(x, y)
F = lambdify((x, y), f, modules='math')
assert abs(beta(1.3, 2.3) - F(1.3, 2.3)) <= 1e-10
def test_betainc_scipy():
if not scipy:
skip("scipy not installed")
f = betainc(w, x, y, z)
F = lambdify((w, x, y, z), f, modules='scipy')
assert abs(betainc(1.4, 3.1, 0.1, 0.5) - F(1.4, 3.1, 0.1, 0.5)) <= 1e-10
def test_betainc_regularized_scipy():
if not scipy:
skip("scipy not installed")
f = betainc_regularized(w, x, y, z)
F = lambdify((w, x, y, z), f, modules='scipy')
assert abs(betainc_regularized(0.2, 3.5, 0.1, 1) - F(0.2, 3.5, 0.1, 1)) <= 1e-10
def test_numpy_special_math():
if not numpy:
skip("numpy not installed")
funcs = [expm1, log1p, exp2, log2, log10, hypot, logaddexp, logaddexp2]
for func in funcs:
if 2 in func.nargs:
expr = func(x, y)
args = (x, y)
num_args = (0.3, 0.4)
elif 1 in func.nargs:
expr = func(x)
args = (x,)
num_args = (0.3,)
else:
raise NotImplementedError("Need to handle other than unary & binary functions in test")
f = lambdify(args, expr)
result = f(*num_args)
reference = expr.subs(dict(zip(args, num_args))).evalf()
assert numpy.allclose(result, float(reference))
lae2 = lambdify((x, y), logaddexp2(log2(x), log2(y)))
assert abs(2.0**lae2(1e-50, 2.5e-50) - 3.5e-50) < 1e-62 # from NumPy's docstring
def test_scipy_special_math():
if not scipy:
skip("scipy not installed")
cm1 = lambdify((x,), cosm1(x), modules='scipy')
assert abs(cm1(1e-20) + 5e-41) < 1e-200
def test_cupy_array_arg():
if not cupy:
skip("CuPy not installed")
f = lambdify([[x, y]], x*x + y, 'cupy')
result = f(cupy.array([2.0, 1.0]))
assert result == 5
assert "cupy" in str(type(result))
def test_cupy_array_arg_using_numpy():
# numpy functions can be run on cupy arrays
# unclear if we can "officialy" support this,
# depends on numpy __array_function__ support
if not cupy:
skip("CuPy not installed")
f = lambdify([[x, y]], x*x + y, 'numpy')
result = f(cupy.array([2.0, 1.0]))
assert result == 5
assert "cupy" in str(type(result))
def test_cupy_dotproduct():
if not cupy:
skip("CuPy not installed")
A = Matrix([x, y, z])
f1 = lambdify([x, y, z], DotProduct(A, A), modules='cupy')
f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='cupy')
f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='cupy')
f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='cupy')
assert f1(1, 2, 3) == \
f2(1, 2, 3) == \
f3(1, 2, 3) == \
f4(1, 2, 3) == \
cupy.array([14])
def test_jax_array_arg():
if not jax:
skip("JAX not installed")
f = lambdify([[x, y]], x*x + y, 'jax')
result = f(jax.numpy.array([2.0, 1.0]))
assert result == 5
assert "jax" in str(type(result))
def test_jax_array_arg_using_numpy():
if not jax:
skip("JAX not installed")
f = lambdify([[x, y]], x*x + y, 'numpy')
result = f(jax.numpy.array([2.0, 1.0]))
assert result == 5
assert "jax" in str(type(result))
def test_jax_dotproduct():
if not jax:
skip("JAX not installed")
A = Matrix([x, y, z])
f1 = lambdify([x, y, z], DotProduct(A, A), modules='jax')
f2 = lambdify([x, y, z], DotProduct(A, A.T), modules='jax')
f3 = lambdify([x, y, z], DotProduct(A.T, A), modules='jax')
f4 = lambdify([x, y, z], DotProduct(A, A.T), modules='jax')
assert f1(1, 2, 3) == \
f2(1, 2, 3) == \
f3(1, 2, 3) == \
f4(1, 2, 3) == \
jax.numpy.array([14])
def test_lambdify_cse():
def dummy_cse(exprs):
return (), exprs
def minmem(exprs):
from sympy.simplify.cse_main import cse_release_variables, cse
return cse(exprs, postprocess=cse_release_variables)
class Case:
def __init__(self, *, args, exprs, num_args, requires_numpy=False):
self.args = args
self.exprs = exprs
self.num_args = num_args
subs_dict = dict(zip(self.args, self.num_args))
self.ref = [e.subs(subs_dict).evalf() for e in exprs]
self.requires_numpy = requires_numpy
def lambdify(self, *, cse):
return lambdify(self.args, self.exprs, cse=cse)
def assertAllClose(self, result, *, abstol=1e-15, reltol=1e-15):
if self.requires_numpy:
assert all(numpy.allclose(result[i], numpy.asarray(r, dtype=float),
rtol=reltol, atol=abstol)
for i, r in enumerate(self.ref))
return
for i, r in enumerate(self.ref):
abs_err = abs(result[i] - r)
if r == 0:
assert abs_err < abstol
else:
assert abs_err/abs(r) < reltol
cases = [
Case(
args=(x, y, z),
exprs=[
x + y + z,
x + y - z,
2*x + 2*y - z,
(x+y)**2 + (y+z)**2,
],
num_args=(2., 3., 4.)
),
Case(
args=(x, y, z),
exprs=[
x + sympy.Heaviside(x),
y + sympy.Heaviside(x),
z + sympy.Heaviside(x, 1),
z/sympy.Heaviside(x, 1)
],
num_args=(0., 3., 4.)
),
Case(
args=(x, y, z),
exprs=[
x + sinc(y),
y + sinc(y),
z - sinc(y)
],
num_args=(0.1, 0.2, 0.3)
),
Case(
args=(x, y, z),
exprs=[
Matrix([[x, x*y], [sin(z) + 4, x**z]]),
x*y+sin(z)-x**z,
Matrix([x*x, sin(z), x**z])
],
num_args=(1.,2.,3.),
requires_numpy=True
),
Case(
args=(x, y),
exprs=[(x + y - 1)**2, x, x + y,
(x + y)/(2*x + 1) + (x + y - 1)**2, (2*x + 1)**(x + y)],
num_args=(1,2)
)
]
for case in cases:
if not numpy and case.requires_numpy:
continue
for cse in [False, True, minmem, dummy_cse]:
f = case.lambdify(cse=cse)
result = f(*case.num_args)
case.assertAllClose(result)
def test_deprecated_set():
with warns_deprecated_sympy():
lambdify({x, y}, x + y)
def test_23536_lambdify_cse_dummy():
f = Function('x')(y)
g = Function('w')(y)
expr = z + (f**4 + g**5)*(f**3 + (g*f)**3)
expr = expr.expand()
eval_expr = lambdify(((f, g), z), expr, cse=True)
eval_expr((1.0, 2.0), 3.0)
|
f87291532c63e16684d98057ca6cd27d5f2c67db3708703e5bff47e4865ed59c | """ Tests from Michael Wester's 1999 paper "Review of CAS mathematical
capabilities".
http://www.math.unm.edu/~wester/cas/book/Wester.pdf
See also http://math.unm.edu/~wester/cas_review.html for detailed output of
each tested system.
"""
from sympy.assumptions.ask import Q, ask
from sympy.assumptions.refine import refine
from sympy.concrete.products import product
from sympy.core import EulerGamma
from sympy.core.evalf import N
from sympy.core.function import (Derivative, Function, Lambda, Subs,
diff, expand, expand_func)
from sympy.core.mul import Mul
from sympy.core.numbers import (AlgebraicNumber, E, I, Rational, igcd,
nan, oo, pi, zoo)
from sympy.core.relational import Eq, Lt
from sympy.core.singleton import S
from sympy.core.symbol import Dummy, Symbol, symbols
from sympy.functions.combinatorial.factorials import (rf, binomial,
factorial, factorial2)
from sympy.functions.combinatorial.numbers import bernoulli, fibonacci
from sympy.functions.elementary.complexes import (conjugate, im, re,
sign)
from sympy.functions.elementary.exponential import LambertW, exp, log
from sympy.functions.elementary.hyperbolic import (asinh, cosh, sinh,
tanh)
from sympy.functions.elementary.integers import ceiling, floor
from sympy.functions.elementary.miscellaneous import Max, Min, sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, acot, asin,
atan, cos, cot, csc, sec, sin, tan)
from sympy.functions.special.bessel import besselj
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.elliptic_integrals import (elliptic_e,
elliptic_f)
from sympy.functions.special.gamma_functions import gamma, polygamma
from sympy.functions.special.hyper import hyper
from sympy.functions.special.polynomials import (assoc_legendre,
chebyshevt)
from sympy.functions.special.zeta_functions import polylog
from sympy.geometry.util import idiff
from sympy.logic.boolalg import And
from sympy.matrices.dense import hessian, wronskian
from sympy.matrices.expressions.matmul import MatMul
from sympy.ntheory.continued_fraction import (
continued_fraction_convergents as cf_c,
continued_fraction_iterator as cf_i, continued_fraction_periodic as
cf_p, continued_fraction_reduce as cf_r)
from sympy.ntheory.factor_ import factorint, totient
from sympy.ntheory.generate import primerange
from sympy.ntheory.partitions_ import npartitions
from sympy.polys.domains.integerring import ZZ
from sympy.polys.orthopolys import legendre_poly
from sympy.polys.partfrac import apart
from sympy.polys.polytools import Poly, factor, gcd, resultant
from sympy.series.limits import limit
from sympy.series.order import O
from sympy.series.residues import residue
from sympy.series.series import series
from sympy.sets.fancysets import ImageSet
from sympy.sets.sets import FiniteSet, Intersection, Interval, Union
from sympy.simplify.combsimp import combsimp
from sympy.simplify.hyperexpand import hyperexpand
from sympy.simplify.powsimp import powdenest, powsimp
from sympy.simplify.radsimp import radsimp
from sympy.simplify.simplify import logcombine, simplify
from sympy.simplify.sqrtdenest import sqrtdenest
from sympy.simplify.trigsimp import trigsimp
from sympy.solvers.solvers import solve
import mpmath
from sympy.functions.combinatorial.numbers import stirling
from sympy.functions.special.delta_functions import Heaviside
from sympy.functions.special.error_functions import Ci, Si, erf
from sympy.functions.special.zeta_functions import zeta
from sympy.testing.pytest import (XFAIL, slow, SKIP, skip, ON_TRAVIS,
raises)
from sympy.utilities.iterables import partitions
from mpmath import mpi, mpc
from sympy.matrices import Matrix, GramSchmidt, eye
from sympy.matrices.expressions.blockmatrix import BlockMatrix, block_collapse
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
from sympy.physics.quantum import Commutator
from sympy.polys.rings import PolyRing
from sympy.polys.fields import FracField
from sympy.polys.solvers import solve_lin_sys
from sympy.concrete import Sum
from sympy.concrete.products import Product
from sympy.integrals import integrate
from sympy.integrals.transforms import laplace_transform,\
inverse_laplace_transform, LaplaceTransform, fourier_transform,\
mellin_transform
from sympy.solvers.recurr import rsolve
from sympy.solvers.solveset import solveset, solveset_real, linsolve
from sympy.solvers.ode import dsolve
from sympy.core.relational import Equality
from itertools import islice, takewhile
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
from sympy.calculus.util import minimum
EmptySet = S.EmptySet
R = Rational
x, y, z = symbols('x y z')
i, j, k, l, m, n = symbols('i j k l m n', integer=True)
f = Function('f')
g = Function('g')
# A. Boolean Logic and Quantifier Elimination
# Not implemented.
# B. Set Theory
def test_B1():
assert (FiniteSet(i, j, j, k, k, k) | FiniteSet(l, k, j) |
FiniteSet(j, m, j)) == FiniteSet(i, j, k, l, m)
def test_B2():
assert (FiniteSet(i, j, j, k, k, k) & FiniteSet(l, k, j) &
FiniteSet(j, m, j)) == Intersection({j, m}, {i, j, k}, {j, k, l})
# Previous output below. Not sure why that should be the expected output.
# There should probably be a way to rewrite Intersections that way but I
# don't see why an Intersection should evaluate like that:
#
# == Union({j}, Intersection({m}, Union({j, k}, Intersection({i}, {l}))))
def test_B3():
assert (FiniteSet(i, j, k, l, m) - FiniteSet(j) ==
FiniteSet(i, k, l, m))
def test_B4():
assert (FiniteSet(*(FiniteSet(i, j)*FiniteSet(k, l))) ==
FiniteSet((i, k), (i, l), (j, k), (j, l)))
# C. Numbers
def test_C1():
assert (factorial(50) ==
30414093201713378043612608166064768844377641568960512000000000000)
def test_C2():
assert (factorint(factorial(50)) == {2: 47, 3: 22, 5: 12, 7: 8,
11: 4, 13: 3, 17: 2, 19: 2, 23: 2, 29: 1, 31: 1, 37: 1,
41: 1, 43: 1, 47: 1})
def test_C3():
assert (factorial2(10), factorial2(9)) == (3840, 945)
# Base conversions; not really implemented by SymPy
# Whatever. Take credit!
def test_C4():
assert 0xABC == 2748
def test_C5():
assert 123 == int('234', 7)
def test_C6():
assert int('677', 8) == int('1BF', 16) == 447
def test_C7():
assert log(32768, 8) == 5
def test_C8():
# Modular multiplicative inverse. Would be nice if divmod could do this.
assert ZZ.invert(5, 7) == 3
assert ZZ.invert(5, 6) == 5
def test_C9():
assert igcd(igcd(1776, 1554), 5698) == 74
def test_C10():
x = 0
for n in range(2, 11):
x += R(1, n)
assert x == R(4861, 2520)
def test_C11():
assert R(1, 7) == S('0.[142857]')
def test_C12():
assert R(7, 11) * R(22, 7) == 2
def test_C13():
test = R(10, 7) * (1 + R(29, 1000)) ** R(1, 3)
good = 3 ** R(1, 3)
assert test == good
def test_C14():
assert sqrtdenest(sqrt(2*sqrt(3) + 4)) == 1 + sqrt(3)
def test_C15():
test = sqrtdenest(sqrt(14 + 3*sqrt(3 + 2*sqrt(5 - 12*sqrt(3 - 2*sqrt(2))))))
good = sqrt(2) + 3
assert test == good
def test_C16():
test = sqrtdenest(sqrt(10 + 2*sqrt(6) + 2*sqrt(10) + 2*sqrt(15)))
good = sqrt(2) + sqrt(3) + sqrt(5)
assert test == good
def test_C17():
test = radsimp((sqrt(3) + sqrt(2)) / (sqrt(3) - sqrt(2)))
good = 5 + 2*sqrt(6)
assert test == good
def test_C18():
assert simplify((sqrt(-2 + sqrt(-5)) * sqrt(-2 - sqrt(-5))).expand(complex=True)) == 3
@XFAIL
def test_C19():
assert radsimp(simplify((90 + 34*sqrt(7)) ** R(1, 3))) == 3 + sqrt(7)
def test_C20():
inside = (135 + 78*sqrt(3))
test = AlgebraicNumber((inside**R(2, 3) + 3) * sqrt(3) / inside**R(1, 3))
assert simplify(test) == AlgebraicNumber(12)
def test_C21():
assert simplify(AlgebraicNumber((41 + 29*sqrt(2)) ** R(1, 5))) == \
AlgebraicNumber(1 + sqrt(2))
@XFAIL
def test_C22():
test = simplify(((6 - 4*sqrt(2))*log(3 - 2*sqrt(2)) + (3 - 2*sqrt(2))*log(17
- 12*sqrt(2)) + 32 - 24*sqrt(2)) / (48*sqrt(2) - 72))
good = sqrt(2)/3 - log(sqrt(2) - 1)/3
assert test == good
def test_C23():
assert 2 * oo - 3 is oo
@XFAIL
def test_C24():
raise NotImplementedError("2**aleph_null == aleph_1")
# D. Numerical Analysis
def test_D1():
assert 0.0 / sqrt(2) == 0.0
def test_D2():
assert str(exp(-1000000).evalf()) == '3.29683147808856e-434295'
def test_D3():
assert exp(pi*sqrt(163)).evalf(50).num.ae(262537412640768744)
def test_D4():
assert floor(R(-5, 3)) == -2
assert ceiling(R(-5, 3)) == -1
@XFAIL
def test_D5():
raise NotImplementedError("cubic_spline([1, 2, 4, 5], [1, 4, 2, 3], x)(3) == 27/8")
@XFAIL
def test_D6():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to FORTRAN")
@XFAIL
def test_D7():
raise NotImplementedError("translate sum(a[i]*x**i, (i,1,n)) to C")
@XFAIL
def test_D8():
# One way is to cheat by converting the sum to a string,
# and replacing the '[' and ']' with ''.
# E.g., horner(S(str(_).replace('[','').replace(']','')))
raise NotImplementedError("apply Horner's rule to sum(a[i]*x**i, (i,1,5))")
@XFAIL
def test_D9():
raise NotImplementedError("translate D8 to FORTRAN")
@XFAIL
def test_D10():
raise NotImplementedError("translate D8 to C")
@XFAIL
def test_D11():
#Is there a way to use count_ops?
raise NotImplementedError("flops(sum(product(f[i][k], (i,1,k)), (k,1,n)))")
@XFAIL
def test_D12():
assert (mpi(-4, 2) * x + mpi(1, 3)) ** 2 == mpi(-8, 16)*x**2 + mpi(-24, 12)*x + mpi(1, 9)
@XFAIL
def test_D13():
raise NotImplementedError("discretize a PDE: diff(f(x,t),t) == diff(diff(f(x,t),x),x)")
# E. Statistics
# See scipy; all of this is numerical.
# F. Combinatorial Theory.
def test_F1():
assert rf(x, 3) == x*(1 + x)*(2 + x)
def test_F2():
assert expand_func(binomial(n, 3)) == n*(n - 1)*(n - 2)/6
@XFAIL
def test_F3():
assert combsimp(2**n * factorial(n) * factorial2(2*n - 1)) == factorial(2*n)
@XFAIL
def test_F4():
assert combsimp(2**n * factorial(n) * product(2*k - 1, (k, 1, n))) == factorial(2*n)
@XFAIL
def test_F5():
assert gamma(n + R(1, 2)) / sqrt(pi) / factorial(n) == factorial(2*n)/2**(2*n)/factorial(n)**2
def test_F6():
partTest = [p.copy() for p in partitions(4)]
partDesired = [{4: 1}, {1: 1, 3: 1}, {2: 2}, {1: 2, 2:1}, {1: 4}]
assert partTest == partDesired
def test_F7():
assert npartitions(4) == 5
def test_F8():
assert stirling(5, 2, signed=True) == -50 # if signed, then kind=1
def test_F9():
assert totient(1776) == 576
# G. Number Theory
def test_G1():
assert list(primerange(999983, 1000004)) == [999983, 1000003]
@XFAIL
def test_G2():
raise NotImplementedError("find the primitive root of 191 == 19")
@XFAIL
def test_G3():
raise NotImplementedError("(a+b)**p mod p == a**p + b**p mod p; p prime")
# ... G14 Modular equations are not implemented.
def test_G15():
assert Rational(sqrt(3).evalf()).limit_denominator(15) == R(26, 15)
assert list(takewhile(lambda x: x.q <= 15, cf_c(cf_i(sqrt(3)))))[-1] == \
R(26, 15)
def test_G16():
assert list(islice(cf_i(pi),10)) == [3, 7, 15, 1, 292, 1, 1, 1, 2, 1]
def test_G17():
assert cf_p(0, 1, 23) == [4, [1, 3, 1, 8]]
def test_G18():
assert cf_p(1, 2, 5) == [[1]]
assert cf_r([[1]]).expand() == S.Half + sqrt(5)/2
@XFAIL
def test_G19():
s = symbols('s', integer=True, positive=True)
it = cf_i((exp(1/s) - 1)/(exp(1/s) + 1))
assert list(islice(it, 5)) == [0, 2*s, 6*s, 10*s, 14*s]
def test_G20():
s = symbols('s', integer=True, positive=True)
# Wester erroneously has this as -s + sqrt(s**2 + 1)
assert cf_r([[2*s]]) == s + sqrt(s**2 + 1)
@XFAIL
def test_G20b():
s = symbols('s', integer=True, positive=True)
assert cf_p(s, 1, s**2 + 1) == [[2*s]]
# H. Algebra
def test_H1():
assert simplify(2*2**n) == simplify(2**(n + 1))
assert powdenest(2*2**n) == simplify(2**(n + 1))
def test_H2():
assert powsimp(4 * 2**n) == 2**(n + 2)
def test_H3():
assert (-1)**(n*(n + 1)) == 1
def test_H4():
expr = factor(6*x - 10)
assert type(expr) is Mul
assert expr.args[0] == 2
assert expr.args[1] == 3*x - 5
p1 = 64*x**34 - 21*x**47 - 126*x**8 - 46*x**5 - 16*x**60 - 81
p2 = 72*x**60 - 25*x**25 - 19*x**23 - 22*x**39 - 83*x**52 + 54*x**10 + 81
q = 34*x**19 - 25*x**16 + 70*x**7 + 20*x**3 - 91*x - 86
def test_H5():
assert gcd(p1, p2, x) == 1
def test_H6():
assert gcd(expand(p1 * q), expand(p2 * q)) == q
def test_H7():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
assert gcd(p1, p2, x, y, z) == 1
def test_H8():
p1 = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
p2 = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
q = 11*x**12*y**7*z**13 - 23*x**2*y**8*z**10 + 47*x**17*y**5*z**8
assert gcd(p1 * q, p2 * q, x, y, z) == q
def test_H9():
p1 = 2*x**(n + 4) - x**(n + 2)
p2 = 4*x**(n + 1) + 3*x**n
assert gcd(p1, p2) == x**n
def test_H10():
p1 = 3*x**4 + 3*x**3 + x**2 - x - 2
p2 = x**3 - 3*x**2 + x + 5
assert resultant(p1, p2, x) == 0
def test_H11():
assert resultant(p1 * q, p2 * q, x) == 0
def test_H12():
num = x**2 - 4
den = x**2 + 4*x + 4
assert simplify(num/den) == (x - 2)/(x + 2)
@XFAIL
def test_H13():
assert simplify((exp(x) - 1) / (exp(x/2) + 1)) == exp(x/2) - 1
def test_H14():
p = (x + 1) ** 20
ep = expand(p)
assert ep == (1 + 20*x + 190*x**2 + 1140*x**3 + 4845*x**4 + 15504*x**5
+ 38760*x**6 + 77520*x**7 + 125970*x**8 + 167960*x**9 + 184756*x**10
+ 167960*x**11 + 125970*x**12 + 77520*x**13 + 38760*x**14 + 15504*x**15
+ 4845*x**16 + 1140*x**17 + 190*x**18 + 20*x**19 + x**20)
dep = diff(ep, x)
assert dep == (20 + 380*x + 3420*x**2 + 19380*x**3 + 77520*x**4
+ 232560*x**5 + 542640*x**6 + 1007760*x**7 + 1511640*x**8 + 1847560*x**9
+ 1847560*x**10 + 1511640*x**11 + 1007760*x**12 + 542640*x**13
+ 232560*x**14 + 77520*x**15 + 19380*x**16 + 3420*x**17 + 380*x**18
+ 20*x**19)
assert factor(dep) == 20*(1 + x)**19
def test_H15():
assert simplify(Mul(*[x - r for r in solveset(x**3 + x**2 - 7)])) == x**3 + x**2 - 7
def test_H16():
assert factor(x**100 - 1) == ((x - 1)*(x + 1)*(x**2 + 1)*(x**4 - x**3
+ x**2 - x + 1)*(x**4 + x**3 + x**2 + x + 1)*(x**8 - x**6 + x**4
- x**2 + 1)*(x**20 - x**15 + x**10 - x**5 + 1)*(x**20 + x**15 + x**10
+ x**5 + 1)*(x**40 - x**30 + x**20 - x**10 + 1))
def test_H17():
assert simplify(factor(expand(p1 * p2)) - p1*p2) == 0
@XFAIL
def test_H18():
# Factor over complex rationals.
test = factor(4*x**4 + 8*x**3 + 77*x**2 + 18*x + 153)
good = (2*x + 3*I)*(2*x - 3*I)*(x + 1 - 4*I)*(x + 1 + 4*I)
assert test == good
def test_H19():
a = symbols('a')
# The idea is to let a**2 == 2, then solve 1/(a-1). Answer is a+1")
assert Poly(a - 1).invert(Poly(a**2 - 2)) == a + 1
@XFAIL
def test_H20():
raise NotImplementedError("let a**2==2; (x**3 + (a-2)*x**2 - "
+ "(2*a+3)*x - 3*a) / (x**2-2) = (x**2 - 2*x - 3) / (x-a)")
@XFAIL
def test_H21():
raise NotImplementedError("evaluate (b+c)**4 assuming b**3==2, c**2==3. \
Answer is 2*b + 8*c + 18*b**2 + 12*b*c + 9")
def test_H22():
assert factor(x**4 - 3*x**2 + 1, modulus=5) == (x - 2)**2 * (x + 2)**2
def test_H23():
f = x**11 + x + 1
g = (x**2 + x + 1) * (x**9 - x**8 + x**6 - x**5 + x**3 - x**2 + 1)
assert factor(f, modulus=65537) == g
def test_H24():
phi = AlgebraicNumber(S.GoldenRatio.expand(func=True), alias='phi')
assert factor(x**4 - 3*x**2 + 1, extension=phi) == \
(x - phi)*(x + 1 - phi)*(x - 1 + phi)*(x + phi)
def test_H25():
e = (x - 2*y**2 + 3*z**3) ** 20
assert factor(expand(e)) == e
def test_H26():
g = expand((sin(x) - 2*cos(y)**2 + 3*tan(z)**3)**20)
assert factor(g, expand=False) == (-sin(x) + 2*cos(y)**2 - 3*tan(z)**3)**20
def test_H27():
f = 24*x*y**19*z**8 - 47*x**17*y**5*z**8 + 6*x**15*y**9*z**2 - 3*x**22 + 5
g = 34*x**5*y**8*z**13 + 20*x**7*y**7*z**7 + 12*x**9*y**16*z**4 + 80*y**14*z
h = -2*z*y**7 \
*(6*x**9*y**9*z**3 + 10*x**7*z**6 + 17*y*x**5*z**12 + 40*y**7) \
*(3*x**22 + 47*x**17*y**5*z**8 - 6*x**15*y**9*z**2 - 24*x*y**19*z**8 - 5)
assert factor(expand(f*g)) == h
@XFAIL
def test_H28():
raise NotImplementedError("expand ((1 - c**2)**5 * (1 - s**2)**5 * "
+ "(c**2 + s**2)**10) with c**2 + s**2 = 1. Answer is c**10*s**10.")
@XFAIL
def test_H29():
assert factor(4*x**2 - 21*x*y + 20*y**2, modulus=3) == (x + y)*(x - y)
def test_H30():
test = factor(x**3 + y**3, extension=sqrt(-3))
answer = (x + y)*(x + y*(-R(1, 2) - sqrt(3)/2*I))*(x + y*(-R(1, 2) + sqrt(3)/2*I))
assert answer == test
def test_H31():
f = (x**2 + 2*x + 3)/(x**3 + 4*x**2 + 5*x + 2)
g = 2 / (x + 1)**2 - 2 / (x + 1) + 3 / (x + 2)
assert apart(f) == g
@XFAIL
def test_H32(): # issue 6558
raise NotImplementedError("[A*B*C - (A*B*C)**(-1)]*A*C*B (product \
of a non-commuting product and its inverse)")
def test_H33():
A, B, C = symbols('A, B, C', commutative=False)
assert (Commutator(A, Commutator(B, C))
+ Commutator(B, Commutator(C, A))
+ Commutator(C, Commutator(A, B))).doit().expand() == 0
# I. Trigonometry
def test_I1():
assert tan(pi*R(7, 10)) == -sqrt(1 + 2/sqrt(5))
@XFAIL
def test_I2():
assert sqrt((1 + cos(6))/2) == -cos(3)
def test_I3():
assert cos(n*pi) + sin((4*n - 1)*pi/2) == (-1)**n - 1
def test_I4():
assert refine(cos(pi*cos(n*pi)) + sin(pi/2*cos(n*pi)), Q.integer(n)) == (-1)**n - 1
@XFAIL
def test_I5():
assert sin((n**5/5 + n**4/2 + n**3/3 - n/30) * pi) == 0
@XFAIL
def test_I6():
raise NotImplementedError("assuming -3*pi<x<-5*pi/2, abs(cos(x)) == -cos(x), abs(sin(x)) == -sin(x)")
@XFAIL
def test_I7():
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
@XFAIL
def test_I8():
assert cos(3*x)/cos(x) == 2*cos(2*x) - 1
@XFAIL
def test_I9():
# Supposed to do this with rewrite rules.
assert cos(3*x)/cos(x) == cos(x)**2 - 3*sin(x)**2
def test_I10():
assert trigsimp((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1)) is nan
@SKIP("hangs")
@XFAIL
def test_I11():
assert limit((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x, 0) != 0
@XFAIL
def test_I12():
# This should fail or return nan or something.
res = diff((tan(x)**2 + 1 - cos(x)**-2) / (sin(x)**2 + cos(x)**2 - 1), x)
assert res is nan # trigsimp(res) gives nan
# J. Special functions.
def test_J1():
assert bernoulli(16) == R(-3617, 510)
def test_J2():
assert diff(elliptic_e(x, y**2), y) == (elliptic_e(x, y**2) - elliptic_f(x, y**2))/y
@XFAIL
def test_J3():
raise NotImplementedError("Jacobi elliptic functions: diff(dn(u,k), u) == -k**2*sn(u,k)*cn(u,k)")
def test_J4():
assert gamma(R(-1, 2)) == -2*sqrt(pi)
def test_J5():
assert polygamma(0, R(1, 3)) == -log(3) - sqrt(3)*pi/6 - EulerGamma - log(sqrt(3))
def test_J6():
assert mpmath.besselj(2, 1 + 1j).ae(mpc('0.04157988694396212', '0.24739764151330632'))
def test_J7():
assert simplify(besselj(R(-5,2), pi/2)) == 12/(pi**2)
def test_J8():
p = besselj(R(3,2), z)
q = (sin(z)/z - cos(z))/sqrt(pi*z/2)
assert simplify(expand_func(p) -q) == 0
def test_J9():
assert besselj(0, z).diff(z) == - besselj(1, z)
def test_J10():
mu, nu = symbols('mu, nu', integer=True)
assert assoc_legendre(nu, mu, 0) == 2**mu*sqrt(pi)/gamma((nu - mu)/2 + 1)/gamma((-nu - mu + 1)/2)
def test_J11():
assert simplify(assoc_legendre(3, 1, x)) == simplify(-R(3, 2)*sqrt(1 - x**2)*(5*x**2 - 1))
@slow
def test_J12():
assert simplify(chebyshevt(1008, x) - 2*x*chebyshevt(1007, x) + chebyshevt(1006, x)) == 0
def test_J13():
a = symbols('a', integer=True, negative=False)
assert chebyshevt(a, -1) == (-1)**a
def test_J14():
p = hyper([S.Half, S.Half], [R(3, 2)], z**2)
assert hyperexpand(p) == asin(z)/z
@XFAIL
def test_J15():
raise NotImplementedError("F((n+2)/2,-(n-2)/2,R(3,2),sin(z)**2) == sin(n*z)/(n*sin(z)*cos(z)); F(.) is hypergeometric function")
@XFAIL
def test_J16():
raise NotImplementedError("diff(zeta(x), x) @ x=0 == -log(2*pi)/2")
def test_J17():
assert integrate(f((x + 2)/5)*DiracDelta((x - 2)/3) - g(x)*diff(DiracDelta(x - 1), x), (x, 0, 3)) == 3*f(R(4, 5)) + Subs(Derivative(g(x), x), x, 1)
@XFAIL
def test_J18():
raise NotImplementedError("define an antisymmetric function")
# K. The Complex Domain
def test_K1():
z1, z2 = symbols('z1, z2', complex=True)
assert re(z1 + I*z2) == -im(z2) + re(z1)
assert im(z1 + I*z2) == im(z1) + re(z2)
def test_K2():
assert abs(3 - sqrt(7) + I*sqrt(6*sqrt(7) - 15)) == 1
@XFAIL
def test_K3():
a, b = symbols('a, b', real=True)
assert simplify(abs(1/(a + I/a + I*b))) == 1/sqrt(a**2 + (I/a + b)**2)
def test_K4():
assert log(3 + 4*I).expand(complex=True) == log(5) + I*atan(R(4, 3))
def test_K5():
x, y = symbols('x, y', real=True)
assert tan(x + I*y).expand(complex=True) == (sin(2*x)/(cos(2*x) +
cosh(2*y)) + I*sinh(2*y)/(cos(2*x) + cosh(2*y)))
def test_K6():
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) == sqrt(x*y)/sqrt(x)
assert sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z)) != sqrt(y)
def test_K7():
y = symbols('y', real=True, negative=False)
expr = sqrt(x*y*abs(z)**2)/(sqrt(x)*abs(z))
sexpr = simplify(expr)
assert sexpr == sqrt(y)
def test_K8():
z = symbols('z', complex=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) != 0 # Passes
z = symbols('z', complex=True, negative=False)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0 # Fails
def test_K9():
z = symbols('z', positive=True)
assert simplify(sqrt(1/z) - 1/sqrt(z)) == 0
def test_K10():
z = symbols('z', negative=True)
assert simplify(sqrt(1/z) + 1/sqrt(z)) == 0
# This goes up to K25
# L. Determining Zero Equivalence
def test_L1():
assert sqrt(997) - (997**3)**R(1, 6) == 0
def test_L2():
assert sqrt(999983) - (999983**3)**R(1, 6) == 0
def test_L3():
assert simplify((2**R(1, 3) + 4**R(1, 3))**3 - 6*(2**R(1, 3) + 4**R(1, 3)) - 6) == 0
def test_L4():
assert trigsimp(cos(x)**3 + cos(x)*sin(x)**2 - cos(x)) == 0
@XFAIL
def test_L5():
assert log(tan(R(1, 2)*x + pi/4)) - asinh(tan(x)) == 0
def test_L6():
assert (log(tan(x/2 + pi/4)) - asinh(tan(x))).diff(x).subs({x: 0}) == 0
@XFAIL
def test_L7():
assert simplify(log((2*sqrt(x) + 1)/(sqrt(4*x + 4*sqrt(x) + 1)))) == 0
@XFAIL
def test_L8():
assert simplify((4*x + 4*sqrt(x) + 1)**(sqrt(x)/(2*sqrt(x) + 1)) \
*(2*sqrt(x) + 1)**(1/(2*sqrt(x) + 1)) - 2*sqrt(x) - 1) == 0
@XFAIL
def test_L9():
z = symbols('z', complex=True)
assert simplify(2**(1 - z)*gamma(z)*zeta(z)*cos(z*pi/2) - pi**2*zeta(1 - z)) == 0
# M. Equations
@XFAIL
def test_M1():
assert Equality(x, 2)/2 + Equality(1, 1) == Equality(x/2 + 1, 2)
def test_M2():
# The roots of this equation should all be real. Note that this
# doesn't test that they are correct.
sol = solveset(3*x**3 - 18*x**2 + 33*x - 19, x)
assert all(s.expand(complex=True).is_real for s in sol)
@XFAIL
def test_M5():
assert solveset(x**6 - 9*x**4 - 4*x**3 + 27*x**2 - 36*x - 23, x) == FiniteSet(2**(1/3) + sqrt(3), 2**(1/3) - sqrt(3), +sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), +sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) + I*sqrt(3)/2**(2/3), -sqrt(3) - 1/2**(2/3) - I*sqrt(3)/2**(2/3))
def test_M6():
assert set(solveset(x**7 - 1, x)) == \
{cos(n*pi*R(2, 7)) + I*sin(n*pi*R(2, 7)) for n in range(0, 7)}
# The paper asks for exp terms, but sin's and cos's may be acceptable;
# if the results are simplified, exp terms appear for all but
# -sin(pi/14) - I*cos(pi/14) and -sin(pi/14) + I*cos(pi/14) which
# will simplify if you apply the transformation foo.rewrite(exp).expand()
def test_M7():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert set(solve(x**8 - 8*x**7 + 34*x**6 - 92*x**5 + 175*x**4 - 236*x**3 +
226*x**2 - 140*x + 46, x)) == set([
1 - sqrt(2)*I*sqrt(-sqrt(-3 + 4*sqrt(3)) + 3)/2,
1 - sqrt(2)*sqrt(-3 + I*sqrt(3 + 4*sqrt(3)))/2,
1 - sqrt(2)*I*sqrt(sqrt(-3 + 4*sqrt(3)) + 3)/2,
1 - sqrt(2)*sqrt(-3 - I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(2)*I*sqrt(sqrt(-3 + 4*sqrt(3)) + 3)/2,
1 + sqrt(2)*sqrt(-3 - I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(2)*sqrt(-3 + I*sqrt(3 + 4*sqrt(3)))/2,
1 + sqrt(2)*I*sqrt(-sqrt(-3 + 4*sqrt(3)) + 3)/2,
])
@XFAIL # There are an infinite number of solutions.
def test_M8():
x = Symbol('x')
z = symbols('z', complex=True)
assert solveset(exp(2*x) + 2*exp(x) + 1 - z, x, S.Reals) == \
FiniteSet(log(1 + z - 2*sqrt(z))/2, log(1 + z + 2*sqrt(z))/2)
# This one could be simplified better (the 1/2 could be pulled into the log
# as a sqrt, and the function inside the log can be factored as a square,
# giving [log(sqrt(z) - 1), log(sqrt(z) + 1)]). Also, there should be an
# infinite number of solutions.
# x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i]
# where n is an arbitrary integer. See url of detailed output above.
@XFAIL
def test_M9():
# x = symbols('x')
raise NotImplementedError("solveset(exp(2-x**2)-exp(-x),x) has complex solutions.")
def test_M10():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(exp(x) - x, x) == [-LambertW(-1)]
@XFAIL
def test_M11():
assert solveset(x**x - x, x) == FiniteSet(-1, 1)
def test_M12():
# TODO: x = [-1, 2*(+/-asinh(1)*I + n*pi}, 3*(pi/6 + n*pi/3)]
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x + 1)*(sin(x)**2 + 1)**2*cos(3*x)**3, x) == [
-1, pi/6, pi/2,
- I*log(1 + sqrt(2)), I*log(1 + sqrt(2)),
pi - I*log(1 + sqrt(2)), pi + I*log(1 + sqrt(2)),
]
@XFAIL
def test_M13():
n = Dummy('n')
assert solveset_real(sin(x) - cos(x), x) == ImageSet(Lambda(n, n*pi - pi*R(7, 4)), S.Integers)
@XFAIL
def test_M14():
n = Dummy('n')
assert solveset_real(tan(x) - 1, x) == ImageSet(Lambda(n, n*pi + pi/4), S.Integers)
def test_M15():
n = Dummy('n')
got = solveset(sin(x) - S.Half)
assert any(got.dummy_eq(i) for i in (
Union(ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi*R(5, 6)), S.Integers)),
Union(ImageSet(Lambda(n, 2*n*pi + pi*R(5, 6)), S.Integers),
ImageSet(Lambda(n, 2*n*pi + pi/6), S.Integers))))
@XFAIL
def test_M16():
n = Dummy('n')
assert solveset(sin(x) - tan(x), x) == ImageSet(Lambda(n, n*pi), S.Integers)
@XFAIL
def test_M17():
assert solveset_real(asin(x) - atan(x), x) == FiniteSet(0)
@XFAIL
def test_M18():
assert solveset_real(acos(x) - atan(x), x) == FiniteSet(sqrt((sqrt(5) - 1)/2))
def test_M19():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve((x - 2)/x**R(1, 3), x) == [2]
def test_M20():
assert solveset(sqrt(x**2 + 1) - x + 2, x) == EmptySet
def test_M21():
assert solveset(x + sqrt(x) - 2) == FiniteSet(1)
def test_M22():
assert solveset(2*sqrt(x) + 3*x**R(1, 4) - 2) == FiniteSet(R(1, 16))
def test_M23():
x = symbols('x', complex=True)
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(x - 1/sqrt(1 + x**2)) == [
-I*sqrt(S.Half + sqrt(5)/2), sqrt(Rational(-1, 2) + sqrt(5)/2)]
def test_M24():
# TODO: Replace solve with solveset, as of now test fails for solveset
solution = solve(1 - binomial(m, 2)*2**k, k)
answer = log(2/(m*(m - 1)), 2)
assert solution[0].expand() == answer.expand()
def test_M25():
a, b, c, d = symbols(':d', positive=True)
x = symbols('x')
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(a*b**x - c*d**x, x)[0].expand() == (log(c/a)/log(b/d)).expand()
def test_M26():
# TODO: Replace solve with solveset, as of now test fails for solveset
assert solve(sqrt(log(x)) - log(sqrt(x))) == [1, exp(4)]
def test_M27():
x = symbols('x', real=True)
b = symbols('b', real=True)
# TODO: Replace solve with solveset which gives both [+/- current answer]
# note that there is a typo in this test in the wester.pdf; there is no
# real solution for the equation as it appears in wester.pdf
assert solve(log(acos(asin(x**R(2, 3) - b)) - 1) + 2, x
) == [(b + sin(cos(exp(-2) + 1)))**R(3, 2)]
@XFAIL
def test_M28():
assert solveset_real(5*x + exp((x - 5)/2) - 8*x**3, x, assume=Q.real(x)) == [-0.784966, -0.016291, 0.802557]
def test_M29():
x = symbols('x')
assert solveset(abs(x - 1) - 2, domain=S.Reals) == FiniteSet(-1, 3)
def test_M30():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(abs(2*x + 5) - abs(x - 2),x, assume=Q.real(x)) == [-1, -7]
assert solveset_real(abs(2*x + 5) - abs(x - 2), x) == FiniteSet(-1, -7)
def test_M31():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# assert solve(1 - abs(x) - max(-x - 2, x - 2),x, assume=Q.real(x)) == [-3/2, 3/2]
assert solveset_real(1 - abs(x) - Max(-x - 2, x - 2), x) == FiniteSet(R(-3, 2), R(3, 2))
@XFAIL
def test_M32():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
assert solveset_real(Max(2 - x**2, x)- Max(-x, (x**3)/9), x) == FiniteSet(-1, 3)
@XFAIL
def test_M33():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports assumptions
# Second answer can be written in another form. The second answer is the root of x**3 + 9*x**2 - 18 = 0 in the interval (-2, -1).
assert solveset_real(Max(2 - x**2, x) - x**3/9, x) == FiniteSet(-3, -1.554894, 3)
@XFAIL
def test_M34():
z = symbols('z', complex=True)
assert solveset((1 + I) * z + (2 - I) * conjugate(z) + 3*I, z) == FiniteSet(2 + 3*I)
def test_M35():
x, y = symbols('x y', real=True)
assert linsolve((3*x - 2*y - I*y + 3*I).as_real_imag(), y, x) == FiniteSet((3, 2))
def test_M36():
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports solving for function
# assert solve(f**2 + f - 2, x) == [Eq(f(x), 1), Eq(f(x), -2)]
assert solveset(f(x)**2 + f(x) - 2, f(x)) == FiniteSet(-2, 1)
def test_M37():
assert linsolve([x + y + z - 6, 2*x + y + 2*z - 10, x + 3*y + z - 10 ], x, y, z) == \
FiniteSet((-z + 4, 2, z))
def test_M38():
a, b, c = symbols('a, b, c')
domain = FracField([a, b, c], ZZ).to_domain()
ring = PolyRing('k1:50', domain)
(k1, k2, k3, k4, k5, k6, k7, k8, k9, k10,
k11, k12, k13, k14, k15, k16, k17, k18, k19, k20,
k21, k22, k23, k24, k25, k26, k27, k28, k29, k30,
k31, k32, k33, k34, k35, k36, k37, k38, k39, k40,
k41, k42, k43, k44, k45, k46, k47, k48, k49) = ring.gens
system = [
-b*k8/a + c*k8/a, -b*k11/a + c*k11/a, -b*k10/a + c*k10/a + k2, -k3 - b*k9/a + c*k9/a,
-b*k14/a + c*k14/a, -b*k15/a + c*k15/a, -b*k18/a + c*k18/a - k2, -b*k17/a + c*k17/a,
-b*k16/a + c*k16/a + k4, -b*k13/a + c*k13/a - b*k21/a + c*k21/a + b*k5/a - c*k5/a,
b*k44/a - c*k44/a, -b*k45/a + c*k45/a, -b*k20/a + c*k20/a, -b*k44/a + c*k44/a,
b*k46/a - c*k46/a, b**2*k47/a**2 - 2*b*c*k47/a**2 + c**2*k47/a**2, k3, -k4,
-b*k12/a + c*k12/a - a*k6/b + c*k6/b, -b*k19/a + c*k19/a + a*k7/c - b*k7/c,
b*k45/a - c*k45/a, -b*k46/a + c*k46/a, -k48 + c*k48/a + c*k48/b - c**2*k48/(a*b),
-k49 + b*k49/a + b*k49/c - b**2*k49/(a*c), a*k1/b - c*k1/b, a*k4/b - c*k4/b,
a*k3/b - c*k3/b + k9, -k10 + a*k2/b - c*k2/b, a*k7/b - c*k7/b, -k9, k11,
b*k12/a - c*k12/a + a*k6/b - c*k6/b, a*k15/b - c*k15/b, k10 + a*k18/b - c*k18/b,
-k11 + a*k17/b - c*k17/b, a*k16/b - c*k16/b, -a*k13/b + c*k13/b + a*k21/b - c*k21/b + a*k5/b - c*k5/b,
-a*k44/b + c*k44/b, a*k45/b - c*k45/b, a*k14/c - b*k14/c + a*k20/b - c*k20/b,
a*k44/b - c*k44/b, -a*k46/b + c*k46/b, -k47 + c*k47/a + c*k47/b - c**2*k47/(a*b),
a*k19/b - c*k19/b, -a*k45/b + c*k45/b, a*k46/b - c*k46/b, a**2*k48/b**2 - 2*a*c*k48/b**2 + c**2*k48/b**2,
-k49 + a*k49/b + a*k49/c - a**2*k49/(b*c), k16, -k17, -a*k1/c + b*k1/c,
-k16 - a*k4/c + b*k4/c, -a*k3/c + b*k3/c, k18 - a*k2/c + b*k2/c, b*k19/a - c*k19/a - a*k7/c + b*k7/c,
-a*k6/c + b*k6/c, -a*k8/c + b*k8/c, -a*k11/c + b*k11/c + k17, -a*k10/c + b*k10/c - k18,
-a*k9/c + b*k9/c, -a*k14/c + b*k14/c - a*k20/b + c*k20/b, -a*k13/c + b*k13/c + a*k21/c - b*k21/c - a*k5/c + b*k5/c,
a*k44/c - b*k44/c, -a*k45/c + b*k45/c, -a*k44/c + b*k44/c, a*k46/c - b*k46/c,
-k47 + b*k47/a + b*k47/c - b**2*k47/(a*c), -a*k12/c + b*k12/c, a*k45/c - b*k45/c,
-a*k46/c + b*k46/c, -k48 + a*k48/b + a*k48/c - a**2*k48/(b*c),
a**2*k49/c**2 - 2*a*b*k49/c**2 + b**2*k49/c**2, k8, k11, -k15, k10 - k18,
-k17, k9, -k16, -k29, k14 - k32, -k21 + k23 - k31, -k24 - k30, -k35, k44,
-k45, k36, k13 - k23 + k39, -k20 + k38, k25 + k37, b*k26/a - c*k26/a - k34 + k42,
-2*k44, k45, k46, b*k47/a - c*k47/a, k41, k44, -k46, -b*k47/a + c*k47/a,
k12 + k24, -k19 - k25, -a*k27/b + c*k27/b - k33, k45, -k46, -a*k48/b + c*k48/b,
a*k28/c - b*k28/c + k40, -k45, k46, a*k48/b - c*k48/b, a*k49/c - b*k49/c,
-a*k49/c + b*k49/c, -k1, -k4, -k3, k15, k18 - k2, k17, k16, k22, k25 - k7,
k24 + k30, k21 + k23 - k31, k28, -k44, k45, -k30 - k6, k20 + k32, k27 + b*k33/a - c*k33/a,
k44, -k46, -b*k47/a + c*k47/a, -k36, k31 - k39 - k5, -k32 - k38, k19 - k37,
k26 - a*k34/b + c*k34/b - k42, k44, -2*k45, k46, a*k48/b - c*k48/b,
a*k35/c - b*k35/c - k41, -k44, k46, b*k47/a - c*k47/a, -a*k49/c + b*k49/c,
-k40, k45, -k46, -a*k48/b + c*k48/b, a*k49/c - b*k49/c, k1, k4, k3, -k8,
-k11, -k10 + k2, -k9, k37 + k7, -k14 - k38, -k22, -k25 - k37, -k24 + k6,
-k13 - k23 + k39, -k28 + b*k40/a - c*k40/a, k44, -k45, -k27, -k44, k46,
b*k47/a - c*k47/a, k29, k32 + k38, k31 - k39 + k5, -k12 + k30, k35 - a*k41/b + c*k41/b,
-k44, k45, -k26 + k34 + a*k42/c - b*k42/c, k44, k45, -2*k46, -b*k47/a + c*k47/a,
-a*k48/b + c*k48/b, a*k49/c - b*k49/c, k33, -k45, k46, a*k48/b - c*k48/b,
-a*k49/c + b*k49/c
]
solution = {
k49: 0, k48: 0, k47: 0, k46: 0, k45: 0, k44: 0, k41: 0, k40: 0,
k38: 0, k37: 0, k36: 0, k35: 0, k33: 0, k32: 0, k30: 0, k29: 0,
k28: 0, k27: 0, k25: 0, k24: 0, k22: 0, k21: 0, k20: 0, k19: 0,
k18: 0, k17: 0, k16: 0, k15: 0, k14: 0, k13: 0, k12: 0, k11: 0,
k10: 0, k9: 0, k8: 0, k7: 0, k6: 0, k5: 0, k4: 0, k3: 0,
k2: 0, k1: 0,
k34: b/c*k42, k31: k39, k26: a/c*k42, k23: k39
}
assert solve_lin_sys(system, ring) == solution
def test_M39():
x, y, z = symbols('x y z', complex=True)
# TODO: Replace solve with solveset, as of now
# solveset doesn't supports non-linear multivariate
assert solve([x**2*y + 3*y*z - 4, -3*x**2*z + 2*y**2 + 1, 2*y*z**2 - z**2 - 1 ]) ==\
[{y: 1, z: 1, x: -1}, {y: 1, z: 1, x: 1},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: -sqrt(-1 - sqrt(2)*I)},\
{y: sqrt(2)*I, z: R(1,3) - sqrt(2)*I/3, x: sqrt(-1 - sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: -sqrt(-1 + sqrt(2)*I)},\
{y: -sqrt(2)*I, z: R(1,3) + sqrt(2)*I/3, x: sqrt(-1 + sqrt(2)*I)}]
# N. Inequalities
def test_N1():
assert ask(E**pi > pi**E)
@XFAIL
def test_N2():
x = symbols('x', real=True)
assert ask(x**4 - x + 1 > 0) is True
assert ask(x**4 - x + 1 > 1) is False
@XFAIL
def test_N3():
x = symbols('x', real=True)
assert ask(And(Lt(-1, x), Lt(x, 1)), abs(x) < 1 )
@XFAIL
def test_N4():
x, y = symbols('x y', real=True)
assert ask(2*x**2 > 2*y**2, (x > y) & (y > 0)) is True
@XFAIL
def test_N5():
x, y, k = symbols('x y k', real=True)
assert ask(k*x**2 > k*y**2, (x > y) & (y > 0) & (k > 0)) is True
@slow
@XFAIL
def test_N6():
x, y, k, n = symbols('x y k n', real=True)
assert ask(k*x**n > k*y**n, (x > y) & (y > 0) & (k > 0) & (n > 0)) is True
@XFAIL
def test_N7():
x, y = symbols('x y', real=True)
assert ask(y > 0, (x > 1) & (y >= x - 1)) is True
@XFAIL
@slow
def test_N8():
x, y, z = symbols('x y z', real=True)
assert ask(Eq(x, y) & Eq(y, z),
(x >= y) & (y >= z) & (z >= x))
def test_N9():
x = Symbol('x')
assert solveset(abs(x - 1) > 2, domain=S.Reals) == Union(Interval(-oo, -1, False, True),
Interval(3, oo, True))
def test_N10():
x = Symbol('x')
p = (x - 1)*(x - 2)*(x - 3)*(x - 4)*(x - 5)
assert solveset(expand(p) < 0, domain=S.Reals) == Union(Interval(-oo, 1, True, True),
Interval(2, 3, True, True),
Interval(4, 5, True, True))
def test_N11():
x = Symbol('x')
assert solveset(6/(x - 3) <= 3, domain=S.Reals) == Union(Interval(-oo, 3, True, True), Interval(5, oo))
def test_N12():
x = Symbol('x')
assert solveset(sqrt(x) < 2, domain=S.Reals) == Interval(0, 4, False, True)
def test_N13():
x = Symbol('x')
assert solveset(sin(x) < 2, domain=S.Reals) == S.Reals
@XFAIL
def test_N14():
x = Symbol('x')
# Gives 'Union(Interval(Integer(0), Mul(Rational(1, 2), pi), false, true),
# Interval(Mul(Rational(1, 2), pi), Mul(Integer(2), pi), true, false))'
# which is not the correct answer, but the provided also seems wrong.
assert solveset(sin(x) < 1, x, domain=S.Reals) == Union(Interval(-oo, pi/2, True, True),
Interval(pi/2, oo, True, True))
def test_N15():
r, t = symbols('r t')
# raises NotImplementedError: only univariate inequalities are supported
solveset(abs(2*r*(cos(t) - 1) + 1) <= 1, r, S.Reals)
def test_N16():
r, t = symbols('r t')
solveset((r**2)*((cos(t) - 4)**2)*sin(t)**2 < 9, r, S.Reals)
@XFAIL
def test_N17():
# currently only univariate inequalities are supported
assert solveset((x + y > 0, x - y < 0), (x, y)) == (abs(x) < y)
def test_O1():
M = Matrix((1 + I, -2, 3*I))
assert sqrt(expand(M.dot(M.H))) == sqrt(15)
def test_O2():
assert Matrix((2, 2, -3)).cross(Matrix((1, 3, 1))) == Matrix([[11],
[-5],
[4]])
# The vector module has no way of representing vectors symbolically (without
# respect to a basis)
@XFAIL
def test_O3():
# assert (va ^ vb) | (vc ^ vd) == -(va | vc)*(vb | vd) + (va | vd)*(vb | vc)
raise NotImplementedError("""The vector module has no way of representing
vectors symbolically (without respect to a basis)""")
def test_O4():
from sympy.vector import CoordSys3D, Del
N = CoordSys3D("N")
delop = Del()
i, j, k = N.base_vectors()
x, y, z = N.base_scalars()
F = i*(x*y*z) + j*((x*y*z)**2) + k*((y**2)*(z**3))
assert delop.cross(F).doit() == (-2*x**2*y**2*z + 2*y*z**3)*i + x*y*j + (2*x*y**2*z**2 - x*z)*k
@XFAIL
def test_O5():
#assert grad|(f^g)-g|(grad^f)+f|(grad^g) == 0
raise NotImplementedError("""The vector module has no way of representing
vectors symbolically (without respect to a basis)""")
#testO8-O9 MISSING!!
def test_O10():
L = [Matrix([2, 3, 5]), Matrix([3, 6, 2]), Matrix([8, 3, 6])]
assert GramSchmidt(L) == [Matrix([
[2],
[3],
[5]]),
Matrix([
[R(23, 19)],
[R(63, 19)],
[R(-47, 19)]]),
Matrix([
[R(1692, 353)],
[R(-1551, 706)],
[R(-423, 706)]])]
def test_P1():
assert Matrix(3, 3, lambda i, j: j - i).diagonal(-1) == Matrix(
1, 2, [-1, -1])
def test_P2():
M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
M.row_del(1)
M.col_del(2)
assert M == Matrix([[1, 2],
[7, 8]])
def test_P3():
A = Matrix([
[11, 12, 13, 14],
[21, 22, 23, 24],
[31, 32, 33, 34],
[41, 42, 43, 44]])
A11 = A[0:3, 1:4]
A12 = A[(0, 1, 3), (2, 0, 3)]
A21 = A
A221 = -A[0:2, 2:4]
A222 = -A[(3, 0), (2, 1)]
A22 = BlockMatrix([[A221, A222]]).T
rows = [[-A11, A12], [A21, A22]]
raises(ValueError, lambda: BlockMatrix(rows))
B = Matrix(rows)
assert B == Matrix([
[-12, -13, -14, 13, 11, 14],
[-22, -23, -24, 23, 21, 24],
[-32, -33, -34, 43, 41, 44],
[11, 12, 13, 14, -13, -23],
[21, 22, 23, 24, -14, -24],
[31, 32, 33, 34, -43, -13],
[41, 42, 43, 44, -42, -12]])
@XFAIL
def test_P4():
raise NotImplementedError("Block matrix diagonalization not supported")
def test_P5():
M = Matrix([[7, 11],
[3, 8]])
assert M % 2 == Matrix([[1, 1],
[1, 0]])
def test_P6():
M = Matrix([[cos(x), sin(x)],
[-sin(x), cos(x)]])
assert M.diff(x, 2) == Matrix([[-cos(x), -sin(x)],
[sin(x), -cos(x)]])
def test_P7():
M = Matrix([[x, y]])*(
z*Matrix([[1, 3, 5],
[2, 4, 6]]) + Matrix([[7, -9, 11],
[-8, 10, -12]]))
assert M == Matrix([[x*(z + 7) + y*(2*z - 8), x*(3*z - 9) + y*(4*z + 10),
x*(5*z + 11) + y*(6*z - 12)]])
def test_P8():
M = Matrix([[1, -2*I],
[-3*I, 4]])
assert M.norm(ord=S.Infinity) == 7
def test_P9():
a, b, c = symbols('a b c', nonzero=True)
M = Matrix([[a/(b*c), 1/c, 1/b],
[1/c, b/(a*c), 1/a],
[1/b, 1/a, c/(a*b)]])
assert factor(M.norm('fro')) == (a**2 + b**2 + c**2)/(abs(a)*abs(b)*abs(c))
@XFAIL
def test_P10():
M = Matrix([[1, 2 + 3*I],
[f(4 - 5*I), 6]])
# conjugate(f(4 - 5*i)) is not simplified to f(4+5*I)
assert M.H == Matrix([[1, f(4 + 5*I)],
[2 + 3*I, 6]])
@XFAIL
def test_P11():
# raises NotImplementedError("Matrix([[x,y],[1,x*y]]).inv()
# not simplifying to extract common factor")
assert Matrix([[x, y],
[1, x*y]]).inv() == (1/(x**2 - 1))*Matrix([[x, -1],
[-1/y, x/y]])
def test_P11_workaround():
# This test was changed to inverse method ADJ because it depended on the
# specific form of inverse returned from the 'GE' method which has changed.
M = Matrix([[x, y], [1, x*y]]).inv('ADJ')
c = gcd(tuple(M))
assert MatMul(c, M/c, evaluate=False) == MatMul(c, Matrix([
[x*y, -y],
[ -1, x]]), evaluate=False)
def test_P12():
A11 = MatrixSymbol('A11', n, n)
A12 = MatrixSymbol('A12', n, n)
A22 = MatrixSymbol('A22', n, n)
B = BlockMatrix([[A11, A12],
[ZeroMatrix(n, n), A22]])
assert block_collapse(B.I) == BlockMatrix([[A11.I, (-1)*A11.I*A12*A22.I],
[ZeroMatrix(n, n), A22.I]])
def test_P13():
M = Matrix([[1, x - 2, x - 3],
[x - 1, x**2 - 3*x + 6, x**2 - 3*x - 2],
[x - 2, x**2 - 8, 2*(x**2) - 12*x + 14]])
L, U, _ = M.LUdecomposition()
assert simplify(L) == Matrix([[1, 0, 0],
[x - 1, 1, 0],
[x - 2, x - 3, 1]])
assert simplify(U) == Matrix([[1, x - 2, x - 3],
[0, 4, x - 5],
[0, 0, x - 7]])
def test_P14():
M = Matrix([[1, 2, 3, 1, 3],
[3, 2, 1, 1, 7],
[0, 2, 4, 1, 1],
[1, 1, 1, 1, 4]])
R, _ = M.rref()
assert R == Matrix([[1, 0, -1, 0, 2],
[0, 1, 2, 0, -1],
[0, 0, 0, 1, 3],
[0, 0, 0, 0, 0]])
def test_P15():
M = Matrix([[-1, 3, 7, -5],
[4, -2, 1, 3],
[2, 4, 15, -7]])
assert M.rank() == 2
def test_P16():
M = Matrix([[2*sqrt(2), 8],
[6*sqrt(6), 24*sqrt(3)]])
assert M.rank() == 1
def test_P17():
t = symbols('t', real=True)
M=Matrix([
[sin(2*t), cos(2*t)],
[2*(1 - (cos(t)**2))*cos(t), (1 - 2*(sin(t)**2))*sin(t)]])
assert M.rank() == 1
def test_P18():
M = Matrix([[1, 0, -2, 0],
[-2, 1, 0, 3],
[-1, 2, -6, 6]])
assert M.nullspace() == [Matrix([[2],
[4],
[1],
[0]]),
Matrix([[0],
[-3],
[0],
[1]])]
def test_P19():
w = symbols('w')
M = Matrix([[1, 1, 1, 1],
[w, x, y, z],
[w**2, x**2, y**2, z**2],
[w**3, x**3, y**3, z**3]])
assert M.det() == (w**3*x**2*y - w**3*x**2*z - w**3*x*y**2 + w**3*x*z**2
+ w**3*y**2*z - w**3*y*z**2 - w**2*x**3*y + w**2*x**3*z
+ w**2*x*y**3 - w**2*x*z**3 - w**2*y**3*z + w**2*y*z**3
+ w*x**3*y**2 - w*x**3*z**2 - w*x**2*y**3 + w*x**2*z**3
+ w*y**3*z**2 - w*y**2*z**3 - x**3*y**2*z + x**3*y*z**2
+ x**2*y**3*z - x**2*y*z**3 - x*y**3*z**2 + x*y**2*z**3
)
@XFAIL
def test_P20():
raise NotImplementedError("Matrix minimal polynomial not supported")
def test_P21():
M = Matrix([[5, -3, -7],
[-2, 1, 2],
[2, -3, -4]])
assert M.charpoly(x).as_expr() == x**3 - 2*x**2 - 5*x + 6
def test_P22():
d = 100
M = (2 - x)*eye(d)
assert M.eigenvals() == {-x + 2: d}
def test_P23():
M = Matrix([
[2, 1, 0, 0, 0],
[1, 2, 1, 0, 0],
[0, 1, 2, 1, 0],
[0, 0, 1, 2, 1],
[0, 0, 0, 1, 2]])
assert M.eigenvals() == {
S('1'): 1,
S('2'): 1,
S('3'): 1,
S('sqrt(3) + 2'): 1,
S('-sqrt(3) + 2'): 1}
def test_P24():
M = Matrix([[611, 196, -192, 407, -8, -52, -49, 29],
[196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]])
assert M.eigenvals() == {
S('0'): 1,
S('10*sqrt(10405)'): 1,
S('100*sqrt(26) + 510'): 1,
S('1000'): 2,
S('-100*sqrt(26) + 510'): 1,
S('-10*sqrt(10405)'): 1,
S('1020'): 1}
def test_P25():
MF = N(Matrix([[ 611, 196, -192, 407, -8, -52, -49, 29],
[ 196, 899, 113, -192, -71, -43, -8, -44],
[-192, 113, 899, 196, 61, 49, 8, 52],
[ 407, -192, 196, 611, 8, 44, 59, -23],
[ -8, -71, 61, 8, 411, -599, 208, 208],
[ -52, -43, 49, 44, -599, 411, 208, 208],
[ -49, -8, 8, 59, 208, 208, 99, -911],
[ 29, -44, 52, -23, 208, 208, -911, 99]]))
ev_1 = sorted(MF.eigenvals(multiple=True))
ev_2 = sorted(
[-1020.0490184299969, 0.0, 0.09804864072151699, 1000.0, 1000.0,
1019.9019513592784, 1020.0, 1020.0490184299969])
for x, y in zip(ev_1, ev_2):
assert abs(x - y) < 1e-12
def test_P26():
a0, a1, a2, a3, a4 = symbols('a0 a1 a2 a3 a4')
M = Matrix([[-a4, -a3, -a2, -a1, -a0, 0, 0, 0, 0],
[ 1, 0, 0, 0, 0, 0, 0, 0, 0],
[ 0, 1, 0, 0, 0, 0, 0, 0, 0],
[ 0, 0, 1, 0, 0, 0, 0, 0, 0],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0, -1, -1, 0, 0],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0],
[ 0, 0, 0, 0, 0, 0, 1, -1, -1],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0]])
assert M.eigenvals(error_when_incomplete=False) == {
S('-1/2 - sqrt(3)*I/2'): 2,
S('-1/2 + sqrt(3)*I/2'): 2}
def test_P27():
a = symbols('a')
M = Matrix([[a, 0, 0, 0, 0],
[0, 0, 0, 0, 1],
[0, 0, a, 0, 0],
[0, 0, 0, a, 0],
[0, -2, 0, 0, 2]])
assert M.eigenvects() == [
(a, 3, [
Matrix([1, 0, 0, 0, 0]),
Matrix([0, 0, 1, 0, 0]),
Matrix([0, 0, 0, 1, 0])
]),
(1 - I, 1, [
Matrix([0, (1 + I)/2, 0, 0, 1])
]),
(1 + I, 1, [
Matrix([0, (1 - I)/2, 0, 0, 1])
]),
]
@XFAIL
def test_P28():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
@XFAIL
def test_P29():
raise NotImplementedError("Generalized eigenvectors not supported \
https://github.com/sympy/sympy/issues/5293")
def test_P30():
M = Matrix([[1, 0, 0, 1, -1],
[0, 1, -2, 3, -3],
[0, 0, -1, 2, -2],
[1, -1, 1, 0, 1],
[1, -1, 1, -1, 2]])
_, J = M.jordan_form()
assert J == Matrix([[-1, 0, 0, 0, 0],
[0, 1, 1, 0, 0],
[0, 0, 1, 0, 0],
[0, 0, 0, 1, 1],
[0, 0, 0, 0, 1]])
@XFAIL
def test_P31():
raise NotImplementedError("Smith normal form not implemented")
def test_P32():
M = Matrix([[1, -2],
[2, 1]])
assert exp(M).rewrite(cos).simplify() == Matrix([[E*cos(2), -E*sin(2)],
[E*sin(2), E*cos(2)]])
def test_P33():
w, t = symbols('w t')
M = Matrix([[0, 1, 0, 0],
[0, 0, 0, 2*w],
[0, 0, 0, 1],
[0, -2*w, 3*w**2, 0]])
assert exp(M*t).rewrite(cos).expand() == Matrix([
[1, -3*t + 4*sin(t*w)/w, 6*t*w - 6*sin(t*w), -2*cos(t*w)/w + 2/w],
[0, 4*cos(t*w) - 3, -6*w*cos(t*w) + 6*w, 2*sin(t*w)],
[0, 2*cos(t*w)/w - 2/w, -3*cos(t*w) + 4, sin(t*w)/w],
[0, -2*sin(t*w), 3*w*sin(t*w), cos(t*w)]])
@XFAIL
def test_P34():
a, b, c = symbols('a b c', real=True)
M = Matrix([[a, 1, 0, 0, 0, 0],
[0, a, 0, 0, 0, 0],
[0, 0, b, 0, 0, 0],
[0, 0, 0, c, 1, 0],
[0, 0, 0, 0, c, 1],
[0, 0, 0, 0, 0, c]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == Matrix([[sin(a), cos(a), 0, 0, 0, 0],
[0, sin(a), 0, 0, 0, 0],
[0, 0, sin(b), 0, 0, 0],
[0, 0, 0, sin(c), cos(c), -sin(c)/2],
[0, 0, 0, 0, sin(c), cos(c)],
[0, 0, 0, 0, 0, sin(c)]])
@XFAIL
def test_P35():
M = pi/2*Matrix([[2, 1, 1],
[2, 3, 2],
[1, 1, 2]])
# raises exception, sin(M) not supported. exp(M*I) also not supported
# https://github.com/sympy/sympy/issues/6218
assert sin(M) == eye(3)
@XFAIL
def test_P36():
M = Matrix([[10, 7],
[7, 17]])
assert sqrt(M) == Matrix([[3, 1],
[1, 4]])
def test_P37():
M = Matrix([[1, 1, 0],
[0, 1, 0],
[0, 0, 1]])
assert M**S.Half == Matrix([[1, R(1, 2), 0],
[0, 1, 0],
[0, 0, 1]])
@XFAIL
def test_P38():
M=Matrix([[0, 1, 0],
[0, 0, 0],
[0, 0, 0]])
with raises(AssertionError):
# raises ValueError: Matrix det == 0; not invertible
M**S.Half
# if it doesn't raise then this assertion will be
# raised and the test will be flagged as not XFAILing
assert None
@XFAIL
def test_P39():
"""
M=Matrix([
[1, 1],
[2, 2],
[3, 3]])
M.SVD()
"""
raise NotImplementedError("Singular value decomposition not implemented")
def test_P40():
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert M.jacobian(Matrix([r, t])) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P41():
r, t = symbols('r t', real=True)
assert hessian(r**2*sin(t),(r,t)) == Matrix([[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P42():
assert wronskian([cos(x), sin(x)], x).simplify() == 1
def test_P43():
def __my_jacobian(M, Y):
return Matrix([M.diff(v).T for v in Y]).T
r, t = symbols('r t', real=True)
M = Matrix([r*cos(t), r*sin(t)])
assert __my_jacobian(M,[r,t]) == Matrix([[cos(t), -r*sin(t)],
[sin(t), r*cos(t)]])
def test_P44():
def __my_hessian(f, Y):
V = Matrix([diff(f, v) for v in Y])
return Matrix([V.T.diff(v) for v in Y])
r, t = symbols('r t', real=True)
assert __my_hessian(r**2*sin(t), (r, t)) == Matrix([
[ 2*sin(t), 2*r*cos(t)],
[2*r*cos(t), -r**2*sin(t)]])
def test_P45():
def __my_wronskian(Y, v):
M = Matrix([Matrix(Y).T.diff(x, n) for n in range(0, len(Y))])
return M.det()
assert __my_wronskian([cos(x), sin(x)], x).simplify() == 1
# Q1-Q6 Tensor tests missing
@XFAIL
def test_R1():
i, j, n = symbols('i j n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
Sm = Sum((xn[i, 0] - Sum(xn[j, 0], (j, 0, n - 1))/n)**2, (i, 0, n - 1))
# sum does not calculate
# Unknown result
Sm.doit()
raise NotImplementedError('Unknown result')
@XFAIL
def test_R2():
m, b = symbols('m b')
i, n = symbols('i n', integer=True, positive=True)
xn = MatrixSymbol('xn', n, 1)
yn = MatrixSymbol('yn', n, 1)
f = Sum((yn[i, 0] - m*xn[i, 0] - b)**2, (i, 0, n - 1))
f1 = diff(f, m)
f2 = diff(f, b)
# raises TypeError: solveset() takes at most 2 arguments (3 given)
solveset((f1, f2), (m, b), domain=S.Reals)
@XFAIL
def test_R3():
n, k = symbols('n k', integer=True, positive=True)
sk = ((-1)**k) * (binomial(2*n, k))**2
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit()
T2 = T.combsimp()
# returns -((-1)**n*factorial(2*n)
# - (factorial(n))**2)*exp_polar(-I*pi)/(factorial(n))**2
assert T2 == (-1)**n*binomial(2*n, n)
@XFAIL
def test_R4():
# Macsyma indefinite sum test case:
#(c15) /* Check whether the full Gosper algorithm is implemented
# => 1/2^(n + 1) binomial(n, k - 1) */
#closedform(indefsum(binomial(n, k)/2^n - binomial(n + 1, k)/2^(n + 1), k));
#Time= 2690 msecs
# (- n + k - 1) binomial(n + 1, k)
#(d15) - --------------------------------
# n
# 2 2 (n + 1)
#
#(c16) factcomb(makefact(%));
#Time= 220 msecs
# n!
#(d16) ----------------
# n
# 2 k! 2 (n - k)!
# Might be possible after fixing https://github.com/sympy/sympy/pull/1879
raise NotImplementedError("Indefinite sum not supported")
@XFAIL
def test_R5():
a, b, c, n, k = symbols('a b c n k', integer=True, positive=True)
sk = ((-1)**k)*(binomial(a + b, a + k)
*binomial(b + c, b + k)*binomial(c + a, c + k))
Sm = Sum(sk, (k, 1, oo))
T = Sm.doit() # hypergeometric series not calculated
assert T == factorial(a+b+c)/(factorial(a)*factorial(b)*factorial(c))
def test_R6():
n, k = symbols('n k', integer=True, positive=True)
gn = MatrixSymbol('gn', n + 2, 1)
Sm = Sum(gn[k, 0] - gn[k - 1, 0], (k, 1, n + 1))
assert Sm.doit() == -gn[0, 0] + gn[n + 1, 0]
def test_R7():
n, k = symbols('n k', integer=True, positive=True)
T = Sum(k**3,(k,1,n)).doit()
assert T.factor() == n**2*(n + 1)**2/4
@XFAIL
def test_R8():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(k**2*binomial(n, k), (k, 1, n))
T = Sm.doit() #returns Piecewise function
assert T.combsimp() == n*(n + 1)*2**(n - 2)
def test_R9():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, k - 1)/k, (k, 1, n + 1))
assert Sm.doit().simplify() == (2**(n + 1) - 1)/(n + 1)
@XFAIL
def test_R10():
n, m, r, k = symbols('n m r k', integer=True, positive=True)
Sm = Sum(binomial(n, k)*binomial(m, r - k), (k, 0, r))
T = Sm.doit()
T2 = T.combsimp().rewrite(factorial)
assert T2 == factorial(m + n)/(factorial(r)*factorial(m + n - r))
assert T2 == binomial(m + n, r).rewrite(factorial)
# rewrite(binomial) is not working.
# https://github.com/sympy/sympy/issues/7135
T3 = T2.rewrite(binomial)
assert T3 == binomial(m + n, r)
@XFAIL
def test_R11():
n, k = symbols('n k', integer=True, positive=True)
sk = binomial(n, k)*fibonacci(k)
Sm = Sum(sk, (k, 0, n))
T = Sm.doit()
# Fibonacci simplification not implemented
# https://github.com/sympy/sympy/issues/7134
assert T == fibonacci(2*n)
@XFAIL
def test_R12():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(fibonacci(k)**2, (k, 0, n))
T = Sm.doit()
assert T == fibonacci(n)*fibonacci(n + 1)
@XFAIL
def test_R13():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin(k*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == cot(x/2)/2 - cos(x*(2*n + 1)/2)/(2*sin(x/2))
@XFAIL
def test_R14():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(sin((2*k - 1)*x), (k, 1, n))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == sin(n*x)**2/sin(x)
@XFAIL
def test_R15():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n - k, k), (k, 0, floor(n/2)))
T = Sm.doit() # Sum is not calculated
assert T.simplify() == fibonacci(n + 1)
def test_R16():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/k**2 + 1/k**3, (k, 1, oo))
assert Sm.doit() == zeta(3) + pi**2/6
def test_R17():
k = symbols('k', integer=True, positive=True)
assert abs(float(Sum(1/k**2 + 1/k**3, (k, 1, oo)))
- 2.8469909700078206) < 1e-15
def test_R18():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(2**k*k**2), (k, 1, oo))
T = Sm.doit()
assert T.simplify() == -log(2)**2/2 + pi**2/12
@slow
@XFAIL
def test_R19():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/((3*k + 1)*(3*k + 2)*(3*k + 3)), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == -log(3)/4 + sqrt(3)*pi/12
@XFAIL
def test_R20():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(binomial(n, 4*k), (k, 0, oo))
T = Sm.doit()
# assert fails, T not simplified
assert T.simplify() == 2**(n/2)*cos(pi*n/4)/2 + 2**(n - 1)/2
@XFAIL
def test_R21():
k = symbols('k', integer=True, positive=True)
Sm = Sum(1/(sqrt(k*(k + 1)) * (sqrt(k) + sqrt(k + 1))), (k, 1, oo))
T = Sm.doit() # Sum not calculated
assert T.simplify() == 1
# test_R22 answer not available in Wester samples
# Sum(Sum(binomial(n, k)*binomial(n - k, n - 2*k)*x**n*y**(n - 2*k),
# (k, 0, floor(n/2))), (n, 0, oo)) with abs(x*y)<1?
@XFAIL
def test_R23():
n, k = symbols('n k', integer=True, positive=True)
Sm = Sum(Sum((factorial(n)/(factorial(k)**2*factorial(n - 2*k)))*
(x/y)**k*(x*y)**(n - k), (n, 2*k, oo)), (k, 0, oo))
# Missing how to express constraint abs(x*y)<1?
T = Sm.doit() # Sum not calculated
assert T == -1/sqrt(x**2*y**2 - 4*x**2 - 2*x*y + 1)
def test_R24():
m, k = symbols('m k', integer=True, positive=True)
Sm = Sum(Product(k/(2*k - 1), (k, 1, m)), (m, 2, oo))
assert Sm.doit() == pi/2
def test_S1():
k = symbols('k', integer=True, positive=True)
Pr = Product(gamma(k/3), (k, 1, 8))
assert Pr.doit().simplify() == 640*sqrt(3)*pi**3/6561
def test_S2():
n, k = symbols('n k', integer=True, positive=True)
assert Product(k, (k, 1, n)).doit() == factorial(n)
def test_S3():
n, k = symbols('n k', integer=True, positive=True)
assert Product(x**k, (k, 1, n)).doit().simplify() == x**(n*(n + 1)/2)
def test_S4():
n, k = symbols('n k', integer=True, positive=True)
assert Product(1 + 1/k, (k, 1, n -1)).doit().simplify() == n
def test_S5():
n, k = symbols('n k', integer=True, positive=True)
assert (Product((2*k - 1)/(2*k), (k, 1, n)).doit().gammasimp() ==
gamma(n + S.Half)/(sqrt(pi)*gamma(n + 1)))
@XFAIL
def test_S6():
n, k = symbols('n k', integer=True, positive=True)
# Product does not evaluate
assert (Product(x**2 -2*x*cos(k*pi/n) + 1, (k, 1, n - 1)).doit().simplify()
== (x**(2*n) - 1)/(x**2 - 1))
@XFAIL
def test_S7():
k = symbols('k', integer=True, positive=True)
Pr = Product((k**3 - 1)/(k**3 + 1), (k, 2, oo))
T = Pr.doit() # Product does not evaluate
assert T.simplify() == R(2, 3)
@XFAIL
def test_S8():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 - 1/(2*k)**2, (k, 1, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == 2/pi
@XFAIL
def test_S9():
k = symbols('k', integer=True, positive=True)
Pr = Product(1 + (-1)**(k + 1)/(2*k - 1), (k, 1, oo))
T = Pr.doit()
# Product produces 0
# https://github.com/sympy/sympy/issues/7133
assert T.simplify() == sqrt(2)
@XFAIL
def test_S10():
k = symbols('k', integer=True, positive=True)
Pr = Product((k*(k + 1) + 1 + I)/(k*(k + 1) + 1 - I), (k, 0, oo))
T = Pr.doit()
# Product does not evaluate
assert T.simplify() == -1
def test_T1():
assert limit((1 + 1/n)**n, n, oo) == E
assert limit((1 - cos(x))/x**2, x, 0) == S.Half
def test_T2():
assert limit((3**x + 5**x)**(1/x), x, oo) == 5
def test_T3():
assert limit(log(x)/(log(x) + sin(x)), x, oo) == 1
def test_T4():
assert limit((exp(x*exp(-x)/(exp(-x) + exp(-2*x**2/(x + 1))))
- exp(x))/x, x, oo) == -exp(2)
def test_T5():
assert limit(x*log(x)*log(x*exp(x) - x**2)**2/log(log(x**2
+ 2*exp(exp(3*x**3*log(x))))), x, oo) == R(1, 3)
def test_T6():
assert limit(1/n * factorial(n)**(1/n), n, oo) == exp(-1)
def test_T7():
limit(1/n * gamma(n + 1)**(1/n), n, oo)
def test_T8():
a, z = symbols('a z', positive=True)
assert limit(gamma(z + a)/gamma(z)*exp(-a*log(z)), z, oo) == 1
@XFAIL
def test_T9():
z, k = symbols('z k', positive=True)
# raises NotImplementedError:
# Don't know how to calculate the mrv of '(1, k)'
assert limit(hyper((1, k), (1,), z/k), k, oo) == exp(z)
@XFAIL
def test_T10():
# No longer raises PoleError, but should return euler-mascheroni constant
assert limit(zeta(x) - 1/(x - 1), x, 1) == integrate(-1/x + 1/floor(x), (x, 1, oo))
@XFAIL
def test_T11():
n, k = symbols('n k', integer=True, positive=True)
# evaluates to 0
assert limit(n**x/(x*product((1 + x/k), (k, 1, n))), n, oo) == gamma(x)
def test_T12():
x, t = symbols('x t', real=True)
# Does not evaluate the limit but returns an expression with erf
assert limit(x * integrate(exp(-t**2), (t, 0, x))/(1 - exp(-x**2)),
x, 0) == 1
def test_T13():
x = symbols('x', real=True)
assert [limit(x/abs(x), x, 0, dir='-'),
limit(x/abs(x), x, 0, dir='+')] == [-1, 1]
def test_T14():
x = symbols('x', real=True)
assert limit(atan(-log(x)), x, 0, dir='+') == pi/2
def test_U1():
x = symbols('x', real=True)
assert diff(abs(x), x) == sign(x)
def test_U2():
f = Lambda(x, Piecewise((-x, x < 0), (x, x >= 0)))
assert diff(f(x), x) == Piecewise((-1, x < 0), (1, x >= 0))
def test_U3():
f = Lambda(x, Piecewise((x**2 - 1, x == 1), (x**3, x != 1)))
f1 = Lambda(x, diff(f(x), x))
assert f1(x) == 3*x**2
assert f1(1) == 3
@XFAIL
def test_U4():
n = symbols('n', integer=True, positive=True)
x = symbols('x', real=True)
d = diff(x**n, x, n)
assert d.rewrite(factorial) == factorial(n)
def test_U5():
# issue 6681
t = symbols('t')
ans = (
Derivative(f(g(t)), g(t))*Derivative(g(t), (t, 2)) +
Derivative(f(g(t)), (g(t), 2))*Derivative(g(t), t)**2)
assert f(g(t)).diff(t, 2) == ans
assert ans.doit() == ans
def test_U6():
h = Function('h')
T = integrate(f(y), (y, h(x), g(x)))
assert T.diff(x) == (
f(g(x))*Derivative(g(x), x) - f(h(x))*Derivative(h(x), x))
@XFAIL
def test_U7():
p, t = symbols('p t', real=True)
# Exact differential => d(V(P, T)) => dV/dP DP + dV/dT DT
# raises ValueError: Since there is more than one variable in the
# expression, the variable(s) of differentiation must be supplied to
# differentiate f(p,t)
diff(f(p, t))
def test_U8():
x, y = symbols('x y', real=True)
eq = cos(x*y) + x
# If SymPy had implicit_diff() function this hack could be avoided
# TODO: Replace solve with solveset, current test fails for solveset
assert idiff(y - eq, y, x) == (-y*sin(x*y) + 1)/(x*sin(x*y) + 1)
def test_U9():
# Wester sample case for Maple:
# O29 := diff(f(x, y), x) + diff(f(x, y), y);
# /d \ /d \
# |-- f(x, y)| + |-- f(x, y)|
# \dx / \dy /
#
# O30 := factor(subs(f(x, y) = g(x^2 + y^2), %));
# 2 2
# 2 D(g)(x + y ) (x + y)
x, y = symbols('x y', real=True)
su = diff(f(x, y), x) + diff(f(x, y), y)
s2 = su.subs(f(x, y), g(x**2 + y**2))
s3 = s2.doit().factor()
# Subs not performed, s3 = 2*(x + y)*Subs(Derivative(
# g(_xi_1), _xi_1), _xi_1, x**2 + y**2)
# Derivative(g(x*2 + y**2), x**2 + y**2) is not valid in SymPy,
# and probably will remain that way. You can take derivatives with respect
# to other expressions only if they are atomic, like a symbol or a
# function.
# D operator should be added to SymPy
# See https://github.com/sympy/sympy/issues/4719.
assert s3 == (x + y)*Subs(Derivative(g(x), x), x, x**2 + y**2)*2
def test_U10():
# see issue 2519:
assert residue((z**3 + 5)/((z**4 - 1)*(z + 1)), z, -1) == R(-9, 4)
@XFAIL
def test_U11():
# assert (2*dx + dz) ^ (3*dx + dy + dz) ^ (dx + dy + 4*dz) == 8*dx ^ dy ^dz
raise NotImplementedError
@XFAIL
def test_U12():
# Wester sample case:
# (c41) /* d(3 x^5 dy /\ dz + 5 x y^2 dz /\ dx + 8 z dx /\ dy)
# => (15 x^4 + 10 x y + 8) dx /\ dy /\ dz */
# factor(ext_diff(3*x^5 * dy ~ dz + 5*x*y^2 * dz ~ dx + 8*z * dx ~ dy));
# 4
# (d41) (10 x y + 15 x + 8) dx dy dz
raise NotImplementedError(
"External diff of differential form not supported")
def test_U13():
assert minimum(x**4 - x + 1, x) == -3*2**R(1,3)/8 + 1
@XFAIL
def test_U14():
#f = 1/(x**2 + y**2 + 1)
#assert [minimize(f), maximize(f)] == [0,1]
raise NotImplementedError("minimize(), maximize() not supported")
@XFAIL
def test_U15():
raise NotImplementedError("minimize() not supported and also solve does \
not support multivariate inequalities")
@XFAIL
def test_U16():
raise NotImplementedError("minimize() not supported in SymPy and also \
solve does not support multivariate inequalities")
@XFAIL
def test_U17():
raise NotImplementedError("Linear programming, symbolic simplex not \
supported in SymPy")
def test_V1():
x = symbols('x', real=True)
assert integrate(abs(x), x) == Piecewise((-x**2/2, x <= 0), (x**2/2, True))
def test_V2():
assert integrate(Piecewise((-x, x < 0), (x, x >= 0)), x
) == Piecewise((-x**2/2, x < 0), (x**2/2, True))
def test_V3():
assert integrate(1/(x**3 + 2),x).diff().simplify() == 1/(x**3 + 2)
def test_V4():
assert integrate(2**x/sqrt(1 + 4**x), x) == asinh(2**x)/log(2)
@XFAIL
def test_V5():
# Returns (-45*x**2 + 80*x - 41)/(5*sqrt(2*x - 1)*(4*x**2 - 4*x + 1))
assert (integrate((3*x - 5)**2/(2*x - 1)**R(7, 2), x).simplify() ==
(-41 + 80*x - 45*x**2)/(5*(2*x - 1)**R(5, 2)))
@XFAIL
def test_V6():
# returns RootSum(40*_z**2 - 1, Lambda(_i, _i*log(-4*_i + exp(-m*x))))/m
assert (integrate(1/(2*exp(m*x) - 5*exp(-m*x)), x) == sqrt(10)*(
log(2*exp(m*x) - sqrt(10)) - log(2*exp(m*x) + sqrt(10)))/(20*m))
def test_V7():
r1 = integrate(sinh(x)**4/cosh(x)**2)
assert r1.simplify() == x*R(-3, 2) + sinh(x)**3/(2*cosh(x)) + 3*tanh(x)/2
@XFAIL
def test_V8_V9():
#Macsyma test case:
#(c27) /* This example involves several symbolic parameters
# => 1/sqrt(b^2 - a^2) log([sqrt(b^2 - a^2) tan(x/2) + a + b]/
# [sqrt(b^2 - a^2) tan(x/2) - a - b]) (a^2 < b^2)
# [Gradshteyn and Ryzhik 2.553(3)] */
#assume(b^2 > a^2)$
#(c28) integrate(1/(a + b*cos(x)), x);
#(c29) trigsimp(ratsimp(diff(%, x)));
# 1
#(d29) ------------
# b cos(x) + a
raise NotImplementedError(
"Integrate with assumption not supported")
def test_V10():
assert integrate(1/(3 + 3*cos(x) + 4*sin(x)), x) == log(4*tan(x/2) + 3)/4
def test_V11():
r1 = integrate(1/(4 + 3*cos(x) + 4*sin(x)), x)
r2 = factor(r1)
assert (logcombine(r2, force=True) ==
log(((tan(x/2) + 1)/(tan(x/2) + 7))**R(1, 3)))
def test_V12():
r1 = integrate(1/(5 + 3*cos(x) + 4*sin(x)), x)
assert r1 == -1/(tan(x/2) + 2)
@XFAIL
def test_V13():
r1 = integrate(1/(6 + 3*cos(x) + 4*sin(x)), x)
# expression not simplified, returns: -sqrt(11)*I*log(tan(x/2) + 4/3
# - sqrt(11)*I/3)/11 + sqrt(11)*I*log(tan(x/2) + 4/3 + sqrt(11)*I/3)/11
assert r1.simplify() == 2*sqrt(11)*atan(sqrt(11)*(3*tan(x/2) + 4)/11)/11
@slow
@XFAIL
def test_V14():
r1 = integrate(log(abs(x**2 - y**2)), x)
# Piecewise result does not simplify to the desired result.
assert (r1.simplify() == x*log(abs(x**2 - y**2))
+ y*log(x + y) - y*log(x - y) - 2*x)
def test_V15():
r1 = integrate(x*acot(x/y), x)
assert simplify(r1 - (x*y + (x**2 + y**2)*acot(x/y))/2) == 0
@XFAIL
def test_V16():
# Integral not calculated
assert integrate(cos(5*x)*Ci(2*x), x) == Ci(2*x)*sin(5*x)/5 - (Si(3*x) + Si(7*x))/10
@XFAIL
def test_V17():
r1 = integrate((diff(f(x), x)*g(x)
- f(x)*diff(g(x), x))/(f(x)**2 - g(x)**2), x)
# integral not calculated
assert simplify(r1 - (f(x) - g(x))/(f(x) + g(x))/2) == 0
@XFAIL
def test_W1():
# The function has a pole at y.
# The integral has a Cauchy principal value of zero but SymPy returns -I*pi
# https://github.com/sympy/sympy/issues/7159
assert integrate(1/(x - y), (x, y - 1, y + 1)) == 0
@XFAIL
def test_W2():
# The function has a pole at y.
# The integral is divergent but SymPy returns -2
# https://github.com/sympy/sympy/issues/7160
# Test case in Macsyma:
# (c6) errcatch(integrate(1/(x - a)^2, x, a - 1, a + 1));
# Integral is divergent
assert integrate(1/(x - y)**2, (x, y - 1, y + 1)) is zoo
@XFAIL
@slow
def test_W3():
# integral is not calculated
# https://github.com/sympy/sympy/issues/7161
assert integrate(sqrt(x + 1/x - 2), (x, 0, 1)) == R(4, 3)
@XFAIL
@slow
def test_W4():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 1, 2)) == -2*sqrt(2)/3 + R(4, 3)
@XFAIL
@slow
def test_W5():
# integral is not calculated
assert integrate(sqrt(x + 1/x - 2), (x, 0, 2)) == -2*sqrt(2)/3 + R(8, 3)
@XFAIL
@slow
def test_W6():
# integral is not calculated
assert integrate(sqrt(2 - 2*cos(2*x))/2, (x, pi*R(-3, 4), -pi/4)) == sqrt(2)
def test_W7():
a = symbols('a', positive=True)
r1 = integrate(cos(x)/(x**2 + a**2), (x, -oo, oo))
assert r1.simplify() == pi*exp(-a)/a
@XFAIL
def test_W8():
# Test case in Mathematica:
# In[19]:= Integrate[t^(a - 1)/(1 + t), {t, 0, Infinity},
# Assumptions -> 0 < a < 1]
# Out[19]= Pi Csc[a Pi]
raise NotImplementedError(
"Integrate with assumption 0 < a < 1 not supported")
@XFAIL
@slow
def test_W9():
# Integrand with a residue at infinity => -2 pi [sin(pi/5) + sin(2pi/5)]
# (principal value) [Levinson and Redheffer, p. 234] *)
r1 = integrate(5*x**3/(1 + x + x**2 + x**3 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == -2*pi*(sqrt(-sqrt(5)/8 + 5/8) + sqrt(sqrt(5)/8 + 5/8))
@XFAIL
def test_W10():
# integrate(1/[1 + x + x^2 + ... + x^(2 n)], x = -infinity..infinity) =
# 2 pi/(2 n + 1) [1 + cos(pi/[2 n + 1])] csc(2 pi/[2 n + 1])
# [Levinson and Redheffer, p. 255] => 2 pi/5 [1 + cos(pi/5)] csc(2 pi/5) */
r1 = integrate(x/(1 + x + x**2 + x**4), (x, -oo, oo))
r2 = r1.doit()
assert r2 == 2*pi*(sqrt(5)/4 + 5/4)*csc(pi*R(2, 5))/5
@XFAIL
def test_W11():
# integral not calculated
assert (integrate(sqrt(1 - x**2)/(1 + x**2), (x, -1, 1)) ==
pi*(-1 + sqrt(2)))
def test_W12():
p = symbols('p', positive=True)
q = symbols('q', real=True)
r1 = integrate(x*exp(-p*x**2 + 2*q*x), (x, -oo, oo))
assert r1.simplify() == sqrt(pi)*q*exp(q**2/p)/p**R(3, 2)
@XFAIL
def test_W13():
# Integral not calculated. Expected result is 2*(Euler_mascheroni_constant)
r1 = integrate(1/log(x) + 1/(1 - x) - log(log(1/x)), (x, 0, 1))
assert r1 == 2*EulerGamma
def test_W14():
assert integrate(sin(x)/x*exp(2*I*x), (x, -oo, oo)) == 0
@XFAIL
def test_W15():
# integral not calculated
assert integrate(log(gamma(x))*cos(6*pi*x), (x, 0, 1)) == R(1, 12)
def test_W16():
assert integrate((1 + x)**3*legendre_poly(1, x)*legendre_poly(2, x),
(x, -1, 1)) == R(36, 35)
def test_W17():
a, b = symbols('a b', positive=True)
assert integrate(exp(-a*x)*besselj(0, b*x),
(x, 0, oo)) == 1/(b*sqrt(a**2/b**2 + 1))
def test_W18():
assert integrate((besselj(1, x)/x)**2, (x, 0, oo)) == 4/(3*pi)
@XFAIL
def test_W19():
# Integral not calculated
# Expected result is (cos 7 - 1)/7 [Gradshteyn and Ryzhik 6.782(3)]
assert integrate(Ci(x)*besselj(0, 2*sqrt(7*x)), (x, 0, oo)) == (cos(7) - 1)/7
@XFAIL
def test_W20():
# integral not calculated
assert (integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)) ==
-pi**2/36 - R(17, 108) + zeta(3)/4 +
(-pi**2/2 - 4*log(2) + log(2)**2 + 35/3)*log(2)/9)
def test_W21():
assert abs(N(integrate(x**2*polylog(3, 1/(x + 1)), (x, 0, 1)))
- 0.210882859565594) < 1e-15
def test_W22():
t, u = symbols('t u', real=True)
s = Lambda(x, Piecewise((1, And(x >= 1, x <= 2)), (0, True)))
assert integrate(s(t)*cos(t), (t, 0, u)) == Piecewise(
(0, u < 0),
(-sin(Min(1, u)) + sin(Min(2, u)), True))
@slow
def test_W23():
a, b = symbols('a b', positive=True)
r1 = integrate(integrate(x/(x**2 + y**2), (x, a, b)), (y, -oo, oo))
assert r1.collect(pi).cancel() == -pi*a + pi*b
def test_W23b():
# like W23 but limits are reversed
a, b = symbols('a b', positive=True)
r2 = integrate(integrate(x/(x**2 + y**2), (y, -oo, oo)), (x, a, b))
assert r2.collect(pi) == pi*(-a + b)
@XFAIL
@slow
def test_W24():
if ON_TRAVIS:
skip("Too slow for travis.")
# Not that slow, but does not fully evaluate so simplify is slow.
# Maybe also require doit()
x, y = symbols('x y', real=True)
r1 = integrate(integrate(sqrt(x**2 + y**2), (x, 0, 1)), (y, 0, 1))
assert (r1 - (sqrt(2) + asinh(1))/3).simplify() == 0
@XFAIL
@slow
def test_W25():
if ON_TRAVIS:
skip("Too slow for travis.")
a, x, y = symbols('a x y', real=True)
i1 = integrate(
sin(a)*sin(y)/sqrt(1 - sin(a)**2*sin(x)**2*sin(y)**2),
(x, 0, pi/2))
i2 = integrate(i1, (y, 0, pi/2))
assert (i2 - pi*a/2).simplify() == 0
def test_W26():
x, y = symbols('x y', real=True)
assert integrate(integrate(abs(y - x**2), (y, 0, 2)),
(x, -1, 1)) == R(46, 15)
def test_W27():
a, b, c = symbols('a b c')
assert integrate(integrate(integrate(1, (z, 0, c*(1 - x/a - y/b))),
(y, 0, b*(1 - x/a))),
(x, 0, a)) == a*b*c/6
def test_X1():
v, c = symbols('v c', real=True)
assert (series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8) ==
5*v**6/(16*c**6) + 3*v**4/(8*c**4) + v**2/(2*c**2) + 1 + O(v**8))
def test_X2():
v, c = symbols('v c', real=True)
s1 = series(1/sqrt(1 - (v/c)**2), v, x0=0, n=8)
assert (1/s1**2).series(v, x0=0, n=8) == -v**2/c**2 + 1 + O(v**8)
def test_X3():
s1 = (sin(x).series()/cos(x).series()).series()
s2 = tan(x).series()
assert s2 == x + x**3/3 + 2*x**5/15 + O(x**6)
assert s1 == s2
def test_X4():
s1 = log(sin(x)/x).series()
assert s1 == -x**2/6 - x**4/180 + O(x**6)
assert log(series(sin(x)/x)).series() == s1
@XFAIL
def test_X5():
# test case in Mathematica syntax:
# In[21]:= (* => [a f'(a d) + g(b d) + integrate(h(c y), y = 0..d)]
# + [a^2 f''(a d) + b g'(b d) + h(c d)] (x - d) *)
# In[22]:= D[f[a*x], x] + g[b*x] + Integrate[h[c*y], {y, 0, x}]
# Out[22]= g[b x] + Integrate[h[c y], {y, 0, x}] + a f'[a x]
# In[23]:= Series[%, {x, d, 1}]
# Out[23]= (g[b d] + Integrate[h[c y], {y, 0, d}] + a f'[a d]) +
# 2 2
# (h[c d] + b g'[b d] + a f''[a d]) (-d + x) + O[-d + x]
h = Function('h')
a, b, c, d = symbols('a b c d', real=True)
# series() raises NotImplementedError:
# The _eval_nseries method should be added to <class
# 'sympy.core.function.Subs'> to give terms up to O(x**n) at x=0
series(diff(f(a*x), x) + g(b*x) + integrate(h(c*y), (y, 0, x)),
x, x0=d, n=2)
# assert missing, until exception is removed
def test_X6():
# Taylor series of nonscalar objects (noncommutative multiplication)
# expected result => (B A - A B) t^2/2 + O(t^3) [Stanly Steinberg]
a, b = symbols('a b', commutative=False, scalar=False)
assert (series(exp((a + b)*x) - exp(a*x) * exp(b*x), x, x0=0, n=3) ==
x**2*(-a*b/2 + b*a/2) + O(x**3))
def test_X7():
# => sum( Bernoulli[k]/k! x^(k - 2), k = 1..infinity )
# = 1/x^2 - 1/(2 x) + 1/12 - x^2/720 + x^4/30240 + O(x^6)
# [Levinson and Redheffer, p. 173]
assert (series(1/(x*(exp(x) - 1)), x, 0, 7) == x**(-2) - 1/(2*x) +
R(1, 12) - x**2/720 + x**4/30240 - x**6/1209600 + O(x**7))
def test_X8():
# Puiseux series (terms with fractional degree):
# => 1/sqrt(x - 3/2 pi) + (x - 3/2 pi)^(3/2) / 12 + O([x - 3/2 pi]^(7/2))
# see issue 7167:
x = symbols('x', real=True)
assert (series(sqrt(sec(x)), x, x0=pi*3/2, n=4) ==
1/sqrt(x - pi*R(3, 2)) + (x - pi*R(3, 2))**R(3, 2)/12 +
(x - pi*R(3, 2))**R(7, 2)/160 + O((x - pi*R(3, 2))**4, (x, pi*R(3, 2))))
def test_X9():
assert (series(x**x, x, x0=0, n=4) == 1 + x*log(x) + x**2*log(x)**2/2 +
x**3*log(x)**3/6 + O(x**4*log(x)**4))
def test_X10():
z, w = symbols('z w')
assert (series(log(sinh(z)) + log(cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
def test_X11():
z, w = symbols('z w')
assert (series(log(sinh(z) * cosh(z + w)), z, x0=0, n=2) ==
log(cosh(w)) + log(z) + z*sinh(w)/cosh(w) + O(z**2))
@XFAIL
def test_X12():
# Look at the generalized Taylor series around x = 1
# Result => (x - 1)^a/e^b [1 - (a + 2 b) (x - 1) / 2 + O((x - 1)^2)]
a, b, x = symbols('a b x', real=True)
# series returns O(log(x-1)**2)
# https://github.com/sympy/sympy/issues/7168
assert (series(log(x)**a*exp(-b*x), x, x0=1, n=2) ==
(x - 1)**a/exp(b)*(1 - (a + 2*b)*(x - 1)/2 + O((x - 1)**2)))
def test_X13():
assert series(sqrt(2*x**2 + 1), x, x0=oo, n=1) == sqrt(2)*x + O(1/x, (x, oo))
@XFAIL
def test_X14():
# Wallis' product => 1/sqrt(pi n) + ... [Knopp, p. 385]
assert series(1/2**(2*n)*binomial(2*n, n),
n, x==oo, n=1) == 1/(sqrt(pi)*sqrt(n)) + O(1/x, (x, oo))
@SKIP("https://github.com/sympy/sympy/issues/7164")
def test_X15():
# => 0!/x - 1!/x^2 + 2!/x^3 - 3!/x^4 + O(1/x^5) [Knopp, p. 544]
x, t = symbols('x t', real=True)
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7164
# 2019-02-17: Raises
# PoleError:
# Asymptotic expansion of Ei around [-oo] is not implemented.
e1 = integrate(exp(-t)/t, (t, x, oo))
assert (series(e1, x, x0=oo, n=5) ==
6/x**4 + 2/x**3 - 1/x**2 + 1/x + O(x**(-5), (x, oo)))
def test_X16():
# Multivariate Taylor series expansion => 1 - (x^2 + 2 x y + y^2)/2 + O(x^4)
assert (series(cos(x + y), x + y, x0=0, n=4) == 1 - (x + y)**2/2 +
O(x**4 + x**3*y + x**2*y**2 + x*y**3 + y**4, x, y))
@XFAIL
def test_X17():
# Power series (compute the general formula)
# (c41) powerseries(log(sin(x)/x), x, 0);
# /aquarius/data2/opt/local/macsyma_422/library1/trgred.so being loaded.
# inf
# ==== i1 2 i1 2 i1
# \ (- 1) 2 bern(2 i1) x
# (d41) > ------------------------------
# / 2 i1 (2 i1)!
# ====
# i1 = 1
# fps does not calculate
assert fps(log(sin(x)/x)) == \
Sum((-1)**k*2**(2*k - 1)*bernoulli(2*k)*x**(2*k)/(k*factorial(2*k)), (k, 1, oo))
@XFAIL
def test_X18():
# Power series (compute the general formula). Maple FPS:
# > FormalPowerSeries(exp(-x)*sin(x), x = 0);
# infinity
# ----- (1/2 k) k
# \ 2 sin(3/4 k Pi) x
# ) -------------------------
# / k!
# -----
#
# Now, SymPy returns
# oo
# _____
# \ `
# \ / k k\
# \ k |I*(-1 - I) I*(-1 + I) |
# \ x *|----------- - -----------|
# / \ 2 2 /
# / ------------------------------
# / k!
# /____,
# k = 0
k = Dummy('k')
assert fps(exp(-x)*sin(x)) == \
Sum(2**(S.Half*k)*sin(R(3, 4)*k*pi)*x**k/factorial(k), (k, 0, oo))
@XFAIL
def test_X19():
# (c45) /* Derive an explicit Taylor series solution of y as a function of
# x from the following implicit relation:
# y = x - 1 + (x - 1)^2/2 + 2/3 (x - 1)^3 + (x - 1)^4 +
# 17/10 (x - 1)^5 + ...
# */
# x = sin(y) + cos(y);
# Time= 0 msecs
# (d45) x = sin(y) + cos(y)
#
# (c46) taylor_revert(%, y, 7);
raise NotImplementedError("Solve using series not supported. \
Inverse Taylor series expansion also not supported")
@XFAIL
def test_X20():
# Pade (rational function) approximation => (2 - x)/(2 + x)
# > numapprox[pade](exp(-x), x = 0, [1, 1]);
# bytes used=9019816, alloc=3669344, time=13.12
# 1 - 1/2 x
# ---------
# 1 + 1/2 x
# mpmath support numeric Pade approximant but there is
# no symbolic implementation in SymPy
# https://en.wikipedia.org/wiki/Pad%C3%A9_approximant
raise NotImplementedError("Symbolic Pade approximant not supported")
def test_X21():
"""
Test whether `fourier_series` of x periodical on the [-p, p] interval equals
`- (2 p / pi) sum( (-1)^n / n sin(n pi x / p), n = 1..infinity )`.
"""
p = symbols('p', positive=True)
n = symbols('n', positive=True, integer=True)
s = fourier_series(x, (x, -p, p))
# All cosine coefficients are equal to 0
assert s.an.formula == 0
# Check for sine coefficients
assert s.bn.formula.subs(s.bn.variables[0], 0) == 0
assert s.bn.formula.subs(s.bn.variables[0], n) == \
-2*p/pi * (-1)**n / n * sin(n*pi*x/p)
@XFAIL
def test_X22():
# (c52) /* => p / 2
# - (2 p / pi^2) sum( [1 - (-1)^n] cos(n pi x / p) / n^2,
# n = 1..infinity ) */
# fourier_series(abs(x), x, p);
# p
# (e52) a = -
# 0 2
#
# %nn
# (2 (- 1) - 2) p
# (e53) a = ------------------
# %nn 2 2
# %pi %nn
#
# (e54) b = 0
# %nn
#
# Time= 5290 msecs
# inf %nn %pi %nn x
# ==== (2 (- 1) - 2) cos(---------)
# \ p
# p > -------------------------------
# / 2
# ==== %nn
# %nn = 1 p
# (d54) ----------------------------------------- + -
# 2 2
# %pi
raise NotImplementedError("Fourier series not supported")
def test_Y1():
t = symbols('t', positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(cos((w - 1)*t), t, s)
assert F == s/(s**2 + (w - 1)**2)
def test_Y2():
t = symbols('t', positive=True)
w = symbols('w', real=True)
s = symbols('s')
f = inverse_laplace_transform(s/(s**2 + (w - 1)**2), s, t)
assert f == cos(t*(w - 1))
def test_Y3():
t = symbols('t', positive=True)
w = symbols('w', real=True)
s = symbols('s')
F, _, _ = laplace_transform(sinh(w*t)*cosh(w*t), t, s)
assert F == w/(s**2 - 4*w**2)
def test_Y4():
t = symbols('t', positive=True)
s = symbols('s')
F, _, _ = laplace_transform(erf(3/sqrt(t)), t, s)
assert F == (1 - exp(-6*sqrt(s)))/s
def test_Y5_Y6():
# Solve y'' + y = 4 [H(t - 1) - H(t - 2)], y(0) = 1, y'(0) = 0 where H is the
# Heaviside (unit step) function (the RHS describes a pulse of magnitude 4 and
# duration 1). See David A. Sanchez, Richard C. Allen, Jr. and Walter T.
# Kyner, _Differential Equations: An Introduction_, Addison-Wesley Publishing
# Company, 1983, p. 211. First, take the Laplace transform of the ODE
# => s^2 Y(s) - s + Y(s) = 4/s [e^(-s) - e^(-2 s)]
# where Y(s) is the Laplace transform of y(t)
t = symbols('t', positive=True)
s = symbols('s')
y = Function('y')
F, _, _ = laplace_transform(diff(y(t), t, 2)
+ y(t)
- 4*(Heaviside(t - 1)
- Heaviside(t - 2)), t, s)
assert (F == s**2*LaplaceTransform(y(t), t, s) - s*y(0) +
LaplaceTransform(y(t), t, s) - Subs(Derivative(y(t), t), t, 0) -
4*exp(-s)/s + 4*exp(-2*s)/s)
# TODO implement second part of test case
# Now, solve for Y(s) and then take the inverse Laplace transform
# => Y(s) = s/(s^2 + 1) + 4 [1/s - s/(s^2 + 1)] [e^(-s) - e^(-2 s)]
# => y(t) = cos t + 4 {[1 - cos(t - 1)] H(t - 1) - [1 - cos(t - 2)] H(t - 2)}
@XFAIL
def test_Y7():
# What is the Laplace transform of an infinite square wave?
# => 1/s + 2 sum( (-1)^n e^(- s n a)/s, n = 1..infinity )
# [Sanchez, Allen and Kyner, p. 213]
t = symbols('t', positive=True)
a = symbols('a', real=True)
s = symbols('s')
F, _, _ = laplace_transform(1 + 2*Sum((-1)**n*Heaviside(t - n*a),
(n, 1, oo)), t, s)
# returns 2*LaplaceTransform(Sum((-1)**n*Heaviside(-a*n + t),
# (n, 1, oo)), t, s) + 1/s
# https://github.com/sympy/sympy/issues/7177
assert F == 2*Sum((-1)**n*exp(-a*n*s)/s, (n, 1, oo)) + 1/s
@XFAIL
def test_Y8():
assert fourier_transform(1, x, z) == DiracDelta(z)
def test_Y9():
assert (fourier_transform(exp(-9*x**2), x, z) ==
sqrt(pi)*exp(-pi**2*z**2/9)/3)
def test_Y10():
assert (fourier_transform(abs(x)*exp(-3*abs(x)), x, z).cancel() ==
(-8*pi**2*z**2 + 18)/(16*pi**4*z**4 + 72*pi**2*z**2 + 81))
@SKIP("https://github.com/sympy/sympy/issues/7181")
@slow
def test_Y11():
# => pi cot(pi s) (0 < Re s < 1) [Gradshteyn and Ryzhik 17.43(5)]
x, s = symbols('x s')
# raises RuntimeError: maximum recursion depth exceeded
# https://github.com/sympy/sympy/issues/7181
# Update 2019-02-17 raises:
# TypeError: cannot unpack non-iterable MellinTransform object
F, _, _ = mellin_transform(1/(1 - x), x, s)
assert F == pi*cot(pi*s)
@XFAIL
def test_Y12():
# => 2^(s - 4) gamma(s/2)/gamma(4 - s/2) (0 < Re s < 1)
# [Gradshteyn and Ryzhik 17.43(16)]
x, s = symbols('x s')
# returns Wrong value -2**(s - 4)*gamma(s/2 - 3)/gamma(-s/2 + 1)
# https://github.com/sympy/sympy/issues/7182
F, _, _ = mellin_transform(besselj(3, x)/x**3, x, s)
assert F == -2**(s - 4)*gamma(s/2)/gamma(-s/2 + 4)
@XFAIL
def test_Y13():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function) z
raise NotImplementedError("z-transform not supported")
@XFAIL
def test_Y14():
# Z[H(t - m T)] => z/[z^m (z - 1)] (H is the Heaviside (unit step) function)
raise NotImplementedError("z-transform not supported")
def test_Z1():
r = Function('r')
assert (rsolve(r(n + 2) - 2*r(n + 1) + r(n) - 2, r(n),
{r(0): 1, r(1): m}).simplify() == n**2 + n*(m - 2) + 1)
def test_Z2():
r = Function('r')
assert (rsolve(r(n) - (5*r(n - 1) - 6*r(n - 2)), r(n), {r(0): 0, r(1): 1})
== -2**n + 3**n)
def test_Z3():
# => r(n) = Fibonacci[n + 1] [Cohen, p. 83]
r = Function('r')
# recurrence solution is correct, Wester expects it to be simplified to
# fibonacci(n+1), but that is quite hard
expected = ((S(1)/2 - sqrt(5)/2)**n*(S(1)/2 - sqrt(5)/10)
+ (S(1)/2 + sqrt(5)/2)**n*(sqrt(5)/10 + S(1)/2))
sol = rsolve(r(n) - (r(n - 1) + r(n - 2)), r(n), {r(1): 1, r(2): 2})
assert sol == expected
@XFAIL
def test_Z4():
# => [c^(n+1) [c^(n+1) - 2 c - 2] + (n+1) c^2 + 2 c - n] / [(c-1)^3 (c+1)]
# [Joan Z. Yu and Robert Israel in sci.math.symbolic]
r = Function('r')
c = symbols('c')
# raises ValueError: Polynomial or rational function expected,
# got '(c**2 - c**n)/(c - c**n)
s = rsolve(r(n) - ((1 + c - c**(n-1) - c**(n+1))/(1 - c**n)*r(n - 1)
- c*(1 - c**(n-2))/(1 - c**(n-1))*r(n - 2) + 1),
r(n), {r(1): 1, r(2): (2 + 2*c + c**2)/(1 + c)})
assert (s - (c*(n + 1)*(c*(n + 1) - 2*c - 2) +
(n + 1)*c**2 + 2*c - n)/((c-1)**3*(c+1)) == 0)
@XFAIL
def test_Z5():
# Second order ODE with initial conditions---solve directly
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
C1, C2 = symbols('C1 C2')
# initial conditions not supported, this is a manual workaround
# https://github.com/sympy/sympy/issues/4720
eq = Derivative(f(x), x, 2) + 4*f(x) - sin(2*x)
sol = dsolve(eq, f(x))
f0 = Lambda(x, sol.rhs)
assert f0(x) == C2*sin(2*x) + (C1 - x/4)*cos(2*x)
f1 = Lambda(x, diff(f0(x), x))
# TODO: Replace solve with solveset, when it works for solveset
const_dict = solve((f0(0), f1(0)))
result = f0(x).subs(C1, const_dict[C1]).subs(C2, const_dict[C2])
assert result == -x*cos(2*x)/4 + sin(2*x)/8
# Result is OK, but ODE solving with initial conditions should be
# supported without all this manual work
raise NotImplementedError('ODE solving with initial conditions \
not supported')
@XFAIL
def test_Z6():
# Second order ODE with initial conditions---solve using Laplace
# transform: f(t) = sin(2 t)/8 - t cos(2 t)/4
t = symbols('t', positive=True)
s = symbols('s')
eq = Derivative(f(t), t, 2) + 4*f(t) - sin(2*t)
F, _, _ = laplace_transform(eq, t, s)
# Laplace transform for diff() not calculated
# https://github.com/sympy/sympy/issues/7176
assert (F == s**2*LaplaceTransform(f(t), t, s) +
4*LaplaceTransform(f(t), t, s) - 2/(s**2 + 4))
# rest of test case not implemented
|
267c376c17e31cca0747ade8e1d18d3ec154104f18cf92a580f812d69a932e9c | from sympy.core import (
S, pi, oo, Symbol, symbols, Rational, Integer, Float, Function, Mod, GoldenRatio, EulerGamma, Catalan,
Lambda, Dummy, nan, Mul, Pow, UnevaluatedExpr
)
from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
from sympy.functions import (
Abs, acos, acosh, asin, asinh, atan, atanh, atan2, ceiling, cos, cosh, erf,
erfc, exp, floor, gamma, log, loggamma, Max, Min, Piecewise, sign, sin, sinh,
sqrt, tan, tanh, fibonacci, lucas
)
from sympy.sets import Range
from sympy.logic import ITE, Implies, Equivalent
from sympy.codegen import For, aug_assign, Assignment
from sympy.testing.pytest import raises, XFAIL
from sympy.printing.c import C89CodePrinter, C99CodePrinter, get_math_macros
from sympy.codegen.ast import (
AddAugmentedAssignment, Element, Type, FloatType, Declaration, Pointer, Variable, value_const, pointer_const,
While, Scope, Print, FunctionPrototype, FunctionDefinition, FunctionCall, Return,
real, float32, float64, float80, float128, intc, Comment, CodeBlock
)
from sympy.codegen.cfunctions import expm1, log1p, exp2, log2, fma, log10, Cbrt, hypot, Sqrt
from sympy.codegen.cnodes import restrict
from sympy.utilities.lambdify import implemented_function
from sympy.tensor import IndexedBase, Idx
from sympy.matrices import Matrix, MatrixSymbol, SparseMatrix
from sympy.printing.codeprinter import ccode
x, y, z = symbols('x,y,z')
def test_printmethod():
class fabs(Abs):
def _ccode(self, printer):
return "fabs(%s)" % printer._print(self.args[0])
assert ccode(fabs(x)) == "fabs(x)"
def test_ccode_sqrt():
assert ccode(sqrt(x)) == "sqrt(x)"
assert ccode(x**0.5) == "sqrt(x)"
assert ccode(sqrt(x)) == "sqrt(x)"
def test_ccode_Pow():
assert ccode(x**3) == "pow(x, 3)"
assert ccode(x**(y**3)) == "pow(x, pow(y, 3))"
g = implemented_function('g', Lambda(x, 2*x))
assert ccode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"pow(3.5*2*x, -x + pow(y, x))/(pow(x, 2) + y)"
assert ccode(x**-1.0) == '1.0/x'
assert ccode(x**Rational(2, 3)) == 'pow(x, 2.0/3.0)'
assert ccode(x**Rational(2, 3), type_aliases={real: float80}) == 'powl(x, 2.0L/3.0L)'
_cond_cfunc = [(lambda base, exp: exp.is_integer, "dpowi"),
(lambda base, exp: not exp.is_integer, "pow")]
assert ccode(x**3, user_functions={'Pow': _cond_cfunc}) == 'dpowi(x, 3)'
assert ccode(x**0.5, user_functions={'Pow': _cond_cfunc}) == 'pow(x, 0.5)'
assert ccode(x**Rational(16, 5), user_functions={'Pow': _cond_cfunc}) == 'pow(x, 16.0/5.0)'
_cond_cfunc2 = [(lambda base, exp: base == 2, lambda base, exp: 'exp2(%s)' % exp),
(lambda base, exp: base != 2, 'pow')]
# Related to gh-11353
assert ccode(2**x, user_functions={'Pow': _cond_cfunc2}) == 'exp2(x)'
assert ccode(x**2, user_functions={'Pow': _cond_cfunc2}) == 'pow(x, 2)'
# For issue 14160
assert ccode(Mul(-2, x, Pow(Mul(y,y,evaluate=False), -1, evaluate=False),
evaluate=False)) == '-2*x/(y*y)'
def test_ccode_Max():
# Test for gh-11926
assert ccode(Max(x,x*x),user_functions={"Max":"my_max", "Pow":"my_pow"}) == 'my_max(x, my_pow(x, 2))'
def test_ccode_Min_performance():
#Shouldn't take more than a few seconds
big_min = Min(*symbols('a[0:50]'))
for curr_standard in ('c89', 'c99', 'c11'):
output = ccode(big_min, standard=curr_standard)
assert output.count('(') == output.count(')')
def test_ccode_constants_mathh():
assert ccode(exp(1)) == "M_E"
assert ccode(pi) == "M_PI"
assert ccode(oo, standard='c89') == "HUGE_VAL"
assert ccode(-oo, standard='c89') == "-HUGE_VAL"
assert ccode(oo) == "INFINITY"
assert ccode(-oo, standard='c99') == "-INFINITY"
assert ccode(pi, type_aliases={real: float80}) == "M_PIl"
def test_ccode_constants_other():
assert ccode(2*GoldenRatio) == "const double GoldenRatio = %s;\n2*GoldenRatio" % GoldenRatio.evalf(17)
assert ccode(
2*Catalan) == "const double Catalan = %s;\n2*Catalan" % Catalan.evalf(17)
assert ccode(2*EulerGamma) == "const double EulerGamma = %s;\n2*EulerGamma" % EulerGamma.evalf(17)
def test_ccode_Rational():
assert ccode(Rational(3, 7)) == "3.0/7.0"
assert ccode(Rational(3, 7), type_aliases={real: float80}) == "3.0L/7.0L"
assert ccode(Rational(18, 9)) == "2"
assert ccode(Rational(3, -7)) == "-3.0/7.0"
assert ccode(Rational(3, -7), type_aliases={real: float80}) == "-3.0L/7.0L"
assert ccode(Rational(-3, -7)) == "3.0/7.0"
assert ccode(Rational(-3, -7), type_aliases={real: float80}) == "3.0L/7.0L"
assert ccode(x + Rational(3, 7)) == "x + 3.0/7.0"
assert ccode(x + Rational(3, 7), type_aliases={real: float80}) == "x + 3.0L/7.0L"
assert ccode(Rational(3, 7)*x) == "(3.0/7.0)*x"
assert ccode(Rational(3, 7)*x, type_aliases={real: float80}) == "(3.0L/7.0L)*x"
def test_ccode_Integer():
assert ccode(Integer(67)) == "67"
assert ccode(Integer(-1)) == "-1"
def test_ccode_functions():
assert ccode(sin(x) ** cos(x)) == "pow(sin(x), cos(x))"
def test_ccode_inline_function():
x = symbols('x')
g = implemented_function('g', Lambda(x, 2*x))
assert ccode(g(x)) == "2*x"
g = implemented_function('g', Lambda(x, 2*x/Catalan))
assert ccode(
g(x)) == "const double Catalan = %s;\n2*x/Catalan" % Catalan.evalf(17)
A = IndexedBase('A')
i = Idx('i', symbols('n', integer=True))
g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x)))
assert ccode(g(A[i]), assign_to=A[i]) == (
"for (int i=0; i<n; i++){\n"
" A[i] = (A[i] + 1)*(A[i] + 2)*A[i];\n"
"}"
)
def test_ccode_exceptions():
assert ccode(gamma(x), standard='C99') == "tgamma(x)"
gamma_c89 = ccode(gamma(x), standard='C89')
assert 'not supported in c' in gamma_c89.lower()
gamma_c89 = ccode(gamma(x), standard='C89', allow_unknown_functions=False)
assert 'not supported in c' in gamma_c89.lower()
gamma_c89 = ccode(gamma(x), standard='C89', allow_unknown_functions=True)
assert 'not supported in c' not in gamma_c89.lower()
def test_ccode_functions2():
assert ccode(ceiling(x)) == "ceil(x)"
assert ccode(Abs(x)) == "fabs(x)"
assert ccode(gamma(x)) == "tgamma(x)"
r, s = symbols('r,s', real=True)
assert ccode(Mod(ceiling(r), ceiling(s))) == '((ceil(r) % ceil(s)) + '\
'ceil(s)) % ceil(s)'
assert ccode(Mod(r, s)) == "fmod(r, s)"
p1, p2 = symbols('p1 p2', integer=True, positive=True)
assert ccode(Mod(p1, p2)) == 'p1 % p2'
assert ccode(Mod(p1, p2 + 3)) == 'p1 % (p2 + 3)'
assert ccode(Mod(-3, -7, evaluate=False)) == '(-3) % (-7)'
assert ccode(-Mod(3, 7, evaluate=False)) == '-(3 % 7)'
assert ccode(r*Mod(p1, p2)) == 'r*(p1 % p2)'
assert ccode(Mod(p1, p2)**s) == 'pow(p1 % p2, s)'
n = symbols('n', integer=True, negative=True)
assert ccode(Mod(-n, p2)) == '(-n) % p2'
assert ccode(fibonacci(n)) == '(1.0/5.0)*pow(2, -n)*sqrt(5)*(-pow(1 - sqrt(5), n) + pow(1 + sqrt(5), n))'
assert ccode(lucas(n)) == 'pow(2, -n)*(pow(1 - sqrt(5), n) + pow(1 + sqrt(5), n))'
def test_ccode_user_functions():
x = symbols('x', integer=False)
n = symbols('n', integer=True)
custom_functions = {
"ceiling": "ceil",
"Abs": [(lambda x: not x.is_integer, "fabs"), (lambda x: x.is_integer, "abs")],
}
assert ccode(ceiling(x), user_functions=custom_functions) == "ceil(x)"
assert ccode(Abs(x), user_functions=custom_functions) == "fabs(x)"
assert ccode(Abs(n), user_functions=custom_functions) == "abs(n)"
expr = Symbol('a')
muladd = Function('muladd')
for i in range(0, 100):
# the large number of terms acts as a regression test for gh-23839
expr = muladd(Rational(1, 2), Symbol(f'a{i}'), expr)
out = ccode(expr, user_functions={'muladd':'muladd'})
assert 'a99' in out
assert out.count('muladd') == 100
def test_ccode_boolean():
assert ccode(True) == "true"
assert ccode(S.true) == "true"
assert ccode(False) == "false"
assert ccode(S.false) == "false"
assert ccode(x & y) == "x && y"
assert ccode(x | y) == "x || y"
assert ccode(~x) == "!x"
assert ccode(x & y & z) == "x && y && z"
assert ccode(x | y | z) == "x || y || z"
assert ccode((x & y) | z) == "z || x && y"
assert ccode((x | y) & z) == "z && (x || y)"
# Automatic rewrites
assert ccode(x ^ y) == '(x || y) && (!x || !y)'
assert ccode((x ^ y) ^ z) == '(x || y || z) && (x || !y || !z) && (y || !x || !z) && (z || !x || !y)'
assert ccode(Implies(x, y)) == 'y || !x'
assert ccode(Equivalent(x, z ^ y, Implies(z, x))) == '(x || (y || !z) && (z || !y)) && (z && !x || (y || z) && (!y || !z))'
def test_ccode_Relational():
assert ccode(Eq(x, y)) == "x == y"
assert ccode(Ne(x, y)) == "x != y"
assert ccode(Le(x, y)) == "x <= y"
assert ccode(Lt(x, y)) == "x < y"
assert ccode(Gt(x, y)) == "x > y"
assert ccode(Ge(x, y)) == "x >= y"
def test_ccode_Piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
assert ccode(expr) == (
"((x < 1) ? (\n"
" x\n"
")\n"
": (\n"
" pow(x, 2)\n"
"))")
assert ccode(expr, assign_to="c") == (
"if (x < 1) {\n"
" c = x;\n"
"}\n"
"else {\n"
" c = pow(x, 2);\n"
"}")
expr = Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True))
assert ccode(expr) == (
"((x < 1) ? (\n"
" x\n"
")\n"
": ((x < 2) ? (\n"
" x + 1\n"
")\n"
": (\n"
" pow(x, 2)\n"
")))")
assert ccode(expr, assign_to='c') == (
"if (x < 1) {\n"
" c = x;\n"
"}\n"
"else if (x < 2) {\n"
" c = x + 1;\n"
"}\n"
"else {\n"
" c = pow(x, 2);\n"
"}")
# Check that Piecewise without a True (default) condition error
expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0))
raises(ValueError, lambda: ccode(expr))
def test_ccode_sinc():
from sympy.functions.elementary.trigonometric import sinc
expr = sinc(x)
assert ccode(expr) == (
"((x != 0) ? (\n"
" sin(x)/x\n"
")\n"
": (\n"
" 1\n"
"))")
def test_ccode_Piecewise_deep():
p = ccode(2*Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True)))
assert p == (
"2*((x < 1) ? (\n"
" x\n"
")\n"
": ((x < 2) ? (\n"
" x + 1\n"
")\n"
": (\n"
" pow(x, 2)\n"
")))")
expr = x*y*z + x**2 + y**2 + Piecewise((0, x < 0.5), (1, True)) + cos(z) - 1
assert ccode(expr) == (
"pow(x, 2) + x*y*z + pow(y, 2) + ((x < 0.5) ? (\n"
" 0\n"
")\n"
": (\n"
" 1\n"
")) + cos(z) - 1")
assert ccode(expr, assign_to='c') == (
"c = pow(x, 2) + x*y*z + pow(y, 2) + ((x < 0.5) ? (\n"
" 0\n"
")\n"
": (\n"
" 1\n"
")) + cos(z) - 1;")
def test_ccode_ITE():
expr = ITE(x < 1, y, z)
assert ccode(expr) == (
"((x < 1) ? (\n"
" y\n"
")\n"
": (\n"
" z\n"
"))")
def test_ccode_settings():
raises(TypeError, lambda: ccode(sin(x), method="garbage"))
def test_ccode_Indexed():
s, n, m, o = symbols('s n m o', integer=True)
i, j, k = Idx('i', n), Idx('j', m), Idx('k', o)
x = IndexedBase('x')[j]
A = IndexedBase('A')[i, j]
B = IndexedBase('B')[i, j, k]
p = C99CodePrinter()
assert p._print_Indexed(x) == 'x[j]'
assert p._print_Indexed(A) == 'A[%s]' % (m*i+j)
assert p._print_Indexed(B) == 'B[%s]' % (i*o*m+j*o+k)
A = IndexedBase('A', shape=(5,3))[i, j]
assert p._print_Indexed(A) == 'A[%s]' % (3*i + j)
A = IndexedBase('A', shape=(5,3), strides='F')[i, j]
assert ccode(A) == 'A[%s]' % (i + 5*j)
A = IndexedBase('A', shape=(29,29), strides=(1, s), offset=o)[i, j]
assert ccode(A) == 'A[o + s*j + i]'
Abase = IndexedBase('A', strides=(s, m, n), offset=o)
assert ccode(Abase[i, j, k]) == 'A[m*j + n*k + o + s*i]'
assert ccode(Abase[2, 3, k]) == 'A[3*m + n*k + o + 2*s]'
def test_Element():
assert ccode(Element('x', 'ij')) == 'x[i][j]'
assert ccode(Element('x', 'ij', strides='kl', offset='o')) == 'x[i*k + j*l + o]'
assert ccode(Element('x', (3,))) == 'x[3]'
assert ccode(Element('x', (3,4,5))) == 'x[3][4][5]'
def test_ccode_Indexed_without_looking_for_contraction():
len_y = 5
y = IndexedBase('y', shape=(len_y,))
x = IndexedBase('x', shape=(len_y,))
Dy = IndexedBase('Dy', shape=(len_y-1,))
i = Idx('i', len_y-1)
e = Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i]))
code0 = ccode(e.rhs, assign_to=e.lhs, contract=False)
assert code0 == 'Dy[i] = (y[%s] - y[i])/(x[%s] - x[i]);' % (i + 1, i + 1)
def test_ccode_loops_matrix_vector():
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = A[%s]*x[j] + y[i];\n' % (i*n + j) +\
' }\n'
'}'
)
assert ccode(A[i, j]*x[j], assign_to=y[i]) == s
def test_dummy_loops():
i, m = symbols('i m', integer=True, cls=Dummy)
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx(i, m)
expected = (
'for (int i_%(icount)i=0; i_%(icount)i<m_%(mcount)i; i_%(icount)i++){\n'
' y[i_%(icount)i] = x[i_%(icount)i];\n'
'}'
) % {'icount': i.label.dummy_index, 'mcount': m.dummy_index}
assert ccode(x[i], assign_to=y[i]) == expected
def test_ccode_loops_add():
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
z = IndexedBase('z')
i = Idx('i', m)
j = Idx('j', n)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = x[i] + z[i];\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = A[%s]*x[j] + y[i];\n' % (i*n + j) +\
' }\n'
'}'
)
assert ccode(A[i, j]*x[j] + x[i] + z[i], assign_to=y[i]) == s
def test_ccode_loops_multiple_contractions():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' for (int l=0; l<p; l++){\n'
' y[i] = a[%s]*b[%s] + y[i];\n' % (i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +\
' }\n'
' }\n'
' }\n'
'}'
)
assert ccode(b[j, k, l]*a[i, j, k, l], assign_to=y[i]) == s
def test_ccode_loops_addfactor():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' for (int l=0; l<p; l++){\n'
' y[i] = (a[%s] + b[%s])*c[%s] + y[i];\n' % (i*n*o*p + j*o*p + k*p + l, i*n*o*p + j*o*p + k*p + l, j*o*p + k*p + l) +\
' }\n'
' }\n'
' }\n'
'}'
)
assert ccode((a[i, j, k, l] + b[i, j, k, l])*c[j, k, l], assign_to=y[i]) == s
def test_ccode_loops_multiple_terms():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
s0 = (
'for (int i=0; i<m; i++){\n'
' y[i] = 0;\n'
'}\n'
)
s1 = (
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' for (int k=0; k<o; k++){\n'
' y[i] = b[j]*b[k]*c[%s] + y[i];\n' % (i*n*o + j*o + k) +\
' }\n'
' }\n'
'}\n'
)
s2 = (
'for (int i=0; i<m; i++){\n'
' for (int k=0; k<o; k++){\n'
' y[i] = a[%s]*b[k] + y[i];\n' % (i*o + k) +\
' }\n'
'}\n'
)
s3 = (
'for (int i=0; i<m; i++){\n'
' for (int j=0; j<n; j++){\n'
' y[i] = a[%s]*b[j] + y[i];\n' % (i*n + j) +\
' }\n'
'}\n'
)
c = ccode(b[j]*a[i, j] + b[k]*a[i, k] + b[j]*b[k]*c[i, j, k], assign_to=y[i])
assert (c == s0 + s1 + s2 + s3[:-1] or
c == s0 + s1 + s3 + s2[:-1] or
c == s0 + s2 + s1 + s3[:-1] or
c == s0 + s2 + s3 + s1[:-1] or
c == s0 + s3 + s1 + s2[:-1] or
c == s0 + s3 + s2 + s1[:-1])
def test_dereference_printing():
expr = x + y + sin(z) + z
assert ccode(expr, dereference=[z]) == "x + y + (*z) + sin((*z))"
def test_Matrix_printing():
# Test returning a Matrix
mat = Matrix([x*y, Piecewise((2 + x, y>0), (y, True)), sin(z)])
A = MatrixSymbol('A', 3, 1)
assert ccode(mat, A) == (
"A[0] = x*y;\n"
"if (y > 0) {\n"
" A[1] = x + 2;\n"
"}\n"
"else {\n"
" A[1] = y;\n"
"}\n"
"A[2] = sin(z);")
# Test using MatrixElements in expressions
expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0]
assert ccode(expr) == (
"((x > 0) ? (\n"
" 2*A[2]\n"
")\n"
": (\n"
" A[2]\n"
")) + sin(A[1]) + A[0]")
# Test using MatrixElements in a Matrix
q = MatrixSymbol('q', 5, 1)
M = MatrixSymbol('M', 3, 3)
m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])],
[q[1,0] + q[2,0], q[3, 0], 5],
[2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]])
assert ccode(m, M) == (
"M[0] = sin(q[1]);\n"
"M[1] = 0;\n"
"M[2] = cos(q[2]);\n"
"M[3] = q[1] + q[2];\n"
"M[4] = q[3];\n"
"M[5] = 5;\n"
"M[6] = 2*q[4]/q[1];\n"
"M[7] = sqrt(q[0]) + 4;\n"
"M[8] = 0;")
def test_sparse_matrix():
# gh-15791
assert 'Not supported in C' in ccode(SparseMatrix([[1, 2, 3]]))
def test_ccode_reserved_words():
x, y = symbols('x, if')
with raises(ValueError):
ccode(y**2, error_on_reserved=True, standard='C99')
assert ccode(y**2) == 'pow(if_, 2)'
assert ccode(x * y**2, dereference=[y]) == 'pow((*if_), 2)*x'
assert ccode(y**2, reserved_word_suffix='_unreserved') == 'pow(if_unreserved, 2)'
def test_ccode_sign():
expr1, ref1 = sign(x) * y, 'y*(((x) > 0) - ((x) < 0))'
expr2, ref2 = sign(cos(x)), '(((cos(x)) > 0) - ((cos(x)) < 0))'
expr3, ref3 = sign(2 * x + x**2) * x + x**2, 'pow(x, 2) + x*(((pow(x, 2) + 2*x) > 0) - ((pow(x, 2) + 2*x) < 0))'
assert ccode(expr1) == ref1
assert ccode(expr1, 'z') == 'z = %s;' % ref1
assert ccode(expr2) == ref2
assert ccode(expr3) == ref3
def test_ccode_Assignment():
assert ccode(Assignment(x, y + z)) == 'x = y + z;'
assert ccode(aug_assign(x, '+', y + z)) == 'x += y + z;'
def test_ccode_For():
f = For(x, Range(0, 10, 2), [aug_assign(y, '*', x)])
assert ccode(f) == ("for (x = 0; x < 10; x += 2) {\n"
" y *= x;\n"
"}")
def test_ccode_Max_Min():
assert ccode(Max(x, 0), standard='C89') == '((0 > x) ? 0 : x)'
assert ccode(Max(x, 0), standard='C99') == 'fmax(0, x)'
assert ccode(Min(x, 0, sqrt(x)), standard='c89') == (
'((0 < ((x < sqrt(x)) ? x : sqrt(x))) ? 0 : ((x < sqrt(x)) ? x : sqrt(x)))'
)
def test_ccode_standard():
assert ccode(expm1(x), standard='c99') == 'expm1(x)'
assert ccode(nan, standard='c99') == 'NAN'
assert ccode(float('nan'), standard='c99') == 'NAN'
def test_C89CodePrinter():
c89printer = C89CodePrinter()
assert c89printer.language == 'C'
assert c89printer.standard == 'C89'
assert 'void' in c89printer.reserved_words
assert 'template' not in c89printer.reserved_words
def test_C99CodePrinter():
assert C99CodePrinter().doprint(expm1(x)) == 'expm1(x)'
assert C99CodePrinter().doprint(log1p(x)) == 'log1p(x)'
assert C99CodePrinter().doprint(exp2(x)) == 'exp2(x)'
assert C99CodePrinter().doprint(log2(x)) == 'log2(x)'
assert C99CodePrinter().doprint(fma(x, y, -z)) == 'fma(x, y, -z)'
assert C99CodePrinter().doprint(log10(x)) == 'log10(x)'
assert C99CodePrinter().doprint(Cbrt(x)) == 'cbrt(x)' # note Cbrt due to cbrt already taken.
assert C99CodePrinter().doprint(hypot(x, y)) == 'hypot(x, y)'
assert C99CodePrinter().doprint(loggamma(x)) == 'lgamma(x)'
assert C99CodePrinter().doprint(Max(x, 3, x**2)) == 'fmax(3, fmax(x, pow(x, 2)))'
assert C99CodePrinter().doprint(Min(x, 3)) == 'fmin(3, x)'
c99printer = C99CodePrinter()
assert c99printer.language == 'C'
assert c99printer.standard == 'C99'
assert 'restrict' in c99printer.reserved_words
assert 'using' not in c99printer.reserved_words
@XFAIL
def test_C99CodePrinter__precision_f80():
f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
assert f80_printer.doprint(sin(x+Float('2.1'))) == 'sinl(x + 2.1L)'
def test_C99CodePrinter__precision():
n = symbols('n', integer=True)
p = symbols('p', integer=True, positive=True)
f32_printer = C99CodePrinter(dict(type_aliases={real: float32}))
f64_printer = C99CodePrinter(dict(type_aliases={real: float64}))
f80_printer = C99CodePrinter(dict(type_aliases={real: float80}))
assert f32_printer.doprint(sin(x+2.1)) == 'sinf(x + 2.1F)'
assert f64_printer.doprint(sin(x+2.1)) == 'sin(x + 2.1000000000000001)'
assert f80_printer.doprint(sin(x+Float('2.0'))) == 'sinl(x + 2.0L)'
for printer, suffix in zip([f32_printer, f64_printer, f80_printer], ['f', '', 'l']):
def check(expr, ref):
assert printer.doprint(expr) == ref.format(s=suffix, S=suffix.upper())
check(Abs(n), 'abs(n)')
check(Abs(x + 2.0), 'fabs{s}(x + 2.0{S})')
check(sin(x + 4.0)**cos(x - 2.0), 'pow{s}(sin{s}(x + 4.0{S}), cos{s}(x - 2.0{S}))')
check(exp(x*8.0), 'exp{s}(8.0{S}*x)')
check(exp2(x), 'exp2{s}(x)')
check(expm1(x*4.0), 'expm1{s}(4.0{S}*x)')
check(Mod(p, 2), 'p % 2')
check(Mod(2*p + 3, 3*p + 5, evaluate=False), '(2*p + 3) % (3*p + 5)')
check(Mod(x + 2.0, 3.0), 'fmod{s}(1.0{S}*x + 2.0{S}, 3.0{S})')
check(Mod(x, 2.0*x + 3.0), 'fmod{s}(1.0{S}*x, 2.0{S}*x + 3.0{S})')
check(log(x/2), 'log{s}((1.0{S}/2.0{S})*x)')
check(log10(3*x/2), 'log10{s}((3.0{S}/2.0{S})*x)')
check(log2(x*8.0), 'log2{s}(8.0{S}*x)')
check(log1p(x), 'log1p{s}(x)')
check(2**x, 'pow{s}(2, x)')
check(2.0**x, 'pow{s}(2.0{S}, x)')
check(x**3, 'pow{s}(x, 3)')
check(x**4.0, 'pow{s}(x, 4.0{S})')
check(sqrt(3+x), 'sqrt{s}(x + 3)')
check(Cbrt(x-2.0), 'cbrt{s}(x - 2.0{S})')
check(hypot(x, y), 'hypot{s}(x, y)')
check(sin(3.*x + 2.), 'sin{s}(3.0{S}*x + 2.0{S})')
check(cos(3.*x - 1.), 'cos{s}(3.0{S}*x - 1.0{S})')
check(tan(4.*y + 2.), 'tan{s}(4.0{S}*y + 2.0{S})')
check(asin(3.*x + 2.), 'asin{s}(3.0{S}*x + 2.0{S})')
check(acos(3.*x + 2.), 'acos{s}(3.0{S}*x + 2.0{S})')
check(atan(3.*x + 2.), 'atan{s}(3.0{S}*x + 2.0{S})')
check(atan2(3.*x, 2.*y), 'atan2{s}(3.0{S}*x, 2.0{S}*y)')
check(sinh(3.*x + 2.), 'sinh{s}(3.0{S}*x + 2.0{S})')
check(cosh(3.*x - 1.), 'cosh{s}(3.0{S}*x - 1.0{S})')
check(tanh(4.0*y + 2.), 'tanh{s}(4.0{S}*y + 2.0{S})')
check(asinh(3.*x + 2.), 'asinh{s}(3.0{S}*x + 2.0{S})')
check(acosh(3.*x + 2.), 'acosh{s}(3.0{S}*x + 2.0{S})')
check(atanh(3.*x + 2.), 'atanh{s}(3.0{S}*x + 2.0{S})')
check(erf(42.*x), 'erf{s}(42.0{S}*x)')
check(erfc(42.*x), 'erfc{s}(42.0{S}*x)')
check(gamma(x), 'tgamma{s}(x)')
check(loggamma(x), 'lgamma{s}(x)')
check(ceiling(x + 2.), "ceil{s}(x + 2.0{S})")
check(floor(x + 2.), "floor{s}(x + 2.0{S})")
check(fma(x, y, -z), 'fma{s}(x, y, -z)')
check(Max(x, 8.0, x**4.0), 'fmax{s}(8.0{S}, fmax{s}(x, pow{s}(x, 4.0{S})))')
check(Min(x, 2.0), 'fmin{s}(2.0{S}, x)')
def test_get_math_macros():
macros = get_math_macros()
assert macros[exp(1)] == 'M_E'
assert macros[1/Sqrt(2)] == 'M_SQRT1_2'
def test_ccode_Declaration():
i = symbols('i', integer=True)
var1 = Variable(i, type=Type.from_expr(i))
dcl1 = Declaration(var1)
assert ccode(dcl1) == 'int i'
var2 = Variable(x, type=float32, attrs={value_const})
dcl2a = Declaration(var2)
assert ccode(dcl2a) == 'const float x'
dcl2b = var2.as_Declaration(value=pi)
assert ccode(dcl2b) == 'const float x = M_PI'
var3 = Variable(y, type=Type('bool'))
dcl3 = Declaration(var3)
printer = C89CodePrinter()
assert 'stdbool.h' not in printer.headers
assert printer.doprint(dcl3) == 'bool y'
assert 'stdbool.h' in printer.headers
u = symbols('u', real=True)
ptr4 = Pointer.deduced(u, attrs={pointer_const, restrict})
dcl4 = Declaration(ptr4)
assert ccode(dcl4) == 'double * const restrict u'
var5 = Variable(x, Type('__float128'), attrs={value_const})
dcl5a = Declaration(var5)
assert ccode(dcl5a) == 'const __float128 x'
var5b = Variable(var5.symbol, var5.type, pi, attrs=var5.attrs)
dcl5b = Declaration(var5b)
assert ccode(dcl5b) == 'const __float128 x = M_PI'
def test_C99CodePrinter_custom_type():
# We will look at __float128 (new in glibc 2.26)
f128 = FloatType('_Float128', float128.nbits, float128.nmant, float128.nexp)
p128 = C99CodePrinter(dict(
type_aliases={real: f128},
type_literal_suffixes={f128: 'Q'},
type_func_suffixes={f128: 'f128'},
type_math_macro_suffixes={
real: 'f128',
f128: 'f128'
},
type_macros={
f128: ('__STDC_WANT_IEC_60559_TYPES_EXT__',)
}
))
assert p128.doprint(x) == 'x'
assert not p128.headers
assert not p128.libraries
assert not p128.macros
assert p128.doprint(2.0) == '2.0Q'
assert not p128.headers
assert not p128.libraries
assert p128.macros == {'__STDC_WANT_IEC_60559_TYPES_EXT__'}
assert p128.doprint(Rational(1, 2)) == '1.0Q/2.0Q'
assert p128.doprint(sin(x)) == 'sinf128(x)'
assert p128.doprint(cos(2., evaluate=False)) == 'cosf128(2.0Q)'
assert p128.doprint(x**-1.0) == '1.0Q/x'
var5 = Variable(x, f128, attrs={value_const})
dcl5a = Declaration(var5)
assert ccode(dcl5a) == 'const _Float128 x'
var5b = Variable(x, f128, pi, attrs={value_const})
dcl5b = Declaration(var5b)
assert p128.doprint(dcl5b) == 'const _Float128 x = M_PIf128'
var5b = Variable(x, f128, value=Catalan.evalf(38), attrs={value_const})
dcl5c = Declaration(var5b)
assert p128.doprint(dcl5c) == 'const _Float128 x = %sQ' % Catalan.evalf(f128.decimal_dig)
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert(ccode(A[0, 0]) == "A[0]")
assert(ccode(3 * A[0, 0]) == "3*A[0]")
F = C[0, 0].subs(C, A - B)
assert(ccode(F) == "(A - B)[0]")
def test_ccode_math_macros():
assert ccode(z + exp(1)) == 'z + M_E'
assert ccode(z + log2(exp(1))) == 'z + M_LOG2E'
assert ccode(z + 1/log(2)) == 'z + M_LOG2E'
assert ccode(z + log(2)) == 'z + M_LN2'
assert ccode(z + log(10)) == 'z + M_LN10'
assert ccode(z + pi) == 'z + M_PI'
assert ccode(z + pi/2) == 'z + M_PI_2'
assert ccode(z + pi/4) == 'z + M_PI_4'
assert ccode(z + 1/pi) == 'z + M_1_PI'
assert ccode(z + 2/pi) == 'z + M_2_PI'
assert ccode(z + 2/sqrt(pi)) == 'z + M_2_SQRTPI'
assert ccode(z + 2/Sqrt(pi)) == 'z + M_2_SQRTPI'
assert ccode(z + sqrt(2)) == 'z + M_SQRT2'
assert ccode(z + Sqrt(2)) == 'z + M_SQRT2'
assert ccode(z + 1/sqrt(2)) == 'z + M_SQRT1_2'
assert ccode(z + 1/Sqrt(2)) == 'z + M_SQRT1_2'
def test_ccode_Type():
assert ccode(Type('float')) == 'float'
assert ccode(intc) == 'int'
def test_ccode_codegen_ast():
# Note that C only allows comments of the form /* ... */, double forward
# slash is not standard C, and some C compilers will grind to a halt upon
# encountering them.
assert ccode(Comment("this is a comment")) == "/* this is a comment */" # not //
assert ccode(While(abs(x) > 1, [aug_assign(x, '-', 1)])) == (
'while (fabs(x) > 1) {\n'
' x -= 1;\n'
'}'
)
assert ccode(Scope([AddAugmentedAssignment(x, 1)])) == (
'{\n'
' x += 1;\n'
'}'
)
inp_x = Declaration(Variable(x, type=real))
assert ccode(FunctionPrototype(real, 'pwer', [inp_x])) == 'double pwer(double x)'
assert ccode(FunctionDefinition(real, 'pwer', [inp_x], [Assignment(x, x**2)])) == (
'double pwer(double x){\n'
' x = pow(x, 2);\n'
'}'
)
# Elements of CodeBlock are formatted as statements:
block = CodeBlock(
x,
Print([x, y], "%d %d"),
FunctionCall('pwer', [x]),
Return(x),
)
assert ccode(block) == '\n'.join([
'x;',
'printf("%d %d", x, y);',
'pwer(x);',
'return x;',
])
def test_ccode_UnevaluatedExpr():
assert ccode(UnevaluatedExpr(y * x) + z) == "z + x*y"
assert ccode(UnevaluatedExpr(y + x) + z) == "z + (x + y)" # gh-21955
w = symbols('w')
assert ccode(UnevaluatedExpr(y + x) + UnevaluatedExpr(z + w)) == "(w + z) + (x + y)"
p, q, r = symbols("p q r", real=True)
q_r = UnevaluatedExpr(q + r)
expr = abs(exp(p+q_r))
assert ccode(expr) == "exp(p + (q + r))"
def test_ccode_array_like_containers():
assert ccode([2,3,4]) == "{2, 3, 4}"
assert ccode((2,3,4)) == "{2, 3, 4}"
|
55b04eb1cd00534b3840c8c0fd4fea519442532a76b56430ec4cce03c945a265 | from sympy import MatAdd, MatMul, Array
from sympy.algebras.quaternion import Quaternion
from sympy.calculus.accumulationbounds import AccumBounds
from sympy.combinatorics.permutations import Cycle, Permutation, AppliedPermutation
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.containers import Tuple, Dict
from sympy.core.expr import UnevaluatedExpr
from sympy.core.function import (Derivative, Function, Lambda, Subs, diff)
from sympy.core.mod import Mod
from sympy.core.mul import Mul
from sympy.core.numbers import (AlgebraicNumber, Float, I, Integer, Rational, oo, pi)
from sympy.core.power import Pow
from sympy.core.relational import Eq, Ne
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, Wild, symbols)
from sympy.functions.combinatorial.factorials import (FallingFactorial, RisingFactorial, binomial, factorial, factorial2, subfactorial)
from sympy.functions.combinatorial.numbers import bernoulli, bell, catalan, euler, lucas, fibonacci, tribonacci
from sympy.functions.elementary.complexes import (Abs, arg, conjugate, im, polar_lift, re)
from sympy.functions.elementary.exponential import (LambertW, exp, log)
from sympy.functions.elementary.hyperbolic import (asinh, coth)
from sympy.functions.elementary.integers import (ceiling, floor, frac)
from sympy.functions.elementary.miscellaneous import (Max, Min, root, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acsc, asin, cos, cot, sin, tan)
from sympy.functions.special.beta_functions import beta
from sympy.functions.special.delta_functions import (DiracDelta, Heaviside)
from sympy.functions.special.elliptic_integrals import (elliptic_e, elliptic_f, elliptic_k, elliptic_pi)
from sympy.functions.special.error_functions import (Chi, Ci, Ei, Shi, Si, expint)
from sympy.functions.special.gamma_functions import (gamma, uppergamma)
from sympy.functions.special.hyper import (hyper, meijerg)
from sympy.functions.special.mathieu_functions import (mathieuc, mathieucprime, mathieus, mathieusprime)
from sympy.functions.special.polynomials import (assoc_laguerre, assoc_legendre, chebyshevt, chebyshevu, gegenbauer, hermite, jacobi, laguerre, legendre)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.spherical_harmonics import (Ynm, Znm)
from sympy.functions.special.tensor_functions import (KroneckerDelta, LeviCivita)
from sympy.functions.special.zeta_functions import (dirichlet_eta, lerchphi, polylog, stieltjes, zeta)
from sympy.integrals.integrals import Integral
from sympy.integrals.transforms import (CosineTransform, FourierTransform, InverseCosineTransform, InverseFourierTransform, InverseLaplaceTransform, InverseMellinTransform, InverseSineTransform, LaplaceTransform, MellinTransform, SineTransform)
from sympy.logic import Implies
from sympy.logic.boolalg import (And, Or, Xor, Equivalent, false, Not, true)
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.kronecker import KroneckerProduct
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.permutation import PermutationMatrix
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.physics.control.lti import TransferFunction, Series, Parallel, Feedback, TransferFunctionMatrix, MIMOSeries, MIMOParallel, MIMOFeedback
from sympy.ntheory.factor_ import (divisor_sigma, primenu, primeomega, reduced_totient, totient, udivisor_sigma)
from sympy.physics.quantum import Commutator, Operator
from sympy.physics.quantum.trace import Tr
from sympy.physics.units import meter, gibibyte, gram, microgram, second, milli, micro
from sympy.polys.domains.integerring import ZZ
from sympy.polys.fields import field
from sympy.polys.polytools import Poly
from sympy.polys.rings import ring
from sympy.polys.rootoftools import (RootSum, rootof)
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
from sympy.series.limits import Limit
from sympy.series.order import Order
from sympy.series.sequences import (SeqAdd, SeqFormula, SeqMul, SeqPer)
from sympy.sets.conditionset import ConditionSet
from sympy.sets.contains import Contains
from sympy.sets.fancysets import (ComplexRegion, ImageSet, Range)
from sympy.sets.ordinals import Ordinal, OrdinalOmega, OmegaPower
from sympy.sets.powerset import PowerSet
from sympy.sets.sets import (FiniteSet, Interval, Union, Intersection, Complement, SymmetricDifference, ProductSet)
from sympy.sets.setexpr import SetExpr
from sympy.stats.crv_types import Normal
from sympy.stats.symbolic_probability import (Covariance, Expectation,
Probability, Variance)
from sympy.tensor.array import (ImmutableDenseNDimArray,
ImmutableSparseNDimArray,
MutableSparseNDimArray,
MutableDenseNDimArray,
tensorproduct)
from sympy.tensor.array.expressions.array_expressions import ArraySymbol, ArrayElement
from sympy.tensor.indexed import (Idx, Indexed, IndexedBase)
from sympy.tensor.toperators import PartialDerivative
from sympy.vector import CoordSys3D, Cross, Curl, Dot, Divergence, Gradient, Laplacian
from sympy.testing.pytest import (XFAIL, raises, _both_exp_pow,
warns_deprecated_sympy)
from sympy.printing.latex import (latex, translate, greek_letters_set,
tex_greek_dictionary, multiline_latex,
latex_escape, LatexPrinter)
import sympy as sym
from sympy.abc import mu, tau
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
x, y, z, t, w, a, b, c, s, p = symbols('x y z t w a b c s p')
k, m, n = symbols('k m n', integer=True)
def test_printmethod():
class R(Abs):
def _latex(self, printer):
return "foo(%s)" % printer._print(self.args[0])
assert latex(R(x)) == r"foo(x)"
class R(Abs):
def _latex(self, printer):
return "foo"
assert latex(R(x)) == r"foo"
def test_latex_basic():
assert latex(1 + x) == r"x + 1"
assert latex(x**2) == r"x^{2}"
assert latex(x**(1 + x)) == r"x^{x + 1}"
assert latex(x**3 + x + 1 + x**2) == r"x^{3} + x^{2} + x + 1"
assert latex(2*x*y) == r"2 x y"
assert latex(2*x*y, mul_symbol='dot') == r"2 \cdot x \cdot y"
assert latex(3*x**2*y, mul_symbol='\\,') == r"3\,x^{2}\,y"
assert latex(1.5*3**x, mul_symbol='\\,') == r"1.5 \cdot 3^{x}"
assert latex(x**S.Half**5) == r"\sqrt[32]{x}"
assert latex(Mul(S.Half, x**2, -5, evaluate=False)) == r"\frac{1}{2} x^{2} \left(-5\right)"
assert latex(Mul(S.Half, x**2, 5, evaluate=False)) == r"\frac{1}{2} x^{2} \cdot 5"
assert latex(Mul(-5, -5, evaluate=False)) == r"\left(-5\right) \left(-5\right)"
assert latex(Mul(5, -5, evaluate=False)) == r"5 \left(-5\right)"
assert latex(Mul(S.Half, -5, S.Half, evaluate=False)) == r"\frac{1}{2} \left(-5\right) \frac{1}{2}"
assert latex(Mul(5, I, 5, evaluate=False)) == r"5 i 5"
assert latex(Mul(5, I, -5, evaluate=False)) == r"5 i \left(-5\right)"
assert latex(Mul(0, 1, evaluate=False)) == r'0 \cdot 1'
assert latex(Mul(1, 0, evaluate=False)) == r'1 \cdot 0'
assert latex(Mul(1, 1, evaluate=False)) == r'1 \cdot 1'
assert latex(Mul(-1, 1, evaluate=False)) == r'\left(-1\right) 1'
assert latex(Mul(1, 1, 1, evaluate=False)) == r'1 \cdot 1 \cdot 1'
assert latex(Mul(1, 2, evaluate=False)) == r'1 \cdot 2'
assert latex(Mul(1, S.Half, evaluate=False)) == r'1 \cdot \frac{1}{2}'
assert latex(Mul(1, 1, S.Half, evaluate=False)) == \
r'1 \cdot 1 \cdot \frac{1}{2}'
assert latex(Mul(1, 1, 2, 3, x, evaluate=False)) == \
r'1 \cdot 1 \cdot 2 \cdot 3 x'
assert latex(Mul(1, -1, evaluate=False)) == r'1 \left(-1\right)'
assert latex(Mul(4, 3, 2, 1, 0, y, x, evaluate=False)) == \
r'4 \cdot 3 \cdot 2 \cdot 1 \cdot 0 y x'
assert latex(Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)) == \
r'4 \cdot 3 \cdot 2 \left(z + 1\right) 0 y x'
assert latex(Mul(Rational(2, 3), Rational(5, 7), evaluate=False)) == \
r'\frac{2}{3} \cdot \frac{5}{7}'
assert latex(1/x) == r"\frac{1}{x}"
assert latex(1/x, fold_short_frac=True) == r"1 / x"
assert latex(-S(3)/2) == r"- \frac{3}{2}"
assert latex(-S(3)/2, fold_short_frac=True) == r"- 3 / 2"
assert latex(1/x**2) == r"\frac{1}{x^{2}}"
assert latex(1/(x + y)/2) == r"\frac{1}{2 \left(x + y\right)}"
assert latex(x/2) == r"\frac{x}{2}"
assert latex(x/2, fold_short_frac=True) == r"x / 2"
assert latex((x + y)/(2*x)) == r"\frac{x + y}{2 x}"
assert latex((x + y)/(2*x), fold_short_frac=True) == \
r"\left(x + y\right) / 2 x"
assert latex((x + y)/(2*x), long_frac_ratio=0) == \
r"\frac{1}{2 x} \left(x + y\right)"
assert latex((x + y)/x) == r"\frac{x + y}{x}"
assert latex((x + y)/x, long_frac_ratio=3) == r"\frac{x + y}{x}"
assert latex((2*sqrt(2)*x)/3) == r"\frac{2 \sqrt{2} x}{3}"
assert latex((2*sqrt(2)*x)/3, long_frac_ratio=2) == \
r"\frac{2 x}{3} \sqrt{2}"
assert latex(binomial(x, y)) == r"{\binom{x}{y}}"
x_star = Symbol('x^*')
f = Function('f')
assert latex(x_star**2) == r"\left(x^{*}\right)^{2}"
assert latex(x_star**2, parenthesize_super=False) == r"{x^{*}}^{2}"
assert latex(Derivative(f(x_star), x_star,2)) == r"\frac{d^{2}}{d \left(x^{*}\right)^{2}} f{\left(x^{*} \right)}"
assert latex(Derivative(f(x_star), x_star,2), parenthesize_super=False) == r"\frac{d^{2}}{d {x^{*}}^{2}} f{\left(x^{*} \right)}"
assert latex(2*Integral(x, x)/3) == r"\frac{2 \int x\, dx}{3}"
assert latex(2*Integral(x, x)/3, fold_short_frac=True) == \
r"\left(2 \int x\, dx\right) / 3"
assert latex(sqrt(x)) == r"\sqrt{x}"
assert latex(x**Rational(1, 3)) == r"\sqrt[3]{x}"
assert latex(x**Rational(1, 3), root_notation=False) == r"x^{\frac{1}{3}}"
assert latex(sqrt(x)**3) == r"x^{\frac{3}{2}}"
assert latex(sqrt(x), itex=True) == r"\sqrt{x}"
assert latex(x**Rational(1, 3), itex=True) == r"\root{3}{x}"
assert latex(sqrt(x)**3, itex=True) == r"x^{\frac{3}{2}}"
assert latex(x**Rational(3, 4)) == r"x^{\frac{3}{4}}"
assert latex(x**Rational(3, 4), fold_frac_powers=True) == r"x^{3/4}"
assert latex((x + 1)**Rational(3, 4)) == \
r"\left(x + 1\right)^{\frac{3}{4}}"
assert latex((x + 1)**Rational(3, 4), fold_frac_powers=True) == \
r"\left(x + 1\right)^{3/4}"
assert latex(AlgebraicNumber(sqrt(2))) == r"\sqrt{2}"
assert latex(AlgebraicNumber(sqrt(2), [3, -7])) == r"-7 + 3 \sqrt{2}"
assert latex(AlgebraicNumber(sqrt(2), alias='alpha')) == r"\alpha"
assert latex(AlgebraicNumber(sqrt(2), [3, -7], alias='alpha')) == \
r"3 \alpha - 7"
assert latex(AlgebraicNumber(2**(S(1)/3), [1, 3, -7], alias='beta')) == \
r"\beta^{2} + 3 \beta - 7"
k = ZZ.cyclotomic_field(5)
assert latex(k.ext.field_element([1, 2, 3, 4])) == \
r"\zeta^{3} + 2 \zeta^{2} + 3 \zeta + 4"
assert latex(k.ext.field_element([1, 2, 3, 4]), order='old') == \
r"4 + 3 \zeta + 2 \zeta^{2} + \zeta^{3}"
assert latex(k.primes_above(19)[0]) == \
r"\left(19, \zeta^{2} + 5 \zeta + 1\right)"
assert latex(k.primes_above(19)[0], order='old') == \
r"\left(19, 1 + 5 \zeta + \zeta^{2}\right)"
assert latex(k.primes_above(7)[0]) == r"\left(7\right)"
assert latex(1.5e20*x) == r"1.5 \cdot 10^{20} x"
assert latex(1.5e20*x, mul_symbol='dot') == r"1.5 \cdot 10^{20} \cdot x"
assert latex(1.5e20*x, mul_symbol='times') == \
r"1.5 \times 10^{20} \times x"
assert latex(1/sin(x)) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**-1) == r"\frac{1}{\sin{\left(x \right)}}"
assert latex(sin(x)**Rational(3, 2)) == \
r"\sin^{\frac{3}{2}}{\left(x \right)}"
assert latex(sin(x)**Rational(3, 2), fold_frac_powers=True) == \
r"\sin^{3/2}{\left(x \right)}"
assert latex(~x) == r"\neg x"
assert latex(x & y) == r"x \wedge y"
assert latex(x & y & z) == r"x \wedge y \wedge z"
assert latex(x | y) == r"x \vee y"
assert latex(x | y | z) == r"x \vee y \vee z"
assert latex((x & y) | z) == r"z \vee \left(x \wedge y\right)"
assert latex(Implies(x, y)) == r"x \Rightarrow y"
assert latex(~(x >> ~y)) == r"x \not\Rightarrow \neg y"
assert latex(Implies(Or(x,y), z)) == r"\left(x \vee y\right) \Rightarrow z"
assert latex(Implies(z, Or(x,y))) == r"z \Rightarrow \left(x \vee y\right)"
assert latex(~(x & y)) == r"\neg \left(x \wedge y\right)"
assert latex(~x, symbol_names={x: "x_i"}) == r"\neg x_i"
assert latex(x & y, symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \wedge y_i"
assert latex(x & y & z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \wedge y_i \wedge z_i"
assert latex(x | y, symbol_names={x: "x_i", y: "y_i"}) == r"x_i \vee y_i"
assert latex(x | y | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"x_i \vee y_i \vee z_i"
assert latex((x & y) | z, symbol_names={x: "x_i", y: "y_i", z: "z_i"}) == \
r"z_i \vee \left(x_i \wedge y_i\right)"
assert latex(Implies(x, y), symbol_names={x: "x_i", y: "y_i"}) == \
r"x_i \Rightarrow y_i"
assert latex(Pow(Rational(1, 3), -1, evaluate=False)) == r"\frac{1}{\frac{1}{3}}"
assert latex(Pow(Rational(1, 3), -2, evaluate=False)) == r"\frac{1}{(\frac{1}{3})^{2}}"
assert latex(Pow(Integer(1)/100, -1, evaluate=False)) == r"\frac{1}{\frac{1}{100}}"
p = Symbol('p', positive=True)
assert latex(exp(-p)*log(p)) == r"e^{- p} \log{\left(p \right)}"
def test_latex_builtins():
assert latex(True) == r"\text{True}"
assert latex(False) == r"\text{False}"
assert latex(None) == r"\text{None}"
assert latex(true) == r"\text{True}"
assert latex(false) == r'\text{False}'
def test_latex_SingularityFunction():
assert latex(SingularityFunction(x, 4, 5)) == \
r"{\left\langle x - 4 \right\rangle}^{5}"
assert latex(SingularityFunction(x, -3, 4)) == \
r"{\left\langle x + 3 \right\rangle}^{4}"
assert latex(SingularityFunction(x, 0, 4)) == \
r"{\left\langle x \right\rangle}^{4}"
assert latex(SingularityFunction(x, a, n)) == \
r"{\left\langle - a + x \right\rangle}^{n}"
assert latex(SingularityFunction(x, 4, -2)) == \
r"{\left\langle x - 4 \right\rangle}^{-2}"
assert latex(SingularityFunction(x, 4, -1)) == \
r"{\left\langle x - 4 \right\rangle}^{-1}"
assert latex(SingularityFunction(x, 4, 5)**3) == \
r"{\left({\langle x - 4 \rangle}^{5}\right)}^{3}"
assert latex(SingularityFunction(x, -3, 4)**3) == \
r"{\left({\langle x + 3 \rangle}^{4}\right)}^{3}"
assert latex(SingularityFunction(x, 0, 4)**3) == \
r"{\left({\langle x \rangle}^{4}\right)}^{3}"
assert latex(SingularityFunction(x, a, n)**3) == \
r"{\left({\langle - a + x \rangle}^{n}\right)}^{3}"
assert latex(SingularityFunction(x, 4, -2)**3) == \
r"{\left({\langle x - 4 \rangle}^{-2}\right)}^{3}"
assert latex((SingularityFunction(x, 4, -1)**3)**3) == \
r"{\left({\langle x - 4 \rangle}^{-1}\right)}^{9}"
def test_latex_cycle():
assert latex(Cycle(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Cycle(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Cycle()) == r"\left( \right)"
def test_latex_permutation():
assert latex(Permutation(1, 2, 4)) == r"\left( 1\; 2\; 4\right)"
assert latex(Permutation(1, 2)(4, 5, 6)) == \
r"\left( 1\; 2\right)\left( 4\; 5\; 6\right)"
assert latex(Permutation()) == r"\left( \right)"
assert latex(Permutation(2, 4)*Permutation(5)) == \
r"\left( 2\; 4\right)\left( 5\right)"
assert latex(Permutation(5)) == r"\left( 5\right)"
assert latex(Permutation(0, 1), perm_cyclic=False) == \
r"\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}"
assert latex(Permutation(0, 1)(2, 3), perm_cyclic=False) == \
r"\begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \end{pmatrix}"
assert latex(Permutation(), perm_cyclic=False) == \
r"\left( \right)"
with warns_deprecated_sympy():
old_print_cyclic = Permutation.print_cyclic
Permutation.print_cyclic = False
assert latex(Permutation(0, 1)(2, 3)) == \
r"\begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \end{pmatrix}"
Permutation.print_cyclic = old_print_cyclic
def test_latex_Float():
assert latex(Float(1.0e100)) == r"1.0 \cdot 10^{100}"
assert latex(Float(1.0e-100)) == r"1.0 \cdot 10^{-100}"
assert latex(Float(1.0e-100), mul_symbol="times") == \
r"1.0 \times 10^{-100}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=2) == \
r"1.0 \cdot 10^{4}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=4) == \
r"1.0 \cdot 10^{4}"
assert latex(Float('10000.0'), full_prec=False, min=-2, max=5) == \
r"10000.0"
assert latex(Float('0.099999'), full_prec=True, min=-2, max=5) == \
r"9.99990000000000 \cdot 10^{-2}"
def test_latex_vector_expressions():
A = CoordSys3D('A')
assert latex(Cross(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Cross(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}"
assert latex(x*Cross(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \times \mathbf{\hat{j}_{A}}\right)"
assert latex(Cross(x*A.i, A.j)) == \
r'- \mathbf{\hat{j}_{A}} \times \left(\left(x\right)\mathbf{\hat{i}_{A}}\right)'
assert latex(Curl(3*A.x*A.j)) == \
r"\nabla\times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*A.x*A.j+A.i)) == \
r"\nabla\times \left(\mathbf{\hat{i}_{A}} + \left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(Curl(3*x*A.x*A.j)) == \
r"\nabla\times \left(\left(3 \mathbf{{x}_{A}} x\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Curl(3*A.x*A.j)) == \
r"x \left(\nabla\times \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Divergence(3*A.x*A.j+A.i)) == \
r"\nabla\cdot \left(\mathbf{\hat{i}_{A}} + \left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(Divergence(3*A.x*A.j)) == \
r"\nabla\cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)"
assert latex(x*Divergence(3*A.x*A.j)) == \
r"x \left(\nabla\cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}}\right)\right)"
assert latex(Dot(A.i, A.j*A.x*3+A.k)) == \
r"\mathbf{\hat{i}_{A}} \cdot \left(\left(3 \mathbf{{x}_{A}}\right)\mathbf{\hat{j}_{A}} + \mathbf{\hat{k}_{A}}\right)"
assert latex(Dot(A.i, A.j)) == \
r"\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}"
assert latex(Dot(x*A.i, A.j)) == \
r"\mathbf{\hat{j}_{A}} \cdot \left(\left(x\right)\mathbf{\hat{i}_{A}}\right)"
assert latex(x*Dot(A.i, A.j)) == \
r"x \left(\mathbf{\hat{i}_{A}} \cdot \mathbf{\hat{j}_{A}}\right)"
assert latex(Gradient(A.x)) == r"\nabla \mathbf{{x}_{A}}"
assert latex(Gradient(A.x + 3*A.y)) == \
r"\nabla \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Gradient(A.x)) == r"x \left(\nabla \mathbf{{x}_{A}}\right)"
assert latex(Gradient(x*A.x)) == r"\nabla \left(\mathbf{{x}_{A}} x\right)"
assert latex(Laplacian(A.x)) == r"\Delta \mathbf{{x}_{A}}"
assert latex(Laplacian(A.x + 3*A.y)) == \
r"\Delta \left(\mathbf{{x}_{A}} + 3 \mathbf{{y}_{A}}\right)"
assert latex(x*Laplacian(A.x)) == r"x \left(\Delta \mathbf{{x}_{A}}\right)"
assert latex(Laplacian(x*A.x)) == r"\Delta \left(\mathbf{{x}_{A}} x\right)"
def test_latex_symbols():
Gamma, lmbda, rho = symbols('Gamma, lambda, rho')
tau, Tau, TAU, taU = symbols('tau, Tau, TAU, taU')
assert latex(tau) == r"\tau"
assert latex(Tau) == r"\mathrm{T}"
assert latex(TAU) == r"\tau"
assert latex(taU) == r"\tau"
# Check that all capitalized greek letters are handled explicitly
capitalized_letters = {l.capitalize() for l in greek_letters_set}
assert len(capitalized_letters - set(tex_greek_dictionary.keys())) == 0
assert latex(Gamma + lmbda) == r"\Gamma + \lambda"
assert latex(Gamma * lmbda) == r"\Gamma \lambda"
assert latex(Symbol('q1')) == r"q_{1}"
assert latex(Symbol('q21')) == r"q_{21}"
assert latex(Symbol('epsilon0')) == r"\epsilon_{0}"
assert latex(Symbol('omega1')) == r"\omega_{1}"
assert latex(Symbol('91')) == r"91"
assert latex(Symbol('alpha_new')) == r"\alpha_{new}"
assert latex(Symbol('C^orig')) == r"C^{orig}"
assert latex(Symbol('x^alpha')) == r"x^{\alpha}"
assert latex(Symbol('beta^alpha')) == r"\beta^{\alpha}"
assert latex(Symbol('e^Alpha')) == r"e^{\mathrm{A}}"
assert latex(Symbol('omega_alpha^beta')) == r"\omega^{\beta}_{\alpha}"
assert latex(Symbol('omega') ** Symbol('beta')) == r"\omega^{\beta}"
@XFAIL
def test_latex_symbols_failing():
rho, mass, volume = symbols('rho, mass, volume')
assert latex(
volume * rho == mass) == r"\rho \mathrm{volume} = \mathrm{mass}"
assert latex(volume / mass * rho == 1) == \
r"\rho \mathrm{volume} {\mathrm{mass}}^{(-1)} = 1"
assert latex(mass**3 * volume**3) == \
r"{\mathrm{mass}}^{3} \cdot {\mathrm{volume}}^{3}"
@_both_exp_pow
def test_latex_functions():
assert latex(exp(x)) == r"e^{x}"
assert latex(exp(1) + exp(2)) == r"e + e^{2}"
f = Function('f')
assert latex(f(x)) == r'f{\left(x \right)}'
assert latex(f) == r'f'
g = Function('g')
assert latex(g(x, y)) == r'g{\left(x,y \right)}'
assert latex(g) == r'g'
h = Function('h')
assert latex(h(x, y, z)) == r'h{\left(x,y,z \right)}'
assert latex(h) == r'h'
Li = Function('Li')
assert latex(Li) == r'\operatorname{Li}'
assert latex(Li(x)) == r'\operatorname{Li}{\left(x \right)}'
mybeta = Function('beta')
# not to be confused with the beta function
assert latex(mybeta(x, y, z)) == r"\beta{\left(x,y,z \right)}"
assert latex(beta(x, y)) == r'\operatorname{B}\left(x, y\right)'
assert latex(beta(x, y)**2) == r'\operatorname{B}^{2}\left(x, y\right)'
assert latex(mybeta(x)) == r"\beta{\left(x \right)}"
assert latex(mybeta) == r"\beta"
g = Function('gamma')
# not to be confused with the gamma function
assert latex(g(x, y, z)) == r"\gamma{\left(x,y,z \right)}"
assert latex(g(x)) == r"\gamma{\left(x \right)}"
assert latex(g) == r"\gamma"
a_1 = Function('a_1')
assert latex(a_1) == r"a_{1}"
assert latex(a_1(x)) == r"a_{1}{\left(x \right)}"
assert latex(Function('a_1')) == r"a_{1}"
# Issue #16925
# multi letter function names
# > simple
assert latex(Function('ab')) == r"\operatorname{ab}"
assert latex(Function('ab1')) == r"\operatorname{ab}_{1}"
assert latex(Function('ab12')) == r"\operatorname{ab}_{12}"
assert latex(Function('ab_1')) == r"\operatorname{ab}_{1}"
assert latex(Function('ab_12')) == r"\operatorname{ab}_{12}"
assert latex(Function('ab_c')) == r"\operatorname{ab}_{c}"
assert latex(Function('ab_cd')) == r"\operatorname{ab}_{cd}"
# > with argument
assert latex(Function('ab')(Symbol('x'))) == r"\operatorname{ab}{\left(x \right)}"
assert latex(Function('ab1')(Symbol('x'))) == r"\operatorname{ab}_{1}{\left(x \right)}"
assert latex(Function('ab12')(Symbol('x'))) == r"\operatorname{ab}_{12}{\left(x \right)}"
assert latex(Function('ab_1')(Symbol('x'))) == r"\operatorname{ab}_{1}{\left(x \right)}"
assert latex(Function('ab_c')(Symbol('x'))) == r"\operatorname{ab}_{c}{\left(x \right)}"
assert latex(Function('ab_cd')(Symbol('x'))) == r"\operatorname{ab}_{cd}{\left(x \right)}"
# > with power
# does not work on functions without brackets
# > with argument and power combined
assert latex(Function('ab')()**2) == r"\operatorname{ab}^{2}{\left( \right)}"
assert latex(Function('ab1')()**2) == r"\operatorname{ab}_{1}^{2}{\left( \right)}"
assert latex(Function('ab12')()**2) == r"\operatorname{ab}_{12}^{2}{\left( \right)}"
assert latex(Function('ab_1')()**2) == r"\operatorname{ab}_{1}^{2}{\left( \right)}"
assert latex(Function('ab_12')()**2) == r"\operatorname{ab}_{12}^{2}{\left( \right)}"
assert latex(Function('ab')(Symbol('x'))**2) == r"\operatorname{ab}^{2}{\left(x \right)}"
assert latex(Function('ab1')(Symbol('x'))**2) == r"\operatorname{ab}_{1}^{2}{\left(x \right)}"
assert latex(Function('ab12')(Symbol('x'))**2) == r"\operatorname{ab}_{12}^{2}{\left(x \right)}"
assert latex(Function('ab_1')(Symbol('x'))**2) == r"\operatorname{ab}_{1}^{2}{\left(x \right)}"
assert latex(Function('ab_12')(Symbol('x'))**2) == \
r"\operatorname{ab}_{12}^{2}{\left(x \right)}"
# single letter function names
# > simple
assert latex(Function('a')) == r"a"
assert latex(Function('a1')) == r"a_{1}"
assert latex(Function('a12')) == r"a_{12}"
assert latex(Function('a_1')) == r"a_{1}"
assert latex(Function('a_12')) == r"a_{12}"
# > with argument
assert latex(Function('a')()) == r"a{\left( \right)}"
assert latex(Function('a1')()) == r"a_{1}{\left( \right)}"
assert latex(Function('a12')()) == r"a_{12}{\left( \right)}"
assert latex(Function('a_1')()) == r"a_{1}{\left( \right)}"
assert latex(Function('a_12')()) == r"a_{12}{\left( \right)}"
# > with power
# does not work on functions without brackets
# > with argument and power combined
assert latex(Function('a')()**2) == r"a^{2}{\left( \right)}"
assert latex(Function('a1')()**2) == r"a_{1}^{2}{\left( \right)}"
assert latex(Function('a12')()**2) == r"a_{12}^{2}{\left( \right)}"
assert latex(Function('a_1')()**2) == r"a_{1}^{2}{\left( \right)}"
assert latex(Function('a_12')()**2) == r"a_{12}^{2}{\left( \right)}"
assert latex(Function('a')(Symbol('x'))**2) == r"a^{2}{\left(x \right)}"
assert latex(Function('a1')(Symbol('x'))**2) == r"a_{1}^{2}{\left(x \right)}"
assert latex(Function('a12')(Symbol('x'))**2) == r"a_{12}^{2}{\left(x \right)}"
assert latex(Function('a_1')(Symbol('x'))**2) == r"a_{1}^{2}{\left(x \right)}"
assert latex(Function('a_12')(Symbol('x'))**2) == r"a_{12}^{2}{\left(x \right)}"
assert latex(Function('a')()**32) == r"a^{32}{\left( \right)}"
assert latex(Function('a1')()**32) == r"a_{1}^{32}{\left( \right)}"
assert latex(Function('a12')()**32) == r"a_{12}^{32}{\left( \right)}"
assert latex(Function('a_1')()**32) == r"a_{1}^{32}{\left( \right)}"
assert latex(Function('a_12')()**32) == r"a_{12}^{32}{\left( \right)}"
assert latex(Function('a')(Symbol('x'))**32) == r"a^{32}{\left(x \right)}"
assert latex(Function('a1')(Symbol('x'))**32) == r"a_{1}^{32}{\left(x \right)}"
assert latex(Function('a12')(Symbol('x'))**32) == r"a_{12}^{32}{\left(x \right)}"
assert latex(Function('a_1')(Symbol('x'))**32) == r"a_{1}^{32}{\left(x \right)}"
assert latex(Function('a_12')(Symbol('x'))**32) == r"a_{12}^{32}{\left(x \right)}"
assert latex(Function('a')()**a) == r"a^{a}{\left( \right)}"
assert latex(Function('a1')()**a) == r"a_{1}^{a}{\left( \right)}"
assert latex(Function('a12')()**a) == r"a_{12}^{a}{\left( \right)}"
assert latex(Function('a_1')()**a) == r"a_{1}^{a}{\left( \right)}"
assert latex(Function('a_12')()**a) == r"a_{12}^{a}{\left( \right)}"
assert latex(Function('a')(Symbol('x'))**a) == r"a^{a}{\left(x \right)}"
assert latex(Function('a1')(Symbol('x'))**a) == r"a_{1}^{a}{\left(x \right)}"
assert latex(Function('a12')(Symbol('x'))**a) == r"a_{12}^{a}{\left(x \right)}"
assert latex(Function('a_1')(Symbol('x'))**a) == r"a_{1}^{a}{\left(x \right)}"
assert latex(Function('a_12')(Symbol('x'))**a) == r"a_{12}^{a}{\left(x \right)}"
ab = Symbol('ab')
assert latex(Function('a')()**ab) == r"a^{ab}{\left( \right)}"
assert latex(Function('a1')()**ab) == r"a_{1}^{ab}{\left( \right)}"
assert latex(Function('a12')()**ab) == r"a_{12}^{ab}{\left( \right)}"
assert latex(Function('a_1')()**ab) == r"a_{1}^{ab}{\left( \right)}"
assert latex(Function('a_12')()**ab) == r"a_{12}^{ab}{\left( \right)}"
assert latex(Function('a')(Symbol('x'))**ab) == r"a^{ab}{\left(x \right)}"
assert latex(Function('a1')(Symbol('x'))**ab) == r"a_{1}^{ab}{\left(x \right)}"
assert latex(Function('a12')(Symbol('x'))**ab) == r"a_{12}^{ab}{\left(x \right)}"
assert latex(Function('a_1')(Symbol('x'))**ab) == r"a_{1}^{ab}{\left(x \right)}"
assert latex(Function('a_12')(Symbol('x'))**ab) == r"a_{12}^{ab}{\left(x \right)}"
assert latex(Function('a^12')(x)) == R"a^{12}{\left(x \right)}"
assert latex(Function('a^12')(x) ** ab) == R"\left(a^{12}\right)^{ab}{\left(x \right)}"
assert latex(Function('a__12')(x)) == R"a^{12}{\left(x \right)}"
assert latex(Function('a__12')(x) ** ab) == R"\left(a^{12}\right)^{ab}{\left(x \right)}"
assert latex(Function('a_1__1_2')(x)) == R"a^{1}_{1 2}{\left(x \right)}"
# issue 5868
omega1 = Function('omega1')
assert latex(omega1) == r"\omega_{1}"
assert latex(omega1(x)) == r"\omega_{1}{\left(x \right)}"
assert latex(sin(x)) == r"\sin{\left(x \right)}"
assert latex(sin(x), fold_func_brackets=True) == r"\sin {x}"
assert latex(sin(2*x**2), fold_func_brackets=True) == \
r"\sin {2 x^{2}}"
assert latex(sin(x**2), fold_func_brackets=True) == \
r"\sin {x^{2}}"
assert latex(asin(x)**2) == r"\operatorname{asin}^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="full") == \
r"\arcsin^{2}{\left(x \right)}"
assert latex(asin(x)**2, inv_trig_style="power") == \
r"\sin^{-1}{\left(x \right)}^{2}"
assert latex(asin(x**2), inv_trig_style="power",
fold_func_brackets=True) == \
r"\sin^{-1} {x^{2}}"
assert latex(acsc(x), inv_trig_style="full") == \
r"\operatorname{arccsc}{\left(x \right)}"
assert latex(asinh(x), inv_trig_style="full") == \
r"\operatorname{arsinh}{\left(x \right)}"
assert latex(factorial(k)) == r"k!"
assert latex(factorial(-k)) == r"\left(- k\right)!"
assert latex(factorial(k)**2) == r"k!^{2}"
assert latex(subfactorial(k)) == r"!k"
assert latex(subfactorial(-k)) == r"!\left(- k\right)"
assert latex(subfactorial(k)**2) == r"\left(!k\right)^{2}"
assert latex(factorial2(k)) == r"k!!"
assert latex(factorial2(-k)) == r"\left(- k\right)!!"
assert latex(factorial2(k)**2) == r"k!!^{2}"
assert latex(binomial(2, k)) == r"{\binom{2}{k}}"
assert latex(binomial(2, k)**2) == r"{\binom{2}{k}}^{2}"
assert latex(FallingFactorial(3, k)) == r"{\left(3\right)}_{k}"
assert latex(RisingFactorial(3, k)) == r"{3}^{\left(k\right)}"
assert latex(floor(x)) == r"\left\lfloor{x}\right\rfloor"
assert latex(ceiling(x)) == r"\left\lceil{x}\right\rceil"
assert latex(frac(x)) == r"\operatorname{frac}{\left(x\right)}"
assert latex(floor(x)**2) == r"\left\lfloor{x}\right\rfloor^{2}"
assert latex(ceiling(x)**2) == r"\left\lceil{x}\right\rceil^{2}"
assert latex(frac(x)**2) == r"\operatorname{frac}{\left(x\right)}^{2}"
assert latex(Min(x, 2, x**3)) == r"\min\left(2, x, x^{3}\right)"
assert latex(Min(x, y)**2) == r"\min\left(x, y\right)^{2}"
assert latex(Max(x, 2, x**3)) == r"\max\left(2, x, x^{3}\right)"
assert latex(Max(x, y)**2) == r"\max\left(x, y\right)^{2}"
assert latex(Abs(x)) == r"\left|{x}\right|"
assert latex(Abs(x)**2) == r"\left|{x}\right|^{2}"
assert latex(re(x)) == r"\operatorname{re}{\left(x\right)}"
assert latex(re(x + y)) == \
r"\operatorname{re}{\left(x\right)} + \operatorname{re}{\left(y\right)}"
assert latex(im(x)) == r"\operatorname{im}{\left(x\right)}"
assert latex(conjugate(x)) == r"\overline{x}"
assert latex(conjugate(x)**2) == r"\overline{x}^{2}"
assert latex(conjugate(x**2)) == r"\overline{x}^{2}"
assert latex(gamma(x)) == r"\Gamma\left(x\right)"
w = Wild('w')
assert latex(gamma(w)) == r"\Gamma\left(w\right)"
assert latex(Order(x)) == r"O\left(x\right)"
assert latex(Order(x, x)) == r"O\left(x\right)"
assert latex(Order(x, (x, 0))) == r"O\left(x\right)"
assert latex(Order(x, (x, oo))) == r"O\left(x; x\rightarrow \infty\right)"
assert latex(Order(x - y, (x, y))) == \
r"O\left(x - y; x\rightarrow y\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, x, y)) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( 0, \ 0\right)\right)"
assert latex(Order(x, (x, oo), (y, oo))) == \
r"O\left(x; \left( x, \ y\right)\rightarrow \left( \infty, \ \infty\right)\right)"
assert latex(lowergamma(x, y)) == r'\gamma\left(x, y\right)'
assert latex(lowergamma(x, y)**2) == r'\gamma^{2}\left(x, y\right)'
assert latex(uppergamma(x, y)) == r'\Gamma\left(x, y\right)'
assert latex(uppergamma(x, y)**2) == r'\Gamma^{2}\left(x, y\right)'
assert latex(cot(x)) == r'\cot{\left(x \right)}'
assert latex(coth(x)) == r'\coth{\left(x \right)}'
assert latex(re(x)) == r'\operatorname{re}{\left(x\right)}'
assert latex(im(x)) == r'\operatorname{im}{\left(x\right)}'
assert latex(root(x, y)) == r'x^{\frac{1}{y}}'
assert latex(arg(x)) == r'\arg{\left(x \right)}'
assert latex(zeta(x)) == r"\zeta\left(x\right)"
assert latex(zeta(x)**2) == r"\zeta^{2}\left(x\right)"
assert latex(zeta(x, y)) == r"\zeta\left(x, y\right)"
assert latex(zeta(x, y)**2) == r"\zeta^{2}\left(x, y\right)"
assert latex(dirichlet_eta(x)) == r"\eta\left(x\right)"
assert latex(dirichlet_eta(x)**2) == r"\eta^{2}\left(x\right)"
assert latex(polylog(x, y)) == r"\operatorname{Li}_{x}\left(y\right)"
assert latex(
polylog(x, y)**2) == r"\operatorname{Li}_{x}^{2}\left(y\right)"
assert latex(lerchphi(x, y, n)) == r"\Phi\left(x, y, n\right)"
assert latex(lerchphi(x, y, n)**2) == r"\Phi^{2}\left(x, y, n\right)"
assert latex(stieltjes(x)) == r"\gamma_{x}"
assert latex(stieltjes(x)**2) == r"\gamma_{x}^{2}"
assert latex(stieltjes(x, y)) == r"\gamma_{x}\left(y\right)"
assert latex(stieltjes(x, y)**2) == r"\gamma_{x}\left(y\right)^{2}"
assert latex(elliptic_k(z)) == r"K\left(z\right)"
assert latex(elliptic_k(z)**2) == r"K^{2}\left(z\right)"
assert latex(elliptic_f(x, y)) == r"F\left(x\middle| y\right)"
assert latex(elliptic_f(x, y)**2) == r"F^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)) == r"E\left(x\middle| y\right)"
assert latex(elliptic_e(x, y)**2) == r"E^{2}\left(x\middle| y\right)"
assert latex(elliptic_e(z)) == r"E\left(z\right)"
assert latex(elliptic_e(z)**2) == r"E^{2}\left(z\right)"
assert latex(elliptic_pi(x, y, z)) == r"\Pi\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y, z)**2) == \
r"\Pi^{2}\left(x; y\middle| z\right)"
assert latex(elliptic_pi(x, y)) == r"\Pi\left(x\middle| y\right)"
assert latex(elliptic_pi(x, y)**2) == r"\Pi^{2}\left(x\middle| y\right)"
assert latex(Ei(x)) == r'\operatorname{Ei}{\left(x \right)}'
assert latex(Ei(x)**2) == r'\operatorname{Ei}^{2}{\left(x \right)}'
assert latex(expint(x, y)) == r'\operatorname{E}_{x}\left(y\right)'
assert latex(expint(x, y)**2) == r'\operatorname{E}_{x}^{2}\left(y\right)'
assert latex(Shi(x)**2) == r'\operatorname{Shi}^{2}{\left(x \right)}'
assert latex(Si(x)**2) == r'\operatorname{Si}^{2}{\left(x \right)}'
assert latex(Ci(x)**2) == r'\operatorname{Ci}^{2}{\left(x \right)}'
assert latex(Chi(x)**2) == r'\operatorname{Chi}^{2}\left(x\right)'
assert latex(Chi(x)) == r'\operatorname{Chi}\left(x\right)'
assert latex(jacobi(n, a, b, x)) == \
r'P_{n}^{\left(a,b\right)}\left(x\right)'
assert latex(jacobi(n, a, b, x)**2) == \
r'\left(P_{n}^{\left(a,b\right)}\left(x\right)\right)^{2}'
assert latex(gegenbauer(n, a, x)) == \
r'C_{n}^{\left(a\right)}\left(x\right)'
assert latex(gegenbauer(n, a, x)**2) == \
r'\left(C_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(chebyshevt(n, x)) == r'T_{n}\left(x\right)'
assert latex(chebyshevt(n, x)**2) == \
r'\left(T_{n}\left(x\right)\right)^{2}'
assert latex(chebyshevu(n, x)) == r'U_{n}\left(x\right)'
assert latex(chebyshevu(n, x)**2) == \
r'\left(U_{n}\left(x\right)\right)^{2}'
assert latex(legendre(n, x)) == r'P_{n}\left(x\right)'
assert latex(legendre(n, x)**2) == r'\left(P_{n}\left(x\right)\right)^{2}'
assert latex(assoc_legendre(n, a, x)) == \
r'P_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_legendre(n, a, x)**2) == \
r'\left(P_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(laguerre(n, x)) == r'L_{n}\left(x\right)'
assert latex(laguerre(n, x)**2) == r'\left(L_{n}\left(x\right)\right)^{2}'
assert latex(assoc_laguerre(n, a, x)) == \
r'L_{n}^{\left(a\right)}\left(x\right)'
assert latex(assoc_laguerre(n, a, x)**2) == \
r'\left(L_{n}^{\left(a\right)}\left(x\right)\right)^{2}'
assert latex(hermite(n, x)) == r'H_{n}\left(x\right)'
assert latex(hermite(n, x)**2) == r'\left(H_{n}\left(x\right)\right)^{2}'
theta = Symbol("theta", real=True)
phi = Symbol("phi", real=True)
assert latex(Ynm(n, m, theta, phi)) == r'Y_{n}^{m}\left(\theta,\phi\right)'
assert latex(Ynm(n, m, theta, phi)**3) == \
r'\left(Y_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
assert latex(Znm(n, m, theta, phi)) == r'Z_{n}^{m}\left(\theta,\phi\right)'
assert latex(Znm(n, m, theta, phi)**3) == \
r'\left(Z_{n}^{m}\left(\theta,\phi\right)\right)^{3}'
# Test latex printing of function names with "_"
assert latex(polar_lift(0)) == \
r"\operatorname{polar\_lift}{\left(0 \right)}"
assert latex(polar_lift(0)**3) == \
r"\operatorname{polar\_lift}^{3}{\left(0 \right)}"
assert latex(totient(n)) == r'\phi\left(n\right)'
assert latex(totient(n) ** 2) == r'\left(\phi\left(n\right)\right)^{2}'
assert latex(reduced_totient(n)) == r'\lambda\left(n\right)'
assert latex(reduced_totient(n) ** 2) == \
r'\left(\lambda\left(n\right)\right)^{2}'
assert latex(divisor_sigma(x)) == r"\sigma\left(x\right)"
assert latex(divisor_sigma(x)**2) == r"\sigma^{2}\left(x\right)"
assert latex(divisor_sigma(x, y)) == r"\sigma_y\left(x\right)"
assert latex(divisor_sigma(x, y)**2) == r"\sigma^{2}_y\left(x\right)"
assert latex(udivisor_sigma(x)) == r"\sigma^*\left(x\right)"
assert latex(udivisor_sigma(x)**2) == r"\sigma^*^{2}\left(x\right)"
assert latex(udivisor_sigma(x, y)) == r"\sigma^*_y\left(x\right)"
assert latex(udivisor_sigma(x, y)**2) == r"\sigma^*^{2}_y\left(x\right)"
assert latex(primenu(n)) == r'\nu\left(n\right)'
assert latex(primenu(n) ** 2) == r'\left(\nu\left(n\right)\right)^{2}'
assert latex(primeomega(n)) == r'\Omega\left(n\right)'
assert latex(primeomega(n) ** 2) == \
r'\left(\Omega\left(n\right)\right)^{2}'
assert latex(LambertW(n)) == r'W\left(n\right)'
assert latex(LambertW(n, -1)) == r'W_{-1}\left(n\right)'
assert latex(LambertW(n, k)) == r'W_{k}\left(n\right)'
assert latex(LambertW(n) * LambertW(n)) == r"W^{2}\left(n\right)"
assert latex(Pow(LambertW(n), 2)) == r"W^{2}\left(n\right)"
assert latex(LambertW(n)**k) == r"W^{k}\left(n\right)"
assert latex(LambertW(n, k)**p) == r"W^{p}_{k}\left(n\right)"
assert latex(Mod(x, 7)) == r'x \bmod 7'
assert latex(Mod(x + 1, 7)) == r'\left(x + 1\right) \bmod 7'
assert latex(Mod(7, x + 1)) == r'7 \bmod \left(x + 1\right)'
assert latex(Mod(2 * x, 7)) == r'2 x \bmod 7'
assert latex(Mod(7, 2 * x)) == r'7 \bmod 2 x'
assert latex(Mod(x, 7) + 1) == r'\left(x \bmod 7\right) + 1'
assert latex(2 * Mod(x, 7)) == r'2 \left(x \bmod 7\right)'
assert latex(Mod(7, 2 * x)**n) == r'\left(7 \bmod 2 x\right)^{n}'
# some unknown function name should get rendered with \operatorname
fjlkd = Function('fjlkd')
assert latex(fjlkd(x)) == r'\operatorname{fjlkd}{\left(x \right)}'
# even when it is referred to without an argument
assert latex(fjlkd) == r'\operatorname{fjlkd}'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert latex(mygamma) == r"\operatorname{mygamma}"
assert latex(mygamma(x)) == r"\operatorname{mygamma}{\left(x \right)}"
def test_hyper_printing():
from sympy.abc import x, z
assert latex(meijerg(Tuple(pi, pi, x), Tuple(1),
(0, 1), Tuple(1, 2, 3/pi), z)) == \
r'{G_{4, 5}^{2, 3}\left(\begin{matrix} \pi, \pi, x & 1 \\0, 1 & 1, 2, '\
r'\frac{3}{\pi} \end{matrix} \middle| {z} \right)}'
assert latex(meijerg(Tuple(), Tuple(1), (0,), Tuple(), z)) == \
r'{G_{1, 1}^{1, 0}\left(\begin{matrix} & 1 \\0 & \end{matrix} \middle| {z} \right)}'
assert latex(hyper((x, 2), (3,), z)) == \
r'{{}_{2}F_{1}\left(\begin{matrix} x, 2 ' \
r'\\ 3 \end{matrix}\middle| {z} \right)}'
assert latex(hyper(Tuple(), Tuple(1), z)) == \
r'{{}_{0}F_{1}\left(\begin{matrix} ' \
r'\\ 1 \end{matrix}\middle| {z} \right)}'
def test_latex_bessel():
from sympy.functions.special.bessel import (besselj, bessely, besseli,
besselk, hankel1, hankel2,
jn, yn, hn1, hn2)
from sympy.abc import z
assert latex(besselj(n, z**2)**k) == r'J^{k}_{n}\left(z^{2}\right)'
assert latex(bessely(n, z)) == r'Y_{n}\left(z\right)'
assert latex(besseli(n, z)) == r'I_{n}\left(z\right)'
assert latex(besselk(n, z)) == r'K_{n}\left(z\right)'
assert latex(hankel1(n, z**2)**2) == \
r'\left(H^{(1)}_{n}\left(z^{2}\right)\right)^{2}'
assert latex(hankel2(n, z)) == r'H^{(2)}_{n}\left(z\right)'
assert latex(jn(n, z)) == r'j_{n}\left(z\right)'
assert latex(yn(n, z)) == r'y_{n}\left(z\right)'
assert latex(hn1(n, z)) == r'h^{(1)}_{n}\left(z\right)'
assert latex(hn2(n, z)) == r'h^{(2)}_{n}\left(z\right)'
def test_latex_fresnel():
from sympy.functions.special.error_functions import (fresnels, fresnelc)
from sympy.abc import z
assert latex(fresnels(z)) == r'S\left(z\right)'
assert latex(fresnelc(z)) == r'C\left(z\right)'
assert latex(fresnels(z)**2) == r'S^{2}\left(z\right)'
assert latex(fresnelc(z)**2) == r'C^{2}\left(z\right)'
def test_latex_brackets():
assert latex((-1)**x) == r"\left(-1\right)^{x}"
def test_latex_indexed():
Psi_symbol = Symbol('Psi_0', complex=True, real=False)
Psi_indexed = IndexedBase(Symbol('Psi', complex=True, real=False))
symbol_latex = latex(Psi_symbol * conjugate(Psi_symbol))
indexed_latex = latex(Psi_indexed[0] * conjugate(Psi_indexed[0]))
# \\overline{{\\Psi}_{0}} {\\Psi}_{0} vs. \\Psi_{0} \\overline{\\Psi_{0}}
assert symbol_latex == r'\Psi_{0} \overline{\Psi_{0}}'
assert indexed_latex == r'\overline{{\Psi}_{0}} {\Psi}_{0}'
# Symbol('gamma') gives r'\gamma'
interval = '\\mathrel{..}\\nobreak '
assert latex(Indexed('x1', Symbol('i'))) == r'{x_{1}}_{i}'
assert latex(Indexed('x2', Idx('i'))) == r'{x_{2}}_{i}'
assert latex(Indexed('x3', Idx('i', Symbol('N')))) == r'{x_{3}}_{{i}_{0'+interval+'N - 1}}'
assert latex(Indexed('x3', Idx('i', Symbol('N')+1))) == r'{x_{3}}_{{i}_{0'+interval+'N}}'
assert latex(Indexed('x4', Idx('i', (Symbol('a'),Symbol('b'))))) == r'{x_{4}}_{{i}_{a'+interval+'b}}'
assert latex(IndexedBase('gamma')) == r'\gamma'
assert latex(IndexedBase('a b')) == r'a b'
assert latex(IndexedBase('a_b')) == r'a_{b}'
def test_latex_derivatives():
# regular "d" for ordinary derivatives
assert latex(diff(x**3, x, evaluate=False)) == \
r"\frac{d}{d x} x^{3}"
assert latex(diff(sin(x) + x**2, x, evaluate=False)) == \
r"\frac{d}{d x} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False))\
== \
r"\frac{d^{2}}{d x^{2}} \left(x^{2} + \sin{\left(x \right)}\right)"
assert latex(diff(diff(diff(sin(x) + x**2, x, evaluate=False), evaluate=False), evaluate=False)) == \
r"\frac{d^{3}}{d x^{3}} \left(x^{2} + \sin{\left(x \right)}\right)"
# \partial for partial derivatives
assert latex(diff(sin(x * y), x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \sin{\left(x y \right)}"
assert latex(diff(sin(x * y) + x**2, x, evaluate=False)) == \
r"\frac{\partial}{\partial x} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial x^{2}} \left(x^{2} + \sin{\left(x y \right)}\right)"
assert latex(diff(diff(diff(sin(x*y) + x**2, x, evaluate=False), x, evaluate=False), x, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial x^{3}} \left(x^{2} + \sin{\left(x y \right)}\right)"
# mixed partial derivatives
f = Function("f")
assert latex(diff(diff(f(x, y), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{2}}{\partial y\partial x} " + latex(f(x, y))
assert latex(diff(diff(diff(f(x, y), x, evaluate=False), x, evaluate=False), y, evaluate=False)) == \
r"\frac{\partial^{3}}{\partial y\partial x^{2}} " + latex(f(x, y))
# for negative nested Derivative
assert latex(diff(-diff(y**2,x,evaluate=False),x,evaluate=False)) == r'\frac{d}{d x} \left(- \frac{d}{d x} y^{2}\right)'
assert latex(diff(diff(-diff(diff(y,x,evaluate=False),x,evaluate=False),x,evaluate=False),x,evaluate=False)) == \
r'\frac{d^{2}}{d x^{2}} \left(- \frac{d^{2}}{d x^{2}} y\right)'
# use ordinary d when one of the variables has been integrated out
assert latex(diff(Integral(exp(-x*y), (x, 0, oo)), y, evaluate=False)) == \
r"\frac{d}{d y} \int\limits_{0}^{\infty} e^{- x y}\, dx"
# Derivative wrapped in power:
assert latex(diff(x, x, evaluate=False)**2) == \
r"\left(\frac{d}{d x} x\right)^{2}"
assert latex(diff(f(x), x)**2) == \
r"\left(\frac{d}{d x} f{\left(x \right)}\right)^{2}"
assert latex(diff(f(x), (x, n))) == \
r"\frac{d^{n}}{d x^{n}} f{\left(x \right)}"
x1 = Symbol('x1')
x2 = Symbol('x2')
assert latex(diff(f(x1, x2), x1)) == r'\frac{\partial}{\partial x_{1}} f{\left(x_{1},x_{2} \right)}'
n1 = Symbol('n1')
assert latex(diff(f(x), (x, n1))) == r'\frac{d^{n_{1}}}{d x^{n_{1}}} f{\left(x \right)}'
n2 = Symbol('n2')
assert latex(diff(f(x), (x, Max(n1, n2)))) == \
r'\frac{d^{\max\left(n_{1}, n_{2}\right)}}{d x^{\max\left(n_{1}, n_{2}\right)}} f{\left(x \right)}'
# set diff operator
assert latex(diff(f(x), x), diff_operator="rd") == r'\frac{\mathrm{d}}{\mathrm{d} x} f{\left(x \right)}'
def test_latex_subs():
assert latex(Subs(x*y, (x, y), (1, 2))) == r'\left. x y \right|_{\substack{ x=1\\ y=2 }}'
def test_latex_integrals():
assert latex(Integral(log(x), x)) == r"\int \log{\left(x \right)}\, dx"
assert latex(Integral(x**2, (x, 0, 1))) == \
r"\int\limits_{0}^{1} x^{2}\, dx"
assert latex(Integral(x**2, (x, 10, 20))) == \
r"\int\limits_{10}^{20} x^{2}\, dx"
assert latex(Integral(y*x**2, (x, 0, 1), y)) == \
r"\int\int\limits_{0}^{1} x^{2} y\, dx\, dy"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*') == \
r"\begin{equation*}\int\int\limits_{0}^{1} x^{2} y\, dx\, dy\end{equation*}"
assert latex(Integral(y*x**2, (x, 0, 1), y), mode='equation*', itex=True) \
== r"$$\int\int_{0}^{1} x^{2} y\, dx\, dy$$"
assert latex(Integral(x, (x, 0))) == r"\int\limits^{0} x\, dx"
assert latex(Integral(x*y, x, y)) == r"\iint x y\, dx\, dy"
assert latex(Integral(x*y*z, x, y, z)) == r"\iiint x y z\, dx\, dy\, dz"
assert latex(Integral(x*y*z*t, x, y, z, t)) == \
r"\iiiint t x y z\, dx\, dy\, dz\, dt"
assert latex(Integral(x, x, x, x, x, x, x)) == \
r"\int\int\int\int\int\int x\, dx\, dx\, dx\, dx\, dx\, dx"
assert latex(Integral(x, x, y, (z, 0, 1))) == \
r"\int\limits_{0}^{1}\int\int x\, dx\, dy\, dz"
# for negative nested Integral
assert latex(Integral(-Integral(y**2,x),x)) == \
r'\int \left(- \int y^{2}\, dx\right)\, dx'
assert latex(Integral(-Integral(-Integral(y,x),x),x)) == \
r'\int \left(- \int \left(- \int y\, dx\right)\, dx\right)\, dx'
# fix issue #10806
assert latex(Integral(z, z)**2) == r"\left(\int z\, dz\right)^{2}"
assert latex(Integral(x + z, z)) == r"\int \left(x + z\right)\, dz"
assert latex(Integral(x+z/2, z)) == \
r"\int \left(x + \frac{z}{2}\right)\, dz"
assert latex(Integral(x**y, z)) == r"\int x^{y}\, dz"
# set diff operator
assert latex(Integral(x, x), diff_operator="rd") == r'\int x\, \mathrm{d}x'
assert latex(Integral(x, (x, 0, 1)), diff_operator="rd") == r'\int\limits_{0}^{1} x\, \mathrm{d}x'
def test_latex_sets():
for s in (frozenset, set):
assert latex(s([x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
s = FiniteSet
assert latex(s(*[x*y, x**2])) == r"\left\{x^{2}, x y\right\}"
assert latex(s(*range(1, 6))) == r"\left\{1, 2, 3, 4, 5\right\}"
assert latex(s(*range(1, 13))) == \
r"\left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\right\}"
def test_latex_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
assert latex(se) == r"SetExpr\left(\left[1, 3\right]\right)"
def test_latex_Range():
assert latex(Range(1, 51)) == r'\left\{1, 2, \ldots, 50\right\}'
assert latex(Range(1, 4)) == r'\left\{1, 2, 3\right\}'
assert latex(Range(0, 3, 1)) == r'\left\{0, 1, 2\right\}'
assert latex(Range(0, 30, 1)) == r'\left\{0, 1, \ldots, 29\right\}'
assert latex(Range(30, 1, -1)) == r'\left\{30, 29, \ldots, 2\right\}'
assert latex(Range(0, oo, 2)) == r'\left\{0, 2, \ldots\right\}'
assert latex(Range(oo, -2, -2)) == r'\left\{\ldots, 2, 0\right\}'
assert latex(Range(-2, -oo, -1)) == r'\left\{-2, -3, \ldots\right\}'
assert latex(Range(-oo, oo)) == r'\left\{\ldots, -1, 0, 1, \ldots\right\}'
assert latex(Range(oo, -oo, -1)) == r'\left\{\ldots, 1, 0, -1, \ldots\right\}'
a, b, c = symbols('a:c')
assert latex(Range(a, b, c)) == r'\text{Range}\left(a, b, c\right)'
assert latex(Range(a, 10, 1)) == r'\text{Range}\left(a, 10\right)'
assert latex(Range(0, b, 1)) == r'\text{Range}\left(b\right)'
assert latex(Range(0, 10, c)) == r'\text{Range}\left(0, 10, c\right)'
i = Symbol('i', integer=True)
n = Symbol('n', negative=True, integer=True)
p = Symbol('p', positive=True, integer=True)
assert latex(Range(i, i + 3)) == r'\left\{i, i + 1, i + 2\right\}'
assert latex(Range(-oo, n, 2)) == r'\left\{\ldots, n - 4, n - 2\right\}'
assert latex(Range(p, oo)) == r'\left\{p, p + 1, \ldots\right\}'
# The following will work if __iter__ is improved
# assert latex(Range(-3, p + 7)) == r'\left\{-3, -2, \ldots, p + 6\right\}'
# Must have integer assumptions
assert latex(Range(a, a + 3)) == r'\text{Range}\left(a, a + 3\right)'
def test_latex_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
latex_str = r'\left[0, 1, 4, 9, \ldots\right]'
assert latex(s1) == latex_str
latex_str = r'\left[1, 2, 1, 2, \ldots\right]'
assert latex(s2) == latex_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
latex_str = r'\left[0, 1, 4\right]'
assert latex(s3) == latex_str
latex_str = r'\left[1, 2, 1\right]'
assert latex(s4) == latex_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
latex_str = r'\left[\ldots, 9, 4, 1, 0\right]'
assert latex(s5) == latex_str
latex_str = r'\left[\ldots, 2, 1, 2, 1\right]'
assert latex(s6) == latex_str
latex_str = r'\left[1, 3, 5, 11, \ldots\right]'
assert latex(SeqAdd(s1, s2)) == latex_str
latex_str = r'\left[1, 3, 5\right]'
assert latex(SeqAdd(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 11, 5, 3, 1\right]'
assert latex(SeqAdd(s5, s6)) == latex_str
latex_str = r'\left[0, 2, 4, 18, \ldots\right]'
assert latex(SeqMul(s1, s2)) == latex_str
latex_str = r'\left[0, 2, 4\right]'
assert latex(SeqMul(s3, s4)) == latex_str
latex_str = r'\left[\ldots, 18, 4, 2, 0\right]'
assert latex(SeqMul(s5, s6)) == latex_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
latex_str = r'\left\{a^{2}\right\}_{a=0}^{x}'
assert latex(s7) == latex_str
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
latex_str = r'\left[0, b, 4 b\right]'
assert latex(s8) == latex_str
def test_latex_FourierSeries():
latex_str = \
r'2 \sin{\left(x \right)} - \sin{\left(2 x \right)} + \frac{2 \sin{\left(3 x \right)}}{3} + \ldots'
assert latex(fourier_series(x, (x, -pi, pi))) == latex_str
def test_latex_FormalPowerSeries():
latex_str = r'\sum_{k=1}^{\infty} - \frac{\left(-1\right)^{- k} x^{k}}{k}'
assert latex(fps(log(1 + x))) == latex_str
def test_latex_intervals():
a = Symbol('a', real=True)
assert latex(Interval(0, 0)) == r"\left\{0\right\}"
assert latex(Interval(0, a)) == r"\left[0, a\right]"
assert latex(Interval(0, a, False, False)) == r"\left[0, a\right]"
assert latex(Interval(0, a, True, False)) == r"\left(0, a\right]"
assert latex(Interval(0, a, False, True)) == r"\left[0, a\right)"
assert latex(Interval(0, a, True, True)) == r"\left(0, a\right)"
def test_latex_AccumuBounds():
a = Symbol('a', real=True)
assert latex(AccumBounds(0, 1)) == r"\left\langle 0, 1\right\rangle"
assert latex(AccumBounds(0, a)) == r"\left\langle 0, a\right\rangle"
assert latex(AccumBounds(a + 1, a + 2)) == \
r"\left\langle a + 1, a + 2\right\rangle"
def test_latex_emptyset():
assert latex(S.EmptySet) == r"\emptyset"
def test_latex_universalset():
assert latex(S.UniversalSet) == r"\mathbb{U}"
def test_latex_commutator():
A = Operator('A')
B = Operator('B')
comm = Commutator(B, A)
assert latex(comm.doit()) == r"- (A B - B A)"
def test_latex_union():
assert latex(Union(Interval(0, 1), Interval(2, 3))) == \
r"\left[0, 1\right] \cup \left[2, 3\right]"
assert latex(Union(Interval(1, 1), Interval(2, 2), Interval(3, 4))) == \
r"\left\{1, 2\right\} \cup \left[3, 4\right]"
def test_latex_intersection():
assert latex(Intersection(Interval(0, 1), Interval(x, y))) == \
r"\left[0, 1\right] \cap \left[x, y\right]"
def test_latex_symmetric_difference():
assert latex(SymmetricDifference(Interval(2, 5), Interval(4, 7),
evaluate=False)) == \
r'\left[2, 5\right] \triangle \left[4, 7\right]'
def test_latex_Complement():
assert latex(Complement(S.Reals, S.Naturals)) == \
r"\mathbb{R} \setminus \mathbb{N}"
def test_latex_productset():
line = Interval(0, 1)
bigline = Interval(0, 10)
fset = FiniteSet(1, 2, 3)
assert latex(line**2) == r"%s^{2}" % latex(line)
assert latex(line**10) == r"%s^{10}" % latex(line)
assert latex((line * bigline * fset).flatten()) == r"%s \times %s \times %s" % (
latex(line), latex(bigline), latex(fset))
def test_latex_powerset():
fset = FiniteSet(1, 2, 3)
assert latex(PowerSet(fset)) == r'\mathcal{P}\left(\left\{1, 2, 3\right\}\right)'
def test_latex_ordinals():
w = OrdinalOmega()
assert latex(w) == r"\omega"
wp = OmegaPower(2, 3)
assert latex(wp) == r'3 \omega^{2}'
assert latex(Ordinal(wp, OmegaPower(1, 1))) == r'3 \omega^{2} + \omega'
assert latex(Ordinal(OmegaPower(2, 1), OmegaPower(1, 2))) == r'\omega^{2} + 2 \omega'
def test_set_operators_parenthesis():
a, b, c, d = symbols('a:d')
A = FiniteSet(a)
B = FiniteSet(b)
C = FiniteSet(c)
D = FiniteSet(d)
U1 = Union(A, B, evaluate=False)
U2 = Union(C, D, evaluate=False)
I1 = Intersection(A, B, evaluate=False)
I2 = Intersection(C, D, evaluate=False)
C1 = Complement(A, B, evaluate=False)
C2 = Complement(C, D, evaluate=False)
D1 = SymmetricDifference(A, B, evaluate=False)
D2 = SymmetricDifference(C, D, evaluate=False)
# XXX ProductSet does not support evaluate keyword
P1 = ProductSet(A, B)
P2 = ProductSet(C, D)
assert latex(Intersection(A, U2, evaluate=False)) == \
r'\left\{a\right\} \cap ' \
r'\left(\left\{c\right\} \cup \left\{d\right\}\right)'
assert latex(Intersection(U1, U2, evaluate=False)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\cap \left(\left\{c\right\} \cup \left\{d\right\}\right)'
assert latex(Intersection(C1, C2, evaluate=False)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \cap \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(Intersection(D1, D2, evaluate=False)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \cap \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
assert latex(Intersection(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \
r'\cap \left(\left\{c\right\} \times ' \
r'\left\{d\right\}\right)'
assert latex(Union(A, I2, evaluate=False)) == \
r'\left\{a\right\} \cup ' \
r'\left(\left\{c\right\} \cap \left\{d\right\}\right)'
assert latex(Union(I1, I2, evaluate=False)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\cup \left(\left\{c\right\} \cap \left\{d\right\}\right)'
assert latex(Union(C1, C2, evaluate=False)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \cup \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(Union(D1, D2, evaluate=False)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \cup \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
assert latex(Union(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \
r'\cup \left(\left\{c\right\} \times ' \
r'\left\{d\right\}\right)'
assert latex(Complement(A, C2, evaluate=False)) == \
r'\left\{a\right\} \setminus \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(Complement(U1, U2, evaluate=False)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\setminus \left(\left\{c\right\} \cup ' \
r'\left\{d\right\}\right)'
assert latex(Complement(I1, I2, evaluate=False)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\setminus \left(\left\{c\right\} \cap ' \
r'\left\{d\right\}\right)'
assert latex(Complement(D1, D2, evaluate=False)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \setminus ' \
r'\left(\left\{c\right\} \triangle \left\{d\right\}\right)'
assert latex(Complement(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) '\
r'\setminus \left(\left\{c\right\} \times '\
r'\left\{d\right\}\right)'
assert latex(SymmetricDifference(A, D2, evaluate=False)) == \
r'\left\{a\right\} \triangle \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
assert latex(SymmetricDifference(U1, U2, evaluate=False)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\triangle \left(\left\{c\right\} \cup ' \
r'\left\{d\right\}\right)'
assert latex(SymmetricDifference(I1, I2, evaluate=False)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\triangle \left(\left\{c\right\} \cap ' \
r'\left\{d\right\}\right)'
assert latex(SymmetricDifference(C1, C2, evaluate=False)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \triangle ' \
r'\left(\left\{c\right\} \setminus \left\{d\right\}\right)'
assert latex(SymmetricDifference(P1, P2, evaluate=False)) == \
r'\left(\left\{a\right\} \times \left\{b\right\}\right) ' \
r'\triangle \left(\left\{c\right\} \times ' \
r'\left\{d\right\}\right)'
# XXX This can be incorrect since cartesian product is not associative
assert latex(ProductSet(A, P2).flatten()) == \
r'\left\{a\right\} \times \left\{c\right\} \times ' \
r'\left\{d\right\}'
assert latex(ProductSet(U1, U2)) == \
r'\left(\left\{a\right\} \cup \left\{b\right\}\right) ' \
r'\times \left(\left\{c\right\} \cup ' \
r'\left\{d\right\}\right)'
assert latex(ProductSet(I1, I2)) == \
r'\left(\left\{a\right\} \cap \left\{b\right\}\right) ' \
r'\times \left(\left\{c\right\} \cap ' \
r'\left\{d\right\}\right)'
assert latex(ProductSet(C1, C2)) == \
r'\left(\left\{a\right\} \setminus ' \
r'\left\{b\right\}\right) \times \left(\left\{c\right\} ' \
r'\setminus \left\{d\right\}\right)'
assert latex(ProductSet(D1, D2)) == \
r'\left(\left\{a\right\} \triangle ' \
r'\left\{b\right\}\right) \times \left(\left\{c\right\} ' \
r'\triangle \left\{d\right\}\right)'
def test_latex_Complexes():
assert latex(S.Complexes) == r"\mathbb{C}"
def test_latex_Naturals():
assert latex(S.Naturals) == r"\mathbb{N}"
def test_latex_Naturals0():
assert latex(S.Naturals0) == r"\mathbb{N}_0"
def test_latex_Integers():
assert latex(S.Integers) == r"\mathbb{Z}"
def test_latex_ImageSet():
x = Symbol('x')
assert latex(ImageSet(Lambda(x, x**2), S.Naturals)) == \
r"\left\{x^{2}\; \middle|\; x \in \mathbb{N}\right\}"
y = Symbol('y')
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
assert latex(imgset) == \
r"\left\{x + y\; \middle|\; x \in \left\{1, 2, 3\right\}, y \in \left\{3, 4\right\}\right\}"
imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4}))
assert latex(imgset) == \
r"\left\{x + y\; \middle|\; \left( x, \ y\right) \in \left\{1, 2, 3\right\} \times \left\{3, 4\right\}\right\}"
def test_latex_ConditionSet():
x = Symbol('x')
assert latex(ConditionSet(x, Eq(x**2, 1), S.Reals)) == \
r"\left\{x\; \middle|\; x \in \mathbb{R} \wedge x^{2} = 1 \right\}"
assert latex(ConditionSet(x, Eq(x**2, 1), S.UniversalSet)) == \
r"\left\{x\; \middle|\; x^{2} = 1 \right\}"
def test_latex_ComplexRegion():
assert latex(ComplexRegion(Interval(3, 5)*Interval(4, 6))) == \
r"\left\{x + y i\; \middle|\; x, y \in \left[3, 5\right] \times \left[4, 6\right] \right\}"
assert latex(ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)) == \
r"\left\{r \left(i \sin{\left(\theta \right)} + \cos{\left(\theta "\
r"\right)}\right)\; \middle|\; r, \theta \in \left[0, 1\right] \times \left[0, 2 \pi\right) \right\}"
def test_latex_Contains():
x = Symbol('x')
assert latex(Contains(x, S.Naturals)) == r"x \in \mathbb{N}"
def test_latex_sum():
assert latex(Sum(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\sum_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Sum(x**2, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} x^{2}"
assert latex(Sum(x**2 + y, (x, -2, 2))) == \
r"\sum_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Sum(x**2 + y, (x, -2, 2))**2) == \
r"\left(\sum_{x=-2}^{2} \left(x^{2} + y\right)\right)^{2}"
def test_latex_product():
assert latex(Product(x*y**2, (x, -2, 2), (y, -5, 5))) == \
r"\prod_{\substack{-2 \leq x \leq 2\\-5 \leq y \leq 5}} x y^{2}"
assert latex(Product(x**2, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} x^{2}"
assert latex(Product(x**2 + y, (x, -2, 2))) == \
r"\prod_{x=-2}^{2} \left(x^{2} + y\right)"
assert latex(Product(x, (x, -2, 2))**2) == \
r"\left(\prod_{x=-2}^{2} x\right)^{2}"
def test_latex_limits():
assert latex(Limit(x, x, oo)) == r"\lim_{x \to \infty} x"
# issue 8175
f = Function('f')
assert latex(Limit(f(x), x, 0)) == r"\lim_{x \to 0^+} f{\left(x \right)}"
assert latex(Limit(f(x), x, 0, "-")) == \
r"\lim_{x \to 0^-} f{\left(x \right)}"
# issue #10806
assert latex(Limit(f(x), x, 0)**2) == \
r"\left(\lim_{x \to 0^+} f{\left(x \right)}\right)^{2}"
# bi-directional limit
assert latex(Limit(f(x), x, 0, dir='+-')) == \
r"\lim_{x \to 0} f{\left(x \right)}"
def test_latex_log():
assert latex(log(x)) == r"\log{\left(x \right)}"
assert latex(log(x), ln_notation=True) == r"\ln{\left(x \right)}"
assert latex(log(x) + log(y)) == \
r"\log{\left(x \right)} + \log{\left(y \right)}"
assert latex(log(x) + log(y), ln_notation=True) == \
r"\ln{\left(x \right)} + \ln{\left(y \right)}"
assert latex(pow(log(x), x)) == r"\log{\left(x \right)}^{x}"
assert latex(pow(log(x), x), ln_notation=True) == \
r"\ln{\left(x \right)}^{x}"
def test_issue_3568():
beta = Symbol(r'\beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
beta = Symbol(r'beta')
y = beta + x
assert latex(y) in [r'\beta + x', r'x + \beta']
def test_latex():
assert latex((2*tau)**Rational(7, 2)) == r"8 \sqrt{2} \tau^{\frac{7}{2}}"
assert latex((2*mu)**Rational(7, 2), mode='equation*') == \
r"\begin{equation*}8 \sqrt{2} \mu^{\frac{7}{2}}\end{equation*}"
assert latex((2*mu)**Rational(7, 2), mode='equation', itex=True) == \
r"$$8 \sqrt{2} \mu^{\frac{7}{2}}$$"
assert latex([2/x, y]) == r"\left[ \frac{2}{x}, \ y\right]"
def test_latex_dict():
d = {Rational(1): 1, x**2: 2, x: 3, x**3: 4}
assert latex(d) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
D = Dict(d)
assert latex(D) == \
r'\left\{ 1 : 1, \ x : 3, \ x^{2} : 2, \ x^{3} : 4\right\}'
def test_latex_list():
ll = [Symbol('omega1'), Symbol('a'), Symbol('alpha')]
assert latex(ll) == r'\left[ \omega_{1}, \ a, \ \alpha\right]'
def test_latex_NumberSymbols():
assert latex(S.Catalan) == "G"
assert latex(S.EulerGamma) == r"\gamma"
assert latex(S.Exp1) == "e"
assert latex(S.GoldenRatio) == r"\phi"
assert latex(S.Pi) == r"\pi"
assert latex(S.TribonacciConstant) == r"\text{TribonacciConstant}"
def test_latex_rational():
# tests issue 3973
assert latex(-Rational(1, 2)) == r"- \frac{1}{2}"
assert latex(Rational(-1, 2)) == r"- \frac{1}{2}"
assert latex(Rational(1, -2)) == r"- \frac{1}{2}"
assert latex(-Rational(-1, 2)) == r"\frac{1}{2}"
assert latex(-Rational(1, 2)*x) == r"- \frac{x}{2}"
assert latex(-Rational(1, 2)*x + Rational(-2, 3)*y) == \
r"- \frac{x}{2} - \frac{2 y}{3}"
def test_latex_inverse():
# tests issue 4129
assert latex(1/x) == r"\frac{1}{x}"
assert latex(1/(x + y)) == r"\frac{1}{x + y}"
def test_latex_DiracDelta():
assert latex(DiracDelta(x)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x)**2) == r"\left(\delta\left(x\right)\right)^{2}"
assert latex(DiracDelta(x, 0)) == r"\delta\left(x\right)"
assert latex(DiracDelta(x, 5)) == \
r"\delta^{\left( 5 \right)}\left( x \right)"
assert latex(DiracDelta(x, 5)**2) == \
r"\left(\delta^{\left( 5 \right)}\left( x \right)\right)^{2}"
def test_latex_Heaviside():
assert latex(Heaviside(x)) == r"\theta\left(x\right)"
assert latex(Heaviside(x)**2) == r"\left(\theta\left(x\right)\right)^{2}"
def test_latex_KroneckerDelta():
assert latex(KroneckerDelta(x, y)) == r"\delta_{x y}"
assert latex(KroneckerDelta(x, y + 1)) == r"\delta_{x, y + 1}"
# issue 6578
assert latex(KroneckerDelta(x + 1, y)) == r"\delta_{y, x + 1}"
assert latex(Pow(KroneckerDelta(x, y), 2, evaluate=False)) == \
r"\left(\delta_{x y}\right)^{2}"
def test_latex_LeviCivita():
assert latex(LeviCivita(x, y, z)) == r"\varepsilon_{x y z}"
assert latex(LeviCivita(x, y, z)**2) == \
r"\left(\varepsilon_{x y z}\right)^{2}"
assert latex(LeviCivita(x, y, z + 1)) == r"\varepsilon_{x, y, z + 1}"
assert latex(LeviCivita(x, y + 1, z)) == r"\varepsilon_{x, y + 1, z}"
assert latex(LeviCivita(x + 1, y, z)) == r"\varepsilon_{x + 1, y, z}"
def test_mode():
expr = x + y
assert latex(expr) == r'x + y'
assert latex(expr, mode='plain') == r'x + y'
assert latex(expr, mode='inline') == r'$x + y$'
assert latex(
expr, mode='equation*') == r'\begin{equation*}x + y\end{equation*}'
assert latex(
expr, mode='equation') == r'\begin{equation}x + y\end{equation}'
raises(ValueError, lambda: latex(expr, mode='foo'))
def test_latex_mathieu():
assert latex(mathieuc(x, y, z)) == r"C\left(x, y, z\right)"
assert latex(mathieus(x, y, z)) == r"S\left(x, y, z\right)"
assert latex(mathieuc(x, y, z)**2) == r"C\left(x, y, z\right)^{2}"
assert latex(mathieus(x, y, z)**2) == r"S\left(x, y, z\right)^{2}"
assert latex(mathieucprime(x, y, z)) == r"C^{\prime}\left(x, y, z\right)"
assert latex(mathieusprime(x, y, z)) == r"S^{\prime}\left(x, y, z\right)"
assert latex(mathieucprime(x, y, z)**2) == r"C^{\prime}\left(x, y, z\right)^{2}"
assert latex(mathieusprime(x, y, z)**2) == r"S^{\prime}\left(x, y, z\right)^{2}"
def test_latex_Piecewise():
p = Piecewise((x, x < 1), (x**2, True))
assert latex(p) == r"\begin{cases} x & \text{for}\: x < 1 \\x^{2} &" \
r" \text{otherwise} \end{cases}"
assert latex(p, itex=True) == \
r"\begin{cases} x & \text{for}\: x \lt 1 \\x^{2} &" \
r" \text{otherwise} \end{cases}"
p = Piecewise((x, x < 0), (0, x >= 0))
assert latex(p) == r'\begin{cases} x & \text{for}\: x < 0 \\0 &' \
r' \text{otherwise} \end{cases}'
A, B = symbols("A B", commutative=False)
p = Piecewise((A**2, Eq(A, B)), (A*B, True))
s = r"\begin{cases} A^{2} & \text{for}\: A = B \\A B & \text{otherwise} \end{cases}"
assert latex(p) == s
assert latex(A*p) == r"A \left(%s\right)" % s
assert latex(p*A) == r"\left(%s\right) A" % s
assert latex(Piecewise((x, x < 1), (x**2, x < 2))) == \
r'\begin{cases} x & ' \
r'\text{for}\: x < 1 \\x^{2} & \text{for}\: x < 2 \end{cases}'
def test_latex_Matrix():
M = Matrix([[1 + x, y], [y, x - 1]])
assert latex(M) == \
r'\left[\begin{matrix}x + 1 & y\\y & x - 1\end{matrix}\right]'
assert latex(M, mode='inline') == \
r'$\left[\begin{smallmatrix}x + 1 & y\\' \
r'y & x - 1\end{smallmatrix}\right]$'
assert latex(M, mat_str='array') == \
r'\left[\begin{array}{cc}x + 1 & y\\y & x - 1\end{array}\right]'
assert latex(M, mat_str='bmatrix') == \
r'\left[\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}\right]'
assert latex(M, mat_delim=None, mat_str='bmatrix') == \
r'\begin{bmatrix}x + 1 & y\\y & x - 1\end{bmatrix}'
M2 = Matrix(1, 11, range(11))
assert latex(M2) == \
r'\left[\begin{array}{ccccccccccc}' \
r'0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]'
def test_latex_matrix_with_functions():
t = symbols('t')
theta1 = symbols('theta1', cls=Function)
M = Matrix([[sin(theta1(t)), cos(theta1(t))],
[cos(theta1(t).diff(t)), sin(theta1(t).diff(t))]])
expected = (r'\left[\begin{matrix}\sin{\left('
r'\theta_{1}{\left(t \right)} \right)} & '
r'\cos{\left(\theta_{1}{\left(t \right)} \right)'
r'}\\\cos{\left(\frac{d}{d t} \theta_{1}{\left(t '
r'\right)} \right)} & \sin{\left(\frac{d}{d t} '
r'\theta_{1}{\left(t \right)} \right'
r')}\end{matrix}\right]')
assert latex(M) == expected
def test_latex_NDimArray():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert latex(M) == r"x"
M = ArrayType([[1 / x, y], [z, w]])
M1 = ArrayType([1 / x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
assert latex(M) == \
r'\left[\begin{matrix}\frac{1}{x} & y\\z & w\end{matrix}\right]'
assert latex(M1) == \
r"\left[\begin{matrix}\frac{1}{x} & y & z\end{matrix}\right]"
assert latex(M2) == \
r"\left[\begin{matrix}" \
r"\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right] & " \
r"\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right]" \
r"\end{matrix}\right]"
assert latex(M3) == \
r"""\left[\begin{matrix}"""\
r"""\left[\begin{matrix}\frac{1}{x^{2}} & \frac{y}{x}\\\frac{z}{x} & \frac{w}{x}\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{y}{x} & y^{2}\\y z & w y\end{matrix}\right]\\"""\
r"""\left[\begin{matrix}\frac{z}{x} & y z\\z^{2} & w z\end{matrix}\right] & """\
r"""\left[\begin{matrix}\frac{w}{x} & w y\\w z & w^{2}\end{matrix}\right]"""\
r"""\end{matrix}\right]"""
Mrow = ArrayType([[x, y, 1/z]])
Mcolumn = ArrayType([[x], [y], [1/z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
assert latex(Mrow) == \
r"\left[\left[\begin{matrix}x & y & \frac{1}{z}\end{matrix}\right]\right]"
assert latex(Mcolumn) == \
r"\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]"
assert latex(Mcol2) == \
r'\left[\begin{matrix}\left[\begin{matrix}x\\y\\\frac{1}{z}\end{matrix}\right]\end{matrix}\right]'
def test_latex_mul_symbol():
assert latex(4*4**x, mul_symbol='times') == r"4 \times 4^{x}"
assert latex(4*4**x, mul_symbol='dot') == r"4 \cdot 4^{x}"
assert latex(4*4**x, mul_symbol='ldot') == r"4 \,.\, 4^{x}"
assert latex(4*x, mul_symbol='times') == r"4 \times x"
assert latex(4*x, mul_symbol='dot') == r"4 \cdot x"
assert latex(4*x, mul_symbol='ldot') == r"4 \,.\, x"
def test_latex_issue_4381():
y = 4*4**log(2)
assert latex(y) == r'4 \cdot 4^{\log{\left(2 \right)}}'
assert latex(1/y) == r'\frac{1}{4 \cdot 4^{\log{\left(2 \right)}}}'
def test_latex_issue_4576():
assert latex(Symbol("beta_13_2")) == r"\beta_{13 2}"
assert latex(Symbol("beta_132_20")) == r"\beta_{132 20}"
assert latex(Symbol("beta_13")) == r"\beta_{13}"
assert latex(Symbol("x_a_b")) == r"x_{a b}"
assert latex(Symbol("x_1_2_3")) == r"x_{1 2 3}"
assert latex(Symbol("x_a_b1")) == r"x_{a b1}"
assert latex(Symbol("x_a_1")) == r"x_{a 1}"
assert latex(Symbol("x_1_a")) == r"x_{1 a}"
assert latex(Symbol("x_1^aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_1__aa")) == r"x^{aa}_{1}"
assert latex(Symbol("x_11^a")) == r"x^{a}_{11}"
assert latex(Symbol("x_11__a")) == r"x^{a}_{11}"
assert latex(Symbol("x_a_a_a_a")) == r"x_{a a a a}"
assert latex(Symbol("x_a_a^a^a")) == r"x^{a a}_{a a}"
assert latex(Symbol("x_a_a__a__a")) == r"x^{a a}_{a a}"
assert latex(Symbol("alpha_11")) == r"\alpha_{11}"
assert latex(Symbol("alpha_11_11")) == r"\alpha_{11 11}"
assert latex(Symbol("alpha_alpha")) == r"\alpha_{\alpha}"
assert latex(Symbol("alpha^aleph")) == r"\alpha^{\aleph}"
assert latex(Symbol("alpha__aleph")) == r"\alpha^{\aleph}"
def test_latex_pow_fraction():
x = Symbol('x')
# Testing exp
assert r'e^{-x}' in latex(exp(-x)/2).replace(' ', '') # Remove Whitespace
# Testing e^{-x} in case future changes alter behavior of muls or fracs
# In particular current output is \frac{1}{2}e^{- x} but perhaps this will
# change to \frac{e^{-x}}{2}
# Testing general, non-exp, power
assert r'3^{-x}' in latex(3**-x/2).replace(' ', '')
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
assert latex(A*B*C**-1) == r"A B C^{-1}"
assert latex(C**-1*A*B) == r"C^{-1} A B"
assert latex(A*C**-1*B) == r"A C^{-1} B"
def test_latex_order():
expr = x**3 + x**2*y + y**4 + 3*x*y**3
assert latex(expr, order='lex') == r"x^{3} + x^{2} y + 3 x y^{3} + y^{4}"
assert latex(
expr, order='rev-lex') == r"y^{4} + 3 x y^{3} + x^{2} y + x^{3}"
assert latex(expr, order='none') == r"x^{3} + y^{4} + y x^{2} + 3 x y^{3}"
def test_latex_Lambda():
assert latex(Lambda(x, x + 1)) == r"\left( x \mapsto x + 1 \right)"
assert latex(Lambda((x, y), x + 1)) == r"\left( \left( x, \ y\right) \mapsto x + 1 \right)"
assert latex(Lambda(x, x)) == r"\left( x \mapsto x \right)"
def test_latex_PolyElement():
Ruv, u, v = ring("u,v", ZZ)
Rxyz, x, y, z = ring("x,y,z", Ruv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + u + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + u + 1"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x"
assert latex((u**2 + 3*u*v + 1)*x**2*y + (u + 1)*x + 1) == \
r"\left({u}^{2} + 3 u v + 1\right) {x}^{2} y + \left(u + 1\right) x + 1"
assert latex((-u**2 + 3*u*v - 1)*x**2*y - (u + 1)*x - 1) == \
r"-\left({u}^{2} - 3 u v + 1\right) {x}^{2} y - \left(u + 1\right) x - 1"
assert latex(-(v**2 + v + 1)*x + 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x + 3 u v + 1"
assert latex(-(v**2 + v + 1)*x - 3*u*v + 1) == \
r"-\left({v}^{2} + v + 1\right) x - 3 u v + 1"
def test_latex_FracElement():
Fuv, u, v = field("u,v", ZZ)
Fxyzt, x, y, z, t = field("x,y,z,t", Fuv)
assert latex(x - x) == r"0"
assert latex(x - 1) == r"x - 1"
assert latex(x + 1) == r"x + 1"
assert latex(x/3) == r"\frac{x}{3}"
assert latex(x/z) == r"\frac{x}{z}"
assert latex(x*y/z) == r"\frac{x y}{z}"
assert latex(x/(z*t)) == r"\frac{x}{z t}"
assert latex(x*y/(z*t)) == r"\frac{x y}{z t}"
assert latex((x - 1)/y) == r"\frac{x - 1}{y}"
assert latex((x + 1)/y) == r"\frac{x + 1}{y}"
assert latex((-x - 1)/y) == r"\frac{-x - 1}{y}"
assert latex((x + 1)/(y*z)) == r"\frac{x + 1}{y z}"
assert latex(-y/(x + 1)) == r"\frac{-y}{x + 1}"
assert latex(y*z/(x + 1)) == r"\frac{y z}{x + 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - 1}"
assert latex(((u + 1)*x*y + 1)/((v - 1)*z - t*u*v - 1)) == \
r"\frac{\left(u + 1\right) x y + 1}{\left(v - 1\right) z - u v t - 1}"
def test_latex_Poly():
assert latex(Poly(x**2 + 2 * x, x)) == \
r"\operatorname{Poly}{\left( x^{2} + 2 x, x, domain=\mathbb{Z} \right)}"
assert latex(Poly(x/y, x)) == \
r"\operatorname{Poly}{\left( \frac{1}{y} x, x, domain=\mathbb{Z}\left(y\right) \right)}"
assert latex(Poly(2.0*x + y)) == \
r"\operatorname{Poly}{\left( 2.0 x + 1.0 y, x, y, domain=\mathbb{R} \right)}"
def test_latex_Poly_order():
assert latex(Poly([a, 1, b, 2, c, 3], x)) == \
r'\operatorname{Poly}{\left( a x^{5} + x^{4} + b x^{3} + 2 x^{2} + c'\
r' x + 3, x, domain=\mathbb{Z}\left[a, b, c\right] \right)}'
assert latex(Poly([a, 1, b+c, 2, 3], x)) == \
r'\operatorname{Poly}{\left( a x^{4} + x^{3} + \left(b + c\right) '\
r'x^{2} + 2 x + 3, x, domain=\mathbb{Z}\left[a, b, c\right] \right)}'
assert latex(Poly(a*x**3 + x**2*y - x*y - c*y**3 - b*x*y**2 + y - a*x + b,
(x, y))) == \
r'\operatorname{Poly}{\left( a x^{3} + x^{2}y - b xy^{2} - xy - '\
r'a x - c y^{3} + y + b, x, y, domain=\mathbb{Z}\left[a, b, c\right] \right)}'
def test_latex_ComplexRootOf():
assert latex(rootof(x**5 + x + 3, 0)) == \
r"\operatorname{CRootOf} {\left(x^{5} + x + 3, 0\right)}"
def test_latex_RootSum():
assert latex(RootSum(x**5 + x + 3, sin)) == \
r"\operatorname{RootSum} {\left(x^{5} + x + 3, \left( x \mapsto \sin{\left(x \right)} \right)\right)}"
def test_settings():
raises(TypeError, lambda: latex(x*y, method="garbage"))
def test_latex_numbers():
assert latex(catalan(n)) == r"C_{n}"
assert latex(catalan(n)**2) == r"C_{n}^{2}"
assert latex(bernoulli(n)) == r"B_{n}"
assert latex(bernoulli(n, x)) == r"B_{n}\left(x\right)"
assert latex(bernoulli(n)**2) == r"B_{n}^{2}"
assert latex(bernoulli(n, x)**2) == r"B_{n}^{2}\left(x\right)"
assert latex(bell(n)) == r"B_{n}"
assert latex(bell(n, x)) == r"B_{n}\left(x\right)"
assert latex(bell(n, m, (x, y))) == r"B_{n, m}\left(x, y\right)"
assert latex(bell(n)**2) == r"B_{n}^{2}"
assert latex(bell(n, x)**2) == r"B_{n}^{2}\left(x\right)"
assert latex(bell(n, m, (x, y))**2) == r"B_{n, m}^{2}\left(x, y\right)"
assert latex(fibonacci(n)) == r"F_{n}"
assert latex(fibonacci(n, x)) == r"F_{n}\left(x\right)"
assert latex(fibonacci(n)**2) == r"F_{n}^{2}"
assert latex(fibonacci(n, x)**2) == r"F_{n}^{2}\left(x\right)"
assert latex(lucas(n)) == r"L_{n}"
assert latex(lucas(n)**2) == r"L_{n}^{2}"
assert latex(tribonacci(n)) == r"T_{n}"
assert latex(tribonacci(n, x)) == r"T_{n}\left(x\right)"
assert latex(tribonacci(n)**2) == r"T_{n}^{2}"
assert latex(tribonacci(n, x)**2) == r"T_{n}^{2}\left(x\right)"
def test_latex_euler():
assert latex(euler(n)) == r"E_{n}"
assert latex(euler(n, x)) == r"E_{n}\left(x\right)"
assert latex(euler(n, x)**2) == r"E_{n}^{2}\left(x\right)"
def test_lamda():
assert latex(Symbol('lamda')) == r"\lambda"
assert latex(Symbol('Lamda')) == r"\Lambda"
def test_custom_symbol_names():
x = Symbol('x')
y = Symbol('y')
assert latex(x) == r"x"
assert latex(x, symbol_names={x: "x_i"}) == r"x_i"
assert latex(x + y, symbol_names={x: "x_i"}) == r"x_i + y"
assert latex(x**2, symbol_names={x: "x_i"}) == r"x_i^{2}"
assert latex(x + y, symbol_names={x: "x_i", y: "y_j"}) == r"x_i + y_j"
def test_matAdd():
C = MatrixSymbol('C', 5, 5)
B = MatrixSymbol('B', 5, 5)
n = symbols("n")
h = MatrixSymbol("h", 1, 1)
assert latex(C - 2*B) in [r'- 2 B + C', r'C -2 B']
assert latex(C + 2*B) in [r'2 B + C', r'C + 2 B']
assert latex(B - 2*C) in [r'B - 2 C', r'- 2 C + B']
assert latex(B + 2*C) in [r'B + 2 C', r'2 C + B']
assert latex(n * h - (-h + h.T) * (h + h.T)) == 'n h - \\left(- h + h^{T}\\right) \\left(h + h^{T}\\right)'
assert latex(MatAdd(MatAdd(h, h), MatAdd(h, h))) == '\\left(h + h\\right) + \\left(h + h\\right)'
assert latex(MatMul(MatMul(h, h), MatMul(h, h))) == '\\left(h h\\right) \\left(h h\\right)'
def test_matMul():
A = MatrixSymbol('A', 5, 5)
B = MatrixSymbol('B', 5, 5)
x = Symbol('x')
assert latex(2*A) == r'2 A'
assert latex(2*x*A) == r'2 x A'
assert latex(-2*A) == r'- 2 A'
assert latex(1.5*A) == r'1.5 A'
assert latex(sqrt(2)*A) == r'\sqrt{2} A'
assert latex(-sqrt(2)*A) == r'- \sqrt{2} A'
assert latex(2*sqrt(2)*x*A) == r'2 \sqrt{2} x A'
assert latex(-2*A*(A + 2*B)) in [r'- 2 A \left(A + 2 B\right)',
r'- 2 A \left(2 B + A\right)']
def test_latex_MatrixSlice():
n = Symbol('n', integer=True)
x, y, z, w, t, = symbols('x y z w t')
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
assert latex(MatrixSlice(X, (None, None, None), (None, None, None))) == r'X\left[:, :\right]'
assert latex(X[x:x + 1, y:y + 1]) == r'X\left[x:x + 1, y:y + 1\right]'
assert latex(X[x:x + 1:2, y:y + 1:2]) == r'X\left[x:x + 1:2, y:y + 1:2\right]'
assert latex(X[:x, y:]) == r'X\left[:x, y:\right]'
assert latex(X[:x, y:]) == r'X\left[:x, y:\right]'
assert latex(X[x:, :y]) == r'X\left[x:, :y\right]'
assert latex(X[x:y, z:w]) == r'X\left[x:y, z:w\right]'
assert latex(X[x:y:t, w:t:x]) == r'X\left[x:y:t, w:t:x\right]'
assert latex(X[x::y, t::w]) == r'X\left[x::y, t::w\right]'
assert latex(X[:x:y, :t:w]) == r'X\left[:x:y, :t:w\right]'
assert latex(X[::x, ::y]) == r'X\left[::x, ::y\right]'
assert latex(MatrixSlice(X, (0, None, None), (0, None, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (None, n, None), (None, n, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (0, n, None), (0, n, None))) == r'X\left[:, :\right]'
assert latex(MatrixSlice(X, (0, n, 2), (0, n, 2))) == r'X\left[::2, ::2\right]'
assert latex(X[1:2:3, 4:5:6]) == r'X\left[1:2:3, 4:5:6\right]'
assert latex(X[1:3:5, 4:6:8]) == r'X\left[1:3:5, 4:6:8\right]'
assert latex(X[1:10:2]) == r'X\left[1:10:2, :\right]'
assert latex(Y[:5, 1:9:2]) == r'Y\left[:5, 1:9:2\right]'
assert latex(Y[:5, 1:10:2]) == r'Y\left[:5, 1::2\right]'
assert latex(Y[5, :5:2]) == r'Y\left[5:6, :5:2\right]'
assert latex(X[0:1, 0:1]) == r'X\left[:1, :1\right]'
assert latex(X[0:1:2, 0:1:2]) == r'X\left[:1:2, :1:2\right]'
assert latex((Y + Z)[2:, 2:]) == r'\left(Y + Z\right)\left[2:, 2:\right]'
def test_latex_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
from sympy.stats.rv import RandomDomain
X = Normal('x1', 0, 1)
assert latex(where(X > 0)) == r"\text{Domain: }0 < x_{1} \wedge x_{1} < \infty"
D = Die('d1', 6)
assert latex(where(D > 4)) == r"\text{Domain: }d_{1} = 5 \vee d_{1} = 6"
A = Exponential('a', 1)
B = Exponential('b', 1)
assert latex(
pspace(Tuple(A, B)).domain) == \
r"\text{Domain: }0 \leq a \wedge 0 \leq b \wedge a < \infty \wedge b < \infty"
assert latex(RandomDomain(FiniteSet(x), FiniteSet(1, 2))) == \
r'\text{Domain: }\left\{x\right\} \in \left\{1, 2\right\}'
def test_PrettyPoly():
from sympy.polys.domains import QQ
F = QQ.frac_field(x, y)
R = QQ[x, y]
assert latex(F.convert(x/(x + y))) == latex(x/(x + y))
assert latex(R.convert(x + y)) == latex(x + y)
def test_integral_transforms():
x = Symbol("x")
k = Symbol("k")
f = Function("f")
a = Symbol("a")
b = Symbol("b")
assert latex(MellinTransform(f(x), x, k)) == \
r"\mathcal{M}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseMellinTransform(f(k), k, x, a, b)) == \
r"\mathcal{M}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(LaplaceTransform(f(x), x, k)) == \
r"\mathcal{L}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseLaplaceTransform(f(k), k, x, (a, b))) == \
r"\mathcal{L}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(FourierTransform(f(x), x, k)) == \
r"\mathcal{F}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseFourierTransform(f(k), k, x)) == \
r"\mathcal{F}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(CosineTransform(f(x), x, k)) == \
r"\mathcal{COS}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseCosineTransform(f(k), k, x)) == \
r"\mathcal{COS}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
assert latex(SineTransform(f(x), x, k)) == \
r"\mathcal{SIN}_{x}\left[f{\left(x \right)}\right]\left(k\right)"
assert latex(InverseSineTransform(f(k), k, x)) == \
r"\mathcal{SIN}^{-1}_{k}\left[f{\left(k \right)}\right]\left(x\right)"
def test_PolynomialRingBase():
from sympy.polys.domains import QQ
assert latex(QQ.old_poly_ring(x, y)) == r"\mathbb{Q}\left[x, y\right]"
assert latex(QQ.old_poly_ring(x, y, order="ilex")) == \
r"S_<^{-1}\mathbb{Q}\left[x, y\right]"
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram,
DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert latex(A1) == r"A_{1}"
assert latex(f1) == r"f_{1}:A_{1}\rightarrow A_{2}"
assert latex(id_A1) == r"id:A_{1}\rightarrow A_{1}"
assert latex(f2*f1) == r"f_{2}\circ f_{1}:A_{1}\rightarrow A_{3}"
assert latex(K1) == r"\mathbf{K_{1}}"
d = Diagram()
assert latex(d) == r"\emptyset"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}, " \
r"\ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}"
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert latex(d) == r"\left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \emptyset, \ id:A_{1}\rightarrow " \
r"A_{1} : \emptyset, \ id:A_{2}\rightarrow A_{2} : " \
r"\emptyset, \ id:A_{3}\rightarrow A_{3} : \emptyset, " \
r"\ f_{1}:A_{1}\rightarrow A_{2} : \left\{unique\right\}," \
r" \ f_{2}:A_{2}\rightarrow A_{3} : \emptyset\right\}" \
r"\Longrightarrow \left\{ f_{2}\circ f_{1}:A_{1}" \
r"\rightarrow A_{3} : \left\{unique\right\}\right\}"
# A linear diagram.
A = Object("A")
B = Object("B")
C = Object("C")
f = NamedMorphism(A, B, "f")
g = NamedMorphism(B, C, "g")
d = Diagram([f, g])
grid = DiagramGrid(d)
assert latex(grid) == r"\begin{array}{cc}" + "\n" \
r"A & B \\" + "\n" \
r" & C " + "\n" \
r"\end{array}" + "\n"
def test_Modules():
from sympy.polys.domains import QQ
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
assert latex(F) == r"{\mathbb{Q}\left[x, y\right]}^{2}"
assert latex(M) == \
r"\left\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle"
I = R.ideal(x**2, y)
assert latex(I) == r"\left\langle {x^{2}},{y} \right\rangle"
Q = F / M
assert latex(Q) == \
r"\frac{{\mathbb{Q}\left[x, y\right]}^{2}}{\left\langle {\left[ {x},"\
r"{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}"
assert latex(Q.submodule([1, x**3/2], [2, y])) == \
r"\left\langle {{\left[ {1},{\frac{x^{3}}{2}} \right]} + {\left"\
r"\langle {\left[ {x},{y} \right]},{\left[ {1},{x^{2}} \right]} "\
r"\right\rangle}},{{\left[ {2},{y} \right]} + {\left\langle {\left[ "\
r"{x},{y} \right]},{\left[ {1},{x^{2}} \right]} \right\rangle}} \right\rangle"
h = homomorphism(QQ.old_poly_ring(x).free_module(2),
QQ.old_poly_ring(x).free_module(2), [0, 0])
assert latex(h) == \
r"{\left[\begin{matrix}0 & 0\\0 & 0\end{matrix}\right]} : "\
r"{{\mathbb{Q}\left[x\right]}^{2}} \to {{\mathbb{Q}\left[x\right]}^{2}}"
def test_QuotientRing():
from sympy.polys.domains import QQ
R = QQ.old_poly_ring(x)/[x**2 + 1]
assert latex(R) == \
r"\frac{\mathbb{Q}\left[x\right]}{\left\langle {x^{2} + 1} \right\rangle}"
assert latex(R.one) == r"{1} + {\left\langle {x^{2} + 1} \right\rangle}"
def test_Tr():
#TODO: Handle indices
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert latex(t) == r'\operatorname{tr}\left(A B\right)'
def test_Determinant():
from sympy.matrices import Determinant, Inverse, BlockMatrix, OneMatrix, ZeroMatrix
m = Matrix(((1, 2), (3, 4)))
assert latex(Determinant(m)) == '\\left|{\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}}\\right|'
assert latex(Determinant(Inverse(m))) == \
'\\left|{\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{-1}}\\right|'
X = MatrixSymbol('X', 2, 2)
assert latex(Determinant(X)) == '\\left|{X}\\right|'
assert latex(Determinant(X + m)) == \
'\\left|{\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X}\\right|'
assert latex(Determinant(BlockMatrix(((OneMatrix(2, 2), X),
(m, ZeroMatrix(2, 2)))))) == \
'\\left|{\\begin{matrix}1 & X\\\\\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] & 0\\end{matrix}}\\right|'
def test_Adjoint():
from sympy.matrices import Adjoint, Inverse, Transpose
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Adjoint(X)) == r'X^{\dagger}'
assert latex(Adjoint(X + Y)) == r'\left(X + Y\right)^{\dagger}'
assert latex(Adjoint(X) + Adjoint(Y)) == r'X^{\dagger} + Y^{\dagger}'
assert latex(Adjoint(X*Y)) == r'\left(X Y\right)^{\dagger}'
assert latex(Adjoint(Y)*Adjoint(X)) == r'Y^{\dagger} X^{\dagger}'
assert latex(Adjoint(X**2)) == r'\left(X^{2}\right)^{\dagger}'
assert latex(Adjoint(X)**2) == r'\left(X^{\dagger}\right)^{2}'
assert latex(Adjoint(Inverse(X))) == r'\left(X^{-1}\right)^{\dagger}'
assert latex(Inverse(Adjoint(X))) == r'\left(X^{\dagger}\right)^{-1}'
assert latex(Adjoint(Transpose(X))) == r'\left(X^{T}\right)^{\dagger}'
assert latex(Transpose(Adjoint(X))) == r'\left(X^{\dagger}\right)^{T}'
assert latex(Transpose(Adjoint(X) + Y)) == r'\left(X^{\dagger} + Y\right)^{T}'
m = Matrix(((1, 2), (3, 4)))
assert latex(Adjoint(m)) == '\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{\\dagger}'
assert latex(Adjoint(m+X)) == \
'\\left(\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X\\right)^{\\dagger}'
from sympy.matrices import BlockMatrix, OneMatrix, ZeroMatrix
assert latex(Adjoint(BlockMatrix(((OneMatrix(2, 2), X),
(m, ZeroMatrix(2, 2)))))) == \
'\\left[\\begin{matrix}1 & X\\\\\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] & 0\\end{matrix}\\right]^{\\dagger}'
# Issue 20959
Mx = MatrixSymbol('M^x', 2, 2)
assert latex(Adjoint(Mx)) == r'\left(M^{x}\right)^{\dagger}'
def test_Transpose():
from sympy.matrices import Transpose, MatPow, HadamardPower
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(Transpose(X)) == r'X^{T}'
assert latex(Transpose(X + Y)) == r'\left(X + Y\right)^{T}'
assert latex(Transpose(HadamardPower(X, 2))) == r'\left(X^{\circ {2}}\right)^{T}'
assert latex(HadamardPower(Transpose(X), 2)) == r'\left(X^{T}\right)^{\circ {2}}'
assert latex(Transpose(MatPow(X, 2))) == r'\left(X^{2}\right)^{T}'
assert latex(MatPow(Transpose(X), 2)) == r'\left(X^{T}\right)^{2}'
m = Matrix(((1, 2), (3, 4)))
assert latex(Transpose(m)) == '\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]^{T}'
assert latex(Transpose(m+X)) == \
'\\left(\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] + X\\right)^{T}'
from sympy.matrices import BlockMatrix, OneMatrix, ZeroMatrix
assert latex(Transpose(BlockMatrix(((OneMatrix(2, 2), X),
(m, ZeroMatrix(2, 2)))))) == \
'\\left[\\begin{matrix}1 & X\\\\\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right] & 0\\end{matrix}\\right]^{T}'
# Issue 20959
Mx = MatrixSymbol('M^x', 2, 2)
assert latex(Transpose(Mx)) == r'\left(M^{x}\right)^{T}'
def test_Hadamard():
from sympy.matrices import HadamardProduct, HadamardPower
from sympy.matrices.expressions import MatAdd, MatMul, MatPow
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(HadamardProduct(X, Y*Y)) == r'X \circ Y^{2}'
assert latex(HadamardProduct(X, Y)*Y) == r'\left(X \circ Y\right) Y'
assert latex(HadamardPower(X, 2)) == r'X^{\circ {2}}'
assert latex(HadamardPower(X, -1)) == r'X^{\circ \left({-1}\right)}'
assert latex(HadamardPower(MatAdd(X, Y), 2)) == \
r'\left(X + Y\right)^{\circ {2}}'
assert latex(HadamardPower(MatMul(X, Y), 2)) == \
r'\left(X Y\right)^{\circ {2}}'
assert latex(HadamardPower(MatPow(X, -1), -1)) == \
r'\left(X^{-1}\right)^{\circ \left({-1}\right)}'
assert latex(MatPow(HadamardPower(X, -1), -1)) == \
r'\left(X^{\circ \left({-1}\right)}\right)^{-1}'
assert latex(HadamardPower(X, n+1)) == \
r'X^{\circ \left({n + 1}\right)}'
def test_MatPow():
from sympy.matrices.expressions import MatPow
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert latex(MatPow(X, 2)) == 'X^{2}'
assert latex(MatPow(X*X, 2)) == '\\left(X^{2}\\right)^{2}'
assert latex(MatPow(X*Y, 2)) == '\\left(X Y\\right)^{2}'
assert latex(MatPow(X + Y, 2)) == '\\left(X + Y\\right)^{2}'
assert latex(MatPow(X + X, 2)) == '\\left(2 X\\right)^{2}'
# Issue 20959
Mx = MatrixSymbol('M^x', 2, 2)
assert latex(MatPow(Mx, 2)) == r'\left(M^{x}\right)^{2}'
def test_ElementwiseApplyFunction():
X = MatrixSymbol('X', 2, 2)
expr = (X.T*X).applyfunc(sin)
assert latex(expr) == r"{\left( d \mapsto \sin{\left(d \right)} \right)}_{\circ}\left({X^{T} X}\right)"
expr = X.applyfunc(Lambda(x, 1/x))
assert latex(expr) == r'{\left( x \mapsto \frac{1}{x} \right)}_{\circ}\left({X}\right)'
def test_ZeroMatrix():
from sympy.matrices.expressions.special import ZeroMatrix
assert latex(ZeroMatrix(1, 1), mat_symbol_style='plain') == r"0"
assert latex(ZeroMatrix(1, 1), mat_symbol_style='bold') == r"\mathbf{0}"
def test_OneMatrix():
from sympy.matrices.expressions.special import OneMatrix
assert latex(OneMatrix(3, 4), mat_symbol_style='plain') == r"1"
assert latex(OneMatrix(3, 4), mat_symbol_style='bold') == r"\mathbf{1}"
def test_Identity():
from sympy.matrices.expressions.special import Identity
assert latex(Identity(1), mat_symbol_style='plain') == r"\mathbb{I}"
assert latex(Identity(1), mat_symbol_style='bold') == r"\mathbf{I}"
def test_latex_DFT_IDFT():
from sympy.matrices.expressions.fourier import DFT, IDFT
assert latex(DFT(13)) == r"\text{DFT}_{13}"
assert latex(IDFT(x)) == r"\text{IDFT}_{x}"
def test_boolean_args_order():
syms = symbols('a:f')
expr = And(*syms)
assert latex(expr) == r'a \wedge b \wedge c \wedge d \wedge e \wedge f'
expr = Or(*syms)
assert latex(expr) == r'a \vee b \vee c \vee d \vee e \vee f'
expr = Equivalent(*syms)
assert latex(expr) == \
r'a \Leftrightarrow b \Leftrightarrow c \Leftrightarrow d \Leftrightarrow e \Leftrightarrow f'
expr = Xor(*syms)
assert latex(expr) == \
r'a \veebar b \veebar c \veebar d \veebar e \veebar f'
def test_imaginary():
i = sqrt(-1)
assert latex(i) == r'i'
def test_builtins_without_args():
assert latex(sin) == r'\sin'
assert latex(cos) == r'\cos'
assert latex(tan) == r'\tan'
assert latex(log) == r'\log'
assert latex(Ei) == r'\operatorname{Ei}'
assert latex(zeta) == r'\zeta'
def test_latex_greek_functions():
# bug because capital greeks that have roman equivalents should not use
# \Alpha, \Beta, \Eta, etc.
s = Function('Alpha')
assert latex(s) == r'\mathrm{A}'
assert latex(s(x)) == r'\mathrm{A}{\left(x \right)}'
s = Function('Beta')
assert latex(s) == r'\mathrm{B}'
s = Function('Eta')
assert latex(s) == r'\mathrm{H}'
assert latex(s(x)) == r'\mathrm{H}{\left(x \right)}'
# bug because sympy.core.numbers.Pi is special
p = Function('Pi')
# assert latex(p(x)) == r'\Pi{\left(x \right)}'
assert latex(p) == r'\Pi'
# bug because not all greeks are included
c = Function('chi')
assert latex(c(x)) == r'\chi{\left(x \right)}'
assert latex(c) == r'\chi'
def test_translate():
s = 'Alpha'
assert translate(s) == r'\mathrm{A}'
s = 'Beta'
assert translate(s) == r'\mathrm{B}'
s = 'Eta'
assert translate(s) == r'\mathrm{H}'
s = 'omicron'
assert translate(s) == r'o'
s = 'Pi'
assert translate(s) == r'\Pi'
s = 'pi'
assert translate(s) == r'\pi'
s = 'LamdaHatDOT'
assert translate(s) == r'\dot{\hat{\Lambda}}'
def test_other_symbols():
from sympy.printing.latex import other_symbols
for s in other_symbols:
assert latex(symbols(s)) == r"" "\\" + s
def test_modifiers():
# Test each modifier individually in the simplest case
# (with funny capitalizations)
assert latex(symbols("xMathring")) == r"\mathring{x}"
assert latex(symbols("xCheck")) == r"\check{x}"
assert latex(symbols("xBreve")) == r"\breve{x}"
assert latex(symbols("xAcute")) == r"\acute{x}"
assert latex(symbols("xGrave")) == r"\grave{x}"
assert latex(symbols("xTilde")) == r"\tilde{x}"
assert latex(symbols("xPrime")) == r"{x}'"
assert latex(symbols("xddDDot")) == r"\ddddot{x}"
assert latex(symbols("xDdDot")) == r"\dddot{x}"
assert latex(symbols("xDDot")) == r"\ddot{x}"
assert latex(symbols("xBold")) == r"\boldsymbol{x}"
assert latex(symbols("xnOrM")) == r"\left\|{x}\right\|"
assert latex(symbols("xAVG")) == r"\left\langle{x}\right\rangle"
assert latex(symbols("xHat")) == r"\hat{x}"
assert latex(symbols("xDot")) == r"\dot{x}"
assert latex(symbols("xBar")) == r"\bar{x}"
assert latex(symbols("xVec")) == r"\vec{x}"
assert latex(symbols("xAbs")) == r"\left|{x}\right|"
assert latex(symbols("xMag")) == r"\left|{x}\right|"
assert latex(symbols("xPrM")) == r"{x}'"
assert latex(symbols("xBM")) == r"\boldsymbol{x}"
# Test strings that are *only* the names of modifiers
assert latex(symbols("Mathring")) == r"Mathring"
assert latex(symbols("Check")) == r"Check"
assert latex(symbols("Breve")) == r"Breve"
assert latex(symbols("Acute")) == r"Acute"
assert latex(symbols("Grave")) == r"Grave"
assert latex(symbols("Tilde")) == r"Tilde"
assert latex(symbols("Prime")) == r"Prime"
assert latex(symbols("DDot")) == r"\dot{D}"
assert latex(symbols("Bold")) == r"Bold"
assert latex(symbols("NORm")) == r"NORm"
assert latex(symbols("AVG")) == r"AVG"
assert latex(symbols("Hat")) == r"Hat"
assert latex(symbols("Dot")) == r"Dot"
assert latex(symbols("Bar")) == r"Bar"
assert latex(symbols("Vec")) == r"Vec"
assert latex(symbols("Abs")) == r"Abs"
assert latex(symbols("Mag")) == r"Mag"
assert latex(symbols("PrM")) == r"PrM"
assert latex(symbols("BM")) == r"BM"
assert latex(symbols("hbar")) == r"\hbar"
# Check a few combinations
assert latex(symbols("xvecdot")) == r"\dot{\vec{x}}"
assert latex(symbols("xDotVec")) == r"\vec{\dot{x}}"
assert latex(symbols("xHATNorm")) == r"\left\|{\hat{x}}\right\|"
# Check a couple big, ugly combinations
assert latex(symbols('xMathringBm_yCheckPRM__zbreveAbs')) == \
r"\boldsymbol{\mathring{x}}^{\left|{\breve{z}}\right|}_{{\check{y}}'}"
assert latex(symbols('alphadothat_nVECDOT__tTildePrime')) == \
r"\hat{\dot{\alpha}}^{{\tilde{t}}'}_{\dot{\vec{n}}}"
def test_greek_symbols():
assert latex(Symbol('alpha')) == r'\alpha'
assert latex(Symbol('beta')) == r'\beta'
assert latex(Symbol('gamma')) == r'\gamma'
assert latex(Symbol('delta')) == r'\delta'
assert latex(Symbol('epsilon')) == r'\epsilon'
assert latex(Symbol('zeta')) == r'\zeta'
assert latex(Symbol('eta')) == r'\eta'
assert latex(Symbol('theta')) == r'\theta'
assert latex(Symbol('iota')) == r'\iota'
assert latex(Symbol('kappa')) == r'\kappa'
assert latex(Symbol('lambda')) == r'\lambda'
assert latex(Symbol('mu')) == r'\mu'
assert latex(Symbol('nu')) == r'\nu'
assert latex(Symbol('xi')) == r'\xi'
assert latex(Symbol('omicron')) == r'o'
assert latex(Symbol('pi')) == r'\pi'
assert latex(Symbol('rho')) == r'\rho'
assert latex(Symbol('sigma')) == r'\sigma'
assert latex(Symbol('tau')) == r'\tau'
assert latex(Symbol('upsilon')) == r'\upsilon'
assert latex(Symbol('phi')) == r'\phi'
assert latex(Symbol('chi')) == r'\chi'
assert latex(Symbol('psi')) == r'\psi'
assert latex(Symbol('omega')) == r'\omega'
assert latex(Symbol('Alpha')) == r'\mathrm{A}'
assert latex(Symbol('Beta')) == r'\mathrm{B}'
assert latex(Symbol('Gamma')) == r'\Gamma'
assert latex(Symbol('Delta')) == r'\Delta'
assert latex(Symbol('Epsilon')) == r'\mathrm{E}'
assert latex(Symbol('Zeta')) == r'\mathrm{Z}'
assert latex(Symbol('Eta')) == r'\mathrm{H}'
assert latex(Symbol('Theta')) == r'\Theta'
assert latex(Symbol('Iota')) == r'\mathrm{I}'
assert latex(Symbol('Kappa')) == r'\mathrm{K}'
assert latex(Symbol('Lambda')) == r'\Lambda'
assert latex(Symbol('Mu')) == r'\mathrm{M}'
assert latex(Symbol('Nu')) == r'\mathrm{N}'
assert latex(Symbol('Xi')) == r'\Xi'
assert latex(Symbol('Omicron')) == r'\mathrm{O}'
assert latex(Symbol('Pi')) == r'\Pi'
assert latex(Symbol('Rho')) == r'\mathrm{P}'
assert latex(Symbol('Sigma')) == r'\Sigma'
assert latex(Symbol('Tau')) == r'\mathrm{T}'
assert latex(Symbol('Upsilon')) == r'\Upsilon'
assert latex(Symbol('Phi')) == r'\Phi'
assert latex(Symbol('Chi')) == r'\mathrm{X}'
assert latex(Symbol('Psi')) == r'\Psi'
assert latex(Symbol('Omega')) == r'\Omega'
assert latex(Symbol('varepsilon')) == r'\varepsilon'
assert latex(Symbol('varkappa')) == r'\varkappa'
assert latex(Symbol('varphi')) == r'\varphi'
assert latex(Symbol('varpi')) == r'\varpi'
assert latex(Symbol('varrho')) == r'\varrho'
assert latex(Symbol('varsigma')) == r'\varsigma'
assert latex(Symbol('vartheta')) == r'\vartheta'
def test_fancyset_symbols():
assert latex(S.Rationals) == r'\mathbb{Q}'
assert latex(S.Naturals) == r'\mathbb{N}'
assert latex(S.Naturals0) == r'\mathbb{N}_0'
assert latex(S.Integers) == r'\mathbb{Z}'
assert latex(S.Reals) == r'\mathbb{R}'
assert latex(S.Complexes) == r'\mathbb{C}'
@XFAIL
def test_builtin_without_args_mismatched_names():
assert latex(CosineTransform) == r'\mathcal{COS}'
def test_builtin_no_args():
assert latex(Chi) == r'\operatorname{Chi}'
assert latex(beta) == r'\operatorname{B}'
assert latex(gamma) == r'\Gamma'
assert latex(KroneckerDelta) == r'\delta'
assert latex(DiracDelta) == r'\delta'
assert latex(lowergamma) == r'\gamma'
def test_issue_6853():
p = Function('Pi')
assert latex(p(x)) == r"\Pi{\left(x \right)}"
def test_Mul():
e = Mul(-2, x + 1, evaluate=False)
assert latex(e) == r'- 2 \left(x + 1\right)'
e = Mul(2, x + 1, evaluate=False)
assert latex(e) == r'2 \left(x + 1\right)'
e = Mul(S.Half, x + 1, evaluate=False)
assert latex(e) == r'\frac{x + 1}{2}'
e = Mul(y, x + 1, evaluate=False)
assert latex(e) == r'y \left(x + 1\right)'
e = Mul(-y, x + 1, evaluate=False)
assert latex(e) == r'- y \left(x + 1\right)'
e = Mul(-2, x + 1)
assert latex(e) == r'- 2 x - 2'
e = Mul(2, x + 1)
assert latex(e) == r'2 x + 2'
def test_Pow():
e = Pow(2, 2, evaluate=False)
assert latex(e) == r'2^{2}'
assert latex(x**(Rational(-1, 3))) == r'\frac{1}{\sqrt[3]{x}}'
x2 = Symbol(r'x^2')
assert latex(x2**2) == r'\left(x^{2}\right)^{2}'
def test_issue_7180():
assert latex(Equivalent(x, y)) == r"x \Leftrightarrow y"
assert latex(Not(Equivalent(x, y))) == r"x \not\Leftrightarrow y"
def test_issue_8409():
assert latex(S.Half**n) == r"\left(\frac{1}{2}\right)^{n}"
def test_issue_8470():
from sympy.parsing.sympy_parser import parse_expr
e = parse_expr("-B*A", evaluate=False)
assert latex(e) == r"A \left(- B\right)"
def test_issue_15439():
x = MatrixSymbol('x', 2, 2)
y = MatrixSymbol('y', 2, 2)
assert latex((x * y).subs(y, -y)) == r"x \left(- y\right)"
assert latex((x * y).subs(y, -2*y)) == r"x \left(- 2 y\right)"
assert latex((x * y).subs(x, -x)) == r"\left(- x\right) y"
def test_issue_2934():
assert latex(Symbol(r'\frac{a_1}{b_1}')) == r'\frac{a_1}{b_1}'
def test_issue_10489():
latexSymbolWithBrace = r'C_{x_{0}}'
s = Symbol(latexSymbolWithBrace)
assert latex(s) == latexSymbolWithBrace
assert latex(cos(s)) == r'\cos{\left(C_{x_{0}} \right)}'
def test_issue_12886():
m__1, l__1 = symbols('m__1, l__1')
assert latex(m__1**2 + l__1**2) == \
r'\left(l^{1}\right)^{2} + \left(m^{1}\right)^{2}'
def test_issue_13559():
from sympy.parsing.sympy_parser import parse_expr
expr = parse_expr('5/1', evaluate=False)
assert latex(expr) == r"\frac{5}{1}"
def test_issue_13651():
expr = c + Mul(-1, a + b, evaluate=False)
assert latex(expr) == r"c - \left(a + b\right)"
def test_latex_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
assert latex(he) == latex(1/x) == r"\frac{1}{x}"
assert latex(he**2) == r"\left(\frac{1}{x}\right)^{2}"
assert latex(he + 1) == r"1 + \frac{1}{x}"
assert latex(x*he) == r"x \frac{1}{x}"
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert latex(A[0, 0]) == r"A_{0, 0}"
assert latex(3 * A[0, 0]) == r"3 A_{0, 0}"
F = C[0, 0].subs(C, A - B)
assert latex(F) == r"\left(A - B\right)_{0, 0}"
i, j, k = symbols("i j k")
M = MatrixSymbol("M", k, k)
N = MatrixSymbol("N", k, k)
assert latex((M*N)[i, j]) == \
r'\sum_{i_{1}=0}^{k - 1} M_{i, i_{1}} N_{i_{1}, j}'
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A) == r"- A"
assert latex(A - A*B - B) == r"A - A B - B"
assert latex(-A*B - A*B*C - B) == r"- A B - A B C - B"
def test_KroneckerProduct_printing():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 2, 2)
assert latex(KroneckerProduct(A, B)) == r'A \otimes B'
def test_Series_printing():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(t*x**2 - t**w*x + w, t - y, y)
assert latex(Series(tf1, tf2)) == \
r'\left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right) \left(\frac{x - y}{x + y}\right)'
assert latex(Series(tf1, tf2, tf3)) == \
r'\left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right) \left(\frac{x - y}{x + y}\right) \left(\frac{t x^{2} - t^{w} x + w}{t - y}\right)'
assert latex(Series(-tf2, tf1)) == \
r'\left(\frac{- x + y}{x + y}\right) \left(\frac{x y^{2} - z}{- t^{3} + y^{3}}\right)'
M_1 = Matrix([[5/s], [5/(2*s)]])
T_1 = TransferFunctionMatrix.from_Matrix(M_1, s)
M_2 = Matrix([[5, 6*s**3]])
T_2 = TransferFunctionMatrix.from_Matrix(M_2, s)
# Brackets
assert latex(T_1*(T_2 + T_2)) == \
r'\left[\begin{matrix}\frac{5}{s}\\\frac{5}{2 s}\end{matrix}\right]_\tau\cdot\left(\left[\begin{matrix}\frac{5}{1} &' \
r' \frac{6 s^{3}}{1}\end{matrix}\right]_\tau + \left[\begin{matrix}\frac{5}{1} & \frac{6 s^{3}}{1}\end{matrix}\right]_\tau\right)' \
== latex(MIMOSeries(MIMOParallel(T_2, T_2), T_1))
# No Brackets
M_3 = Matrix([[5, 6], [6, 5/s]])
T_3 = TransferFunctionMatrix.from_Matrix(M_3, s)
assert latex(T_1*T_2 + T_3) == r'\left[\begin{matrix}\frac{5}{s}\\\frac{5}{2 s}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}' \
r'\frac{5}{1} & \frac{6 s^{3}}{1}\end{matrix}\right]_\tau + \left[\begin{matrix}\frac{5}{1} & \frac{6}{1}\\\frac{6}{1} & ' \
r'\frac{5}{s}\end{matrix}\right]_\tau' == latex(MIMOParallel(MIMOSeries(T_2, T_1), T_3))
def test_TransferFunction_printing():
tf1 = TransferFunction(x - 1, x + 1, x)
assert latex(tf1) == r"\frac{x - 1}{x + 1}"
tf2 = TransferFunction(x + 1, 2 - y, x)
assert latex(tf2) == r"\frac{x + 1}{2 - y}"
tf3 = TransferFunction(y, y**2 + 2*y + 3, y)
assert latex(tf3) == r"\frac{y}{y^{2} + 2 y + 3}"
def test_Parallel_printing():
tf1 = TransferFunction(x*y**2 - z, y**3 - t**3, y)
tf2 = TransferFunction(x - y, x + y, y)
assert latex(Parallel(tf1, tf2)) == \
r'\frac{x y^{2} - z}{- t^{3} + y^{3}} + \frac{x - y}{x + y}'
assert latex(Parallel(-tf2, tf1)) == \
r'\frac{- x + y}{x + y} + \frac{x y^{2} - z}{- t^{3} + y^{3}}'
M_1 = Matrix([[5, 6], [6, 5/s]])
T_1 = TransferFunctionMatrix.from_Matrix(M_1, s)
M_2 = Matrix([[5/s, 6], [6, 5/(s - 1)]])
T_2 = TransferFunctionMatrix.from_Matrix(M_2, s)
M_3 = Matrix([[6, 5/(s*(s - 1))], [5, 6]])
T_3 = TransferFunctionMatrix.from_Matrix(M_3, s)
assert latex(T_1 + T_2 + T_3) == r'\left[\begin{matrix}\frac{5}{1} & \frac{6}{1}\\\frac{6}{1} & \frac{5}{s}\end{matrix}\right]' \
r'_\tau + \left[\begin{matrix}\frac{5}{s} & \frac{6}{1}\\\frac{6}{1} & \frac{5}{s - 1}\end{matrix}\right]_\tau + \left[\begin{matrix}' \
r'\frac{6}{1} & \frac{5}{s \left(s - 1\right)}\\\frac{5}{1} & \frac{6}{1}\end{matrix}\right]_\tau' \
== latex(MIMOParallel(T_1, T_2, T_3)) == latex(MIMOParallel(T_1, MIMOParallel(T_2, T_3))) == latex(MIMOParallel(MIMOParallel(T_1, T_2), T_3))
def test_TransferFunctionMatrix_printing():
tf1 = TransferFunction(p, p + x, p)
tf2 = TransferFunction(-s + p, p + s, p)
tf3 = TransferFunction(p, y**2 + 2*y + 3, p)
assert latex(TransferFunctionMatrix([[tf1], [tf2]])) == \
r'\left[\begin{matrix}\frac{p}{p + x}\\\frac{p - s}{p + s}\end{matrix}\right]_\tau'
assert latex(TransferFunctionMatrix([[tf1, tf2], [tf3, -tf1]])) == \
r'\left[\begin{matrix}\frac{p}{p + x} & \frac{p - s}{p + s}\\\frac{p}{y^{2} + 2 y + 3} & \frac{\left(-1\right) p}{p + x}\end{matrix}\right]_\tau'
def test_Feedback_printing():
tf1 = TransferFunction(p, p + x, p)
tf2 = TransferFunction(-s + p, p + s, p)
# Negative Feedback (Default)
assert latex(Feedback(tf1, tf2)) == \
r'\frac{\frac{p}{p + x}}{\frac{1}{1} + \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
assert latex(Feedback(tf1*tf2, TransferFunction(1, 1, p))) == \
r'\frac{\left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}{\frac{1}{1} + \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
# Positive Feedback
assert latex(Feedback(tf1, tf2, 1)) == \
r'\frac{\frac{p}{p + x}}{\frac{1}{1} - \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
assert latex(Feedback(tf1*tf2, sign=1)) == \
r'\frac{\left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}{\frac{1}{1} - \left(\frac{p}{p + x}\right) \left(\frac{p - s}{p + s}\right)}'
def test_MIMOFeedback_printing():
tf1 = TransferFunction(1, s, s)
tf2 = TransferFunction(s, s**2 - 1, s)
tf3 = TransferFunction(s, s - 1, s)
tf4 = TransferFunction(s**2, s**2 - 1, s)
tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]])
tfm_2 = TransferFunctionMatrix([[tf4, tf3], [tf2, tf1]])
# Negative Feedback (Default)
assert latex(MIMOFeedback(tfm_1, tfm_2)) == \
r'\left(I_{\tau} + \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left[' \
r'\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1} & \frac{1}{s}\end{matrix}\right]_\tau\right)^{-1} \cdot \left[\begin{matrix}' \
r'\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau'
# Positive Feedback
assert latex(MIMOFeedback(tfm_1*tfm_2, tfm_1, 1)) == \
r'\left(I_{\tau} - \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left' \
r'[\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1} & \frac{1}{s}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}' \
r'\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\right)^{-1} \cdot \left[\begin{matrix}\frac{1}{s} & \frac{s}{s^{2} - 1}' \
r'\\\frac{s}{s - 1} & \frac{s^{2}}{s^{2} - 1}\end{matrix}\right]_\tau\cdot\left[\begin{matrix}\frac{s^{2}}{s^{2} - 1} & \frac{s}{s - 1}\\\frac{s}{s^{2} - 1}' \
r' & \frac{1}{s}\end{matrix}\right]_\tau'
def test_Quaternion_latex_printing():
q = Quaternion(x, y, z, t)
assert latex(q) == r"x + y i + z j + t k"
q = Quaternion(x, y, z, x*t)
assert latex(q) == r"x + y i + z j + t x k"
q = Quaternion(x, y, z, x + t)
assert latex(q) == r"x + y i + z j + \left(t + x\right) k"
def test_TensorProduct_printing():
from sympy.tensor.functions import TensorProduct
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert latex(TensorProduct(A, B)) == r"A \otimes B"
def test_WedgeProduct_printing():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert latex(wp) == r"\operatorname{d}x \wedge \operatorname{d}y"
def test_issue_9216():
expr_1 = Pow(1, -1, evaluate=False)
assert latex(expr_1) == r"1^{-1}"
expr_2 = Pow(1, Pow(1, -1, evaluate=False), evaluate=False)
assert latex(expr_2) == r"1^{1^{-1}}"
expr_3 = Pow(3, -2, evaluate=False)
assert latex(expr_3) == r"\frac{1}{9}"
expr_4 = Pow(1, -2, evaluate=False)
assert latex(expr_4) == r"1^{-2}"
def test_latex_printer_tensor():
from sympy.tensor.tensor import TensorIndexType, tensor_indices, TensorHead, tensor_heads
L = TensorIndexType("L")
i, j, k, l = tensor_indices("i j k l", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
K = TensorHead("K", [L, L, L, L])
assert latex(i) == r"{}^{i}"
assert latex(-i) == r"{}_{i}"
expr = A(i)
assert latex(expr) == r"A{}^{i}"
expr = A(i0)
assert latex(expr) == r"A{}^{i_{0}}"
expr = A(-i)
assert latex(expr) == r"A{}_{i}"
expr = -3*A(i)
assert latex(expr) == r"-3A{}^{i}"
expr = K(i, j, -k, -i0)
assert latex(expr) == r"K{}^{ij}{}_{ki_{0}}"
expr = K(i, -j, -k, i0)
assert latex(expr) == r"K{}^{i}{}_{jk}{}^{i_{0}}"
expr = K(i, -j, k, -i0)
assert latex(expr) == r"K{}^{i}{}_{j}{}^{k}{}_{i_{0}}"
expr = H(i, -j)
assert latex(expr) == r"H{}^{i}{}_{j}"
expr = H(i, j)
assert latex(expr) == r"H{}^{ij}"
expr = H(-i, -j)
assert latex(expr) == r"H{}_{ij}"
expr = (1+x)*A(i)
assert latex(expr) == r"\left(x + 1\right)A{}^{i}"
expr = H(i, -i)
assert latex(expr) == r"H{}^{L_{0}}{}_{L_{0}}"
expr = H(i, -j)*A(j)*B(k)
assert latex(expr) == r"H{}^{i}{}_{L_{0}}A{}^{L_{0}}B{}^{k}"
expr = A(i) + 3*B(i)
assert latex(expr) == r"3B{}^{i} + A{}^{i}"
# Test ``TensorElement``:
from sympy.tensor.tensor import TensorElement
expr = TensorElement(K(i, j, k, l), {i: 3, k: 2})
assert latex(expr) == r'K{}^{i=3,j,k=2,l}'
expr = TensorElement(K(i, j, k, l), {i: 3})
assert latex(expr) == r'K{}^{i=3,jkl}'
expr = TensorElement(K(i, -j, k, l), {i: 3, k: 2})
assert latex(expr) == r'K{}^{i=3}{}_{j}{}^{k=2,l}'
expr = TensorElement(K(i, -j, k, -l), {i: 3, k: 2})
assert latex(expr) == r'K{}^{i=3}{}_{j}{}^{k=2}{}_{l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3, -k: 2})
assert latex(expr) == r'K{}^{i=3,j}{}_{k=2,l}'
expr = TensorElement(K(i, j, -k, -l), {i: 3})
assert latex(expr) == r'K{}^{i=3,j}{}_{kl}'
expr = PartialDerivative(A(i), A(i))
assert latex(expr) == r"\frac{\partial}{\partial {A{}^{L_{0}}}}{A{}^{L_{0}}}"
expr = PartialDerivative(A(-i), A(-j))
assert latex(expr) == r"\frac{\partial}{\partial {A{}_{j}}}{A{}_{i}}"
expr = PartialDerivative(K(i, j, -k, -l), A(m), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}^{m}} \partial {A{}_{n}}}{K{}^{ij}{}_{kl}}"
expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(A{}_{i} + B{}_{i}\right)}"
expr = PartialDerivative(3*A(-i), A(-j), A(-n))
assert latex(expr) == r"\frac{\partial^{2}}{\partial {A{}_{j}} \partial {A{}_{n}}}{\left(3A{}_{i}\right)}"
def test_multiline_latex():
a, b, c, d, e, f = symbols('a b c d e f')
expr = -a + 2*b -3*c +4*d -5*e
expected = r"\begin{eqnarray}" + "\n"\
r"f & = &- a \nonumber\\" + "\n"\
r"& & + 2 b \nonumber\\" + "\n"\
r"& & - 3 c \nonumber\\" + "\n"\
r"& & + 4 d \nonumber\\" + "\n"\
r"& & - 5 e " + "\n"\
r"\end{eqnarray}"
assert multiline_latex(f, expr, environment="eqnarray") == expected
expected2 = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b \nonumber\\' + '\n'\
r'& & - 3 c + 4 d \nonumber\\' + '\n'\
r'& & - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 2, environment="eqnarray") == expected2
expected3 = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b - 3 c \nonumber\\'+ '\n'\
r'& & + 4 d - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 3, environment="eqnarray") == expected3
expected3dots = r'\begin{eqnarray}' + '\n'\
r'f & = &- a + 2 b - 3 c \dots\nonumber\\'+ '\n'\
r'& & + 4 d - 5 e ' + '\n'\
r'\end{eqnarray}'
assert multiline_latex(f, expr, 3, environment="eqnarray", use_dots=True) == expected3dots
expected3align = r'\begin{align*}' + '\n'\
r'f = &- a + 2 b - 3 c \\'+ '\n'\
r'& + 4 d - 5 e ' + '\n'\
r'\end{align*}'
assert multiline_latex(f, expr, 3) == expected3align
assert multiline_latex(f, expr, 3, environment='align*') == expected3align
expected2ieee = r'\begin{IEEEeqnarray}{rCl}' + '\n'\
r'f & = &- a + 2 b \nonumber\\' + '\n'\
r'& & - 3 c + 4 d \nonumber\\' + '\n'\
r'& & - 5 e ' + '\n'\
r'\end{IEEEeqnarray}'
assert multiline_latex(f, expr, 2, environment="IEEEeqnarray") == expected2ieee
raises(ValueError, lambda: multiline_latex(f, expr, environment="foo"))
def test_issue_15353():
a, x = symbols('a x')
# Obtained from nonlinsolve([(sin(a*x)),cos(a*x)],[x,a])
sol = ConditionSet(
Tuple(x, a), Eq(sin(a*x), 0) & Eq(cos(a*x), 0), S.Complexes**2)
assert latex(sol) == \
r'\left\{\left( x, \ a\right)\; \middle|\; \left( x, \ a\right) \in ' \
r'\mathbb{C}^{2} \wedge \sin{\left(a x \right)} = 0 \wedge ' \
r'\cos{\left(a x \right)} = 0 \right\}'
def test_latex_symbolic_probability():
mu = symbols("mu")
sigma = symbols("sigma", positive=True)
X = Normal("X", mu, sigma)
assert latex(Expectation(X)) == r'\operatorname{E}\left[X\right]'
assert latex(Variance(X)) == r'\operatorname{Var}\left(X\right)'
assert latex(Probability(X > 0)) == r'\operatorname{P}\left(X > 0\right)'
Y = Normal("Y", mu, sigma)
assert latex(Covariance(X, Y)) == r'\operatorname{Cov}\left(X, Y\right)'
def test_trace():
# Issue 15303
from sympy.matrices.expressions.trace import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A)) == r"\operatorname{tr}\left(A \right)"
assert latex(trace(A**2)) == r"\operatorname{tr}\left(A^{2} \right)"
def test_print_basic():
# Issue 15303
from sympy.core.basic import Basic
from sympy.core.expr import Expr
# dummy class for testing printing where the function is not
# implemented in latex.py
class UnimplementedExpr(Expr):
def __new__(cls, e):
return Basic.__new__(cls, e)
# dummy function for testing
def unimplemented_expr(expr):
return UnimplementedExpr(expr).doit()
# override class name to use superscript / subscript
def unimplemented_expr_sup_sub(expr):
result = UnimplementedExpr(expr)
result.__class__.__name__ = 'UnimplementedExpr_x^1'
return result
assert latex(unimplemented_expr(x)) == r'\operatorname{UnimplementedExpr}\left(x\right)'
assert latex(unimplemented_expr(x**2)) == \
r'\operatorname{UnimplementedExpr}\left(x^{2}\right)'
assert latex(unimplemented_expr_sup_sub(x)) == \
r'\operatorname{UnimplementedExpr^{1}_{x}}\left(x\right)'
def test_MatrixSymbol_bold():
# Issue #15871
from sympy.matrices.expressions.trace import trace
A = MatrixSymbol("A", 2, 2)
assert latex(trace(A), mat_symbol_style='bold') == \
r"\operatorname{tr}\left(\mathbf{A} \right)"
assert latex(trace(A), mat_symbol_style='plain') == \
r"\operatorname{tr}\left(A \right)"
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert latex(-A, mat_symbol_style='bold') == r"- \mathbf{A}"
assert latex(A - A*B - B, mat_symbol_style='bold') == \
r"\mathbf{A} - \mathbf{A} \mathbf{B} - \mathbf{B}"
assert latex(-A*B - A*B*C - B, mat_symbol_style='bold') == \
r"- \mathbf{A} \mathbf{B} - \mathbf{A} \mathbf{B} \mathbf{C} - \mathbf{B}"
A_k = MatrixSymbol("A_k", 3, 3)
assert latex(A_k, mat_symbol_style='bold') == r"\mathbf{A}_{k}"
A = MatrixSymbol(r"\nabla_k", 3, 3)
assert latex(A, mat_symbol_style='bold') == r"\mathbf{\nabla}_{k}"
def test_AppliedPermutation():
p = Permutation(0, 1, 2)
x = Symbol('x')
assert latex(AppliedPermutation(p, x)) == \
r'\sigma_{\left( 0\; 1\; 2\right)}(x)'
def test_PermutationMatrix():
p = Permutation(0, 1, 2)
assert latex(PermutationMatrix(p)) == r'P_{\left( 0\; 1\; 2\right)}'
p = Permutation(0, 3)(1, 2)
assert latex(PermutationMatrix(p)) == \
r'P_{\left( 0\; 3\right)\left( 1\; 2\right)}'
def test_issue_21758():
from sympy.functions.elementary.piecewise import piecewise_fold
from sympy.series.fourier import FourierSeries
x = Symbol('x')
k, n = symbols('k n')
fo = FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), SeqFormula(
Piecewise((-2*pi*cos(n*pi)/n + 2*sin(n*pi)/n**2, (n > -oo) & (n < oo) & Ne(n, 0)),
(0, True))*sin(n*x)/pi, (n, 1, oo))))
assert latex(piecewise_fold(fo)) == '\\begin{cases} 2 \\sin{\\left(x \\right)}' \
' - \\sin{\\left(2 x \\right)} + \\frac{2 \\sin{\\left(3 x \\right)}}{3} +' \
' \\ldots & \\text{for}\\: n > -\\infty \\wedge n < \\infty \\wedge ' \
'n \\neq 0 \\\\0 & \\text{otherwise} \\end{cases}'
assert latex(FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)),
SeqFormula(0, (n, 1, oo))))) == '0'
def test_imaginary_unit():
assert latex(1 + I) == r'1 + i'
assert latex(1 + I, imaginary_unit='i') == r'1 + i'
assert latex(1 + I, imaginary_unit='j') == r'1 + j'
assert latex(1 + I, imaginary_unit='foo') == r'1 + foo'
assert latex(I, imaginary_unit="ti") == r'\text{i}'
assert latex(I, imaginary_unit="tj") == r'\text{j}'
def test_text_re_im():
assert latex(im(x), gothic_re_im=True) == r'\Im{\left(x\right)}'
assert latex(im(x), gothic_re_im=False) == r'\operatorname{im}{\left(x\right)}'
assert latex(re(x), gothic_re_im=True) == r'\Re{\left(x\right)}'
assert latex(re(x), gothic_re_im=False) == r'\operatorname{re}{\left(x\right)}'
def test_latex_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField, Differential
from sympy.diffgeom.rn import R2
x,y = symbols('x y', real=True)
m = Manifold('M', 2)
assert latex(m) == r'\text{M}'
p = Patch('P', m)
assert latex(p) == r'\text{P}_{\text{M}}'
rect = CoordSystem('rect', p, [x, y])
assert latex(rect) == r'\text{rect}^{\text{P}}_{\text{M}}'
b = BaseScalarField(rect, 0)
assert latex(b) == r'\mathbf{x}'
g = Function('g')
s_field = g(R2.x, R2.y)
assert latex(Differential(s_field)) == \
r'\operatorname{d}\left(g{\left(\mathbf{x},\mathbf{y} \right)}\right)'
def test_unit_printing():
assert latex(5*meter) == r'5 \text{m}'
assert latex(3*gibibyte) == r'3 \text{gibibyte}'
assert latex(4*microgram/second) == r'\frac{4 \mu\text{g}}{\text{s}}'
assert latex(4*micro*gram/second) == r'\frac{4 \mu \text{g}}{\text{s}}'
assert latex(5*milli*meter) == r'5 \text{m} \text{m}'
assert latex(milli) == r'\text{m}'
def test_issue_17092():
x_star = Symbol('x^*')
assert latex(Derivative(x_star, x_star,2)) == r'\frac{d^{2}}{d \left(x^{*}\right)^{2}} x^{*}'
def test_latex_decimal_separator():
x, y, z, t = symbols('x y z t')
k, m, n = symbols('k m n', integer=True)
f, g, h = symbols('f g h', cls=Function)
# comma decimal_separator
assert(latex([1, 2.3, 4.5], decimal_separator='comma') == r'\left[ 1; \ 2{,}3; \ 4{,}5\right]')
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='comma') == r'\left\{1; 2{,}3; 4{,}5\right\}')
assert(latex((1, 2.3, 4.6), decimal_separator = 'comma') == r'\left( 1; \ 2{,}3; \ 4{,}6\right)')
assert(latex((1,), decimal_separator='comma') == r'\left( 1;\right)')
# period decimal_separator
assert(latex([1, 2.3, 4.5], decimal_separator='period') == r'\left[ 1, \ 2.3, \ 4.5\right]' )
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period') == r'\left\{1, 2.3, 4.5\right\}')
assert(latex((1, 2.3, 4.6), decimal_separator = 'period') == r'\left( 1, \ 2.3, \ 4.6\right)')
assert(latex((1,), decimal_separator='period') == r'\left( 1,\right)')
# default decimal_separator
assert(latex([1, 2.3, 4.5]) == r'\left[ 1, \ 2.3, \ 4.5\right]')
assert(latex(FiniteSet(1, 2.3, 4.5)) == r'\left\{1, 2.3, 4.5\right\}')
assert(latex((1, 2.3, 4.6)) == r'\left( 1, \ 2.3, \ 4.6\right)')
assert(latex((1,)) == r'\left( 1,\right)')
assert(latex(Mul(3.4,5.3), decimal_separator = 'comma') == r'18{,}02')
assert(latex(3.4*5.3, decimal_separator = 'comma') == r'18{,}02')
x = symbols('x')
y = symbols('y')
z = symbols('z')
assert(latex(x*5.3 + 2**y**3.4 + 4.5 + z, decimal_separator = 'comma') == r'2^{y^{3{,}4}} + 5{,}3 x + z + 4{,}5')
assert(latex(0.987, decimal_separator='comma') == r'0{,}987')
assert(latex(S(0.987), decimal_separator='comma') == r'0{,}987')
assert(latex(.3, decimal_separator='comma') == r'0{,}3')
assert(latex(S(.3), decimal_separator='comma') == r'0{,}3')
assert(latex(5.8*10**(-7), decimal_separator='comma') == r'5{,}8 \cdot 10^{-7}')
assert(latex(S(5.7)*10**(-7), decimal_separator='comma') == r'5{,}7 \cdot 10^{-7}')
assert(latex(S(5.7*10**(-7)), decimal_separator='comma') == r'5{,}7 \cdot 10^{-7}')
x = symbols('x')
assert(latex(1.2*x+3.4, decimal_separator='comma') == r'1{,}2 x + 3{,}4')
assert(latex(FiniteSet(1, 2.3, 4.5), decimal_separator='period') == r'\left\{1, 2.3, 4.5\right\}')
# Error Handling tests
raises(ValueError, lambda: latex([1,2.3,4.5], decimal_separator='non_existing_decimal_separator_in_list'))
raises(ValueError, lambda: latex(FiniteSet(1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_set'))
raises(ValueError, lambda: latex((1,2.3,4.5), decimal_separator='non_existing_decimal_separator_in_tuple'))
def test_Str():
from sympy.core.symbol import Str
assert str(Str('x')) == r'x'
def test_latex_escape():
assert latex_escape(r"~^\&%$#_{}") == "".join([
r'\textasciitilde',
r'\textasciicircum',
r'\textbackslash',
r'\&',
r'\%',
r'\$',
r'\#',
r'\_',
r'\{',
r'\}',
])
def test_emptyPrinter():
class MyObject:
def __repr__(self):
return "<MyObject with {...}>"
# unknown objects are monospaced
assert latex(MyObject()) == r"\mathtt{\text{<MyObject with \{...\}>}}"
# even if they are nested within other objects
assert latex((MyObject(),)) == r"\left( \mathtt{\text{<MyObject with \{...\}>}},\right)"
def test_global_settings():
import inspect
# settings should be visible in the signature of `latex`
assert inspect.signature(latex).parameters['imaginary_unit'].default == r'i'
assert latex(I) == r'i'
try:
# but changing the defaults...
LatexPrinter.set_global_settings(imaginary_unit='j')
# ... should change the signature
assert inspect.signature(latex).parameters['imaginary_unit'].default == r'j'
assert latex(I) == r'j'
finally:
# there's no public API to undo this, but we need to make sure we do
# so as not to impact other tests
del LatexPrinter._global_settings['imaginary_unit']
# check we really did undo it
assert inspect.signature(latex).parameters['imaginary_unit'].default == r'i'
assert latex(I) == r'i'
def test_pickleable():
# this tests that the _PrintFunction instance is pickleable
import pickle
assert pickle.loads(pickle.dumps(latex)) is latex
def test_printing_latex_array_expressions():
assert latex(ArraySymbol("A", (2, 3, 4))) == "A"
assert latex(ArrayElement("A", (2, 1/(1-x), 0))) == "{{A}_{2, \\frac{1}{1 - x}, 0}}"
M = MatrixSymbol("M", 3, 3)
N = MatrixSymbol("N", 3, 3)
assert latex(ArrayElement(M*N, [x, 0])) == "{{\\left(M N\\right)}_{x, 0}}"
def test_Array():
arr = Array(range(10))
assert latex(arr) == r'\left[\begin{matrix}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\end{matrix}\right]'
arr = Array(range(11))
# added empty arguments {}
assert latex(arr) == r'\left[\begin{array}{}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\end{array}\right]'
|
2a5d1119d340111d7e7507cef821f6c442297037c0d11c266339dbf80b352031 | from sympy.core import (S, pi, oo, Symbol, symbols, Rational, Integer,
GoldenRatio, EulerGamma, Catalan, Lambda, Dummy)
from sympy.functions import (Piecewise, sin, cos, Abs, exp, ceiling, sqrt,
gamma, sign, Max, Min, factorial, beta)
from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
from sympy.sets import Range
from sympy.logic import ITE
from sympy.codegen import For, aug_assign, Assignment
from sympy.testing.pytest import raises
from sympy.printing.rcode import RCodePrinter
from sympy.utilities.lambdify import implemented_function
from sympy.tensor import IndexedBase, Idx
from sympy.matrices import Matrix, MatrixSymbol
from sympy.printing.rcode import rcode
x, y, z = symbols('x,y,z')
def test_printmethod():
class fabs(Abs):
def _rcode(self, printer):
return "abs(%s)" % printer._print(self.args[0])
assert rcode(fabs(x)) == "abs(x)"
def test_rcode_sqrt():
assert rcode(sqrt(x)) == "sqrt(x)"
assert rcode(x**0.5) == "sqrt(x)"
assert rcode(sqrt(x)) == "sqrt(x)"
def test_rcode_Pow():
assert rcode(x**3) == "x^3"
assert rcode(x**(y**3)) == "x^(y^3)"
g = implemented_function('g', Lambda(x, 2*x))
assert rcode(1/(g(x)*3.5)**(x - y**x)/(x**2 + y)) == \
"(3.5*2*x)^(-x + y^x)/(x^2 + y)"
assert rcode(x**-1.0) == '1.0/x'
assert rcode(x**Rational(2, 3)) == 'x^(2.0/3.0)'
_cond_cfunc = [(lambda base, exp: exp.is_integer, "dpowi"),
(lambda base, exp: not exp.is_integer, "pow")]
assert rcode(x**3, user_functions={'Pow': _cond_cfunc}) == 'dpowi(x, 3)'
assert rcode(x**3.2, user_functions={'Pow': _cond_cfunc}) == 'pow(x, 3.2)'
def test_rcode_Max():
# Test for gh-11926
assert rcode(Max(x,x*x),user_functions={"Max":"my_max", "Pow":"my_pow"}) == 'my_max(x, my_pow(x, 2))'
def test_rcode_constants_mathh():
assert rcode(exp(1)) == "exp(1)"
assert rcode(pi) == "pi"
assert rcode(oo) == "Inf"
assert rcode(-oo) == "-Inf"
def test_rcode_constants_other():
assert rcode(2*GoldenRatio) == "GoldenRatio = 1.61803398874989;\n2*GoldenRatio"
assert rcode(
2*Catalan) == "Catalan = 0.915965594177219;\n2*Catalan"
assert rcode(2*EulerGamma) == "EulerGamma = 0.577215664901533;\n2*EulerGamma"
def test_rcode_Rational():
assert rcode(Rational(3, 7)) == "3.0/7.0"
assert rcode(Rational(18, 9)) == "2"
assert rcode(Rational(3, -7)) == "-3.0/7.0"
assert rcode(Rational(-3, -7)) == "3.0/7.0"
assert rcode(x + Rational(3, 7)) == "x + 3.0/7.0"
assert rcode(Rational(3, 7)*x) == "(3.0/7.0)*x"
def test_rcode_Integer():
assert rcode(Integer(67)) == "67"
assert rcode(Integer(-1)) == "-1"
def test_rcode_functions():
assert rcode(sin(x) ** cos(x)) == "sin(x)^cos(x)"
assert rcode(factorial(x) + gamma(y)) == "factorial(x) + gamma(y)"
assert rcode(beta(Min(x, y), Max(x, y))) == "beta(min(x, y), max(x, y))"
def test_rcode_inline_function():
x = symbols('x')
g = implemented_function('g', Lambda(x, 2*x))
assert rcode(g(x)) == "2*x"
g = implemented_function('g', Lambda(x, 2*x/Catalan))
assert rcode(
g(x)) == "Catalan = %s;\n2*x/Catalan" % Catalan.n()
A = IndexedBase('A')
i = Idx('i', symbols('n', integer=True))
g = implemented_function('g', Lambda(x, x*(1 + x)*(2 + x)))
res=rcode(g(A[i]), assign_to=A[i])
ref=(
"for (i in 1:n){\n"
" A[i] = (A[i] + 1)*(A[i] + 2)*A[i];\n"
"}"
)
assert res == ref
def test_rcode_exceptions():
assert rcode(ceiling(x)) == "ceiling(x)"
assert rcode(Abs(x)) == "abs(x)"
assert rcode(gamma(x)) == "gamma(x)"
def test_rcode_user_functions():
x = symbols('x', integer=False)
n = symbols('n', integer=True)
custom_functions = {
"ceiling": "myceil",
"Abs": [(lambda x: not x.is_integer, "fabs"), (lambda x: x.is_integer, "abs")],
}
assert rcode(ceiling(x), user_functions=custom_functions) == "myceil(x)"
assert rcode(Abs(x), user_functions=custom_functions) == "fabs(x)"
assert rcode(Abs(n), user_functions=custom_functions) == "abs(n)"
def test_rcode_boolean():
assert rcode(True) == "True"
assert rcode(S.true) == "True"
assert rcode(False) == "False"
assert rcode(S.false) == "False"
assert rcode(x & y) == "x & y"
assert rcode(x | y) == "x | y"
assert rcode(~x) == "!x"
assert rcode(x & y & z) == "x & y & z"
assert rcode(x | y | z) == "x | y | z"
assert rcode((x & y) | z) == "z | x & y"
assert rcode((x | y) & z) == "z & (x | y)"
def test_rcode_Relational():
assert rcode(Eq(x, y)) == "x == y"
assert rcode(Ne(x, y)) == "x != y"
assert rcode(Le(x, y)) == "x <= y"
assert rcode(Lt(x, y)) == "x < y"
assert rcode(Gt(x, y)) == "x > y"
assert rcode(Ge(x, y)) == "x >= y"
def test_rcode_Piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
res=rcode(expr)
ref="ifelse(x < 1,x,x^2)"
assert res == ref
tau=Symbol("tau")
res=rcode(expr,tau)
ref="tau = ifelse(x < 1,x,x^2);"
assert res == ref
expr = 2*Piecewise((x, x < 1), (x**2, x<2), (x**3,True))
assert rcode(expr) == "2*ifelse(x < 1,x,ifelse(x < 2,x^2,x^3))"
res = rcode(expr, assign_to='c')
assert res == "c = 2*ifelse(x < 1,x,ifelse(x < 2,x^2,x^3));"
# Check that Piecewise without a True (default) condition error
#expr = Piecewise((x, x < 1), (x**2, x > 1), (sin(x), x > 0))
#raises(ValueError, lambda: rcode(expr))
expr = 2*Piecewise((x, x < 1), (x**2, x<2))
assert(rcode(expr))== "2*ifelse(x < 1,x,ifelse(x < 2,x^2,NA))"
def test_rcode_sinc():
from sympy.functions.elementary.trigonometric import sinc
expr = sinc(x)
res = rcode(expr)
ref = "ifelse(x != 0,sin(x)/x,1)"
assert res == ref
def test_rcode_Piecewise_deep():
p = rcode(2*Piecewise((x, x < 1), (x + 1, x < 2), (x**2, True)))
assert p == "2*ifelse(x < 1,x,ifelse(x < 2,x + 1,x^2))"
expr = x*y*z + x**2 + y**2 + Piecewise((0, x < 0.5), (1, True)) + cos(z) - 1
p = rcode(expr)
ref="x^2 + x*y*z + y^2 + ifelse(x < 0.5,0,1) + cos(z) - 1"
assert p == ref
ref="c = x^2 + x*y*z + y^2 + ifelse(x < 0.5,0,1) + cos(z) - 1;"
p = rcode(expr, assign_to='c')
assert p == ref
def test_rcode_ITE():
expr = ITE(x < 1, y, z)
p = rcode(expr)
ref="ifelse(x < 1,y,z)"
assert p == ref
def test_rcode_settings():
raises(TypeError, lambda: rcode(sin(x), method="garbage"))
def test_rcode_Indexed():
n, m, o = symbols('n m o', integer=True)
i, j, k = Idx('i', n), Idx('j', m), Idx('k', o)
p = RCodePrinter()
p._not_r = set()
x = IndexedBase('x')[j]
assert p._print_Indexed(x) == 'x[j]'
A = IndexedBase('A')[i, j]
assert p._print_Indexed(A) == 'A[i, j]'
B = IndexedBase('B')[i, j, k]
assert p._print_Indexed(B) == 'B[i, j, k]'
assert p._not_r == set()
def test_rcode_Indexed_without_looking_for_contraction():
len_y = 5
y = IndexedBase('y', shape=(len_y,))
x = IndexedBase('x', shape=(len_y,))
Dy = IndexedBase('Dy', shape=(len_y-1,))
i = Idx('i', len_y-1)
e=Eq(Dy[i], (y[i+1]-y[i])/(x[i+1]-x[i]))
code0 = rcode(e.rhs, assign_to=e.lhs, contract=False)
assert code0 == 'Dy[i] = (y[%s] - y[i])/(x[%s] - x[i]);' % (i + 1, i + 1)
def test_rcode_loops_matrix_vector():
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
s = (
'for (i in 1:m){\n'
' y[i] = 0;\n'
'}\n'
'for (i in 1:m){\n'
' for (j in 1:n){\n'
' y[i] = A[i, j]*x[j] + y[i];\n'
' }\n'
'}'
)
c = rcode(A[i, j]*x[j], assign_to=y[i])
assert c == s
def test_dummy_loops():
# the following line could also be
# [Dummy(s, integer=True) for s in 'im']
# or [Dummy(integer=True) for s in 'im']
i, m = symbols('i m', integer=True, cls=Dummy)
x = IndexedBase('x')
y = IndexedBase('y')
i = Idx(i, m)
expected = (
'for (i_%(icount)i in 1:m_%(mcount)i){\n'
' y[i_%(icount)i] = x[i_%(icount)i];\n'
'}'
) % {'icount': i.label.dummy_index, 'mcount': m.dummy_index}
code = rcode(x[i], assign_to=y[i])
assert code == expected
def test_rcode_loops_add():
n, m = symbols('n m', integer=True)
A = IndexedBase('A')
x = IndexedBase('x')
y = IndexedBase('y')
z = IndexedBase('z')
i = Idx('i', m)
j = Idx('j', n)
s = (
'for (i in 1:m){\n'
' y[i] = x[i] + z[i];\n'
'}\n'
'for (i in 1:m){\n'
' for (j in 1:n){\n'
' y[i] = A[i, j]*x[j] + y[i];\n'
' }\n'
'}'
)
c = rcode(A[i, j]*x[j] + x[i] + z[i], assign_to=y[i])
assert c == s
def test_rcode_loops_multiple_contractions():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (i in 1:m){\n'
' y[i] = 0;\n'
'}\n'
'for (i in 1:m){\n'
' for (j in 1:n){\n'
' for (k in 1:o){\n'
' for (l in 1:p){\n'
' y[i] = a[i, j, k, l]*b[j, k, l] + y[i];\n'
' }\n'
' }\n'
' }\n'
'}'
)
c = rcode(b[j, k, l]*a[i, j, k, l], assign_to=y[i])
assert c == s
def test_rcode_loops_addfactor():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
l = Idx('l', p)
s = (
'for (i in 1:m){\n'
' y[i] = 0;\n'
'}\n'
'for (i in 1:m){\n'
' for (j in 1:n){\n'
' for (k in 1:o){\n'
' for (l in 1:p){\n'
' y[i] = (a[i, j, k, l] + b[i, j, k, l])*c[j, k, l] + y[i];\n'
' }\n'
' }\n'
' }\n'
'}'
)
c = rcode((a[i, j, k, l] + b[i, j, k, l])*c[j, k, l], assign_to=y[i])
assert c == s
def test_rcode_loops_multiple_terms():
n, m, o, p = symbols('n m o p', integer=True)
a = IndexedBase('a')
b = IndexedBase('b')
c = IndexedBase('c')
y = IndexedBase('y')
i = Idx('i', m)
j = Idx('j', n)
k = Idx('k', o)
s0 = (
'for (i in 1:m){\n'
' y[i] = 0;\n'
'}\n'
)
s1 = (
'for (i in 1:m){\n'
' for (j in 1:n){\n'
' for (k in 1:o){\n'
' y[i] = b[j]*b[k]*c[i, j, k] + y[i];\n'
' }\n'
' }\n'
'}\n'
)
s2 = (
'for (i in 1:m){\n'
' for (k in 1:o){\n'
' y[i] = a[i, k]*b[k] + y[i];\n'
' }\n'
'}\n'
)
s3 = (
'for (i in 1:m){\n'
' for (j in 1:n){\n'
' y[i] = a[i, j]*b[j] + y[i];\n'
' }\n'
'}\n'
)
c = rcode(
b[j]*a[i, j] + b[k]*a[i, k] + b[j]*b[k]*c[i, j, k], assign_to=y[i])
ref=dict()
ref[0] = s0 + s1 + s2 + s3[:-1]
ref[1] = s0 + s1 + s3 + s2[:-1]
ref[2] = s0 + s2 + s1 + s3[:-1]
ref[3] = s0 + s2 + s3 + s1[:-1]
ref[4] = s0 + s3 + s1 + s2[:-1]
ref[5] = s0 + s3 + s2 + s1[:-1]
assert (c == ref[0] or
c == ref[1] or
c == ref[2] or
c == ref[3] or
c == ref[4] or
c == ref[5])
def test_dereference_printing():
expr = x + y + sin(z) + z
assert rcode(expr, dereference=[z]) == "x + y + (*z) + sin((*z))"
def test_Matrix_printing():
# Test returning a Matrix
mat = Matrix([x*y, Piecewise((2 + x, y>0), (y, True)), sin(z)])
A = MatrixSymbol('A', 3, 1)
p = rcode(mat, A)
assert p == (
"A[0] = x*y;\n"
"A[1] = ifelse(y > 0,x + 2,y);\n"
"A[2] = sin(z);")
# Test using MatrixElements in expressions
expr = Piecewise((2*A[2, 0], x > 0), (A[2, 0], True)) + sin(A[1, 0]) + A[0, 0]
p = rcode(expr)
assert p == ("ifelse(x > 0,2*A[2],A[2]) + sin(A[1]) + A[0]")
# Test using MatrixElements in a Matrix
q = MatrixSymbol('q', 5, 1)
M = MatrixSymbol('M', 3, 3)
m = Matrix([[sin(q[1,0]), 0, cos(q[2,0])],
[q[1,0] + q[2,0], q[3, 0], 5],
[2*q[4, 0]/q[1,0], sqrt(q[0,0]) + 4, 0]])
assert rcode(m, M) == (
"M[0] = sin(q[1]);\n"
"M[1] = 0;\n"
"M[2] = cos(q[2]);\n"
"M[3] = q[1] + q[2];\n"
"M[4] = q[3];\n"
"M[5] = 5;\n"
"M[6] = 2*q[4]/q[1];\n"
"M[7] = sqrt(q[0]) + 4;\n"
"M[8] = 0;")
def test_rcode_sgn():
expr = sign(x) * y
assert rcode(expr) == 'y*sign(x)'
p = rcode(expr, 'z')
assert p == 'z = y*sign(x);'
p = rcode(sign(2 * x + x**2) * x + x**2)
assert p == "x^2 + x*sign(x^2 + 2*x)"
expr = sign(cos(x))
p = rcode(expr)
assert p == 'sign(cos(x))'
def test_rcode_Assignment():
assert rcode(Assignment(x, y + z)) == 'x = y + z;'
assert rcode(aug_assign(x, '+', y + z)) == 'x += y + z;'
def test_rcode_For():
f = For(x, Range(0, 10, 2), [aug_assign(y, '*', x)])
sol = rcode(f)
assert sol == ("for (x = 0; x < 10; x += 2) {\n"
" y *= x;\n"
"}")
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
assert(rcode(A[0, 0]) == "A[0]")
assert(rcode(3 * A[0, 0]) == "3*A[0]")
F = C[0, 0].subs(C, A - B)
assert(rcode(F) == "(A - B)[0]")
|
137e858fbd87c4c2b47b5a4a4e4a13a6ff7023852fc07755b9fa3e240e9d62ad | """
Important note on tests in this module - the Theano printing functions use a
global cache by default, which means that tests using it will modify global
state and thus not be independent from each other. Instead of using the "cache"
keyword argument each time, this module uses the theano_code_ and
theano_function_ functions defined below which default to using a new, empty
cache instead.
"""
import logging
from sympy.external import import_module
from sympy.testing.pytest import raises, SKIP, warns_deprecated_sympy
theanologger = logging.getLogger('theano.configdefaults')
theanologger.setLevel(logging.CRITICAL)
theano = import_module('theano')
theanologger.setLevel(logging.WARNING)
if theano:
import numpy as np
ts = theano.scalar
tt = theano.tensor
xt, yt, zt = [tt.scalar(name, 'floatX') for name in 'xyz']
Xt, Yt, Zt = [tt.tensor('floatX', (False, False), name=n) for n in 'XYZ']
else:
#bin/test will not execute any tests now
disabled = True
import sympy as sy
from sympy.core.singleton import S
from sympy.abc import x, y, z, t
from sympy.printing.theanocode import (theano_code, dim_handling,
theano_function)
# Default set of matrix symbols for testing - make square so we can both
# multiply and perform elementwise operations between them.
X, Y, Z = [sy.MatrixSymbol(n, 4, 4) for n in 'XYZ']
# For testing AppliedUndef
f_t = sy.Function('f')(t)
def theano_code_(expr, **kwargs):
""" Wrapper for theano_code that uses a new, empty cache by default. """
kwargs.setdefault('cache', {})
with warns_deprecated_sympy():
return theano_code(expr, **kwargs)
def theano_function_(inputs, outputs, **kwargs):
""" Wrapper for theano_function that uses a new, empty cache by default. """
kwargs.setdefault('cache', {})
with warns_deprecated_sympy():
return theano_function(inputs, outputs, **kwargs)
def fgraph_of(*exprs):
""" Transform SymPy expressions into Theano Computation.
Parameters
==========
exprs
SymPy expressions
Returns
=======
theano.gof.FunctionGraph
"""
outs = list(map(theano_code_, exprs))
ins = theano.gof.graph.inputs(outs)
ins, outs = theano.gof.graph.clone(ins, outs)
return theano.gof.FunctionGraph(ins, outs)
def theano_simplify(fgraph):
""" Simplify a Theano Computation.
Parameters
==========
fgraph : theano.gof.FunctionGraph
Returns
=======
theano.gof.FunctionGraph
"""
mode = theano.compile.get_default_mode().excluding("fusion")
fgraph = fgraph.clone()
mode.optimizer.optimize(fgraph)
return fgraph
def theq(a, b):
""" Test two Theano objects for equality.
Also accepts numeric types and lists/tuples of supported types.
Note - debugprint() has a bug where it will accept numeric types but does
not respect the "file" argument and in this case and instead prints the number
to stdout and returns an empty string. This can lead to tests passing where
they should fail because any two numbers will always compare as equal. To
prevent this we treat numbers as a separate case.
"""
numeric_types = (int, float, np.number)
a_is_num = isinstance(a, numeric_types)
b_is_num = isinstance(b, numeric_types)
# Compare numeric types using regular equality
if a_is_num or b_is_num:
if not (a_is_num and b_is_num):
return False
return a == b
# Compare sequences element-wise
a_is_seq = isinstance(a, (tuple, list))
b_is_seq = isinstance(b, (tuple, list))
if a_is_seq or b_is_seq:
if not (a_is_seq and b_is_seq) or type(a) != type(b):
return False
return list(map(theq, a)) == list(map(theq, b))
# Otherwise, assume debugprint() can handle it
astr = theano.printing.debugprint(a, file='str')
bstr = theano.printing.debugprint(b, file='str')
# Check for bug mentioned above
for argname, argval, argstr in [('a', a, astr), ('b', b, bstr)]:
if argstr == '':
raise TypeError(
'theano.printing.debugprint(%s) returned empty string '
'(%s is instance of %r)'
% (argname, argname, type(argval))
)
return astr == bstr
def test_example_symbols():
"""
Check that the example symbols in this module print to their Theano
equivalents, as many of the other tests depend on this.
"""
assert theq(xt, theano_code_(x))
assert theq(yt, theano_code_(y))
assert theq(zt, theano_code_(z))
assert theq(Xt, theano_code_(X))
assert theq(Yt, theano_code_(Y))
assert theq(Zt, theano_code_(Z))
def test_Symbol():
""" Test printing a Symbol to a theano variable. """
xx = theano_code_(x)
assert isinstance(xx, (tt.TensorVariable, ts.ScalarVariable))
assert xx.broadcastable == ()
assert xx.name == x.name
xx2 = theano_code_(x, broadcastables={x: (False,)})
assert xx2.broadcastable == (False,)
assert xx2.name == x.name
def test_MatrixSymbol():
""" Test printing a MatrixSymbol to a theano variable. """
XX = theano_code_(X)
assert isinstance(XX, tt.TensorVariable)
assert XX.broadcastable == (False, False)
@SKIP # TODO - this is currently not checked but should be implemented
def test_MatrixSymbol_wrong_dims():
""" Test MatrixSymbol with invalid broadcastable. """
bcs = [(), (False,), (True,), (True, False), (False, True,), (True, True)]
for bc in bcs:
with raises(ValueError):
theano_code_(X, broadcastables={X: bc})
def test_AppliedUndef():
""" Test printing AppliedUndef instance, which works similarly to Symbol. """
ftt = theano_code_(f_t)
assert isinstance(ftt, tt.TensorVariable)
assert ftt.broadcastable == ()
assert ftt.name == 'f_t'
def test_add():
expr = x + y
comp = theano_code_(expr)
assert comp.owner.op == theano.tensor.add
def test_trig():
assert theq(theano_code_(sy.sin(x)), tt.sin(xt))
assert theq(theano_code_(sy.tan(x)), tt.tan(xt))
def test_many():
""" Test printing a complex expression with multiple symbols. """
expr = sy.exp(x**2 + sy.cos(y)) * sy.log(2*z)
comp = theano_code_(expr)
expected = tt.exp(xt**2 + tt.cos(yt)) * tt.log(2*zt)
assert theq(comp, expected)
def test_dtype():
""" Test specifying specific data types through the dtype argument. """
for dtype in ['float32', 'float64', 'int8', 'int16', 'int32', 'int64']:
assert theano_code_(x, dtypes={x: dtype}).type.dtype == dtype
# "floatX" type
assert theano_code_(x, dtypes={x: 'floatX'}).type.dtype in ('float32', 'float64')
# Type promotion
assert theano_code_(x + 1, dtypes={x: 'float32'}).type.dtype == 'float32'
assert theano_code_(x + y, dtypes={x: 'float64', y: 'float32'}).type.dtype == 'float64'
def test_broadcastables():
""" Test the "broadcastables" argument when printing symbol-like objects. """
# No restrictions on shape
for s in [x, f_t]:
for bc in [(), (False,), (True,), (False, False), (True, False)]:
assert theano_code_(s, broadcastables={s: bc}).broadcastable == bc
# TODO - matrix broadcasting?
def test_broadcasting():
""" Test "broadcastable" attribute after applying element-wise binary op. """
expr = x + y
cases = [
[(), (), ()],
[(False,), (False,), (False,)],
[(True,), (False,), (False,)],
[(False, True), (False, False), (False, False)],
[(True, False), (False, False), (False, False)],
]
for bc1, bc2, bc3 in cases:
comp = theano_code_(expr, broadcastables={x: bc1, y: bc2})
assert comp.broadcastable == bc3
def test_MatMul():
expr = X*Y*Z
expr_t = theano_code_(expr)
assert isinstance(expr_t.owner.op, tt.Dot)
assert theq(expr_t, Xt.dot(Yt).dot(Zt))
def test_Transpose():
assert isinstance(theano_code_(X.T).owner.op, tt.DimShuffle)
def test_MatAdd():
expr = X+Y+Z
assert isinstance(theano_code_(expr).owner.op, tt.Elemwise)
def test_Rationals():
assert theq(theano_code_(sy.Integer(2) / 3), tt.true_div(2, 3))
assert theq(theano_code_(S.Half), tt.true_div(1, 2))
def test_Integers():
assert theano_code_(sy.Integer(3)) == 3
def test_factorial():
n = sy.Symbol('n')
assert theano_code_(sy.factorial(n))
def test_Derivative():
simp = lambda expr: theano_simplify(fgraph_of(expr))
assert theq(simp(theano_code_(sy.Derivative(sy.sin(x), x, evaluate=False))),
simp(theano.grad(tt.sin(xt), xt)))
def test_theano_function_simple():
""" Test theano_function() with single output. """
f = theano_function_([x, y], [x+y])
assert f(2, 3) == 5
def test_theano_function_multi():
""" Test theano_function() with multiple outputs. """
f = theano_function_([x, y], [x+y, x-y])
o1, o2 = f(2, 3)
assert o1 == 5
assert o2 == -1
def test_theano_function_numpy():
""" Test theano_function() vs Numpy implementation. """
f = theano_function_([x, y], [x+y], dim=1,
dtypes={x: 'float64', y: 'float64'})
assert np.linalg.norm(f([1, 2], [3, 4]) - np.asarray([4, 6])) < 1e-9
f = theano_function_([x, y], [x+y], dtypes={x: 'float64', y: 'float64'},
dim=1)
xx = np.arange(3).astype('float64')
yy = 2*np.arange(3).astype('float64')
assert np.linalg.norm(f(xx, yy) - 3*np.arange(3)) < 1e-9
def test_theano_function_matrix():
m = sy.Matrix([[x, y], [z, x + y + z]])
expected = np.array([[1.0, 2.0], [3.0, 1.0 + 2.0 + 3.0]])
f = theano_function_([x, y, z], [m])
np.testing.assert_allclose(f(1.0, 2.0, 3.0), expected)
f = theano_function_([x, y, z], [m], scalar=True)
np.testing.assert_allclose(f(1.0, 2.0, 3.0), expected)
f = theano_function_([x, y, z], [m, m])
assert isinstance(f(1.0, 2.0, 3.0), type([]))
np.testing.assert_allclose(f(1.0, 2.0, 3.0)[0], expected)
np.testing.assert_allclose(f(1.0, 2.0, 3.0)[1], expected)
def test_dim_handling():
assert dim_handling([x], dim=2) == {x: (False, False)}
assert dim_handling([x, y], dims={x: 1, y: 2}) == {x: (False, True),
y: (False, False)}
assert dim_handling([x], broadcastables={x: (False,)}) == {x: (False,)}
def test_theano_function_kwargs():
"""
Test passing additional kwargs from theano_function() to theano.function().
"""
import numpy as np
f = theano_function_([x, y, z], [x+y], dim=1, on_unused_input='ignore',
dtypes={x: 'float64', y: 'float64', z: 'float64'})
assert np.linalg.norm(f([1, 2], [3, 4], [0, 0]) - np.asarray([4, 6])) < 1e-9
f = theano_function_([x, y, z], [x+y],
dtypes={x: 'float64', y: 'float64', z: 'float64'},
dim=1, on_unused_input='ignore')
xx = np.arange(3).astype('float64')
yy = 2*np.arange(3).astype('float64')
zz = 2*np.arange(3).astype('float64')
assert np.linalg.norm(f(xx, yy, zz) - 3*np.arange(3)) < 1e-9
def test_theano_function_scalar():
""" Test the "scalar" argument to theano_function(). """
args = [
([x, y], [x + y], None, [0]), # Single 0d output
([X, Y], [X + Y], None, [2]), # Single 2d output
([x, y], [x + y], {x: 0, y: 1}, [1]), # Single 1d output
([x, y], [x + y, x - y], None, [0, 0]), # Two 0d outputs
([x, y, X, Y], [x + y, X + Y], None, [0, 2]), # One 0d output, one 2d
]
# Create and test functions with and without the scalar setting
for inputs, outputs, in_dims, out_dims in args:
for scalar in [False, True]:
f = theano_function_(inputs, outputs, dims=in_dims, scalar=scalar)
# Check the theano_function attribute is set whether wrapped or not
assert isinstance(f.theano_function, theano.compile.function_module.Function)
# Feed in inputs of the appropriate size and get outputs
in_values = [
np.ones([1 if bc else 5 for bc in i.type.broadcastable])
for i in f.theano_function.input_storage
]
out_values = f(*in_values)
if not isinstance(out_values, list):
out_values = [out_values]
# Check output types and shapes
assert len(out_dims) == len(out_values)
for d, value in zip(out_dims, out_values):
if scalar and d == 0:
# Should have been converted to a scalar value
assert isinstance(value, np.number)
else:
# Otherwise should be an array
assert isinstance(value, np.ndarray)
assert value.ndim == d
def test_theano_function_bad_kwarg():
"""
Passing an unknown keyword argument to theano_function() should raise an
exception.
"""
raises(Exception, lambda : theano_function_([x], [x+1], foobar=3))
def test_slice():
assert theano_code_(slice(1, 2, 3)) == slice(1, 2, 3)
def theq_slice(s1, s2):
for attr in ['start', 'stop', 'step']:
a1 = getattr(s1, attr)
a2 = getattr(s2, attr)
if a1 is None or a2 is None:
if not (a1 is None or a2 is None):
return False
elif not theq(a1, a2):
return False
return True
dtypes = {x: 'int32', y: 'int32'}
assert theq_slice(theano_code_(slice(x, y), dtypes=dtypes), slice(xt, yt))
assert theq_slice(theano_code_(slice(1, x, 3), dtypes=dtypes), slice(1, xt, 3))
def test_MatrixSlice():
from theano import Constant
cache = {}
n = sy.Symbol('n', integer=True)
X = sy.MatrixSymbol('X', n, n)
Y = X[1:2:3, 4:5:6]
Yt = theano_code_(Y, cache=cache)
s = ts.Scalar('int64')
assert tuple(Yt.owner.op.idx_list) == (slice(s, s, s), slice(s, s, s))
assert Yt.owner.inputs[0] == theano_code_(X, cache=cache)
# == doesn't work in theano like it does in SymPy. You have to use
# equals.
assert all(Yt.owner.inputs[i].equals(Constant(s, i)) for i in range(1, 7))
k = sy.Symbol('k')
theano_code_(k, dtypes={k: 'int32'})
start, stop, step = 4, k, 2
Y = X[start:stop:step]
Yt = theano_code_(Y, dtypes={n: 'int32', k: 'int32'})
# assert Yt.owner.op.idx_list[0].stop == kt
def test_BlockMatrix():
n = sy.Symbol('n', integer=True)
A, B, C, D = [sy.MatrixSymbol(name, n, n) for name in 'ABCD']
At, Bt, Ct, Dt = map(theano_code_, (A, B, C, D))
Block = sy.BlockMatrix([[A, B], [C, D]])
Blockt = theano_code_(Block)
solutions = [tt.join(0, tt.join(1, At, Bt), tt.join(1, Ct, Dt)),
tt.join(1, tt.join(0, At, Ct), tt.join(0, Bt, Dt))]
assert any(theq(Blockt, solution) for solution in solutions)
@SKIP
def test_BlockMatrix_Inverse_execution():
k, n = 2, 4
dtype = 'float32'
A = sy.MatrixSymbol('A', n, k)
B = sy.MatrixSymbol('B', n, n)
inputs = A, B
output = B.I*A
cutsizes = {A: [(n//2, n//2), (k//2, k//2)],
B: [(n//2, n//2), (n//2, n//2)]}
cutinputs = [sy.blockcut(i, *cutsizes[i]) for i in inputs]
cutoutput = output.subs(dict(zip(inputs, cutinputs)))
dtypes = dict(zip(inputs, [dtype]*len(inputs)))
f = theano_function_(inputs, [output], dtypes=dtypes, cache={})
fblocked = theano_function_(inputs, [sy.block_collapse(cutoutput)],
dtypes=dtypes, cache={})
ninputs = [np.random.rand(*x.shape).astype(dtype) for x in inputs]
ninputs = [np.arange(n*k).reshape(A.shape).astype(dtype),
np.eye(n).astype(dtype)]
ninputs[1] += np.ones(B.shape)*1e-5
assert np.allclose(f(*ninputs), fblocked(*ninputs), rtol=1e-5)
def test_DenseMatrix():
t = sy.Symbol('theta')
for MatrixType in [sy.Matrix, sy.ImmutableMatrix]:
X = MatrixType([[sy.cos(t), -sy.sin(t)], [sy.sin(t), sy.cos(t)]])
tX = theano_code_(X)
assert isinstance(tX, tt.TensorVariable)
assert tX.owner.op == tt.join_
def test_cache_basic():
""" Test single symbol-like objects are cached when printed by themselves. """
# Pairs of objects which should be considered equivalent with respect to caching
pairs = [
(x, sy.Symbol('x')),
(X, sy.MatrixSymbol('X', *X.shape)),
(f_t, sy.Function('f')(sy.Symbol('t'))),
]
for s1, s2 in pairs:
cache = {}
st = theano_code_(s1, cache=cache)
# Test hit with same instance
assert theano_code_(s1, cache=cache) is st
# Test miss with same instance but new cache
assert theano_code_(s1, cache={}) is not st
# Test hit with different but equivalent instance
assert theano_code_(s2, cache=cache) is st
def test_global_cache():
""" Test use of the global cache. """
from sympy.printing.theanocode import global_cache
backup = dict(global_cache)
try:
# Temporarily empty global cache
global_cache.clear()
for s in [x, X, f_t]:
with warns_deprecated_sympy():
st = theano_code(s)
assert theano_code(s) is st
finally:
# Restore global cache
global_cache.update(backup)
def test_cache_types_distinct():
"""
Test that symbol-like objects of different types (Symbol, MatrixSymbol,
AppliedUndef) are distinguished by the cache even if they have the same
name.
"""
symbols = [sy.Symbol('f_t'), sy.MatrixSymbol('f_t', 4, 4), f_t]
cache = {} # Single shared cache
printed = {}
for s in symbols:
st = theano_code_(s, cache=cache)
assert st not in printed.values()
printed[s] = st
# Check all printed objects are distinct
assert len(set(map(id, printed.values()))) == len(symbols)
# Check retrieving
for s, st in printed.items():
with warns_deprecated_sympy():
assert theano_code(s, cache=cache) is st
def test_symbols_are_created_once():
"""
Test that a symbol is cached and reused when it appears in an expression
more than once.
"""
expr = sy.Add(x, x, evaluate=False)
comp = theano_code_(expr)
assert theq(comp, xt + xt)
assert not theq(comp, xt + theano_code_(x))
def test_cache_complex():
"""
Test caching on a complicated expression with multiple symbols appearing
multiple times.
"""
expr = x ** 2 + (y - sy.exp(x)) * sy.sin(z - x * y)
symbol_names = {s.name for s in expr.free_symbols}
expr_t = theano_code_(expr)
# Iterate through variables in the Theano computational graph that the
# printed expression depends on
seen = set()
for v in theano.gof.graph.ancestors([expr_t]):
# Owner-less, non-constant variables should be our symbols
if v.owner is None and not isinstance(v, theano.gof.graph.Constant):
# Check it corresponds to a symbol and appears only once
assert v.name in symbol_names
assert v.name not in seen
seen.add(v.name)
# Check all were present
assert seen == symbol_names
def test_Piecewise():
# A piecewise linear
expr = sy.Piecewise((0, x<0), (x, x<2), (1, True)) # ___/III
result = theano_code_(expr)
assert result.owner.op == tt.switch
expected = tt.switch(xt<0, 0, tt.switch(xt<2, xt, 1))
assert theq(result, expected)
expr = sy.Piecewise((x, x < 0))
result = theano_code_(expr)
expected = tt.switch(xt < 0, xt, np.nan)
assert theq(result, expected)
expr = sy.Piecewise((0, sy.And(x>0, x<2)), \
(x, sy.Or(x>2, x<0)))
result = theano_code_(expr)
expected = tt.switch(tt.and_(xt>0,xt<2), 0, \
tt.switch(tt.or_(xt>2, xt<0), xt, np.nan))
assert theq(result, expected)
def test_Relationals():
assert theq(theano_code_(sy.Eq(x, y)), tt.eq(xt, yt))
# assert theq(theano_code_(sy.Ne(x, y)), tt.neq(xt, yt)) # TODO - implement
assert theq(theano_code_(x > y), xt > yt)
assert theq(theano_code_(x < y), xt < yt)
assert theq(theano_code_(x >= y), xt >= yt)
assert theq(theano_code_(x <= y), xt <= yt)
def test_complexfunctions():
with warns_deprecated_sympy():
xt, yt = theano_code_(x, dtypes={x:'complex128'}), theano_code_(y, dtypes={y: 'complex128'})
from sympy.functions.elementary.complexes import conjugate
from theano.tensor import as_tensor_variable as atv
from theano.tensor import complex as cplx
with warns_deprecated_sympy():
assert theq(theano_code_(y*conjugate(x)), yt*(xt.conj()))
assert theq(theano_code_((1+2j)*x), xt*(atv(1.0)+atv(2.0)*cplx(0,1)))
def test_constantfunctions():
with warns_deprecated_sympy():
tf = theano_function_([],[1+1j])
assert(tf()==1+1j)
def test_Exp1():
"""
Test that exp(1) prints without error and evaluates close to SymPy's E
"""
# sy.exp(1) should yield same instance of E as sy.E (singleton), but extra
# check added for sanity
e_a = sy.exp(1)
e_b = sy.E
np.testing.assert_allclose(float(e_a), np.e)
np.testing.assert_allclose(float(e_b), np.e)
e = theano_code_(e_a)
np.testing.assert_allclose(float(e_a), e.eval())
e = theano_code_(e_b)
np.testing.assert_allclose(float(e_b), e.eval())
|
b5eda5fb25967e49451e76b0f85da293cab676a112d9cc6b9c4d6fa09c4c18bd | from sympy.core import (S, pi, oo, symbols, Function, Rational, Integer,
Tuple, Symbol, Eq, Ne, Le, Lt, Gt, Ge)
from sympy.core import EulerGamma, GoldenRatio, Catalan, Lambda, Mul, Pow
from sympy.functions import Piecewise, sqrt, ceiling, exp, sin, cos, sinc, lucas
from sympy.testing.pytest import raises
from sympy.utilities.lambdify import implemented_function
from sympy.matrices import (eye, Matrix, MatrixSymbol, Identity,
HadamardProduct, SparseMatrix)
from sympy.functions.special.bessel import besseli
from sympy.printing.maple import maple_code
x, y, z = symbols('x,y,z')
def test_Integer():
assert maple_code(Integer(67)) == "67"
assert maple_code(Integer(-1)) == "-1"
def test_Rational():
assert maple_code(Rational(3, 7)) == "3/7"
assert maple_code(Rational(18, 9)) == "2"
assert maple_code(Rational(3, -7)) == "-3/7"
assert maple_code(Rational(-3, -7)) == "3/7"
assert maple_code(x + Rational(3, 7)) == "x + 3/7"
assert maple_code(Rational(3, 7) * x) == '(3/7)*x'
def test_Relational():
assert maple_code(Eq(x, y)) == "x = y"
assert maple_code(Ne(x, y)) == "x <> y"
assert maple_code(Le(x, y)) == "x <= y"
assert maple_code(Lt(x, y)) == "x < y"
assert maple_code(Gt(x, y)) == "x > y"
assert maple_code(Ge(x, y)) == "x >= y"
def test_Function():
assert maple_code(sin(x) ** cos(x)) == "sin(x)^cos(x)"
assert maple_code(abs(x)) == "abs(x)"
assert maple_code(ceiling(x)) == "ceil(x)"
def test_Pow():
assert maple_code(x ** 3) == "x^3"
assert maple_code(x ** (y ** 3)) == "x^(y^3)"
assert maple_code((x ** 3) ** y) == "(x^3)^y"
assert maple_code(x ** Rational(2, 3)) == 'x^(2/3)'
g = implemented_function('g', Lambda(x, 2 * x))
assert maple_code(1 / (g(x) * 3.5) ** (x - y ** x) / (x ** 2 + y)) == \
"(3.5*2*x)^(-x + y^x)/(x^2 + y)"
# For issue 14160
assert maple_code(Mul(-2, x, Pow(Mul(y, y, evaluate=False), -1, evaluate=False),
evaluate=False)) == '-2*x/(y*y)'
def test_basic_ops():
assert maple_code(x * y) == "x*y"
assert maple_code(x + y) == "x + y"
assert maple_code(x - y) == "x - y"
assert maple_code(-x) == "-x"
def test_1_over_x_and_sqrt():
# 1.0 and 0.5 would do something different in regular StrPrinter,
# but these are exact in IEEE floating point so no different here.
assert maple_code(1 / x) == '1/x'
assert maple_code(x ** -1) == maple_code(x ** -1.0) == '1/x'
assert maple_code(1 / sqrt(x)) == '1/sqrt(x)'
assert maple_code(x ** -S.Half) == maple_code(x ** -0.5) == '1/sqrt(x)'
assert maple_code(sqrt(x)) == 'sqrt(x)'
assert maple_code(x ** S.Half) == maple_code(x ** 0.5) == 'sqrt(x)'
assert maple_code(1 / pi) == '1/Pi'
assert maple_code(pi ** -1) == maple_code(pi ** -1.0) == '1/Pi'
assert maple_code(pi ** -0.5) == '1/sqrt(Pi)'
def test_mix_number_mult_symbols():
assert maple_code(3 * x) == "3*x"
assert maple_code(pi * x) == "Pi*x"
assert maple_code(3 / x) == "3/x"
assert maple_code(pi / x) == "Pi/x"
assert maple_code(x / 3) == '(1/3)*x'
assert maple_code(x / pi) == "x/Pi"
assert maple_code(x * y) == "x*y"
assert maple_code(3 * x * y) == "3*x*y"
assert maple_code(3 * pi * x * y) == "3*Pi*x*y"
assert maple_code(x / y) == "x/y"
assert maple_code(3 * x / y) == "3*x/y"
assert maple_code(x * y / z) == "x*y/z"
assert maple_code(x / y * z) == "x*z/y"
assert maple_code(1 / x / y) == "1/(x*y)"
assert maple_code(2 * pi * x / y / z) == "2*Pi*x/(y*z)"
assert maple_code(3 * pi / x) == "3*Pi/x"
assert maple_code(S(3) / 5) == "3/5"
assert maple_code(S(3) / 5 * x) == '(3/5)*x'
assert maple_code(x / y / z) == "x/(y*z)"
assert maple_code((x + y) / z) == "(x + y)/z"
assert maple_code((x + y) / (z + x)) == "(x + y)/(x + z)"
assert maple_code((x + y) / EulerGamma) == '(x + y)/gamma'
assert maple_code(x / 3 / pi) == '(1/3)*x/Pi'
assert maple_code(S(3) / 5 * x * y / pi) == '(3/5)*x*y/Pi'
def test_mix_number_pow_symbols():
assert maple_code(pi ** 3) == 'Pi^3'
assert maple_code(x ** 2) == 'x^2'
assert maple_code(x ** (pi ** 3)) == 'x^(Pi^3)'
assert maple_code(x ** y) == 'x^y'
assert maple_code(x ** (y ** z)) == 'x^(y^z)'
assert maple_code((x ** y) ** z) == '(x^y)^z'
def test_imag():
I = S('I')
assert maple_code(I) == "I"
assert maple_code(5 * I) == "5*I"
assert maple_code((S(3) / 2) * I) == "(3/2)*I"
assert maple_code(3 + 4 * I) == "3 + 4*I"
def test_constants():
assert maple_code(pi) == "Pi"
assert maple_code(oo) == "infinity"
assert maple_code(-oo) == "-infinity"
assert maple_code(S.NegativeInfinity) == "-infinity"
assert maple_code(S.NaN) == "undefined"
assert maple_code(S.Exp1) == "exp(1)"
assert maple_code(exp(1)) == "exp(1)"
def test_constants_other():
assert maple_code(2 * GoldenRatio) == '2*(1/2 + (1/2)*sqrt(5))'
assert maple_code(2 * Catalan) == '2*Catalan'
assert maple_code(2 * EulerGamma) == "2*gamma"
def test_boolean():
assert maple_code(x & y) == "x && y"
assert maple_code(x | y) == "x || y"
assert maple_code(~x) == "!x"
assert maple_code(x & y & z) == "x && y && z"
assert maple_code(x | y | z) == "x || y || z"
assert maple_code((x & y) | z) == "z || x && y"
assert maple_code((x | y) & z) == "z && (x || y)"
def test_Matrices():
assert maple_code(Matrix(1, 1, [10])) == \
'Matrix([[10]], storage = rectangular)'
A = Matrix([[1, sin(x / 2), abs(x)],
[0, 1, pi],
[0, exp(1), ceiling(x)]])
expected = \
'Matrix(' \
'[[1, sin((1/2)*x), abs(x)],' \
' [0, 1, Pi],' \
' [0, exp(1), ceil(x)]], ' \
'storage = rectangular)'
assert maple_code(A) == expected
# row and columns
assert maple_code(A[:, 0]) == \
'Matrix([[1], [0], [0]], storage = rectangular)'
assert maple_code(A[0, :]) == \
'Matrix([[1, sin((1/2)*x), abs(x)]], storage = rectangular)'
assert maple_code(Matrix([[x, x - y, -y]])) == \
'Matrix([[x, x - y, -y]], storage = rectangular)'
# empty matrices
assert maple_code(Matrix(0, 0, [])) == \
'Matrix([], storage = rectangular)'
assert maple_code(Matrix(0, 3, [])) == \
'Matrix([], storage = rectangular)'
def test_SparseMatrices():
assert maple_code(SparseMatrix(Identity(2))) == 'Matrix([[1, 0], [0, 1]], storage = sparse)'
def test_vector_entries_hadamard():
# For a row or column, user might to use the other dimension
A = Matrix([[1, sin(2 / x), 3 * pi / x / 5]])
assert maple_code(A) == \
'Matrix([[1, sin(2/x), (3/5)*Pi/x]], storage = rectangular)'
assert maple_code(A.T) == \
'Matrix([[1], [sin(2/x)], [(3/5)*Pi/x]], storage = rectangular)'
def test_Matrices_entries_not_hadamard():
A = Matrix([[1, sin(2 / x), 3 * pi / x / 5], [1, 2, x * y]])
expected = \
'Matrix([[1, sin(2/x), (3/5)*Pi/x], [1, 2, x*y]], ' \
'storage = rectangular)'
assert maple_code(A) == expected
def test_MatrixSymbol():
n = Symbol('n', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
assert maple_code(A * B) == "A.B"
assert maple_code(B * A) == "B.A"
assert maple_code(2 * A * B) == "2*A.B"
assert maple_code(B * 2 * A) == "2*B.A"
assert maple_code(
A * (B + 3 * Identity(n))) == "A.(3*Matrix(n, shape = identity) + B)"
assert maple_code(A ** (x ** 2)) == "MatrixPower(A, x^2)"
assert maple_code(A ** 3) == "MatrixPower(A, 3)"
assert maple_code(A ** (S.Half)) == "MatrixPower(A, 1/2)"
def test_special_matrices():
assert maple_code(6 * Identity(3)) == "6*Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], storage = sparse)"
assert maple_code(Identity(x)) == 'Matrix(x, shape = identity)'
def test_containers():
assert maple_code([1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]) == \
"[1, 2, 3, [4, 5, [6, 7]], 8, [9, 10], 11]"
assert maple_code((1, 2, (3, 4))) == "[1, 2, [3, 4]]"
assert maple_code([1]) == "[1]"
assert maple_code((1,)) == "[1]"
assert maple_code(Tuple(*[1, 2, 3])) == "[1, 2, 3]"
assert maple_code((1, x * y, (3, x ** 2))) == "[1, x*y, [3, x^2]]"
# scalar, matrix, empty matrix and empty list
assert maple_code((1, eye(3), Matrix(0, 0, []), [])) == \
"[1, Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]], storage = rectangular), Matrix([], storage = rectangular), []]"
def test_maple_noninline():
source = maple_code((x + y)/Catalan, assign_to='me', inline=False)
expected = "me := (x + y)/Catalan"
assert source == expected
def test_maple_matrix_assign_to():
A = Matrix([[1, 2, 3]])
assert maple_code(A, assign_to='a') == "a := Matrix([[1, 2, 3]], storage = rectangular)"
A = Matrix([[1, 2], [3, 4]])
assert maple_code(A, assign_to='A') == "A := Matrix([[1, 2], [3, 4]], storage = rectangular)"
def test_maple_matrix_assign_to_more():
# assigning to Symbol or MatrixSymbol requires lhs/rhs match
A = Matrix([[1, 2, 3]])
B = MatrixSymbol('B', 1, 3)
C = MatrixSymbol('C', 2, 3)
assert maple_code(A, assign_to=B) == "B := Matrix([[1, 2, 3]], storage = rectangular)"
raises(ValueError, lambda: maple_code(A, assign_to=x))
raises(ValueError, lambda: maple_code(A, assign_to=C))
def test_maple_matrix_1x1():
A = Matrix([[3]])
assert maple_code(A, assign_to='B') == "B := Matrix([[3]], storage = rectangular)"
def test_maple_matrix_elements():
A = Matrix([[x, 2, x * y]])
assert maple_code(A[0, 0] ** 2 + A[0, 1] + A[0, 2]) == "x^2 + x*y + 2"
AA = MatrixSymbol('AA', 1, 3)
assert maple_code(AA) == "AA"
assert maple_code(AA[0, 0] ** 2 + sin(AA[0, 1]) + AA[0, 2]) == \
"sin(AA[1, 2]) + AA[1, 1]^2 + AA[1, 3]"
assert maple_code(sum(AA)) == "AA[1, 1] + AA[1, 2] + AA[1, 3]"
def test_maple_boolean():
assert maple_code(True) == "true"
assert maple_code(S.true) == "true"
assert maple_code(False) == "false"
assert maple_code(S.false) == "false"
def test_sparse():
M = SparseMatrix(5, 6, {})
M[2, 2] = 10
M[1, 2] = 20
M[1, 3] = 22
M[0, 3] = 30
M[3, 0] = x * y
assert maple_code(M) == \
'Matrix([[0, 0, 0, 30, 0, 0],' \
' [0, 0, 20, 22, 0, 0],' \
' [0, 0, 10, 0, 0, 0],' \
' [x*y, 0, 0, 0, 0, 0],' \
' [0, 0, 0, 0, 0, 0]], ' \
'storage = sparse)'
# Not an important point.
def test_maple_not_supported():
assert maple_code(S.ComplexInfinity) == (
"# Not supported in maple:\n"
"# ComplexInfinity\n"
"zoo"
) # PROBLEM
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
assert (maple_code(A[0, 0]) == "A[1, 1]")
assert (maple_code(3 * A[0, 0]) == "3*A[1, 1]")
F = A-B
assert (maple_code(F[0,0]) == "A[1, 1] - B[1, 1]")
def test_hadamard():
A = MatrixSymbol('A', 3, 3)
B = MatrixSymbol('B', 3, 3)
v = MatrixSymbol('v', 3, 1)
h = MatrixSymbol('h', 1, 3)
C = HadamardProduct(A, B)
assert maple_code(C) == "A*B"
assert maple_code(C * v) == "(A*B).v"
# HadamardProduct is higher than dot product.
assert maple_code(h * C * v) == "h.(A*B).v"
assert maple_code(C * A) == "(A*B).A"
# mixing Hadamard and scalar strange b/c we vectorize scalars
assert maple_code(C * x * y) == "x*y*(A*B)"
def test_maple_piecewise():
expr = Piecewise((x, x < 1), (x ** 2, True))
assert maple_code(expr) == "piecewise(x < 1, x, x^2)"
assert maple_code(expr, assign_to="r") == (
"r := piecewise(x < 1, x, x^2)")
expr = Piecewise((x ** 2, x < 1), (x ** 3, x < 2), (x ** 4, x < 3), (x ** 5, True))
expected = "piecewise(x < 1, x^2, x < 2, x^3, x < 3, x^4, x^5)"
assert maple_code(expr) == expected
assert maple_code(expr, assign_to="r") == "r := " + expected
# Check that Piecewise without a True (default) condition error
expr = Piecewise((x, x < 1), (x ** 2, x > 1), (sin(x), x > 0))
raises(ValueError, lambda: maple_code(expr))
def test_maple_piecewise_times_const():
pw = Piecewise((x, x < 1), (x ** 2, True))
assert maple_code(2 * pw) == "2*piecewise(x < 1, x, x^2)"
assert maple_code(pw / x) == "piecewise(x < 1, x, x^2)/x"
assert maple_code(pw / (x * y)) == "piecewise(x < 1, x, x^2)/(x*y)"
assert maple_code(pw / 3) == "(1/3)*piecewise(x < 1, x, x^2)"
def test_maple_derivatives():
f = Function('f')
assert maple_code(f(x).diff(x)) == 'diff(f(x), x)'
assert maple_code(f(x).diff(x, 2)) == 'diff(f(x), x$2)'
def test_automatic_rewrites():
assert maple_code(lucas(x)) == '2^(-x)*((1 - sqrt(5))^x + (1 + sqrt(5))^x)'
assert maple_code(sinc(x)) == 'piecewise(x <> 0, sin(x)/x, 1)'
def test_specfun():
assert maple_code('asin(x)') == 'arcsin(x)'
assert maple_code(besseli(x, y)) == 'BesselI(x, y)'
|
202ec1394674d7ff96aea06f9f79c400bf3d51eccacee712f1d34302f0f30076 | # -*- coding: utf-8 -*-
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import (Dict, Tuple)
from sympy.core.function import (Derivative, Function, Lambda, Subs)
from sympy.core.mul import Mul
from sympy.core import (EulerGamma, GoldenRatio, Catalan)
from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.power import Pow
from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import conjugate
from sympy.functions.elementary.exponential import LambertW
from sympy.functions.special.bessel import (airyai, airyaiprime, airybi, airybiprime)
from sympy.functions.special.delta_functions import Heaviside
from sympy.functions.special.error_functions import (fresnelc, fresnels)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.zeta_functions import dirichlet_eta
from sympy.geometry.line import (Ray, Segment)
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import (And, Equivalent, ITE, Implies, Nand, Nor, Not, Or, Xor)
from sympy.matrices.dense import (Matrix, diag)
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions.trace import Trace
from sympy.polys.domains.finitefield import FF
from sympy.polys.domains.integerring import ZZ
from sympy.polys.domains.rationalfield import QQ
from sympy.polys.domains.realfield import RR
from sympy.polys.orderings import (grlex, ilex)
from sympy.polys.polytools import groebner
from sympy.polys.rootoftools import (RootSum, rootof)
from sympy.series.formal import fps
from sympy.series.fourier import fourier_series
from sympy.series.limits import Limit
from sympy.series.order import O
from sympy.series.sequences import (SeqAdd, SeqFormula, SeqMul, SeqPer)
from sympy.sets.contains import Contains
from sympy.sets.fancysets import Range
from sympy.sets.sets import (Complement, FiniteSet, Intersection, Interval, Union)
from sympy.codegen.ast import (Assignment, AddAugmentedAssignment,
SubAugmentedAssignment, MulAugmentedAssignment, DivAugmentedAssignment, ModAugmentedAssignment)
from sympy.core.expr import UnevaluatedExpr
from sympy.physics.quantum.trace import Tr
from sympy.functions import (Abs, Chi, Ci, Ei, KroneckerDelta,
Piecewise, Shi, Si, atan2, beta, binomial, catalan, ceiling, cos,
euler, exp, expint, factorial, factorial2, floor, gamma, hyper, log,
meijerg, sin, sqrt, subfactorial, tan, uppergamma, lerchphi,
elliptic_k, elliptic_f, elliptic_e, elliptic_pi, DiracDelta, bell,
bernoulli, fibonacci, tribonacci, lucas, stieltjes, mathieuc, mathieus,
mathieusprime, mathieucprime)
from sympy.matrices import (Adjoint, Inverse, MatrixSymbol, Transpose,
KroneckerProduct, BlockMatrix, OneMatrix, ZeroMatrix)
from sympy.matrices.expressions import hadamard_power
from sympy.physics import mechanics
from sympy.physics.control.lti import (TransferFunction, Feedback, TransferFunctionMatrix,
Series, Parallel, MIMOSeries, MIMOParallel, MIMOFeedback)
from sympy.physics.units import joule, degree
from sympy.printing.pretty import pprint, pretty as xpretty
from sympy.printing.pretty.pretty_symbology import center_accent, is_combining
from sympy.sets.conditionset import ConditionSet
from sympy.sets import ImageSet, ProductSet
from sympy.sets.setexpr import SetExpr
from sympy.stats.crv_types import Normal
from sympy.stats.symbolic_probability import (Covariance, Expectation,
Probability, Variance)
from sympy.tensor.array import (ImmutableDenseNDimArray, ImmutableSparseNDimArray,
MutableDenseNDimArray, MutableSparseNDimArray, tensorproduct)
from sympy.tensor.functions import TensorProduct
from sympy.tensor.tensor import (TensorIndexType, tensor_indices, TensorHead,
TensorElement, tensor_heads)
from sympy.testing.pytest import raises, _both_exp_pow, warns_deprecated_sympy
from sympy.vector import CoordSys3D, Gradient, Curl, Divergence, Dot, Cross, Laplacian
import sympy as sym
class lowergamma(sym.lowergamma):
pass # testing notation inheritance by a subclass with same name
a, b, c, d, x, y, z, k, n, s, p = symbols('a,b,c,d,x,y,z,k,n,s,p')
f = Function("f")
th = Symbol('theta')
ph = Symbol('phi')
"""
Expressions whose pretty-printing is tested here:
(A '#' to the right of an expression indicates that its various acceptable
orderings are accounted for by the tests.)
BASIC EXPRESSIONS:
oo
(x**2)
1/x
y*x**-2
x**Rational(-5,2)
(-2)**x
Pow(3, 1, evaluate=False)
(x**2 + x + 1) #
1-x #
1-2*x #
x/y
-x/y
(x+2)/y #
(1+x)*y #3
-5*x/(x+10) # correct placement of negative sign
1 - Rational(3,2)*(x+1)
-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5) # issue 5524
ORDERING:
x**2 + x + 1
1 - x
1 - 2*x
2*x**4 + y**2 - x**2 + y**3
RELATIONAL:
Eq(x, y)
Lt(x, y)
Gt(x, y)
Le(x, y)
Ge(x, y)
Ne(x/(y+1), y**2) #
RATIONAL NUMBERS:
y*x**-2
y**Rational(3,2) * x**Rational(-5,2)
sin(x)**3/tan(x)**2
FUNCTIONS (ABS, CONJ, EXP, FUNCTION BRACES, FACTORIAL, FLOOR, CEILING):
(2*x + exp(x)) #
Abs(x)
Abs(x/(x**2+1)) #
Abs(1 / (y - Abs(x)))
factorial(n)
factorial(2*n)
subfactorial(n)
subfactorial(2*n)
factorial(factorial(factorial(n)))
factorial(n+1) #
conjugate(x)
conjugate(f(x+1)) #
f(x)
f(x, y)
f(x/(y+1), y) #
f(x**x**x**x**x**x)
sin(x)**2
conjugate(a+b*I)
conjugate(exp(a+b*I))
conjugate( f(1 + conjugate(f(x))) ) #
f(x/(y+1), y) # denom of first arg
floor(1 / (y - floor(x)))
ceiling(1 / (y - ceiling(x)))
SQRT:
sqrt(2)
2**Rational(1,3)
2**Rational(1,1000)
sqrt(x**2 + 1)
(1 + sqrt(5))**Rational(1,3)
2**(1/x)
sqrt(2+pi)
(2+(1+x**2)/(2+x))**Rational(1,4)+(1+x**Rational(1,1000))/sqrt(3+x**2)
DERIVATIVES:
Derivative(log(x), x, evaluate=False)
Derivative(log(x), x, evaluate=False) + x #
Derivative(log(x) + x**2, x, y, evaluate=False)
Derivative(2*x*y, y, x, evaluate=False) + x**2 #
beta(alpha).diff(alpha)
INTEGRALS:
Integral(log(x), x)
Integral(x**2, x)
Integral((sin(x))**2 / (tan(x))**2)
Integral(x**(2**x), x)
Integral(x**2, (x,1,2))
Integral(x**2, (x,Rational(1,2),10))
Integral(x**2*y**2, x,y)
Integral(x**2, (x, None, 1))
Integral(x**2, (x, 1, None))
Integral(sin(th)/cos(ph), (th,0,pi), (ph, 0, 2*pi))
MATRICES:
Matrix([[x**2+1, 1], [y, x+y]]) #
Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
PIECEWISE:
Piecewise((x,x<1),(x**2,True))
ITE:
ITE(x, y, z)
SEQUENCES (TUPLES, LISTS, DICTIONARIES):
()
[]
{}
(1/x,)
[x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
{x: sin(x)}
{1/x: 1/y, x: sin(x)**2} #
[x**2]
(x**2,)
{x**2: 1}
LIMITS:
Limit(x, x, oo)
Limit(x**2, x, 0)
Limit(1/x, x, 0)
Limit(sin(x)/x, x, 0)
UNITS:
joule => kg*m**2/s
SUBS:
Subs(f(x), x, ph**2)
Subs(f(x).diff(x), x, 0)
Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ORDER:
O(1)
O(1/x)
O(x**2 + y**2)
"""
def pretty(expr, order=None):
"""ASCII pretty-printing"""
return xpretty(expr, order=order, use_unicode=False, wrap_line=False)
def upretty(expr, order=None):
"""Unicode pretty-printing"""
return xpretty(expr, order=order, use_unicode=True, wrap_line=False)
def test_pretty_ascii_str():
assert pretty( 'xxx' ) == 'xxx'
assert pretty( "xxx" ) == 'xxx'
assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx'
assert pretty( 'xxx"xxx' ) == 'xxx\"xxx'
assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx'
assert pretty( "xxx'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\"xxx" ) == 'xxx\"xxx'
assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx'
assert pretty( "xxx\nxxx" ) == 'xxx\nxxx'
def test_pretty_unicode_str():
assert pretty( 'xxx' ) == 'xxx'
assert pretty( 'xxx' ) == 'xxx'
assert pretty( 'xxx\'xxx' ) == 'xxx\'xxx'
assert pretty( 'xxx"xxx' ) == 'xxx\"xxx'
assert pretty( 'xxx\"xxx' ) == 'xxx\"xxx'
assert pretty( "xxx'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\'xxx" ) == 'xxx\'xxx'
assert pretty( "xxx\"xxx" ) == 'xxx\"xxx'
assert pretty( "xxx\"xxx\'xxx" ) == 'xxx"xxx\'xxx'
assert pretty( "xxx\nxxx" ) == 'xxx\nxxx'
def test_upretty_greek():
assert upretty( oo ) == '∞'
assert upretty( Symbol('alpha^+_1') ) == 'α⁺₁'
assert upretty( Symbol('beta') ) == 'β'
assert upretty(Symbol('lambda')) == 'λ'
def test_upretty_multiindex():
assert upretty( Symbol('beta12') ) == 'β₁₂'
assert upretty( Symbol('Y00') ) == 'Y₀₀'
assert upretty( Symbol('Y_00') ) == 'Y₀₀'
assert upretty( Symbol('F^+-') ) == 'F⁺⁻'
def test_upretty_sub_super():
assert upretty( Symbol('beta_1_2') ) == 'β₁ ₂'
assert upretty( Symbol('beta^1^2') ) == 'β¹ ²'
assert upretty( Symbol('beta_1^2') ) == 'β²₁'
assert upretty( Symbol('beta_10_20') ) == 'β₁₀ ₂₀'
assert upretty( Symbol('beta_ax_gamma^i') ) == 'βⁱₐₓ ᵧ'
assert upretty( Symbol("F^1^2_3_4") ) == 'F¹ ²₃ ₄'
assert upretty( Symbol("F_1_2^3^4") ) == 'F³ ⁴₁ ₂'
assert upretty( Symbol("F_1_2_3_4") ) == 'F₁ ₂ ₃ ₄'
assert upretty( Symbol("F^1^2^3^4") ) == 'F¹ ² ³ ⁴'
def test_upretty_subs_missing_in_24():
assert upretty( Symbol('F_beta') ) == 'Fᵦ'
assert upretty( Symbol('F_gamma') ) == 'Fᵧ'
assert upretty( Symbol('F_rho') ) == 'Fᵨ'
assert upretty( Symbol('F_phi') ) == 'Fᵩ'
assert upretty( Symbol('F_chi') ) == 'Fᵪ'
assert upretty( Symbol('F_a') ) == 'Fₐ'
assert upretty( Symbol('F_e') ) == 'Fₑ'
assert upretty( Symbol('F_i') ) == 'Fᵢ'
assert upretty( Symbol('F_o') ) == 'Fₒ'
assert upretty( Symbol('F_u') ) == 'Fᵤ'
assert upretty( Symbol('F_r') ) == 'Fᵣ'
assert upretty( Symbol('F_v') ) == 'Fᵥ'
assert upretty( Symbol('F_x') ) == 'Fₓ'
def test_missing_in_2X_issue_9047():
assert upretty( Symbol('F_h') ) == 'Fₕ'
assert upretty( Symbol('F_k') ) == 'Fₖ'
assert upretty( Symbol('F_l') ) == 'Fₗ'
assert upretty( Symbol('F_m') ) == 'Fₘ'
assert upretty( Symbol('F_n') ) == 'Fₙ'
assert upretty( Symbol('F_p') ) == 'Fₚ'
assert upretty( Symbol('F_s') ) == 'Fₛ'
assert upretty( Symbol('F_t') ) == 'Fₜ'
def test_upretty_modifiers():
# Accents
assert upretty( Symbol('Fmathring') ) == 'F̊'
assert upretty( Symbol('Fddddot') ) == 'F⃜'
assert upretty( Symbol('Fdddot') ) == 'F⃛'
assert upretty( Symbol('Fddot') ) == 'F̈'
assert upretty( Symbol('Fdot') ) == 'Ḟ'
assert upretty( Symbol('Fcheck') ) == 'F̌'
assert upretty( Symbol('Fbreve') ) == 'F̆'
assert upretty( Symbol('Facute') ) == 'F́'
assert upretty( Symbol('Fgrave') ) == 'F̀'
assert upretty( Symbol('Ftilde') ) == 'F̃'
assert upretty( Symbol('Fhat') ) == 'F̂'
assert upretty( Symbol('Fbar') ) == 'F̅'
assert upretty( Symbol('Fvec') ) == 'F⃗'
assert upretty( Symbol('Fprime') ) == 'F′'
assert upretty( Symbol('Fprm') ) == 'F′'
# No faces are actually implemented, but test to make sure the modifiers are stripped
assert upretty( Symbol('Fbold') ) == 'Fbold'
assert upretty( Symbol('Fbm') ) == 'Fbm'
assert upretty( Symbol('Fcal') ) == 'Fcal'
assert upretty( Symbol('Fscr') ) == 'Fscr'
assert upretty( Symbol('Ffrak') ) == 'Ffrak'
# Brackets
assert upretty( Symbol('Fnorm') ) == '‖F‖'
assert upretty( Symbol('Favg') ) == '⟨F⟩'
assert upretty( Symbol('Fabs') ) == '|F|'
assert upretty( Symbol('Fmag') ) == '|F|'
# Combinations
assert upretty( Symbol('xvecdot') ) == 'x⃗̇'
assert upretty( Symbol('xDotVec') ) == 'ẋ⃗'
assert upretty( Symbol('xHATNorm') ) == '‖x̂‖'
assert upretty( Symbol('xMathring_yCheckPRM__zbreveAbs') ) == 'x̊_y̌′__|z̆|'
assert upretty( Symbol('alphadothat_nVECDOT__tTildePrime') ) == 'α̇̂_n⃗̇__t̃′'
assert upretty( Symbol('x_dot') ) == 'x_dot'
assert upretty( Symbol('x__dot') ) == 'x__dot'
def test_pretty_Cycle():
from sympy.combinatorics.permutations import Cycle
assert pretty(Cycle(1, 2)) == '(1 2)'
assert pretty(Cycle(2)) == '(2)'
assert pretty(Cycle(1, 3)(4, 5)) == '(1 3)(4 5)'
assert pretty(Cycle()) == '()'
def test_pretty_Permutation():
from sympy.combinatorics.permutations import Permutation
p1 = Permutation(1, 2)(3, 4)
assert xpretty(p1, perm_cyclic=True, use_unicode=True) == "(1 2)(3 4)"
assert xpretty(p1, perm_cyclic=True, use_unicode=False) == "(1 2)(3 4)"
assert xpretty(p1, perm_cyclic=False, use_unicode=True) == \
'⎛0 1 2 3 4⎞\n'\
'⎝0 2 1 4 3⎠'
assert xpretty(p1, perm_cyclic=False, use_unicode=False) == \
"/0 1 2 3 4\\\n"\
"\\0 2 1 4 3/"
with warns_deprecated_sympy():
old_print_cyclic = Permutation.print_cyclic
Permutation.print_cyclic = False
assert xpretty(p1, use_unicode=True) == \
'⎛0 1 2 3 4⎞\n'\
'⎝0 2 1 4 3⎠'
assert xpretty(p1, use_unicode=False) == \
"/0 1 2 3 4\\\n"\
"\\0 2 1 4 3/"
Permutation.print_cyclic = old_print_cyclic
def test_pretty_basic():
assert pretty( -Rational(1)/2 ) == '-1/2'
assert pretty( -Rational(13)/22 ) == \
"""\
-13 \n\
----\n\
22 \
"""
expr = oo
ascii_str = \
"""\
oo\
"""
ucode_str = \
"""\
∞\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2)
ascii_str = \
"""\
2\n\
x \
"""
ucode_str = \
"""\
2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 1/x
ascii_str = \
"""\
1\n\
-\n\
x\
"""
ucode_str = \
"""\
1\n\
─\n\
x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# not the same as 1/x
expr = x**-1.0
ascii_str = \
"""\
-1.0\n\
x \
"""
ucode_str = \
"""\
-1.0\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# see issue #2860
expr = Pow(S(2), -1.0, evaluate=False)
ascii_str = \
"""\
-1.0\n\
2 \
"""
ucode_str = \
"""\
-1.0\n\
2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
"""\
y \n\
──\n\
2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
#see issue #14033
expr = x**Rational(1, 3)
ascii_str = \
"""\
1/3\n\
x \
"""
ucode_str = \
"""\
1/3\n\
x \
"""
assert xpretty(expr, use_unicode=False, wrap_line=False,\
root_notation = False) == ascii_str
assert xpretty(expr, use_unicode=True, wrap_line=False,\
root_notation = False) == ucode_str
expr = x**Rational(-5, 2)
ascii_str = \
"""\
1 \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
"""\
1 \n\
────\n\
5/2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (-2)**x
ascii_str = \
"""\
x\n\
(-2) \
"""
ucode_str = \
"""\
x\n\
(-2) \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# See issue 4923
expr = Pow(3, 1, evaluate=False)
ascii_str = \
"""\
1\n\
3 \
"""
ucode_str = \
"""\
1\n\
3 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2 + x + 1)
ascii_str_1 = \
"""\
2\n\
1 + x + x \
"""
ascii_str_2 = \
"""\
2 \n\
x + x + 1\
"""
ascii_str_3 = \
"""\
2 \n\
x + 1 + x\
"""
ucode_str_1 = \
"""\
2\n\
1 + x + x \
"""
ucode_str_2 = \
"""\
2 \n\
x + x + 1\
"""
ucode_str_3 = \
"""\
2 \n\
x + 1 + x\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
expr = 1 - x
ascii_str_1 = \
"""\
1 - x\
"""
ascii_str_2 = \
"""\
-x + 1\
"""
ucode_str_1 = \
"""\
1 - x\
"""
ucode_str_2 = \
"""\
-x + 1\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 1 - 2*x
ascii_str_1 = \
"""\
1 - 2*x\
"""
ascii_str_2 = \
"""\
-2*x + 1\
"""
ucode_str_1 = \
"""\
1 - 2⋅x\
"""
ucode_str_2 = \
"""\
-2⋅x + 1\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = x/y
ascii_str = \
"""\
x\n\
-\n\
y\
"""
ucode_str = \
"""\
x\n\
─\n\
y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/y
ascii_str = \
"""\
-x \n\
---\n\
y \
"""
ucode_str = \
"""\
-x \n\
───\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x + 2)/y
ascii_str_1 = \
"""\
2 + x\n\
-----\n\
y \
"""
ascii_str_2 = \
"""\
x + 2\n\
-----\n\
y \
"""
ucode_str_1 = \
"""\
2 + x\n\
─────\n\
y \
"""
ucode_str_2 = \
"""\
x + 2\n\
─────\n\
y \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = (1 + x)*y
ascii_str_1 = \
"""\
y*(1 + x)\
"""
ascii_str_2 = \
"""\
(1 + x)*y\
"""
ascii_str_3 = \
"""\
y*(x + 1)\
"""
ucode_str_1 = \
"""\
y⋅(1 + x)\
"""
ucode_str_2 = \
"""\
(1 + x)⋅y\
"""
ucode_str_3 = \
"""\
y⋅(x + 1)\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2, ascii_str_3]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
# Test for correct placement of the negative sign
expr = -5*x/(x + 10)
ascii_str_1 = \
"""\
-5*x \n\
------\n\
10 + x\
"""
ascii_str_2 = \
"""\
-5*x \n\
------\n\
x + 10\
"""
ucode_str_1 = \
"""\
-5⋅x \n\
──────\n\
10 + x\
"""
ucode_str_2 = \
"""\
-5⋅x \n\
──────\n\
x + 10\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = -S.Half - 3*x
ascii_str = \
"""\
-3*x - 1/2\
"""
ucode_str = \
"""\
-3⋅x - 1/2\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S.Half - 3*x
ascii_str = \
"""\
1/2 - 3*x\
"""
ucode_str = \
"""\
1/2 - 3⋅x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -S.Half - 3*x/2
ascii_str = \
"""\
3*x 1\n\
- --- - -\n\
2 2\
"""
ucode_str = \
"""\
3⋅x 1\n\
- ─── - ─\n\
2 2\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = S.Half - 3*x/2
ascii_str = \
"""\
1 3*x\n\
- - ---\n\
2 2 \
"""
ucode_str = \
"""\
1 3⋅x\n\
─ - ───\n\
2 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_negative_fractions():
expr = -x/y
ascii_str =\
"""\
-x \n\
---\n\
y \
"""
ucode_str =\
"""\
-x \n\
───\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x*z/y
ascii_str =\
"""\
-x*z \n\
-----\n\
y \
"""
ucode_str =\
"""\
-x⋅z \n\
─────\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x**2/y
ascii_str =\
"""\
2\n\
x \n\
--\n\
y \
"""
ucode_str =\
"""\
2\n\
x \n\
──\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x**2/y
ascii_str =\
"""\
2 \n\
-x \n\
----\n\
y \
"""
ucode_str =\
"""\
2 \n\
-x \n\
────\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -x/(y*z)
ascii_str =\
"""\
-x \n\
---\n\
y*z\
"""
ucode_str =\
"""\
-x \n\
───\n\
y⋅z\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -a/y**2
ascii_str =\
"""\
-a \n\
---\n\
2\n\
y \
"""
ucode_str =\
"""\
-a \n\
───\n\
2\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**(-a/b)
ascii_str =\
"""\
-a \n\
---\n\
b \n\
y \
"""
ucode_str =\
"""\
-a \n\
───\n\
b \n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -1/y**2
ascii_str =\
"""\
-1 \n\
---\n\
2\n\
y \
"""
ucode_str =\
"""\
-1 \n\
───\n\
2\n\
y \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -10/b**2
ascii_str =\
"""\
-10 \n\
----\n\
2 \n\
b \
"""
ucode_str =\
"""\
-10 \n\
────\n\
2 \n\
b \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Rational(-200, 37)
ascii_str =\
"""\
-200 \n\
-----\n\
37 \
"""
ucode_str =\
"""\
-200 \n\
─────\n\
37 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_Mul():
expr = Mul(0, 1, evaluate=False)
assert pretty(expr) == "0*1"
assert upretty(expr) == "0⋅1"
expr = Mul(1, 0, evaluate=False)
assert pretty(expr) == "1*0"
assert upretty(expr) == "1⋅0"
expr = Mul(1, 1, evaluate=False)
assert pretty(expr) == "1*1"
assert upretty(expr) == "1⋅1"
expr = Mul(1, 1, 1, evaluate=False)
assert pretty(expr) == "1*1*1"
assert upretty(expr) == "1⋅1⋅1"
expr = Mul(1, 2, evaluate=False)
assert pretty(expr) == "1*2"
assert upretty(expr) == "1⋅2"
expr = Add(0, 1, evaluate=False)
assert pretty(expr) == "0 + 1"
assert upretty(expr) == "0 + 1"
expr = Mul(1, 1, 2, evaluate=False)
assert pretty(expr) == "1*1*2"
assert upretty(expr) == "1⋅1⋅2"
expr = Add(0, 0, 1, evaluate=False)
assert pretty(expr) == "0 + 0 + 1"
assert upretty(expr) == "0 + 0 + 1"
expr = Mul(1, -1, evaluate=False)
assert pretty(expr) == "1*-1"
assert upretty(expr) == "1⋅-1"
expr = Mul(1.0, x, evaluate=False)
assert pretty(expr) == "1.0*x"
assert upretty(expr) == "1.0⋅x"
expr = Mul(1, 1, 2, 3, x, evaluate=False)
assert pretty(expr) == "1*1*2*3*x"
assert upretty(expr) == "1⋅1⋅2⋅3⋅x"
expr = Mul(-1, 1, evaluate=False)
assert pretty(expr) == "-1*1"
assert upretty(expr) == "-1⋅1"
expr = Mul(4, 3, 2, 1, 0, y, x, evaluate=False)
assert pretty(expr) == "4*3*2*1*0*y*x"
assert upretty(expr) == "4⋅3⋅2⋅1⋅0⋅y⋅x"
expr = Mul(4, 3, 2, 1+z, 0, y, x, evaluate=False)
assert pretty(expr) == "4*3*2*(z + 1)*0*y*x"
assert upretty(expr) == "4⋅3⋅2⋅(z + 1)⋅0⋅y⋅x"
expr = Mul(Rational(2, 3), Rational(5, 7), evaluate=False)
assert pretty(expr) == "2/3*5/7"
assert upretty(expr) == "2/3⋅5/7"
expr = Mul(x + y, Rational(1, 2), evaluate=False)
assert pretty(expr) == "(x + y)*1/2"
assert upretty(expr) == "(x + y)⋅1/2"
expr = Mul(Rational(1, 2), x + y, evaluate=False)
assert pretty(expr) == "x + y\n-----\n 2 "
assert upretty(expr) == "x + y\n─────\n 2 "
expr = Mul(S.One, x + y, evaluate=False)
assert pretty(expr) == "1*(x + y)"
assert upretty(expr) == "1⋅(x + y)"
expr = Mul(x - y, S.One, evaluate=False)
assert pretty(expr) == "(x - y)*1"
assert upretty(expr) == "(x - y)⋅1"
expr = Mul(Rational(1, 2), x - y, S.One, x + y, evaluate=False)
assert pretty(expr) == "1/2*(x - y)*1*(x + y)"
assert upretty(expr) == "1/2⋅(x - y)⋅1⋅(x + y)"
expr = Mul(x + y, Rational(3, 4), S.One, y - z, evaluate=False)
assert pretty(expr) == "(x + y)*3/4*1*(y - z)"
assert upretty(expr) == "(x + y)⋅3/4⋅1⋅(y - z)"
expr = Mul(x + y, Rational(1, 1), Rational(3, 4), Rational(5, 6),evaluate=False)
assert pretty(expr) == "(x + y)*1*3/4*5/6"
assert upretty(expr) == "(x + y)⋅1⋅3/4⋅5/6"
expr = Mul(Rational(3, 4), x + y, S.One, y - z, evaluate=False)
assert pretty(expr) == "3/4*(x + y)*1*(y - z)"
assert upretty(expr) == "3/4⋅(x + y)⋅1⋅(y - z)"
def test_issue_5524():
assert pretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
"""\
2 / ___ \\\n\
- (5 - y) + (x - 5)*\\-x - 2*\\/ 2 + 5/\
"""
assert upretty(-(-x + 5)*(-x - 2*sqrt(2) + 5) - (-y + 5)*(-y + 5)) == \
"""\
2 \n\
- (5 - y) + (x - 5)⋅(-x - 2⋅√2 + 5)\
"""
def test_pretty_ordering():
assert pretty(x**2 + x + 1, order='lex') == \
"""\
2 \n\
x + x + 1\
"""
assert pretty(x**2 + x + 1, order='rev-lex') == \
"""\
2\n\
1 + x + x \
"""
assert pretty(1 - x, order='lex') == '-x + 1'
assert pretty(1 - x, order='rev-lex') == '1 - x'
assert pretty(1 - 2*x, order='lex') == '-2*x + 1'
assert pretty(1 - 2*x, order='rev-lex') == '1 - 2*x'
f = 2*x**4 + y**2 - x**2 + y**3
assert pretty(f, order=None) == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='lex') == \
"""\
4 2 3 2\n\
2*x - x + y + y \
"""
assert pretty(f, order='rev-lex') == \
"""\
2 3 2 4\n\
y + y - x + 2*x \
"""
expr = x - x**3/6 + x**5/120 + O(x**6)
ascii_str = \
"""\
3 5 \n\
x x / 6\\\n\
x - -- + --- + O\\x /\n\
6 120 \
"""
ucode_str = \
"""\
3 5 \n\
x x ⎛ 6⎞\n\
x - ── + ─── + O⎝x ⎠\n\
6 120 \
"""
assert pretty(expr, order=None) == ascii_str
assert upretty(expr, order=None) == ucode_str
assert pretty(expr, order='lex') == ascii_str
assert upretty(expr, order='lex') == ucode_str
assert pretty(expr, order='rev-lex') == ascii_str
assert upretty(expr, order='rev-lex') == ucode_str
def test_EulerGamma():
assert pretty(EulerGamma) == str(EulerGamma) == "EulerGamma"
assert upretty(EulerGamma) == "γ"
def test_GoldenRatio():
assert pretty(GoldenRatio) == str(GoldenRatio) == "GoldenRatio"
assert upretty(GoldenRatio) == "φ"
def test_Catalan():
assert pretty(Catalan) == upretty(Catalan) == "G"
def test_pretty_relational():
expr = Eq(x, y)
ascii_str = \
"""\
x = y\
"""
ucode_str = \
"""\
x = y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lt(x, y)
ascii_str = \
"""\
x < y\
"""
ucode_str = \
"""\
x < y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Gt(x, y)
ascii_str = \
"""\
x > y\
"""
ucode_str = \
"""\
x > y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Le(x, y)
ascii_str = \
"""\
x <= y\
"""
ucode_str = \
"""\
x ≤ y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ge(x, y)
ascii_str = \
"""\
x >= y\
"""
ucode_str = \
"""\
x ≥ y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Ne(x/(y + 1), y**2)
ascii_str_1 = \
"""\
x 2\n\
----- != y \n\
1 + y \
"""
ascii_str_2 = \
"""\
x 2\n\
----- != y \n\
y + 1 \
"""
ucode_str_1 = \
"""\
x 2\n\
───── ≠ y \n\
1 + y \
"""
ucode_str_2 = \
"""\
x 2\n\
───── ≠ y \n\
y + 1 \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
def test_Assignment():
expr = Assignment(x, y)
ascii_str = \
"""\
x := y\
"""
ucode_str = \
"""\
x := y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_AugmentedAssignment():
expr = AddAugmentedAssignment(x, y)
ascii_str = \
"""\
x += y\
"""
ucode_str = \
"""\
x += y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = SubAugmentedAssignment(x, y)
ascii_str = \
"""\
x -= y\
"""
ucode_str = \
"""\
x -= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = MulAugmentedAssignment(x, y)
ascii_str = \
"""\
x *= y\
"""
ucode_str = \
"""\
x *= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = DivAugmentedAssignment(x, y)
ascii_str = \
"""\
x /= y\
"""
ucode_str = \
"""\
x /= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ModAugmentedAssignment(x, y)
ascii_str = \
"""\
x %= y\
"""
ucode_str = \
"""\
x %= y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_rational():
expr = y*x**-2
ascii_str = \
"""\
y \n\
--\n\
2\n\
x \
"""
ucode_str = \
"""\
y \n\
──\n\
2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = y**Rational(3, 2) * x**Rational(-5, 2)
ascii_str = \
"""\
3/2\n\
y \n\
----\n\
5/2\n\
x \
"""
ucode_str = \
"""\
3/2\n\
y \n\
────\n\
5/2\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**3/tan(x)**2
ascii_str = \
"""\
3 \n\
sin (x)\n\
-------\n\
2 \n\
tan (x)\
"""
ucode_str = \
"""\
3 \n\
sin (x)\n\
───────\n\
2 \n\
tan (x)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
@_both_exp_pow
def test_pretty_functions():
"""Tests for Abs, conjugate, exp, function braces, and factorial."""
expr = (2*x + exp(x))
ascii_str_1 = \
"""\
x\n\
2*x + e \
"""
ascii_str_2 = \
"""\
x \n\
e + 2*x\
"""
ucode_str_1 = \
"""\
x\n\
2⋅x + ℯ \
"""
ucode_str_2 = \
"""\
x \n\
ℯ + 2⋅x\
"""
ucode_str_3 = \
"""\
x \n\
ℯ + 2⋅x\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2, ucode_str_3]
expr = Abs(x)
ascii_str = \
"""\
|x|\
"""
ucode_str = \
"""\
│x│\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Abs(x/(x**2 + 1))
ascii_str_1 = \
"""\
| x |\n\
|------|\n\
| 2|\n\
|1 + x |\
"""
ascii_str_2 = \
"""\
| x |\n\
|------|\n\
| 2 |\n\
|x + 1|\
"""
ucode_str_1 = \
"""\
│ x │\n\
│──────│\n\
│ 2│\n\
│1 + x │\
"""
ucode_str_2 = \
"""\
│ x │\n\
│──────│\n\
│ 2 │\n\
│x + 1│\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Abs(1 / (y - Abs(x)))
ascii_str = \
"""\
1 \n\
---------\n\
|y - |x||\
"""
ucode_str = \
"""\
1 \n\
─────────\n\
│y - │x││\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial(n)
ascii_str = \
"""\
n!\
"""
ucode_str = \
"""\
n!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(2*n)
ascii_str = \
"""\
(2*n)!\
"""
ucode_str = \
"""\
(2⋅n)!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(factorial(factorial(n)))
ascii_str = \
"""\
((n!)!)!\
"""
ucode_str = \
"""\
((n!)!)!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial(n + 1)
ascii_str_1 = \
"""\
(1 + n)!\
"""
ascii_str_2 = \
"""\
(n + 1)!\
"""
ucode_str_1 = \
"""\
(1 + n)!\
"""
ucode_str_2 = \
"""\
(n + 1)!\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = subfactorial(n)
ascii_str = \
"""\
!n\
"""
ucode_str = \
"""\
!n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = subfactorial(2*n)
ascii_str = \
"""\
!(2*n)\
"""
ucode_str = \
"""\
!(2⋅n)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
n = Symbol('n', integer=True)
expr = factorial2(n)
ascii_str = \
"""\
n!!\
"""
ucode_str = \
"""\
n!!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(2*n)
ascii_str = \
"""\
(2*n)!!\
"""
ucode_str = \
"""\
(2⋅n)!!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(factorial2(factorial2(n)))
ascii_str = \
"""\
((n!!)!!)!!\
"""
ucode_str = \
"""\
((n!!)!!)!!\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = factorial2(n + 1)
ascii_str_1 = \
"""\
(1 + n)!!\
"""
ascii_str_2 = \
"""\
(n + 1)!!\
"""
ucode_str_1 = \
"""\
(1 + n)!!\
"""
ucode_str_2 = \
"""\
(n + 1)!!\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = 2*binomial(n, k)
ascii_str = \
"""\
/n\\\n\
2*| |\n\
\\k/\
"""
ucode_str = \
"""\
⎛n⎞\n\
2⋅⎜ ⎟\n\
⎝k⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(2*n, k)
ascii_str = \
"""\
/2*n\\\n\
2*| |\n\
\\ k /\
"""
ucode_str = \
"""\
⎛2⋅n⎞\n\
2⋅⎜ ⎟\n\
⎝ k ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*binomial(n**2, k)
ascii_str = \
"""\
/ 2\\\n\
|n |\n\
2*| |\n\
\\k /\
"""
ucode_str = \
"""\
⎛ 2⎞\n\
⎜n ⎟\n\
2⋅⎜ ⎟\n\
⎝k ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
"""\
C \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = catalan(n)
ascii_str = \
"""\
C \n\
n\
"""
ucode_str = \
"""\
C \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bell(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
"""\
B \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n)
ascii_str = \
"""\
B \n\
n\
"""
ucode_str = \
"""\
B \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = bernoulli(n, x)
ascii_str = \
"""\
B (x)\n\
n \
"""
ucode_str = \
"""\
B (x)\n\
n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = fibonacci(n)
ascii_str = \
"""\
F \n\
n\
"""
ucode_str = \
"""\
F \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = lucas(n)
ascii_str = \
"""\
L \n\
n\
"""
ucode_str = \
"""\
L \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = tribonacci(n)
ascii_str = \
"""\
T \n\
n\
"""
ucode_str = \
"""\
T \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = stieltjes(n)
ascii_str = \
"""\
stieltjes \n\
n\
"""
ucode_str = \
"""\
γ \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = stieltjes(n, x)
ascii_str = \
"""\
stieltjes (x)\n\
n \
"""
ucode_str = \
"""\
γ (x)\n\
n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieuc(x, y, z)
ascii_str = 'C(x, y, z)'
ucode_str = 'C(x, y, z)'
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieus(x, y, z)
ascii_str = 'S(x, y, z)'
ucode_str = 'S(x, y, z)'
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieucprime(x, y, z)
ascii_str = "C'(x, y, z)"
ucode_str = "C'(x, y, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = mathieusprime(x, y, z)
ascii_str = "S'(x, y, z)"
ucode_str = "S'(x, y, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(x)
ascii_str = \
"""\
_\n\
x\
"""
ucode_str = \
"""\
_\n\
x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
f = Function('f')
expr = conjugate(f(x + 1))
ascii_str_1 = \
"""\
________\n\
f(1 + x)\
"""
ascii_str_2 = \
"""\
________\n\
f(x + 1)\
"""
ucode_str_1 = \
"""\
________\n\
f(1 + x)\
"""
ucode_str_2 = \
"""\
________\n\
f(x + 1)\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x)
ascii_str = \
"""\
f(x)\
"""
ucode_str = \
"""\
f(x)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x, y)
ascii_str = \
"""\
f(x, y)\
"""
ucode_str = \
"""\
f(x, y)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
"""
ucode_str_2 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x**x**x**x**x**x)
ascii_str = \
"""\
/ / / / / x\\\\\\\\\\
| | | | \\x /||||
| | | \\x /|||
| | \\x /||
| \\x /|
f\\x /\
"""
ucode_str = \
"""\
⎛ ⎛ ⎛ ⎛ ⎛ x⎞⎞⎞⎞⎞
⎜ ⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟⎟
⎜ ⎜ ⎜ ⎝x ⎠⎟⎟⎟
⎜ ⎜ ⎝x ⎠⎟⎟
⎜ ⎝x ⎠⎟
f⎝x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sin(x)**2
ascii_str = \
"""\
2 \n\
sin (x)\
"""
ucode_str = \
"""\
2 \n\
sin (x)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(a + b*I)
ascii_str = \
"""\
_ _\n\
a - I*b\
"""
ucode_str = \
"""\
_ _\n\
a - ⅈ⋅b\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate(exp(a + b*I))
ascii_str = \
"""\
_ _\n\
a - I*b\n\
e \
"""
ucode_str = \
"""\
_ _\n\
a - ⅈ⋅b\n\
ℯ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = conjugate( f(1 + conjugate(f(x))) )
ascii_str_1 = \
"""\
___________\n\
/ ____\\\n\
f\\1 + f(x)/\
"""
ascii_str_2 = \
"""\
___________\n\
/____ \\\n\
f\\f(x) + 1/\
"""
ucode_str_1 = \
"""\
___________\n\
⎛ ____⎞\n\
f⎝1 + f(x)⎠\
"""
ucode_str_2 = \
"""\
___________\n\
⎛____ ⎞\n\
f⎝f(x) + 1⎠\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = f(x/(y + 1), y)
ascii_str_1 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\1 + y /\
"""
ascii_str_2 = \
"""\
/ x \\\n\
f|-----, y|\n\
\\y + 1 /\
"""
ucode_str_1 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝1 + y ⎠\
"""
ucode_str_2 = \
"""\
⎛ x ⎞\n\
f⎜─────, y⎟\n\
⎝y + 1 ⎠\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = floor(1 / (y - floor(x)))
ascii_str = \
"""\
/ 1 \\\n\
floor|------------|\n\
\\y - floor(x)/\
"""
ucode_str = \
"""\
⎢ 1 ⎥\n\
⎢───────⎥\n\
⎣y - ⌊x⌋⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = ceiling(1 / (y - ceiling(x)))
ascii_str = \
"""\
/ 1 \\\n\
ceiling|--------------|\n\
\\y - ceiling(x)/\
"""
ucode_str = \
"""\
⎡ 1 ⎤\n\
⎢───────⎥\n\
⎢y - ⌈x⌉⎥\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n)
ascii_str = \
"""\
E \n\
n\
"""
ucode_str = \
"""\
E \n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(1/(1 + 1/(1 + 1/n)))
ascii_str = \
"""\
E \n\
1 \n\
---------\n\
1 \n\
1 + -----\n\
1\n\
1 + -\n\
n\
"""
ucode_str = \
"""\
E \n\
1 \n\
─────────\n\
1 \n\
1 + ─────\n\
1\n\
1 + ─\n\
n\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x)
ascii_str = \
"""\
E (x)\n\
n \
"""
ucode_str = \
"""\
E (x)\n\
n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = euler(n, x/2)
ascii_str = \
"""\
/x\\\n\
E |-|\n\
n\\2/\
"""
ucode_str = \
"""\
⎛x⎞\n\
E ⎜─⎟\n\
n⎝2⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt():
expr = sqrt(2)
ascii_str = \
"""\
___\n\
\\/ 2 \
"""
ucode_str = \
"√2"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 3)
ascii_str = \
"""\
3 ___\n\
\\/ 2 \
"""
ucode_str = \
"""\
3 ___\n\
╲╱ 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**Rational(1, 1000)
ascii_str = \
"""\
1000___\n\
\\/ 2 \
"""
ucode_str = \
"""\
1000___\n\
╲╱ 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(x**2 + 1)
ascii_str = \
"""\
________\n\
/ 2 \n\
\\/ x + 1 \
"""
ucode_str = \
"""\
________\n\
╱ 2 \n\
╲╱ x + 1 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1 + sqrt(5))**Rational(1, 3)
ascii_str = \
"""\
___________\n\
3 / ___ \n\
\\/ 1 + \\/ 5 \
"""
ucode_str = \
"""\
3 ________\n\
╲╱ 1 + √5 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2**(1/x)
ascii_str = \
"""\
x ___\n\
\\/ 2 \
"""
ucode_str = \
"""\
x ___\n\
╲╱ 2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = sqrt(2 + pi)
ascii_str = \
"""\
________\n\
\\/ 2 + pi \
"""
ucode_str = \
"""\
_______\n\
╲╱ 2 + π \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (2 + (
1 + x**2)/(2 + x))**Rational(1, 4) + (1 + x**Rational(1, 1000))/sqrt(3 + x**2)
ascii_str = \
"""\
____________ \n\
/ 2 1000___ \n\
/ x + 1 \\/ x + 1\n\
4 / 2 + ------ + -----------\n\
\\/ x + 2 ________\n\
/ 2 \n\
\\/ x + 3 \
"""
ucode_str = \
"""\
____________ \n\
╱ 2 1000___ \n\
╱ x + 1 ╲╱ x + 1\n\
4 ╱ 2 + ────── + ───────────\n\
╲╱ x + 2 ________\n\
╱ 2 \n\
╲╱ x + 3 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_sqrt_char_knob():
# See PR #9234.
expr = sqrt(2)
ucode_str1 = \
"""\
___\n\
╲╱ 2 \
"""
ucode_str2 = \
"√2"
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=False) == ucode_str1
assert xpretty(expr, use_unicode=True,
use_unicode_sqrt_char=True) == ucode_str2
def test_pretty_sqrt_longsymbol_no_sqrt_char():
# Do not use unicode sqrt char for long symbols (see PR #9234).
expr = sqrt(Symbol('C1'))
ucode_str = \
"""\
____\n\
╲╱ C₁ \
"""
assert upretty(expr) == ucode_str
def test_pretty_KroneckerDelta():
x, y = symbols("x, y")
expr = KroneckerDelta(x, y)
ascii_str = \
"""\
d \n\
x,y\
"""
ucode_str = \
"""\
δ \n\
x,y\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_product():
n, m, k, l = symbols('n m k l')
f = symbols('f', cls=Function)
expr = Product(f((n/3)**2), (n, k**2, l))
unicode_str = \
"""\
l \n\
─┬──────┬─ \n\
│ │ ⎛ 2⎞\n\
│ │ ⎜n ⎟\n\
│ │ f⎜──⎟\n\
│ │ ⎝9 ⎠\n\
│ │ \n\
2 \n\
n = k """
ascii_str = \
"""\
l \n\
__________ \n\
| | / 2\\\n\
| | |n |\n\
| | f|--|\n\
| | \\9 /\n\
| | \n\
2 \n\
n = k """
expr = Product(f((n/3)**2), (n, k**2, l), (l, 1, m))
unicode_str = \
"""\
m l \n\
─┬──────┬─ ─┬──────┬─ \n\
│ │ │ │ ⎛ 2⎞\n\
│ │ │ │ ⎜n ⎟\n\
│ │ │ │ f⎜──⎟\n\
│ │ │ │ ⎝9 ⎠\n\
│ │ │ │ \n\
l = 1 2 \n\
n = k """
ascii_str = \
"""\
m l \n\
__________ __________ \n\
| | | | / 2\\\n\
| | | | |n |\n\
| | | | f|--|\n\
| | | | \\9 /\n\
| | | | \n\
l = 1 2 \n\
n = k """
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_pretty_Lambda():
# S.IdentityFunction is a special case
expr = Lambda(y, y)
assert pretty(expr) == "x -> x"
assert upretty(expr) == "x ↦ x"
expr = Lambda(x, x+1)
assert pretty(expr) == "x -> x + 1"
assert upretty(expr) == "x ↦ x + 1"
expr = Lambda(x, x**2)
ascii_str = \
"""\
2\n\
x -> x \
"""
ucode_str = \
"""\
2\n\
x ↦ x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(x, x**2)**2
ascii_str = \
"""\
2
/ 2\\ \n\
\\x -> x / \
"""
ucode_str = \
"""\
2
⎛ 2⎞ \n\
⎝x ↦ x ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x)
ascii_str = "(x, y) -> x"
ucode_str = "(x, y) ↦ x"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda((x, y), x**2)
ascii_str = \
"""\
2\n\
(x, y) -> x \
"""
ucode_str = \
"""\
2\n\
(x, y) ↦ x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Lambda(((x, y),), x**2)
ascii_str = \
"""\
2\n\
((x, y),) -> x \
"""
ucode_str = \
"""\
2\n\
((x, y),) ↦ x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_TransferFunction():
tf1 = TransferFunction(s - 1, s + 1, s)
assert upretty(tf1) == "s - 1\n─────\ns + 1"
tf2 = TransferFunction(2*s + 1, 3 - p, s)
assert upretty(tf2) == "2⋅s + 1\n───────\n 3 - p "
tf3 = TransferFunction(p, p + 1, p)
assert upretty(tf3) == " p \n─────\np + 1"
def test_pretty_Series():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(x**2 + y, y - x, y)
tf4 = TransferFunction(2, 3, y)
tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]])
tfm2 = TransferFunctionMatrix([[tf3], [-tf4]])
tfm3 = TransferFunctionMatrix([[tf1, -tf2, -tf3], [tf3, -tf4, tf2]])
tfm4 = TransferFunctionMatrix([[tf1, tf2], [tf3, -tf4], [-tf2, -tf1]])
tfm5 = TransferFunctionMatrix([[-tf2, -tf1], [tf4, -tf3], [tf1, tf2]])
expected1 = \
"""\
⎛ 2 ⎞\n\
⎛ x + y ⎞ ⎜x + y⎟\n\
⎜───────⎟⋅⎜──────⎟\n\
⎝x - 2⋅y⎠ ⎝-x + y⎠\
"""
expected2 = \
"""\
⎛-x + y⎞ ⎛ -x - y⎞\n\
⎜──────⎟⋅⎜───────⎟\n\
⎝x + y ⎠ ⎝x - 2⋅y⎠\
"""
expected3 = \
"""\
⎛ 2 ⎞ \n\
⎜x + y⎟ ⎛ x + y ⎞ ⎛ -x - y x - y⎞\n\
⎜──────⎟⋅⎜───────⎟⋅⎜─────── + ─────⎟\n\
⎝-x + y⎠ ⎝x - 2⋅y⎠ ⎝x - 2⋅y x + y⎠\
"""
expected4 = \
"""\
⎛ 2 ⎞\n\
⎛ x + y x - y⎞ ⎜x - y x + y⎟\n\
⎜─────── + ─────⎟⋅⎜───── + ──────⎟\n\
⎝x - 2⋅y x + y⎠ ⎝x + y -x + y⎠\
"""
expected5 = \
"""\
⎡ x + y x - y⎤ ⎡ 2 ⎤ \n\
⎢─────── ─────⎥ ⎢x + y⎥ \n\
⎢x - 2⋅y x + y⎥ ⎢──────⎥ \n\
⎢ ⎥ ⎢-x + y⎥ \n\
⎢ 2 ⎥ ⋅⎢ ⎥ \n\
⎢x + y 2 ⎥ ⎢ -2 ⎥ \n\
⎢────── ─ ⎥ ⎢ ─── ⎥ \n\
⎣-x + y 3 ⎦τ ⎣ 3 ⎦τ\
"""
expected6 = \
"""\
⎛⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎞\n\
⎜⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎟\n\
⎡ x + y x - y⎤ ⎡ 2 ⎤ ⎜⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎟\n\
⎢─────── ─────⎥ ⎢ x + y -x + y - x - y⎥ ⎜⎢ ⎥ ⎢ ⎥ ⎟\n\
⎢x - 2⋅y x + y⎥ ⎢─────── ────── ────────⎥ ⎜⎢ 2 ⎥ ⎢ 2 ⎥ ⎟\n\
⎢ ⎥ ⎢x - 2⋅y x + y -x + y ⎥ ⎜⎢x + y -2 ⎥ ⎢ -2 x + y ⎥ ⎟\n\
⎢ 2 ⎥ ⋅⎢ ⎥ ⋅⎜⎢────── ─── ⎥ + ⎢ ─── ────── ⎥ ⎟\n\
⎢x + y 2 ⎥ ⎢ 2 ⎥ ⎜⎢-x + y 3 ⎥ ⎢ 3 -x + y ⎥ ⎟\n\
⎢────── ─ ⎥ ⎢x + y -2 x - y ⎥ ⎜⎢ ⎥ ⎢ ⎥ ⎟\n\
⎣-x + y 3 ⎦τ ⎢────── ─── ───── ⎥ ⎜⎢-x + y -x - y⎥ ⎢ -x - y -x + y ⎥ ⎟\n\
⎣-x + y 3 x + y ⎦τ ⎜⎢────── ───────⎥ ⎢─────── ────── ⎥ ⎟\n\
⎝⎣x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ⎠\
"""
assert upretty(Series(tf1, tf3)) == expected1
assert upretty(Series(-tf2, -tf1)) == expected2
assert upretty(Series(tf3, tf1, Parallel(-tf1, tf2))) == expected3
assert upretty(Series(Parallel(tf1, tf2), Parallel(tf2, tf3))) == expected4
assert upretty(MIMOSeries(tfm2, tfm1)) == expected5
assert upretty(MIMOSeries(MIMOParallel(tfm4, -tfm5), tfm3, tfm1)) == expected6
def test_pretty_Parallel():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(x**2 + y, y - x, y)
tf4 = TransferFunction(y**2 - x, x**3 + x, y)
tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf3, -tf4], [-tf2, -tf1]])
tfm2 = TransferFunctionMatrix([[-tf2, -tf1], [tf4, -tf3], [tf1, tf2]])
tfm3 = TransferFunctionMatrix([[-tf1, tf2], [-tf3, tf4], [tf2, tf1]])
tfm4 = TransferFunctionMatrix([[-tf1, -tf2], [-tf3, -tf4]])
expected1 = \
"""\
x + y x - y\n\
─────── + ─────\n\
x - 2⋅y x + y\
"""
expected2 = \
"""\
-x + y -x - y\n\
────── + ───────\n\
x + y x - 2⋅y\
"""
expected3 = \
"""\
2 \n\
x + y x + y ⎛ -x - y⎞ ⎛x - y⎞\n\
────── + ─────── + ⎜───────⎟⋅⎜─────⎟\n\
-x + y x - 2⋅y ⎝x - 2⋅y⎠ ⎝x + y⎠\
"""
expected4 = \
"""\
⎛ 2 ⎞\n\
⎛ x + y ⎞ ⎛x - y⎞ ⎛x - y⎞ ⎜x + y⎟\n\
⎜───────⎟⋅⎜─────⎟ + ⎜─────⎟⋅⎜──────⎟\n\
⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x + y⎠ ⎝-x + y⎠\
"""
expected5 = \
"""\
⎡ x + y -x + y ⎤ ⎡ x - y x + y ⎤ ⎡ x + y x - y ⎤ \n\
⎢─────── ────── ⎥ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ \n\
⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎢x - 2⋅y x + y ⎥ \n\
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ \n\
⎢ 2 2 ⎥ ⎢ 2 2 ⎥ ⎢ 2 2 ⎥ \n\
⎢x + y x - y ⎥ ⎢x - y x + y ⎥ ⎢x + y x - y ⎥ \n\
⎢────── ────── ⎥ + ⎢────── ────── ⎥ + ⎢────── ────── ⎥ \n\
⎢-x + y 3 ⎥ ⎢ 3 -x + y ⎥ ⎢-x + y 3 ⎥ \n\
⎢ x + x ⎥ ⎢x + x ⎥ ⎢ x + x ⎥ \n\
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ \n\
⎢-x + y -x - y⎥ ⎢ -x - y -x + y ⎥ ⎢-x + y -x - y⎥ \n\
⎢────── ───────⎥ ⎢─────── ────── ⎥ ⎢────── ───────⎥ \n\
⎣x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ ⎣x + y x - 2⋅y⎦τ\
"""
expected6 = \
"""\
⎡ x - y x + y ⎤ ⎡-x + y -x - y ⎤ \n\
⎢ ───── ───────⎥ ⎢────── ─────── ⎥ \n\
⎢ x + y x - 2⋅y⎥ ⎡ -x - y -x + y⎤ ⎢x + y x - 2⋅y ⎥ \n\
⎢ ⎥ ⎢─────── ──────⎥ ⎢ ⎥ \n\
⎢ 2 2 ⎥ ⎢x - 2⋅y x + y ⎥ ⎢ 2 2 ⎥ \n\
⎢x - y x + y ⎥ ⎢ ⎥ ⎢-x + y - x - y⎥ \n\
⎢────── ────── ⎥ ⋅⎢ 2 2⎥ + ⎢─────── ────────⎥ \n\
⎢ 3 -x + y ⎥ ⎢- x - y x - y ⎥ ⎢ 3 -x + y ⎥ \n\
⎢x + x ⎥ ⎢──────── ──────⎥ ⎢ x + x ⎥ \n\
⎢ ⎥ ⎢ -x + y 3 ⎥ ⎢ ⎥ \n\
⎢ -x - y -x + y ⎥ ⎣ x + x⎦τ ⎢ x + y x - y ⎥ \n\
⎢─────── ────── ⎥ ⎢─────── ───── ⎥ \n\
⎣x - 2⋅y x + y ⎦τ ⎣x - 2⋅y x + y ⎦τ\
"""
assert upretty(Parallel(tf1, tf2)) == expected1
assert upretty(Parallel(-tf2, -tf1)) == expected2
assert upretty(Parallel(tf3, tf1, Series(-tf1, tf2))) == expected3
assert upretty(Parallel(Series(tf1, tf2), Series(tf2, tf3))) == expected4
assert upretty(MIMOParallel(-tfm3, -tfm2, tfm1)) == expected5
assert upretty(MIMOParallel(MIMOSeries(tfm4, -tfm2), tfm2)) == expected6
def test_pretty_Feedback():
tf = TransferFunction(1, 1, y)
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(y**2 - 2*y + 1, y + 5, y)
tf4 = TransferFunction(x - 2*y**3, x + y, x)
tf5 = TransferFunction(1 - x, x - y, y)
tf6 = TransferFunction(2, 2, x)
expected1 = \
"""\
⎛1⎞ \n\
⎜─⎟ \n\
⎝1⎠ \n\
─────────────\n\
1 ⎛ x + y ⎞\n\
─ + ⎜───────⎟\n\
1 ⎝x - 2⋅y⎠\
"""
expected2 = \
"""\
⎛1⎞ \n\
⎜─⎟ \n\
⎝1⎠ \n\
────────────────────────────────────\n\
⎛ 2 ⎞\n\
1 ⎛x - y⎞ ⎛ x + y ⎞ ⎜y - 2⋅y + 1⎟\n\
─ + ⎜─────⎟⋅⎜───────⎟⋅⎜────────────⎟\n\
1 ⎝x + y⎠ ⎝x - 2⋅y⎠ ⎝ y + 5 ⎠\
"""
expected3 = \
"""\
⎛ x + y ⎞ \n\
⎜───────⎟ \n\
⎝x - 2⋅y⎠ \n\
────────────────────────────────────────────\n\
⎛ 2 ⎞ \n\
1 ⎛ x + y ⎞ ⎛x - y⎞ ⎜y - 2⋅y + 1⎟ ⎛1 - x⎞\n\
─ + ⎜───────⎟⋅⎜─────⎟⋅⎜────────────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝ y + 5 ⎠ ⎝x - y⎠\
"""
expected4 = \
"""\
⎛ x + y ⎞ ⎛x - y⎞ \n\
⎜───────⎟⋅⎜─────⎟ \n\
⎝x - 2⋅y⎠ ⎝x + y⎠ \n\
─────────────────────\n\
1 ⎛ x + y ⎞ ⎛x - y⎞\n\
─ + ⎜───────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠\
"""
expected5 = \
"""\
⎛ x + y ⎞ ⎛x - y⎞ \n\
⎜───────⎟⋅⎜─────⎟ \n\
⎝x - 2⋅y⎠ ⎝x + y⎠ \n\
─────────────────────────────\n\
1 ⎛ x + y ⎞ ⎛x - y⎞ ⎛1 - x⎞\n\
─ + ⎜───────⎟⋅⎜─────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x - y⎠\
"""
expected6 = \
"""\
⎛ 2 ⎞ \n\
⎜y - 2⋅y + 1⎟ ⎛1 - x⎞ \n\
⎜────────────⎟⋅⎜─────⎟ \n\
⎝ y + 5 ⎠ ⎝x - y⎠ \n\
────────────────────────────────────────────\n\
⎛ 2 ⎞ \n\
1 ⎜y - 2⋅y + 1⎟ ⎛1 - x⎞ ⎛x - y⎞ ⎛ x + y ⎞\n\
─ + ⎜────────────⎟⋅⎜─────⎟⋅⎜─────⎟⋅⎜───────⎟\n\
1 ⎝ y + 5 ⎠ ⎝x - y⎠ ⎝x + y⎠ ⎝x - 2⋅y⎠\
"""
expected7 = \
"""\
⎛ 3⎞ \n\
⎜x - 2⋅y ⎟ \n\
⎜────────⎟ \n\
⎝ x + y ⎠ \n\
──────────────────\n\
⎛ 3⎞ \n\
1 ⎜x - 2⋅y ⎟ ⎛2⎞\n\
─ + ⎜────────⎟⋅⎜─⎟\n\
1 ⎝ x + y ⎠ ⎝2⎠\
"""
expected8 = \
"""\
⎛1 - x⎞ \n\
⎜─────⎟ \n\
⎝x - y⎠ \n\
───────────\n\
1 ⎛1 - x⎞\n\
─ + ⎜─────⎟\n\
1 ⎝x - y⎠\
"""
expected9 = \
"""\
⎛ x + y ⎞ ⎛x - y⎞ \n\
⎜───────⎟⋅⎜─────⎟ \n\
⎝x - 2⋅y⎠ ⎝x + y⎠ \n\
─────────────────────────────\n\
1 ⎛ x + y ⎞ ⎛x - y⎞ ⎛1 - x⎞\n\
─ - ⎜───────⎟⋅⎜─────⎟⋅⎜─────⎟\n\
1 ⎝x - 2⋅y⎠ ⎝x + y⎠ ⎝x - y⎠\
"""
expected10 = \
"""\
⎛1 - x⎞ \n\
⎜─────⎟ \n\
⎝x - y⎠ \n\
───────────\n\
1 ⎛1 - x⎞\n\
─ - ⎜─────⎟\n\
1 ⎝x - y⎠\
"""
assert upretty(Feedback(tf, tf1)) == expected1
assert upretty(Feedback(tf, tf2*tf1*tf3)) == expected2
assert upretty(Feedback(tf1, tf2*tf3*tf5)) == expected3
assert upretty(Feedback(tf1*tf2, tf)) == expected4
assert upretty(Feedback(tf1*tf2, tf5)) == expected5
assert upretty(Feedback(tf3*tf5, tf2*tf1)) == expected6
assert upretty(Feedback(tf4, tf6)) == expected7
assert upretty(Feedback(tf5, tf)) == expected8
assert upretty(Feedback(tf1*tf2, tf5, 1)) == expected9
assert upretty(Feedback(tf5, tf, 1)) == expected10
def test_pretty_MIMOFeedback():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]])
tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]])
tfm_3 = TransferFunctionMatrix([[tf1, tf1], [tf2, tf2]])
expected1 = \
"""\
⎛ ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎞-1 ⎡ x + y x - y ⎤ \n\
⎜ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎟ ⎢─────── ───── ⎥ \n\
⎜ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎟ ⎢x - 2⋅y x + y ⎥ \n\
⎜I - ⎢ ⎥ ⋅⎢ ⎥ ⎟ ⋅ ⎢ ⎥ \n\
⎜ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ ⎟ ⎢ x - y x + y ⎥ \n\
⎜ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ ⎟ ⎢ ───── ───────⎥ \n\
⎝ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ⎠ ⎣ x + y x - 2⋅y⎦τ\
"""
expected2 = \
"""\
⎛ ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ ⎡ x + y x + y ⎤ ⎞-1 ⎡ x + y x - y ⎤ ⎡ x - y x + y ⎤ \n\
⎜ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ ⎢─────── ───────⎥ ⎟ ⎢─────── ───── ⎥ ⎢ ───── ───────⎥ \n\
⎜ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ ⎢x - 2⋅y x - 2⋅y⎥ ⎟ ⎢x - 2⋅y x + y ⎥ ⎢ x + y x - 2⋅y⎥ \n\
⎜I + ⎢ ⎥ ⋅⎢ ⎥ ⋅⎢ ⎥ ⎟ ⋅ ⎢ ⎥ ⋅⎢ ⎥ \n\
⎜ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ ⎢ x - y x - y ⎥ ⎟ ⎢ x - y x + y ⎥ ⎢ x + y x - y ⎥ \n\
⎜ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ ⎢ ───── ───── ⎥ ⎟ ⎢ ───── ───────⎥ ⎢─────── ───── ⎥ \n\
⎝ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ ⎣ x + y x + y ⎦τ⎠ ⎣ x + y x - 2⋅y⎦τ ⎣x - 2⋅y x + y ⎦τ\
"""
assert upretty(MIMOFeedback(tfm_1, tfm_2, 1)) == \
expected1 # Positive MIMOFeedback
assert upretty(MIMOFeedback(tfm_1*tfm_2, tfm_3)) == \
expected2 # Negative MIMOFeedback (Default)
def test_pretty_TransferFunctionMatrix():
tf1 = TransferFunction(x + y, x - 2*y, y)
tf2 = TransferFunction(x - y, x + y, y)
tf3 = TransferFunction(y**2 - 2*y + 1, y + 5, y)
tf4 = TransferFunction(y, x**2 + x + 1, y)
tf5 = TransferFunction(1 - x, x - y, y)
tf6 = TransferFunction(2, 2, y)
expected1 = \
"""\
⎡ x + y ⎤ \n\
⎢───────⎥ \n\
⎢x - 2⋅y⎥ \n\
⎢ ⎥ \n\
⎢ x - y ⎥ \n\
⎢ ───── ⎥ \n\
⎣ x + y ⎦τ\
"""
expected2 = \
"""\
⎡ x + y ⎤ \n\
⎢ ─────── ⎥ \n\
⎢ x - 2⋅y ⎥ \n\
⎢ ⎥ \n\
⎢ x - y ⎥ \n\
⎢ ───── ⎥ \n\
⎢ x + y ⎥ \n\
⎢ ⎥ \n\
⎢ 2 ⎥ \n\
⎢- y + 2⋅y - 1⎥ \n\
⎢──────────────⎥ \n\
⎣ y + 5 ⎦τ\
"""
expected3 = \
"""\
⎡ x + y x - y ⎤ \n\
⎢ ─────── ───── ⎥ \n\
⎢ x - 2⋅y x + y ⎥ \n\
⎢ ⎥ \n\
⎢ 2 ⎥ \n\
⎢y - 2⋅y + 1 y ⎥ \n\
⎢──────────── ──────────⎥ \n\
⎢ y + 5 2 ⎥ \n\
⎢ x + x + 1⎥ \n\
⎢ ⎥ \n\
⎢ 1 - x 2 ⎥ \n\
⎢ ───── ─ ⎥ \n\
⎣ x - y 2 ⎦τ\
"""
expected4 = \
"""\
⎡ x - y x + y y ⎤ \n\
⎢ ───── ─────── ──────────⎥ \n\
⎢ x + y x - 2⋅y 2 ⎥ \n\
⎢ x + x + 1⎥ \n\
⎢ ⎥ \n\
⎢ 2 ⎥ \n\
⎢- y + 2⋅y - 1 x - 1 -2 ⎥ \n\
⎢────────────── ───── ─── ⎥ \n\
⎣ y + 5 x - y 2 ⎦τ\
"""
expected5 = \
"""\
⎡ x + y x - y x + y y ⎤ \n\
⎢───────⋅───── ─────── ──────────⎥ \n\
⎢x - 2⋅y x + y x - 2⋅y 2 ⎥ \n\
⎢ x + x + 1⎥ \n\
⎢ ⎥ \n\
⎢ 1 - x 2 x + y -2 ⎥ \n\
⎢ ───── + ─ ─────── ─── ⎥ \n\
⎣ x - y 2 x - 2⋅y 2 ⎦τ\
"""
assert upretty(TransferFunctionMatrix([[tf1], [tf2]])) == expected1
assert upretty(TransferFunctionMatrix([[tf1], [tf2], [-tf3]])) == expected2
assert upretty(TransferFunctionMatrix([[tf1, tf2], [tf3, tf4], [tf5, tf6]])) == expected3
assert upretty(TransferFunctionMatrix([[tf2, tf1, tf4], [-tf3, -tf5, -tf6]])) == expected4
assert upretty(TransferFunctionMatrix([[Series(tf2, tf1), tf1, tf4], [Parallel(tf6, tf5), tf1, -tf6]])) == \
expected5
def test_pretty_order():
expr = O(1)
ascii_str = \
"""\
O(1)\
"""
ucode_str = \
"""\
O(1)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x)
ascii_str = \
"""\
/1\\\n\
O|-|\n\
\\x/\
"""
ucode_str = \
"""\
⎛1⎞\n\
O⎜─⎟\n\
⎝x⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2)
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (0, 0)/\
"""
ucode_str = \
"""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (0, 0)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1, (x, oo))
ascii_str = \
"""\
O(1; x -> oo)\
"""
ucode_str = \
"""\
O(1; x → ∞)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(1/x, (x, oo))
ascii_str = \
"""\
/1 \\\n\
O|-; x -> oo|\n\
\\x /\
"""
ucode_str = \
"""\
⎛1 ⎞\n\
O⎜─; x → ∞⎟\n\
⎝x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = O(x**2 + y**2, (x, oo), (y, oo))
ascii_str = \
"""\
/ 2 2 \\\n\
O\\x + y ; (x, y) -> (oo, oo)/\
"""
ucode_str = \
"""\
⎛ 2 2 ⎞\n\
O⎝x + y ; (x, y) → (∞, ∞)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_derivatives():
# Simple
expr = Derivative(log(x), x, evaluate=False)
ascii_str = \
"""\
d \n\
--(log(x))\n\
dx \
"""
ucode_str = \
"""\
d \n\
──(log(x))\n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(log(x), x, evaluate=False) + x
ascii_str_1 = \
"""\
d \n\
x + --(log(x))\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(log(x)) + x\n\
dx \
"""
ucode_str_1 = \
"""\
d \n\
x + ──(log(x))\n\
dx \
"""
ucode_str_2 = \
"""\
d \n\
──(log(x)) + x\n\
dx \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
# basic partial derivatives
expr = Derivative(log(x + y) + x, x)
ascii_str_1 = \
"""\
d \n\
--(log(x + y) + x)\n\
dx \
"""
ascii_str_2 = \
"""\
d \n\
--(x + log(x + y))\n\
dx \
"""
ucode_str_1 = \
"""\
∂ \n\
──(log(x + y) + x)\n\
∂x \
"""
ucode_str_2 = \
"""\
∂ \n\
──(x + log(x + y))\n\
∂x \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2], upretty(expr)
# Multiple symbols
expr = Derivative(log(x) + x**2, x, y)
ascii_str_1 = \
"""\
2 \n\
d / 2\\\n\
-----\\log(x) + x /\n\
dy dx \
"""
ascii_str_2 = \
"""\
2 \n\
d / 2 \\\n\
-----\\x + log(x)/\n\
dy dx \
"""
ucode_str_1 = \
"""\
2 \n\
d ⎛ 2⎞\n\
─────⎝log(x) + x ⎠\n\
dy dx \
"""
ucode_str_2 = \
"""\
2 \n\
d ⎛ 2 ⎞\n\
─────⎝x + log(x)⎠\n\
dy dx \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, y, x) + x**2
ascii_str_1 = \
"""\
2 \n\
d 2\n\
-----(2*x*y) + x \n\
dx dy \
"""
ascii_str_2 = \
"""\
2 \n\
2 d \n\
x + -----(2*x*y)\n\
dx dy \
"""
ucode_str_1 = \
"""\
2 \n\
∂ 2\n\
─────(2⋅x⋅y) + x \n\
∂x ∂y \
"""
ucode_str_2 = \
"""\
2 \n\
2 ∂ \n\
x + ─────(2⋅x⋅y)\n\
∂x ∂y \
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Derivative(2*x*y, x, x)
ascii_str = \
"""\
2 \n\
d \n\
---(2*x*y)\n\
2 \n\
dx \
"""
ucode_str = \
"""\
2 \n\
∂ \n\
───(2⋅x⋅y)\n\
2 \n\
∂x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, 17)
ascii_str = \
"""\
17 \n\
d \n\
----(2*x*y)\n\
17 \n\
dx \
"""
ucode_str = \
"""\
17 \n\
∂ \n\
────(2⋅x⋅y)\n\
17 \n\
∂x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(2*x*y, x, x, y)
ascii_str = \
"""\
3 \n\
d \n\
------(2*x*y)\n\
2 \n\
dy dx \
"""
ucode_str = \
"""\
3 \n\
∂ \n\
──────(2⋅x⋅y)\n\
2 \n\
∂y ∂x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# Greek letters
alpha = Symbol('alpha')
beta = Function('beta')
expr = beta(alpha).diff(alpha)
ascii_str = \
"""\
d \n\
------(beta(alpha))\n\
dalpha \
"""
ucode_str = \
"""\
d \n\
──(β(α))\n\
dα \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Derivative(f(x), (x, n))
ascii_str = \
"""\
n \n\
d \n\
---(f(x))\n\
n \n\
dx \
"""
ucode_str = \
"""\
n \n\
d \n\
───(f(x))\n\
n \n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_integrals():
expr = Integral(log(x), x)
ascii_str = \
"""\
/ \n\
| \n\
| log(x) dx\n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ log(x) dx\n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, x)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral((sin(x))**2 / (tan(x))**2)
ascii_str = \
"""\
/ \n\
| \n\
| 2 \n\
| sin (x) \n\
| ------- dx\n\
| 2 \n\
| tan (x) \n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ 2 \n\
⎮ sin (x) \n\
⎮ ─────── dx\n\
⎮ 2 \n\
⎮ tan (x) \n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**(2**x), x)
ascii_str = \
"""\
/ \n\
| \n\
| / x\\ \n\
| \\2 / \n\
| x dx\n\
| \n\
/ \
"""
ucode_str = \
"""\
⌠ \n\
⎮ ⎛ x⎞ \n\
⎮ ⎝2 ⎠ \n\
⎮ x dx\n\
⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, 1, 2))
ascii_str = \
"""\
2 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1 \
"""
ucode_str = \
"""\
2 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2, (x, Rational(1, 2), 10))
ascii_str = \
"""\
10 \n\
/ \n\
| \n\
| 2 \n\
| x dx\n\
| \n\
/ \n\
1/2 \
"""
ucode_str = \
"""\
10 \n\
⌠ \n\
⎮ 2 \n\
⎮ x dx\n\
⌡ \n\
1/2 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(x**2*y**2, x, y)
ascii_str = \
"""\
/ / \n\
| | \n\
| | 2 2 \n\
| | x *y dx dy\n\
| | \n\
/ / \
"""
ucode_str = \
"""\
⌠ ⌠ \n\
⎮ ⎮ 2 2 \n\
⎮ ⎮ x ⋅y dx dy\n\
⌡ ⌡ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(sin(th)/cos(ph), (th, 0, pi), (ph, 0, 2*pi))
ascii_str = \
"""\
2*pi pi \n\
/ / \n\
| | \n\
| | sin(theta) \n\
| | ---------- d(theta) d(phi)\n\
| | cos(phi) \n\
| | \n\
/ / \n\
0 0 \
"""
ucode_str = \
"""\
2⋅π π \n\
⌠ ⌠ \n\
⎮ ⎮ sin(θ) \n\
⎮ ⎮ ────── dθ dφ\n\
⎮ ⎮ cos(φ) \n\
⌡ ⌡ \n\
0 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_matrix():
# Empty Matrix
expr = Matrix()
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(2, 0, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix(0, 2, lambda i, j: 0)
ascii_str = "[]"
unicode_str = "[]"
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Matrix([[x**2 + 1, 1], [y, x + y]])
ascii_str_1 = \
"""\
[ 2 ]
[1 + x 1 ]
[ ]
[ y x + y]\
"""
ascii_str_2 = \
"""\
[ 2 ]
[x + 1 1 ]
[ ]
[ y x + y]\
"""
ucode_str_1 = \
"""\
⎡ 2 ⎤
⎢1 + x 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
"""
ucode_str_2 = \
"""\
⎡ 2 ⎤
⎢x + 1 1 ⎥
⎢ ⎥
⎣ y x + y⎦\
"""
assert pretty(expr) in [ascii_str_1, ascii_str_2]
assert upretty(expr) in [ucode_str_1, ucode_str_2]
expr = Matrix([[x/y, y, th], [0, exp(I*k*ph), 1]])
ascii_str = \
"""\
[x ]
[- y theta]
[y ]
[ ]
[ I*k*phi ]
[0 e 1 ]\
"""
ucode_str = \
"""\
⎡x ⎤
⎢─ y θ⎥
⎢y ⎥
⎢ ⎥
⎢ ⅈ⋅k⋅φ ⎥
⎣0 ℯ 1⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
unicode_str = \
"""\
⎡v̇_msc_00 0 0 ⎤
⎢ ⎥
⎢ 0 v̇_msc_01 0 ⎥
⎢ ⎥
⎣ 0 0 v̇_msc_02⎦\
"""
expr = diag(*MatrixSymbol('vdot_msc',1,3))
assert upretty(expr) == unicode_str
def test_pretty_ndim_arrays():
x, y, z, w = symbols("x y z w")
for ArrayType in (ImmutableDenseNDimArray, ImmutableSparseNDimArray, MutableDenseNDimArray, MutableSparseNDimArray):
# Basic: scalar array
M = ArrayType(x)
assert pretty(M) == "x"
assert upretty(M) == "x"
M = ArrayType([[1/x, y], [z, w]])
M1 = ArrayType([1/x, y, z])
M2 = tensorproduct(M1, M)
M3 = tensorproduct(M, M)
ascii_str = \
"""\
[1 ]\n\
[- y]\n\
[x ]\n\
[ ]\n\
[z w]\
"""
ucode_str = \
"""\
⎡1 ⎤\n\
⎢─ y⎥\n\
⎢x ⎥\n\
⎢ ⎥\n\
⎣z w⎦\
"""
assert pretty(M) == ascii_str
assert upretty(M) == ucode_str
ascii_str = \
"""\
[1 ]\n\
[- y z]\n\
[x ]\
"""
ucode_str = \
"""\
⎡1 ⎤\n\
⎢─ y z⎥\n\
⎣x ⎦\
"""
assert pretty(M1) == ascii_str
assert upretty(M1) == ucode_str
ascii_str = \
"""\
[[1 y] ]\n\
[[-- -] [z ]]\n\
[[ 2 x] [ y 2 ] [- y*z]]\n\
[[x ] [ - y ] [x ]]\n\
[[ ] [ x ] [ ]]\n\
[[z w] [ ] [ 2 ]]\n\
[[- -] [y*z w*y] [z w*z]]\n\
[[x x] ]\
"""
ucode_str = \
"""\
⎡⎡1 y⎤ ⎤\n\
⎢⎢── ─⎥ ⎡z ⎤⎥\n\
⎢⎢ 2 x⎥ ⎡ y 2 ⎤ ⎢─ y⋅z⎥⎥\n\
⎢⎢x ⎥ ⎢ ─ y ⎥ ⎢x ⎥⎥\n\
⎢⎢ ⎥ ⎢ x ⎥ ⎢ ⎥⎥\n\
⎢⎢z w⎥ ⎢ ⎥ ⎢ 2 ⎥⎥\n\
⎢⎢─ ─⎥ ⎣y⋅z w⋅y⎦ ⎣z w⋅z⎦⎥\n\
⎣⎣x x⎦ ⎦\
"""
assert pretty(M2) == ascii_str
assert upretty(M2) == ucode_str
ascii_str = \
"""\
[ [1 y] ]\n\
[ [-- -] ]\n\
[ [ 2 x] [ y 2 ]]\n\
[ [x ] [ - y ]]\n\
[ [ ] [ x ]]\n\
[ [z w] [ ]]\n\
[ [- -] [y*z w*y]]\n\
[ [x x] ]\n\
[ ]\n\
[[z ] [ w ]]\n\
[[- y*z] [ - w*y]]\n\
[[x ] [ x ]]\n\
[[ ] [ ]]\n\
[[ 2 ] [ 2 ]]\n\
[[z w*z] [w*z w ]]\
"""
ucode_str = \
"""\
⎡ ⎡1 y⎤ ⎤\n\
⎢ ⎢── ─⎥ ⎥\n\
⎢ ⎢ 2 x⎥ ⎡ y 2 ⎤⎥\n\
⎢ ⎢x ⎥ ⎢ ─ y ⎥⎥\n\
⎢ ⎢ ⎥ ⎢ x ⎥⎥\n\
⎢ ⎢z w⎥ ⎢ ⎥⎥\n\
⎢ ⎢─ ─⎥ ⎣y⋅z w⋅y⎦⎥\n\
⎢ ⎣x x⎦ ⎥\n\
⎢ ⎥\n\
⎢⎡z ⎤ ⎡ w ⎤⎥\n\
⎢⎢─ y⋅z⎥ ⎢ ─ w⋅y⎥⎥\n\
⎢⎢x ⎥ ⎢ x ⎥⎥\n\
⎢⎢ ⎥ ⎢ ⎥⎥\n\
⎢⎢ 2 ⎥ ⎢ 2 ⎥⎥\n\
⎣⎣z w⋅z⎦ ⎣w⋅z w ⎦⎦\
"""
assert pretty(M3) == ascii_str
assert upretty(M3) == ucode_str
Mrow = ArrayType([[x, y, 1 / z]])
Mcolumn = ArrayType([[x], [y], [1 / z]])
Mcol2 = ArrayType([Mcolumn.tolist()])
ascii_str = \
"""\
[[ 1]]\n\
[[x y -]]\n\
[[ z]]\
"""
ucode_str = \
"""\
⎡⎡ 1⎤⎤\n\
⎢⎢x y ─⎥⎥\n\
⎣⎣ z⎦⎦\
"""
assert pretty(Mrow) == ascii_str
assert upretty(Mrow) == ucode_str
ascii_str = \
"""\
[x]\n\
[ ]\n\
[y]\n\
[ ]\n\
[1]\n\
[-]\n\
[z]\
"""
ucode_str = \
"""\
⎡x⎤\n\
⎢ ⎥\n\
⎢y⎥\n\
⎢ ⎥\n\
⎢1⎥\n\
⎢─⎥\n\
⎣z⎦\
"""
assert pretty(Mcolumn) == ascii_str
assert upretty(Mcolumn) == ucode_str
ascii_str = \
"""\
[[x]]\n\
[[ ]]\n\
[[y]]\n\
[[ ]]\n\
[[1]]\n\
[[-]]\n\
[[z]]\
"""
ucode_str = \
"""\
⎡⎡x⎤⎤\n\
⎢⎢ ⎥⎥\n\
⎢⎢y⎥⎥\n\
⎢⎢ ⎥⎥\n\
⎢⎢1⎥⎥\n\
⎢⎢─⎥⎥\n\
⎣⎣z⎦⎦\
"""
assert pretty(Mcol2) == ascii_str
assert upretty(Mcol2) == ucode_str
def test_tensor_TensorProduct():
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
assert upretty(TensorProduct(A, B)) == "A\u2297B"
assert upretty(TensorProduct(A, B, A)) == "A\u2297B\u2297A"
def test_diffgeom_print_WedgeProduct():
from sympy.diffgeom.rn import R2
from sympy.diffgeom import WedgeProduct
wp = WedgeProduct(R2.dx, R2.dy)
assert upretty(wp) == "ⅆ x∧ⅆ y"
assert pretty(wp) == r"d x/\d y"
def test_Adjoint():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert pretty(Adjoint(X)) == " +\nX "
assert pretty(Adjoint(X + Y)) == " +\n(X + Y) "
assert pretty(Adjoint(X) + Adjoint(Y)) == " + +\nX + Y "
assert pretty(Adjoint(X*Y)) == " +\n(X*Y) "
assert pretty(Adjoint(Y)*Adjoint(X)) == " + +\nY *X "
assert pretty(Adjoint(X**2)) == " +\n/ 2\\ \n\\X / "
assert pretty(Adjoint(X)**2) == " 2\n/ +\\ \n\\X / "
assert pretty(Adjoint(Inverse(X))) == " +\n/ -1\\ \n\\X / "
assert pretty(Inverse(Adjoint(X))) == " -1\n/ +\\ \n\\X / "
assert pretty(Adjoint(Transpose(X))) == " +\n/ T\\ \n\\X / "
assert pretty(Transpose(Adjoint(X))) == " T\n/ +\\ \n\\X / "
assert upretty(Adjoint(X)) == " †\nX "
assert upretty(Adjoint(X + Y)) == " †\n(X + Y) "
assert upretty(Adjoint(X) + Adjoint(Y)) == " † †\nX + Y "
assert upretty(Adjoint(X*Y)) == " †\n(X⋅Y) "
assert upretty(Adjoint(Y)*Adjoint(X)) == " † †\nY ⋅X "
assert upretty(Adjoint(X**2)) == \
" †\n⎛ 2⎞ \n⎝X ⎠ "
assert upretty(Adjoint(X)**2) == \
" 2\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Inverse(X))) == \
" †\n⎛ -1⎞ \n⎝X ⎠ "
assert upretty(Inverse(Adjoint(X))) == \
" -1\n⎛ †⎞ \n⎝X ⎠ "
assert upretty(Adjoint(Transpose(X))) == \
" †\n⎛ T⎞ \n⎝X ⎠ "
assert upretty(Transpose(Adjoint(X))) == \
" T\n⎛ †⎞ \n⎝X ⎠ "
m = Matrix(((1, 2), (3, 4)))
assert upretty(Adjoint(m)) == \
' †\n'\
'⎡1 2⎤ \n'\
'⎢ ⎥ \n'\
'⎣3 4⎦ '
assert upretty(Adjoint(m+X)) == \
' †\n'\
'⎛⎡1 2⎤ ⎞ \n'\
'⎜⎢ ⎥ + X⎟ \n'\
'⎝⎣3 4⎦ ⎠ '
assert upretty(Adjoint(BlockMatrix(((OneMatrix(2, 2), X),
(m, ZeroMatrix(2, 2)))))) == \
' †\n'\
'⎡ 𝟙 X⎤ \n'\
'⎢ ⎥ \n'\
'⎢⎡1 2⎤ ⎥ \n'\
'⎢⎢ ⎥ 𝟘⎥ \n'\
'⎣⎣3 4⎦ ⎦ '
def test_Transpose():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
assert pretty(Transpose(X)) == " T\nX "
assert pretty(Transpose(X + Y)) == " T\n(X + Y) "
assert pretty(Transpose(X) + Transpose(Y)) == " T T\nX + Y "
assert pretty(Transpose(X*Y)) == " T\n(X*Y) "
assert pretty(Transpose(Y)*Transpose(X)) == " T T\nY *X "
assert pretty(Transpose(X**2)) == " T\n/ 2\\ \n\\X / "
assert pretty(Transpose(X)**2) == " 2\n/ T\\ \n\\X / "
assert pretty(Transpose(Inverse(X))) == " T\n/ -1\\ \n\\X / "
assert pretty(Inverse(Transpose(X))) == " -1\n/ T\\ \n\\X / "
assert upretty(Transpose(X)) == " T\nX "
assert upretty(Transpose(X + Y)) == " T\n(X + Y) "
assert upretty(Transpose(X) + Transpose(Y)) == " T T\nX + Y "
assert upretty(Transpose(X*Y)) == " T\n(X⋅Y) "
assert upretty(Transpose(Y)*Transpose(X)) == " T T\nY ⋅X "
assert upretty(Transpose(X**2)) == \
" T\n⎛ 2⎞ \n⎝X ⎠ "
assert upretty(Transpose(X)**2) == \
" 2\n⎛ T⎞ \n⎝X ⎠ "
assert upretty(Transpose(Inverse(X))) == \
" T\n⎛ -1⎞ \n⎝X ⎠ "
assert upretty(Inverse(Transpose(X))) == \
" -1\n⎛ T⎞ \n⎝X ⎠ "
m = Matrix(((1, 2), (3, 4)))
assert upretty(Transpose(m)) == \
' T\n'\
'⎡1 2⎤ \n'\
'⎢ ⎥ \n'\
'⎣3 4⎦ '
assert upretty(Transpose(m+X)) == \
' T\n'\
'⎛⎡1 2⎤ ⎞ \n'\
'⎜⎢ ⎥ + X⎟ \n'\
'⎝⎣3 4⎦ ⎠ '
assert upretty(Transpose(BlockMatrix(((OneMatrix(2, 2), X),
(m, ZeroMatrix(2, 2)))))) == \
' T\n'\
'⎡ 𝟙 X⎤ \n'\
'⎢ ⎥ \n'\
'⎢⎡1 2⎤ ⎥ \n'\
'⎢⎢ ⎥ 𝟘⎥ \n'\
'⎣⎣3 4⎦ ⎦ '
def test_pretty_Trace_issue_9044():
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[2, 4], [6, 8]])
ascii_str_1 = \
"""\
/[1 2]\\
tr|[ ]|
\\[3 4]/\
"""
ucode_str_1 = \
"""\
⎛⎡1 2⎤⎞
tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠\
"""
ascii_str_2 = \
"""\
/[1 2]\\ /[2 4]\\
tr|[ ]| + tr|[ ]|
\\[3 4]/ \\[6 8]/\
"""
ucode_str_2 = \
"""\
⎛⎡1 2⎤⎞ ⎛⎡2 4⎤⎞
tr⎜⎢ ⎥⎟ + tr⎜⎢ ⎥⎟
⎝⎣3 4⎦⎠ ⎝⎣6 8⎦⎠\
"""
assert pretty(Trace(X)) == ascii_str_1
assert upretty(Trace(X)) == ucode_str_1
assert pretty(Trace(X) + Trace(Y)) == ascii_str_2
assert upretty(Trace(X) + Trace(Y)) == ucode_str_2
def test_MatrixSlice():
n = Symbol('n', integer=True)
x, y, z, w, t, = symbols('x y z w t')
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', 10, 10)
Z = MatrixSymbol('Z', 10, 10)
expr = MatrixSlice(X, (None, None, None), (None, None, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = X[x:x + 1, y:y + 1]
assert pretty(expr) == upretty(expr) == 'X[x:x + 1, y:y + 1]'
expr = X[x:x + 1:2, y:y + 1:2]
assert pretty(expr) == upretty(expr) == 'X[x:x + 1:2, y:y + 1:2]'
expr = X[:x, y:]
assert pretty(expr) == upretty(expr) == 'X[:x, y:]'
expr = X[:x, y:]
assert pretty(expr) == upretty(expr) == 'X[:x, y:]'
expr = X[x:, :y]
assert pretty(expr) == upretty(expr) == 'X[x:, :y]'
expr = X[x:y, z:w]
assert pretty(expr) == upretty(expr) == 'X[x:y, z:w]'
expr = X[x:y:t, w:t:x]
assert pretty(expr) == upretty(expr) == 'X[x:y:t, w:t:x]'
expr = X[x::y, t::w]
assert pretty(expr) == upretty(expr) == 'X[x::y, t::w]'
expr = X[:x:y, :t:w]
assert pretty(expr) == upretty(expr) == 'X[:x:y, :t:w]'
expr = X[::x, ::y]
assert pretty(expr) == upretty(expr) == 'X[::x, ::y]'
expr = MatrixSlice(X, (0, None, None), (0, None, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (None, n, None), (None, n, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (0, n, None), (0, n, None))
assert pretty(expr) == upretty(expr) == 'X[:, :]'
expr = MatrixSlice(X, (0, n, 2), (0, n, 2))
assert pretty(expr) == upretty(expr) == 'X[::2, ::2]'
expr = X[1:2:3, 4:5:6]
assert pretty(expr) == upretty(expr) == 'X[1:2:3, 4:5:6]'
expr = X[1:3:5, 4:6:8]
assert pretty(expr) == upretty(expr) == 'X[1:3:5, 4:6:8]'
expr = X[1:10:2]
assert pretty(expr) == upretty(expr) == 'X[1:10:2, :]'
expr = Y[:5, 1:9:2]
assert pretty(expr) == upretty(expr) == 'Y[:5, 1:9:2]'
expr = Y[:5, 1:10:2]
assert pretty(expr) == upretty(expr) == 'Y[:5, 1::2]'
expr = Y[5, :5:2]
assert pretty(expr) == upretty(expr) == 'Y[5:6, :5:2]'
expr = X[0:1, 0:1]
assert pretty(expr) == upretty(expr) == 'X[:1, :1]'
expr = X[0:1:2, 0:1:2]
assert pretty(expr) == upretty(expr) == 'X[:1:2, :1:2]'
expr = (Y + Z)[2:, 2:]
assert pretty(expr) == upretty(expr) == '(Y + Z)[2:, 2:]'
def test_MatrixExpressions():
n = Symbol('n', integer=True)
X = MatrixSymbol('X', n, n)
assert pretty(X) == upretty(X) == "X"
# Apply function elementwise (`ElementwiseApplyFunc`):
expr = (X.T*X).applyfunc(sin)
ascii_str = """\
/ T \\\n\
(d -> sin(d)).\\X *X/\
"""
ucode_str = """\
⎛ T ⎞\n\
(d ↦ sin(d))˳⎝X ⋅X⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
lamda = Lambda(x, 1/x)
expr = (n*X).applyfunc(lamda)
ascii_str = """\
/ 1\\ \n\
|x -> -|.(n*X)\n\
\\ x/ \
"""
ucode_str = """\
⎛ 1⎞ \n\
⎜x ↦ ─⎟˳(n⋅X)\n\
⎝ x⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_dotproduct():
from sympy.matrices.expressions.dotproduct import DotProduct
n = symbols("n", integer=True)
A = MatrixSymbol('A', n, 1)
B = MatrixSymbol('B', n, 1)
C = Matrix(1, 3, [1, 2, 3])
D = Matrix(1, 3, [1, 3, 4])
assert pretty(DotProduct(A, B)) == "A*B"
assert pretty(DotProduct(C, D)) == "[1 2 3]*[1 3 4]"
assert upretty(DotProduct(A, B)) == "A⋅B"
assert upretty(DotProduct(C, D)) == "[1 2 3]⋅[1 3 4]"
def test_pretty_Determinant():
from sympy.matrices import Determinant, Inverse, BlockMatrix, OneMatrix, ZeroMatrix
m = Matrix(((1, 2), (3, 4)))
assert upretty(Determinant(m)) == '│1 2│\n│ │\n│3 4│'
assert upretty(Determinant(Inverse(m))) == \
'│ -1│\n'\
'│⎡1 2⎤ │\n'\
'│⎢ ⎥ │\n'\
'│⎣3 4⎦ │'
X = MatrixSymbol('X', 2, 2)
assert upretty(Determinant(X)) == '│X│'
assert upretty(Determinant(X + m)) == \
'│⎡1 2⎤ │\n'\
'│⎢ ⎥ + X│\n'\
'│⎣3 4⎦ │'
assert upretty(Determinant(BlockMatrix(((OneMatrix(2, 2), X),
(m, ZeroMatrix(2, 2)))))) == \
'│ 𝟙 X│\n'\
'│ │\n'\
'│⎡1 2⎤ │\n'\
'│⎢ ⎥ 𝟘│\n'\
'│⎣3 4⎦ │'
def test_pretty_piecewise():
expr = Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
/x for x < 1\n\
| \n\
< 2 \n\
|x otherwise\n\
\\ \
"""
ucode_str = \
"""\
⎧x for x < 1\n\
⎪ \n\
⎨ 2 \n\
⎪x otherwise\n\
⎩ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x < 1), (x**2, True))
ascii_str = \
"""\
//x for x < 1\\\n\
|| |\n\
-|< 2 |\n\
||x otherwise|\n\
\\\\ /\
"""
ucode_str = \
"""\
⎛⎧x for x < 1⎞\n\
⎜⎪ ⎟\n\
-⎜⎨ 2 ⎟\n\
⎜⎪x otherwise⎟\n\
⎝⎩ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x + Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x + |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
"""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x + ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x - Piecewise((x, x > 0), (y, True)) + Piecewise((x/y, x < 2),
(y**2, x > 2), (1, True)) + 1
ascii_str = \
"""\
//x \\ \n\
||- for x < 2| \n\
||y | \n\
//x for x > 0\\ || | \n\
x - |< | + |< 2 | + 1\n\
\\\\y otherwise/ ||y for x > 2| \n\
|| | \n\
||1 otherwise| \n\
\\\\ / \
"""
ucode_str = \
"""\
⎛⎧x ⎞ \n\
⎜⎪─ for x < 2⎟ \n\
⎜⎪y ⎟ \n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟ \n\
x - ⎜⎨ ⎟ + ⎜⎨ 2 ⎟ + 1\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟ \n\
⎜⎪ ⎟ \n\
⎜⎪1 otherwise⎟ \n\
⎝⎩ ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = x*Piecewise((x, x > 0), (y, True))
ascii_str = \
"""\
//x for x > 0\\\n\
x*|< |\n\
\\\\y otherwise/\
"""
ucode_str = \
"""\
⎛⎧x for x > 0⎞\n\
x⋅⎜⎨ ⎟\n\
⎝⎩y otherwise⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x >
2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
"""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -Piecewise((x, x > 0), (y, True))*Piecewise((x/y, x < 2), (y**2, x
> 2), (1, True))
ascii_str = \
"""\
//x \\\n\
||- for x < 2|\n\
||y |\n\
//x for x > 0\\ || |\n\
-|< |*|< 2 |\n\
\\\\y otherwise/ ||y for x > 2|\n\
|| |\n\
||1 otherwise|\n\
\\\\ /\
"""
ucode_str = \
"""\
⎛⎧x ⎞\n\
⎜⎪─ for x < 2⎟\n\
⎜⎪y ⎟\n\
⎛⎧x for x > 0⎞ ⎜⎪ ⎟\n\
-⎜⎨ ⎟⋅⎜⎨ 2 ⎟\n\
⎝⎩y otherwise⎠ ⎜⎪y for x > 2⎟\n\
⎜⎪ ⎟\n\
⎜⎪1 otherwise⎟\n\
⎝⎩ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Piecewise((0, Abs(1/y) < 1), (1, Abs(y) < 1), (y*meijerg(((2, 1),
()), ((), (1, 0)), 1/y), True))
ascii_str = \
"""\
/ 1 \n\
| 0 for --- < 1\n\
| |y| \n\
| \n\
< 1 for |y| < 1\n\
| \n\
| __0, 2 /2, 1 | 1\\ \n\
|y*/__ | | -| otherwise \n\
\\ \\_|2, 2 \\ 1, 0 | y/ \
"""
ucode_str = \
"""\
⎧ 1 \n\
⎪ 0 for ─── < 1\n\
⎪ │y│ \n\
⎪ \n\
⎨ 1 for │y│ < 1\n\
⎪ \n\
⎪ ╭─╮0, 2 ⎛2, 1 │ 1⎞ \n\
⎪y⋅│╶┐ ⎜ │ ─⎟ otherwise \n\
⎩ ╰─╯2, 2 ⎝ 1, 0 │ y⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
# XXX: We have to use evaluate=False here because Piecewise._eval_power
# denests the power.
expr = Pow(Piecewise((x, x > 0), (y, True)), 2, evaluate=False)
ascii_str = \
"""\
2\n\
//x for x > 0\\ \n\
|< | \n\
\\\\y otherwise/ \
"""
ucode_str = \
"""\
2\n\
⎛⎧x for x > 0⎞ \n\
⎜⎨ ⎟ \n\
⎝⎩y otherwise⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ITE():
expr = ITE(x, y, z)
assert pretty(expr) == (
'/y for x \n'
'< \n'
'\\z otherwise'
)
assert upretty(expr) == """\
⎧y for x \n\
⎨ \n\
⎩z otherwise\
"""
def test_pretty_seq():
expr = ()
ascii_str = \
"""\
()\
"""
ucode_str = \
"""\
()\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = []
ascii_str = \
"""\
[]\
"""
ucode_str = \
"""\
[]\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {}
expr_2 = {}
ascii_str = \
"""\
{}\
"""
ucode_str = \
"""\
{}\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = (1/x,)
ascii_str = \
"""\
1 \n\
(-,)\n\
x \
"""
ucode_str = \
"""\
⎛1 ⎞\n\
⎜─,⎟\n\
⎝x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = [x**2, 1/x, x, y, sin(th)**2/cos(ph)**2]
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
[x , -, x, y, -----------]\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
"""\
⎡ 2 ⎤\n\
⎢ 2 1 sin (θ)⎥\n\
⎢x , ─, x, y, ───────⎥\n\
⎢ x 2 ⎥\n\
⎣ cos (φ)⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2, 1/x, x, y, sin(th)**2/cos(ph)**2)
ascii_str = \
"""\
2 \n\
2 1 sin (theta) \n\
(x , -, x, y, -----------)\n\
x 2 \n\
cos (phi) \
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎜ 2 1 sin (θ)⎟\n\
⎜x , ─, x, y, ───────⎟\n\
⎜ x 2 ⎟\n\
⎝ cos (φ)⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x: sin(x)}
expr_2 = Dict({x: sin(x)})
ascii_str = \
"""\
{x: sin(x)}\
"""
ucode_str = \
"""\
{x: sin(x)}\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
expr = {1/x: 1/y, x: sin(x)**2}
expr_2 = Dict({1/x: 1/y, x: sin(x)**2})
ascii_str = \
"""\
1 1 2 \n\
{-: -, x: sin (x)}\n\
x y \
"""
ucode_str = \
"""\
⎧1 1 2 ⎫\n\
⎨─: ─, x: sin (x)⎬\n\
⎩x y ⎭\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
# There used to be a bug with pretty-printing sequences of even height.
expr = [x**2]
ascii_str = \
"""\
2 \n\
[x ]\
"""
ucode_str = \
"""\
⎡ 2⎤\n\
⎣x ⎦\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (x**2,)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎝x ,⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Tuple(x**2)
ascii_str = \
"""\
2 \n\
(x ,)\
"""
ucode_str = \
"""\
⎛ 2 ⎞\n\
⎝x ,⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = {x**2: 1}
expr_2 = Dict({x**2: 1})
ascii_str = \
"""\
2 \n\
{x : 1}\
"""
ucode_str = \
"""\
⎧ 2 ⎫\n\
⎨x : 1⎬\n\
⎩ ⎭\
"""
assert pretty(expr) == ascii_str
assert pretty(expr_2) == ascii_str
assert upretty(expr) == ucode_str
assert upretty(expr_2) == ucode_str
def test_any_object_in_sequence():
# Cf. issue 5306
b1 = Basic()
b2 = Basic(Basic())
expr = [b2, b1]
assert pretty(expr) == "[Basic(Basic()), Basic()]"
assert upretty(expr) == "[Basic(Basic()), Basic()]"
expr = {b2, b1}
assert pretty(expr) == "{Basic(), Basic(Basic())}"
assert upretty(expr) == "{Basic(), Basic(Basic())}"
expr = {b2: b1, b1: b2}
expr2 = Dict({b2: b1, b1: b2})
assert pretty(expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert pretty(
expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
assert upretty(
expr2) == "{Basic(): Basic(Basic()), Basic(Basic()): Basic()}"
def test_print_builtin_set():
assert pretty(set()) == 'set()'
assert upretty(set()) == 'set()'
assert pretty(frozenset()) == 'frozenset()'
assert upretty(frozenset()) == 'frozenset()'
s1 = {1/x, x}
s2 = frozenset(s1)
assert pretty(s1) == \
"""\
1 \n\
{-, x}
x \
"""
assert upretty(s1) == \
"""\
⎧1 ⎫
⎨─, x⎬
⎩x ⎭\
"""
assert pretty(s2) == \
"""\
1 \n\
frozenset({-, x})
x \
"""
assert upretty(s2) == \
"""\
⎛⎧1 ⎫⎞
frozenset⎜⎨─, x⎬⎟
⎝⎩x ⎭⎠\
"""
def test_pretty_sets():
s = FiniteSet
assert pretty(s(*[x*y, x**2])) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(s(*range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(s(*range(1, 13))) == "{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty({x*y, x**2}) == \
"""\
2 \n\
{x , x*y}\
"""
assert pretty(set(range(1, 6))) == "{1, 2, 3, 4, 5}"
assert pretty(set(range(1, 13))) == \
"{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"
assert pretty(frozenset([x*y, x**2])) == \
"""\
2 \n\
frozenset({x , x*y})\
"""
assert pretty(frozenset(range(1, 6))) == "frozenset({1, 2, 3, 4, 5})"
assert pretty(frozenset(range(1, 13))) == \
"frozenset({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12})"
assert pretty(Range(0, 3, 1)) == '{0, 1, 2}'
ascii_str = '{0, 1, ..., 29}'
ucode_str = '{0, 1, …, 29}'
assert pretty(Range(0, 30, 1)) == ascii_str
assert upretty(Range(0, 30, 1)) == ucode_str
ascii_str = '{30, 29, ..., 2}'
ucode_str = '{30, 29, …, 2}'
assert pretty(Range(30, 1, -1)) == ascii_str
assert upretty(Range(30, 1, -1)) == ucode_str
ascii_str = '{0, 2, ...}'
ucode_str = '{0, 2, …}'
assert pretty(Range(0, oo, 2)) == ascii_str
assert upretty(Range(0, oo, 2)) == ucode_str
ascii_str = '{..., 2, 0}'
ucode_str = '{…, 2, 0}'
assert pretty(Range(oo, -2, -2)) == ascii_str
assert upretty(Range(oo, -2, -2)) == ucode_str
ascii_str = '{-2, -3, ...}'
ucode_str = '{-2, -3, …}'
assert pretty(Range(-2, -oo, -1)) == ascii_str
assert upretty(Range(-2, -oo, -1)) == ucode_str
def test_pretty_SetExpr():
iv = Interval(1, 3)
se = SetExpr(iv)
ascii_str = "SetExpr([1, 3])"
ucode_str = "SetExpr([1, 3])"
assert pretty(se) == ascii_str
assert upretty(se) == ucode_str
def test_pretty_ImageSet():
imgset = ImageSet(Lambda((x, y), x + y), {1, 2, 3}, {3, 4})
ascii_str = '{x + y | x in {1, 2, 3}, y in {3, 4}}'
ucode_str = '{x + y │ x ∊ {1, 2, 3}, y ∊ {3, 4}}'
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(((x, y),), x + y), ProductSet({1, 2, 3}, {3, 4}))
ascii_str = '{x + y | (x, y) in {1, 2, 3} x {3, 4}}'
ucode_str = '{x + y │ (x, y) ∊ {1, 2, 3} × {3, 4}}'
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda(x, x**2), S.Naturals)
ascii_str = '''\
2 \n\
{x | x in Naturals}'''
ucode_str = '''\
⎧ 2 │ ⎫\n\
⎨x │ x ∊ ℕ⎬\n\
⎩ │ ⎭'''
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
# TODO: The "x in N" parts below should be centered independently of the
# 1/x**2 fraction
imgset = ImageSet(Lambda(x, 1/x**2), S.Naturals)
ascii_str = '''\
1 \n\
{-- | x in Naturals}
2 \n\
x '''
ucode_str = '''\
⎧1 │ ⎫\n\
⎪── │ x ∊ ℕ⎪\n\
⎨ 2 │ ⎬\n\
⎪x │ ⎪\n\
⎩ │ ⎭'''
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
imgset = ImageSet(Lambda((x, y), 1/(x + y)**2), S.Naturals, S.Naturals)
ascii_str = '''\
1 \n\
{-------- | x in Naturals, y in Naturals}
2 \n\
(x + y) '''
ucode_str = '''\
⎧ 1 │ ⎫
⎪──────── │ x ∊ ℕ, y ∊ ℕ⎪
⎨ 2 │ ⎬
⎪(x + y) │ ⎪
⎩ │ ⎭'''
assert pretty(imgset) == ascii_str
assert upretty(imgset) == ucode_str
def test_pretty_ConditionSet():
ascii_str = '{x | x in (-oo, oo) and sin(x) = 0}'
ucode_str = '{x │ x ∊ ℝ ∧ (sin(x) = 0)}'
assert pretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ascii_str
assert upretty(ConditionSet(x, Eq(sin(x), 0), S.Reals)) == ucode_str
assert pretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}'
assert upretty(ConditionSet(x, Contains(x, S.Reals, evaluate=False), FiniteSet(1))) == '{1}'
assert pretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "EmptySet"
assert upretty(ConditionSet(x, And(x > 1, x < -1), FiniteSet(1, 2, 3))) == "∅"
assert pretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}'
assert upretty(ConditionSet(x, Or(x > 1, x < -1), FiniteSet(1, 2))) == '{2}'
condset = ConditionSet(x, 1/x**2 > 0)
ascii_str = '''\
1 \n\
{x | -- > 0}
2 \n\
x '''
ucode_str = '''\
⎧ │ ⎛1 ⎞⎫
⎪x │ ⎜── > 0⎟⎪
⎨ │ ⎜ 2 ⎟⎬
⎪ │ ⎝x ⎠⎪
⎩ │ ⎭'''
assert pretty(condset) == ascii_str
assert upretty(condset) == ucode_str
condset = ConditionSet(x, 1/x**2 > 0, S.Reals)
ascii_str = '''\
1 \n\
{x | x in (-oo, oo) and -- > 0}
2 \n\
x '''
ucode_str = '''\
⎧ │ ⎛1 ⎞⎫
⎪x │ x ∊ ℝ ∧ ⎜── > 0⎟⎪
⎨ │ ⎜ 2 ⎟⎬
⎪ │ ⎝x ⎠⎪
⎩ │ ⎭'''
assert pretty(condset) == ascii_str
assert upretty(condset) == ucode_str
def test_pretty_ComplexRegion():
from sympy.sets.fancysets import ComplexRegion
cregion = ComplexRegion(Interval(3, 5)*Interval(4, 6))
ascii_str = '{x + y*I | x, y in [3, 5] x [4, 6]}'
ucode_str = '{x + y⋅ⅈ │ x, y ∊ [3, 5] × [4, 6]}'
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
cregion = ComplexRegion(Interval(0, 1)*Interval(0, 2*pi), polar=True)
ascii_str = '{r*(I*sin(theta) + cos(theta)) | r, theta in [0, 1] x [0, 2*pi)}'
ucode_str = '{r⋅(ⅈ⋅sin(θ) + cos(θ)) │ r, θ ∊ [0, 1] × [0, 2⋅π)}'
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
cregion = ComplexRegion(Interval(3, 1/a**2)*Interval(4, 6))
ascii_str = '''\
1 \n\
{x + y*I | x, y in [3, --] x [4, 6]}
2 \n\
a '''
ucode_str = '''\
⎧ │ ⎡ 1 ⎤ ⎫
⎪x + y⋅ⅈ │ x, y ∊ ⎢3, ──⎥ × [4, 6]⎪
⎨ │ ⎢ 2⎥ ⎬
⎪ │ ⎣ a ⎦ ⎪
⎩ │ ⎭'''
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
cregion = ComplexRegion(Interval(0, 1/a**2)*Interval(0, 2*pi), polar=True)
ascii_str = '''\
1 \n\
{r*(I*sin(theta) + cos(theta)) | r, theta in [0, --] x [0, 2*pi)}
2 \n\
a '''
ucode_str = '''\
⎧ │ ⎡ 1 ⎤ ⎫
⎪r⋅(ⅈ⋅sin(θ) + cos(θ)) │ r, θ ∊ ⎢0, ──⎥ × [0, 2⋅π)⎪
⎨ │ ⎢ 2⎥ ⎬
⎪ │ ⎣ a ⎦ ⎪
⎩ │ ⎭'''
assert pretty(cregion) == ascii_str
assert upretty(cregion) == ucode_str
def test_pretty_Union_issue_10414():
a, b = Interval(2, 3), Interval(4, 7)
ucode_str = '[2, 3] ∪ [4, 7]'
ascii_str = '[2, 3] U [4, 7]'
assert upretty(Union(a, b)) == ucode_str
assert pretty(Union(a, b)) == ascii_str
def test_pretty_Intersection_issue_10414():
x, y, z, w = symbols('x, y, z, w')
a, b = Interval(x, y), Interval(z, w)
ucode_str = '[x, y] ∩ [z, w]'
ascii_str = '[x, y] n [z, w]'
assert upretty(Intersection(a, b)) == ucode_str
assert pretty(Intersection(a, b)) == ascii_str
def test_ProductSet_exponent():
ucode_str = ' 1\n[0, 1] '
assert upretty(Interval(0, 1)**1) == ucode_str
ucode_str = ' 2\n[0, 1] '
assert upretty(Interval(0, 1)**2) == ucode_str
def test_ProductSet_parenthesis():
ucode_str = '([4, 7] × {1, 2}) ∪ ([2, 3] × [4, 7])'
a, b = Interval(2, 3), Interval(4, 7)
assert upretty(Union(a*b, b*FiniteSet(1, 2))) == ucode_str
def test_ProductSet_prod_char_issue_10413():
ascii_str = '[2, 3] x [4, 7]'
ucode_str = '[2, 3] × [4, 7]'
a, b = Interval(2, 3), Interval(4, 7)
assert pretty(a*b) == ascii_str
assert upretty(a*b) == ucode_str
def test_pretty_sequences():
s1 = SeqFormula(a**2, (0, oo))
s2 = SeqPer((1, 2))
ascii_str = '[0, 1, 4, 9, ...]'
ucode_str = '[0, 1, 4, 9, …]'
assert pretty(s1) == ascii_str
assert upretty(s1) == ucode_str
ascii_str = '[1, 2, 1, 2, ...]'
ucode_str = '[1, 2, 1, 2, …]'
assert pretty(s2) == ascii_str
assert upretty(s2) == ucode_str
s3 = SeqFormula(a**2, (0, 2))
s4 = SeqPer((1, 2), (0, 2))
ascii_str = '[0, 1, 4]'
ucode_str = '[0, 1, 4]'
assert pretty(s3) == ascii_str
assert upretty(s3) == ucode_str
ascii_str = '[1, 2, 1]'
ucode_str = '[1, 2, 1]'
assert pretty(s4) == ascii_str
assert upretty(s4) == ucode_str
s5 = SeqFormula(a**2, (-oo, 0))
s6 = SeqPer((1, 2), (-oo, 0))
ascii_str = '[..., 9, 4, 1, 0]'
ucode_str = '[…, 9, 4, 1, 0]'
assert pretty(s5) == ascii_str
assert upretty(s5) == ucode_str
ascii_str = '[..., 2, 1, 2, 1]'
ucode_str = '[…, 2, 1, 2, 1]'
assert pretty(s6) == ascii_str
assert upretty(s6) == ucode_str
ascii_str = '[1, 3, 5, 11, ...]'
ucode_str = '[1, 3, 5, 11, …]'
assert pretty(SeqAdd(s1, s2)) == ascii_str
assert upretty(SeqAdd(s1, s2)) == ucode_str
ascii_str = '[1, 3, 5]'
ucode_str = '[1, 3, 5]'
assert pretty(SeqAdd(s3, s4)) == ascii_str
assert upretty(SeqAdd(s3, s4)) == ucode_str
ascii_str = '[..., 11, 5, 3, 1]'
ucode_str = '[…, 11, 5, 3, 1]'
assert pretty(SeqAdd(s5, s6)) == ascii_str
assert upretty(SeqAdd(s5, s6)) == ucode_str
ascii_str = '[0, 2, 4, 18, ...]'
ucode_str = '[0, 2, 4, 18, …]'
assert pretty(SeqMul(s1, s2)) == ascii_str
assert upretty(SeqMul(s1, s2)) == ucode_str
ascii_str = '[0, 2, 4]'
ucode_str = '[0, 2, 4]'
assert pretty(SeqMul(s3, s4)) == ascii_str
assert upretty(SeqMul(s3, s4)) == ucode_str
ascii_str = '[..., 18, 4, 2, 0]'
ucode_str = '[…, 18, 4, 2, 0]'
assert pretty(SeqMul(s5, s6)) == ascii_str
assert upretty(SeqMul(s5, s6)) == ucode_str
# Sequences with symbolic limits, issue 12629
s7 = SeqFormula(a**2, (a, 0, x))
raises(NotImplementedError, lambda: pretty(s7))
raises(NotImplementedError, lambda: upretty(s7))
b = Symbol('b')
s8 = SeqFormula(b*a**2, (a, 0, 2))
ascii_str = '[0, b, 4*b]'
ucode_str = '[0, b, 4⋅b]'
assert pretty(s8) == ascii_str
assert upretty(s8) == ucode_str
def test_pretty_FourierSeries():
f = fourier_series(x, (x, -pi, pi))
ascii_str = \
"""\
2*sin(3*x) \n\
2*sin(x) - sin(2*x) + ---------- + ...\n\
3 \
"""
ucode_str = \
"""\
2⋅sin(3⋅x) \n\
2⋅sin(x) - sin(2⋅x) + ────────── + …\n\
3 \
"""
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_FormalPowerSeries():
f = fps(log(1 + x))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -k k \n\
\\ -(-1) *x \n\
/ -----------\n\
/ k \n\
/___, \n\
k = 1 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ -k k \n\
╲ -(-1) ⋅x \n\
╱ ───────────\n\
╱ k \n\
╱ \n\
‾‾‾‾ \n\
k = 1 \
"""
assert pretty(f) == ascii_str
assert upretty(f) == ucode_str
def test_pretty_limits():
expr = Limit(x, x, oo)
ascii_str = \
"""\
lim x\n\
x->oo \
"""
ucode_str = \
"""\
lim x\n\
x─→∞ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x**2, x, 0)
ascii_str = \
"""\
2\n\
lim x \n\
x->0+ \
"""
ucode_str = \
"""\
2\n\
lim x \n\
x─→0⁺ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(1/x, x, 0)
ascii_str = \
"""\
1\n\
lim -\n\
x->0+x\
"""
ucode_str = \
"""\
1\n\
lim ─\n\
x─→0⁺x\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0)
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0+\\ x /\
"""
ucode_str = \
"""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁺⎝ x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x)/x, x, 0, "-")
ascii_str = \
"""\
/sin(x)\\\n\
lim |------|\n\
x->0-\\ x /\
"""
ucode_str = \
"""\
⎛sin(x)⎞\n\
lim ⎜──────⎟\n\
x─→0⁻⎝ x ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x + sin(x), x, 0)
ascii_str = \
"""\
lim (x + sin(x))\n\
x->0+ \
"""
ucode_str = \
"""\
lim (x + sin(x))\n\
x─→0⁺ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x, x, 0)**2
ascii_str = \
"""\
2\n\
/ lim x\\ \n\
\\x->0+ / \
"""
ucode_str = \
"""\
2\n\
⎛ lim x⎞ \n\
⎝x─→0⁺ ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
"""\
⎛ ⎛y⎞⎞\n\
lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = 2*Limit(x*Limit(y/2,y,0), x, 0)
ascii_str = \
"""\
/ /y\\\\\n\
2* lim |x* lim |-||\n\
x->0+\\ y->0+\\2//\
"""
ucode_str = \
"""\
⎛ ⎛y⎞⎞\n\
2⋅ lim ⎜x⋅ lim ⎜─⎟⎟\n\
x─→0⁺⎝ y─→0⁺⎝2⎠⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Limit(sin(x), x, 0, dir='+-')
ascii_str = \
"""\
lim sin(x)\n\
x->0 \
"""
ucode_str = \
"""\
lim sin(x)\n\
x─→0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_ComplexRootOf():
expr = rootof(x**5 + 11*x - 2, 0)
ascii_str = \
"""\
/ 5 \\\n\
CRootOf\\x + 11*x - 2, 0/\
"""
ucode_str = \
"""\
⎛ 5 ⎞\n\
CRootOf⎝x + 11⋅x - 2, 0⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_RootSum():
expr = RootSum(x**5 + 11*x - 2, auto=False)
ascii_str = \
"""\
/ 5 \\\n\
RootSum\\x + 11*x - 2/\
"""
ucode_str = \
"""\
⎛ 5 ⎞\n\
RootSum⎝x + 11⋅x - 2⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = RootSum(x**5 + 11*x - 2, Lambda(z, exp(z)))
ascii_str = \
"""\
/ 5 z\\\n\
RootSum\\x + 11*x - 2, z -> e /\
"""
ucode_str = \
"""\
⎛ 5 z⎞\n\
RootSum⎝x + 11⋅x - 2, z ↦ ℯ ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_GroebnerBasis():
expr = groebner([], x, y)
ascii_str = \
"""\
GroebnerBasis([], x, y, domain=ZZ, order=lex)\
"""
ucode_str = \
"""\
GroebnerBasis([], x, y, domain=ℤ, order=lex)\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
F = [x**2 - 3*y - x + 1, y**2 - 2*x + y - 1]
expr = groebner(F, x, y, order='grlex')
ascii_str = \
"""\
/[ 2 2 ] \\\n\
GroebnerBasis\\[x - x - 3*y + 1, y - 2*x + y - 1], x, y, domain=ZZ, order=grlex/\
"""
ucode_str = \
"""\
⎛⎡ 2 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣x - x - 3⋅y + 1, y - 2⋅x + y - 1⎦, x, y, domain=ℤ, order=grlex⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = expr.fglm('lex')
ascii_str = \
"""\
/[ 2 4 3 2 ] \\\n\
GroebnerBasis\\[2*x - y - y + 1, y + 2*y - 3*y - 16*y + 7], x, y, domain=ZZ, order=lex/\
"""
ucode_str = \
"""\
⎛⎡ 2 4 3 2 ⎤ ⎞\n\
GroebnerBasis⎝⎣2⋅x - y - y + 1, y + 2⋅y - 3⋅y - 16⋅y + 7⎦, x, y, domain=ℤ, order=lex⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_UniversalSet():
assert pretty(S.UniversalSet) == "UniversalSet"
assert upretty(S.UniversalSet) == '𝕌'
def test_pretty_Boolean():
expr = Not(x, evaluate=False)
assert pretty(expr) == "Not(x)"
assert upretty(expr) == "¬x"
expr = And(x, y)
assert pretty(expr) == "And(x, y)"
assert upretty(expr) == "x ∧ y"
expr = Or(x, y)
assert pretty(expr) == "Or(x, y)"
assert upretty(expr) == "x ∨ y"
syms = symbols('a:f')
expr = And(*syms)
assert pretty(expr) == "And(a, b, c, d, e, f)"
assert upretty(expr) == "a ∧ b ∧ c ∧ d ∧ e ∧ f"
expr = Or(*syms)
assert pretty(expr) == "Or(a, b, c, d, e, f)"
assert upretty(expr) == "a ∨ b ∨ c ∨ d ∨ e ∨ f"
expr = Xor(x, y, evaluate=False)
assert pretty(expr) == "Xor(x, y)"
assert upretty(expr) == "x ⊻ y"
expr = Nand(x, y, evaluate=False)
assert pretty(expr) == "Nand(x, y)"
assert upretty(expr) == "x ⊼ y"
expr = Nor(x, y, evaluate=False)
assert pretty(expr) == "Nor(x, y)"
assert upretty(expr) == "x ⊽ y"
expr = Implies(x, y, evaluate=False)
assert pretty(expr) == "Implies(x, y)"
assert upretty(expr) == "x → y"
# don't sort args
expr = Implies(y, x, evaluate=False)
assert pretty(expr) == "Implies(y, x)"
assert upretty(expr) == "y → x"
expr = Equivalent(x, y, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == "x ⇔ y"
expr = Equivalent(y, x, evaluate=False)
assert pretty(expr) == "Equivalent(x, y)"
assert upretty(expr) == "x ⇔ y"
def test_pretty_Domain():
expr = FF(23)
assert pretty(expr) == "GF(23)"
assert upretty(expr) == "ℤ₂₃"
expr = ZZ
assert pretty(expr) == "ZZ"
assert upretty(expr) == "ℤ"
expr = QQ
assert pretty(expr) == "QQ"
assert upretty(expr) == "ℚ"
expr = RR
assert pretty(expr) == "RR"
assert upretty(expr) == "ℝ"
expr = QQ[x]
assert pretty(expr) == "QQ[x]"
assert upretty(expr) == "ℚ[x]"
expr = QQ[x, y]
assert pretty(expr) == "QQ[x, y]"
assert upretty(expr) == "ℚ[x, y]"
expr = ZZ.frac_field(x)
assert pretty(expr) == "ZZ(x)"
assert upretty(expr) == "ℤ(x)"
expr = ZZ.frac_field(x, y)
assert pretty(expr) == "ZZ(x, y)"
assert upretty(expr) == "ℤ(x, y)"
expr = QQ.poly_ring(x, y, order=grlex)
assert pretty(expr) == "QQ[x, y, order=grlex]"
assert upretty(expr) == "ℚ[x, y, order=grlex]"
expr = QQ.poly_ring(x, y, order=ilex)
assert pretty(expr) == "QQ[x, y, order=ilex]"
assert upretty(expr) == "ℚ[x, y, order=ilex]"
def test_pretty_prec():
assert xpretty(S("0.3"), full_prec=True, wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec="auto", wrap_line=False) == "0.300000000000000"
assert xpretty(S("0.3"), full_prec=False, wrap_line=False) == "0.3"
assert xpretty(S("0.3")*x, full_prec=True, use_unicode=False, wrap_line=False) in [
"0.300000000000000*x",
"x*0.300000000000000"
]
assert xpretty(S("0.3")*x, full_prec="auto", use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
assert xpretty(S("0.3")*x, full_prec=False, use_unicode=False, wrap_line=False) in [
"0.3*x",
"x*0.3"
]
def test_pprint():
import sys
from io import StringIO
fd = StringIO()
sso = sys.stdout
sys.stdout = fd
try:
pprint(pi, use_unicode=False, wrap_line=False)
finally:
sys.stdout = sso
assert fd.getvalue() == 'pi\n'
def test_pretty_class():
"""Test that the printer dispatcher correctly handles classes."""
class C:
pass # C has no .__class__ and this was causing problems
class D:
pass
assert pretty( C ) == str( C )
assert pretty( D ) == str( D )
def test_pretty_no_wrap_line():
huge_expr = 0
for i in range(20):
huge_expr += i*sin(i + x)
assert xpretty(huge_expr ).find('\n') != -1
assert xpretty(huge_expr, wrap_line=False).find('\n') == -1
def test_settings():
raises(TypeError, lambda: pretty(S(4), method="garbage"))
def test_pretty_sum():
from sympy.abc import x, a, b, k, m, n
expr = Sum(k**k, (k, 0, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = 0 \
"""
ucode_str = \
"""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**k, (k, oo, n))
ascii_str = \
"""\
n \n\
___ \n\
\\ ` \n\
\\ k\n\
/ k \n\
/__, \n\
k = oo \
"""
ucode_str = \
"""\
n \n\
___ \n\
╲ \n\
╲ k\n\
╱ k \n\
╱ \n\
‾‾‾ \n\
k = ∞ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (k, 0, n**n))
ascii_str = \
"""\
n \n\
n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
"""\
n \n\
n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
"""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(Integral(x**n, (x, -oo, oo))), (
k, x + n + x**2 + n**2 + (x/n) + (1/x), Integral(x**x, (x, -oo, oo))))
ascii_str = \
"""\
oo \n\
/ \n\
| \n\
| x \n\
| x dx \n\
| \n\
/ \n\
-oo \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
2 2 1 x \n\
k = n + n + x + x + - + - \n\
x n \
"""
ucode_str = \
"""\
∞ \n\
⌠ \n\
⎮ x \n\
⎮ x dx \n\
⌡ \n\
-∞ \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
2 2 1 x \n\
k = n + n + x + x + ─ + ─ \n\
x n \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(k**(
Integral(x**n, (x, -oo, oo))), (k, 0, x + n + x**2 + n**2 + (x/n) + (1/x)))
ascii_str = \
"""\
2 2 1 x \n\
n + n + x + x + - + - \n\
x n \n\
______ \n\
\\ ` \n\
\\ oo \n\
\\ / \n\
\\ | \n\
\\ | n \n\
) | x dx\n\
/ | \n\
/ / \n\
/ -oo \n\
/ k \n\
/_____, \n\
k = 0 \
"""
ucode_str = \
"""\
2 2 1 x \n\
n + n + x + x + ─ + ─ \n\
x n \n\
______ \n\
╲ \n\
╲ \n\
╲ ∞ \n\
╲ ⌠ \n\
╲ ⎮ n \n\
╱ ⎮ x dx\n\
╱ ⌡ \n\
╱ -∞ \n\
╱ k \n\
╱ \n\
‾‾‾‾‾‾ \n\
k = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x, (x, 0, oo))
ascii_str = \
"""\
oo \n\
__ \n\
\\ ` \n\
) x\n\
/_, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
___ \n\
╲ \n\
╲ \n\
╱ x\n\
╱ \n\
‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
___ \n\
\\ ` \n\
\\ 2\n\
/ x \n\
/__, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
___ \n\
╲ \n\
╲ 2\n\
╱ x \n\
╱ \n\
‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
___ \n\
\\ ` \n\
\\ x\n\
) -\n\
/ 2\n\
/__, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ \n\
╲ x\n\
╱ ─\n\
╱ 2\n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(x**3/2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 3\n\
\\ x \n\
/ --\n\
/ 2 \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ 3\n\
╲ x \n\
╱ ──\n\
╱ 2 \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum((x**3*y**(x/2))**n, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ n\n\
\\ / x\\ \n\
) | -| \n\
/ | 3 2| \n\
/ \\x *y / \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
_____ \n\
╲ \n\
╲ \n\
╲ n\n\
╲ ⎛ x⎞ \n\
╱ ⎜ ─⎟ \n\
╱ ⎜ 3 2⎟ \n\
╱ ⎝x ⋅y ⎠ \n\
╱ \n\
‾‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/x**2, (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ 1 \n\
\\ --\n\
/ 2\n\
/ x \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ 1 \n\
╲ ──\n\
╱ 2\n\
╱ x \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo))
ascii_str = \
"""\
oo \n\
____ \n\
\\ ` \n\
\\ -a \n\
\\ ---\n\
/ b \n\
/ y \n\
/___, \n\
x = 0 \
"""
ucode_str = \
"""\
∞ \n\
____ \n\
╲ \n\
╲ -a \n\
╲ ───\n\
╱ b \n\
╱ y \n\
╱ \n\
‾‾‾‾ \n\
x = 0 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Sum(1/y**(a/b), (x, 0, oo), (y, 1, 2))
ascii_str = \
"""\
2 oo \n\
____ ____ \n\
\\ ` \\ ` \n\
\\ \\ -a\n\
\\ \\ --\n\
/ / b \n\
/ / y \n\
/___, /___, \n\
y = 1 x = 0 \
"""
ucode_str = \
"""\
2 ∞ \n\
____ ____ \n\
╲ ╲ \n\
╲ ╲ -a\n\
╲ ╲ ──\n\
╱ ╱ b \n\
╱ ╱ y \n\
╱ ╱ \n\
‾‾‾‾ ‾‾‾‾ \n\
y = 1 x = 0 \
"""
expr = Sum(1/(1 + 1/(
1 + 1/k)) + 1, (k, 111, 1 + 1/n), (k, 1/(1 + m), oo)) + 1/(1 + 1/k)
ascii_str = \
"""\
1 \n\
1 + - \n\
oo n \n\
_____ _____ \n\
\\ ` \\ ` \n\
\\ \\ / 1 \\ \n\
\\ \\ |1 + ---------| \n\
\\ \\ | 1 | 1 \n\
) ) | 1 + -----| + -----\n\
/ / | 1| 1\n\
/ / | 1 + -| 1 + -\n\
/ / \\ k/ k\n\
/____, /____, \n\
1 k = 111 \n\
k = ----- \n\
m + 1 \
"""
ucode_str = \
"""\
1 \n\
1 + ─ \n\
∞ n \n\
______ ______ \n\
╲ ╲ \n\
╲ ╲ \n\
╲ ╲ ⎛ 1 ⎞ \n\
╲ ╲ ⎜1 + ─────────⎟ \n\
╲ ╲ ⎜ 1 ⎟ 1 \n\
╱ ╱ ⎜ 1 + ─────⎟ + ─────\n\
╱ ╱ ⎜ 1⎟ 1\n\
╱ ╱ ⎜ 1 + ─⎟ 1 + ─\n\
╱ ╱ ⎝ k⎠ k\n\
╱ ╱ \n\
‾‾‾‾‾‾ ‾‾‾‾‾‾ \n\
1 k = 111 \n\
k = ───── \n\
m + 1 \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_units():
expr = joule
ascii_str1 = \
"""\
2\n\
kilogram*meter \n\
---------------\n\
2 \n\
second \
"""
unicode_str1 = \
"""\
2\n\
kilogram⋅meter \n\
───────────────\n\
2 \n\
second \
"""
ascii_str2 = \
"""\
2\n\
3*x*y*kilogram*meter \n\
---------------------\n\
2 \n\
second \
"""
unicode_str2 = \
"""\
2\n\
3⋅x⋅y⋅kilogram⋅meter \n\
─────────────────────\n\
2 \n\
second \
"""
from sympy.physics.units import kg, m, s
assert upretty(expr) == "joule"
assert pretty(expr) == "joule"
assert upretty(expr.convert_to(kg*m**2/s**2)) == unicode_str1
assert pretty(expr.convert_to(kg*m**2/s**2)) == ascii_str1
assert upretty(3*kg*x*m**2*y/s**2) == unicode_str2
assert pretty(3*kg*x*m**2*y/s**2) == ascii_str2
def test_pretty_Subs():
f = Function('f')
expr = Subs(f(x), x, ph**2)
ascii_str = \
"""\
(f(x))| 2\n\
|x=phi \
"""
unicode_str = \
"""\
(f(x))│ 2\n\
│x=φ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x), x, 0)
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
\\dx /|x=0\
"""
unicode_str = \
"""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎝dx ⎠│x=0\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
expr = Subs(f(x).diff(x)/y, (x, y), (0, Rational(1, 2)))
ascii_str = \
"""\
/d \\| \n\
|--(f(x))|| \n\
|dx || \n\
|--------|| \n\
\\ y /|x=0, y=1/2\
"""
unicode_str = \
"""\
⎛d ⎞│ \n\
⎜──(f(x))⎟│ \n\
⎜dx ⎟│ \n\
⎜────────⎟│ \n\
⎝ y ⎠│x=0, y=1/2\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == unicode_str
def test_gammas():
assert upretty(lowergamma(x, y)) == "γ(x, y)"
assert upretty(uppergamma(x, y)) == "Γ(x, y)"
assert xpretty(gamma(x), use_unicode=True) == 'Γ(x)'
assert xpretty(gamma, use_unicode=True) == 'Γ'
assert xpretty(symbols('gamma', cls=Function)(x), use_unicode=True) == 'γ(x)'
assert xpretty(symbols('gamma', cls=Function), use_unicode=True) == 'γ'
def test_beta():
assert xpretty(beta(x,y), use_unicode=True) == 'Β(x, y)'
assert xpretty(beta(x,y), use_unicode=False) == 'B(x, y)'
assert xpretty(beta, use_unicode=True) == 'Β'
assert xpretty(beta, use_unicode=False) == 'B'
mybeta = Function('beta')
assert xpretty(mybeta(x), use_unicode=True) == 'β(x)'
assert xpretty(mybeta(x, y, z), use_unicode=False) == 'beta(x, y, z)'
assert xpretty(mybeta, use_unicode=True) == 'β'
# test that notation passes to subclasses of the same name only
def test_function_subclass_different_name():
class mygamma(gamma):
pass
assert xpretty(mygamma, use_unicode=True) == r"mygamma"
assert xpretty(mygamma(x), use_unicode=True) == r"mygamma(x)"
def test_SingularityFunction():
assert xpretty(SingularityFunction(x, 0, n), use_unicode=True) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=True) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=True) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=True) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=True) == (
"""\
n\n\
<x - y> \
""")
assert xpretty(SingularityFunction(x, 0, n), use_unicode=False) == (
"""\
n\n\
<x> \
""")
assert xpretty(SingularityFunction(x, 1, n), use_unicode=False) == (
"""\
n\n\
<x - 1> \
""")
assert xpretty(SingularityFunction(x, -1, n), use_unicode=False) == (
"""\
n\n\
<x + 1> \
""")
assert xpretty(SingularityFunction(x, a, n), use_unicode=False) == (
"""\
n\n\
<-a + x> \
""")
assert xpretty(SingularityFunction(x, y, n), use_unicode=False) == (
"""\
n\n\
<x - y> \
""")
def test_deltas():
assert xpretty(DiracDelta(x), use_unicode=True) == 'δ(x)'
assert xpretty(DiracDelta(x, 1), use_unicode=True) == \
"""\
(1) \n\
δ (x)\
"""
assert xpretty(x*DiracDelta(x, 1), use_unicode=True) == \
"""\
(1) \n\
x⋅δ (x)\
"""
def test_hyper():
expr = hyper((), (), z)
ucode_str = \
"""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ z⎟\n\
0╵ 0 ⎝ │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | z|\n\
0 0 \\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((), (1,), x)
ucode_str = \
"""\
┌─ ⎛ │ ⎞\n\
├─ ⎜ │ x⎟\n\
0╵ 1 ⎝1 │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ / | \\\n\
| | | x|\n\
0 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([2], [1], x)
ucode_str = \
"""\
┌─ ⎛2 │ ⎞\n\
├─ ⎜ │ x⎟\n\
1╵ 1 ⎝1 │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ /2 | \\\n\
| | | x|\n\
1 1 \\1 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi/3, -2*k), (3, 4, 5, -3), x)
ucode_str = \
"""\
⎛ π │ ⎞\n\
┌─ ⎜ ─, -2⋅k │ ⎟\n\
├─ ⎜ 3 │ x⎟\n\
2╵ 4 ⎜ │ ⎟\n\
⎝3, 4, 5, -3 │ ⎠\
"""
ascii_str = \
"""\
\n\
_ / pi | \\\n\
|_ | --, -2*k | |\n\
| | 3 | x|\n\
2 4 | | |\n\
\\3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper((pi, S('2/3'), -2*k), (3, 4, 5, -3), x**2)
ucode_str = \
"""\
┌─ ⎛π, 2/3, -2⋅k │ 2⎞\n\
├─ ⎜ │ x ⎟\n\
3╵ 4 ⎝3, 4, 5, -3 │ ⎠\
"""
ascii_str = \
"""\
_ \n\
|_ /pi, 2/3, -2*k | 2\\\n\
| | | x |\n\
3 4 \\ 3, 4, 5, -3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hyper([1, 2], [3, 4], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
┌─ ⎜1, 2 │ 1 + ─────────⎟\n\
├─ ⎜ │ 1 ⎟\n\
2╵ 2 ⎜3, 4 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
"""
ascii_str = \
"""\
\n\
/ | 1 \\\n\
| | -------------|\n\
_ | | 1 |\n\
|_ |1, 2 | 1 + ---------|\n\
| | | 1 |\n\
2 2 |3, 4 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_meijerg():
expr = meijerg([pi, pi, x], [1], [0, 1], [1, 2, 3], z)
ucode_str = \
"""\
╭─╮2, 3 ⎛π, π, x 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯4, 5 ⎝ 0, 1 1, 2, 3 │ ⎠\
"""
ascii_str = \
"""\
__2, 3 /pi, pi, x 1 | \\\n\
/__ | | z|\n\
\\_|4, 5 \\ 0, 1 1, 2, 3 | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, pi/7], [2, pi, 5], [], [], z**2)
ucode_str = \
"""\
⎛ π │ ⎞\n\
╭─╮0, 2 ⎜1, ─ 2, π, 5 │ 2⎟\n\
│╶┐ ⎜ 7 │ z ⎟\n\
╰─╯5, 0 ⎜ │ ⎟\n\
⎝ │ ⎠\
"""
ascii_str = \
"""\
/ pi | \\\n\
__0, 2 |1, -- 2, pi, 5 | 2|\n\
/__ | 7 | z |\n\
\\_|5, 0 | | |\n\
\\ | /\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ucode_str = \
"""\
╭─╮ 1, 10 ⎛1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 │ ⎞\n\
│╶┐ ⎜ │ z⎟\n\
╰─╯11, 2 ⎝ 1 1 │ ⎠\
"""
ascii_str = \
"""\
__ 1, 10 /1, 1, 1, 1, 1, 1, 1, 1, 1, 1 1 | \\\n\
/__ | | z|\n\
\\_|11, 2 \\ 1 1 | /\
"""
expr = meijerg([1]*10, [1], [1], [1], z)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = meijerg([1, 2, ], [4, 3], [3], [4, 5], 1/(1/(1/(1/x + 1) + 1) + 1))
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
⎜ │ ─────────────⎟\n\
⎜ │ 1 ⎟\n\
╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟\n\
│╶┐ ⎜ │ 1 ⎟\n\
╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟\n\
⎜ │ 1⎟\n\
⎜ │ 1 + ─⎟\n\
⎝ │ x⎠\
"""
ascii_str = \
"""\
/ | 1 \\\n\
| | -------------|\n\
| | 1 |\n\
__1, 2 |1, 2 4, 3 | 1 + ---------|\n\
/__ | | 1 |\n\
\\_|4, 3 | 3 4, 5 | 1 + -----|\n\
| | 1|\n\
| | 1 + -|\n\
\\ | x/\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = Integral(expr, x)
ucode_str = \
"""\
⌠ \n\
⎮ ⎛ │ 1 ⎞ \n\
⎮ ⎜ │ ─────────────⎟ \n\
⎮ ⎜ │ 1 ⎟ \n\
⎮ ╭─╮1, 2 ⎜1, 2 4, 3 │ 1 + ─────────⎟ \n\
⎮ │╶┐ ⎜ │ 1 ⎟ dx\n\
⎮ ╰─╯4, 3 ⎜ 3 4, 5 │ 1 + ─────⎟ \n\
⎮ ⎜ │ 1⎟ \n\
⎮ ⎜ │ 1 + ─⎟ \n\
⎮ ⎝ │ x⎠ \n\
⌡ \
"""
ascii_str = \
"""\
/ \n\
| \n\
| / | 1 \\ \n\
| | | -------------| \n\
| | | 1 | \n\
| __1, 2 |1, 2 4, 3 | 1 + ---------| \n\
| /__ | | 1 | dx\n\
| \\_|4, 3 | 3 4, 5 | 1 + -----| \n\
| | | 1| \n\
| | | 1 + -| \n\
| \\ | x/ \n\
| \n\
/ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_noncommutative():
A, B, C = symbols('A,B,C', commutative=False)
expr = A*B*C**-1
ascii_str = \
"""\
-1\n\
A*B*C \
"""
ucode_str = \
"""\
-1\n\
A⋅B⋅C \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = C**-1*A*B
ascii_str = \
"""\
-1 \n\
C *A*B\
"""
ucode_str = \
"""\
-1 \n\
C ⋅A⋅B\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B
ascii_str = \
"""\
-1 \n\
A*C *B\
"""
ucode_str = \
"""\
-1 \n\
A⋅C ⋅B\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A*C**-1*B/x
ascii_str = \
"""\
-1 \n\
A*C *B\n\
-------\n\
x \
"""
ucode_str = \
"""\
-1 \n\
A⋅C ⋅B\n\
───────\n\
x \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_special_functions():
x, y = symbols("x y")
# atan2
expr = atan2(y/sqrt(200), sqrt(x))
ascii_str = \
"""\
/ ___ \\\n\
|\\/ 2 *y ___|\n\
atan2|-------, \\/ x |\n\
\\ 20 /\
"""
ucode_str = \
"""\
⎛√2⋅y ⎞\n\
atan2⎜────, √x⎟\n\
⎝ 20 ⎠\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_geometry():
e = Segment((0, 1), (0, 2))
assert pretty(e) == 'Segment2D(Point2D(0, 1), Point2D(0, 2))'
e = Ray((1, 1), angle=4.02*pi)
assert pretty(e) == 'Ray2D(Point2D(1, 1), Point2D(2, tan(pi/50) + 1))'
def test_expint():
expr = Ei(x)
string = 'Ei(x)'
assert pretty(expr) == string
assert upretty(expr) == string
expr = expint(1, z)
ucode_str = "E₁(z)"
ascii_str = "expint(1, z)"
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
assert pretty(Shi(x)) == 'Shi(x)'
assert pretty(Si(x)) == 'Si(x)'
assert pretty(Ci(x)) == 'Ci(x)'
assert pretty(Chi(x)) == 'Chi(x)'
assert upretty(Shi(x)) == 'Shi(x)'
assert upretty(Si(x)) == 'Si(x)'
assert upretty(Ci(x)) == 'Ci(x)'
assert upretty(Chi(x)) == 'Chi(x)'
def test_elliptic_functions():
ascii_str = \
"""\
/ 1 \\\n\
K|-----|\n\
\\z + 1/\
"""
ucode_str = \
"""\
⎛ 1 ⎞\n\
K⎜─────⎟\n\
⎝z + 1⎠\
"""
expr = elliptic_k(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
F|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
F⎜1│─────⎟\n\
⎝ │z + 1⎠\
"""
expr = elliptic_f(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 1 \\\n\
E|-----|\n\
\\z + 1/\
"""
ucode_str = \
"""\
⎛ 1 ⎞\n\
E⎜─────⎟\n\
⎝z + 1⎠\
"""
expr = elliptic_e(1/(z + 1))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ | 1 \\\n\
E|1|-----|\n\
\\ |z + 1/\
"""
ucode_str = \
"""\
⎛ │ 1 ⎞\n\
E⎜1│─────⎟\n\
⎝ │z + 1⎠\
"""
expr = elliptic_e(1, 1/(1 + z))
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ |4\\\n\
Pi|3|-|\n\
\\ |x/\
"""
ucode_str = \
"""\
⎛ │4⎞\n\
Π⎜3│─⎟\n\
⎝ │x⎠\
"""
expr = elliptic_pi(3, 4/x)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
ascii_str = \
"""\
/ 4| \\\n\
Pi|3; -|6|\n\
\\ x| /\
"""
ucode_str = \
"""\
⎛ 4│ ⎞\n\
Π⎜3; ─│6⎟\n\
⎝ x│ ⎠\
"""
expr = elliptic_pi(3, 4/x, 6)
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_RandomDomain():
from sympy.stats import Normal, Die, Exponential, pspace, where
X = Normal('x1', 0, 1)
assert upretty(where(X > 0)) == "Domain: 0 < x₁ ∧ x₁ < ∞"
D = Die('d1', 6)
assert upretty(where(D > 4)) == 'Domain: d₁ = 5 ∨ d₁ = 6'
A = Exponential('a', 1)
B = Exponential('b', 1)
assert upretty(pspace(Tuple(A, B)).domain) == \
'Domain: 0 ≤ a ∧ 0 ≤ b ∧ a < ∞ ∧ b < ∞'
def test_PrettyPoly():
F = QQ.frac_field(x, y)
R = QQ.poly_ring(x, y)
expr = F.convert(x/(x + y))
assert pretty(expr) == "x/(x + y)"
assert upretty(expr) == "x/(x + y)"
expr = R.convert(x + y)
assert pretty(expr) == "x + y"
assert upretty(expr) == "x + y"
def test_issue_6285():
assert pretty(Pow(2, -5, evaluate=False)) == '1 \n--\n 5\n2 '
assert pretty(Pow(x, (1/pi))) == \
' 1 \n'\
' --\n'\
' pi\n'\
'x '
def test_issue_6359():
assert pretty(Integral(x**2, x)**2) == \
"""\
2
/ / \\ \n\
| | | \n\
| | 2 | \n\
| | x dx| \n\
| | | \n\
\\/ / \
"""
assert upretty(Integral(x**2, x)**2) == \
"""\
2
⎛⌠ ⎞ \n\
⎜⎮ 2 ⎟ \n\
⎜⎮ x dx⎟ \n\
⎝⌡ ⎠ \
"""
assert pretty(Sum(x**2, (x, 0, 1))**2) == \
"""\
2
/ 1 \\ \n\
| ___ | \n\
| \\ ` | \n\
| \\ 2| \n\
| / x | \n\
| /__, | \n\
\\x = 0 / \
"""
assert upretty(Sum(x**2, (x, 0, 1))**2) == \
"""\
2
⎛ 1 ⎞ \n\
⎜ ___ ⎟ \n\
⎜ ╲ ⎟ \n\
⎜ ╲ 2⎟ \n\
⎜ ╱ x ⎟ \n\
⎜ ╱ ⎟ \n\
⎜ ‾‾‾ ⎟ \n\
⎝x = 0 ⎠ \
"""
assert pretty(Product(x**2, (x, 1, 2))**2) == \
"""\
2
/ 2 \\ \n\
|______ | \n\
| | | 2| \n\
| | | x | \n\
| | | | \n\
\\x = 1 / \
"""
assert upretty(Product(x**2, (x, 1, 2))**2) == \
"""\
2
⎛ 2 ⎞ \n\
⎜─┬──┬─ ⎟ \n\
⎜ │ │ 2⎟ \n\
⎜ │ │ x ⎟ \n\
⎜ │ │ ⎟ \n\
⎝x = 1 ⎠ \
"""
f = Function('f')
assert pretty(Derivative(f(x), x)**2) == \
"""\
2
/d \\ \n\
|--(f(x))| \n\
\\dx / \
"""
assert upretty(Derivative(f(x), x)**2) == \
"""\
2
⎛d ⎞ \n\
⎜──(f(x))⎟ \n\
⎝dx ⎠ \
"""
def test_issue_6739():
ascii_str = \
"""\
1 \n\
-----\n\
___\n\
\\/ x \
"""
ucode_str = \
"""\
1 \n\
──\n\
√x\
"""
assert pretty(1/sqrt(x)) == ascii_str
assert upretty(1/sqrt(x)) == ucode_str
def test_complicated_symbol_unchanged():
for symb_name in ["dexpr2_d1tau", "dexpr2^d1tau"]:
assert pretty(Symbol(symb_name)) == symb_name
def test_categories():
from sympy.categories import (Object, IdentityMorphism,
NamedMorphism, Category, Diagram, DiagramGrid)
A1 = Object("A1")
A2 = Object("A2")
A3 = Object("A3")
f1 = NamedMorphism(A1, A2, "f1")
f2 = NamedMorphism(A2, A3, "f2")
id_A1 = IdentityMorphism(A1)
K1 = Category("K1")
assert pretty(A1) == "A1"
assert upretty(A1) == "A₁"
assert pretty(f1) == "f1:A1-->A2"
assert upretty(f1) == "f₁:A₁——▶A₂"
assert pretty(id_A1) == "id:A1-->A1"
assert upretty(id_A1) == "id:A₁——▶A₁"
assert pretty(f2*f1) == "f2*f1:A1-->A3"
assert upretty(f2*f1) == "f₂∘f₁:A₁——▶A₃"
assert pretty(K1) == "K1"
assert upretty(K1) == "K₁"
# Test how diagrams are printed.
d = Diagram()
assert pretty(d) == "EmptySet"
assert upretty(d) == "∅"
d = Diagram({f1: "unique", f2: S.EmptySet})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \
"EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \
"EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}"
assert upretty(d) == "{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, " \
"id:A₂——▶A₂: ∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}"
d = Diagram({f1: "unique", f2: S.EmptySet}, {f2 * f1: "unique"})
assert pretty(d) == "{f2*f1:A1-->A3: EmptySet, id:A1-->A1: " \
"EmptySet, id:A2-->A2: EmptySet, id:A3-->A3: " \
"EmptySet, f1:A1-->A2: {unique}, f2:A2-->A3: EmptySet}" \
" ==> {f2*f1:A1-->A3: {unique}}"
assert upretty(d) == "{f₂∘f₁:A₁——▶A₃: ∅, id:A₁——▶A₁: ∅, id:A₂——▶A₂: " \
"∅, id:A₃——▶A₃: ∅, f₁:A₁——▶A₂: {unique}, f₂:A₂——▶A₃: ∅}" \
" ══▶ {f₂∘f₁:A₁——▶A₃: {unique}}"
grid = DiagramGrid(d)
assert pretty(grid) == "A1 A2\n \nA3 "
assert upretty(grid) == "A₁ A₂\n \nA₃ "
def test_PrettyModules():
R = QQ.old_poly_ring(x, y)
F = R.free_module(2)
M = F.submodule([x, y], [1, x**2])
ucode_str = \
"""\
2\n\
ℚ[x, y] \
"""
ascii_str = \
"""\
2\n\
QQ[x, y] \
"""
assert upretty(F) == ucode_str
assert pretty(F) == ascii_str
ucode_str = \
"""\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
"""
ascii_str = \
"""\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(M) == ucode_str
assert pretty(M) == ascii_str
I = R.ideal(x**2, y)
ucode_str = \
"""\
╱ 2 ╲\n\
╲x , y╱\
"""
ascii_str = \
"""\
2 \n\
<x , y>\
"""
assert upretty(I) == ucode_str
assert pretty(I) == ascii_str
Q = F / M
ucode_str = \
"""\
2 \n\
ℚ[x, y] \n\
─────────────────\n\
╱ ⎡ 2⎤╲\n\
╲[x, y], ⎣1, x ⎦╱\
"""
ascii_str = \
"""\
2 \n\
QQ[x, y] \n\
-----------------\n\
2 \n\
<[x, y], [1, x ]>\
"""
assert upretty(Q) == ucode_str
assert pretty(Q) == ascii_str
ucode_str = \
"""\
╱⎡ 3⎤ ╲\n\
│⎢ x ⎥ ╱ ⎡ 2⎤╲ ╱ ⎡ 2⎤╲│\n\
│⎢1, ──⎥ + ╲[x, y], ⎣1, x ⎦╱, [2, y] + ╲[x, y], ⎣1, x ⎦╱│\n\
╲⎣ 2 ⎦ ╱\
"""
ascii_str = \
"""\
3 \n\
x 2 2 \n\
<[1, --] + <[x, y], [1, x ]>, [2, y] + <[x, y], [1, x ]>>\n\
2 \
"""
def test_QuotientRing():
R = QQ.old_poly_ring(x)/[x**2 + 1]
ucode_str = \
"""\
ℚ[x] \n\
────────\n\
╱ 2 ╲\n\
╲x + 1╱\
"""
ascii_str = \
"""\
QQ[x] \n\
--------\n\
2 \n\
<x + 1>\
"""
assert upretty(R) == ucode_str
assert pretty(R) == ascii_str
ucode_str = \
"""\
╱ 2 ╲\n\
1 + ╲x + 1╱\
"""
ascii_str = \
"""\
2 \n\
1 + <x + 1>\
"""
assert upretty(R.one) == ucode_str
assert pretty(R.one) == ascii_str
def test_Homomorphism():
from sympy.polys.agca import homomorphism
R = QQ.old_poly_ring(x)
expr = homomorphism(R.free_module(1), R.free_module(1), [0])
ucode_str = \
"""\
1 1\n\
[0] : ℚ[x] ──> ℚ[x] \
"""
ascii_str = \
"""\
1 1\n\
[0] : QQ[x] --> QQ[x] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(2), R.free_module(2), [0, 0])
ucode_str = \
"""\
⎡0 0⎤ 2 2\n\
⎢ ⎥ : ℚ[x] ──> ℚ[x] \n\
⎣0 0⎦ \
"""
ascii_str = \
"""\
[0 0] 2 2\n\
[ ] : QQ[x] --> QQ[x] \n\
[0 0] \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
expr = homomorphism(R.free_module(1), R.free_module(1) / [[x]], [0])
ucode_str = \
"""\
1\n\
1 ℚ[x] \n\
[0] : ℚ[x] ──> ─────\n\
<[x]>\
"""
ascii_str = \
"""\
1\n\
1 QQ[x] \n\
[0] : QQ[x] --> ------\n\
<[x]> \
"""
assert upretty(expr) == ucode_str
assert pretty(expr) == ascii_str
def test_Tr():
A, B = symbols('A B', commutative=False)
t = Tr(A*B)
assert pretty(t) == r'Tr(A*B)'
assert upretty(t) == 'Tr(A⋅B)'
def test_pretty_Add():
eq = Mul(-2, x - 2, evaluate=False) + 5
assert pretty(eq) == '5 - 2*(x - 2)'
def test_issue_7179():
assert upretty(Not(Equivalent(x, y))) == 'x ⇎ y'
assert upretty(Not(Implies(x, y))) == 'x ↛ y'
def test_issue_7180():
assert upretty(Equivalent(x, y)) == 'x ⇔ y'
def test_pretty_Complement():
assert pretty(S.Reals - S.Naturals) == '(-oo, oo) \\ Naturals'
assert upretty(S.Reals - S.Naturals) == 'ℝ \\ ℕ'
assert pretty(S.Reals - S.Naturals0) == '(-oo, oo) \\ Naturals0'
assert upretty(S.Reals - S.Naturals0) == 'ℝ \\ ℕ₀'
def test_pretty_SymmetricDifference():
from sympy.sets.sets import SymmetricDifference
assert upretty(SymmetricDifference(Interval(2,3), Interval(3,5), \
evaluate = False)) == '[2, 3] ∆ [3, 5]'
with raises(NotImplementedError):
pretty(SymmetricDifference(Interval(2,3), Interval(3,5), evaluate = False))
def test_pretty_Contains():
assert pretty(Contains(x, S.Integers)) == 'Contains(x, Integers)'
assert upretty(Contains(x, S.Integers)) == 'x ∈ ℤ'
def test_issue_8292():
from sympy.core import sympify
e = sympify('((x+x**4)/(x-1))-(2*(x-1)**4/(x-1)**4)', evaluate=False)
ucode_str = \
"""\
4 4 \n\
2⋅(x - 1) x + x\n\
- ────────── + ──────\n\
4 x - 1 \n\
(x - 1) \
"""
ascii_str = \
"""\
4 4 \n\
2*(x - 1) x + x\n\
- ---------- + ------\n\
4 x - 1 \n\
(x - 1) \
"""
assert pretty(e) == ascii_str
assert upretty(e) == ucode_str
def test_issue_4335():
y = Function('y')
expr = -y(x).diff(x)
ucode_str = \
"""\
d \n\
-──(y(x))\n\
dx \
"""
ascii_str = \
"""\
d \n\
- --(y(x))\n\
dx \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_8344():
from sympy.core import sympify
e = sympify('2*x*y**2/1**2 + 1', evaluate=False)
ucode_str = \
"""\
2 \n\
2⋅x⋅y \n\
────── + 1\n\
2 \n\
1 \
"""
assert upretty(e) == ucode_str
def test_issue_6324():
x = Pow(2, 3, evaluate=False)
y = Pow(10, -2, evaluate=False)
e = Mul(x, y, evaluate=False)
ucode_str = \
"""\
3\n\
2 \n\
───\n\
2\n\
10 \
"""
assert upretty(e) == ucode_str
def test_issue_7927():
e = sin(x/2)**cos(x/2)
ucode_str = \
"""\
⎛x⎞\n\
cos⎜─⎟\n\
⎝2⎠\n\
⎛ ⎛x⎞⎞ \n\
⎜sin⎜─⎟⎟ \n\
⎝ ⎝2⎠⎠ \
"""
assert upretty(e) == ucode_str
e = sin(x)**(S(11)/13)
ucode_str = \
"""\
11\n\
──\n\
13\n\
(sin(x)) \
"""
assert upretty(e) == ucode_str
def test_issue_6134():
from sympy.abc import lamda, t
phi = Function('phi')
e = lamda*x*Integral(phi(t)*pi*sin(pi*t), (t, 0, 1)) + lamda*x**2*Integral(phi(t)*2*pi*sin(2*pi*t), (t, 0, 1))
ucode_str = \
"""\
1 1 \n\
2 ⌠ ⌠ \n\
λ⋅x ⋅⎮ 2⋅π⋅φ(t)⋅sin(2⋅π⋅t) dt + λ⋅x⋅⎮ π⋅φ(t)⋅sin(π⋅t) dt\n\
⌡ ⌡ \n\
0 0 \
"""
assert upretty(e) == ucode_str
def test_issue_9877():
ucode_str1 = '(2, 3) ∪ ([1, 2] \\ {x})'
a, b, c = Interval(2, 3, True, True), Interval(1, 2), FiniteSet(x)
assert upretty(Union(a, Complement(b, c))) == ucode_str1
ucode_str2 = '{x} ∩ {y} ∩ ({z} \\ [1, 2])'
d, e, f, g = FiniteSet(x), FiniteSet(y), FiniteSet(z), Interval(1, 2)
assert upretty(Intersection(d, e, Complement(f, g))) == ucode_str2
def test_issue_13651():
expr1 = c + Mul(-1, a + b, evaluate=False)
assert pretty(expr1) == 'c - (a + b)'
expr2 = c + Mul(-1, a - b + d, evaluate=False)
assert pretty(expr2) == 'c - (a - b + d)'
def test_pretty_primenu():
from sympy.ntheory.factor_ import primenu
ascii_str1 = "nu(n)"
ucode_str1 = "ν(n)"
n = symbols('n', integer=True)
assert pretty(primenu(n)) == ascii_str1
assert upretty(primenu(n)) == ucode_str1
def test_pretty_primeomega():
from sympy.ntheory.factor_ import primeomega
ascii_str1 = "Omega(n)"
ucode_str1 = "Ω(n)"
n = symbols('n', integer=True)
assert pretty(primeomega(n)) == ascii_str1
assert upretty(primeomega(n)) == ucode_str1
def test_pretty_Mod():
from sympy.core import Mod
ascii_str1 = "x mod 7"
ucode_str1 = "x mod 7"
ascii_str2 = "(x + 1) mod 7"
ucode_str2 = "(x + 1) mod 7"
ascii_str3 = "2*x mod 7"
ucode_str3 = "2⋅x mod 7"
ascii_str4 = "(x mod 7) + 1"
ucode_str4 = "(x mod 7) + 1"
ascii_str5 = "2*(x mod 7)"
ucode_str5 = "2⋅(x mod 7)"
x = symbols('x', integer=True)
assert pretty(Mod(x, 7)) == ascii_str1
assert upretty(Mod(x, 7)) == ucode_str1
assert pretty(Mod(x + 1, 7)) == ascii_str2
assert upretty(Mod(x + 1, 7)) == ucode_str2
assert pretty(Mod(2 * x, 7)) == ascii_str3
assert upretty(Mod(2 * x, 7)) == ucode_str3
assert pretty(Mod(x, 7) + 1) == ascii_str4
assert upretty(Mod(x, 7) + 1) == ucode_str4
assert pretty(2 * Mod(x, 7)) == ascii_str5
assert upretty(2 * Mod(x, 7)) == ucode_str5
def test_issue_11801():
assert pretty(Symbol("")) == ""
assert upretty(Symbol("")) == ""
def test_pretty_UnevaluatedExpr():
x = symbols('x')
he = UnevaluatedExpr(1/x)
ucode_str = \
"""\
1\n\
─\n\
x\
"""
assert upretty(he) == ucode_str
ucode_str = \
"""\
2\n\
⎛1⎞ \n\
⎜─⎟ \n\
⎝x⎠ \
"""
assert upretty(he**2) == ucode_str
ucode_str = \
"""\
1\n\
1 + ─\n\
x\
"""
assert upretty(he + 1) == ucode_str
ucode_str = \
('''\
1\n\
x⋅─\n\
x\
''')
assert upretty(x*he) == ucode_str
def test_issue_10472():
M = (Matrix([[0, 0], [0, 0]]), Matrix([0, 0]))
ucode_str = \
"""\
⎛⎡0 0⎤ ⎡0⎤⎞
⎜⎢ ⎥, ⎢ ⎥⎟
⎝⎣0 0⎦ ⎣0⎦⎠\
"""
assert upretty(M) == ucode_str
def test_MatrixElement_printing():
# test cases for issue #11821
A = MatrixSymbol("A", 1, 3)
B = MatrixSymbol("B", 1, 3)
C = MatrixSymbol("C", 1, 3)
ascii_str1 = "A_00"
ucode_str1 = "A₀₀"
assert pretty(A[0, 0]) == ascii_str1
assert upretty(A[0, 0]) == ucode_str1
ascii_str1 = "3*A_00"
ucode_str1 = "3⋅A₀₀"
assert pretty(3*A[0, 0]) == ascii_str1
assert upretty(3*A[0, 0]) == ucode_str1
ascii_str1 = "(-B + A)[0, 0]"
ucode_str1 = "(-B + A)[0, 0]"
F = C[0, 0].subs(C, A - B)
assert pretty(F) == ascii_str1
assert upretty(F) == ucode_str1
def test_issue_12675():
x, y, t, j = symbols('x y t j')
e = CoordSys3D('e')
ucode_str = \
"""\
⎛ t⎞ \n\
⎜⎛x⎞ ⎟ j_e\n\
⎜⎜─⎟ ⎟ \n\
⎝⎝y⎠ ⎠ \
"""
assert upretty((x/y)**t*e.j) == ucode_str
ucode_str = \
"""\
⎛1⎞ \n\
⎜─⎟ j_e\n\
⎝y⎠ \
"""
assert upretty((1/y)*e.j) == ucode_str
def test_MatrixSymbol_printing():
# test cases for issue #14237
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert pretty(-A*B*C) == "-A*B*C"
assert pretty(A - B) == "-B + A"
assert pretty(A*B*C - A*B - B*C) == "-A*B -B*C + A*B*C"
# issue #14814
x = MatrixSymbol('x', n, n)
y = MatrixSymbol('y*', n, n)
assert pretty(x + y) == "x + y*"
ascii_str = \
"""\
2 \n\
-2*y* -a*x\
"""
assert pretty(-a*x + -2*y*y) == ascii_str
def test_degree_printing():
expr1 = 90*degree
assert pretty(expr1) == '90°'
expr2 = x*degree
assert pretty(expr2) == 'x°'
expr3 = cos(x*degree + 90*degree)
assert pretty(expr3) == 'cos(x° + 90°)'
def test_vector_expr_pretty_printing():
A = CoordSys3D('A')
assert upretty(Cross(A.i, A.x*A.i+3*A.y*A.j)) == "(i_A)×((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(x*Cross(A.i, A.j)) == 'x⋅(i_A)×(j_A)'
assert upretty(Curl(A.x*A.i + 3*A.y*A.j)) == "∇×((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(Divergence(A.x*A.i + 3*A.y*A.j)) == "∇⋅((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(Dot(A.i, A.x*A.i+3*A.y*A.j)) == "(i_A)⋅((x_A) i_A + (3⋅y_A) j_A)"
assert upretty(Gradient(A.x+3*A.y)) == "∇(x_A + 3⋅y_A)"
assert upretty(Laplacian(A.x+3*A.y)) == "∆(x_A + 3⋅y_A)"
# TODO: add support for ASCII pretty.
def test_pretty_print_tensor_expr():
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
i0 = tensor_indices("i_0", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
expr = -i
ascii_str = \
"""\
-i\
"""
ucode_str = \
"""\
-i\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)
ascii_str = \
"""\
i\n\
A \n\
\
"""
ucode_str = \
"""\
i\n\
A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i0)
ascii_str = \
"""\
i_0\n\
A \n\
\
"""
ucode_str = \
"""\
i₀\n\
A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(-i)
ascii_str = \
"""\
\n\
A \n\
i\
"""
ucode_str = \
"""\
\n\
A \n\
i\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = -3*A(-i)
ascii_str = \
"""\
\n\
-3*A \n\
i\
"""
ucode_str = \
"""\
\n\
-3⋅A \n\
i\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)
ascii_str = \
"""\
i \n\
H \n\
j\
"""
ucode_str = \
"""\
i \n\
H \n\
j\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -i)
ascii_str = \
"""\
L_0 \n\
H \n\
L_0\
"""
ucode_str = \
"""\
L₀ \n\
H \n\
L₀\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = H(i, -j)*A(j)*B(k)
ascii_str = \
"""\
i L_0 k\n\
H *A *B \n\
L_0 \
"""
ucode_str = \
"""\
i L₀ k\n\
H ⋅A ⋅B \n\
L₀ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (1+x)*A(i)
ascii_str = \
"""\
i\n\
(x + 1)*A \n\
\
"""
ucode_str = \
"""\
i\n\
(x + 1)⋅A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i) + 3*B(i)
ascii_str = \
"""\
i i\n\
3*B + A \n\
\
"""
ucode_str = \
"""\
i i\n\
3⋅B + A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_pretty_print_tensor_partial_deriv():
from sympy.tensor.toperators import PartialDerivative
L = TensorIndexType("L")
i, j, k = tensor_indices("i j k", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
expr = PartialDerivative(A(i), A(j))
ascii_str = \
"""\
d / i\\\n\
---|A |\n\
j\\ /\n\
dA \n\
\
"""
ucode_str = \
"""\
∂ ⎛ i⎞\n\
───⎜A ⎟\n\
j⎝ ⎠\n\
∂A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k \\\n\
A *---|H |\n\
j\\ L_0/\n\
dA \n\
\
"""
ucode_str = \
"""\
L₀ ∂ ⎛ k ⎞\n\
A ⋅───⎜H ⎟\n\
j⎝ L₀⎠\n\
∂A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j))
ascii_str = \
"""\
L_0 d / k k \\\n\
A *---|3*H + B *C |\n\
j\\ L_0 L_0/\n\
dA \n\
\
"""
ucode_str = \
"""\
L₀ ∂ ⎛ k k ⎞\n\
A ⋅───⎜3⋅H + B ⋅C ⎟\n\
j⎝ L₀ L₀⎠\n\
∂A \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(j), D(j))
ascii_str = \
"""\
/ i i\\ d / L_0\\\n\
|A + B |*-----|C |\n\
\\ / L_0\\ /\n\
dD \n\
\
"""
ucode_str = \
"""\
⎛ i i⎞ ∂ ⎛ L₀⎞\n\
⎜A + B ⎟⋅────⎜C ⎟\n\
⎝ ⎠ L₀⎝ ⎠\n\
∂D \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = (A(i) + B(i))*PartialDerivative(C(-i), D(j))
ascii_str = \
"""\
/ L_0 L_0\\ d / \\\n\
|A + B |*---|C |\n\
\\ / j\\ L_0/\n\
dD \n\
\
"""
ucode_str = \
"""\
⎛ L₀ L₀⎞ ∂ ⎛ ⎞\n\
⎜A + B ⎟⋅───⎜C ⎟\n\
⎝ ⎠ j⎝ L₀⎠\n\
∂D \n\
\
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = PartialDerivative(B(-i) + A(-i), A(-j), A(-n))
ucode_str = """\
2 \n\
∂ ⎛ ⎞\n\
───────⎜A + B ⎟\n\
⎝ i i⎠\n\
∂A ∂A \n\
n j \
"""
assert upretty(expr) == ucode_str
expr = PartialDerivative(3*A(-i), A(-j), A(-n))
ucode_str = """\
2 \n\
∂ ⎛ ⎞\n\
───────⎜3⋅A ⎟\n\
⎝ i⎠\n\
∂A ∂A \n\
n j \
"""
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i:1})
ascii_str = \
"""\
i=1,j\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {i: 1, j: 1})
ascii_str = \
"""\
i=1,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = TensorElement(H(i, j), {j: 1})
ascii_str = \
"""\
i,j=1\n\
H \n\
\
"""
ucode_str = ascii_str
expr = TensorElement(H(-i, j), {-i: 1})
ascii_str = \
"""\
j\n\
H \n\
i=1 \
"""
ucode_str = ascii_str
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_15560():
a = MatrixSymbol('a', 1, 1)
e = pretty(a*(KroneckerProduct(a, a)))
result = 'a*(a x a)'
assert e == result
def test_print_lerchphi():
# Part of issue 6013
a = Symbol('a')
pretty(lerchphi(a, 1, 2))
uresult = 'Φ(a, 1, 2)'
aresult = 'lerchphi(a, 1, 2)'
assert pretty(lerchphi(a, 1, 2)) == aresult
assert upretty(lerchphi(a, 1, 2)) == uresult
def test_issue_15583():
N = mechanics.ReferenceFrame('N')
result = '(n_x, n_y, n_z)'
e = pretty((N.x, N.y, N.z))
assert e == result
def test_matrixSymbolBold():
# Issue 15871
def boldpretty(expr):
return xpretty(expr, use_unicode=True, wrap_line=False, mat_symbol_style="bold")
from sympy.matrices.expressions.trace import trace
A = MatrixSymbol("A", 2, 2)
assert boldpretty(trace(A)) == 'tr(𝐀)'
A = MatrixSymbol("A", 3, 3)
B = MatrixSymbol("B", 3, 3)
C = MatrixSymbol("C", 3, 3)
assert boldpretty(-A) == '-𝐀'
assert boldpretty(A - A*B - B) == '-𝐁 -𝐀⋅𝐁 + 𝐀'
assert boldpretty(-A*B - A*B*C - B) == '-𝐁 -𝐀⋅𝐁 -𝐀⋅𝐁⋅𝐂'
A = MatrixSymbol("Addot", 3, 3)
assert boldpretty(A) == '𝐀̈'
omega = MatrixSymbol("omega", 3, 3)
assert boldpretty(omega) == 'ω'
omega = MatrixSymbol("omeganorm", 3, 3)
assert boldpretty(omega) == '‖ω‖'
a = Symbol('alpha')
b = Symbol('b')
c = MatrixSymbol("c", 3, 1)
d = MatrixSymbol("d", 3, 1)
assert boldpretty(a*B*c+b*d) == 'b⋅𝐝 + α⋅𝐁⋅𝐜'
d = MatrixSymbol("delta", 3, 1)
B = MatrixSymbol("Beta", 3, 3)
assert boldpretty(a*B*c+b*d) == 'b⋅δ + α⋅Β⋅𝐜'
A = MatrixSymbol("A_2", 3, 3)
assert boldpretty(A) == '𝐀₂'
def test_center_accent():
assert center_accent('a', '\N{COMBINING TILDE}') == 'ã'
assert center_accent('aa', '\N{COMBINING TILDE}') == 'aã'
assert center_accent('aaa', '\N{COMBINING TILDE}') == 'aãa'
assert center_accent('aaaa', '\N{COMBINING TILDE}') == 'aaãa'
assert center_accent('aaaaa', '\N{COMBINING TILDE}') == 'aaãaa'
assert center_accent('abcdefg', '\N{COMBINING FOUR DOTS ABOVE}') == 'abcd⃜efg'
def test_imaginary_unit():
from sympy.printing.pretty import pretty # b/c it was redefined above
assert pretty(1 + I, use_unicode=False) == '1 + I'
assert pretty(1 + I, use_unicode=True) == '1 + ⅈ'
assert pretty(1 + I, use_unicode=False, imaginary_unit='j') == '1 + I'
assert pretty(1 + I, use_unicode=True, imaginary_unit='j') == '1 + ⅉ'
raises(TypeError, lambda: pretty(I, imaginary_unit=I))
raises(ValueError, lambda: pretty(I, imaginary_unit="kkk"))
def test_str_special_matrices():
from sympy.matrices import Identity, ZeroMatrix, OneMatrix
assert pretty(Identity(4)) == 'I'
assert upretty(Identity(4)) == '𝕀'
assert pretty(ZeroMatrix(2, 2)) == '0'
assert upretty(ZeroMatrix(2, 2)) == '𝟘'
assert pretty(OneMatrix(2, 2)) == '1'
assert upretty(OneMatrix(2, 2)) == '𝟙'
def test_pretty_misc_functions():
assert pretty(LambertW(x)) == 'W(x)'
assert upretty(LambertW(x)) == 'W(x)'
assert pretty(LambertW(x, y)) == 'W(x, y)'
assert upretty(LambertW(x, y)) == 'W(x, y)'
assert pretty(airyai(x)) == 'Ai(x)'
assert upretty(airyai(x)) == 'Ai(x)'
assert pretty(airybi(x)) == 'Bi(x)'
assert upretty(airybi(x)) == 'Bi(x)'
assert pretty(airyaiprime(x)) == "Ai'(x)"
assert upretty(airyaiprime(x)) == "Ai'(x)"
assert pretty(airybiprime(x)) == "Bi'(x)"
assert upretty(airybiprime(x)) == "Bi'(x)"
assert pretty(fresnelc(x)) == 'C(x)'
assert upretty(fresnelc(x)) == 'C(x)'
assert pretty(fresnels(x)) == 'S(x)'
assert upretty(fresnels(x)) == 'S(x)'
assert pretty(Heaviside(x)) == 'Heaviside(x)'
assert upretty(Heaviside(x)) == 'θ(x)'
assert pretty(Heaviside(x, y)) == 'Heaviside(x, y)'
assert upretty(Heaviside(x, y)) == 'θ(x, y)'
assert pretty(dirichlet_eta(x)) == 'dirichlet_eta(x)'
assert upretty(dirichlet_eta(x)) == 'η(x)'
def test_hadamard_power():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
B = MatrixSymbol('B', m, n)
# Testing printer:
expr = hadamard_power(A, n)
ascii_str = \
"""\
.n\n\
A \
"""
ucode_str = \
"""\
∘n\n\
A \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hadamard_power(A, 1+n)
ascii_str = \
"""\
.(n + 1)\n\
A \
"""
ucode_str = \
"""\
∘(n + 1)\n\
A \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
expr = hadamard_power(A*B.T, 1+n)
ascii_str = \
"""\
.(n + 1)\n\
/ T\\ \n\
\\A*B / \
"""
ucode_str = \
"""\
∘(n + 1)\n\
⎛ T⎞ \n\
⎝A⋅B ⎠ \
"""
assert pretty(expr) == ascii_str
assert upretty(expr) == ucode_str
def test_issue_17258():
n = Symbol('n', integer=True)
assert pretty(Sum(n, (n, -oo, 1))) == \
' 1 \n'\
' __ \n'\
' \\ ` \n'\
' ) n\n'\
' /_, \n'\
'n = -oo '
assert upretty(Sum(n, (n, -oo, 1))) == \
"""\
1 \n\
___ \n\
╲ \n\
╲ \n\
╱ n\n\
╱ \n\
‾‾‾ \n\
n = -∞ \
"""
def test_is_combining():
line = "v̇_m"
assert [is_combining(sym) for sym in line] == \
[False, True, False, False]
def test_issue_17616():
assert pretty(pi**(1/exp(1))) == \
' / -1\\\n'\
' \\e /\n'\
'pi '
assert upretty(pi**(1/exp(1))) == \
' ⎛ -1⎞\n'\
' ⎝ℯ ⎠\n'\
'π '
assert pretty(pi**(1/pi)) == \
' 1 \n'\
' --\n'\
' pi\n'\
'pi '
assert upretty(pi**(1/pi)) == \
' 1\n'\
' ─\n'\
' π\n'\
'π '
assert pretty(pi**(1/EulerGamma)) == \
' 1 \n'\
' ----------\n'\
' EulerGamma\n'\
'pi '
assert upretty(pi**(1/EulerGamma)) == \
' 1\n'\
' ─\n'\
' γ\n'\
'π '
z = Symbol("x_17")
assert upretty(7**(1/z)) == \
'x₁₇___\n'\
' ╲╱ 7 '
assert pretty(7**(1/z)) == \
'x_17___\n'\
' \\/ 7 '
def test_issue_17857():
assert pretty(Range(-oo, oo)) == '{..., -1, 0, 1, ...}'
assert pretty(Range(oo, -oo, -1)) == '{..., 1, 0, -1, ...}'
def test_issue_18272():
x = Symbol('x')
n = Symbol('n')
assert upretty(ConditionSet(x, Eq(-x + exp(x), 0), S.Complexes)) == \
'⎧ │ ⎛ x ⎞⎫\n'\
'⎨x │ x ∊ ℂ ∧ ⎝-x + ℯ = 0⎠⎬\n'\
'⎩ │ ⎭'
assert upretty(ConditionSet(x, Contains(n/2, Interval(0, oo)), FiniteSet(-n/2, n/2))) == \
'⎧ │ ⎧-n n⎫ ⎛n ⎞⎫\n'\
'⎨x │ x ∊ ⎨───, ─⎬ ∧ ⎜─ ∈ [0, ∞)⎟⎬\n'\
'⎩ │ ⎩ 2 2⎭ ⎝2 ⎠⎭'
assert upretty(ConditionSet(x, Eq(Piecewise((1, x >= 3), (x/2 - 1/2, x >= 2), (1/2, x >= 1),
(x/2, True)) - 1/2, 0), Interval(0, 3))) == \
'⎧ │ ⎛⎛⎧ 1 for x ≥ 3⎞ ⎞⎫\n'\
'⎪ │ ⎜⎜⎪ ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪x ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪─ - 0.5 for x ≥ 2⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪2 ⎟ ⎟⎪\n'\
'⎨x │ x ∊ [0, 3] ∧ ⎜⎜⎨ ⎟ - 0.5 = 0⎟⎬\n'\
'⎪ │ ⎜⎜⎪ 0.5 for x ≥ 1⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪ ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪ x ⎟ ⎟⎪\n'\
'⎪ │ ⎜⎜⎪ ─ otherwise⎟ ⎟⎪\n'\
'⎩ │ ⎝⎝⎩ 2 ⎠ ⎠⎭'
def test_Str():
from sympy.core.symbol import Str
assert pretty(Str('x')) == 'x'
def test_symbolic_probability():
mu = symbols("mu")
sigma = symbols("sigma", positive=True)
X = Normal("X", mu, sigma)
assert pretty(Expectation(X)) == r'E[X]'
assert pretty(Variance(X)) == r'Var(X)'
assert pretty(Probability(X > 0)) == r'P(X > 0)'
Y = Normal("Y", mu, sigma)
assert pretty(Covariance(X, Y)) == 'Cov(X, Y)'
def test_issue_21758():
from sympy.functions.elementary.piecewise import piecewise_fold
from sympy.series.fourier import FourierSeries
x = Symbol('x')
k, n = symbols('k n')
fo = FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)), SeqFormula(
Piecewise((-2*pi*cos(n*pi)/n + 2*sin(n*pi)/n**2, (n > -oo) & (n < oo) & Ne(n, 0)),
(0, True))*sin(n*x)/pi, (n, 1, oo))))
assert upretty(piecewise_fold(fo)) == \
'⎧ 2⋅sin(3⋅x) \n'\
'⎪2⋅sin(x) - sin(2⋅x) + ────────── + … for n > -∞ ∧ n < ∞ ∧ n ≠ 0\n'\
'⎨ 3 \n'\
'⎪ \n'\
'⎩ 0 otherwise '
assert pretty(FourierSeries(x, (x, -pi, pi), (0, SeqFormula(0, (k, 1, oo)),
SeqFormula(0, (n, 1, oo))))) == '0'
def test_diffgeom():
from sympy.diffgeom import Manifold, Patch, CoordSystem, BaseScalarField
x,y = symbols('x y', real=True)
m = Manifold('M', 2)
assert pretty(m) == 'M'
p = Patch('P', m)
assert pretty(p) == "P"
rect = CoordSystem('rect', p, [x, y])
assert pretty(rect) == "rect"
b = BaseScalarField(rect, 0)
assert pretty(b) == "x"
def test_deprecated_prettyForm():
with warns_deprecated_sympy():
from sympy.printing.pretty.pretty_symbology import xstr
assert xstr(1) == '1'
with warns_deprecated_sympy():
from sympy.printing.pretty.stringpict import prettyForm
p = prettyForm('s', unicode='s')
with warns_deprecated_sympy():
assert p.unicode == p.s == 's'
|
cfe91205afd300de8ed3b481ff632a9f9fec29824a00e50e6f3db28386b990b5 | import math
from sympy.concrete.summations import (Sum, summation)
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import (Derivative, Function, Lambda, diff)
from sympy.core import EulerGamma
from sympy.core.numbers import (E, Float, I, Rational, nan, oo, pi, zoo)
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.core.sympify import sympify
from sympy.functions.elementary.complexes import (Abs, im, polar_lift, re, sign)
from sympy.functions.elementary.exponential import (LambertW, exp, exp_polar, log)
from sympy.functions.elementary.hyperbolic import (acosh, asinh, cosh, coth, csch, sinh, tanh, sech)
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, asin, atan, cos, sin, sinc, tan, sec)
from sympy.functions.special.delta_functions import DiracDelta, Heaviside
from sympy.functions.special.error_functions import (Ci, Ei, Si, erf, erfc, erfi, fresnelc, li)
from sympy.functions.special.gamma_functions import (gamma, polygamma)
from sympy.functions.special.hyper import (hyper, meijerg)
from sympy.functions.special.singularity_functions import SingularityFunction
from sympy.functions.special.zeta_functions import lerchphi
from sympy.integrals.integrals import integrate
from sympy.logic.boolalg import And
from sympy.matrices.dense import Matrix
from sympy.polys.polytools import (Poly, factor)
from sympy.printing.str import sstr
from sympy.series.order import O
from sympy.sets.sets import Interval
from sympy.simplify.gammasimp import gammasimp
from sympy.simplify.simplify import simplify
from sympy.simplify.trigsimp import trigsimp
from sympy.tensor.indexed import (Idx, IndexedBase)
from sympy.core.expr import unchanged
from sympy.functions.elementary.integers import floor
from sympy.integrals.integrals import Integral
from sympy.integrals.risch import NonElementaryIntegral
from sympy.physics import units
from sympy.testing.pytest import (raises, slow, skip, ON_TRAVIS,
warns_deprecated_sympy, warns)
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.core.random import verify_numerically
x, y, z, a, b, c, d, e, s, t, x_1, x_2 = symbols('x y z a b c d e s t x_1 x_2')
n = Symbol('n', integer=True)
f = Function('f')
def NS(e, n=15, **options):
return sstr(sympify(e).evalf(n, **options), full_prec=True)
def test_poly_deprecated():
p = Poly(2*x, x)
assert p.integrate(x) == Poly(x**2, x, domain='QQ')
# The stacklevel is based on Integral(Poly)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
integrate(p, x)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Integral(p, (x,))
@slow
def test_principal_value():
g = 1 / x
assert Integral(g, (x, -oo, oo)).principal_value() == 0
assert Integral(g, (y, -oo, oo)).principal_value() == oo * sign(1 / x)
raises(ValueError, lambda: Integral(g, (x)).principal_value())
raises(ValueError, lambda: Integral(g).principal_value())
l = 1 / ((x ** 3) - 1)
assert Integral(l, (x, -oo, oo)).principal_value().together() == -sqrt(3)*pi/3
raises(ValueError, lambda: Integral(l, (x, -oo, 1)).principal_value())
d = 1 / (x ** 2 - 1)
assert Integral(d, (x, -oo, oo)).principal_value() == 0
assert Integral(d, (x, -2, 2)).principal_value() == -log(3)
v = x / (x ** 2 - 1)
assert Integral(v, (x, -oo, oo)).principal_value() == 0
assert Integral(v, (x, -2, 2)).principal_value() == 0
s = x ** 2 / (x ** 2 - 1)
assert Integral(s, (x, -oo, oo)).principal_value() is oo
assert Integral(s, (x, -2, 2)).principal_value() == -log(3) + 4
f = 1 / ((x ** 2 - 1) * (1 + x ** 2))
assert Integral(f, (x, -oo, oo)).principal_value() == -pi / 2
assert Integral(f, (x, -2, 2)).principal_value() == -atan(2) - log(3) / 2
def diff_test(i):
"""Return the set of symbols, s, which were used in testing that
i.diff(s) agrees with i.doit().diff(s). If there is an error then
the assertion will fail, causing the test to fail."""
syms = i.free_symbols
for s in syms:
assert (i.diff(s).doit() - i.doit().diff(s)).expand() == 0
return syms
def test_improper_integral():
assert integrate(log(x), (x, 0, 1)) == -1
assert integrate(x**(-2), (x, 1, oo)) == 1
assert integrate(1/(1 + exp(x)), (x, 0, oo)) == log(2)
def test_constructor():
# this is shared by Sum, so testing Integral's constructor
# is equivalent to testing Sum's
s1 = Integral(n, n)
assert s1.limits == (Tuple(n),)
s2 = Integral(n, (n,))
assert s2.limits == (Tuple(n),)
s3 = Integral(Sum(x, (x, 1, y)))
assert s3.limits == (Tuple(y),)
s4 = Integral(n, Tuple(n,))
assert s4.limits == (Tuple(n),)
s5 = Integral(n, (n, Interval(1, 2)))
assert s5.limits == (Tuple(n, 1, 2),)
# Testing constructor with inequalities:
s6 = Integral(n, n > 10)
assert s6.limits == (Tuple(n, 10, oo),)
s7 = Integral(n, (n > 2) & (n < 5))
assert s7.limits == (Tuple(n, 2, 5),)
def test_basics():
assert Integral(0, x) != 0
assert Integral(x, (x, 1, 1)) != 0
assert Integral(oo, x) != oo
assert Integral(S.NaN, x) is S.NaN
assert diff(Integral(y, y), x) == 0
assert diff(Integral(x, (x, 0, 1)), x) == 0
assert diff(Integral(x, x), x) == x
assert diff(Integral(t, (t, 0, x)), x) == x
e = (t + 1)**2
assert diff(integrate(e, (t, 0, x)), x) == \
diff(Integral(e, (t, 0, x)), x).doit().expand() == \
((1 + x)**2).expand()
assert diff(integrate(e, (t, 0, x)), t) == \
diff(Integral(e, (t, 0, x)), t) == 0
assert diff(integrate(e, (t, 0, x)), a) == \
diff(Integral(e, (t, 0, x)), a) == 0
assert diff(integrate(e, t), a) == diff(Integral(e, t), a) == 0
assert integrate(e, (t, a, x)).diff(x) == \
Integral(e, (t, a, x)).diff(x).doit().expand()
assert Integral(e, (t, a, x)).diff(x).doit() == ((1 + x)**2)
assert integrate(e, (t, x, a)).diff(x).doit() == (-(1 + x)**2).expand()
assert integrate(t**2, (t, x, 2*x)).diff(x) == 7*x**2
assert Integral(x, x).atoms() == {x}
assert Integral(f(x), (x, 0, 1)).atoms() == {S.Zero, S.One, x}
assert diff_test(Integral(x, (x, 3*y))) == {y}
assert diff_test(Integral(x, (a, 3*y))) == {x, y}
assert integrate(x, (x, oo, oo)) == 0 #issue 8171
assert integrate(x, (x, -oo, -oo)) == 0
# sum integral of terms
assert integrate(y + x + exp(x), x) == x*y + x**2/2 + exp(x)
assert Integral(x).is_commutative
n = Symbol('n', commutative=False)
assert Integral(n + x, x).is_commutative is False
def test_diff_wrt():
class Test(Expr):
_diff_wrt = True
is_commutative = True
t = Test()
assert integrate(t + 1, t) == t**2/2 + t
assert integrate(t + 1, (t, 0, 1)) == Rational(3, 2)
raises(ValueError, lambda: integrate(x + 1, x + 1))
raises(ValueError, lambda: integrate(x + 1, (x + 1, 0, 1)))
def test_basics_multiple():
assert diff_test(Integral(x, (x, 3*x, 5*y), (y, x, 2*x))) == {x}
assert diff_test(Integral(x, (x, 5*y), (y, x, 2*x))) == {x}
assert diff_test(Integral(x, (x, 5*y), (y, y, 2*x))) == {x, y}
assert diff_test(Integral(y, y, x)) == {x, y}
assert diff_test(Integral(y*x, x, y)) == {x, y}
assert diff_test(Integral(x + y, y, (y, 1, x))) == {x}
assert diff_test(Integral(x + y, (x, x, y), (y, y, x))) == {x, y}
def test_conjugate_transpose():
A, B = symbols("A B", commutative=False)
x = Symbol("x", complex=True)
p = Integral(A*B, (x,))
assert p.adjoint().doit() == p.doit().adjoint()
assert p.conjugate().doit() == p.doit().conjugate()
assert p.transpose().doit() == p.doit().transpose()
x = Symbol("x", real=True)
p = Integral(A*B, (x,))
assert p.adjoint().doit() == p.doit().adjoint()
assert p.conjugate().doit() == p.doit().conjugate()
assert p.transpose().doit() == p.doit().transpose()
def test_integration():
assert integrate(0, (t, 0, x)) == 0
assert integrate(3, (t, 0, x)) == 3*x
assert integrate(t, (t, 0, x)) == x**2/2
assert integrate(3*t, (t, 0, x)) == 3*x**2/2
assert integrate(3*t**2, (t, 0, x)) == x**3
assert integrate(1/t, (t, 1, x)) == log(x)
assert integrate(-1/t**2, (t, 1, x)) == 1/x - 1
assert integrate(t**2 + 5*t - 8, (t, 0, x)) == x**3/3 + 5*x**2/2 - 8*x
assert integrate(x**2, x) == x**3/3
assert integrate((3*t*x)**5, x) == (3*t)**5 * x**6 / 6
b = Symbol("b")
c = Symbol("c")
assert integrate(a*t, (t, 0, x)) == a*x**2/2
assert integrate(a*t**4, (t, 0, x)) == a*x**5/5
assert integrate(a*t**2 + b*t + c, (t, 0, x)) == a*x**3/3 + b*x**2/2 + c*x
def test_multiple_integration():
assert integrate((x**2)*(y**2), (x, 0, 1), (y, -1, 2)) == Rational(1)
assert integrate((y**2)*(x**2), x, y) == Rational(1, 9)*(x**3)*(y**3)
assert integrate(1/(x + 3)/(1 + x)**3, x) == \
log(3 + x)*Rational(-1, 8) + log(1 + x)*Rational(1, 8) + x/(4 + 8*x + 4*x**2)
assert integrate(sin(x*y)*y, (x, 0, 1), (y, 0, 1)) == -sin(1) + 1
def test_issue_3532():
assert integrate(exp(-x), (x, 0, oo)) == 1
def test_issue_3560():
assert integrate(sqrt(x)**3, x) == 2*sqrt(x)**5/5
assert integrate(sqrt(x), x) == 2*sqrt(x)**3/3
assert integrate(1/sqrt(x)**3, x) == -2/sqrt(x)
def test_issue_18038():
raises(AttributeError, lambda: integrate((x, x)))
def test_integrate_poly():
p = Poly(x + x**2*y + y**3, x, y)
# The stacklevel is based on Integral(Poly)
with warns_deprecated_sympy():
qx = Integral(p, x)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
qx = integrate(p, x)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
qy = integrate(p, y)
assert isinstance(qx, Poly) is True
assert isinstance(qy, Poly) is True
assert qx.gens == (x, y)
assert qy.gens == (x, y)
assert qx.as_expr() == x**2/2 + x**3*y/3 + x*y**3
assert qy.as_expr() == x*y + x**2*y**2/2 + y**4/4
def test_integrate_poly_definite():
p = Poly(x + x**2*y + y**3, x, y)
with warns_deprecated_sympy():
Qx = Integral(p, (x, 0, 1))
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Qx = integrate(p, (x, 0, 1))
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Qy = integrate(p, (y, 0, pi))
assert isinstance(Qx, Poly) is True
assert isinstance(Qy, Poly) is True
assert Qx.gens == (y,)
assert Qy.gens == (x,)
assert Qx.as_expr() == S.Half + y/3 + y**3
assert Qy.as_expr() == pi**4/4 + pi*x + pi**2*x**2/2
def test_integrate_omit_var():
y = Symbol('y')
assert integrate(x) == x**2/2
raises(ValueError, lambda: integrate(2))
raises(ValueError, lambda: integrate(x*y))
def test_integrate_poly_accurately():
y = Symbol('y')
assert integrate(x*sin(y), x) == x**2*sin(y)/2
# when passed to risch_norman, this will be a CPU hog, so this really
# checks, that integrated function is recognized as polynomial
assert integrate(x**1000*sin(y), x) == x**1001*sin(y)/1001
def test_issue_3635():
y = Symbol('y')
assert integrate(x**2, y) == x**2*y
assert integrate(x**2, (y, -1, 1)) == 2*x**2
# works in SymPy and py.test but hangs in `setup.py test`
def test_integrate_linearterm_pow():
# check integrate((a*x+b)^c, x) -- issue 3499
y = Symbol('y', positive=True)
# TODO: Remove conds='none' below, let the assumption take care of it.
assert integrate(x**y, x, conds='none') == x**(y + 1)/(y + 1)
assert integrate((exp(y)*x + 1/y)**(1 + sin(y)), x, conds='none') == \
exp(-y)*(exp(y)*x + 1/y)**(2 + sin(y)) / (2 + sin(y))
def test_issue_3618():
assert integrate(pi*sqrt(x), x) == 2*pi*sqrt(x)**3/3
assert integrate(pi*sqrt(x) + E*sqrt(x)**3, x) == \
2*pi*sqrt(x)**3/3 + 2*E *sqrt(x)**5/5
def test_issue_3623():
assert integrate(cos((n + 1)*x), x) == Piecewise(
(sin(x*(n + 1))/(n + 1), Ne(n + 1, 0)), (x, True))
assert integrate(cos((n - 1)*x), x) == Piecewise(
(sin(x*(n - 1))/(n - 1), Ne(n - 1, 0)), (x, True))
assert integrate(cos((n + 1)*x) + cos((n - 1)*x), x) == \
Piecewise((sin(x*(n - 1))/(n - 1), Ne(n - 1, 0)), (x, True)) + \
Piecewise((sin(x*(n + 1))/(n + 1), Ne(n + 1, 0)), (x, True))
def test_issue_3664():
n = Symbol('n', integer=True, nonzero=True)
assert integrate(-1./2 * x * sin(n * pi * x/2), [x, -2, 0]) == \
2.0*cos(pi*n)/(pi*n)
assert integrate(x * sin(n * pi * x/2) * Rational(-1, 2), [x, -2, 0]) == \
2*cos(pi*n)/(pi*n)
def test_issue_3679():
# definite integration of rational functions gives wrong answers
assert NS(Integral(1/(x**2 - 8*x + 17), (x, 2, 4))) == '1.10714871779409'
def test_issue_3686(): # remove this when fresnel itegrals are implemented
from sympy.core.function import expand_func
from sympy.functions.special.error_functions import fresnels
assert expand_func(integrate(sin(x**2), x)) == \
sqrt(2)*sqrt(pi)*fresnels(sqrt(2)*x/sqrt(pi))/2
def test_integrate_units():
m = units.m
s = units.s
assert integrate(x * m/s, (x, 1*s, 5*s)) == 12*m*s
def test_transcendental_functions():
assert integrate(LambertW(2*x), x) == \
-x + x*LambertW(2*x) + x/LambertW(2*x)
def test_log_polylog():
assert integrate(log(1 - x)/x, (x, 0, 1)) == -pi**2/6
assert integrate(log(x)*(1 - x)**(-1), (x, 0, 1)) == -pi**2/6
def test_issue_3740():
f = 4*log(x) - 2*log(x)**2
fid = diff(integrate(f, x), x)
assert abs(f.subs(x, 42).evalf() - fid.subs(x, 42).evalf()) < 1e-10
def test_issue_3788():
assert integrate(1/(1 + x**2), x) == atan(x)
def test_issue_3952():
f = sin(x)
assert integrate(f, x) == -cos(x)
raises(ValueError, lambda: integrate(f, 2*x))
def test_issue_4516():
assert integrate(2**x - 2*x, x) == 2**x/log(2) - x**2
def test_issue_7450():
ans = integrate(exp(-(1 + I)*x), (x, 0, oo))
assert re(ans) == S.Half and im(ans) == Rational(-1, 2)
def test_issue_8623():
assert integrate((1 + cos(2*x)) / (3 - 2*cos(2*x)), (x, 0, pi)) == -pi/2 + sqrt(5)*pi/2
assert integrate((1 + cos(2*x))/(3 - 2*cos(2*x))) == -x/2 + sqrt(5)*(atan(sqrt(5)*tan(x)) + \
pi*floor((x - pi/2)/pi))/2
def test_issue_9569():
assert integrate(1 / (2 - cos(x)), (x, 0, pi)) == pi/sqrt(3)
assert integrate(1/(2 - cos(x))) == 2*sqrt(3)*(atan(sqrt(3)*tan(x/2)) + pi*floor((x/2 - pi/2)/pi))/3
def test_issue_13733():
s = Symbol('s', positive=True)
pz = exp(-(z - y)**2/(2*s*s))/sqrt(2*pi*s*s)
pzgx = integrate(pz, (z, x, oo))
assert integrate(pzgx, (x, 0, oo)) == sqrt(2)*s*exp(-y**2/(2*s**2))/(2*sqrt(pi)) + \
y*erf(sqrt(2)*y/(2*s))/2 + y/2
def test_issue_13749():
assert integrate(1 / (2 + cos(x)), (x, 0, pi)) == pi/sqrt(3)
assert integrate(1/(2 + cos(x))) == 2*sqrt(3)*(atan(sqrt(3)*tan(x/2)/3) + pi*floor((x/2 - pi/2)/pi))/3
def test_issue_18133():
assert integrate(exp(x)/(1 + x)**2, x) == NonElementaryIntegral(exp(x)/(x + 1)**2, x)
def test_issue_21741():
a = Float('3999999.9999999995', precision=53)
b = Float('2.5000000000000004e-7', precision=53)
r = Piecewise((b*I*exp(-a*I*pi*t*y)*exp(-a*I*pi*x*z)/(pi*x),
Ne(1.0*pi*x*exp(a*I*pi*t*y), 0)),
(z*exp(-a*I*pi*t*y), True))
fun = E**((-2*I*pi*(z*x+t*y))/(500*10**(-9)))
assert integrate(fun, z) == r
def test_matrices():
M = Matrix(2, 2, lambda i, j: (i + j + 1)*sin((i + j + 1)*x))
assert integrate(M, x) == Matrix([
[-cos(x), -cos(2*x)],
[-cos(2*x), -cos(3*x)],
])
def test_integrate_functions():
# issue 4111
assert integrate(f(x), x) == Integral(f(x), x)
assert integrate(f(x), (x, 0, 1)) == Integral(f(x), (x, 0, 1))
assert integrate(f(x)*diff(f(x), x), x) == f(x)**2/2
assert integrate(diff(f(x), x) / f(x), x) == log(f(x))
def test_integrate_derivatives():
assert integrate(Derivative(f(x), x), x) == f(x)
assert integrate(Derivative(f(y), y), x) == x*Derivative(f(y), y)
assert integrate(Derivative(f(x), x)**2, x) == \
Integral(Derivative(f(x), x)**2, x)
def test_transform():
a = Integral(x**2 + 1, (x, -1, 2))
fx = x
fy = 3*y + 1
assert a.doit() == a.transform(fx, fy).doit()
assert a.transform(fx, fy).transform(fy, fx) == a
fx = 3*x + 1
fy = y
assert a.transform(fx, fy).transform(fy, fx) == a
a = Integral(sin(1/x), (x, 0, 1))
assert a.transform(x, 1/y) == Integral(sin(y)/y**2, (y, 1, oo))
assert a.transform(x, 1/y).transform(y, 1/x) == a
a = Integral(exp(-x**2), (x, -oo, oo))
assert a.transform(x, 2*y) == Integral(2*exp(-4*y**2), (y, -oo, oo))
# < 3 arg limit handled properly
assert Integral(x, x).transform(x, a*y).doit() == \
Integral(y*a**2, y).doit()
_3 = S(3)
assert Integral(x, (x, 0, -_3)).transform(x, 1/y).doit() == \
Integral(-1/x**3, (x, -oo, -1/_3)).doit()
assert Integral(x, (x, 0, _3)).transform(x, 1/y) == \
Integral(y**(-3), (y, 1/_3, oo))
# issue 8400
i = Integral(x + y, (x, 1, 2), (y, 1, 2))
assert i.transform(x, (x + 2*y, x)).doit() == \
i.transform(x, (x + 2*z, x)).doit() == 3
i = Integral(x, (x, a, b))
assert i.transform(x, 2*s) == Integral(4*s, (s, a/2, b/2))
raises(ValueError, lambda: i.transform(x, 1))
raises(ValueError, lambda: i.transform(x, s*t))
raises(ValueError, lambda: i.transform(x, -s))
raises(ValueError, lambda: i.transform(x, (s, t)))
raises(ValueError, lambda: i.transform(2*x, 2*s))
i = Integral(x**2, (x, 1, 2))
raises(ValueError, lambda: i.transform(x**2, s))
am = Symbol('a', negative=True)
bp = Symbol('b', positive=True)
i = Integral(x, (x, bp, am))
i.transform(x, 2*s)
assert i.transform(x, 2*s) == Integral(-4*s, (s, am/2, bp/2))
i = Integral(x, (x, a))
assert i.transform(x, 2*s) == Integral(4*s, (s, a/2))
def test_issue_4052():
f = S.Half*asin(x) + x*sqrt(1 - x**2)/2
assert integrate(cos(asin(x)), x) == f
assert integrate(sin(acos(x)), x) == f
@slow
def test_evalf_integrals():
assert NS(Integral(x, (x, 2, 5)), 15) == '10.5000000000000'
gauss = Integral(exp(-x**2), (x, -oo, oo))
assert NS(gauss, 15) == '1.77245385090552'
assert NS(gauss**2 - pi + E*Rational(
1, 10**20), 15) in ('2.71828182845904e-20', '2.71828182845905e-20')
# A monster of an integral from http://mathworld.wolfram.com/DefiniteIntegral.html
t = Symbol('t')
a = 8*sqrt(3)/(1 + 3*t**2)
b = 16*sqrt(2)*(3*t + 1)*sqrt(4*t**2 + t + 1)**3
c = (3*t**2 + 1)*(11*t**2 + 2*t + 3)**2
d = sqrt(2)*(249*t**2 + 54*t + 65)/(11*t**2 + 2*t + 3)**2
f = a - b/c - d
assert NS(Integral(f, (t, 0, 1)), 50) == \
NS((3*sqrt(2) - 49*pi + 162*atan(sqrt(2)))/12, 50)
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(1/x))/(1 + x + x**2), (x, 0, 1)), 15) == \
NS('pi/sqrt(3) * log(2*pi**(5/6) / gamma(1/6))', 15)
# http://mathworld.wolfram.com/AhmedsIntegral.html
assert NS(Integral(atan(sqrt(x**2 + 2))/(sqrt(x**2 + 2)*(x**2 + 1)), (x,
0, 1)), 15) == NS(5*pi**2/96, 15)
# http://mathworld.wolfram.com/AbelsIntegral.html
assert NS(Integral(x/((exp(pi*x) - exp(
-pi*x))*(x**2 + 1)), (x, 0, oo)), 15) == NS('log(2)/2-1/4', 15)
# Complex part trimming
# http://mathworld.wolfram.com/VardisIntegral.html
assert NS(Integral(log(log(sin(x)/cos(x))), (x, pi/4, pi/2)), 15, chop=True) == \
NS('pi/4*log(4*pi**3/gamma(1/4)**4)', 15)
#
# Endpoints causing trouble (rounding error in integration points -> complex log)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 17, chop=True) == NS(2, 17)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 20, chop=True) == NS(2, 20)
assert NS(
2 + Integral(log(2*cos(x/2)), (x, -pi, pi)), 22, chop=True) == NS(2, 22)
# Needs zero handling
assert NS(pi - 4*Integral(
'sqrt(1-x**2)', (x, 0, 1)), 15, maxn=30, chop=True) in ('0.0', '0')
# Oscillatory quadrature
a = Integral(sin(x)/x**2, (x, 1, oo)).evalf(maxn=15)
assert 0.49 < a < 0.51
assert NS(
Integral(sin(x)/x**2, (x, 1, oo)), quad='osc') == '0.504067061906928'
assert NS(Integral(
cos(pi*x + 1)/x, (x, -oo, -1)), quad='osc') == '0.276374705640365'
# indefinite integrals aren't evaluated
assert NS(Integral(x, x)) == 'Integral(x, x)'
assert NS(Integral(x, (x, y))) == 'Integral(x, (x, y))'
def test_evalf_issue_939():
# https://github.com/sympy/sympy/issues/4038
# The output form of an integral may differ by a step function between
# revisions, making this test a bit useless. This can't be said about
# other two tests. For now, all values of this evaluation are used here,
# but in future this should be reconsidered.
assert NS(integrate(1/(x**5 + 1), x).subs(x, 4), chop=True) in \
['-0.000976138910649103', '0.965906660135753', '1.93278945918216']
assert NS(Integral(1/(x**5 + 1), (x, 2, 4))) == '0.0144361088886740'
assert NS(
integrate(1/(x**5 + 1), (x, 2, 4)), chop=True) == '0.0144361088886740'
def test_double_previously_failing_integrals():
# Double integrals not implemented <- Sure it is!
res = integrate(sqrt(x) + x*y, (x, 1, 2), (y, -1, 1))
# Old numerical test
assert NS(res, 15) == '2.43790283299492'
# Symbolic test
assert res == Rational(-4, 3) + 8*sqrt(2)/3
# double integral + zero detection
assert integrate(sin(x + x*y), (x, -1, 1), (y, -1, 1)) is S.Zero
def test_integrate_SingularityFunction():
in_1 = SingularityFunction(x, a, 3) + SingularityFunction(x, 5, -1)
out_1 = SingularityFunction(x, a, 4)/4 + SingularityFunction(x, 5, 0)
assert integrate(in_1, x) == out_1
in_2 = 10*SingularityFunction(x, 4, 0) - 5*SingularityFunction(x, -6, -2)
out_2 = 10*SingularityFunction(x, 4, 1) - 5*SingularityFunction(x, -6, -1)
assert integrate(in_2, x) == out_2
in_3 = 2*x**2*y -10*SingularityFunction(x, -4, 7) - 2*SingularityFunction(y, 10, -2)
out_3_1 = 2*x**3*y/3 - 2*x*SingularityFunction(y, 10, -2) - 5*SingularityFunction(x, -4, 8)/4
out_3_2 = x**2*y**2 - 10*y*SingularityFunction(x, -4, 7) - 2*SingularityFunction(y, 10, -1)
assert integrate(in_3, x) == out_3_1
assert integrate(in_3, y) == out_3_2
assert unchanged(Integral, in_3, (x,))
assert Integral(in_3, x) == Integral(in_3, (x,))
assert Integral(in_3, x).doit() == out_3_1
in_4 = 10*SingularityFunction(x, -4, 7) - 2*SingularityFunction(x, 10, -2)
out_4 = 5*SingularityFunction(x, -4, 8)/4 - 2*SingularityFunction(x, 10, -1)
assert integrate(in_4, (x, -oo, x)) == out_4
assert integrate(SingularityFunction(x, 5, -1), x) == SingularityFunction(x, 5, 0)
assert integrate(SingularityFunction(x, 0, -1), (x, -oo, oo)) == 1
assert integrate(5*SingularityFunction(x, 5, -1), (x, -oo, oo)) == 5
assert integrate(SingularityFunction(x, 5, -1) * f(x), (x, -oo, oo)) == f(5)
def test_integrate_DiracDelta():
# This is here to check that deltaintegrate is being called, but also
# to test definite integrals. More tests are in test_deltafunctions.py
assert integrate(DiracDelta(x) * f(x), (x, -oo, oo)) == f(0)
assert integrate(DiracDelta(x)**2, (x, -oo, oo)) == DiracDelta(0)
# issue 4522
assert integrate(integrate((4 - 4*x + x*y - 4*y) * \
DiracDelta(x)*DiracDelta(y - 1), (x, 0, 1)), (y, 0, 1)) == 0
# issue 5729
p = exp(-(x**2 + y**2))/pi
assert integrate(p*DiracDelta(x - 10*y), (x, -oo, oo), (y, -oo, oo)) == \
integrate(p*DiracDelta(x - 10*y), (y, -oo, oo), (x, -oo, oo)) == \
integrate(p*DiracDelta(10*x - y), (x, -oo, oo), (y, -oo, oo)) == \
integrate(p*DiracDelta(10*x - y), (y, -oo, oo), (x, -oo, oo)) == \
1/sqrt(101*pi)
def test_integrate_returns_piecewise():
assert integrate(x**y, x) == Piecewise(
(x**(y + 1)/(y + 1), Ne(y, -1)), (log(x), True))
assert integrate(x**y, y) == Piecewise(
(x**y/log(x), Ne(log(x), 0)), (y, True))
assert integrate(exp(n*x), x) == Piecewise(
(exp(n*x)/n, Ne(n, 0)), (x, True))
assert integrate(x*exp(n*x), x) == Piecewise(
((n*x - 1)*exp(n*x)/n**2, Ne(n**2, 0)), (x**2/2, True))
assert integrate(x**(n*y), x) == Piecewise(
(x**(n*y + 1)/(n*y + 1), Ne(n*y, -1)), (log(x), True))
assert integrate(x**(n*y), y) == Piecewise(
(x**(n*y)/(n*log(x)), Ne(n*log(x), 0)), (y, True))
assert integrate(cos(n*x), x) == Piecewise(
(sin(n*x)/n, Ne(n, 0)), (x, True))
assert integrate(cos(n*x)**2, x) == Piecewise(
((n*x/2 + sin(n*x)*cos(n*x)/2)/n, Ne(n, 0)), (x, True))
assert integrate(x*cos(n*x), x) == Piecewise(
(x*sin(n*x)/n + cos(n*x)/n**2, Ne(n, 0)), (x**2/2, True))
assert integrate(sin(n*x), x) == Piecewise(
(-cos(n*x)/n, Ne(n, 0)), (0, True))
assert integrate(sin(n*x)**2, x) == Piecewise(
((n*x/2 - sin(n*x)*cos(n*x)/2)/n, Ne(n, 0)), (0, True))
assert integrate(x*sin(n*x), x) == Piecewise(
(-x*cos(n*x)/n + sin(n*x)/n**2, Ne(n, 0)), (0, True))
assert integrate(exp(x*y), (x, 0, z)) == Piecewise(
(exp(y*z)/y - 1/y, (y > -oo) & (y < oo) & Ne(y, 0)), (z, True))
# https://github.com/sympy/sympy/issues/23707
assert integrate(exp(t)*exp(-t*sqrt(x - y)), t) == Piecewise(
(-exp(t)/(sqrt(x - y)*exp(t*sqrt(x - y)) - exp(t*sqrt(x - y))),
Ne(x, y + 1)), (t, True))
def test_integrate_max_min():
x = symbols('x', real=True)
assert integrate(Min(x, 2), (x, 0, 3)) == 4
assert integrate(Max(x**2, x**3), (x, 0, 2)) == Rational(49, 12)
assert integrate(Min(exp(x), exp(-x))**2, x) == Piecewise( \
(exp(2*x)/2, x <= 0), (1 - exp(-2*x)/2, True))
# issue 7907
c = symbols('c', extended_real=True)
int1 = integrate(Max(c, x)*exp(-x**2), (x, -oo, oo))
int2 = integrate(c*exp(-x**2), (x, -oo, c))
int3 = integrate(x*exp(-x**2), (x, c, oo))
assert int1 == int2 + int3 == sqrt(pi)*c*erf(c)/2 + \
sqrt(pi)*c/2 + exp(-c**2)/2
def test_integrate_Abs_sign():
assert integrate(Abs(x), (x, -2, 1)) == Rational(5, 2)
assert integrate(Abs(x), (x, 0, 1)) == S.Half
assert integrate(Abs(x + 1), (x, 0, 1)) == Rational(3, 2)
assert integrate(Abs(x**2 - 1), (x, -2, 2)) == 4
assert integrate(Abs(x**2 - 3*x), (x, -15, 15)) == 2259
assert integrate(sign(x), (x, -1, 2)) == 1
assert integrate(sign(x)*sin(x), (x, -pi, pi)) == 4
assert integrate(sign(x - 2) * x**2, (x, 0, 3)) == Rational(11, 3)
t, s = symbols('t s', real=True)
assert integrate(Abs(t), t) == Piecewise(
(-t**2/2, t <= 0), (t**2/2, True))
assert integrate(Abs(2*t - 6), t) == Piecewise(
(-t**2 + 6*t, t <= 3), (t**2 - 6*t + 18, True))
assert (integrate(abs(t - s**2), (t, 0, 2)) ==
2*s**2*Min(2, s**2) - 2*s**2 - Min(2, s**2)**2 + 2)
assert integrate(exp(-Abs(t)), t) == Piecewise(
(exp(t), t <= 0), (2 - exp(-t), True))
assert integrate(sign(2*t - 6), t) == Piecewise(
(-t, t < 3), (t - 6, True))
assert integrate(2*t*sign(t**2 - 1), t) == Piecewise(
(t**2, t < -1), (-t**2 + 2, t < 1), (t**2, True))
assert integrate(sign(t), (t, s + 1)) == Piecewise(
(s + 1, s + 1 > 0), (-s - 1, s + 1 < 0), (0, True))
def test_subs1():
e = Integral(exp(x - y), x)
assert e.subs(y, 3) == Integral(exp(x - 3), x)
e = Integral(exp(x - y), (x, 0, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo))
def test_subs2():
e = Integral(exp(x - y), x, t)
assert e.subs(y, 3) == Integral(exp(x - 3), x, t)
e = Integral(exp(x - y), (x, 0, 1), (t, 0, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 1), (t, 0, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo), (t, 0, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs3():
e = Integral(exp(x - y), (x, 0, y), (t, y, 1))
assert e.subs(y, 3) == Integral(exp(x - 3), (x, 0, 3), (t, 3, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(x - y)*f(y), (y, -oo, oo), (t, x, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs4():
e = Integral(exp(x), (x, 0, y), (t, y, 1))
assert e.subs(y, 3) == Integral(exp(x), (x, 0, 3), (t, 3, 1))
f = Lambda(x, exp(-x**2))
conv = Integral(f(y)*f(y), (y, -oo, oo), (t, x, 1))
assert conv.subs({x: 0}) == Integral(exp(-2*y**2), (y, -oo, oo), (t, 0, 1))
def test_subs5():
e = Integral(exp(-x**2), (x, -oo, oo))
assert e.subs(x, 5) == e
e = Integral(exp(-x**2 + y), x)
assert e.subs(y, 5) == Integral(exp(-x**2 + 5), x)
e = Integral(exp(-x**2 + y), (x, x))
assert e.subs(x, 5) == Integral(exp(y - x**2), (x, 5))
assert e.subs(y, 5) == Integral(exp(-x**2 + 5), x)
e = Integral(exp(-x**2 + y), (y, -oo, oo), (x, -oo, oo))
assert e.subs(x, 5) == e
assert e.subs(y, 5) == e
# Test evaluation of antiderivatives
e = Integral(exp(-x**2), (x, x))
assert e.subs(x, 5) == Integral(exp(-x**2), (x, 5))
e = Integral(exp(x), x)
assert (e.subs(x,1) - e.subs(x,0) - Integral(exp(x), (x, 0, 1))
).doit().is_zero
def test_subs6():
a, b = symbols('a b')
e = Integral(x*y, (x, f(x), f(y)))
assert e.subs(x, 1) == Integral(x*y, (x, f(1), f(y)))
assert e.subs(y, 1) == Integral(x, (x, f(x), f(1)))
e = Integral(x*y, (x, f(x), f(y)), (y, f(x), f(y)))
assert e.subs(x, 1) == Integral(x*y, (x, f(1), f(y)), (y, f(1), f(y)))
assert e.subs(y, 1) == Integral(x*y, (x, f(x), f(y)), (y, f(x), f(1)))
e = Integral(x*y, (x, f(x), f(a)), (y, f(x), f(a)))
assert e.subs(a, 1) == Integral(x*y, (x, f(x), f(1)), (y, f(x), f(1)))
def test_subs7():
e = Integral(x, (x, 1, y), (y, 1, 2))
assert e.subs({x: 1, y: 2}) == e
e = Integral(sin(x) + sin(y), (x, sin(x), sin(y)),
(y, 1, 2))
assert e.subs(sin(y), 1) == e
assert e.subs(sin(x), 1) == Integral(sin(x) + sin(y), (x, 1, sin(y)),
(y, 1, 2))
def test_expand():
e = Integral(f(x)+f(x**2), (x, 1, y))
assert e.expand() == Integral(f(x), (x, 1, y)) + Integral(f(x**2), (x, 1, y))
def test_integration_variable():
raises(ValueError, lambda: Integral(exp(-x**2), 3))
raises(ValueError, lambda: Integral(exp(-x**2), (3, -oo, oo)))
def test_expand_integral():
assert Integral(cos(x**2)*(sin(x**2) + 1), (x, 0, 1)).expand() == \
Integral(cos(x**2)*sin(x**2), (x, 0, 1)) + \
Integral(cos(x**2), (x, 0, 1))
assert Integral(cos(x**2)*(sin(x**2) + 1), x).expand() == \
Integral(cos(x**2)*sin(x**2), x) + \
Integral(cos(x**2), x)
def test_as_sum_midpoint1():
e = Integral(sqrt(x**3 + 1), (x, 2, 10))
assert e.as_sum(1, method="midpoint") == 8*sqrt(217)
assert e.as_sum(2, method="midpoint") == 4*sqrt(65) + 12*sqrt(57)
assert e.as_sum(3, method="midpoint") == 8*sqrt(217)/3 + \
8*sqrt(3081)/27 + 8*sqrt(52809)/27
assert e.as_sum(4, method="midpoint") == 2*sqrt(730) + \
4*sqrt(7) + 4*sqrt(86) + 6*sqrt(14)
assert abs(e.as_sum(4, method="midpoint").n() - e.n()) < 0.5
e = Integral(sqrt(x**3 + y**3), (x, 2, 10), (y, 0, 10))
raises(NotImplementedError, lambda: e.as_sum(4))
def test_as_sum_midpoint2():
e = Integral((x + y)**2, (x, 0, 1))
n = Symbol('n', positive=True, integer=True)
assert e.as_sum(1, method="midpoint").expand() == Rational(1, 4) + y + y**2
assert e.as_sum(2, method="midpoint").expand() == Rational(5, 16) + y + y**2
assert e.as_sum(3, method="midpoint").expand() == Rational(35, 108) + y + y**2
assert e.as_sum(4, method="midpoint").expand() == Rational(21, 64) + y + y**2
assert e.as_sum(n, method="midpoint").expand() == \
y**2 + y + Rational(1, 3) - 1/(12*n**2)
def test_as_sum_left():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="left").expand() == y**2
assert e.as_sum(2, method="left").expand() == Rational(1, 8) + y/2 + y**2
assert e.as_sum(3, method="left").expand() == Rational(5, 27) + y*Rational(2, 3) + y**2
assert e.as_sum(4, method="left").expand() == Rational(7, 32) + y*Rational(3, 4) + y**2
assert e.as_sum(n, method="left").expand() == \
y**2 + y + Rational(1, 3) - y/n - 1/(2*n) + 1/(6*n**2)
assert e.as_sum(10, method="left", evaluate=False).has(Sum)
def test_as_sum_right():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="right").expand() == 1 + 2*y + y**2
assert e.as_sum(2, method="right").expand() == Rational(5, 8) + y*Rational(3, 2) + y**2
assert e.as_sum(3, method="right").expand() == Rational(14, 27) + y*Rational(4, 3) + y**2
assert e.as_sum(4, method="right").expand() == Rational(15, 32) + y*Rational(5, 4) + y**2
assert e.as_sum(n, method="right").expand() == \
y**2 + y + Rational(1, 3) + y/n + 1/(2*n) + 1/(6*n**2)
def test_as_sum_trapezoid():
e = Integral((x + y)**2, (x, 0, 1))
assert e.as_sum(1, method="trapezoid").expand() == y**2 + y + S.Half
assert e.as_sum(2, method="trapezoid").expand() == y**2 + y + Rational(3, 8)
assert e.as_sum(3, method="trapezoid").expand() == y**2 + y + Rational(19, 54)
assert e.as_sum(4, method="trapezoid").expand() == y**2 + y + Rational(11, 32)
assert e.as_sum(n, method="trapezoid").expand() == \
y**2 + y + Rational(1, 3) + 1/(6*n**2)
assert Integral(sign(x), (x, 0, 1)).as_sum(1, 'trapezoid') == S.Half
def test_as_sum_raises():
e = Integral((x + y)**2, (x, 0, 1))
raises(ValueError, lambda: e.as_sum(-1))
raises(ValueError, lambda: e.as_sum(0))
raises(ValueError, lambda: Integral(x).as_sum(3))
raises(ValueError, lambda: e.as_sum(oo))
raises(ValueError, lambda: e.as_sum(3, method='xxxx2'))
def test_nested_doit():
e = Integral(Integral(x, x), x)
f = Integral(x, x, x)
assert e.doit() == f.doit()
def test_issue_4665():
# Allow only upper or lower limit evaluation
e = Integral(x**2, (x, None, 1))
f = Integral(x**2, (x, 1, None))
assert e.doit() == Rational(1, 3)
assert f.doit() == Rational(-1, 3)
assert Integral(x*y, (x, None, y)).subs(y, t) == Integral(x*t, (x, None, t))
assert Integral(x*y, (x, y, None)).subs(y, t) == Integral(x*t, (x, t, None))
assert integrate(x**2, (x, None, 1)) == Rational(1, 3)
assert integrate(x**2, (x, 1, None)) == Rational(-1, 3)
assert integrate("x**2", ("x", "1", None)) == Rational(-1, 3)
def test_integral_reconstruct():
e = Integral(x**2, (x, -1, 1))
assert e == Integral(*e.args)
def test_doit_integrals():
e = Integral(Integral(2*x), (x, 0, 1))
assert e.doit() == Rational(1, 3)
assert e.doit(deep=False) == Rational(1, 3)
f = Function('f')
# doesn't matter if the integral can't be performed
assert Integral(f(x), (x, 1, 1)).doit() == 0
# doesn't matter if the limits can't be evaluated
assert Integral(0, (x, 1, Integral(f(x), x))).doit() == 0
assert Integral(x, (a, 0)).doit() == 0
limits = ((a, 1, exp(x)), (x, 0))
assert Integral(a, *limits).doit() == Rational(1, 4)
assert Integral(a, *list(reversed(limits))).doit() == 0
def test_issue_4884():
assert integrate(sqrt(x)*(1 + x)) == \
Piecewise(
(2*sqrt(x)*(x + 1)**2/5 - 2*sqrt(x)*(x + 1)/15 - 4*sqrt(x)/15,
Abs(x + 1) > 1),
(2*I*sqrt(-x)*(x + 1)**2/5 - 2*I*sqrt(-x)*(x + 1)/15 -
4*I*sqrt(-x)/15, True))
assert integrate(x**x*(1 + log(x))) == x**x
def test_issue_18153():
assert integrate(x**n*log(x),x) == \
Piecewise(
(n*x*x**n*log(x)/(n**2 + 2*n + 1) +
x*x**n*log(x)/(n**2 + 2*n + 1) - x*x**n/(n**2 + 2*n + 1)
, Ne(n, -1)), (log(x)**2/2, True)
)
def test_is_number():
from sympy.abc import x, y, z
assert Integral(x).is_number is False
assert Integral(1, x).is_number is False
assert Integral(1, (x, 1)).is_number is True
assert Integral(1, (x, 1, 2)).is_number is True
assert Integral(1, (x, 1, y)).is_number is False
assert Integral(1, (x, y)).is_number is False
assert Integral(x, y).is_number is False
assert Integral(x, (y, 1, x)).is_number is False
assert Integral(x, (y, 1, 2)).is_number is False
assert Integral(x, (x, 1, 2)).is_number is True
# `foo.is_number` should always be equivalent to `not foo.free_symbols`
# in each of these cases, there are pseudo-free symbols
i = Integral(x, (y, 1, 1))
assert i.is_number is False and i.n() == 0
i = Integral(x, (y, z, z))
assert i.is_number is False and i.n() == 0
i = Integral(1, (y, z, z + 2))
assert i.is_number is False and i.n() == 2
assert Integral(x*y, (x, 1, 2), (y, 1, 3)).is_number is True
assert Integral(x*y, (x, 1, 2), (y, 1, z)).is_number is False
assert Integral(x, (x, 1)).is_number is True
assert Integral(x, (x, 1, Integral(y, (y, 1, 2)))).is_number is True
assert Integral(Sum(z, (z, 1, 2)), (x, 1, 2)).is_number is True
# it is possible to get a false negative if the integrand is
# actually an unsimplified zero, but this is true of is_number in general.
assert Integral(sin(x)**2 + cos(x)**2 - 1, x).is_number is False
assert Integral(f(x), (x, 0, 1)).is_number is True
def test_free_symbols():
from sympy.abc import x, y, z
assert Integral(0, x).free_symbols == {x}
assert Integral(x).free_symbols == {x}
assert Integral(x, (x, None, y)).free_symbols == {y}
assert Integral(x, (x, y, None)).free_symbols == {y}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert Integral(x, (x, y, 1)).free_symbols == {y}
assert Integral(x, (x, x, y)).free_symbols == {x, y}
assert Integral(x, x, y).free_symbols == {x, y}
assert Integral(x, (x, 1, 2)).free_symbols == set()
assert Integral(x, (y, 1, 2)).free_symbols == {x}
# pseudo-free in this case
assert Integral(x, (y, z, z)).free_symbols == {x, z}
assert Integral(x, (y, 1, 2), (y, None, None)
).free_symbols == {x, y}
assert Integral(x, (y, 1, 2), (x, 1, y)
).free_symbols == {y}
assert Integral(2, (y, 1, 2), (y, 1, x), (x, 1, 2)
).free_symbols == set()
assert Integral(2, (y, x, 2), (y, 1, x), (x, 1, 2)
).free_symbols == set()
assert Integral(2, (x, 1, 2), (y, x, 2), (y, 1, 2)
).free_symbols == {x}
assert Integral(f(x), (f(x), 1, y)).free_symbols == {y}
assert Integral(f(x), (f(x), 1, x)).free_symbols == {x}
def test_is_zero():
from sympy.abc import x, m
assert Integral(0, (x, 1, x)).is_zero
assert Integral(1, (x, 1, 1)).is_zero
assert Integral(1, (x, 1, 2), (y, 2)).is_zero is False
assert Integral(x, (m, 0)).is_zero
assert Integral(x + m, (m, 0)).is_zero is None
i = Integral(m, (m, 1, exp(x)), (x, 0))
assert i.is_zero is None
assert Integral(m, (x, 0), (m, 1, exp(x))).is_zero is True
assert Integral(x, (x, oo, oo)).is_zero # issue 8171
assert Integral(x, (x, -oo, -oo)).is_zero
# this is zero but is beyond the scope of what is_zero
# should be doing
assert Integral(sin(x), (x, 0, 2*pi)).is_zero is None
def test_series():
from sympy.abc import x
i = Integral(cos(x), (x, x))
e = i.lseries(x)
assert i.nseries(x, n=8).removeO() == Add(*[next(e) for j in range(4)])
def test_trig_nonelementary_integrals():
x = Symbol('x')
assert integrate((1 + sin(x))/x, x) == log(x) + Si(x)
# next one comes out as log(x) + log(x**2)/2 + Ci(x)
# so not hardcoding this log ugliness
assert integrate((cos(x) + 2)/x, x).has(Ci)
def test_issue_4403():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z', positive=True)
assert integrate(sqrt(x**2 + z**2), x) == \
z**2*asinh(x/z)/2 + x*sqrt(x**2 + z**2)/2
assert integrate(sqrt(x**2 - z**2), x) == \
x*sqrt(x**2 - z**2)/2 - z**2*log(x + sqrt(x**2 - z**2))/2
x = Symbol('x', real=True)
y = Symbol('y', positive=True)
assert integrate(1/(x**2 + y**2)**S('3/2'), x) == \
x/(y**2*sqrt(x**2 + y**2))
# If y is real and nonzero, we get x*Abs(y)/(y**3*sqrt(x**2 + y**2)),
# which results from sqrt(1 + x**2/y**2) = sqrt(x**2 + y**2)/|y|.
def test_issue_4403_2():
assert integrate(sqrt(-x**2 - 4), x) == \
-2*atan(x/sqrt(-4 - x**2)) + x*sqrt(-4 - x**2)/2
def test_issue_4100():
R = Symbol('R', positive=True)
assert integrate(sqrt(R**2 - x**2), (x, 0, R)) == pi*R**2/4
def test_issue_5167():
from sympy.abc import w, x, y, z
f = Function('f')
assert Integral(Integral(f(x), x), x) == Integral(f(x), x, x)
assert Integral(f(x)).args == (f(x), Tuple(x))
assert Integral(Integral(f(x))).args == (f(x), Tuple(x), Tuple(x))
assert Integral(Integral(f(x)), y).args == (f(x), Tuple(x), Tuple(y))
assert Integral(Integral(f(x), z), y).args == (f(x), Tuple(z), Tuple(y))
assert Integral(Integral(Integral(f(x), x), y), z).args == \
(f(x), Tuple(x), Tuple(y), Tuple(z))
assert integrate(Integral(f(x), x), x) == Integral(f(x), x, x)
assert integrate(Integral(f(x), y), x) == y*Integral(f(x), x)
assert integrate(Integral(f(x), x), y) in [Integral(y*f(x), x), y*Integral(f(x), x)]
assert integrate(Integral(2, x), x) == x**2
assert integrate(Integral(2, x), y) == 2*x*y
# don't re-order given limits
assert Integral(1, x, y).args != Integral(1, y, x).args
# do as many as possible
assert Integral(f(x), y, x, y, x).doit() == y**2*Integral(f(x), x, x)/2
assert Integral(f(x), (x, 1, 2), (w, 1, x), (z, 1, y)).doit() == \
y*(x - 1)*Integral(f(x), (x, 1, 2)) - (x - 1)*Integral(f(x), (x, 1, 2))
def test_issue_4890():
z = Symbol('z', positive=True)
assert integrate(exp(-log(x)**2), x) == \
sqrt(pi)*exp(Rational(1, 4))*erf(log(x) - S.Half)/2
assert integrate(exp(log(x)**2), x) == \
sqrt(pi)*exp(Rational(-1, 4))*erfi(log(x)+S.Half)/2
assert integrate(exp(-z*log(x)**2), x) == \
sqrt(pi)*exp(1/(4*z))*erf(sqrt(z)*log(x) - 1/(2*sqrt(z)))/(2*sqrt(z))
def test_issue_4551():
assert not integrate(1/(x*sqrt(1 - x**2)), x).has(Integral)
def test_issue_4376():
n = Symbol('n', integer=True, positive=True)
assert simplify(integrate(n*(x**(1/n) - 1), (x, 0, S.Half)) -
(n**2 - 2**(1/n)*n**2 - n*2**(1/n))/(2**(1 + 1/n) + n*2**(1 + 1/n))) == 0
def test_issue_4517():
assert integrate((sqrt(x) - x**3)/x**Rational(1, 3), x) == \
6*x**Rational(7, 6)/7 - 3*x**Rational(11, 3)/11
def test_issue_4527():
k, m = symbols('k m', integer=True)
assert integrate(sin(k*x)*sin(m*x), (x, 0, pi)).simplify() == \
Piecewise((0, Eq(k, 0) | Eq(m, 0)),
(-pi/2, Eq(k, -m) | (Eq(k, 0) & Eq(m, 0))),
(pi/2, Eq(k, m) | (Eq(k, 0) & Eq(m, 0))),
(0, True))
# Should be possible to further simplify to:
# Piecewise(
# (0, Eq(k, 0) | Eq(m, 0)),
# (-pi/2, Eq(k, -m)),
# (pi/2, Eq(k, m)),
# (0, True))
assert integrate(sin(k*x)*sin(m*x), (x,)) == Piecewise(
(0, And(Eq(k, 0), Eq(m, 0))),
(-x*sin(m*x)**2/2 - x*cos(m*x)**2/2 + sin(m*x)*cos(m*x)/(2*m), Eq(k, -m)),
(x*sin(m*x)**2/2 + x*cos(m*x)**2/2 - sin(m*x)*cos(m*x)/(2*m), Eq(k, m)),
(m*sin(k*x)*cos(m*x)/(k**2 - m**2) -
k*sin(m*x)*cos(k*x)/(k**2 - m**2), True))
def test_issue_4199():
ypos = Symbol('y', positive=True)
# TODO: Remove conds='none' below, let the assumption take care of it.
assert integrate(exp(-I*2*pi*ypos*x)*x, (x, -oo, oo), conds='none') == \
Integral(exp(-I*2*pi*ypos*x)*x, (x, -oo, oo))
def test_issue_3940():
a, b, c, d = symbols('a:d', positive=True)
assert integrate(exp(-x**2 + I*c*x), x) == \
-sqrt(pi)*exp(-c**2/4)*erf(I*c/2 - x)/2
assert integrate(exp(a*x**2 + b*x + c), x) == \
sqrt(pi)*exp(c)*exp(-b**2/(4*a))*erfi(sqrt(a)*x + b/(2*sqrt(a)))/(2*sqrt(a))
from sympy.core.function import expand_mul
from sympy.abc import k
assert expand_mul(integrate(exp(-x**2)*exp(I*k*x), (x, -oo, oo))) == \
sqrt(pi)*exp(-k**2/4)
a, d = symbols('a d', positive=True)
assert expand_mul(integrate(exp(-a*x**2 + 2*d*x), (x, -oo, oo))) == \
sqrt(pi)*exp(d**2/a)/sqrt(a)
def test_issue_5413():
# Note that this is not the same as testing ratint() because integrate()
# pulls out the coefficient.
assert integrate(-a/(a**2 + x**2), x) == I*log(-I*a + x)/2 - I*log(I*a + x)/2
def test_issue_4892a():
A, z = symbols('A z')
c = Symbol('c', nonzero=True)
P1 = -A*exp(-z)
P2 = -A/(c*t)*(sin(x)**2 + cos(y)**2)
h1 = -sin(x)**2 - cos(y)**2
h2 = -sin(x)**2 + sin(y)**2 - 1
# there is still some non-deterministic behavior in integrate
# or trigsimp which permits one of the following
assert integrate(c*(P2 - P1), t) in [
c*(-A*(-h1)*log(c*t)/c + A*t*exp(-z)),
c*(-A*(-h2)*log(c*t)/c + A*t*exp(-z)),
c*( A* h1 *log(c*t)/c + A*t*exp(-z)),
c*( A* h2 *log(c*t)/c + A*t*exp(-z)),
(A*c*t - A*(-h1)*log(t)*exp(z))*exp(-z),
(A*c*t - A*(-h2)*log(t)*exp(z))*exp(-z),
]
def test_issue_4892b():
# Issues relating to issue 4596 are making the actual result of this hard
# to test. The answer should be something like
#
# (-sin(y) + sqrt(-72 + 48*cos(y) - 8*cos(y)**2)/2)*log(x + sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/(2*(3 - cos(y)))) + (-sin(y) - sqrt(-72 +
# 48*cos(y) - 8*cos(y)**2)/2)*log(x - sqrt(-72 + 48*cos(y) -
# 8*cos(y)**2)/(2*(3 - cos(y)))) + x**2*sin(y)/2 + 2*x*cos(y)
expr = (sin(y)*x**3 + 2*cos(y)*x**2 + 12)/(x**2 + 2)
assert trigsimp(factor(integrate(expr, x).diff(x) - expr)) == 0
def test_issue_5178():
assert integrate(sin(x)*f(y, z), (x, 0, pi), (y, 0, pi), (z, 0, pi)) == \
2*Integral(f(y, z), (y, 0, pi), (z, 0, pi))
def test_integrate_series():
f = sin(x).series(x, 0, 10)
g = x**2/2 - x**4/24 + x**6/720 - x**8/40320 + x**10/3628800 + O(x**11)
assert integrate(f, x) == g
assert diff(integrate(f, x), x) == f
assert integrate(O(x**5), x) == O(x**6)
def test_atom_bug():
from sympy.integrals.heurisch import heurisch
assert heurisch(meijerg([], [], [1], [], x), x) is None
def test_limit_bug():
z = Symbol('z', zero=False)
assert integrate(sin(x*y*z), (x, 0, pi), (y, 0, pi)).together() == \
(log(z) - Ci(pi**2*z) + EulerGamma + 2*log(pi))/z
def test_issue_4703():
g = Function('g')
assert integrate(exp(x)*g(x), x).has(Integral)
def test_issue_1888():
f = Function('f')
assert integrate(f(x).diff(x)**2, x).has(Integral)
# The following tests work using meijerint.
def test_issue_3558():
assert integrate(cos(x*y), (x, -pi/2, pi/2), (y, 0, pi)) == 2*Si(pi**2/2)
def test_issue_4422():
assert integrate(1/sqrt(16 + 4*x**2), x) == asinh(x/2) / 2
def test_issue_4493():
assert simplify(integrate(x*sqrt(1 + 2*x), x)) == \
sqrt(2*x + 1)*(6*x**2 + x - 1)/15
def test_issue_4737():
assert integrate(sin(x)/x, (x, -oo, oo)) == pi
assert integrate(sin(x)/x, (x, 0, oo)) == pi/2
assert integrate(sin(x)/x, x) == Si(x)
def test_issue_4992():
# Note: psi in _check_antecedents becomes NaN.
from sympy.core.function import expand_func
a = Symbol('a', positive=True)
assert simplify(expand_func(integrate(exp(-x)*log(x)*x**a, (x, 0, oo)))) == \
(a*polygamma(0, a) + 1)*gamma(a)
def test_issue_4487():
from sympy.functions.special.gamma_functions import lowergamma
assert simplify(integrate(exp(-x)*x**y, x)) == lowergamma(y + 1, x)
def test_issue_4215():
x = Symbol("x")
assert integrate(1/(x**2), (x, -1, 1)) is oo
def test_issue_4400():
n = Symbol('n', integer=True, positive=True)
assert integrate((x**n)*log(x), x) == \
n*x*x**n*log(x)/(n**2 + 2*n + 1) + x*x**n*log(x)/(n**2 + 2*n + 1) - \
x*x**n/(n**2 + 2*n + 1)
def test_issue_6253():
# Note: this used to raise NotImplementedError
# Note: psi in _check_antecedents becomes NaN.
assert integrate((sqrt(1 - x) + sqrt(1 + x))**2/x, x, meijerg=True) == \
Integral((sqrt(-x + 1) + sqrt(x + 1))**2/x, x)
def test_issue_4153():
assert integrate(1/(1 + x + y + z), (x, 0, 1), (y, 0, 1), (z, 0, 1)) in [
-12*log(3) - 3*log(6)/2 + 3*log(8)/2 + 5*log(2) + 7*log(4),
6*log(2) + 8*log(4) - 27*log(3)/2, 22*log(2) - 27*log(3)/2,
-12*log(3) - 3*log(6)/2 + 47*log(2)/2]
def test_issue_4326():
R, b, h = symbols('R b h')
# It doesn't matter if we can do the integral. Just make sure the result
# doesn't contain nan. This is really a test against _eval_interval.
e = integrate(((h*(x - R + b))/b)*sqrt(R**2 - x**2), (x, R - b, R))
assert not e.has(nan)
# See that it evaluates
assert not e.has(Integral)
def test_powers():
assert integrate(2**x + 3**x, x) == 2**x/log(2) + 3**x/log(3)
def test_manual_option():
raises(ValueError, lambda: integrate(1/x, x, manual=True, meijerg=True))
# an example of a function that manual integration cannot handle
assert integrate(log(1+x)/x, (x, 0, 1), manual=True).has(Integral)
def test_meijerg_option():
raises(ValueError, lambda: integrate(1/x, x, meijerg=True, risch=True))
# an example of a function that meijerg integration cannot handle
assert integrate(tan(x), x, meijerg=True) == Integral(tan(x), x)
def test_risch_option():
# risch=True only allowed on indefinite integrals
raises(ValueError, lambda: integrate(1/log(x), (x, 0, oo), risch=True))
assert integrate(exp(-x**2), x, risch=True) == NonElementaryIntegral(exp(-x**2), x)
assert integrate(log(1/x)*y, x, y, risch=True) == y**2*(x*log(1/x)/2 + x/2)
assert integrate(erf(x), x, risch=True) == Integral(erf(x), x)
# TODO: How to test risch=False?
@slow
def test_heurisch_option():
raises(ValueError, lambda: integrate(1/x, x, risch=True, heurisch=True))
# an integral that heurisch can handle
assert integrate(exp(x**2), x, heurisch=True) == sqrt(pi)*erfi(x)/2
# an integral that heurisch currently cannot handle
assert integrate(exp(x)/x, x, heurisch=True) == Integral(exp(x)/x, x)
# an integral where heurisch currently hangs, issue 15471
assert integrate(log(x)*cos(log(x))/x**Rational(3, 4), x, heurisch=False) == (
-128*x**Rational(1, 4)*sin(log(x))/289 + 240*x**Rational(1, 4)*cos(log(x))/289 +
(16*x**Rational(1, 4)*sin(log(x))/17 + 4*x**Rational(1, 4)*cos(log(x))/17)*log(x))
def test_issue_6828():
f = 1/(1.08*x**2 - 4.3)
g = integrate(f, x).diff(x)
assert verify_numerically(f, g, tol=1e-12)
def test_issue_4803():
x_max = Symbol("x_max")
assert integrate(y/pi*exp(-(x_max - x)/cos(a)), x) == \
y*exp((x - x_max)/cos(a))*cos(a)/pi
def test_issue_4234():
assert integrate(1/sqrt(1 + tan(x)**2)) == tan(x)/sqrt(1 + tan(x)**2)
def test_issue_4492():
assert simplify(integrate(x**2 * sqrt(5 - x**2), x)).factor(
deep=True) == Piecewise(
(I*(2*x**5 - 15*x**3 + 25*x - 25*sqrt(x**2 - 5)*acosh(sqrt(5)*x/5)) /
(8*sqrt(x**2 - 5)), (x > sqrt(5)) | (x < -sqrt(5))),
((2*x**5 - 15*x**3 + 25*x - 25*sqrt(5 - x**2)*asin(sqrt(5)*x/5)) /
(-8*sqrt(-x**2 + 5)), True))
def test_issue_2708():
# This test needs to use an integration function that can
# not be evaluated in closed form. Update as needed.
f = 1/(a + z + log(z))
integral_f = NonElementaryIntegral(f, (z, 2, 3))
assert Integral(f, (z, 2, 3)).doit() == integral_f
assert integrate(f + exp(z), (z, 2, 3)) == integral_f - exp(2) + exp(3)
assert integrate(2*f + exp(z), (z, 2, 3)) == \
2*integral_f - exp(2) + exp(3)
assert integrate(exp(1.2*n*s*z*(-t + z)/t), (z, 0, x)) == \
NonElementaryIntegral(exp(-1.2*n*s*z)*exp(1.2*n*s*z**2/t),
(z, 0, x))
def test_issue_2884():
f = (4.000002016020*x + 4.000002016020*y + 4.000006024032)*exp(10.0*x)
e = integrate(f, (x, 0.1, 0.2))
assert str(e) == '1.86831064982608*y + 2.16387491480008'
def test_issue_8368i():
from sympy.functions.elementary.complexes import arg, Abs
assert integrate(exp(-s*x)*cosh(x), (x, 0, oo)) == \
Piecewise(
( pi*Piecewise(
( -s/(pi*(-s**2 + 1)),
Abs(s**2) < 1),
( 1/(pi*s*(1 - 1/s**2)),
Abs(s**(-2)) < 1),
( meijerg(
((S.Half,), (0, 0)),
((0, S.Half), (0,)),
polar_lift(s)**2),
True)
),
s**2 > 1
),
(
Integral(exp(-s*x)*cosh(x), (x, 0, oo)),
True))
assert integrate(exp(-s*x)*sinh(x), (x, 0, oo)) == \
Piecewise(
( -1/(s + 1)/2 - 1/(-s + 1)/2,
And(
Abs(s) > 1,
Abs(arg(s)) < pi/2,
Abs(arg(s)) <= pi/2
)),
( Integral(exp(-s*x)*sinh(x), (x, 0, oo)),
True))
def test_issue_8901():
assert integrate(sinh(1.0*x)) == 1.0*cosh(1.0*x)
assert integrate(tanh(1.0*x)) == 1.0*x - 1.0*log(tanh(1.0*x) + 1)
assert integrate(tanh(x)) == x - log(tanh(x) + 1)
@slow
def test_issue_8945():
assert integrate(sin(x)**3/x, (x, 0, 1)) == -Si(3)/4 + 3*Si(1)/4
assert integrate(sin(x)**3/x, (x, 0, oo)) == pi/4
assert integrate(cos(x)**2/x**2, x) == -Si(2*x) - cos(2*x)/(2*x) - 1/(2*x)
@slow
def test_issue_7130():
if ON_TRAVIS:
skip("Too slow for travis.")
i, L, a, b = symbols('i L a b')
integrand = (cos(pi*i*x/L)**2 / (a + b*x)).rewrite(exp)
assert x not in integrate(integrand, (x, 0, L)).free_symbols
def test_issue_10567():
a, b, c, t = symbols('a b c t')
vt = Matrix([a*t, b, c])
assert integrate(vt, t) == Integral(vt, t).doit()
assert integrate(vt, t) == Matrix([[a*t**2/2], [b*t], [c*t]])
def test_issue_11742():
assert integrate(sqrt(-x**2 + 8*x + 48), (x, 4, 12)) == 16*pi
def test_issue_11856():
t = symbols('t')
assert integrate(sinc(pi*t), t) == Si(pi*t)/pi
@slow
def test_issue_11876():
assert integrate(sqrt(log(1/x)), (x, 0, 1)) == sqrt(pi)/2
def test_issue_4950():
assert integrate((-60*exp(x) - 19.2*exp(4*x))*exp(4*x), x) ==\
-2.4*exp(8*x) - 12.0*exp(5*x)
def test_issue_4968():
assert integrate(sin(log(x**2))) == x*sin(log(x**2))/5 - 2*x*cos(log(x**2))/5
def test_singularities():
assert integrate(1/x**2, (x, -oo, oo)) is oo
assert integrate(1/x**2, (x, -1, 1)) is oo
assert integrate(1/(x - 1)**2, (x, -2, 2)) is oo
assert integrate(1/x**2, (x, 1, -1)) is -oo
assert integrate(1/(x - 1)**2, (x, 2, -2)) is -oo
def test_issue_12645():
x, y = symbols('x y', real=True)
assert (integrate(sin(x*x*x + y*y),
(x, -sqrt(pi - y*y), sqrt(pi - y*y)),
(y, -sqrt(pi), sqrt(pi)))
== Integral(sin(x**3 + y**2),
(x, -sqrt(-y**2 + pi), sqrt(-y**2 + pi)),
(y, -sqrt(pi), sqrt(pi))))
def test_issue_12677():
assert integrate(sin(x) / (cos(x)**3), (x, 0, pi/6)) == Rational(1, 6)
def test_issue_14078():
assert integrate((cos(3*x)-cos(x))/x, (x, 0, oo)) == -log(3)
def test_issue_14064():
assert integrate(1/cosh(x), (x, 0, oo)) == pi/2
def test_issue_14027():
assert integrate(1/(1 + exp(x - S.Half)/(1 + exp(x))), x) == \
x - exp(S.Half)*log(exp(x) + exp(S.Half)/(1 + exp(S.Half)))/(exp(S.Half) + E)
def test_issue_8170():
assert integrate(tan(x), (x, 0, pi/2)) is S.Infinity
def test_issue_8440_14040():
assert integrate(1/x, (x, -1, 1)) is S.NaN
assert integrate(1/(x + 1), (x, -2, 3)) is S.NaN
def test_issue_14096():
assert integrate(1/(x + y)**2, (x, 0, 1)) == -1/(y + 1) + 1/y
assert integrate(1/(1 + x + y + z)**2, (x, 0, 1), (y, 0, 1), (z, 0, 1)) == \
-4*log(4) - 6*log(2) + 9*log(3)
def test_issue_14144():
assert Abs(integrate(1/sqrt(1 - x**3), (x, 0, 1)).n() - 1.402182) < 1e-6
assert Abs(integrate(sqrt(1 - x**3), (x, 0, 1)).n() - 0.841309) < 1e-6
def test_issue_14375():
# This raised a TypeError. The antiderivative has exp_polar, which
# may be possible to unpolarify, so the exact output is not asserted here.
assert integrate(exp(I*x)*log(x), x).has(Ei)
def test_issue_14437():
f = Function('f')(x, y, z)
assert integrate(f, (x, 0, 1), (y, 0, 2), (z, 0, 3)) == \
Integral(f, (x, 0, 1), (y, 0, 2), (z, 0, 3))
def test_issue_14470():
assert integrate(1/sqrt(exp(x) + 1), x) == log(sqrt(exp(x) + 1) - 1) - log(sqrt(exp(x) + 1) + 1)
def test_issue_14877():
f = exp(1 - exp(x**2)*x + 2*x**2)*(2*x**3 + x)/(1 - exp(x**2)*x)**2
assert integrate(f, x) == \
-exp(2*x**2 - x*exp(x**2) + 1)/(x*exp(3*x**2) - exp(2*x**2))
def test_issue_14782():
f = sqrt(-x**2 + 1)*(-x**2 + x)
assert integrate(f, [x, -1, 1]) == - pi / 8
@slow
def test_issue_14782_slow():
f = sqrt(-x**2 + 1)*(-x**2 + x)
assert integrate(f, [x, 0, 1]) == S.One / 3 - pi / 16
def test_issue_12081():
f = x**(Rational(-3, 2))*exp(-x)
assert integrate(f, [x, 0, oo]) is oo
def test_issue_15285():
y = 1/x - 1
f = 4*y*exp(-2*y)/x**2
assert integrate(f, [x, 0, 1]) == 1
def test_issue_15432():
assert integrate(x**n * exp(-x) * log(x), (x, 0, oo)).gammasimp() == Piecewise(
(gamma(n + 1)*polygamma(0, n) + gamma(n + 1)/n, re(n) + 1 > 0),
(Integral(x**n*exp(-x)*log(x), (x, 0, oo)), True))
def test_issue_15124():
omega = IndexedBase('omega')
m, p = symbols('m p', cls=Idx)
assert integrate(exp(x*I*(omega[m] + omega[p])), x, conds='none') == \
-I*exp(I*x*omega[m])*exp(I*x*omega[p])/(omega[m] + omega[p])
def test_issue_15218():
with warns_deprecated_sympy():
Integral(Eq(x, y))
with warns_deprecated_sympy():
assert Integral(Eq(x, y), x) == Eq(Integral(x, x), Integral(y, x))
with warns_deprecated_sympy():
assert Integral(Eq(x, y), x).doit() == Eq(x**2/2, x*y)
with warns(SymPyDeprecationWarning, test_stacklevel=False):
# The warning is made in the ExprWithLimits superclass. The stacklevel
# is correct for integrate(Eq) but not Eq.integrate
assert Eq(x, y).integrate(x) == Eq(x**2/2, x*y)
# These are not deprecated because they are definite integrals
assert integrate(Eq(x, y), (x, 0, 1)) == Eq(S.Half, y)
assert Eq(x, y).integrate((x, 0, 1)) == Eq(S.Half, y)
def test_issue_15292():
res = integrate(exp(-x**2*cos(2*t)) * cos(x**2*sin(2*t)), (x, 0, oo))
assert isinstance(res, Piecewise)
assert gammasimp((res - sqrt(pi)/2 * cos(t)).subs(t, pi/6)) == 0
def test_issue_4514():
assert integrate(sin(2*x)/sin(x), x) == 2*sin(x)
def test_issue_15457():
x, a, b = symbols('x a b', real=True)
definite = integrate(exp(Abs(x-2)), (x, a, b))
indefinite = integrate(exp(Abs(x-2)), x)
assert definite.subs({a: 1, b: 3}) == -2 + 2*E
assert indefinite.subs(x, 3) - indefinite.subs(x, 1) == -2 + 2*E
assert definite.subs({a: -3, b: -1}) == -exp(3) + exp(5)
assert indefinite.subs(x, -1) - indefinite.subs(x, -3) == -exp(3) + exp(5)
def test_issue_15431():
assert integrate(x*exp(x)*log(x), x) == \
(x*exp(x) - exp(x))*log(x) - exp(x) + Ei(x)
def test_issue_15640_log_substitutions():
f = x/log(x)
F = Ei(2*log(x))
assert integrate(f, x) == F and F.diff(x) == f
f = x**3/log(x)**2
F = -x**4/log(x) + 4*Ei(4*log(x))
assert integrate(f, x) == F and F.diff(x) == f
f = sqrt(log(x))/x**2
F = -sqrt(pi)*erfc(sqrt(log(x)))/2 - sqrt(log(x))/x
assert integrate(f, x) == F and F.diff(x) == f
def test_issue_15509():
from sympy.vector import CoordSys3D
N = CoordSys3D('N')
x = N.x
assert integrate(cos(a*x + b), (x, x_1, x_2), heurisch=True) == Piecewise(
(-sin(a*x_1 + b)/a + sin(a*x_2 + b)/a, (a > -oo) & (a < oo) & Ne(a, 0)), \
(-x_1*cos(b) + x_2*cos(b), True))
def test_issue_4311_fast():
x = symbols('x', real=True)
assert integrate(x*abs(9-x**2), x) == Piecewise(
(x**4/4 - 9*x**2/2, x <= -3),
(-x**4/4 + 9*x**2/2 - Rational(81, 2), x <= 3),
(x**4/4 - 9*x**2/2, True))
def test_integrate_with_complex_constants():
K = Symbol('K', positive=True)
x = Symbol('x', real=True)
m = Symbol('m', real=True)
t = Symbol('t', real=True)
assert integrate(exp(-I*K*x**2+m*x), x) == sqrt(I)*sqrt(pi)*exp(-I*m**2
/(4*K))*erfi((-2*I*K*x + m)/(2*sqrt(K)*sqrt(-I)))/(2*sqrt(K))
assert integrate(1/(1 + I*x**2), x) == (-I*(sqrt(-I)*log(x - I*sqrt(-I))/2
- sqrt(-I)*log(x + I*sqrt(-I))/2))
assert integrate(exp(-I*x**2), x) == sqrt(pi)*erf(sqrt(I)*x)/(2*sqrt(I))
assert integrate((1/(exp(I*t)-2)), t) == -t/2 - I*log(exp(I*t) - 2)/2
assert integrate((1/(exp(I*t)-2)), (t, 0, 2*pi)) == -pi
def test_issue_14241():
x = Symbol('x')
n = Symbol('n', positive=True, integer=True)
assert integrate(n * x ** (n - 1) / (x + 1), x) == \
n**2*x**n*lerchphi(x*exp_polar(I*pi), 1, n)*gamma(n)/gamma(n + 1)
def test_issue_13112():
assert integrate(sin(t)**2 / (5 - 4*cos(t)), [t, 0, 2*pi]) == pi / 4
def test_issue_14709b():
h = Symbol('h', positive=True)
i = integrate(x*acos(1 - 2*x/h), (x, 0, h))
assert i == 5*h**2*pi/16
def test_issue_8614():
x = Symbol('x')
t = Symbol('t')
assert integrate(exp(t)/t, (t, -oo, x)) == Ei(x)
assert integrate((exp(-x) - exp(-2*x))/x, (x, 0, oo)) == log(2)
@slow
def test_issue_15494():
s = symbols('s', positive=True)
integrand = (exp(s/2) - 2*exp(1.6*s) + exp(s))*exp(s)
solution = integrate(integrand, s)
assert solution != S.NaN
# Not sure how to test this properly as it is a symbolic expression with floats
# assert str(solution) == '0.666666666666667*exp(1.5*s) + 0.5*exp(2.0*s) - 0.769230769230769*exp(2.6*s)'
# Maybe
assert abs(solution.subs(s, 1) - (-3.67440080236188)) <= 1e-8
integrand = (exp(s/2) - 2*exp(S(8)/5*s) + exp(s))*exp(s)
assert integrate(integrand, s) == -10*exp(13*s/5)/13 + 2*exp(3*s/2)/3 + exp(2*s)/2
def test_li_integral():
y = Symbol('y')
assert Integral(li(y*x**2), x).doit() == Piecewise((x*li(x**2*y) - \
x*Ei(3*log(x**2*y)/2)/sqrt(x**2*y),
Ne(y, 0)), (0, True))
def test_issue_17473():
x = Symbol('x')
n = Symbol('n')
assert integrate(sin(x**n), x) == \
x*x**n*gamma(S(1)/2 + 1/(2*n))*hyper((S(1)/2 + 1/(2*n),),
(S(3)/2, S(3)/2 + 1/(2*n)),
-x**(2*n)/4)/(2*n*gamma(S(3)/2 + 1/(2*n)))
def test_issue_17671():
assert integrate(log(log(x)) / x**2, [x, 1, oo]) == -EulerGamma
assert integrate(log(log(x)) / x**3, [x, 1, oo]) == -log(2)/2 - EulerGamma/2
assert integrate(log(log(x)) / x**10, [x, 1, oo]) == -log(9)/9 - EulerGamma/9
def test_issue_2975():
w = Symbol('w')
C = Symbol('C')
y = Symbol('y')
assert integrate(1/(y**2+C)**(S(3)/2), (y, -w/2, w/2)) == w/(C**(S(3)/2)*sqrt(1 + w**2/(4*C)))
def test_issue_7827():
x, n, M = symbols('x n M')
N = Symbol('N', integer=True)
assert integrate(summation(x*n, (n, 1, N)), x) == x**2*(N**2/4 + N/4)
assert integrate(summation(x*sin(n), (n,1,N)), x) == \
Sum(x**2*sin(n)/2, (n, 1, N))
assert integrate(summation(sin(n*x), (n,1,N)), x) == \
Sum(Piecewise((-cos(n*x)/n, Ne(n, 0)), (0, True)), (n, 1, N))
assert integrate(integrate(summation(sin(n*x), (n,1,N)), x), x) == \
Piecewise((Sum(Piecewise((-sin(n*x)/n**2, Ne(n, 0)), (-x/n, True)),
(n, 1, N)), (n > -oo) & (n < oo) & Ne(n, 0)), (0, True))
assert integrate(Sum(x, (n, 1, M)), x) == M*x**2/2
raises(ValueError, lambda: integrate(Sum(x, (x, y, n)), y))
raises(ValueError, lambda: integrate(Sum(x, (x, 1, n)), n))
raises(ValueError, lambda: integrate(Sum(x, (x, 1, y)), x))
def test_issue_4231():
f = (1 + 2*x + sqrt(x + log(x))*(1 + 3*x) + x**2)/(x*(x + sqrt(x + log(x)))*sqrt(x + log(x)))
assert integrate(f, x) == 2*sqrt(x + log(x)) + 2*log(x + sqrt(x + log(x)))
def test_issue_17841():
f = diff(1/(x**2+x+I), x)
assert integrate(f, x) == 1/(x**2 + x + I)
def test_issue_21034():
x = Symbol('x', real=True, nonzero=True)
f1 = x*(-x**4/asin(5)**4 - x*sinh(x + log(asin(5))) + 5)
f2 = (x + cosh(cos(4)))/(x*(x + 1/(12*x)))
assert integrate(f1, x) == \
-x**6/(6*asin(5)**4) - x**2*cosh(x + log(asin(5))) + 5*x**2/2 + 2*x*sinh(x + log(asin(5))) - 2*cosh(x + log(asin(5)))
assert integrate(f2, x) == \
log(x**2 + S(1)/12)/2 + 2*sqrt(3)*cosh(cos(4))*atan(2*sqrt(3)*x)
def test_issue_4187():
assert integrate(log(x)*exp(-x), x) == Ei(-x) - exp(-x)*log(x)
assert integrate(log(x)*exp(-x), (x, 0, oo)) == -EulerGamma
def test_issue_5547():
L = Symbol('L')
z = Symbol('z')
r0 = Symbol('r0')
R0 = Symbol('R0')
assert integrate(r0**2*cos(z)**2, (z, -L/2, L/2)) == -r0**2*(-L/4 -
sin(L/2)*cos(L/2)/2) + r0**2*(L/4 + sin(L/2)*cos(L/2)/2)
assert integrate(r0**2*cos(R0*z)**2, (z, -L/2, L/2)) == Piecewise(
(-r0**2*(-L*R0/4 - sin(L*R0/2)*cos(L*R0/2)/2)/R0 +
r0**2*(L*R0/4 + sin(L*R0/2)*cos(L*R0/2)/2)/R0, (R0 > -oo) & (R0 < oo) & Ne(R0, 0)),
(L*r0**2, True))
w = 2*pi*z/L
sol = sqrt(2)*sqrt(L)*r0**2*fresnelc(sqrt(2)*sqrt(L))*gamma(S.One/4)/(16*gamma(S(5)/4)) + L*r0**2/2
assert integrate(r0**2*cos(w*z)**2, (z, -L/2, L/2)) == sol
def test_issue_15810():
assert integrate(1/(2**(2*x/3) + 1), (x, 0, oo)) == Rational(3, 2)
def test_issue_21024():
x = Symbol('x', real=True, nonzero=True)
f = log(x)*log(4*x) + log(3*x + exp(2))
F = x*log(x)**2 + x*(1 - 2*log(2)) + (-2*x + 2*x*log(2))*log(x) + \
(x + exp(2)/6)*log(3*x + exp(2)) + exp(2)*log(3*x + exp(2))/6
assert F == integrate(f, x)
f = (x + exp(3))/x**2
F = log(x) - exp(3)/x
assert F == integrate(f, x)
f = (x**2 + exp(5))/x
F = x**2/2 + exp(5)*log(x)
assert F == integrate(f, x)
f = x/(2*x + tanh(1))
F = x/2 - log(2*x + tanh(1))*tanh(1)/4
assert F == integrate(f, x)
f = x - sinh(4)/x
F = x**2/2 - log(x)*sinh(4)
assert F == integrate(f, x)
f = log(x + exp(5)/x)
F = x*log(x + exp(5)/x) - x + 2*exp(Rational(5, 2))*atan(x*exp(Rational(-5, 2)))
assert F == integrate(f, x)
f = x**5/(x + E)
F = x**5/5 - E*x**4/4 + x**3*exp(2)/3 - x**2*exp(3)/2 + x*exp(4) - exp(5)*log(x + E)
assert F == integrate(f, x)
f = 4*x/(x + sinh(5))
F = 4*x - 4*log(x + sinh(5))*sinh(5)
assert F == integrate(f, x)
f = x**2/(2*x + sinh(2))
F = x**2/4 - x*sinh(2)/4 + log(2*x + sinh(2))*sinh(2)**2/8
assert F == integrate(f, x)
f = -x**2/(x + E)
F = -x**2/2 + E*x - exp(2)*log(x + E)
assert F == integrate(f, x)
f = (2*x + 3)*exp(5)/x
F = 2*x*exp(5) + 3*exp(5)*log(x)
assert F == integrate(f, x)
f = x + 2 + cosh(3)/x
F = x**2/2 + 2*x + log(x)*cosh(3)
assert F == integrate(f, x)
f = x - tanh(1)/x**3
F = x**2/2 + tanh(1)/(2*x**2)
assert F == integrate(f, x)
f = (3*x - exp(6))/x
F = 3*x - exp(6)*log(x)
assert F == integrate(f, x)
f = x**4/(x + exp(5))**2 + x
F = x**3/3 + x**2*(Rational(1, 2) - exp(5)) + 3*x*exp(10) - 4*exp(15)*log(x + exp(5)) - exp(20)/(x + exp(5))
assert F == integrate(f, x)
f = x*(x + exp(10)/x**2) + x
F = x**3/3 + x**2/2 + exp(10)*log(x)
assert F == integrate(f, x)
f = x + x/(5*x + sinh(3))
F = x**2/2 + x/5 - log(5*x + sinh(3))*sinh(3)/25
assert F == integrate(f, x)
f = (x + exp(3))/(2*x**2 + 2*x)
F = exp(3)*log(x)/2 - exp(3)*log(x + 1)/2 + log(x + 1)/2
assert F == integrate(f, x).expand()
f = log(x + 4*sinh(4))
F = x*log(x + 4*sinh(4)) - x + 4*log(x + 4*sinh(4))*sinh(4)
assert F == integrate(f, x)
f = -x + 20*(exp(-5) - atan(4)/x)**3*sin(4)/x
F = (-x**2*exp(15)/2 + 20*log(x)*sin(4) - (-180*x**2*exp(5)*sin(4)*atan(4) + 90*x*exp(10)*sin(4)*atan(4)**2 - \
20*exp(15)*sin(4)*atan(4)**3)/(3*x**3))*exp(-15)
assert F == integrate(f, x)
f = 2*x**2*exp(-4) + 6/x
F_true = (2*x**3/3 + 6*exp(4)*log(x))*exp(-4)
assert F_true == integrate(f, x)
def test_issue_21721():
a = Symbol('a')
assert integrate(1/(pi*(1+(x-a)**2)),(x,-oo,oo)).expand() == \
-Heaviside(im(a) - 1, 0) + Heaviside(im(a) + 1, 0)
def test_issue_21831():
theta = symbols('theta')
assert integrate(cos(3*theta)/(5-4*cos(theta)), (theta, 0, 2*pi)) == pi/12
integrand = cos(2*theta)/(5 - 4*cos(theta))
assert integrate(integrand, (theta, 0, 2*pi)) == pi/6
@slow
def test_issue_22033_integral():
assert integrate((x**2 - Rational(1, 4))**2 * sqrt(1 - x**2), (x, -1, 1)) == pi/32
@slow
def test_issue_21671():
assert integrate(1,(z,x**2+y**2,2-x**2-y**2),(y,-sqrt(1-x**2),sqrt(1-x**2)),(x,-1,1)) == pi
assert integrate(-4*(1 - x**2)**(S(3)/2)/3 + 2*sqrt(1 - x**2)*(2 - 2*x**2), (x, -1, 1)) == pi
def test_issue_18527():
# The manual integrator can not currently solve this. Assert that it does
# not give an incorrect result involving Abs when x has real assumptions.
xr = symbols('xr', real=True)
expr = (cos(x)/(4+(sin(x))**2))
res_real = integrate(expr.subs(x, xr), xr, manual=True).subs(xr, x)
assert integrate(expr, x, manual=True) == res_real == Integral(expr, x)
def test_issue_23718():
f = 1/(b*cos(x) + a*sin(x))
Fpos = (-log(-a/b + tan(x/2) - sqrt(a**2 + b**2)/b)/sqrt(a**2 + b**2)
+log(-a/b + tan(x/2) + sqrt(a**2 + b**2)/b)/sqrt(a**2 + b**2))
F = Piecewise(
# XXX: The zoo case here is for a=b=0 so it should just be zoo or maybe
# it doesn't really need to be included at all given that the original
# integrand is really undefined in that case anyway.
(zoo*(-log(tan(x/2) - 1) + log(tan(x/2) + 1)), Eq(a, 0) & Eq(b, 0)),
(log(tan(x/2))/a, Eq(b, 0)),
(-I/(-I*b*sin(x) + b*cos(x)), Eq(a, -I*b)),
(I/(I*b*sin(x) + b*cos(x)), Eq(a, I*b)),
(Fpos, True),
)
assert integrate(f, x) == F
ap, bp = symbols('a, b', positive=True)
rep = {a: ap, b: bp}
assert integrate(f.subs(rep), x) == Fpos.subs(rep)
def test_issue_23566():
i = integrate(1/sqrt(x**2-1), (x, -2, -1))
assert i == -log(2 - sqrt(3))
assert math.isclose(i.n(), 1.31695789692482)
def test_pr_23583():
# This result from meijerg is wrong. Check whether new result is correct when this test fail.
assert integrate(1/sqrt((x - I)**2-1)) == Piecewise((acosh(x - I), Abs((x - I)**2) > 1), (-I*asin(x - I), True))
def test_issue_7264():
assert integrate(exp(x)*sqrt(1 + exp(2*x))) == sqrt(exp(2*x) + 1)*exp(x)/2 + asinh(exp(x))/2
def test_issue_11254a():
assert integrate(sech(x), (x, 0, 1)) == 2*atan(tanh(S.Half))
def test_issue_11254b():
assert integrate(csch(x), x) == log(tanh(x/2))
assert integrate(csch(x), (x, 0, 1)) == oo
def test_issue_11254d():
assert integrate((sech(x)**2).rewrite(sinh), x) == 2*tanh(x/2)/(tanh(x/2)**2 + 1)
def test_issue_22863():
i = integrate((3*x**3-x**2+2*x-4)/sqrt(x**2-3*x+2), (x, 0, 1))
assert i == -101*sqrt(2)/8 - 135*log(3 - 2*sqrt(2))/16
assert math.isclose(i.n(), -2.98126694400554)
def test_issue_9723():
assert integrate(sqrt(x + sqrt(x))) == \
2*sqrt(sqrt(x) + x)*(sqrt(x)/12 + x/3 - S(1)/8) + log(2*sqrt(x) + 2*sqrt(sqrt(x) + x) + 1)/8
assert integrate(sqrt(2*x+3+sqrt(4*x+5))**3) == \
sqrt(2*x + sqrt(4*x + 5) + 3) * \
(9*x/10 + 11*(4*x + 5)**(S(3)/2)/40 + sqrt(4*x + 5)/40 + (4*x + 5)**2/10 + S(11)/10)/2
def test_issue_23704():
# XXX: This is testing that an exception is not raised in risch Ideally
# manualintegrate (manual=True) would be able to compute this but
# manualintegrate is very slow for this example so we don't test that here.
assert (integrate(log(x)/x**2/(c*x**2+b*x+a),x, risch=True)
== NonElementaryIntegral(log(x)/(a*x**2 + b*x**3 + c*x**4), x))
def test_exp_substitution():
assert integrate(1/sqrt(1-exp(2*x))) == log(sqrt(1 - exp(2*x)) - 1)/2 - log(sqrt(1 - exp(2*x)) + 1)/2
def test_hyperbolic():
assert integrate(coth(x)) == x - log(tanh(x) + 1) + log(tanh(x))
assert integrate(sech(x)) == 2*atan(tanh(x/2))
assert integrate(csch(x)) == log(tanh(x/2))
def test_nested_pow():
assert integrate(sqrt(x**2)) == x*sqrt(x**2)/2
def test_sqrt_quadratic():
assert integrate(1/sqrt(3*x**2+4*x+5)) == sqrt(3)*asinh(3*sqrt(11)*(x + S(2)/3)/11)/3
assert integrate(1/sqrt(-3*x**2+4*x+5)) == sqrt(3)*asin(3*sqrt(19)*(x - S(2)/3)/19)/3
assert integrate(1/sqrt(3*x**2+4*x-5)) == sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 4*x - 5) + 4)/3
assert integrate(1/sqrt(a+b*x+c*x**2), x) == \
Piecewise((log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(c, 0)),
(2*sqrt(a + b*x)/b, Ne(b, 0)), (x/sqrt(a), True))
assert integrate((7*x+6)/sqrt(3*x**2+4*x+5)) == \
7*sqrt(3*x**2 + 4*x + 5)/3 + 4*sqrt(3)*asinh(3*sqrt(11)*(x + S(2)/3)/11)/9
assert integrate((7*x+6)/sqrt(-3*x**2+4*x+5)) == \
-7*sqrt(-3*x**2 + 4*x + 5)/3 + 32*sqrt(3)*asin(3*sqrt(19)*(x - S(2)/3)/19)/9
assert integrate((7*x+6)/sqrt(3*x**2+4*x-5)) == \
7*sqrt(3*x**2 + 4*x - 5)/3 + 4*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 4*x - 5) + 4)/9
assert integrate((d+e*x)/sqrt(a+b*x+c*x**2), x) == \
Piecewise((e*sqrt(a + b*x + c*x**2)/c +
(-b*e/(2*c) + d)*log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(c, 0)),
((2*d*sqrt(a + b*x) + 2*e*(-a*sqrt(a + b*x) + (a + b*x)**(S(3)/2)/3)/b)/b, Ne(b, 0)),
((d*x + e*x**2/2)/sqrt(a), True))
assert integrate((3*x**3-x**2+2*x-4)/sqrt(x**2-3*x+2)) == \
sqrt(x**2 - 3*x + 2)*(x**2 + 13*x/4 + S(101)/8) + 135*log(2*x + 2*sqrt(x**2 - 3*x + 2) - 3)/16
assert integrate(sqrt(53225*x**2-66732*x+23013)) == \
(x/2 - S(16683)/53225)*sqrt(53225*x**2 - 66732*x + 23013) + \
111576969*sqrt(2129)*asinh(53225*x/10563 - S(11122)/3521)/1133160250
assert integrate(sqrt(a+b*x+c*x**2), x) == \
Piecewise(((x/2 + b/(4*c))*sqrt(a + b*x + c*x**2) +
(a/2 - b**2/(8*c))*log(b + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(c, 0)),
(2*(a + b*x)**(S(3)/2)/(3*b), Ne(b, 0)),
(sqrt(a)*x, True))
assert integrate(x*sqrt(x**2+2*x+4)) == \
(x**2/3 + x/6 + S(5)/6)*sqrt(x**2 + 2*x + 4) - 3*asinh(sqrt(3)*(x + 1)/3)/2
def test_mul_pow_derivative():
assert integrate(x*sec(x)*tan(x)) == x*sec(x) - log(tan(x) + sec(x))
assert integrate(x*sec(x)**2, x) == x*tan(x) + log(cos(x))
assert integrate(x**3*Derivative(f(x), (x, 4))) == \
x**3*Derivative(f(x), (x, 3)) - 3*x**2*Derivative(f(x), (x, 2)) + 6*x*Derivative(f(x), x) - 6*f(x)
|
80351e256f8e9449f3f85f80a150e46b44ecdd7c56b7e59b2a6167c0a1ab356f | # A collection of failing integrals from the issues.
from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.hyperbolic import (sech, sinh)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, atan, cos, sin, tan)
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.gamma_functions import gamma
from sympy.integrals.integrals import (Integral, integrate)
from sympy.testing.pytest import XFAIL, SKIP, slow, skip, ON_TRAVIS
from sympy.abc import x, k, c, y, b, h, a, m, z, n, t
@SKIP("Too slow for @slow")
@XFAIL
def test_issue_3880():
# integrate_hyperexponential(Poly(t*2*(1 - t0**2)*t0*(x**3 + x**2), t), Poly((1 + t0**2)**2*2*(x**2 + x + 1), t), [Poly(1, x), Poly(1 + t0**2, t0), Poly(t, t)], [x, t0, t], [exp, tan])
assert not integrate(exp(x)*cos(2*x)*sin(2*x) * (x**3 + x**2)/(2*(x**2 + x + 1)), x).has(Integral)
@XFAIL
def test_issue_4212():
assert not integrate(sign(x), x).has(Integral)
@XFAIL
def test_issue_4511():
# This works, but gives a complicated answer. The correct answer is x - cos(x).
# If current answer is simplified, 1 - cos(x) + x is obtained.
# The last one is what Maple gives. It is also quite slow.
assert integrate(cos(x)**2 / (1 - sin(x))) in [x - cos(x), 1 - cos(x) + x,
-2/(tan((S.Half)*x)**2 + 1) + x]
@XFAIL
def test_integrate_DiracDelta_fails():
# issue 6427
assert integrate(integrate(integrate(
DiracDelta(x - y - z), (z, 0, oo)), (y, 0, 1)), (x, 0, 1)) == S.Half
@XFAIL
@slow
def test_issue_4525():
# Warning: takes a long time
assert not integrate((x**m * (1 - x)**n * (a + b*x + c*x**2))/(1 + x**2), (x, 0, 1)).has(Integral)
@XFAIL
@slow
def test_issue_4540():
if ON_TRAVIS:
skip("Too slow for travis.")
# Note, this integral is probably nonelementary
assert not integrate(
(sin(1/x) - x*exp(x)) /
((-sin(1/x) + x*exp(x))*x + x*sin(1/x)), x).has(Integral)
@XFAIL
@slow
def test_issue_4891():
# Requires the hypergeometric function.
assert not integrate(cos(x)**y, x).has(Integral)
@XFAIL
@slow
def test_issue_1796a():
assert not integrate(exp(2*b*x)*exp(-a*x**2), x).has(Integral)
@XFAIL
def test_issue_4895b():
assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, 0)).has(Integral)
@XFAIL
def test_issue_4895c():
assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, -oo, oo)).has(Integral)
@XFAIL
def test_issue_4895d():
assert not integrate(exp(2*b*x)*exp(-a*x**2), (x, 0, oo)).has(Integral)
@XFAIL
@slow
def test_issue_4941():
if ON_TRAVIS:
skip("Too slow for travis.")
assert not integrate(sqrt(1 + sinh(x/20)**2), (x, -25, 25)).has(Integral)
@XFAIL
def test_issue_4992():
# Nonelementary integral. Requires hypergeometric/Meijer-G handling.
assert not integrate(log(x) * x**(k - 1) * exp(-x) / gamma(k), (x, 0, oo)).has(Integral)
@XFAIL
def test_issue_16396a():
i = integrate(1/(1+sqrt(tan(x))), (x, pi/3, pi/6))
assert not i.has(Integral)
@XFAIL
def test_issue_16396b():
i = integrate(x*sin(x)/(1+cos(x)**2), (x, 0, pi))
assert not i.has(Integral)
@XFAIL
def test_issue_16046():
assert integrate(exp(exp(I*x)), [x, 0, 2*pi]) == 2*pi
@XFAIL
def test_issue_15925a():
assert not integrate(sqrt((1+sin(x))**2+(cos(x))**2), (x, -pi/2, pi/2)).has(Integral)
@XFAIL
@slow
def test_issue_15925b():
if ON_TRAVIS:
skip("Too slow for travis.")
assert not integrate(sqrt((-12*cos(x)**2*sin(x))**2+(12*cos(x)*sin(x)**2)**2),
(x, 0, pi/6)).has(Integral)
@XFAIL
def test_issue_15925b_manual():
assert not integrate(sqrt((-12*cos(x)**2*sin(x))**2+(12*cos(x)*sin(x)**2)**2),
(x, 0, pi/6), manual=True).has(Integral)
@XFAIL
@slow
def test_issue_15227():
if ON_TRAVIS:
skip("Too slow for travis.")
i = integrate(log(1-x)*log((1+x)**2)/x, (x, 0, 1))
assert not i.has(Integral)
# assert i == -5*zeta(3)/4
@XFAIL
@slow
def test_issue_14716():
i = integrate(log(x + 5)*cos(pi*x),(x, S.Half, 1))
assert not i.has(Integral)
# Mathematica can not solve it either, but
# integrate(log(x + 5)*cos(pi*x),(x, S.Half, 1)).transform(x, y - 5).doit()
# works
# assert i == -log(Rational(11, 2))/pi - Si(pi*Rational(11, 2))/pi + Si(6*pi)/pi
@XFAIL
def test_issue_14709a():
i = integrate(x*acos(1 - 2*x/h), (x, 0, h))
assert not i.has(Integral)
# assert i == 5*h**2*pi/16
@slow
@XFAIL
def test_issue_14398():
assert not integrate(exp(x**2)*cos(x), x).has(Integral)
@XFAIL
def test_issue_14074():
i = integrate(log(sin(x)), (x, 0, pi/2))
assert not i.has(Integral)
# assert i == -pi*log(2)/2
@XFAIL
@slow
def test_issue_14078b():
i = integrate((atan(4*x)-atan(2*x))/x, (x, 0, oo))
assert not i.has(Integral)
# assert i == pi*log(2)/2
@XFAIL
def test_issue_13792():
i = integrate(log(1/x) / (1 - x), (x, 0, 1))
assert not i.has(Integral)
# assert i in [polylog(2, -exp_polar(I*pi)), pi**2/6]
@XFAIL
def test_issue_11845a():
assert not integrate(exp(y - x**3), (x, 0, 1)).has(Integral)
@XFAIL
def test_issue_11845b():
assert not integrate(exp(-y - x**3), (x, 0, 1)).has(Integral)
@XFAIL
def test_issue_11813():
assert not integrate((a - x)**Rational(-1, 2)*x, (x, 0, a)).has(Integral)
@XFAIL
def test_issue_11254c():
assert not integrate(sech(x)**2, (x, 0, 1)).has(Integral)
@XFAIL
def test_issue_10584():
assert not integrate(sqrt(x**2 + 1/x**2), x).has(Integral)
@XFAIL
def test_issue_9101():
assert not integrate(log(x + sqrt(x**2 + y**2 + z**2)), z).has(Integral)
@XFAIL
def test_issue_7147():
assert not integrate(x/sqrt(a*x**2 + b*x + c)**3, x).has(Integral)
@XFAIL
def test_issue_7109():
assert not integrate(sqrt(a**2/(a**2 - x**2)), x).has(Integral)
@XFAIL
def test_integrate_Piecewise_rational_over_reals():
f = Piecewise(
(0, t - 478.515625*pi < 0),
(13.2075145209219*pi/(0.000871222*t + 0.995)**2, t - 478.515625*pi >= 0))
assert abs((integrate(f, (t, 0, oo)) - 15235.9375*pi).evalf()) <= 1e-7
@XFAIL
def test_issue_4311_slow():
# Not slow when bypassing heurish
assert not integrate(x*abs(9-x**2), x).has(Integral)
@XFAIL
def test_issue_20370():
a = symbols('a', positive=True)
assert integrate((1 + a * cos(x))**-1, (x, 0, 2 * pi)) == (2 * pi / sqrt(1 - a**2))
@XFAIL
def test_polylog():
# log(1/x)*log(x+1)-polylog(2, -x)
assert not integrate(log(1/x)/(x + 1), x).has(Integral)
@XFAIL
def test_polylog_manual():
# Make sure _parts_rule does not go into an infinite loop here
assert not integrate(log(1/x)/(x + 1), x, manual=True).has(Integral)
|
65b83f6217e996fea57dd8f18ee35a26f18fe7bb2def5b1c60b1ec854054a62b | from sympy.calculus.accumulationbounds import AccumBounds
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import (Dict, Tuple)
from sympy.core.function import (Derivative, Function, Lambda, Subs)
from sympy.core.mul import Mul
from sympy.core.numbers import (Float, I, Integer, Rational, oo, pi, zoo)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, Wild, symbols)
from sympy.core.sympify import SympifyError
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (atan2, cos, cot, sin, tan)
from sympy.matrices.dense import (Matrix, zeros)
from sympy.matrices.expressions.special import ZeroMatrix
from sympy.polys.polytools import factor
from sympy.polys.rootoftools import RootOf
from sympy.simplify.cse_main import cse
from sympy.simplify.simplify import nsimplify
from sympy.core.basic import _aresame
from sympy.testing.pytest import XFAIL, raises
from sympy.abc import a, x, y, z, t
def test_subs():
n3 = Rational(3)
e = x
e = e.subs(x, n3)
assert e == Rational(3)
e = 2*x
assert e == 2*x
e = e.subs(x, n3)
assert e == Rational(6)
def test_subs_Matrix():
z = zeros(2)
z1 = ZeroMatrix(2, 2)
assert (x*y).subs({x:z, y:0}) in [z, z1]
assert (x*y).subs({y:z, x:0}) == 0
assert (x*y).subs({y:z, x:0}, simultaneous=True) in [z, z1]
assert (x + y).subs({x: z, y: z}, simultaneous=True) in [z, z1]
assert (x + y).subs({x: z, y: z}) in [z, z1]
# Issue #15528
assert Mul(Matrix([[3]]), x).subs(x, 2.0) == Matrix([[6.0]])
# Does not raise a TypeError, see comment on the MatAdd postprocessor
assert Add(Matrix([[3]]), x).subs(x, 2.0) == Add(Matrix([[3]]), 2.0)
def test_subs_AccumBounds():
e = x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(1, 3)
e = 2*x
e = e.subs(x, AccumBounds(1, 3))
assert e == AccumBounds(2, 6)
e = x + x**2
e = e.subs(x, AccumBounds(-1, 1))
assert e == AccumBounds(-1, 2)
def test_trigonometric():
n3 = Rational(3)
e = (sin(x)**2).diff(x)
assert e == 2*sin(x)*cos(x)
e = e.subs(x, n3)
assert e == 2*cos(n3)*sin(n3)
e = (sin(x)**2).diff(x)
assert e == 2*sin(x)*cos(x)
e = e.subs(sin(x), cos(x))
assert e == 2*cos(x)**2
assert exp(pi).subs(exp, sin) == 0
assert cos(exp(pi)).subs(exp, sin) == 1
i = Symbol('i', integer=True)
zoo = S.ComplexInfinity
assert tan(x).subs(x, pi/2) is zoo
assert cot(x).subs(x, pi) is zoo
assert cot(i*x).subs(x, pi) is zoo
assert tan(i*x).subs(x, pi/2) == tan(i*pi/2)
assert tan(i*x).subs(x, pi/2).subs(i, 1) is zoo
o = Symbol('o', odd=True)
assert tan(o*x).subs(x, pi/2) == tan(o*pi/2)
def test_powers():
assert sqrt(1 - sqrt(x)).subs(x, 4) == I
assert (sqrt(1 - x**2)**3).subs(x, 2) == - 3*I*sqrt(3)
assert (x**Rational(1, 3)).subs(x, 27) == 3
assert (x**Rational(1, 3)).subs(x, -27) == 3*(-1)**Rational(1, 3)
assert ((-x)**Rational(1, 3)).subs(x, 27) == 3*(-1)**Rational(1, 3)
n = Symbol('n', negative=True)
assert (x**n).subs(x, 0) is S.ComplexInfinity
assert exp(-1).subs(S.Exp1, 0) is S.ComplexInfinity
assert (x**(4.0*y)).subs(x**(2.0*y), n) == n**2.0
assert (2**(x + 2)).subs(2, 3) == 3**(x + 3)
def test_logexppow(): # no eval()
x = Symbol('x', real=True)
w = Symbol('w')
e = (3**(1 + x) + 2**(1 + x))/(3**x + 2**x)
assert e.subs(2**x, w) != e
assert e.subs(exp(x*log(Rational(2))), w) != e
def test_bug():
x1 = Symbol('x1')
x2 = Symbol('x2')
y = x1*x2
assert y.subs(x1, Float(3.0)) == Float(3.0)*x2
def test_subbug1():
# see that they don't fail
(x**x).subs(x, 1)
(x**x).subs(x, 1.0)
def test_subbug2():
# Ensure this does not cause infinite recursion
assert Float(7.7).epsilon_eq(abs(x).subs(x, -7.7))
def test_dict_set():
a, b, c = map(Wild, 'abc')
f = 3*cos(4*x)
r = f.match(a*cos(b*x))
assert r == {a: 3, b: 4}
e = a/b*sin(b*x)
assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(r) == 3*sin(4*x) / 4
s = set(r.items())
assert e.subs(s) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(s) == 3*sin(4*x) / 4
assert e.subs(r) == r[a]/r[b]*sin(r[b]*x)
assert e.subs(r) == 3*sin(4*x) / 4
assert x.subs(Dict((x, 1))) == 1
def test_dict_ambigous(): # see issue 3566
f = x*exp(x)
g = z*exp(z)
df = {x: y, exp(x): y}
dg = {z: y, exp(z): y}
assert f.subs(df) == y**2
assert g.subs(dg) == y**2
# and this is how order can affect the result
assert f.subs(x, y).subs(exp(x), y) == y*exp(y)
assert f.subs(exp(x), y).subs(x, y) == y**2
# length of args and count_ops are the same so
# default_sort_key resolves ordering...if one
# doesn't want this result then an unordered
# sequence should not be used.
e = 1 + x*y
assert e.subs({x: y, y: 2}) == 5
# here, there are no obviously clashing keys or values
# but the results depend on the order
assert exp(x/2 + y).subs({exp(y + 1): 2, x: 2}) == exp(y + 1)
def test_deriv_sub_bug3():
f = Function('f')
pat = Derivative(f(x), x, x)
assert pat.subs(y, y**2) == Derivative(f(x), x, x)
assert pat.subs(y, y**2) != Derivative(f(x), x)
def test_equality_subs1():
f = Function('f')
eq = Eq(f(x)**2, x)
res = Eq(Integer(16), x)
assert eq.subs(f(x), 4) == res
def test_equality_subs2():
f = Function('f')
eq = Eq(f(x)**2, 16)
assert bool(eq.subs(f(x), 3)) is False
assert bool(eq.subs(f(x), 4)) is True
def test_issue_3742():
e = sqrt(x)*exp(y)
assert e.subs(sqrt(x), 1) == exp(y)
def test_subs_dict1():
assert (1 + x*y).subs(x, pi) == 1 + pi*y
assert (1 + x*y).subs({x: pi, y: 2}) == 1 + 2*pi
c2, c3, q1p, q2p, c1, s1, s2, s3 = symbols('c2 c3 q1p q2p c1 s1 s2 s3')
test = (c2**2*q2p*c3 + c1**2*s2**2*q2p*c3 + s1**2*s2**2*q2p*c3
- c1**2*q1p*c2*s3 - s1**2*q1p*c2*s3)
assert (test.subs({c1**2: 1 - s1**2, c2**2: 1 - s2**2, c3**3: 1 - s3**2})
== c3*q2p*(1 - s2**2) + c3*q2p*s2**2*(1 - s1**2)
- c2*q1p*s3*(1 - s1**2) + c3*q2p*s1**2*s2**2 - c2*q1p*s3*s1**2)
def test_mul():
x, y, z, a, b, c = symbols('x y z a b c')
A, B, C = symbols('A B C', commutative=0)
assert (x*y*z).subs(z*x, y) == y**2
assert (z*x).subs(1/x, z) == 1
assert (x*y/z).subs(1/z, a) == a*x*y
assert (x*y/z).subs(x/z, a) == a*y
assert (x*y/z).subs(y/z, a) == a*x
assert (x*y/z).subs(x/z, 1/a) == y/a
assert (x*y/z).subs(x, 1/a) == y/(z*a)
assert (2*x*y).subs(5*x*y, z) != z*Rational(2, 5)
assert (x*y*A).subs(x*y, a) == a*A
assert (x**2*y**(x*Rational(3, 2))).subs(x*y**(x/2), 2) == 4*y**(x/2)
assert (x*exp(x*2)).subs(x*exp(x), 2) == 2*exp(x)
assert ((x**(2*y))**3).subs(x**y, 2) == 64
assert (x*A*B).subs(x*A, y) == y*B
assert (x*y*(1 + x)*(1 + x*y)).subs(x*y, 2) == 6*(1 + x)
assert ((1 + A*B)*A*B).subs(A*B, x*A*B)
assert (x*a/z).subs(x/z, A) == a*A
assert (x**3*A).subs(x**2*A, a) == a*x
assert (x**2*A*B).subs(x**2*B, a) == a*A
assert (x**2*A*B).subs(x**2*A, a) == a*B
assert (b*A**3/(a**3*c**3)).subs(a**4*c**3*A**3/b**4, z) == \
b*A**3/(a**3*c**3)
assert (6*x).subs(2*x, y) == 3*y
assert (y*exp(x*Rational(3, 2))).subs(y*exp(x), 2) == 2*exp(x/2)
assert (y*exp(x*Rational(3, 2))).subs(y*exp(x), 2) == 2*exp(x/2)
assert (A**2*B*A**2*B*A**2).subs(A*B*A, C) == A*C**2*A
assert (x*A**3).subs(x*A, y) == y*A**2
assert (x**2*A**3).subs(x*A, y) == y**2*A
assert (x*A**3).subs(x*A, B) == B*A**2
assert (x*A*B*A*exp(x*A*B)).subs(x*A, B) == B**2*A*exp(B*B)
assert (x**2*A*B*A*exp(x*A*B)).subs(x*A, B) == B**3*exp(B**2)
assert (x**3*A*exp(x*A*B)*A*exp(x*A*B)).subs(x*A, B) == \
x*B*exp(B**2)*B*exp(B**2)
assert (x*A*B*C*A*B).subs(x*A*B, C) == C**2*A*B
assert (-I*a*b).subs(a*b, 2) == -2*I
# issue 6361
assert (-8*I*a).subs(-2*a, 1) == 4*I
assert (-I*a).subs(-a, 1) == I
# issue 6441
assert (4*x**2).subs(2*x, y) == y**2
assert (2*4*x**2).subs(2*x, y) == 2*y**2
assert (-x**3/9).subs(-x/3, z) == -z**2*x
assert (-x**3/9).subs(x/3, z) == -z**2*x
assert (-2*x**3/9).subs(x/3, z) == -2*x*z**2
assert (-2*x**3/9).subs(-x/3, z) == -2*x*z**2
assert (-2*x**3/9).subs(-2*x, z) == z*x**2/9
assert (-2*x**3/9).subs(2*x, z) == -z*x**2/9
assert (2*(3*x/5/7)**2).subs(3*x/5, z) == 2*(Rational(1, 7))**2*z**2
assert (4*x).subs(-2*x, z) == 4*x # try keep subs literal
def test_subs_simple():
a = symbols('a', commutative=True)
x = symbols('x', commutative=False)
assert (2*a).subs(1, 3) == 2*a
assert (2*a).subs(2, 3) == 3*a
assert (2*a).subs(a, 3) == 6
assert sin(2).subs(1, 3) == sin(2)
assert sin(2).subs(2, 3) == sin(3)
assert sin(a).subs(a, 3) == sin(3)
assert (2*x).subs(1, 3) == 2*x
assert (2*x).subs(2, 3) == 3*x
assert (2*x).subs(x, 3) == 6
assert sin(x).subs(x, 3) == sin(3)
def test_subs_constants():
a, b = symbols('a b', commutative=True)
x, y = symbols('x y', commutative=False)
assert (a*b).subs(2*a, 1) == a*b
assert (1.5*a*b).subs(a, 1) == 1.5*b
assert (2*a*b).subs(2*a, 1) == b
assert (2*a*b).subs(4*a, 1) == 2*a*b
assert (x*y).subs(2*x, 1) == x*y
assert (1.5*x*y).subs(x, 1) == 1.5*y
assert (2*x*y).subs(2*x, 1) == y
assert (2*x*y).subs(4*x, 1) == 2*x*y
def test_subs_commutative():
a, b, c, d, K = symbols('a b c d K', commutative=True)
assert (a*b).subs(a*b, K) == K
assert (a*b*a*b).subs(a*b, K) == K**2
assert (a*a*b*b).subs(a*b, K) == K**2
assert (a*b*c*d).subs(a*b*c, K) == d*K
assert (a*b**c).subs(a, K) == K*b**c
assert (a*b**c).subs(b, K) == a*K**c
assert (a*b**c).subs(c, K) == a*b**K
assert (a*b*c*b*a).subs(a*b, K) == c*K**2
assert (a**3*b**2*a).subs(a*b, K) == a**2*K**2
def test_subs_noncommutative():
w, x, y, z, L = symbols('w x y z L', commutative=False)
alpha = symbols('alpha', commutative=True)
someint = symbols('someint', commutative=True, integer=True)
assert (x*y).subs(x*y, L) == L
assert (w*y*x).subs(x*y, L) == w*y*x
assert (w*x*y*z).subs(x*y, L) == w*L*z
assert (x*y*x*y).subs(x*y, L) == L**2
assert (x*x*y).subs(x*y, L) == x*L
assert (x*x*y*y).subs(x*y, L) == x*L*y
assert (w*x*y).subs(x*y*z, L) == w*x*y
assert (x*y**z).subs(x, L) == L*y**z
assert (x*y**z).subs(y, L) == x*L**z
assert (x*y**z).subs(z, L) == x*y**L
assert (w*x*y*z*x*y).subs(x*y*z, L) == w*L*x*y
assert (w*x*y*y*w*x*x*y*x*y*y*x*y).subs(x*y, L) == w*L*y*w*x*L**2*y*L
# Check fractional power substitutions. It should not do
# substitutions that choose a value for noncommutative log,
# or inverses that don't already appear in the expressions.
assert (x*x*x).subs(x*x, L) == L*x
assert (x*x*x*y*x*x*x*x).subs(x*x, L) == L*x*y*L**2
for p in range(1, 5):
for k in range(10):
assert (y * x**k).subs(x**p, L) == y * L**(k//p) * x**(k % p)
assert (x**Rational(3, 2)).subs(x**S.Half, L) == x**Rational(3, 2)
assert (x**S.Half).subs(x**S.Half, L) == L
assert (x**Rational(-1, 2)).subs(x**S.Half, L) == x**Rational(-1, 2)
assert (x**Rational(-1, 2)).subs(x**Rational(-1, 2), L) == L
assert (x**(2*someint)).subs(x**someint, L) == L**2
assert (x**(2*someint + 3)).subs(x**someint, L) == L**2*x**3
assert (x**(3*someint + 3)).subs(x**someint, L) == L**3*x**3
assert (x**(3*someint)).subs(x**(2*someint), L) == L * x**someint
assert (x**(4*someint)).subs(x**(2*someint), L) == L**2
assert (x**(4*someint + 1)).subs(x**(2*someint), L) == L**2 * x
assert (x**(4*someint)).subs(x**(3*someint), L) == L * x**someint
assert (x**(4*someint + 1)).subs(x**(3*someint), L) == L * x**(someint + 1)
assert (x**(2*alpha)).subs(x**alpha, L) == x**(2*alpha)
assert (x**(2*alpha + 2)).subs(x**2, L) == x**(2*alpha + 2)
assert ((2*z)**alpha).subs(z**alpha, y) == (2*z)**alpha
assert (x**(2*someint*alpha)).subs(x**someint, L) == x**(2*someint*alpha)
assert (x**(2*someint + alpha)).subs(x**someint, L) == x**(2*someint + alpha)
# This could in principle be substituted, but is not currently
# because it requires recognizing that someint**2 is divisible by
# someint.
assert (x**(someint**2 + 3)).subs(x**someint, L) == x**(someint**2 + 3)
# alpha**z := exp(log(alpha) z) is usually well-defined
assert (4**z).subs(2**z, y) == y**2
# Negative powers
assert (x**(-1)).subs(x**3, L) == x**(-1)
assert (x**(-2)).subs(x**3, L) == x**(-2)
assert (x**(-3)).subs(x**3, L) == L**(-1)
assert (x**(-4)).subs(x**3, L) == L**(-1) * x**(-1)
assert (x**(-5)).subs(x**3, L) == L**(-1) * x**(-2)
assert (x**(-1)).subs(x**(-3), L) == x**(-1)
assert (x**(-2)).subs(x**(-3), L) == x**(-2)
assert (x**(-3)).subs(x**(-3), L) == L
assert (x**(-4)).subs(x**(-3), L) == L * x**(-1)
assert (x**(-5)).subs(x**(-3), L) == L * x**(-2)
assert (x**1).subs(x**(-3), L) == x
assert (x**2).subs(x**(-3), L) == x**2
assert (x**3).subs(x**(-3), L) == L**(-1)
assert (x**4).subs(x**(-3), L) == L**(-1) * x
assert (x**5).subs(x**(-3), L) == L**(-1) * x**2
def test_subs_basic_funcs():
a, b, c, d, K = symbols('a b c d K', commutative=True)
w, x, y, z, L = symbols('w x y z L', commutative=False)
assert (x + y).subs(x + y, L) == L
assert (x - y).subs(x - y, L) == L
assert (x/y).subs(x, L) == L/y
assert (x**y).subs(x, L) == L**y
assert (x**y).subs(y, L) == x**L
assert ((a - c)/b).subs(b, K) == (a - c)/K
assert (exp(x*y - z)).subs(x*y, L) == exp(L - z)
assert (a*exp(x*y - w*z) + b*exp(x*y + w*z)).subs(z, 0) == \
a*exp(x*y) + b*exp(x*y)
assert ((a - b)/(c*d - a*b)).subs(c*d - a*b, K) == (a - b)/K
assert (w*exp(a*b - c)*x*y/4).subs(x*y, L) == w*exp(a*b - c)*L/4
def test_subs_wild():
R, S, T, U = symbols('R S T U', cls=Wild)
assert (R*S).subs(R*S, T) == T
assert (S*R).subs(R*S, T) == T
assert (R + S).subs(R + S, T) == T
assert (R**S).subs(R, T) == T**S
assert (R**S).subs(S, T) == R**T
assert (R*S**T).subs(R, U) == U*S**T
assert (R*S**T).subs(S, U) == R*U**T
assert (R*S**T).subs(T, U) == R*S**U
def test_subs_mixed():
a, b, c, d, K = symbols('a b c d K', commutative=True)
w, x, y, z, L = symbols('w x y z L', commutative=False)
R, S, T, U = symbols('R S T U', cls=Wild)
assert (a*x*y).subs(x*y, L) == a*L
assert (a*b*x*y*x).subs(x*y, L) == a*b*L*x
assert (R*x*y*exp(x*y)).subs(x*y, L) == R*L*exp(L)
assert (a*x*y*y*x - x*y*z*exp(a*b)).subs(x*y, L) == a*L*y*x - L*z*exp(a*b)
e = c*y*x*y*x**(R*S - a*b) - T*(a*R*b*S)
assert e.subs(x*y, L).subs(a*b, K).subs(R*S, U) == \
c*y*L*x**(U - K) - T*(U*K)
def test_division():
a, b, c = symbols('a b c', commutative=True)
x, y, z = symbols('x y z', commutative=True)
assert (1/a).subs(a, c) == 1/c
assert (1/a**2).subs(a, c) == 1/c**2
assert (1/a**2).subs(a, -2) == Rational(1, 4)
assert (-(1/a**2)).subs(a, -2) == Rational(-1, 4)
assert (1/x).subs(x, z) == 1/z
assert (1/x**2).subs(x, z) == 1/z**2
assert (1/x**2).subs(x, -2) == Rational(1, 4)
assert (-(1/x**2)).subs(x, -2) == Rational(-1, 4)
#issue 5360
assert (1/x).subs(x, 0) == 1/S.Zero
def test_add():
a, b, c, d, x, y, t = symbols('a b c d x y t')
assert (a**2 - b - c).subs(a**2 - b, d) in [d - c, a**2 - b - c]
assert (a**2 - c).subs(a**2 - c, d) == d
assert (a**2 - b - c).subs(a**2 - c, d) in [d - b, a**2 - b - c]
assert (a**2 - x - c).subs(a**2 - c, d) in [d - x, a**2 - x - c]
assert (a**2 - b - sqrt(a)).subs(a**2 - sqrt(a), c) == c - b
assert (a + b + exp(a + b)).subs(a + b, c) == c + exp(c)
assert (c + b + exp(c + b)).subs(c + b, a) == a + exp(a)
assert (a + b + c + d).subs(b + c, x) == a + d + x
assert (a + b + c + d).subs(-b - c, x) == a + d - x
assert ((x + 1)*y).subs(x + 1, t) == t*y
assert ((-x - 1)*y).subs(x + 1, t) == -t*y
assert ((x - 1)*y).subs(x + 1, t) == y*(t - 2)
assert ((-x + 1)*y).subs(x + 1, t) == y*(-t + 2)
# this should work every time:
e = a**2 - b - c
assert e.subs(Add(*e.args[:2]), d) == d + e.args[2]
assert e.subs(a**2 - c, d) == d - b
# the fallback should recognize when a change has
# been made; while .1 == Rational(1, 10) they are not the same
# and the change should be made
assert (0.1 + a).subs(0.1, Rational(1, 10)) == Rational(1, 10) + a
e = (-x*(-y + 1) - y*(y - 1))
ans = (-x*(x) - y*(-x)).expand()
assert e.subs(-y + 1, x) == ans
#Test issue 18747
assert (exp(x) + cos(x)).subs(x, oo) == oo
assert Add(*[AccumBounds(-1, 1), oo]) == oo
assert Add(*[oo, AccumBounds(-1, 1)]) == oo
def test_subs_issue_4009():
assert (I*Symbol('a')).subs(1, 2) == I*Symbol('a')
def test_functions_subs():
f, g = symbols('f g', cls=Function)
l = Lambda((x, y), sin(x) + y)
assert (g(y, x) + cos(x)).subs(g, l) == sin(y) + x + cos(x)
assert (f(x)**2).subs(f, sin) == sin(x)**2
assert (f(x, y)).subs(f, log) == log(x, y)
assert (f(x, y)).subs(f, sin) == f(x, y)
assert (sin(x) + atan2(x, y)).subs([[atan2, f], [sin, g]]) == \
f(x, y) + g(x)
assert (g(f(x + y, x))).subs([[f, l], [g, exp]]) == exp(x + sin(x + y))
def test_derivative_subs():
f = Function('f')
g = Function('g')
assert Derivative(f(x), x).subs(f(x), y) != 0
# need xreplace to put the function back, see #13803
assert Derivative(f(x), x).subs(f(x), y).xreplace({y: f(x)}) == \
Derivative(f(x), x)
# issues 5085, 5037
assert cse(Derivative(f(x), x) + f(x))[1][0].has(Derivative)
assert cse(Derivative(f(x, y), x) +
Derivative(f(x, y), y))[1][0].has(Derivative)
eq = Derivative(g(x), g(x))
assert eq.subs(g, f) == Derivative(f(x), f(x))
assert eq.subs(g(x), f(x)) == Derivative(f(x), f(x))
assert eq.subs(g, cos) == Subs(Derivative(y, y), y, cos(x))
def test_derivative_subs2():
f_func, g_func = symbols('f g', cls=Function)
f, g = f_func(x, y, z), g_func(x, y, z)
assert Derivative(f, x, y).subs(Derivative(f, x, y), g) == g
assert Derivative(f, y, x).subs(Derivative(f, x, y), g) == g
assert Derivative(f, x, y).subs(Derivative(f, x), g) == Derivative(g, y)
assert Derivative(f, x, y).subs(Derivative(f, y), g) == Derivative(g, x)
assert (Derivative(f, x, y, z).subs(
Derivative(f, x, z), g) == Derivative(g, y))
assert (Derivative(f, x, y, z).subs(
Derivative(f, z, y), g) == Derivative(g, x))
assert (Derivative(f, x, y, z).subs(
Derivative(f, z, y, x), g) == g)
# Issue 9135
assert (Derivative(f, x, x, y).subs(
Derivative(f, y, y), g) == Derivative(f, x, x, y))
assert (Derivative(f, x, y, y, z).subs(
Derivative(f, x, y, y, y), g) == Derivative(f, x, y, y, z))
assert Derivative(f, x, y).subs(Derivative(f_func(x), x, y), g) == Derivative(f, x, y)
def test_derivative_subs3():
dex = Derivative(exp(x), x)
assert Derivative(dex, x).subs(dex, exp(x)) == dex
assert dex.subs(exp(x), dex) == Derivative(exp(x), x, x)
def test_issue_5284():
A, B = symbols('A B', commutative=False)
assert (x*A).subs(x**2*A, B) == x*A
assert (A**2).subs(A**3, B) == A**2
assert (A**6).subs(A**3, B) == B**2
def test_subs_iter():
assert x.subs(reversed([[x, y]])) == y
it = iter([[x, y]])
assert x.subs(it) == y
assert x.subs(Tuple((x, y))) == y
def test_subs_dict():
a, b, c, d, e = symbols('a b c d e')
assert (2*x + y + z).subs(dict(x=1, y=2)) == 4 + z
l = [(sin(x), 2), (x, 1)]
assert (sin(x)).subs(l) == \
(sin(x)).subs(dict(l)) == 2
assert sin(x).subs(reversed(l)) == sin(1)
expr = sin(2*x) + sqrt(sin(2*x))*cos(2*x)*sin(exp(x)*x)
reps = dict([
(sin(2*x), c),
(sqrt(sin(2*x)), a),
(cos(2*x), b),
(exp(x), e),
(x, d),
])
assert expr.subs(reps) == c + a*b*sin(d*e)
l = [(x, 3), (y, x**2)]
assert (x + y).subs(l) == 3 + x**2
assert (x + y).subs(reversed(l)) == 12
# If changes are made to convert lists into dictionaries and do
# a dictionary-lookup replacement, these tests will help to catch
# some logical errors that might occur
l = [(y, z + 2), (1 + z, 5), (z, 2)]
assert (y - 1 + 3*x).subs(l) == 5 + 3*x
l = [(y, z + 2), (z, 3)]
assert (y - 2).subs(l) == 3
def test_no_arith_subs_on_floats():
assert (x + 3).subs(x + 3, a) == a
assert (x + 3).subs(x + 2, a) == a + 1
assert (x + y + 3).subs(x + 3, a) == a + y
assert (x + y + 3).subs(x + 2, a) == a + y + 1
assert (x + 3.0).subs(x + 3.0, a) == a
assert (x + 3.0).subs(x + 2.0, a) == x + 3.0
assert (x + y + 3.0).subs(x + 3.0, a) == a + y
assert (x + y + 3.0).subs(x + 2.0, a) == x + y + 3.0
def test_issue_5651():
a, b, c, K = symbols('a b c K', commutative=True)
assert (a/(b*c)).subs(b*c, K) == a/K
assert (a/(b**2*c**3)).subs(b*c, K) == a/(c*K**2)
assert (1/(x*y)).subs(x*y, 2) == S.Half
assert ((1 + x*y)/(x*y)).subs(x*y, 1) == 2
assert (x*y*z).subs(x*y, 2) == 2*z
assert ((1 + x*y)/(x*y)/z).subs(x*y, 1) == 2/z
def test_issue_6075():
assert Tuple(1, True).subs(1, 2) == Tuple(2, True)
def test_issue_6079():
# since x + 2.0 == x + 2 we can't do a simple equality test
assert _aresame((x + 2.0).subs(2, 3), x + 2.0)
assert _aresame((x + 2.0).subs(2.0, 3), x + 3)
assert not _aresame(x + 2, x + 2.0)
assert not _aresame(Basic(cos(x), S(1)), Basic(cos(x), S(1.)))
assert _aresame(cos, cos)
assert not _aresame(1, S.One)
assert not _aresame(x, symbols('x', positive=True))
def test_issue_4680():
N = Symbol('N')
assert N.subs(dict(N=3)) == 3
def test_issue_6158():
assert (x - 1).subs(1, y) == x - y
assert (x - 1).subs(-1, y) == x + y
assert (x - oo).subs(oo, y) == x - y
assert (x - oo).subs(-oo, y) == x + y
def test_Function_subs():
f, g, h, i = symbols('f g h i', cls=Function)
p = Piecewise((g(f(x, y)), x < -1), (g(x), x <= 1))
assert p.subs(g, h) == Piecewise((h(f(x, y)), x < -1), (h(x), x <= 1))
assert (f(y) + g(x)).subs({f: h, g: i}) == i(x) + h(y)
def test_simultaneous_subs():
reps = {x: 0, y: 0}
assert (x/y).subs(reps) != (y/x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
reps = reps.items()
assert (x/y).subs(reps) != (y/x).subs(reps)
assert (x/y).subs(reps, simultaneous=True) == \
(y/x).subs(reps, simultaneous=True)
assert Derivative(x, y, z).subs(reps, simultaneous=True) == \
Subs(Derivative(0, y, z), y, 0)
def test_issue_6419_6421():
assert (1/(1 + x/y)).subs(x/y, x) == 1/(1 + x)
assert (-2*I).subs(2*I, x) == -x
assert (-I*x).subs(I*x, x) == -x
assert (-3*I*y**4).subs(3*I*y**2, x) == -x*y**2
def test_issue_6559():
assert (-12*x + y).subs(-x, 1) == 12 + y
# though this involves cse it generated a failure in Mul._eval_subs
x0, x1 = symbols('x0 x1')
e = -log(-12*sqrt(2) + 17)/24 - log(-2*sqrt(2) + 3)/12 + sqrt(2)/3
# XXX modify cse so x1 is eliminated and x0 = -sqrt(2)?
assert cse(e) == (
[(x0, sqrt(2))], [x0/3 - log(-12*x0 + 17)/24 - log(-2*x0 + 3)/12])
def test_issue_5261():
x = symbols('x', real=True)
e = I*x
assert exp(e).subs(exp(x), y) == y**I
assert (2**e).subs(2**x, y) == y**I
eq = (-2)**e
assert eq.subs((-2)**x, y) == eq
def test_issue_6923():
assert (-2*x*sqrt(2)).subs(2*x, y) == -sqrt(2)*y
def test_2arg_hack():
N = Symbol('N', commutative=False)
ans = Mul(2, y + 1, evaluate=False)
assert (2*x*(y + 1)).subs(x, 1, hack2=True) == ans
assert (2*(y + 1 + N)).subs(N, 0, hack2=True) == ans
@XFAIL
def test_mul2():
"""When this fails, remove things labelled "2-arg hack"
1) remove special handling in the fallback of subs that
was added in the same commit as this test
2) remove the special handling in Mul.flatten
"""
assert (2*(x + 1)).is_Mul
def test_noncommutative_subs():
x,y = symbols('x,y', commutative=False)
assert (x*y*x).subs([(x, x*y), (y, x)], simultaneous=True) == (x*y*x**2*y)
def test_issue_2877():
f = Float(2.0)
assert (x + f).subs({f: 2}) == x + 2
def r(a, b, c):
return factor(a*x**2 + b*x + c)
e = r(5.0/6, 10, 5)
assert nsimplify(e) == 5*x**2/6 + 10*x + 5
def test_issue_5910():
t = Symbol('t')
assert (1/(1 - t)).subs(t, 1) is zoo
n = t
d = t - 1
assert (n/d).subs(t, 1) is zoo
assert (-n/-d).subs(t, 1) is zoo
def test_issue_5217():
s = Symbol('s')
z = (1 - 2*x*x)
w = (1 + 2*x*x)
q = 2*x*x*2*y*y
sub = {2*x*x: s}
assert w.subs(sub) == 1 + s
assert z.subs(sub) == 1 - s
assert q == 4*x**2*y**2
assert q.subs(sub) == 2*y**2*s
def test_issue_10829():
assert (4**x).subs(2**x, y) == y**2
assert (9**x).subs(3**x, y) == y**2
def test_pow_eval_subs_no_cache():
# Tests pull request 9376 is working
from sympy.core.cache import clear_cache
s = 1/sqrt(x**2)
# This bug only appeared when the cache was turned off.
# We need to approximate running this test without the cache.
# This creates approximately the same situation.
clear_cache()
# This used to fail with a wrong result.
# It incorrectly returned 1/sqrt(x**2) before this pull request.
result = s.subs(sqrt(x**2), y)
assert result == 1/y
def test_RootOf_issue_10092():
x = Symbol('x', real=True)
eq = x**3 - 17*x**2 + 81*x - 118
r = RootOf(eq, 0)
assert (x < r).subs(x, r) is S.false
def test_issue_8886():
from sympy.physics.mechanics import ReferenceFrame as R
# if something can't be sympified we assume that it
# doesn't play well with SymPy and disallow the
# substitution
v = R('A').x
raises(SympifyError, lambda: x.subs(x, v))
raises(SympifyError, lambda: v.subs(v, x))
assert v.__eq__(x) is False
def test_issue_12657():
# treat -oo like the atom that it is
reps = [(-oo, 1), (oo, 2)]
assert (x < -oo).subs(reps) == (x < 1)
assert (x < -oo).subs(list(reversed(reps))) == (x < 1)
reps = [(-oo, 2), (oo, 1)]
assert (x < oo).subs(reps) == (x < 1)
assert (x < oo).subs(list(reversed(reps))) == (x < 1)
def test_recurse_Application_args():
F = Lambda((x, y), exp(2*x + 3*y))
f = Function('f')
A = f(x, f(x, x))
C = F(x, F(x, x))
assert A.subs(f, F) == A.replace(f, F) == C
def test_Subs_subs():
assert Subs(x*y, x, x).subs(x, y) == Subs(x*y, x, y)
assert Subs(x*y, x, x + 1).subs(x, y) == \
Subs(x*y, x, y + 1)
assert Subs(x*y, y, x + 1).subs(x, y) == \
Subs(y**2, y, y + 1)
a = Subs(x*y*z, (y, x, z), (x + 1, x + z, x))
b = Subs(x*y*z, (y, x, z), (x + 1, y + z, y))
assert a.subs(x, y) == b and \
a.doit().subs(x, y) == a.subs(x, y).doit()
f = Function('f')
g = Function('g')
assert Subs(2*f(x, y) + g(x), f(x, y), 1).subs(y, 2) == Subs(
2*f(x, y) + g(x), (f(x, y), y), (1, 2))
def test_issue_13333():
eq = 1/x
assert eq.subs(dict(x='1/2')) == 2
assert eq.subs(dict(x='(1/2)')) == 2
def test_issue_15234():
x, y = symbols('x y', real=True)
p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed
x, y = symbols('x y', complex=True)
p = 6*x**5 + x**4 - 4*x**3 + 4*x**2 - 2*x + 3
p_subbed = 6*x**5 - 4*x**3 - 2*x + y**4 + 4*y**2 + 3
assert p.subs([(x**i, y**i) for i in [2, 4]]) == p_subbed
def test_issue_6976():
x, y = symbols('x y')
assert (sqrt(x)**3 + sqrt(x) + x + x**2).subs(sqrt(x), y) == \
y**4 + y**3 + y**2 + y
assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
sqrt(x) + x**3 + x + y**2 + y
assert x.subs(x**3, y) == x
assert x.subs(x**Rational(1, 3), y) == y**3
# More substitutions are possible with nonnegative symbols
x, y = symbols('x y', nonnegative=True)
assert (x**4 + x**3 + x**2 + x + sqrt(x)).subs(x**2, y) == \
y**Rational(1, 4) + y**Rational(3, 2) + sqrt(y) + y**2 + y
assert x.subs(x**3, y) == y**Rational(1, 3)
def test_issue_11746():
assert (1/x).subs(x**2, 1) == 1/x
assert (1/(x**3)).subs(x**2, 1) == x**(-3)
assert (1/(x**4)).subs(x**2, 1) == 1
assert (1/(x**3)).subs(x**4, 1) == x**(-3)
assert (1/(y**5)).subs(x**5, 1) == y**(-5)
def test_issue_17823():
from sympy.physics.mechanics import dynamicsymbols
q1, q2 = dynamicsymbols('q1, q2')
expr = q1.diff().diff()**2*q1 + q1.diff()*q2.diff()
reps={q1: a, q1.diff(): a*x*y, q1.diff().diff(): z}
assert expr.subs(reps) == a*x*y*Derivative(q2, t) + a*z**2
def test_issue_19326():
x, y = [i(t) for i in map(Function, 'xy')]
assert (x*y).subs({x: 1 + x, y: x}) == (1 + x)*x
def test_issue_19558():
e = (7*x*cos(x) - 12*log(x)**3)*(-log(x)**4 + 2*sin(x) + 1)**2/ \
(2*(x*cos(x) - 2*log(x)**3)*(3*log(x)**4 - 7*sin(x) + 3)**2)
assert e.subs(x, oo) == AccumBounds(-oo, oo)
assert (sin(x) + cos(x)).subs(x, oo) == AccumBounds(-2, 2)
def test_issue_22033():
xr = Symbol('xr', real=True)
e = (1/xr)
assert e.subs(xr**2, y) == e
def test_guard_against_indeterminate_evaluation():
eq = x**y
assert eq.subs([(x, 1), (y, oo)]) == 1 # because 1**y == 1
assert eq.subs([(y, oo), (x, 1)]) is S.NaN
assert eq.subs({x: 1, y: oo}) is S.NaN
assert eq.subs([(x, 1), (y, oo)], simultaneous=True) is S.NaN
|
5b0f8e9c7ad6bcd1c8fb4e3f7c177761a21abb562b647a6057340b43d39b05f3 | from sympy.assumptions.refine import refine
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import (ExprBuilder, unchanged, Expr,
UnevaluatedExpr)
from sympy.core.function import (Function, expand, WildFunction,
AppliedUndef, Derivative, diff, Subs)
from sympy.core.mul import Mul
from sympy.core.numbers import (NumberSymbol, E, zoo, oo, Float, I,
Rational, nan, Integer, Number, pi)
from sympy.core.power import Pow
from sympy.core.relational import Ge, Lt, Gt, Le
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Symbol, symbols, Dummy, Wild
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.exponential import exp_polar, exp, log
from sympy.functions.elementary.miscellaneous import sqrt, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import tan, sin, cos
from sympy.functions.special.delta_functions import (Heaviside,
DiracDelta)
from sympy.functions.special.error_functions import Si
from sympy.functions.special.gamma_functions import gamma
from sympy.integrals.integrals import integrate, Integral
from sympy.physics.secondquant import FockState
from sympy.polys.partfrac import apart
from sympy.polys.polytools import factor, cancel, Poly
from sympy.polys.rationaltools import together
from sympy.series.order import O
from sympy.sets.sets import FiniteSet
from sympy.simplify.combsimp import combsimp
from sympy.simplify.gammasimp import gammasimp
from sympy.simplify.powsimp import powsimp
from sympy.simplify.radsimp import collect, radsimp
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import simplify, nsimplify
from sympy.simplify.trigsimp import trigsimp
from sympy.tensor.indexed import Indexed
from sympy.physics.units import meter
from sympy.testing.pytest import raises, XFAIL
from sympy.abc import a, b, c, n, t, u, x, y, z
f, g, h = symbols('f,g,h', cls=Function)
class DummyNumber:
"""
Minimal implementation of a number that works with SymPy.
If one has a Number class (e.g. Sage Integer, or some other custom class)
that one wants to work well with SymPy, one has to implement at least the
methods of this class DummyNumber, resp. its subclasses I5 and F1_1.
Basically, one just needs to implement either __int__() or __float__() and
then one needs to make sure that the class works with Python integers and
with itself.
"""
def __radd__(self, a):
if isinstance(a, (int, float)):
return a + self.number
return NotImplemented
def __add__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number + a
return NotImplemented
def __rsub__(self, a):
if isinstance(a, (int, float)):
return a - self.number
return NotImplemented
def __sub__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number - a
return NotImplemented
def __rmul__(self, a):
if isinstance(a, (int, float)):
return a * self.number
return NotImplemented
def __mul__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number * a
return NotImplemented
def __rtruediv__(self, a):
if isinstance(a, (int, float)):
return a / self.number
return NotImplemented
def __truediv__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number / a
return NotImplemented
def __rpow__(self, a):
if isinstance(a, (int, float)):
return a ** self.number
return NotImplemented
def __pow__(self, a):
if isinstance(a, (int, float, DummyNumber)):
return self.number ** a
return NotImplemented
def __pos__(self):
return self.number
def __neg__(self):
return - self.number
class I5(DummyNumber):
number = 5
def __int__(self):
return self.number
class F1_1(DummyNumber):
number = 1.1
def __float__(self):
return self.number
i5 = I5()
f1_1 = F1_1()
# basic SymPy objects
basic_objs = [
Rational(2),
Float("1.3"),
x,
y,
pow(x, y)*y,
]
# all supported objects
all_objs = basic_objs + [
5,
5.5,
i5,
f1_1
]
def dotest(s):
for xo in all_objs:
for yo in all_objs:
s(xo, yo)
return True
def test_basic():
def j(a, b):
x = a
x = +a
x = -a
x = a + b
x = a - b
x = a*b
x = a/b
x = a**b
del x
assert dotest(j)
def test_ibasic():
def s(a, b):
x = a
x += b
x = a
x -= b
x = a
x *= b
x = a
x /= b
assert dotest(s)
class NonBasic:
'''This class represents an object that knows how to implement binary
operations like +, -, etc with Expr but is not a subclass of Basic itself.
The NonExpr subclass below does subclass Basic but not Expr.
For both NonBasic and NonExpr it should be possible for them to override
Expr.__add__ etc because Expr.__add__ should be returning NotImplemented
for non Expr classes. Otherwise Expr.__add__ would create meaningless
objects like Add(Integer(1), FiniteSet(2)) and it wouldn't be possible for
other classes to override these operations when interacting with Expr.
'''
def __add__(self, other):
return SpecialOp('+', self, other)
def __radd__(self, other):
return SpecialOp('+', other, self)
def __sub__(self, other):
return SpecialOp('-', self, other)
def __rsub__(self, other):
return SpecialOp('-', other, self)
def __mul__(self, other):
return SpecialOp('*', self, other)
def __rmul__(self, other):
return SpecialOp('*', other, self)
def __truediv__(self, other):
return SpecialOp('/', self, other)
def __rtruediv__(self, other):
return SpecialOp('/', other, self)
def __floordiv__(self, other):
return SpecialOp('//', self, other)
def __rfloordiv__(self, other):
return SpecialOp('//', other, self)
def __mod__(self, other):
return SpecialOp('%', self, other)
def __rmod__(self, other):
return SpecialOp('%', other, self)
def __divmod__(self, other):
return SpecialOp('divmod', self, other)
def __rdivmod__(self, other):
return SpecialOp('divmod', other, self)
def __pow__(self, other):
return SpecialOp('**', self, other)
def __rpow__(self, other):
return SpecialOp('**', other, self)
def __lt__(self, other):
return SpecialOp('<', self, other)
def __gt__(self, other):
return SpecialOp('>', self, other)
def __le__(self, other):
return SpecialOp('<=', self, other)
def __ge__(self, other):
return SpecialOp('>=', self, other)
class NonExpr(Basic, NonBasic):
'''Like NonBasic above except this is a subclass of Basic but not Expr'''
pass
class SpecialOp():
'''Represents the results of operations with NonBasic and NonExpr'''
def __new__(cls, op, arg1, arg2):
obj = object.__new__(cls)
obj.args = (op, arg1, arg2)
return obj
class NonArithmetic(Basic):
'''Represents a Basic subclass that does not support arithmetic operations'''
pass
def test_cooperative_operations():
'''Tests that Expr uses binary operations cooperatively.
In particular it should be possible for non-Expr classes to override
binary operators like +, - etc when used with Expr instances. This should
work for non-Expr classes whether they are Basic subclasses or not. Also
non-Expr classes that do not define binary operators with Expr should give
TypeError.
'''
# A bunch of instances of Expr subclasses
exprs = [
Expr(),
S.Zero,
S.One,
S.Infinity,
S.NegativeInfinity,
S.ComplexInfinity,
S.Half,
Float(0.5),
Integer(2),
Symbol('x'),
Mul(2, Symbol('x')),
Add(2, Symbol('x')),
Pow(2, Symbol('x')),
]
for e in exprs:
# Test that these classes can override arithmetic operations in
# combination with various Expr types.
for ne in [NonBasic(), NonExpr()]:
results = [
(ne + e, ('+', ne, e)),
(e + ne, ('+', e, ne)),
(ne - e, ('-', ne, e)),
(e - ne, ('-', e, ne)),
(ne * e, ('*', ne, e)),
(e * ne, ('*', e, ne)),
(ne / e, ('/', ne, e)),
(e / ne, ('/', e, ne)),
(ne // e, ('//', ne, e)),
(e // ne, ('//', e, ne)),
(ne % e, ('%', ne, e)),
(e % ne, ('%', e, ne)),
(divmod(ne, e), ('divmod', ne, e)),
(divmod(e, ne), ('divmod', e, ne)),
(ne ** e, ('**', ne, e)),
(e ** ne, ('**', e, ne)),
(e < ne, ('>', ne, e)),
(ne < e, ('<', ne, e)),
(e > ne, ('<', ne, e)),
(ne > e, ('>', ne, e)),
(e <= ne, ('>=', ne, e)),
(ne <= e, ('<=', ne, e)),
(e >= ne, ('<=', ne, e)),
(ne >= e, ('>=', ne, e)),
]
for res, args in results:
assert type(res) is SpecialOp and res.args == args
# These classes do not support binary operators with Expr. Every
# operation should raise in combination with any of the Expr types.
for na in [NonArithmetic(), object()]:
raises(TypeError, lambda : e + na)
raises(TypeError, lambda : na + e)
raises(TypeError, lambda : e - na)
raises(TypeError, lambda : na - e)
raises(TypeError, lambda : e * na)
raises(TypeError, lambda : na * e)
raises(TypeError, lambda : e / na)
raises(TypeError, lambda : na / e)
raises(TypeError, lambda : e // na)
raises(TypeError, lambda : na // e)
raises(TypeError, lambda : e % na)
raises(TypeError, lambda : na % e)
raises(TypeError, lambda : divmod(e, na))
raises(TypeError, lambda : divmod(na, e))
raises(TypeError, lambda : e ** na)
raises(TypeError, lambda : na ** e)
raises(TypeError, lambda : e > na)
raises(TypeError, lambda : na > e)
raises(TypeError, lambda : e < na)
raises(TypeError, lambda : na < e)
raises(TypeError, lambda : e >= na)
raises(TypeError, lambda : na >= e)
raises(TypeError, lambda : e <= na)
raises(TypeError, lambda : na <= e)
def test_relational():
from sympy.core.relational import Lt
assert (pi < 3) is S.false
assert (pi <= 3) is S.false
assert (pi > 3) is S.true
assert (pi >= 3) is S.true
assert (-pi < 3) is S.true
assert (-pi <= 3) is S.true
assert (-pi > 3) is S.false
assert (-pi >= 3) is S.false
r = Symbol('r', real=True)
assert (r - 2 < r - 3) is S.false
assert Lt(x + I, x + I + 2).func == Lt # issue 8288
def test_relational_assumptions():
m1 = Symbol("m1", nonnegative=False)
m2 = Symbol("m2", positive=False)
m3 = Symbol("m3", nonpositive=False)
m4 = Symbol("m4", negative=False)
assert (m1 < 0) == Lt(m1, 0)
assert (m2 <= 0) == Le(m2, 0)
assert (m3 > 0) == Gt(m3, 0)
assert (m4 >= 0) == Ge(m4, 0)
m1 = Symbol("m1", nonnegative=False, real=True)
m2 = Symbol("m2", positive=False, real=True)
m3 = Symbol("m3", nonpositive=False, real=True)
m4 = Symbol("m4", negative=False, real=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=True)
m2 = Symbol("m2", nonpositive=True)
m3 = Symbol("m3", positive=True)
m4 = Symbol("m4", nonnegative=True)
assert (m1 < 0) is S.true
assert (m2 <= 0) is S.true
assert (m3 > 0) is S.true
assert (m4 >= 0) is S.true
m1 = Symbol("m1", negative=False, real=True)
m2 = Symbol("m2", nonpositive=False, real=True)
m3 = Symbol("m3", positive=False, real=True)
m4 = Symbol("m4", nonnegative=False, real=True)
assert (m1 < 0) is S.false
assert (m2 <= 0) is S.false
assert (m3 > 0) is S.false
assert (m4 >= 0) is S.false
# See https://github.com/sympy/sympy/issues/17708
#def test_relational_noncommutative():
# from sympy import Lt, Gt, Le, Ge
# A, B = symbols('A,B', commutative=False)
# assert (A < B) == Lt(A, B)
# assert (A <= B) == Le(A, B)
# assert (A > B) == Gt(A, B)
# assert (A >= B) == Ge(A, B)
def test_basic_nostr():
for obj in basic_objs:
raises(TypeError, lambda: obj + '1')
raises(TypeError, lambda: obj - '1')
if obj == 2:
assert obj * '1' == '11'
else:
raises(TypeError, lambda: obj * '1')
raises(TypeError, lambda: obj / '1')
raises(TypeError, lambda: obj ** '1')
def test_series_expansion_for_uniform_order():
assert (1/x + y + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + x).series(x, 0, 1) == 1/x + y + O(x)
assert (1/x + 1 + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + 1 + x).series(x, 0, 1) == 1/x + 1 + O(x)
assert (1/x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 0) == 1/x + O(1, x)
assert (1/x + y + y*x + x).series(x, 0, 1) == 1/x + y + O(x)
def test_leadterm():
assert (3 + 2*x**(log(3)/log(2) - 1)).leadterm(x) == (3, 0)
assert (1/x**2 + 1 + x + x**2).leadterm(x)[1] == -2
assert (1/x + 1 + x + x**2).leadterm(x)[1] == -1
assert (x**2 + 1/x).leadterm(x)[1] == -1
assert (1 + x**2).leadterm(x)[1] == 0
assert (x + 1).leadterm(x)[1] == 0
assert (x + x**2).leadterm(x)[1] == 1
assert (x**2).leadterm(x)[1] == 2
def test_as_leading_term():
assert (3 + 2*x**(log(3)/log(2) - 1)).as_leading_term(x) == 3
assert (1/x**2 + 1 + x + x**2).as_leading_term(x) == 1/x**2
assert (1/x + 1 + x + x**2).as_leading_term(x) == 1/x
assert (x**2 + 1/x).as_leading_term(x) == 1/x
assert (1 + x**2).as_leading_term(x) == 1
assert (x + 1).as_leading_term(x) == 1
assert (x + x**2).as_leading_term(x) == x
assert (x**2).as_leading_term(x) == x**2
assert (x + oo).as_leading_term(x) is oo
raises(ValueError, lambda: (x + 1).as_leading_term(1))
# https://github.com/sympy/sympy/issues/21177
e = -3*x + (x + Rational(3, 2) - sqrt(3)*S.ImaginaryUnit/2)**2\
- Rational(3, 2) + 3*sqrt(3)*S.ImaginaryUnit/2
assert e.as_leading_term(x) == \
(12*sqrt(3)*x - 12*S.ImaginaryUnit*x)/(4*sqrt(3) + 12*S.ImaginaryUnit)
# https://github.com/sympy/sympy/issues/21245
e = 1 - x - x**2
d = (1 + sqrt(5))/2
assert e.subs(x, y + 1/d).as_leading_term(y) == \
(-576*sqrt(5)*y - 1280*y)/(256*sqrt(5) + 576)
def test_leadterm2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).leadterm(x) == \
(sin(1 + sin(1)), 0)
def test_leadterm3():
assert (y + z + x).leadterm(x) == (y + z, 0)
def test_as_leading_term2():
assert (x*cos(1)*cos(1 + sin(1)) + sin(1 + sin(1))).as_leading_term(x) == \
sin(1 + sin(1))
def test_as_leading_term3():
assert (2 + pi + x).as_leading_term(x) == 2 + pi
assert (2*x + pi*x + x**2).as_leading_term(x) == 2*x + pi*x
def test_as_leading_term4():
# see issue 6843
n = Symbol('n', integer=True, positive=True)
r = -n**3/(2*n**2 + 4*n + 2) - n**2/(n**2 + 2*n + 1) + \
n**2/(n + 1) - n/(2*n**2 + 4*n + 2) + n/(n*x + x) + 2*n/(n + 1) - \
1 + 1/(n*x + x) + 1/(n + 1) - 1/x
assert r.as_leading_term(x).cancel() == n/2
def test_as_leading_term_stub():
class foo(Function):
pass
assert foo(1/x).as_leading_term(x) == foo(1/x)
assert foo(1).as_leading_term(x) == foo(1)
raises(NotImplementedError, lambda: foo(x).as_leading_term(x))
def test_as_leading_term_deriv_integral():
# related to issue 11313
assert Derivative(x ** 3, x).as_leading_term(x) == 3*x**2
assert Derivative(x ** 3, y).as_leading_term(x) == 0
assert Integral(x ** 3, x).as_leading_term(x) == x**4/4
assert Integral(x ** 3, y).as_leading_term(x) == y*x**3
assert Derivative(exp(x), x).as_leading_term(x) == 1
assert Derivative(log(x), x).as_leading_term(x) == (1/x).as_leading_term(x)
def test_atoms():
assert x.atoms() == {x}
assert (1 + x).atoms() == {x, S.One}
assert (1 + 2*cos(x)).atoms(Symbol) == {x}
assert (1 + 2*cos(x)).atoms(Symbol, Number) == {S.One, S(2), x}
assert (2*(x**(y**x))).atoms() == {S(2), x, y}
assert S.Half.atoms() == {S.Half}
assert S.Half.atoms(Symbol) == set()
assert sin(oo).atoms(oo) == set()
assert Poly(0, x).atoms() == {S.Zero, x}
assert Poly(1, x).atoms() == {S.One, x}
assert Poly(x, x).atoms() == {x}
assert Poly(x, x, y).atoms() == {x, y}
assert Poly(x + y, x, y).atoms() == {x, y}
assert Poly(x + y, x, y, z).atoms() == {x, y, z}
assert Poly(x + y*t, x, y, z).atoms() == {t, x, y, z}
assert (I*pi).atoms(NumberSymbol) == {pi}
assert (I*pi).atoms(NumberSymbol, I) == \
(I*pi).atoms(I, NumberSymbol) == {pi, I}
assert exp(exp(x)).atoms(exp) == {exp(exp(x)), exp(x)}
assert (1 + x*(2 + y) + exp(3 + z)).atoms(Add) == \
{1 + x*(2 + y) + exp(3 + z), 2 + y, 3 + z}
# issue 6132
e = (f(x) + sin(x) + 2)
assert e.atoms(AppliedUndef) == \
{f(x)}
assert e.atoms(AppliedUndef, Function) == \
{f(x), sin(x)}
assert e.atoms(Function) == \
{f(x), sin(x)}
assert e.atoms(AppliedUndef, Number) == \
{f(x), S(2)}
assert e.atoms(Function, Number) == \
{S(2), sin(x), f(x)}
def test_is_polynomial():
k = Symbol('k', nonnegative=True, integer=True)
assert Rational(2).is_polynomial(x, y, z) is True
assert (S.Pi).is_polynomial(x, y, z) is True
assert x.is_polynomial(x) is True
assert x.is_polynomial(y) is True
assert (x**2).is_polynomial(x) is True
assert (x**2).is_polynomial(y) is True
assert (x**(-2)).is_polynomial(x) is False
assert (x**(-2)).is_polynomial(y) is True
assert (2**x).is_polynomial(x) is False
assert (2**x).is_polynomial(y) is True
assert (x**k).is_polynomial(x) is False
assert (x**k).is_polynomial(k) is False
assert (x**x).is_polynomial(x) is False
assert (k**k).is_polynomial(k) is False
assert (k**x).is_polynomial(k) is False
assert (x**(-k)).is_polynomial(x) is False
assert ((2*x)**k).is_polynomial(x) is False
assert (x**2 + 3*x - 8).is_polynomial(x) is True
assert (x**2 + 3*x - 8).is_polynomial(y) is True
assert (x**2 + 3*x - 8).is_polynomial() is True
assert sqrt(x).is_polynomial(x) is False
assert (sqrt(x)**3).is_polynomial(x) is False
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(x) is True
assert (x**2 + 3*x*sqrt(y) - 8).is_polynomial(y) is False
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial() is True
assert ((x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial() is False
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(2)).is_polynomial(x, y) is True
assert (
(x**2)*(y**2) + x*(y**2) + y*x + exp(x)).is_polynomial(x, y) is False
assert (1/f(x) + 1).is_polynomial(f(x)) is False
def test_is_rational_function():
assert Integer(1).is_rational_function() is True
assert Integer(1).is_rational_function(x) is True
assert Rational(17, 54).is_rational_function() is True
assert Rational(17, 54).is_rational_function(x) is True
assert (12/x).is_rational_function() is True
assert (12/x).is_rational_function(x) is True
assert (x/y).is_rational_function() is True
assert (x/y).is_rational_function(x) is True
assert (x/y).is_rational_function(x, y) is True
assert (x**2 + 1/x/y).is_rational_function() is True
assert (x**2 + 1/x/y).is_rational_function(x) is True
assert (x**2 + 1/x/y).is_rational_function(x, y) is True
assert (sin(y)/x).is_rational_function() is False
assert (sin(y)/x).is_rational_function(y) is False
assert (sin(y)/x).is_rational_function(x) is True
assert (sin(y)/x).is_rational_function(x, y) is False
assert (S.NaN).is_rational_function() is False
assert (S.Infinity).is_rational_function() is False
assert (S.NegativeInfinity).is_rational_function() is False
assert (S.ComplexInfinity).is_rational_function() is False
def test_is_meromorphic():
f = a/x**2 + b + x + c*x**2
assert f.is_meromorphic(x, 0) is True
assert f.is_meromorphic(x, 1) is True
assert f.is_meromorphic(x, zoo) is True
g = 3 + 2*x**(log(3)/log(2) - 1)
assert g.is_meromorphic(x, 0) is False
assert g.is_meromorphic(x, 1) is True
assert g.is_meromorphic(x, zoo) is False
n = Symbol('n', integer=True)
e = sin(1/x)**n*x
assert e.is_meromorphic(x, 0) is False
assert e.is_meromorphic(x, 1) is True
assert e.is_meromorphic(x, zoo) is False
e = log(x)**pi
assert e.is_meromorphic(x, 0) is False
assert e.is_meromorphic(x, 1) is False
assert e.is_meromorphic(x, 2) is True
assert e.is_meromorphic(x, zoo) is False
assert (log(x)**a).is_meromorphic(x, 0) is False
assert (log(x)**a).is_meromorphic(x, 1) is False
assert (a**log(x)).is_meromorphic(x, 0) is None
assert (3**log(x)).is_meromorphic(x, 0) is False
assert (3**log(x)).is_meromorphic(x, 1) is True
def test_is_algebraic_expr():
assert sqrt(3).is_algebraic_expr(x) is True
assert sqrt(3).is_algebraic_expr() is True
eq = ((1 + x**2)/(1 - y**2))**(S.One/3)
assert eq.is_algebraic_expr(x) is True
assert eq.is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(x) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr(y) is True
assert (sqrt(x) + y**(S(2)/3)).is_algebraic_expr() is True
assert (cos(y)/sqrt(x)).is_algebraic_expr() is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x) is True
assert (cos(y)/sqrt(x)).is_algebraic_expr(y) is False
assert (cos(y)/sqrt(x)).is_algebraic_expr(x, y) is False
def test_SAGE1():
#see https://github.com/sympy/sympy/issues/3346
class MyInt:
def _sympy_(self):
return Integer(5)
m = MyInt()
e = Rational(2)*m
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE2():
class MyInt:
def __int__(self):
return 5
assert sympify(MyInt()) == 5
e = Rational(2)*MyInt()
assert e == 10
raises(TypeError, lambda: Rational(2)*MyInt)
def test_SAGE3():
class MySymbol:
def __rmul__(self, other):
return ('mys', other, self)
o = MySymbol()
e = x*o
assert e == ('mys', x, o)
def test_len():
e = x*y
assert len(e.args) == 2
e = x + y + z
assert len(e.args) == 3
def test_doit():
a = Integral(x**2, x)
assert isinstance(a.doit(), Integral) is False
assert isinstance(a.doit(integrals=True), Integral) is False
assert isinstance(a.doit(integrals=False), Integral) is True
assert (2*Integral(x, x)).doit() == x**2
def test_attribute_error():
raises(AttributeError, lambda: x.cos())
raises(AttributeError, lambda: x.sin())
raises(AttributeError, lambda: x.exp())
def test_args():
assert (x*y).args in ((x, y), (y, x))
assert (x + y).args in ((x, y), (y, x))
assert (x*y + 1).args in ((x*y, 1), (1, x*y))
assert sin(x*y).args == (x*y,)
assert sin(x*y).args[0] == x*y
assert (x**y).args == (x, y)
assert (x**y).args[0] == x
assert (x**y).args[1] == y
def test_noncommutative_expand_issue_3757():
A, B, C = symbols('A,B,C', commutative=False)
assert A*B - B*A != 0
assert (A*(A + B)*B).expand() == A**2*B + A*B**2
assert (A*(A + B + C)*B).expand() == A**2*B + A*B**2 + A*C*B
def test_as_numer_denom():
a, b, c = symbols('a, b, c')
assert nan.as_numer_denom() == (nan, 1)
assert oo.as_numer_denom() == (oo, 1)
assert (-oo).as_numer_denom() == (-oo, 1)
assert zoo.as_numer_denom() == (zoo, 1)
assert (-zoo).as_numer_denom() == (zoo, 1)
assert x.as_numer_denom() == (x, 1)
assert (1/x).as_numer_denom() == (1, x)
assert (x/y).as_numer_denom() == (x, y)
assert (x/2).as_numer_denom() == (x, 2)
assert (x*y/z).as_numer_denom() == (x*y, z)
assert (x/(y*z)).as_numer_denom() == (x, y*z)
assert S.Half.as_numer_denom() == (1, 2)
assert (1/y**2).as_numer_denom() == (1, y**2)
assert (x/y**2).as_numer_denom() == (x, y**2)
assert ((x**2 + 1)/y).as_numer_denom() == (x**2 + 1, y)
assert (x*(y + 1)/y**7).as_numer_denom() == (x*(y + 1), y**7)
assert (x**-2).as_numer_denom() == (1, x**2)
assert (a/x + b/2/x + c/3/x).as_numer_denom() == \
(6*a + 3*b + 2*c, 6*x)
assert (a/x + b/2/x + c/3/y).as_numer_denom() == \
(2*c*x + y*(6*a + 3*b), 6*x*y)
assert (a/x + b/2/x + c/.5/x).as_numer_denom() == \
(2*a + b + 4.0*c, 2*x)
# this should take no more than a few seconds
assert int(log(Add(*[Dummy()/i/x for i in range(1, 705)]
).as_numer_denom()[1]/x).n(4)) == 705
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).as_numer_denom() == \
(x + i, 3)
assert (S.Infinity + x/3 + y/4).as_numer_denom() == \
(4*x + 3*y + S.Infinity, 12)
assert (oo*x + zoo*y).as_numer_denom() == \
(zoo*y + oo*x, 1)
A, B, C = symbols('A,B,C', commutative=False)
assert (A*B*C**-1).as_numer_denom() == (A*B*C**-1, 1)
assert (A*B*C**-1/x).as_numer_denom() == (A*B*C**-1, x)
assert (C**-1*A*B).as_numer_denom() == (C**-1*A*B, 1)
assert (C**-1*A*B/x).as_numer_denom() == (C**-1*A*B, x)
assert ((A*B*C)**-1).as_numer_denom() == ((A*B*C)**-1, 1)
assert ((A*B*C)**-1/x).as_numer_denom() == ((A*B*C)**-1, x)
# the following morphs from Add to Mul during processing
assert Add(0, (x + y)/z/-2, evaluate=False).as_numer_denom(
) == (-x - y, 2*z)
def test_trunc():
import math
x, y = symbols('x y')
assert math.trunc(2) == 2
assert math.trunc(4.57) == 4
assert math.trunc(-5.79) == -5
assert math.trunc(pi) == 3
assert math.trunc(log(7)) == 1
assert math.trunc(exp(5)) == 148
assert math.trunc(cos(pi)) == -1
assert math.trunc(sin(5)) == 0
raises(TypeError, lambda: math.trunc(x))
raises(TypeError, lambda: math.trunc(x + y**2))
raises(TypeError, lambda: math.trunc(oo))
def test_as_independent():
assert S.Zero.as_independent(x, as_Add=True) == (0, 0)
assert S.Zero.as_independent(x, as_Add=False) == (0, 0)
assert (2*x*sin(x) + y + x).as_independent(x) == (y, x + 2*x*sin(x))
assert (2*x*sin(x) + y + x).as_independent(y) == (x + 2*x*sin(x), y)
assert (2*x*sin(x) + y + x).as_independent(x, y) == (0, y + x + 2*x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(x) == (cos(y), x*sin(x))
assert (x*sin(x)*cos(y)).as_independent(y) == (x*sin(x), cos(y))
assert (x*sin(x)*cos(y)).as_independent(x, y) == (1, x*sin(x)*cos(y))
assert (sin(x)).as_independent(x) == (1, sin(x))
assert (sin(x)).as_independent(y) == (sin(x), 1)
assert (2*sin(x)).as_independent(x) == (2, sin(x))
assert (2*sin(x)).as_independent(y) == (2*sin(x), 1)
# issue 4903 = 1766b
n1, n2, n3 = symbols('n1 n2 n3', commutative=False)
assert (n1 + n1*n2).as_independent(n2) == (n1, n1*n2)
assert (n2*n1 + n1*n2).as_independent(n2) == (0, n1*n2 + n2*n1)
assert (n1*n2*n1).as_independent(n2) == (n1, n2*n1)
assert (n1*n2*n1).as_independent(n1) == (1, n1*n2*n1)
assert (3*x).as_independent(x, as_Add=True) == (0, 3*x)
assert (3*x).as_independent(x, as_Add=False) == (3, x)
assert (3 + x).as_independent(x, as_Add=True) == (3, x)
assert (3 + x).as_independent(x, as_Add=False) == (1, 3 + x)
# issue 5479
assert (3*x).as_independent(Symbol) == (3, x)
# issue 5648
assert (n1*x*y).as_independent(x) == (n1*y, x)
assert ((x + n1)*(x - y)).as_independent(x) == (1, (x + n1)*(x - y))
assert ((x + n1)*(x - y)).as_independent(y) == (x + n1, x - y)
assert (DiracDelta(x - n1)*DiracDelta(x - y)).as_independent(x) \
== (1, DiracDelta(x - n1)*DiracDelta(x - y))
assert (x*y*n1*n2*n3).as_independent(n2) == (x*y*n1, n2*n3)
assert (x*y*n1*n2*n3).as_independent(n1) == (x*y, n1*n2*n3)
assert (x*y*n1*n2*n3).as_independent(n3) == (x*y*n1*n2, n3)
assert (DiracDelta(x - n1)*DiracDelta(y - n1)*DiracDelta(x - n2)).as_independent(y) == \
(DiracDelta(x - n1)*DiracDelta(x - n2), DiracDelta(y - n1))
# issue 5784
assert (x + Integral(x, (x, 1, 2))).as_independent(x, strict=True) == \
(Integral(x, (x, 1, 2)), x)
eq = Add(x, -x, 2, -3, evaluate=False)
assert eq.as_independent(x) == (-1, Add(x, -x, evaluate=False))
eq = Mul(x, 1/x, 2, -3, evaluate=False)
assert eq.as_independent(x) == (-6, Mul(x, 1/x, evaluate=False))
assert (x*y).as_independent(z, as_Add=True) == (x*y, 0)
@XFAIL
def test_call_2():
# TODO UndefinedFunction does not subclass Expr
assert (2*f)(x) == 2*f(x)
def test_replace():
e = log(sin(x)) + tan(sin(x**2))
assert e.replace(sin, cos) == log(cos(x)) + tan(cos(x**2))
assert e.replace(
sin, lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
a = Wild('a')
b = Wild('b')
assert e.replace(sin(a), cos(a)) == log(cos(x)) + tan(cos(x**2))
assert e.replace(
sin(a), lambda a: sin(2*a)) == log(sin(2*x)) + tan(sin(2*x**2))
# test exact
assert (2*x).replace(a*x + b, b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, b - a) == 2*x
assert (2*x).replace(a*x + b, b - a, exact=False) == 2/x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=True) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a) == 2*x
assert (2*x).replace(a*x + b, lambda a, b: b - a, exact=False) == 2/x
g = 2*sin(x**3)
assert g.replace(
lambda expr: expr.is_Number, lambda expr: expr**2) == 4*sin(x**9)
assert cos(x).replace(cos, sin, map=True) == (sin(x), {cos(x): sin(x)})
assert sin(x).replace(cos, sin) == sin(x)
cond, func = lambda x: x.is_Mul, lambda x: 2*x
assert (x*y).replace(cond, func, map=True) == (2*x*y, {x*y: 2*x*y})
assert (x*(1 + x*y)).replace(cond, func, map=True) == \
(2*x*(2*x*y + 1), {x*(2*x*y + 1): 2*x*(2*x*y + 1), x*y: 2*x*y})
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y, map=True) == \
(sin(x), {sin(x): sin(x)/y})
# if not simultaneous then y*sin(x) -> y*sin(x)/y = sin(x) -> sin(x)/y
assert (y*sin(x)).replace(sin, lambda expr: sin(expr)/y,
simultaneous=False) == sin(x)/y
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e
) == x**2/2 + O(x**3)
assert (x**2 + O(x**3)).replace(Pow, lambda b, e: b**e/e,
simultaneous=False) == x**2/2 + O(x**3)
assert (x*(x*y + 3)).replace(lambda x: x.is_Mul, lambda x: 2 + x) == \
x*(x*y + 5) + 2
e = (x*y + 1)*(2*x*y + 1) + 1
assert e.replace(cond, func, map=True) == (
2*((2*x*y + 1)*(4*x*y + 1)) + 1,
{2*x*y: 4*x*y, x*y: 2*x*y, (2*x*y + 1)*(4*x*y + 1):
2*((2*x*y + 1)*(4*x*y + 1))})
assert x.replace(x, y) == y
assert (x + 1).replace(1, 2) == x + 2
# https://groups.google.com/forum/#!topic/sympy/8wCgeC95tz0
n1, n2, n3 = symbols('n1:4', commutative=False)
assert (n1*f(n2)).replace(f, lambda x: x) == n1*n2
assert (n3*f(n2)).replace(f, lambda x: x) == n3*n2
# issue 16725
assert S.Zero.replace(Wild('x'), 1) == 1
# let the user override the default decision of False
assert S.Zero.replace(Wild('x'), 1, exact=True) == 0
def test_find():
expr = (x + y + 2 + sin(3*x))
assert expr.find(lambda u: u.is_Integer) == {S(2), S(3)}
assert expr.find(lambda u: u.is_Symbol) == {x, y}
assert expr.find(lambda u: u.is_Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(lambda u: u.is_Symbol, group=True) == {x: 2, y: 1}
assert expr.find(Integer) == {S(2), S(3)}
assert expr.find(Symbol) == {x, y}
assert expr.find(Integer, group=True) == {S(2): 1, S(3): 1}
assert expr.find(Symbol, group=True) == {x: 2, y: 1}
a = Wild('a')
expr = sin(sin(x)) + sin(x) + cos(x) + x
assert expr.find(lambda u: type(u) is sin) == {sin(x), sin(sin(x))}
assert expr.find(
lambda u: type(u) is sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin(a)) == {sin(x), sin(sin(x))}
assert expr.find(sin(a), group=True) == {sin(x): 2, sin(sin(x)): 1}
assert expr.find(sin) == {sin(x), sin(sin(x))}
assert expr.find(sin, group=True) == {sin(x): 2, sin(sin(x)): 1}
def test_count():
expr = (x + y + 2 + sin(3*x))
assert expr.count(lambda u: u.is_Integer) == 2
assert expr.count(lambda u: u.is_Symbol) == 3
assert expr.count(Integer) == 2
assert expr.count(Symbol) == 3
assert expr.count(2) == 1
a = Wild('a')
assert expr.count(sin) == 1
assert expr.count(sin(a)) == 1
assert expr.count(lambda u: type(u) is sin) == 1
assert f(x).count(f(x)) == 1
assert f(x).diff(x).count(f(x)) == 1
assert f(x).diff(x).count(x) == 2
def test_has_basics():
p = Wild('p')
assert sin(x).has(x)
assert sin(x).has(sin)
assert not sin(x).has(y)
assert not sin(x).has(cos)
assert f(x).has(x)
assert f(x).has(f)
assert not f(x).has(y)
assert not f(x).has(g)
assert f(x).diff(x).has(x)
assert f(x).diff(x).has(f)
assert f(x).diff(x).has(Derivative)
assert not f(x).diff(x).has(y)
assert not f(x).diff(x).has(g)
assert not f(x).diff(x).has(sin)
assert (x**2).has(Symbol)
assert not (x**2).has(Wild)
assert (2*p).has(Wild)
assert not x.has()
def test_has_multiple():
f = x**2*y + sin(2**t + log(z))
assert f.has(x)
assert f.has(y)
assert f.has(z)
assert f.has(t)
assert not f.has(u)
assert f.has(x, y, z, t)
assert f.has(x, y, z, t, u)
i = Integer(4400)
assert not i.has(x)
assert (i*x**i).has(x)
assert not (i*y**i).has(x)
assert (i*y**i).has(x, y)
assert not (i*y**i).has(x, z)
def test_has_piecewise():
f = (x*y + 3/y)**(3 + 2)
p = Piecewise((g(x), x < -1), (1, x <= 1), (f, True))
assert p.has(x)
assert p.has(y)
assert not p.has(z)
assert p.has(1)
assert p.has(3)
assert not p.has(4)
assert p.has(f)
assert p.has(g)
assert not p.has(h)
def test_has_iterative():
A, B, C = symbols('A,B,C', commutative=False)
f = x*gamma(x)*sin(x)*exp(x*y)*A*B*C*cos(x*A*B)
assert f.has(x)
assert f.has(x*y)
assert f.has(x*sin(x))
assert not f.has(x*sin(y))
assert f.has(x*A)
assert f.has(x*A*B)
assert not f.has(x*A*C)
assert f.has(x*A*B*C)
assert not f.has(x*A*C*B)
assert f.has(x*sin(x)*A*B*C)
assert not f.has(x*sin(x)*A*C*B)
assert not f.has(x*sin(y)*A*B*C)
assert f.has(x*gamma(x))
assert not f.has(x + sin(x))
assert (x & y & z).has(x & z)
def test_has_integrals():
f = Integral(x**2 + sin(x*y*z), (x, 0, x + y + z))
assert f.has(x + y)
assert f.has(x + z)
assert f.has(y + z)
assert f.has(x*y)
assert f.has(x*z)
assert f.has(y*z)
assert not f.has(2*x + y)
assert not f.has(2*x*y)
def test_has_tuple():
assert Tuple(x, y).has(x)
assert not Tuple(x, y).has(z)
assert Tuple(f(x), g(x)).has(x)
assert not Tuple(f(x), g(x)).has(y)
assert Tuple(f(x), g(x)).has(f)
assert Tuple(f(x), g(x)).has(f(x))
# XXX to be deprecated
#assert not Tuple(f, g).has(x)
#assert Tuple(f, g).has(f)
#assert not Tuple(f, g).has(h)
assert Tuple(True).has(True)
assert Tuple(True).has(S.true)
assert not Tuple(True).has(1)
def test_has_units():
from sympy.physics.units import m, s
assert (x*m/s).has(x)
assert (x*m/s).has(y, z) is False
def test_has_polys():
poly = Poly(x**2 + x*y*sin(z), x, y, t)
assert poly.has(x)
assert poly.has(x, y, z)
assert poly.has(x, y, z, t)
def test_has_physics():
assert FockState((x, y)).has(x)
def test_as_poly_as_expr():
f = x**2 + 2*x*y
assert f.as_poly().as_expr() == f
assert f.as_poly(x, y).as_expr() == f
assert (f + sin(x)).as_poly(x, y) is None
p = Poly(f, x, y)
assert p.as_poly() == p
# https://github.com/sympy/sympy/issues/20610
assert S(2).as_poly() is None
assert sqrt(2).as_poly(extension=True) is None
raises(AttributeError, lambda: Tuple(x, x).as_poly(x))
raises(AttributeError, lambda: Tuple(x ** 2, x, y).as_poly(x))
def test_nonzero():
assert bool(S.Zero) is False
assert bool(S.One) is True
assert bool(x) is True
assert bool(x + y) is True
assert bool(x - x) is False
assert bool(x*y) is True
assert bool(x*1) is True
assert bool(x*0) is False
def test_is_number():
assert Float(3.14).is_number is True
assert Integer(737).is_number is True
assert Rational(3, 2).is_number is True
assert Rational(8).is_number is True
assert x.is_number is False
assert (2*x).is_number is False
assert (x + y).is_number is False
assert log(2).is_number is True
assert log(x).is_number is False
assert (2 + log(2)).is_number is True
assert (8 + log(2)).is_number is True
assert (2 + log(x)).is_number is False
assert (8 + log(2) + x).is_number is False
assert (1 + x**2/x - x).is_number is True
assert Tuple(Integer(1)).is_number is False
assert Add(2, x).is_number is False
assert Mul(3, 4).is_number is True
assert Pow(log(2), 2).is_number is True
assert oo.is_number is True
g = WildFunction('g')
assert g.is_number is False
assert (2*g).is_number is False
assert (x**2).subs(x, 3).is_number is True
# test extensibility of .is_number
# on subinstances of Basic
class A(Basic):
pass
a = A()
assert a.is_number is False
def test_as_coeff_add():
assert S(2).as_coeff_add() == (2, ())
assert S(3.0).as_coeff_add() == (0, (S(3.0),))
assert S(-3.0).as_coeff_add() == (0, (S(-3.0),))
assert x.as_coeff_add() == (0, (x,))
assert (x - 1).as_coeff_add() == (-1, (x,))
assert (x + 1).as_coeff_add() == (1, (x,))
assert (x + 2).as_coeff_add() == (2, (x,))
assert (x + y).as_coeff_add(y) == (x, (y,))
assert (3*x).as_coeff_add(y) == (3*x, ())
# don't do expansion
e = (x + y)**2
assert e.as_coeff_add(y) == (0, (e,))
def test_as_coeff_mul():
assert S(2).as_coeff_mul() == (2, ())
assert S(3.0).as_coeff_mul() == (1, (S(3.0),))
assert S(-3.0).as_coeff_mul() == (-1, (S(3.0),))
assert S(-3.0).as_coeff_mul(rational=False) == (-S(3.0), ())
assert x.as_coeff_mul() == (1, (x,))
assert (-x).as_coeff_mul() == (-1, (x,))
assert (2*x).as_coeff_mul() == (2, (x,))
assert (x*y).as_coeff_mul(y) == (x, (y,))
assert (3 + x).as_coeff_mul() == (1, (3 + x,))
assert (3 + x).as_coeff_mul(y) == (3 + x, ())
# don't do expansion
e = exp(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
e = 2**(x + y)
assert e.as_coeff_mul(y) == (1, (e,))
assert (1.1*x).as_coeff_mul(rational=False) == (1.1, (x,))
assert (1.1*x).as_coeff_mul() == (1, (1.1, x))
assert (-oo*x).as_coeff_mul(rational=True) == (-1, (oo, x))
def test_as_coeff_exponent():
assert (3*x**4).as_coeff_exponent(x) == (3, 4)
assert (2*x**3).as_coeff_exponent(x) == (2, 3)
assert (4*x**2).as_coeff_exponent(x) == (4, 2)
assert (6*x**1).as_coeff_exponent(x) == (6, 1)
assert (3*x**0).as_coeff_exponent(x) == (3, 0)
assert (2*x**0).as_coeff_exponent(x) == (2, 0)
assert (1*x**0).as_coeff_exponent(x) == (1, 0)
assert (0*x**0).as_coeff_exponent(x) == (0, 0)
assert (-1*x**0).as_coeff_exponent(x) == (-1, 0)
assert (-2*x**0).as_coeff_exponent(x) == (-2, 0)
assert (2*x**3 + pi*x**3).as_coeff_exponent(x) == (2 + pi, 3)
assert (x*log(2)/(2*x + pi*x)).as_coeff_exponent(x) == \
(log(2)/(2 + pi), 0)
# issue 4784
D = Derivative
fx = D(f(x), x)
assert fx.as_coeff_exponent(f(x)) == (fx, 0)
def test_extractions():
for base in (2, S.Exp1):
assert Pow(base**x, 3, evaluate=False
).extract_multiplicatively(base**x) == base**(2*x)
assert (base**(5*x)).extract_multiplicatively(
base**(3*x)) == base**(2*x)
assert ((x*y)**3).extract_multiplicatively(x**2 * y) == x*y**2
assert ((x*y)**3).extract_multiplicatively(x**4 * y) is None
assert (2*x).extract_multiplicatively(2) == x
assert (2*x).extract_multiplicatively(3) is None
assert (2*x).extract_multiplicatively(-1) is None
assert (S.Half*x).extract_multiplicatively(3) == x/6
assert (sqrt(x)).extract_multiplicatively(x) is None
assert (sqrt(x)).extract_multiplicatively(1/x) is None
assert x.extract_multiplicatively(-x) is None
assert (-2 - 4*I).extract_multiplicatively(-2) == 1 + 2*I
assert (-2 - 4*I).extract_multiplicatively(3) is None
assert (-2*x - 4*y - 8).extract_multiplicatively(-2) == x + 2*y + 4
assert (-2*x*y - 4*x**2*y).extract_multiplicatively(-2*y) == 2*x**2 + x
assert (2*x*y + 4*x**2*y).extract_multiplicatively(2*y) == 2*x**2 + x
assert (-4*y**2*x).extract_multiplicatively(-3*y) is None
assert (2*x).extract_multiplicatively(1) == 2*x
assert (-oo).extract_multiplicatively(5) is -oo
assert (oo).extract_multiplicatively(5) is oo
assert ((x*y)**3).extract_additively(1) is None
assert (x + 1).extract_additively(x) == 1
assert (x + 1).extract_additively(2*x) is None
assert (x + 1).extract_additively(-x) is None
assert (-x + 1).extract_additively(2*x) is None
assert (2*x + 3).extract_additively(x) == x + 3
assert (2*x + 3).extract_additively(2) == 2*x + 1
assert (2*x + 3).extract_additively(3) == 2*x
assert (2*x + 3).extract_additively(-2) is None
assert (2*x + 3).extract_additively(3*x) is None
assert (2*x + 3).extract_additively(2*x) == 3
assert x.extract_additively(0) == x
assert S(2).extract_additively(x) is None
assert S(2.).extract_additively(2) is S.Zero
assert S(2*x + 3).extract_additively(x + 1) == x + 2
assert S(2*x + 3).extract_additively(y + 1) is None
assert S(2*x - 3).extract_additively(x + 1) is None
assert S(2*x - 3).extract_additively(y + z) is None
assert ((a + 1)*x*4 + y).extract_additively(x).expand() == \
4*a*x + 3*x + y
assert ((a + 1)*x*4 + 3*y).extract_additively(x + 2*y).expand() == \
4*a*x + 3*x + y
assert (y*(x + 1)).extract_additively(x + 1) is None
assert ((y + 1)*(x + 1) + 3).extract_additively(x + 1) == \
y*(x + 1) + 3
assert ((x + y)*(x + 1) + x + y + 3).extract_additively(x + y) == \
x*(x + y) + 3
assert (x + y + 2*((x + y)*(x + 1)) + 3).extract_additively((x + y)*(x + 1)) == \
x + y + (x + 1)*(x + y) + 3
assert ((y + 1)*(x + 2*y + 1) + 3).extract_additively(y + 1) == \
(x + 2*y)*(y + 1) + 3
assert (-x - x*I).extract_additively(-x) == -I*x
# extraction does not leave artificats, now
assert (4*x*(y + 1) + y).extract_additively(x) == x*(4*y + 3) + y
n = Symbol("n", integer=True)
assert (Integer(-3)).could_extract_minus_sign() is True
assert (-n*x + x).could_extract_minus_sign() != \
(n*x - x).could_extract_minus_sign()
assert (x - y).could_extract_minus_sign() != \
(-x + y).could_extract_minus_sign()
assert (1 - x - y).could_extract_minus_sign() is True
assert (1 - x + y).could_extract_minus_sign() is False
assert ((-x - x*y)/y).could_extract_minus_sign() is False
assert ((x + x*y)/(-y)).could_extract_minus_sign() is True
assert ((x + x*y)/y).could_extract_minus_sign() is False
assert ((-x - y)/(x + y)).could_extract_minus_sign() is False
class sign_invariant(Function, Expr):
nargs = 1
def __neg__(self):
return self
foo = sign_invariant(x)
assert foo == -foo
assert foo.could_extract_minus_sign() is False
assert (x - y).could_extract_minus_sign() is False
assert (-x + y).could_extract_minus_sign() is True
assert (x - 1).could_extract_minus_sign() is False
assert (1 - x).could_extract_minus_sign() is True
assert (sqrt(2) - 1).could_extract_minus_sign() is True
assert (1 - sqrt(2)).could_extract_minus_sign() is False
# check that result is canonical
eq = (3*x + 15*y).extract_multiplicatively(3)
assert eq.args == eq.func(*eq.args).args
def test_nan_extractions():
for r in (1, 0, I, nan):
assert nan.extract_additively(r) is None
assert nan.extract_multiplicatively(r) is None
def test_coeff():
assert (x + 1).coeff(x + 1) == 1
assert (3*x).coeff(0) == 0
assert (z*(1 + x)*x**2).coeff(1 + x) == z*x**2
assert (1 + 2*x*x**(1 + x)).coeff(x*x**(1 + x)) == 2
assert (1 + 2*x**(y + z)).coeff(x**(y + z)) == 2
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (3 + 2*x + 4*x**2).coeff(-1) == 0
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (-x/8 + x*y).coeff(x) == Rational(-1, 8) + y
assert (-x/8 + x*y).coeff(-x) == S.One/8
assert (4*x).coeff(2*x) == 0
assert (2*x).coeff(2*x) == 1
assert (-oo*x).coeff(x*oo) == -1
assert (10*x).coeff(x, 0) == 0
assert (10*x).coeff(10*x, 0) == 0
n1, n2 = symbols('n1 n2', commutative=False)
assert (n1*n2).coeff(n1) == 1
assert (n1*n2).coeff(n2) == n1
assert (n1*n2 + x*n1).coeff(n1) == 1 # 1*n1*(n2+x)
assert (n2*n1 + x*n1).coeff(n1) == n2 + x
assert (n2*n1 + x*n1**2).coeff(n1) == n2
assert (n1**x).coeff(n1) == 0
assert (n1*n2 + n2*n1).coeff(n1) == 0
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=1) == n2
assert (2*(n1 + n2)*n2).coeff(n1 + n2, right=0) == 2
assert (2*f(x) + 3*f(x).diff(x)).coeff(f(x)) == 2
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr.coeff(x + y) == 0
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
assert (x + y + 3*z).coeff(1) == x + y
assert (-x + 2*y).coeff(-1) == x
assert (x - 2*y).coeff(-1) == 2*y
assert (3 + 2*x + 4*x**2).coeff(1) == 0
assert (-x - 2*y).coeff(2) == -y
assert (x + sqrt(2)*x).coeff(sqrt(2)) == x
assert (3 + 2*x + 4*x**2).coeff(x) == 2
assert (3 + 2*x + 4*x**2).coeff(x**2) == 4
assert (3 + 2*x + 4*x**2).coeff(x**3) == 0
assert (z*(x + y)**2).coeff((x + y)**2) == z
assert (z*(x + y)**2).coeff(x + y) == 0
assert (2 + 2*x + (x + 1)*y).coeff(x + 1) == y
assert (x + 2*y + 3).coeff(1) == x
assert (x + 2*y + 3).coeff(x, 0) == 2*y + 3
assert (x**2 + 2*y + 3*x).coeff(x**2, 0) == 2*y + 3*x
assert x.coeff(0, 0) == 0
assert x.coeff(x, 0) == 0
n, m, o, l = symbols('n m o l', commutative=False)
assert n.coeff(n) == 1
assert y.coeff(n) == 0
assert (3*n).coeff(n) == 3
assert (2 + n).coeff(x*m) == 0
assert (2*x*n*m).coeff(x) == 2*n*m
assert (2 + n).coeff(x*m*n + y) == 0
assert (2*x*n*m).coeff(3*n) == 0
assert (n*m + m*n*m).coeff(n) == 1 + m
assert (n*m + m*n*m).coeff(n, right=True) == m # = (1 + m)*n*m
assert (n*m + m*n).coeff(n) == 0
assert (n*m + o*m*n).coeff(m*n) == o
assert (n*m + o*m*n).coeff(m*n, right=True) == 1
assert (n*m + n*m*n).coeff(n*m, right=True) == 1 + n # = n*m*(n + 1)
assert (x*y).coeff(z, 0) == x*y
assert (x*n + y*n + z*m).coeff(n) == x + y
assert (n*m + n*o + o*l).coeff(n, right=True) == m + o
assert (x*n*m*n + y*n*m*o + z*l).coeff(m, right=True) == x*n + y*o
assert (x*n*m*n + x*n*m*o + z*l).coeff(m, right=True) == n + o
assert (x*n*m*n + x*n*m*o + z*l).coeff(m) == x*n
def test_coeff2():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r)) == 2/r
def test_coeff2_0():
r, kappa = symbols('r, kappa')
psi = Function("psi")
g = 1/r**2 * (2*r*psi(r).diff(r, 1) + r**2 * psi(r).diff(r, 2))
g = g.expand()
assert g.coeff(psi(r).diff(r, 2)) == 1
def test_coeff_expand():
expr = z*(x + y)**2
expr2 = z*(x + y)**2 + z*(2*x + 2*y)**2
assert expr.coeff(z) == (x + y)**2
assert expr2.coeff(z) == (x + y)**2 + (2*x + 2*y)**2
def test_integrate():
assert x.integrate(x) == x**2/2
assert x.integrate((x, 0, 1)) == S.Half
def test_as_base_exp():
assert x.as_base_exp() == (x, S.One)
assert (x*y*z).as_base_exp() == (x*y*z, S.One)
assert (x + y + z).as_base_exp() == (x + y + z, S.One)
assert ((x + y)**z).as_base_exp() == (x + y, z)
def test_issue_4963():
assert hasattr(Mul(x, y), "is_commutative")
assert hasattr(Mul(x, y, evaluate=False), "is_commutative")
assert hasattr(Pow(x, y), "is_commutative")
assert hasattr(Pow(x, y, evaluate=False), "is_commutative")
expr = Mul(Pow(2, 2, evaluate=False), 3, evaluate=False) + 1
assert hasattr(expr, "is_commutative")
def test_action_verbs():
assert nsimplify(1/(exp(3*pi*x/5) + 1)) == \
(1/(exp(3*pi*x/5) + 1)).nsimplify()
assert ratsimp(1/x + 1/y) == (1/x + 1/y).ratsimp()
assert trigsimp(log(x), deep=True) == (log(x)).trigsimp(deep=True)
assert radsimp(1/(2 + sqrt(2))) == (1/(2 + sqrt(2))).radsimp()
assert radsimp(1/(a + b*sqrt(c)), symbolic=False) == \
(1/(a + b*sqrt(c))).radsimp(symbolic=False)
assert powsimp(x**y*x**z*y**z, combine='all') == \
(x**y*x**z*y**z).powsimp(combine='all')
assert (x**t*y**t).powsimp(force=True) == (x*y)**t
assert simplify(x**y*x**z*y**z) == (x**y*x**z*y**z).simplify()
assert together(1/x + 1/y) == (1/x + 1/y).together()
assert collect(a*x**2 + b*x**2 + a*x - b*x + c, x) == \
(a*x**2 + b*x**2 + a*x - b*x + c).collect(x)
assert apart(y/(y + 2)/(y + 1), y) == (y/(y + 2)/(y + 1)).apart(y)
assert combsimp(y/(x + 2)/(x + 1)) == (y/(x + 2)/(x + 1)).combsimp()
assert gammasimp(gamma(x)/gamma(x-5)) == (gamma(x)/gamma(x-5)).gammasimp()
assert factor(x**2 + 5*x + 6) == (x**2 + 5*x + 6).factor()
assert refine(sqrt(x**2)) == sqrt(x**2).refine()
assert cancel((x**2 + 5*x + 6)/(x + 2)) == ((x**2 + 5*x + 6)/(x + 2)).cancel()
def test_as_powers_dict():
assert x.as_powers_dict() == {x: 1}
assert (x**y*z).as_powers_dict() == {x: y, z: 1}
assert Mul(2, 2, evaluate=False).as_powers_dict() == {S(2): S(2)}
assert (x*y).as_powers_dict()[z] == 0
assert (x + y).as_powers_dict()[z] == 0
def test_as_coefficients_dict():
check = [S.One, x, y, x*y, 1]
assert [Add(3*x, 2*x, y, 3).as_coefficients_dict()[i] for i in check] == \
[3, 5, 1, 0, 3]
assert [Add(3*x, 2*x, y, 3, evaluate=False).as_coefficients_dict()[i]
for i in check] == [3, 5, 1, 0, 3]
assert [(3*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3, 0]
assert [(3.0*x*y).as_coefficients_dict()[i] for i in check] == \
[0, 0, 0, 3.0, 0]
assert (3.0*x*y).as_coefficients_dict()[3.0*x*y] == 0
eq = x*(x + 1)*a + x*b + c/x
assert eq.as_coefficients_dict(x) == {x: b, 1/x: c,
x*(x + 1): a}
assert eq.expand().as_coefficients_dict(x) == {x**2: a, x: a + b, 1/x: c}
assert x.as_coefficients_dict() == {x: S.One}
def test_args_cnc():
A = symbols('A', commutative=False)
assert (x + A).args_cnc() == \
[[], [x + A]]
assert (x + a).args_cnc() == \
[[a + x], []]
assert (x*a).args_cnc() == \
[[a, x], []]
assert (x*y*A*(A + 1)).args_cnc(cset=True) == \
[{x, y}, [A, 1 + A]]
assert Mul(x, x, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x}, []]
assert Mul(x, x**2, evaluate=False).args_cnc(cset=True, warn=False) == \
[{x, x**2}, []]
raises(ValueError, lambda: Mul(x, x, evaluate=False).args_cnc(cset=True))
assert Mul(x, y, x, evaluate=False).args_cnc() == \
[[x, y, x], []]
# always split -1 from leading number
assert (-1.*x).args_cnc() == [[-1, 1.0, x], []]
def test_new_rawargs():
n = Symbol('n', commutative=False)
a = x + n
assert a.is_commutative is False
assert a._new_rawargs(x).is_commutative
assert a._new_rawargs(x, y).is_commutative
assert a._new_rawargs(x, n).is_commutative is False
assert a._new_rawargs(x, y, n).is_commutative is False
m = x*n
assert m.is_commutative is False
assert m._new_rawargs(x).is_commutative
assert m._new_rawargs(n).is_commutative is False
assert m._new_rawargs(x, y).is_commutative
assert m._new_rawargs(x, n).is_commutative is False
assert m._new_rawargs(x, y, n).is_commutative is False
assert m._new_rawargs(x, n, reeval=False).is_commutative is False
assert m._new_rawargs(S.One) is S.One
def test_issue_5226():
assert Add(evaluate=False) == 0
assert Mul(evaluate=False) == 1
assert Mul(x + y, evaluate=False).is_Add
def test_free_symbols():
# free_symbols should return the free symbols of an object
assert S.One.free_symbols == set()
assert x.free_symbols == {x}
assert Integral(x, (x, 1, y)).free_symbols == {y}
assert (-Integral(x, (x, 1, y))).free_symbols == {y}
assert meter.free_symbols == set()
assert (meter**x).free_symbols == {x}
def test_has_free():
assert x.has_free(x)
assert not x.has_free(y)
assert (x + y).has_free(x)
assert (x + y).has_free(*(x, z))
assert f(x).has_free(x)
assert f(x).has_free(f(x))
assert Integral(f(x), (f(x), 1, y)).has_free(y)
assert not Integral(f(x), (f(x), 1, y)).has_free(x)
assert not Integral(f(x), (f(x), 1, y)).has_free(f(x))
# simple extraction
assert (x + 1 + y).has_free(x + 1)
assert not (x + 2 + y).has_free(x + 1)
assert (2 + 3*x*y).has_free(3*x)
raises(TypeError, lambda: x.has_free({x, y}))
s = FiniteSet(1, 2)
assert Piecewise((s, x > 3), (4, True)).has_free(s)
assert not Piecewise((1, x > 3), (4, True)).has_free(s)
# can't make set of these, but fallback will handle
raises(TypeError, lambda: x.has_free(y, []))
def test_has_xfree():
assert (x + 1).has_xfree({x})
assert ((x + 1)**2).has_xfree({x + 1})
assert not (x + y + 1).has_xfree({x + 1})
raises(TypeError, lambda: x.has_xfree(x))
raises(TypeError, lambda: x.has_xfree([x]))
def test_issue_5300():
x = Symbol('x', commutative=False)
assert x*sqrt(2)/sqrt(6) == x*sqrt(3)/3
def test_floordiv():
from sympy.functions.elementary.integers import floor
assert x // y == floor(x / y)
def test_as_coeff_Mul():
assert Integer(3).as_coeff_Mul() == (Integer(3), Integer(1))
assert Rational(3, 4).as_coeff_Mul() == (Rational(3, 4), Integer(1))
assert Float(5.0).as_coeff_Mul() == (Float(5.0), Integer(1))
assert (Integer(3)*x).as_coeff_Mul() == (Integer(3), x)
assert (Rational(3, 4)*x).as_coeff_Mul() == (Rational(3, 4), x)
assert (Float(5.0)*x).as_coeff_Mul() == (Float(5.0), x)
assert (Integer(3)*x*y).as_coeff_Mul() == (Integer(3), x*y)
assert (Rational(3, 4)*x*y).as_coeff_Mul() == (Rational(3, 4), x*y)
assert (Float(5.0)*x*y).as_coeff_Mul() == (Float(5.0), x*y)
assert (x).as_coeff_Mul() == (S.One, x)
assert (x*y).as_coeff_Mul() == (S.One, x*y)
assert (-oo*x).as_coeff_Mul(rational=True) == (-1, oo*x)
def test_as_coeff_Add():
assert Integer(3).as_coeff_Add() == (Integer(3), Integer(0))
assert Rational(3, 4).as_coeff_Add() == (Rational(3, 4), Integer(0))
assert Float(5.0).as_coeff_Add() == (Float(5.0), Integer(0))
assert (Integer(3) + x).as_coeff_Add() == (Integer(3), x)
assert (Rational(3, 4) + x).as_coeff_Add() == (Rational(3, 4), x)
assert (Float(5.0) + x).as_coeff_Add() == (Float(5.0), x)
assert (Float(5.0) + x).as_coeff_Add(rational=True) == (0, Float(5.0) + x)
assert (Integer(3) + x + y).as_coeff_Add() == (Integer(3), x + y)
assert (Rational(3, 4) + x + y).as_coeff_Add() == (Rational(3, 4), x + y)
assert (Float(5.0) + x + y).as_coeff_Add() == (Float(5.0), x + y)
assert (x).as_coeff_Add() == (S.Zero, x)
assert (x*y).as_coeff_Add() == (S.Zero, x*y)
def test_expr_sorting():
exprs = [1/x**2, 1/x, sqrt(sqrt(x)), sqrt(x), x, sqrt(x)**3, x**2]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x, 2*x, 2*x**2, 2*x**3, x**n, 2*x**n, sin(x), sin(x)**n,
sin(x**2), cos(x), cos(x**2), tan(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [x + 1, x**2 + x + 1, x**3 + x**2 + x + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [S(4), x - 3*I/2, x + 3*I/2, x - 4*I + 1, x + 4*I + 1]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [f(x), g(x), exp(x), sin(x), cos(x), factorial(x)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [Tuple(x, y), Tuple(x, z), Tuple(x, y, z)]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[3], [1, 2]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [[1, 2], [1, 2, 3]]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{x: -y}, {x: y}]
assert sorted(exprs, key=default_sort_key) == exprs
exprs = [{1}, {1, 2}]
assert sorted(exprs, key=default_sort_key) == exprs
a, b = exprs = [Dummy('x'), Dummy('x')]
assert sorted([b, a], key=default_sort_key) == exprs
def test_as_ordered_factors():
assert x.as_ordered_factors() == [x]
assert (2*x*x**n*sin(x)*cos(x)).as_ordered_factors() \
== [Integer(2), x, x**n, sin(x), cos(x)]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Mul(*args)
assert expr.as_ordered_factors() == args
A, B = symbols('A,B', commutative=False)
assert (A*B).as_ordered_factors() == [A, B]
assert (B*A).as_ordered_factors() == [B, A]
def test_as_ordered_terms():
assert x.as_ordered_terms() == [x]
assert (sin(x)**2*cos(x) + sin(x)*cos(x)**2 + 1).as_ordered_terms() \
== [sin(x)**2*cos(x), sin(x)*cos(x)**2, 1]
args = [f(1), f(2), f(3), f(1, 2, 3), g(1), g(2), g(3), g(1, 2, 3)]
expr = Add(*args)
assert expr.as_ordered_terms() == args
assert (1 + 4*sqrt(3)*pi*x).as_ordered_terms() == [4*pi*x*sqrt(3), 1]
assert ( 2 + 3*I).as_ordered_terms() == [2, 3*I]
assert (-2 + 3*I).as_ordered_terms() == [-2, 3*I]
assert ( 2 - 3*I).as_ordered_terms() == [2, -3*I]
assert (-2 - 3*I).as_ordered_terms() == [-2, -3*I]
assert ( 4 + 3*I).as_ordered_terms() == [4, 3*I]
assert (-4 + 3*I).as_ordered_terms() == [-4, 3*I]
assert ( 4 - 3*I).as_ordered_terms() == [4, -3*I]
assert (-4 - 3*I).as_ordered_terms() == [-4, -3*I]
e = x**2*y**2 + x*y**4 + y + 2
assert e.as_ordered_terms(order="lex") == [x**2*y**2, x*y**4, y, 2]
assert e.as_ordered_terms(order="grlex") == [x*y**4, x**2*y**2, y, 2]
assert e.as_ordered_terms(order="rev-lex") == [2, y, x*y**4, x**2*y**2]
assert e.as_ordered_terms(order="rev-grlex") == [2, y, x**2*y**2, x*y**4]
k = symbols('k')
assert k.as_ordered_terms(data=True) == ([(k, ((1.0, 0.0), (1,), ()))], [k])
def test_sort_key_atomic_expr():
from sympy.physics.units import m, s
assert sorted([-m, s], key=lambda arg: arg.sort_key()) == [-m, s]
def test_eval_interval():
assert exp(x)._eval_interval(*Tuple(x, 0, 1)) == exp(1) - exp(0)
# issue 4199
a = x/y
raises(NotImplementedError, lambda: a._eval_interval(x, S.Zero, oo)._eval_interval(y, oo, S.Zero))
raises(NotImplementedError, lambda: a._eval_interval(x, S.Zero, oo)._eval_interval(y, S.Zero, oo))
a = x - y
raises(NotImplementedError, lambda: a._eval_interval(x, S.One, oo)._eval_interval(y, oo, S.One))
raises(ValueError, lambda: x._eval_interval(x, None, None))
a = -y*Heaviside(x - y)
assert a._eval_interval(x, -oo, oo) == -y
assert a._eval_interval(x, oo, -oo) == y
def test_eval_interval_zoo():
# Test that limit is used when zoo is returned
assert Si(1/x)._eval_interval(x, S.Zero, S.One) == -pi/2 + Si(1)
def test_primitive():
assert (3*(x + 1)**2).primitive() == (3, (x + 1)**2)
assert (6*x + 2).primitive() == (2, 3*x + 1)
assert (x/2 + 3).primitive() == (S.Half, x + 6)
eq = (6*x + 2)*(x/2 + 3)
assert eq.primitive()[0] == 1
eq = (2 + 2*x)**2
assert eq.primitive()[0] == 1
assert (4.0*x).primitive() == (1, 4.0*x)
assert (4.0*x + y/2).primitive() == (S.Half, 8.0*x + y)
assert (-2*x).primitive() == (2, -x)
assert Add(5*z/7, 0.5*x, 3*y/2, evaluate=False).primitive() == \
(S.One/14, 7.0*x + 21*y + 10*z)
for i in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
assert (i + x/3).primitive() == \
(S.One/3, i + x)
assert (S.Infinity + 2*x/3 + 4*y/7).primitive() == \
(S.One/21, 14*x + 12*y + oo)
assert S.Zero.primitive() == (S.One, S.Zero)
def test_issue_5843():
a = 1 + x
assert (2*a).extract_multiplicatively(a) == 2
assert (4*a).extract_multiplicatively(2*a) == 2
assert ((3*a)*(2*a)).extract_multiplicatively(a) == 6*a
def test_is_constant():
from sympy.solvers.solvers import checksol
assert Sum(x, (x, 1, 10)).is_constant() is True
assert Sum(x, (x, 1, n)).is_constant() is False
assert Sum(x, (x, 1, n)).is_constant(y) is True
assert Sum(x, (x, 1, n)).is_constant(n) is False
assert Sum(x, (x, 1, n)).is_constant(x) is True
eq = a*cos(x)**2 + a*sin(x)**2 - a
assert eq.is_constant() is True
assert eq.subs({x: pi, a: 2}) == eq.subs({x: pi, a: 3}) == 0
assert x.is_constant() is False
assert x.is_constant(y) is True
assert log(x/y).is_constant() is False
assert checksol(x, x, Sum(x, (x, 1, n))) is False
assert checksol(x, x, Sum(x, (x, 1, n))) is False
assert f(1).is_constant
assert checksol(x, x, f(x)) is False
assert Pow(x, S.Zero, evaluate=False).is_constant() is True # == 1
assert Pow(S.Zero, x, evaluate=False).is_constant() is False # == 0 or 1
assert (2**x).is_constant() is False
assert Pow(S(2), S(3), evaluate=False).is_constant() is True
z1, z2 = symbols('z1 z2', zero=True)
assert (z1 + 2*z2).is_constant() is True
assert meter.is_constant() is True
assert (3*meter).is_constant() is True
assert (x*meter).is_constant() is False
def test_equals():
assert (-3 - sqrt(5) + (-sqrt(10)/2 - sqrt(2)/2)**2).equals(0)
assert (x**2 - 1).equals((x + 1)*(x - 1))
assert (cos(x)**2 + sin(x)**2).equals(1)
assert (a*cos(x)**2 + a*sin(x)**2).equals(a)
r = sqrt(2)
assert (-1/(r + r*x) + 1/r/(1 + x)).equals(0)
assert factorial(x + 1).equals((x + 1)*factorial(x))
assert sqrt(3).equals(2*sqrt(3)) is False
assert (sqrt(5)*sqrt(3)).equals(sqrt(3)) is False
assert (sqrt(5) + sqrt(3)).equals(0) is False
assert (sqrt(5) + pi).equals(0) is False
assert meter.equals(0) is False
assert (3*meter**2).equals(0) is False
eq = -(-1)**(S(3)/4)*6**(S.One/4) + (-6)**(S.One/4)*I
if eq != 0: # if canonicalization makes this zero, skip the test
assert eq.equals(0)
assert sqrt(x).equals(0) is False
# from integrate(x*sqrt(1 + 2*x), x);
# diff is zero only when assumptions allow
i = 2*sqrt(2)*x**(S(5)/2)*(1 + 1/(2*x))**(S(5)/2)/5 + \
2*sqrt(2)*x**(S(3)/2)*(1 + 1/(2*x))**(S(5)/2)/(-6 - 3/x)
ans = sqrt(2*x + 1)*(6*x**2 + x - 1)/15
diff = i - ans
assert diff.equals(0) is None # should be False, but previously this was False due to wrong intermediate result
assert diff.subs(x, Rational(-1, 2)/2) == 7*sqrt(2)/120
# there are regions for x for which the expression is True, for
# example, when x < -1/2 or x > 0 the expression is zero
p = Symbol('p', positive=True)
assert diff.subs(x, p).equals(0) is True
assert diff.subs(x, -1).equals(0) is True
# prove via minimal_polynomial or self-consistency
eq = sqrt(1 + sqrt(3)) + sqrt(3 + 3*sqrt(3)) - sqrt(10 + 6*sqrt(3))
assert eq.equals(0)
q = 3**Rational(1, 3) + 3
p = expand(q**3)**Rational(1, 3)
assert (p - q).equals(0)
# issue 6829
# eq = q*x + q/4 + x**4 + x**3 + 2*x**2 - S.One/3
# z = eq.subs(x, solve(eq, x)[0])
q = symbols('q')
z = (q*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12)/2 - sqrt((2*q - S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/6)/2 - S.One/4) + q/4 + (-sqrt(-2*(-(q
- S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q
- S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**4 + (-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**3 + 2*(-sqrt(-2*(-(q - S(7)/8)**S(2)/8 -
S(2197)/13824)**(S.One/3) - S(13)/12)/2 - sqrt((2*q -
S(7)/4)/sqrt(-2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/12) + 2*(-(q - S(7)/8)**S(2)/8 - S(2197)/13824)**(S.One/3) -
S(13)/6)/2 - S.One/4)**2 - Rational(1, 3))
assert z.equals(0)
def test_random():
from sympy.functions.combinatorial.numbers import lucas
from sympy.simplify.simplify import posify
assert posify(x)[0]._random() is not None
assert lucas(n)._random(2, -2, 0, -1, 1) is None
# issue 8662
assert Piecewise((Max(x, y), z))._random() is None
def test_round():
assert str(Float('0.1249999').round(2)) == '0.12'
d20 = 12345678901234567890
ans = S(d20).round(2)
assert ans.is_Integer and ans == d20
ans = S(d20).round(-2)
assert ans.is_Integer and ans == 12345678901234567900
assert str(S('1/7').round(4)) == '0.1429'
assert str(S('.[12345]').round(4)) == '0.1235'
assert str(S('.1349').round(2)) == '0.13'
n = S(12345)
ans = n.round()
assert ans.is_Integer
assert ans == n
ans = n.round(1)
assert ans.is_Integer
assert ans == n
ans = n.round(4)
assert ans.is_Integer
assert ans == n
assert n.round(-1) == 12340
r = Float(str(n)).round(-4)
assert r == 10000
assert n.round(-5) == 0
assert str((pi + sqrt(2)).round(2)) == '4.56'
assert (10*(pi + sqrt(2))).round(-1) == 50
raises(TypeError, lambda: round(x + 2, 2))
assert str(S(2.3).round(1)) == '2.3'
# rounding in SymPy (as in Decimal) should be
# exact for the given precision; we check here
# that when a 5 follows the last digit that
# the rounded digit will be even.
for i in range(-99, 100):
# construct a decimal that ends in 5, e.g. 123 -> 0.1235
s = str(abs(i))
p = len(s) # we are going to round to the last digit of i
n = '0.%s5' % s # put a 5 after i's digits
j = p + 2 # 2 for '0.'
if i < 0: # 1 for '-'
j += 1
n = '-' + n
v = str(Float(n).round(p))[:j] # pertinent digits
if v.endswith('.'):
continue # it ends with 0 which is even
L = int(v[-1]) # last digit
assert L % 2 == 0, (n, '->', v)
assert (Float(.3, 3) + 2*pi).round() == 7
assert (Float(.3, 3) + 2*pi*100).round() == 629
assert (pi + 2*E*I).round() == 3 + 5*I
# don't let request for extra precision give more than
# what is known (in this case, only 3 digits)
assert str((Float(.03, 3) + 2*pi/100).round(5)) == '0.0928'
assert str((Float(.03, 3) + 2*pi/100).round(4)) == '0.0928'
assert S.Zero.round() == 0
a = (Add(1, Float('1.' + '9'*27, ''), evaluate=0))
assert a.round(10) == Float('3.0000000000', '')
assert a.round(25) == Float('3.0000000000000000000000000', '')
assert a.round(26) == Float('3.00000000000000000000000000', '')
assert a.round(27) == Float('2.999999999999999999999999999', '')
assert a.round(30) == Float('2.999999999999999999999999999', '')
raises(TypeError, lambda: x.round())
raises(TypeError, lambda: f(1).round())
# exact magnitude of 10
assert str(S.One.round()) == '1'
assert str(S(100).round()) == '100'
# applied to real and imaginary portions
assert (2*pi + E*I).round() == 6 + 3*I
assert (2*pi + I/10).round() == 6
assert (pi/10 + 2*I).round() == 2*I
# the lhs re and im parts are Float with dps of 2
# and those on the right have dps of 15 so they won't compare
# equal unless we use string or compare components (which will
# then coerce the floats to the same precision) or re-create
# the floats
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
assert str((pi/10 + E*I).round(2).as_real_imag()) == '(0.31, 2.72)'
assert str((pi/10 + E*I).round(2)) == '0.31 + 2.72*I'
# issue 6914
assert (I**(I + 3)).round(3) == Float('-0.208', '')*I
# issue 8720
assert S(-123.6).round() == -124
assert S(-1.5).round() == -2
assert S(-100.5).round() == -100
assert S(-1.5 - 10.5*I).round() == -2 - 10*I
# issue 7961
assert str(S(0.006).round(2)) == '0.01'
assert str(S(0.00106).round(4)) == '0.0011'
# issue 8147
assert S.NaN.round() is S.NaN
assert S.Infinity.round() is S.Infinity
assert S.NegativeInfinity.round() is S.NegativeInfinity
assert S.ComplexInfinity.round() is S.ComplexInfinity
# check that types match
for i in range(2):
fi = float(i)
# 2 args
assert all(type(round(i, p)) is int for p in (-1, 0, 1))
assert all(S(i).round(p).is_Integer for p in (-1, 0, 1))
assert all(type(round(fi, p)) is float for p in (-1, 0, 1))
assert all(S(fi).round(p).is_Float for p in (-1, 0, 1))
# 1 arg (p is None)
assert type(round(i)) is int
assert S(i).round().is_Integer
assert type(round(fi)) is int
assert S(fi).round().is_Integer
def test_held_expression_UnevaluatedExpr():
x = symbols("x")
he = UnevaluatedExpr(1/x)
e1 = x*he
assert isinstance(e1, Mul)
assert e1.args == (x, he)
assert e1.doit() == 1
assert UnevaluatedExpr(Derivative(x, x)).doit(deep=False
) == Derivative(x, x)
assert UnevaluatedExpr(Derivative(x, x)).doit() == 1
xx = Mul(x, x, evaluate=False)
assert xx != x**2
ue2 = UnevaluatedExpr(xx)
assert isinstance(ue2, UnevaluatedExpr)
assert ue2.args == (xx,)
assert ue2.doit() == x**2
assert ue2.doit(deep=False) == xx
x2 = UnevaluatedExpr(2)*2
assert type(x2) is Mul
assert x2.args == (2, UnevaluatedExpr(2))
def test_round_exception_nostr():
# Don't use the string form of the expression in the round exception, as
# it's too slow
s = Symbol('bad')
try:
s.round()
except TypeError as e:
assert 'bad' not in str(e)
else:
# Did not raise
raise AssertionError("Did not raise")
def test_extract_branch_factor():
assert exp_polar(2.0*I*pi).extract_branch_factor() == (1, 1)
def test_identity_removal():
assert Add.make_args(x + 0) == (x,)
assert Mul.make_args(x*1) == (x,)
def test_float_0():
assert Float(0.0) + 1 == Float(1.0)
@XFAIL
def test_float_0_fail():
assert Float(0.0)*x == Float(0.0)
assert (x + Float(0.0)).is_Add
def test_issue_6325():
ans = (b**2 + z**2 - (b*(a + b*t) + z*(c + t*z))**2/(
(a + b*t)**2 + (c + t*z)**2))/sqrt((a + b*t)**2 + (c + t*z)**2)
e = sqrt((a + b*t)**2 + (c + z*t)**2)
assert diff(e, t, 2) == ans
assert e.diff(t, 2) == ans
assert diff(e, t, 2, simplify=False) != ans
def test_issue_7426():
f1 = a % c
f2 = x % z
assert f1.equals(f2) is None
def test_issue_11122():
x = Symbol('x', extended_positive=False)
assert unchanged(Gt, x, 0) # (x > 0)
# (x > 0) should remain unevaluated after PR #16956
x = Symbol('x', positive=False, real=True)
assert (x > 0) is S.false
def test_issue_10651():
x = Symbol('x', real=True)
e1 = (-1 + x)/(1 - x)
e3 = (4*x**2 - 4)/((1 - x)*(1 + x))
e4 = 1/(cos(x)**2) - (tan(x))**2
x = Symbol('x', positive=True)
e5 = (1 + x)/x
assert e1.is_constant() is None
assert e3.is_constant() is None
assert e4.is_constant() is None
assert e5.is_constant() is False
def test_issue_10161():
x = symbols('x', real=True)
assert x*abs(x)*abs(x) == x**3
def test_issue_10755():
x = symbols('x')
raises(TypeError, lambda: int(log(x)))
raises(TypeError, lambda: log(x).round(2))
def test_issue_11877():
x = symbols('x')
assert integrate(log(S.Half - x), (x, 0, S.Half)) == Rational(-1, 2) -log(2)/2
def test_normal():
x = symbols('x')
e = Mul(S.Half, 1 + x, evaluate=False)
assert e.normal() == e
def test_expr():
x = symbols('x')
raises(TypeError, lambda: tan(x).series(x, 2, oo, "+"))
def test_ExprBuilder():
eb = ExprBuilder(Mul)
eb.args.extend([x, x])
assert eb.build() == x**2
def test_issue_22020():
from sympy.parsing.sympy_parser import parse_expr
x = parse_expr("log((2*V/3-V)/C)/-(R+r)*C")
y = parse_expr("log((2*V/3-V)/C)/-(R+r)*2")
assert x.equals(y) is False
def test_non_string_equality():
# Expressions should not compare equal to strings
x = symbols('x')
one = sympify(1)
assert (x == 'x') is False
assert (x != 'x') is True
assert (one == '1') is False
assert (one != '1') is True
assert (x + 1 == 'x + 1') is False
assert (x + 1 != 'x + 1') is True
# Make sure == doesn't try to convert the resulting expression to a string
# (e.g., by calling sympify() instead of _sympify())
class BadRepr:
def __repr__(self):
raise RuntimeError
assert (x == BadRepr()) is False
assert (x != BadRepr()) is True
def test_21494():
from sympy.testing.pytest import warns_deprecated_sympy
with warns_deprecated_sympy():
assert x.expr_free_symbols == {x}
with warns_deprecated_sympy():
assert Basic().expr_free_symbols == set()
with warns_deprecated_sympy():
assert S(2).expr_free_symbols == {S(2)}
with warns_deprecated_sympy():
assert Indexed("A", x).expr_free_symbols == {Indexed("A", x)}
with warns_deprecated_sympy():
assert Subs(x, x, 0).expr_free_symbols == set()
def test_Expr__eq__iterable_handling():
assert x != range(3)
def test_format():
assert '{:1.2f}'.format(S.Zero) == '0.00'
assert '{:+3.0f}'.format(S(3)) == ' +3'
assert '{:23.20f}'.format(pi) == ' 3.14159265358979323846'
assert '{:50.48f}'.format(exp(sin(1))) == '2.319776824715853173956590377503266813254904772376'
|
2e9b54587398ac57cd835fbc70d10cc43a10cc2743b60ba286056819bd100bd3 | from sympy.core import (
Basic, Rational, Symbol, S, Float, Integer, Mul, Number, Pow,
Expr, I, nan, pi, symbols, oo, zoo, N)
from sympy.core.parameters import global_parameters
from sympy.core.tests.test_evalf import NS
from sympy.core.function import expand_multinomial
from sympy.functions.elementary.miscellaneous import sqrt, cbrt
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.special.error_functions import erf
from sympy.functions.elementary.trigonometric import (
sin, cos, tan, sec, csc, atan)
from sympy.functions.elementary.hyperbolic import cosh, sinh, tanh
from sympy.polys import Poly
from sympy.series.order import O
from sympy.sets import FiniteSet
from sympy.core.power import power, integer_nthroot
from sympy.testing.pytest import warns, _both_exp_pow
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.abc import a, b, c, x, y
def test_rational():
a = Rational(1, 5)
r = sqrt(5)/5
assert sqrt(a) == r
assert 2*sqrt(a) == 2*r
r = a*a**S.Half
assert a**Rational(3, 2) == r
assert 2*a**Rational(3, 2) == 2*r
r = a**5*a**Rational(2, 3)
assert a**Rational(17, 3) == r
assert 2 * a**Rational(17, 3) == 2*r
def test_large_rational():
e = (Rational(123712**12 - 1, 7) + Rational(1, 7))**Rational(1, 3)
assert e == 234232585392159195136 * (Rational(1, 7)**Rational(1, 3))
def test_negative_real():
def feq(a, b):
return abs(a - b) < 1E-10
assert feq(S.One / Float(-0.5), -Integer(2))
def test_expand():
assert (2**(-1 - x)).expand() == S.Half*2**(-x)
def test_issue_3449():
#test if powers are simplified correctly
#see also issue 3995
assert ((x**Rational(1, 3))**Rational(2)) == x**Rational(2, 3)
assert (
(x**Rational(3))**Rational(2, 5)) == (x**Rational(3))**Rational(2, 5)
a = Symbol('a', real=True)
b = Symbol('b', real=True)
assert (a**2)**b == (abs(a)**b)**2
assert sqrt(1/a) != 1/sqrt(a) # e.g. for a = -1
assert (a**3)**Rational(1, 3) != a
assert (x**a)**b != x**(a*b) # e.g. x = -1, a=2, b=1/2
assert (x**.5)**b == x**(.5*b)
assert (x**.5)**.5 == x**.25
assert (x**2.5)**.5 != x**1.25 # e.g. for x = 5*I
k = Symbol('k', integer=True)
m = Symbol('m', integer=True)
assert (x**k)**m == x**(k*m)
assert Number(5)**Rational(2, 3) == Number(25)**Rational(1, 3)
assert (x**.5)**2 == x**1.0
assert (x**2)**k == (x**k)**2 == x**(2*k)
a = Symbol('a', positive=True)
assert (a**3)**Rational(2, 5) == a**Rational(6, 5)
assert (a**2)**b == (a**b)**2
assert (a**Rational(2, 3))**x == a**(x*Rational(2, 3)) != (a**x)**Rational(2, 3)
def test_issue_3866():
assert --sqrt(sqrt(5) - 1) == sqrt(sqrt(5) - 1)
def test_negative_one():
x = Symbol('x', complex=True)
y = Symbol('y', complex=True)
assert 1/x**y == x**(-y)
def test_issue_4362():
neg = Symbol('neg', negative=True)
nonneg = Symbol('nonneg', nonnegative=True)
any = Symbol('any')
num, den = sqrt(1/neg).as_numer_denom()
assert num == sqrt(-1)
assert den == sqrt(-neg)
num, den = sqrt(1/nonneg).as_numer_denom()
assert num == 1
assert den == sqrt(nonneg)
num, den = sqrt(1/any).as_numer_denom()
assert num == sqrt(1/any)
assert den == 1
def eqn(num, den, pow):
return (num/den)**pow
npos = 1
nneg = -1
dpos = 2 - sqrt(3)
dneg = 1 - sqrt(3)
assert dpos > 0 and dneg < 0 and npos > 0 and nneg < 0
# pos or neg integer
eq = eqn(npos, dpos, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dpos**2)
eq = eqn(npos, dneg, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dneg**2)
eq = eqn(nneg, dpos, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dpos**2)
eq = eqn(nneg, dneg, 2)
assert eq.is_Pow and eq.as_numer_denom() == (1, dneg**2)
eq = eqn(npos, dpos, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**2, 1)
eq = eqn(npos, dneg, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dneg**2, 1)
eq = eqn(nneg, dpos, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**2, 1)
eq = eqn(nneg, dneg, -2)
assert eq.is_Pow and eq.as_numer_denom() == (dneg**2, 1)
# pos or neg rational
pow = S.Half
eq = eqn(npos, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (npos**pow, dpos**pow)
eq = eqn(npos, dneg, pow)
assert eq.is_Pow is False and eq.as_numer_denom() == ((-npos)**pow, (-dneg)**pow)
eq = eqn(nneg, dpos, pow)
assert not eq.is_Pow or eq.as_numer_denom() == (nneg**pow, dpos**pow)
eq = eqn(nneg, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-nneg)**pow, (-dneg)**pow)
eq = eqn(npos, dpos, -pow)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**pow, npos**pow)
eq = eqn(npos, dneg, -pow)
assert eq.is_Pow is False and eq.as_numer_denom() == (-(-npos)**pow*(-dneg)**pow, npos)
eq = eqn(nneg, dpos, -pow)
assert not eq.is_Pow or eq.as_numer_denom() == (dpos**pow, nneg**pow)
eq = eqn(nneg, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-nneg)**pow)
# unknown exponent
pow = 2*any
eq = eqn(npos, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (npos**pow, dpos**pow)
eq = eqn(npos, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-npos)**pow, (-dneg)**pow)
eq = eqn(nneg, dpos, pow)
assert eq.is_Pow and eq.as_numer_denom() == (nneg**pow, dpos**pow)
eq = eqn(nneg, dneg, pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-nneg)**pow, (-dneg)**pow)
eq = eqn(npos, dpos, -pow)
assert eq.as_numer_denom() == (dpos**pow, npos**pow)
eq = eqn(npos, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-npos)**pow)
eq = eqn(nneg, dpos, -pow)
assert eq.is_Pow and eq.as_numer_denom() == (dpos**pow, nneg**pow)
eq = eqn(nneg, dneg, -pow)
assert eq.is_Pow and eq.as_numer_denom() == ((-dneg)**pow, (-nneg)**pow)
assert ((1/(1 + x/3))**(-S.One)).as_numer_denom() == (3 + x, 3)
notp = Symbol('notp', positive=False) # not positive does not imply real
b = ((1 + x/notp)**-2)
assert (b**(-y)).as_numer_denom() == (1, b**y)
assert (b**(-S.One)).as_numer_denom() == ((notp + x)**2, notp**2)
nonp = Symbol('nonp', nonpositive=True)
assert (((1 + x/nonp)**-2)**(-S.One)).as_numer_denom() == ((-nonp -
x)**2, nonp**2)
n = Symbol('n', negative=True)
assert (x**n).as_numer_denom() == (1, x**-n)
assert sqrt(1/n).as_numer_denom() == (S.ImaginaryUnit, sqrt(-n))
n = Symbol('0 or neg', nonpositive=True)
# if x and n are split up without negating each term and n is negative
# then the answer might be wrong; if n is 0 it won't matter since
# 1/oo and 1/zoo are both zero as is sqrt(0)/sqrt(-x) unless x is also
# zero (in which case the negative sign doesn't matter):
# 1/sqrt(1/-1) = -I but sqrt(-1)/sqrt(1) = I
assert (1/sqrt(x/n)).as_numer_denom() == (sqrt(-n), sqrt(-x))
c = Symbol('c', complex=True)
e = sqrt(1/c)
assert e.as_numer_denom() == (e, 1)
i = Symbol('i', integer=True)
assert ((1 + x/y)**i).as_numer_denom() == ((x + y)**i, y**i)
def test_Pow_Expr_args():
bases = [Basic(), Poly(x, x), FiniteSet(x)]
for base in bases:
# The cache can mess with the stacklevel test
with warns(SymPyDeprecationWarning, test_stacklevel=False):
Pow(base, S.One)
def test_Pow_signs():
"""Cf. issues 4595 and 5250"""
n = Symbol('n', even=True)
assert (3 - y)**2 != (y - 3)**2
assert (3 - y)**n != (y - 3)**n
assert (-3 + y - x)**2 != (3 - y + x)**2
assert (y - 3)**3 != -(3 - y)**3
def test_power_with_noncommutative_mul_as_base():
x = Symbol('x', commutative=False)
y = Symbol('y', commutative=False)
assert not (x*y)**3 == x**3*y**3
assert (2*x*y)**3 == 8*(x*y)**3
@_both_exp_pow
def test_power_rewrite_exp():
assert (I**I).rewrite(exp) == exp(-pi/2)
expr = (2 + 3*I)**(4 + 5*I)
assert expr.rewrite(exp) == exp((4 + 5*I)*(log(sqrt(13)) + I*atan(Rational(3, 2))))
assert expr.rewrite(exp).expand() == \
169*exp(5*I*log(13)/2)*exp(4*I*atan(Rational(3, 2)))*exp(-5*atan(Rational(3, 2)))
assert ((6 + 7*I)**5).rewrite(exp) == 7225*sqrt(85)*exp(5*I*atan(Rational(7, 6)))
expr = 5**(6 + 7*I)
assert expr.rewrite(exp) == exp((6 + 7*I)*log(5))
assert expr.rewrite(exp).expand() == 15625*exp(7*I*log(5))
assert Pow(123, 789, evaluate=False).rewrite(exp) == 123**789
assert (1**I).rewrite(exp) == 1**I
assert (0**I).rewrite(exp) == 0**I
expr = (-2)**(2 + 5*I)
assert expr.rewrite(exp) == exp((2 + 5*I)*(log(2) + I*pi))
assert expr.rewrite(exp).expand() == 4*exp(-5*pi)*exp(5*I*log(2))
assert ((-2)**S(-5)).rewrite(exp) == (-2)**S(-5)
x, y = symbols('x y')
assert (x**y).rewrite(exp) == exp(y*log(x))
if global_parameters.exp_is_pow:
assert (7**x).rewrite(exp) == Pow(S.Exp1, x*log(7), evaluate=False)
else:
assert (7**x).rewrite(exp) == exp(x*log(7), evaluate=False)
assert ((2 + 3*I)**x).rewrite(exp) == exp(x*(log(sqrt(13)) + I*atan(Rational(3, 2))))
assert (y**(5 + 6*I)).rewrite(exp) == exp(log(y)*(5 + 6*I))
assert all((1/func(x)).rewrite(exp) == 1/(func(x).rewrite(exp)) for func in
(sin, cos, tan, sec, csc, sinh, cosh, tanh))
def test_zero():
assert 0**x != 0
assert 0**(2*x) == 0**x
assert 0**(1.0*x) == 0**x
assert 0**(2.0*x) == 0**x
assert (0**(2 - x)).as_base_exp() == (0, 2 - x)
assert 0**(x - 2) != S.Infinity**(2 - x)
assert 0**(2*x*y) == 0**(x*y)
assert 0**(-2*x*y) == S.ComplexInfinity**(x*y)
assert Float(0)**2 is not S.Zero
assert Float(0)**2 == 0.0
assert Float(0)**-2 is zoo
assert Float(0)**oo is S.Zero
#Test issue 19572
assert 0 ** -oo is zoo
assert power(0, -oo) is zoo
assert Float(0)**-oo is zoo
def test_pow_as_base_exp():
assert (S.Infinity**(2 - x)).as_base_exp() == (S.Infinity, 2 - x)
assert (S.Infinity**(x - 2)).as_base_exp() == (S.Infinity, x - 2)
p = S.Half**x
assert p.base, p.exp == p.as_base_exp() == (S(2), -x)
p = (S(3)/2)**x
assert p.base, p.exp == p.as_base_exp() == (3*S.Half, x)
p = (S(2)/3)**x
assert p.as_base_exp() == (S(3)/2, -x)
assert p.base, p.exp == (S(2)/3, x)
# issue 8344:
assert Pow(1, 2, evaluate=False).as_base_exp() == (S.One, S(2))
def test_nseries():
assert sqrt(I*x - 1)._eval_nseries(x, 4, None, 1) == I + x/2 + I*x**2/8 - x**3/16 + O(x**4)
assert sqrt(I*x - 1)._eval_nseries(x, 4, None, -1) == -I - x/2 - I*x**2/8 + x**3/16 + O(x**4)
assert cbrt(I*x - 1)._eval_nseries(x, 4, None, 1) == (-1)**(S(1)/3) - (-1)**(S(5)/6)*x/3 + \
(-1)**(S(1)/3)*x**2/9 + 5*(-1)**(S(5)/6)*x**3/81 + O(x**4)
assert cbrt(I*x - 1)._eval_nseries(x, 4, None, -1) == -(-1)**(S(2)/3) - (-1)**(S(1)/6)*x/3 - \
(-1)**(S(2)/3)*x**2/9 + 5*(-1)**(S(1)/6)*x**3/81 + O(x**4)
assert (1 / (exp(-1/x) + 1/x))._eval_nseries(x, 2, None) == x + O(x**2)
# test issue 23752
assert sqrt(-I*x**2 + x - 3)._eval_nseries(x, 4, None, 1) == -sqrt(3)*I + sqrt(3)*I*x/6 - \
sqrt(3)*I*x**2*(-S(1)/72 + I/6) - sqrt(3)*I*x**3*(-S(1)/432 + I/36) + O(x**4)
assert sqrt(-I*x**2 + x - 3)._eval_nseries(x, 4, None, -1) == -sqrt(3)*I + sqrt(3)*I*x/6 - \
sqrt(3)*I*x**2*(-S(1)/72 + I/6) - sqrt(3)*I*x**3*(-S(1)/432 + I/36) + O(x**4)
assert cbrt(-I*x**2 + x - 3)._eval_nseries(x, 4, None, 1) == -(-1)**(S(2)/3)*3**(S(1)/3) + \
(-1)**(S(2)/3)*3**(S(1)/3)*x/9 - (-1)**(S(2)/3)*3**(S(1)/3)*x**2*(-S(1)/81 + I/9) - \
(-1)**(S(2)/3)*3**(S(1)/3)*x**3*(-S(5)/2187 + 2*I/81) + O(x**4)
assert cbrt(-I*x**2 + x - 3)._eval_nseries(x, 4, None, -1) == -(-1)**(S(2)/3)*3**(S(1)/3) + \
(-1)**(S(2)/3)*3**(S(1)/3)*x/9 - (-1)**(S(2)/3)*3**(S(1)/3)*x**2*(-S(1)/81 + I/9) - \
(-1)**(S(2)/3)*3**(S(1)/3)*x**3*(-S(5)/2187 + 2*I/81) + O(x**4)
def test_issue_6100_12942_4473():
assert x**1.0 != x
assert x != x**1.0
assert True != x**1.0
assert x**1.0 is not True
assert x is not True
assert x*y != (x*y)**1.0
# Pow != Symbol
assert (x**1.0)**1.0 != x
assert (x**1.0)**2.0 != x**2
b = Expr()
assert Pow(b, 1.0, evaluate=False) != b
# if the following gets distributed as a Mul (x**1.0*y**1.0 then
# __eq__ methods could be added to Symbol and Pow to detect the
# power-of-1.0 case.
assert ((x*y)**1.0).func is Pow
def test_issue_6208():
from sympy.functions.elementary.miscellaneous import root
assert sqrt(33**(I*9/10)) == -33**(I*9/20)
assert root((6*I)**(2*I), 3).as_base_exp()[1] == Rational(1, 3) # != 2*I/3
assert root((6*I)**(I/3), 3).as_base_exp()[1] == I/9
assert sqrt(exp(3*I)) == exp(3*I/2)
assert sqrt(-sqrt(3)*(1 + 2*I)) == sqrt(sqrt(3))*sqrt(-1 - 2*I)
assert sqrt(exp(5*I)) == -exp(5*I/2)
assert root(exp(5*I), 3).exp == Rational(1, 3)
def test_issue_6990():
assert (sqrt(a + b*x + x**2)).series(x, 0, 3).removeO() == \
sqrt(a)*x**2*(1/(2*a) - b**2/(8*a**2)) + sqrt(a) + b*x/(2*sqrt(a))
def test_issue_6068():
assert sqrt(sin(x)).series(x, 0, 7) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 + O(x**7)
assert sqrt(sin(x)).series(x, 0, 9) == \
sqrt(x) - x**Rational(5, 2)/12 + x**Rational(9, 2)/1440 - \
x**Rational(13, 2)/24192 - 67*x**Rational(17, 2)/29030400 + O(x**9)
assert sqrt(sin(x**3)).series(x, 0, 19) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 + O(x**19)
assert sqrt(sin(x**3)).series(x, 0, 20) == \
x**Rational(3, 2) - x**Rational(15, 2)/12 + x**Rational(27, 2)/1440 - \
x**Rational(39, 2)/24192 + O(x**20)
def test_issue_6782():
assert sqrt(sin(x**3)).series(x, 0, 7) == x**Rational(3, 2) + O(x**7)
assert sqrt(sin(x**4)).series(x, 0, 3) == x**2 + O(x**3)
def test_issue_6653():
assert (1 / sqrt(1 + sin(x**2))).series(x, 0, 3) == 1 - x**2/2 + O(x**3)
def test_issue_6429():
f = (c**2 + x)**(0.5)
assert f.series(x, x0=0, n=1) == (c**2)**0.5 + O(x)
assert f.taylor_term(0, x) == (c**2)**0.5
assert f.taylor_term(1, x) == 0.5*x*(c**2)**(-0.5)
assert f.taylor_term(2, x) == -0.125*x**2*(c**2)**(-1.5)
def test_issue_7638():
f = pi/log(sqrt(2))
assert ((1 + I)**(I*f/2))**0.3 == (1 + I)**(0.15*I*f)
# if 1/3 -> 1.0/3 this should fail since it cannot be shown that the
# sign will be +/-1; for the previous "small arg" case, it didn't matter
# that this could not be proved
assert (1 + I)**(4*I*f) == ((1 + I)**(12*I*f))**Rational(1, 3)
assert (((1 + I)**(I*(1 + 7*f)))**Rational(1, 3)).exp == Rational(1, 3)
r = symbols('r', real=True)
assert sqrt(r**2) == abs(r)
assert cbrt(r**3) != r
assert sqrt(Pow(2*I, 5*S.Half)) != (2*I)**Rational(5, 4)
p = symbols('p', positive=True)
assert cbrt(p**2) == p**Rational(2, 3)
assert NS(((0.2 + 0.7*I)**(0.7 + 1.0*I))**(0.5 - 0.1*I), 1) == '0.4 + 0.2*I'
assert sqrt(1/(1 + I)) == sqrt(1 - I)/sqrt(2) # or 1/sqrt(1 + I)
e = 1/(1 - sqrt(2))
assert sqrt(e) == I/sqrt(-1 + sqrt(2))
assert e**Rational(-1, 2) == -I*sqrt(-1 + sqrt(2))
assert sqrt((cos(1)**2 + sin(1)**2 - 1)**(3 + I)).exp in [S.Half,
Rational(3, 2) + I/2]
assert sqrt(r**Rational(4, 3)) != r**Rational(2, 3)
assert sqrt((p + I)**Rational(4, 3)) == (p + I)**Rational(2, 3)
for q in 1+I, 1-I:
assert sqrt(q**2) == q
for q in -1+I, -1-I:
assert sqrt(q**2) == -q
assert sqrt((p + r*I)**2) != p + r*I
e = (1 + I/5)
assert sqrt(e**5) == e**(5*S.Half)
assert sqrt(e**6) == e**3
assert sqrt((1 + I*r)**6) != (1 + I*r)**3
def test_issue_8582():
assert 1**oo is nan
assert 1**(-oo) is nan
assert 1**zoo is nan
assert 1**(oo + I) is nan
assert 1**(1 + I*oo) is nan
assert 1**(oo + I*oo) is nan
def test_issue_8650():
n = Symbol('n', integer=True, nonnegative=True)
assert (n**n).is_positive is True
x = 5*n + 5
assert (x**(5*(n + 1))).is_positive is True
def test_issue_13914():
b = Symbol('b')
assert (-1)**zoo is nan
assert 2**zoo is nan
assert (S.Half)**(1 + zoo) is nan
assert I**(zoo + I) is nan
assert b**(I + zoo) is nan
def test_better_sqrt():
n = Symbol('n', integer=True, nonnegative=True)
assert sqrt(3 + 4*I) == 2 + I
assert sqrt(3 - 4*I) == 2 - I
assert sqrt(-3 - 4*I) == 1 - 2*I
assert sqrt(-3 + 4*I) == 1 + 2*I
assert sqrt(32 + 24*I) == 6 + 2*I
assert sqrt(32 - 24*I) == 6 - 2*I
assert sqrt(-32 - 24*I) == 2 - 6*I
assert sqrt(-32 + 24*I) == 2 + 6*I
# triple (3, 4, 5):
# parity of 3 matches parity of 5 and
# den, 4, is a square
assert sqrt((3 + 4*I)/4) == 1 + I/2
# triple (8, 15, 17)
# parity of 8 doesn't match parity of 17 but
# den/2, 8/2, is a square
assert sqrt((8 + 15*I)/8) == (5 + 3*I)/4
# handle the denominator
assert sqrt((3 - 4*I)/25) == (2 - I)/5
assert sqrt((3 - 4*I)/26) == (2 - I)/sqrt(26)
# mul
# issue #12739
assert sqrt((3 + 4*I)/(3 - 4*I)) == (3 + 4*I)/5
assert sqrt(2/(3 + 4*I)) == sqrt(2)/5*(2 - I)
assert sqrt(n/(3 + 4*I)).subs(n, 2) == sqrt(2)/5*(2 - I)
assert sqrt(-2/(3 + 4*I)) == sqrt(2)/5*(1 + 2*I)
assert sqrt(-n/(3 + 4*I)).subs(n, 2) == sqrt(2)/5*(1 + 2*I)
# power
assert sqrt(1/(3 + I*4)) == (2 - I)/5
assert sqrt(1/(3 - I)) == sqrt(10)*sqrt(3 + I)/10
# symbolic
i = symbols('i', imaginary=True)
assert sqrt(3/i) == Mul(sqrt(3), 1/sqrt(i), evaluate=False)
# multiples of 1/2; don't make this too automatic
assert sqrt(3 + 4*I)**3 == (2 + I)**3
assert Pow(3 + 4*I, Rational(3, 2)) == 2 + 11*I
assert Pow(6 + 8*I, Rational(3, 2)) == 2*sqrt(2)*(2 + 11*I)
n, d = (3 + 4*I), (3 - 4*I)**3
a = n/d
assert a.args == (1/d, n)
eq = sqrt(a)
assert eq.args == (a, S.Half)
assert expand_multinomial(eq) == sqrt((-117 + 44*I)*(3 + 4*I))/125
assert eq.expand() == (7 - 24*I)/125
# issue 12775
# pos im part
assert sqrt(2*I) == (1 + I)
assert sqrt(2*9*I) == Mul(3, 1 + I, evaluate=False)
assert Pow(2*I, 3*S.Half) == (1 + I)**3
# neg im part
assert sqrt(-I/2) == Mul(S.Half, 1 - I, evaluate=False)
# fractional im part
assert Pow(Rational(-9, 2)*I, Rational(3, 2)) == 27*(1 - I)**3/8
def test_issue_2993():
assert str((2.3*x - 4)**0.3) == '1.5157165665104*(0.575*x - 1)**0.3'
assert str((2.3*x + 4)**0.3) == '1.5157165665104*(0.575*x + 1)**0.3'
assert str((-2.3*x + 4)**0.3) == '1.5157165665104*(1 - 0.575*x)**0.3'
assert str((-2.3*x - 4)**0.3) == '1.5157165665104*(-0.575*x - 1)**0.3'
assert str((2.3*x - 2)**0.3) == '1.28386201800527*(x - 0.869565217391304)**0.3'
assert str((-2.3*x - 2)**0.3) == '1.28386201800527*(-x - 0.869565217391304)**0.3'
assert str((-2.3*x + 2)**0.3) == '1.28386201800527*(0.869565217391304 - x)**0.3'
assert str((2.3*x + 2)**0.3) == '1.28386201800527*(x + 0.869565217391304)**0.3'
assert str((2.3*x - 4)**Rational(1, 3)) == '2**(2/3)*(0.575*x - 1)**(1/3)'
eq = (2.3*x + 4)
assert eq**2 == 16*(0.575*x + 1)**2
assert (1/eq).args == (eq, -1) # don't change trivial power
# issue 17735
q=.5*exp(x) - .5*exp(-x) + 0.1
assert int((q**2).subs(x, 1)) == 1
# issue 17756
y = Symbol('y')
assert len(sqrt(x/(x + y)**2 + Float('0.008', 30)).subs(y, pi.n(25)).atoms(Float)) == 2
# issue 17756
a, b, c, d, e, f, g = symbols('a:g')
expr = sqrt(1 + a*(c**4 + g*d - 2*g*e - f*(-g + d))**2/
(c**3*b**2*(d - 3*e + 2*f)**2))/2
r = [
(a, N('0.0170992456333788667034850458615', 30)),
(b, N('0.0966594956075474769169134801223', 30)),
(c, N('0.390911862903463913632151616184', 30)),
(d, N('0.152812084558656566271750185933', 30)),
(e, N('0.137562344465103337106561623432', 30)),
(f, N('0.174259178881496659302933610355', 30)),
(g, N('0.220745448491223779615401870086', 30))]
tru = expr.n(30, subs=dict(r))
seq = expr.subs(r)
# although `tru` is the right way to evaluate
# expr with numerical values, `seq` will have
# significant loss of precision if extraction of
# the largest coefficient of a power's base's terms
# is done improperly
assert seq == tru
def test_issue_17450():
assert (erf(cosh(1)**7)**I).is_real is None
assert (erf(cosh(1)**7)**I).is_imaginary is False
assert (Pow(exp(1+sqrt(2)), ((1-sqrt(2))*I*pi), evaluate=False)).is_real is None
assert ((-10)**(10*I*pi/3)).is_real is False
assert ((-5)**(4*I*pi)).is_real is False
def test_issue_18190():
assert sqrt(1 / tan(1 + I)) == 1 / sqrt(tan(1 + I))
def test_issue_14815():
x = Symbol('x', real=True)
assert sqrt(x).is_extended_negative is False
x = Symbol('x', real=False)
assert sqrt(x).is_extended_negative is None
x = Symbol('x', complex=True)
assert sqrt(x).is_extended_negative is False
x = Symbol('x', extended_real=True)
assert sqrt(x).is_extended_negative is False
assert sqrt(zoo, evaluate=False).is_extended_negative is None
assert sqrt(nan, evaluate=False).is_extended_negative is None
def test_issue_18509():
x = Symbol('x', prime=True)
assert x**oo is oo
assert (1/x)**oo is S.Zero
assert (-1/x)**oo is S.Zero
assert (-x)**oo is zoo
assert (-oo)**(-1 + I) is S.Zero
assert (-oo)**(1 + I) is zoo
assert (oo)**(-1 + I) is S.Zero
assert (oo)**(1 + I) is zoo
def test_issue_18762():
e, p = symbols('e p')
g0 = sqrt(1 + e**2 - 2*e*cos(p))
assert len(g0.series(e, 1, 3).args) == 4
def test_issue_21860():
e = 3*2**Rational(66666666667,200000000000)*3**Rational(16666666667,50000000000)*x**Rational(66666666667, 200000000000)
ans = Mul(Rational(3, 2),
Pow(Integer(2), Rational(33333333333, 100000000000)),
Pow(Integer(3), Rational(26666666667, 40000000000)))
assert e.xreplace({x: Rational(3,8)}) == ans
def test_issue_21647():
e = log((Integer(567)/500)**(811*(Integer(567)/500)**x/100))
ans = log(Mul(Rational(64701150190720499096094005280169087619821081527,
76293945312500000000000000000000000000000000000),
Pow(Integer(2), Rational(396204892125479941, 781250000000000000)),
Pow(Integer(3), Rational(385045107874520059, 390625000000000000)),
Pow(Integer(5), Rational(407364676376439823, 1562500000000000000)),
Pow(Integer(7), Rational(385045107874520059, 1562500000000000000))))
assert e.xreplace({x: 6}) == ans
def test_issue_21762():
e = (x**2 + 6)**(Integer(33333333333333333)/50000000000000000)
ans = Mul(Rational(5, 4),
Pow(Integer(2), Rational(16666666666666667, 25000000000000000)),
Pow(Integer(5), Rational(8333333333333333, 25000000000000000)))
assert e.xreplace({x: S.Half}) == ans
def test_issue_14704():
a = 144**144
x, xexact = integer_nthroot(a,a)
assert x == 1 and xexact is False
def test_rational_powers_larger_than_one():
assert Rational(2, 3)**Rational(3, 2) == 2*sqrt(6)/9
assert Rational(1, 6)**Rational(9, 4) == 6**Rational(3, 4)/216
assert Rational(3, 7)**Rational(7, 3) == 9*3**Rational(1, 3)*7**Rational(2, 3)/343
def test_power_dispatcher():
class NewBase(Expr):
pass
class NewPow(NewBase, Pow):
pass
a, b = Symbol('a'), NewBase()
@power.register(Expr, NewBase)
@power.register(NewBase, Expr)
@power.register(NewBase, NewBase)
def _(a, b):
return NewPow(a, b)
# Pow called as fallback
assert power(2, 3) == 8*S.One
assert power(a, 2) == Pow(a, 2)
assert power(a, a) == Pow(a, a)
# NewPow called by dispatch
assert power(a, b) == NewPow(a, b)
assert power(b, a) == NewPow(b, a)
assert power(b, b) == NewPow(b, b)
def test_powers_of_I():
assert [sqrt(I)**i for i in range(13)] == [
1, sqrt(I), I, sqrt(I)**3, -1, -sqrt(I), -I, -sqrt(I)**3,
1, sqrt(I), I, sqrt(I)**3, -1]
assert sqrt(I)**(S(9)/2) == -I**(S(1)/4)
def test_issue_23918():
b = S(2)/3
assert (b**x).as_base_exp() == (1/b, -x)
|
36907c95af8fc2faa5959bd2c645f8bd71579a937acce8dd612c0e9941108195 | """Tests for the implementation of RootOf class and related tools. """
from sympy.polys.polytools import Poly
import sympy.polys.rootoftools as rootoftools
from sympy.polys.rootoftools import (rootof, RootOf, CRootOf, RootSum,
_pure_key_dict as D)
from sympy.polys.polyerrors import (
MultivariatePolynomialError,
GeneratorsNeeded,
PolynomialError,
)
from sympy.core.function import (Function, Lambda)
from sympy.core.numbers import (Float, I, Rational)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import tan
from sympy.integrals.integrals import Integral
from sympy.polys.orthopolys import legendre_poly
from sympy.solvers.solvers import solve
from sympy.testing.pytest import raises, slow
from sympy.core.expr import unchanged
from sympy.abc import a, b, x, y, z, r
def test_CRootOf___new__():
assert rootof(x, 0) == 0
assert rootof(x, -1) == 0
assert rootof(x, S.Zero) == 0
assert rootof(x - 1, 0) == 1
assert rootof(x - 1, -1) == 1
assert rootof(x + 1, 0) == -1
assert rootof(x + 1, -1) == -1
assert rootof(x**2 + 2*x + 3, 0) == -1 - I*sqrt(2)
assert rootof(x**2 + 2*x + 3, 1) == -1 + I*sqrt(2)
assert rootof(x**2 + 2*x + 3, -1) == -1 + I*sqrt(2)
assert rootof(x**2 + 2*x + 3, -2) == -1 - I*sqrt(2)
r = rootof(x**2 + 2*x + 3, 0, radicals=False)
assert isinstance(r, RootOf) is True
r = rootof(x**2 + 2*x + 3, 1, radicals=False)
assert isinstance(r, RootOf) is True
r = rootof(x**2 + 2*x + 3, -1, radicals=False)
assert isinstance(r, RootOf) is True
r = rootof(x**2 + 2*x + 3, -2, radicals=False)
assert isinstance(r, RootOf) is True
assert rootof((x - 1)*(x + 1), 0, radicals=False) == -1
assert rootof((x - 1)*(x + 1), 1, radicals=False) == 1
assert rootof((x - 1)*(x + 1), -1, radicals=False) == 1
assert rootof((x - 1)*(x + 1), -2, radicals=False) == -1
assert rootof((x - 1)*(x + 1), 0, radicals=True) == -1
assert rootof((x - 1)*(x + 1), 1, radicals=True) == 1
assert rootof((x - 1)*(x + 1), -1, radicals=True) == 1
assert rootof((x - 1)*(x + 1), -2, radicals=True) == -1
assert rootof((x - 1)*(x**3 + x + 3), 0) == rootof(x**3 + x + 3, 0)
assert rootof((x - 1)*(x**3 + x + 3), 1) == 1
assert rootof((x - 1)*(x**3 + x + 3), 2) == rootof(x**3 + x + 3, 1)
assert rootof((x - 1)*(x**3 + x + 3), 3) == rootof(x**3 + x + 3, 2)
assert rootof((x - 1)*(x**3 + x + 3), -1) == rootof(x**3 + x + 3, 2)
assert rootof((x - 1)*(x**3 + x + 3), -2) == rootof(x**3 + x + 3, 1)
assert rootof((x - 1)*(x**3 + x + 3), -3) == 1
assert rootof((x - 1)*(x**3 + x + 3), -4) == rootof(x**3 + x + 3, 0)
assert rootof(x**4 + 3*x**3, 0) == -3
assert rootof(x**4 + 3*x**3, 1) == 0
assert rootof(x**4 + 3*x**3, 2) == 0
assert rootof(x**4 + 3*x**3, 3) == 0
raises(GeneratorsNeeded, lambda: rootof(0, 0))
raises(GeneratorsNeeded, lambda: rootof(1, 0))
raises(PolynomialError, lambda: rootof(Poly(0, x), 0))
raises(PolynomialError, lambda: rootof(Poly(1, x), 0))
raises(PolynomialError, lambda: rootof(x - y, 0))
# issue 8617
raises(PolynomialError, lambda: rootof(exp(x), 0))
raises(NotImplementedError, lambda: rootof(x**3 - x + sqrt(2), 0))
raises(NotImplementedError, lambda: rootof(x**3 - x + I, 0))
raises(IndexError, lambda: rootof(x**2 - 1, -4))
raises(IndexError, lambda: rootof(x**2 - 1, -3))
raises(IndexError, lambda: rootof(x**2 - 1, 2))
raises(IndexError, lambda: rootof(x**2 - 1, 3))
raises(ValueError, lambda: rootof(x**2 - 1, x))
assert rootof(Poly(x - y, x), 0) == y
assert rootof(Poly(x**2 - y, x), 0) == -sqrt(y)
assert rootof(Poly(x**2 - y, x), 1) == sqrt(y)
assert rootof(Poly(x**3 - y, x), 0) == y**Rational(1, 3)
assert rootof(y*x**3 + y*x + 2*y, x, 0) == -1
raises(NotImplementedError, lambda: rootof(x**3 + x + 2*y, x, 0))
assert rootof(x**3 + x + 1, 0).is_commutative is True
def test_CRootOf_attributes():
r = rootof(x**3 + x + 3, 0)
assert r.is_number
assert r.free_symbols == set()
# if the following assertion fails then multivariate polynomials
# are apparently supported and the RootOf.free_symbols routine
# should be changed to return whatever symbols would not be
# the PurePoly dummy symbol
raises(NotImplementedError, lambda: rootof(Poly(x**3 + y*x + 1, x), 0))
def test_CRootOf___eq__():
assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 0)) is True
assert (rootof(x**3 + x + 3, 0) == rootof(x**3 + x + 3, 1)) is False
assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 1)) is True
assert (rootof(x**3 + x + 3, 1) == rootof(x**3 + x + 3, 2)) is False
assert (rootof(x**3 + x + 3, 2) == rootof(x**3 + x + 3, 2)) is True
assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 0)) is True
assert (rootof(x**3 + x + 3, 0) == rootof(y**3 + y + 3, 1)) is False
assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 1)) is True
assert (rootof(x**3 + x + 3, 1) == rootof(y**3 + y + 3, 2)) is False
assert (rootof(x**3 + x + 3, 2) == rootof(y**3 + y + 3, 2)) is True
def test_CRootOf___eval_Eq__():
f = Function('f')
eq = x**3 + x + 3
r = rootof(eq, 2)
r1 = rootof(eq, 1)
assert Eq(r, r1) is S.false
assert Eq(r, r) is S.true
assert unchanged(Eq, r, x)
assert Eq(r, 0) is S.false
assert Eq(r, S.Infinity) is S.false
assert Eq(r, I) is S.false
assert unchanged(Eq, r, f(0))
sol = solve(eq)
for s in sol:
if s.is_real:
assert Eq(r, s) is S.false
r = rootof(eq, 0)
for s in sol:
if s.is_real:
assert Eq(r, s) is S.true
eq = x**3 + x + 1
sol = solve(eq)
assert [Eq(rootof(eq, i), j) for i in range(3) for j in sol
].count(True) == 3
assert Eq(rootof(eq, 0), 1 + S.ImaginaryUnit) == False
def test_CRootOf_is_real():
assert rootof(x**3 + x + 3, 0).is_real is True
assert rootof(x**3 + x + 3, 1).is_real is False
assert rootof(x**3 + x + 3, 2).is_real is False
def test_CRootOf_is_complex():
assert rootof(x**3 + x + 3, 0).is_complex is True
def test_CRootOf_subs():
assert rootof(x**3 + x + 1, 0).subs(x, y) == rootof(y**3 + y + 1, 0)
def test_CRootOf_diff():
assert rootof(x**3 + x + 1, 0).diff(x) == 0
assert rootof(x**3 + x + 1, 0).diff(y) == 0
@slow
def test_CRootOf_evalf():
real = rootof(x**3 + x + 3, 0).evalf(n=20)
assert real.epsilon_eq(Float("-1.2134116627622296341"))
re, im = rootof(x**3 + x + 3, 1).evalf(n=20).as_real_imag()
assert re.epsilon_eq( Float("0.60670583138111481707"))
assert im.epsilon_eq(-Float("1.45061224918844152650"))
re, im = rootof(x**3 + x + 3, 2).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("0.60670583138111481707"))
assert im.epsilon_eq(Float("1.45061224918844152650"))
p = legendre_poly(4, x, polys=True)
roots = [str(r.n(17)) for r in p.real_roots()]
# magnitudes are given by
# sqrt(3/S(7) - 2*sqrt(6/S(5))/7)
# and
# sqrt(3/S(7) + 2*sqrt(6/S(5))/7)
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
re = rootof(x**5 - 5*x + 12, 0).evalf(n=20)
assert re.epsilon_eq(Float("-1.84208596619025438271"))
re, im = rootof(x**5 - 5*x + 12, 1).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("-1.709561043370328882010"))
re, im = rootof(x**5 - 5*x + 12, 2).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("-0.351854240827371999559"))
assert im.epsilon_eq(Float("+1.709561043370328882010"))
re, im = rootof(x**5 - 5*x + 12, 3).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("-0.719798681483861386681"))
re, im = rootof(x**5 - 5*x + 12, 4).evalf(n=20).as_real_imag()
assert re.epsilon_eq(Float("+1.272897223922499190910"))
assert im.epsilon_eq(Float("+0.719798681483861386681"))
# issue 6393
assert str(rootof(x**5 + 2*x**4 + x**3 - 68719476736, 0).n(3)) == '147.'
eq = (531441*x**11 + 3857868*x**10 + 13730229*x**9 + 32597882*x**8 +
55077472*x**7 + 60452000*x**6 + 32172064*x**5 - 4383808*x**4 -
11942912*x**3 - 1506304*x**2 + 1453312*x + 512)
a, b = rootof(eq, 1).n(2).as_real_imag()
c, d = rootof(eq, 2).n(2).as_real_imag()
assert a == c
assert b < d
assert b == -d
# issue 6451
r = rootof(legendre_poly(64, x), 7)
assert r.n(2) == r.n(100).n(2)
# issue 9019
r0 = rootof(x**2 + 1, 0, radicals=False)
r1 = rootof(x**2 + 1, 1, radicals=False)
assert r0.n(4) == -1.0*I
assert r1.n(4) == 1.0*I
# make sure verification is used in case a max/min traps the "root"
assert str(rootof(4*x**5 + 16*x**3 + 12*x**2 + 7, 0).n(3)) == '-0.976'
# watch out for UnboundLocalError
c = CRootOf(90720*x**6 - 4032*x**4 + 84*x**2 - 1, 0)
assert c._eval_evalf(2) # doesn't fail
# watch out for imaginary parts that don't want to evaluate
assert str(RootOf(x**16 + 32*x**14 + 508*x**12 + 5440*x**10 +
39510*x**8 + 204320*x**6 + 755548*x**4 + 1434496*x**2 +
877969, 10).n(2)) == '-3.4*I'
assert abs(RootOf(x**4 + 10*x**2 + 1, 0).n(2)) < 0.4
# check reset and args
r = [RootOf(x**3 + x + 3, i) for i in range(3)]
r[0]._reset()
for ri in r:
i = ri._get_interval()
ri.n(2)
assert i != ri._get_interval()
ri._reset()
assert i == ri._get_interval()
assert i == i.func(*i.args)
def test_CRootOf_evalf_caching_bug():
r = rootof(x**5 - 5*x + 12, 1)
r.n()
a = r._get_interval()
r = rootof(x**5 - 5*x + 12, 1)
r.n()
b = r._get_interval()
assert a == b
def test_CRootOf_real_roots():
assert Poly(x**5 + x + 1).real_roots() == [rootof(x**3 - x**2 + 1, 0)]
assert Poly(x**5 + x + 1).real_roots(radicals=False) == [rootof(
x**3 - x**2 + 1, 0)]
# https://github.com/sympy/sympy/issues/20902
p = Poly(-3*x**4 - 10*x**3 - 12*x**2 - 6*x - 1, x, domain='ZZ')
assert CRootOf.real_roots(p) == [S(-1), S(-1), S(-1), S(-1)/3]
def test_CRootOf_all_roots():
assert Poly(x**5 + x + 1).all_roots() == [
rootof(x**3 - x**2 + 1, 0),
Rational(-1, 2) - sqrt(3)*I/2,
Rational(-1, 2) + sqrt(3)*I/2,
rootof(x**3 - x**2 + 1, 1),
rootof(x**3 - x**2 + 1, 2),
]
assert Poly(x**5 + x + 1).all_roots(radicals=False) == [
rootof(x**3 - x**2 + 1, 0),
rootof(x**2 + x + 1, 0, radicals=False),
rootof(x**2 + x + 1, 1, radicals=False),
rootof(x**3 - x**2 + 1, 1),
rootof(x**3 - x**2 + 1, 2),
]
def test_CRootOf_eval_rational():
p = legendre_poly(4, x, polys=True)
roots = [r.eval_rational(n=18) for r in p.real_roots()]
for root in roots:
assert isinstance(root, Rational)
roots = [str(root.n(17)) for root in roots]
assert roots == [
"-0.86113631159405258",
"-0.33998104358485626",
"0.33998104358485626",
"0.86113631159405258",
]
def test_CRootOf_lazy():
# irreducible poly with both real and complex roots:
f = Poly(x**3 + 2*x + 2)
# real root:
CRootOf.clear_cache()
r = CRootOf(f, 0)
# Not yet in cache, after construction:
assert r.poly not in rootoftools._reals_cache
assert r.poly not in rootoftools._complexes_cache
r.evalf()
# In cache after evaluation:
assert r.poly in rootoftools._reals_cache
assert r.poly not in rootoftools._complexes_cache
# complex root:
CRootOf.clear_cache()
r = CRootOf(f, 1)
# Not yet in cache, after construction:
assert r.poly not in rootoftools._reals_cache
assert r.poly not in rootoftools._complexes_cache
r.evalf()
# In cache after evaluation:
assert r.poly in rootoftools._reals_cache
assert r.poly in rootoftools._complexes_cache
# composite poly with both real and complex roots:
f = Poly((x**2 - 2)*(x**2 + 1))
# real root:
CRootOf.clear_cache()
r = CRootOf(f, 0)
# In cache immediately after construction:
assert r.poly in rootoftools._reals_cache
assert r.poly not in rootoftools._complexes_cache
# complex root:
CRootOf.clear_cache()
r = CRootOf(f, 2)
# In cache immediately after construction:
assert r.poly in rootoftools._reals_cache
assert r.poly in rootoftools._complexes_cache
def test_RootSum___new__():
f = x**3 + x + 3
g = Lambda(r, log(r*x))
s = RootSum(f, g)
assert isinstance(s, RootSum) is True
assert RootSum(f**2, g) == 2*RootSum(f, g)
assert RootSum((x - 7)*f**3, g) == log(7*x) + 3*RootSum(f, g)
# issue 5571
assert hash(RootSum((x - 7)*f**3, g)) == hash(log(7*x) + 3*RootSum(f, g))
raises(MultivariatePolynomialError, lambda: RootSum(x**3 + x + y))
raises(ValueError, lambda: RootSum(x**2 + 3, lambda x: x))
assert RootSum(f, exp) == RootSum(f, Lambda(x, exp(x)))
assert RootSum(f, log) == RootSum(f, Lambda(x, log(x)))
assert isinstance(RootSum(f, auto=False), RootSum) is True
assert RootSum(f) == 0
assert RootSum(f, Lambda(x, x)) == 0
assert RootSum(f, Lambda(x, x**2)) == -2
assert RootSum(f, Lambda(x, 1)) == 3
assert RootSum(f, Lambda(x, 2)) == 6
assert RootSum(f, auto=False).is_commutative is True
assert RootSum(f, Lambda(x, 1/(x + x**2))) == Rational(11, 3)
assert RootSum(f, Lambda(x, y/(x + x**2))) == Rational(11, 3)*y
assert RootSum(x**2 - 1, Lambda(x, 3*x**2), x) == 6
assert RootSum(x**2 - y, Lambda(x, 3*x**2), x) == 6*y
assert RootSum(x**2 - 1, Lambda(x, z*x**2), x) == 2*z
assert RootSum(x**2 - y, Lambda(x, z*x**2), x) == 2*z*y
assert RootSum(
x**2 - 1, Lambda(x, exp(x)), quadratic=True) == exp(-1) + exp(1)
assert RootSum(x**3 + a*x + a**3, tan, x) == \
RootSum(x**3 + x + 1, Lambda(x, tan(a*x)))
assert RootSum(a**3*x**3 + a*x + 1, tan, x) == \
RootSum(x**3 + x + 1, Lambda(x, tan(x/a)))
def test_RootSum_free_symbols():
assert RootSum(x**3 + x + 3, Lambda(r, exp(r))).free_symbols == set()
assert RootSum(x**3 + x + 3, Lambda(r, exp(a*r))).free_symbols == {a}
assert RootSum(
x**3 + x + y, Lambda(r, exp(a*r)), x).free_symbols == {a, y}
def test_RootSum___eq__():
f = Lambda(x, exp(x))
assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 1, f)) is True
assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 1, f)) is True
assert (RootSum(x**3 + x + 1, f) == RootSum(x**3 + x + 2, f)) is False
assert (RootSum(x**3 + x + 1, f) == RootSum(y**3 + y + 2, f)) is False
def test_RootSum_doit():
rs = RootSum(x**2 + 1, exp)
assert isinstance(rs, RootSum) is True
assert rs.doit() == exp(-I) + exp(I)
rs = RootSum(x**2 + a, exp, x)
assert isinstance(rs, RootSum) is True
assert rs.doit() == exp(-sqrt(-a)) + exp(sqrt(-a))
def test_RootSum_evalf():
rs = RootSum(x**2 + 1, exp)
assert rs.evalf(n=20, chop=True).epsilon_eq(Float("1.0806046117362794348"))
assert rs.evalf(n=15, chop=True).epsilon_eq(Float("1.08060461173628"))
rs = RootSum(x**2 + a, exp, x)
assert rs.evalf() == rs
def test_RootSum_diff():
f = x**3 + x + 3
g = Lambda(r, exp(r*x))
h = Lambda(r, r*exp(r*x))
assert RootSum(f, g).diff(x) == RootSum(f, h)
def test_RootSum_subs():
f = x**3 + x + 3
g = Lambda(r, exp(r*x))
F = y**3 + y + 3
G = Lambda(r, exp(r*y))
assert RootSum(f, g).subs(y, 1) == RootSum(f, g)
assert RootSum(f, g).subs(x, y) == RootSum(F, G)
def test_RootSum_rational():
assert RootSum(
z**5 - z + 1, Lambda(z, z/(x - z))) == (4*x - 5)/(x**5 - x + 1)
f = 161*z**3 + 115*z**2 + 19*z + 1
g = Lambda(z, z*log(
-3381*z**4/4 - 3381*z**3/4 - 625*z**2/2 - z*Rational(125, 2) - 5 + exp(x)))
assert RootSum(f, g).diff(x) == -(
(5*exp(2*x) - 6*exp(x) + 4)*exp(x)/(exp(3*x) - exp(2*x) + 1))/7
def test_RootSum_independent():
f = (x**3 - a)**2*(x**4 - b)**3
g = Lambda(x, 5*tan(x) + 7)
h = Lambda(x, tan(x))
r0 = RootSum(x**3 - a, h, x)
r1 = RootSum(x**4 - b, h, x)
assert RootSum(f, g, x).as_ordered_terms() == [10*r0, 15*r1, 126]
def test_issue_7876():
l1 = Poly(x**6 - x + 1, x).all_roots()
l2 = [rootof(x**6 - x + 1, i) for i in range(6)]
assert frozenset(l1) == frozenset(l2)
def test_issue_8316():
f = Poly(7*x**8 - 9)
assert len(f.all_roots()) == 8
f = Poly(7*x**8 - 10)
assert len(f.all_roots()) == 8
def test__imag_count():
from sympy.polys.rootoftools import _imag_count_of_factor
def imag_count(p):
return sum([_imag_count_of_factor(f)*m for f, m in
p.factor_list()[1]])
assert imag_count(Poly(x**6 + 10*x**2 + 1)) == 2
assert imag_count(Poly(x**2)) == 0
assert imag_count(Poly([1]*3 + [-1], x)) == 0
assert imag_count(Poly(x**3 + 1)) == 0
assert imag_count(Poly(x**2 + 1)) == 2
assert imag_count(Poly(x**2 - 1)) == 0
assert imag_count(Poly(x**4 - 1)) == 2
assert imag_count(Poly(x**4 + 1)) == 0
assert imag_count(Poly([1, 2, 3], x)) == 0
assert imag_count(Poly(x**3 + x + 1)) == 0
assert imag_count(Poly(x**4 + x + 1)) == 0
def q(r1, r2, p):
return Poly(((x - r1)*(x - r2)).subs(x, x**p), x)
assert imag_count(q(-1, -2, 2)) == 4
assert imag_count(q(-1, 2, 2)) == 2
assert imag_count(q(1, 2, 2)) == 0
assert imag_count(q(1, 2, 4)) == 4
assert imag_count(q(-1, 2, 4)) == 2
assert imag_count(q(-1, -2, 4)) == 0
def test_RootOf_is_imaginary():
r = RootOf(x**4 + 4*x**2 + 1, 1)
i = r._get_interval()
assert r.is_imaginary and i.ax*i.bx <= 0
def test_is_disjoint():
eq = x**3 + 5*x + 1
ir = rootof(eq, 0)._get_interval()
ii = rootof(eq, 1)._get_interval()
assert ir.is_disjoint(ii)
assert ii.is_disjoint(ir)
def test_pure_key_dict():
p = D()
assert (x in p) is False
assert (1 in p) is False
p[x] = 1
assert x in p
assert y in p
assert p[y] == 1
raises(KeyError, lambda: p[1])
def dont(k):
p[k] = 2
raises(ValueError, lambda: dont(1))
@slow
def test_eval_approx_relative():
CRootOf.clear_cache()
t = [CRootOf(x**3 + 10*x + 1, i) for i in range(3)]
assert [i.eval_rational(1e-1) for i in t] == [
Rational(-21, 220), Rational(15, 256) - I*805/256,
Rational(15, 256) + I*805/256]
t[0]._reset()
assert [i.eval_rational(1e-1, 1e-4) for i in t] == [
Rational(-21, 220), Rational(3275, 65536) - I*414645/131072,
Rational(3275, 65536) + I*414645/131072]
assert S(t[0]._get_interval().dx) < 1e-1
assert S(t[1]._get_interval().dx) < 1e-1
assert S(t[1]._get_interval().dy) < 1e-4
assert S(t[2]._get_interval().dx) < 1e-1
assert S(t[2]._get_interval().dy) < 1e-4
t[0]._reset()
assert [i.eval_rational(1e-4, 1e-4) for i in t] == [
Rational(-2001, 20020), Rational(6545, 131072) - I*414645/131072,
Rational(6545, 131072) + I*414645/131072]
assert S(t[0]._get_interval().dx) < 1e-4
assert S(t[1]._get_interval().dx) < 1e-4
assert S(t[1]._get_interval().dy) < 1e-4
assert S(t[2]._get_interval().dx) < 1e-4
assert S(t[2]._get_interval().dy) < 1e-4
# in the following, the actual relative precision is
# less than tested, but it should never be greater
t[0]._reset()
assert [i.eval_rational(n=2) for i in t] == [
Rational(-202201, 2024022), Rational(104755, 2097152) - I*6634255/2097152,
Rational(104755, 2097152) + I*6634255/2097152]
assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-2
assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-2
assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-2
assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-2
assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-2
t[0]._reset()
assert [i.eval_rational(n=3) for i in t] == [
Rational(-202201, 2024022), Rational(1676045, 33554432) - I*106148135/33554432,
Rational(1676045, 33554432) + I*106148135/33554432]
assert abs(S(t[0]._get_interval().dx)/t[0]) < 1e-3
assert abs(S(t[1]._get_interval().dx)/t[1]).n() < 1e-3
assert abs(S(t[1]._get_interval().dy)/t[1]).n() < 1e-3
assert abs(S(t[2]._get_interval().dx)/t[2]).n() < 1e-3
assert abs(S(t[2]._get_interval().dy)/t[2]).n() < 1e-3
t[0]._reset()
a = [i.eval_approx(2) for i in t]
assert [str(i) for i in a] == [
'-0.10', '0.05 - 3.2*I', '0.05 + 3.2*I']
assert all(abs(((a[i] - t[i])/t[i]).n()) < 1e-2 for i in range(len(a)))
def test_issue_15920():
r = rootof(x**5 - x + 1, 0)
p = Integral(x, (x, 1, y))
assert unchanged(Eq, r, p)
def test_issue_19113():
eq = y**3 - y + 1
# generator is a canonical x in RootOf
assert str(Poly(eq).real_roots()) == '[CRootOf(x**3 - x + 1, 0)]'
assert str(Poly(eq.subs(y, tan(y))).real_roots()
) == '[CRootOf(x**3 - x + 1, 0)]'
assert str(Poly(eq.subs(y, tan(x))).real_roots()
) == '[CRootOf(x**3 - x + 1, 0)]'
|
90feb98039851229f4786e92bb82b6417573ba79c2fcaedda386585e8a3eab52 | """Tests for algorithms for computing symbolic roots of polynomials. """
from sympy.core.numbers import (I, Rational, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, Wild, symbols)
from sympy.functions.elementary.complexes import (conjugate, im, re)
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import (root, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (acos, cos, sin)
from sympy.polys.domains.integerring import ZZ
from sympy.sets.sets import Interval
from sympy.simplify.powsimp import powsimp
from sympy.polys import Poly, cyclotomic_poly, intervals, nroots, rootof
from sympy.polys.polyroots import (root_factors, roots_linear,
roots_quadratic, roots_cubic, roots_quartic, roots_cyclotomic,
roots_binomial, preprocess_roots, roots)
from sympy.polys.orthopolys import legendre_poly
from sympy.polys.polyerrors import PolynomialError, \
UnsolvableFactorError
from sympy.polys.polyutils import _nsort
from sympy.testing.pytest import raises, slow
from sympy.core.random import verify_numerically
import mpmath
from itertools import product
a, b, c, d, e, q, t, x, y, z = symbols('a,b,c,d,e,q,t,x,y,z')
def _check(roots):
# this is the desired invariant for roots returned
# by all_roots. It is trivially true for linear
# polynomials.
nreal = sum([1 if i.is_real else 0 for i in roots])
assert list(sorted(roots[:nreal])) == list(roots[:nreal])
for ix in range(nreal, len(roots), 2):
if not (
roots[ix + 1] == roots[ix] or
roots[ix + 1] == conjugate(roots[ix])):
return False
return True
def test_roots_linear():
assert roots_linear(Poly(2*x + 1, x)) == [Rational(-1, 2)]
def test_roots_quadratic():
assert roots_quadratic(Poly(2*x**2, x)) == [0, 0]
assert roots_quadratic(Poly(2*x**2 + 3*x, x)) == [Rational(-3, 2), 0]
assert roots_quadratic(Poly(2*x**2 + 3, x)) == [-I*sqrt(6)/2, I*sqrt(6)/2]
assert roots_quadratic(Poly(2*x**2 + 4*x + 3, x)) == [-1 - I*sqrt(2)/2, -1 + I*sqrt(2)/2]
_check(Poly(2*x**2 + 4*x + 3, x).all_roots())
f = x**2 + (2*a*e + 2*c*e)/(a - c)*x + (d - b + a*e**2 - c*e**2)/(a - c)
assert roots_quadratic(Poly(f, x)) == \
[-e*(a + c)/(a - c) - sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c),
-e*(a + c)/(a - c) + sqrt(a*b + c*d - a*d - b*c + 4*a*c*e**2)/(a - c)]
# check for simplification
f = Poly(y*x**2 - 2*x - 2*y, x)
assert roots_quadratic(f) == \
[-sqrt(2*y**2 + 1)/y + 1/y, sqrt(2*y**2 + 1)/y + 1/y]
f = Poly(x**2 + (-y**2 - 2)*x + y**2 + 1, x)
assert roots_quadratic(f) == \
[1,y**2 + 1]
f = Poly(sqrt(2)*x**2 - 1, x)
r = roots_quadratic(f)
assert r == _nsort(r)
# issue 8255
f = Poly(-24*x**2 - 180*x + 264)
assert [w.n(2) for w in f.all_roots(radicals=True)] == \
[w.n(2) for w in f.all_roots(radicals=False)]
for _a, _b, _c in product((-2, 2), (-2, 2), (0, -1)):
f = Poly(_a*x**2 + _b*x + _c)
roots = roots_quadratic(f)
assert roots == _nsort(roots)
def test_issue_7724():
eq = Poly(x**4*I + x**2 + I, x)
assert roots(eq) == {
sqrt(I/2 + sqrt(5)*I/2): 1,
sqrt(-sqrt(5)*I/2 + I/2): 1,
-sqrt(I/2 + sqrt(5)*I/2): 1,
-sqrt(-sqrt(5)*I/2 + I/2): 1}
def test_issue_8438():
p = Poly([1, y, -2, -3], x).as_expr()
roots = roots_cubic(Poly(p, x), x)
z = Rational(-3, 2) - I*7/2 # this will fail in code given in commit msg
post = [r.subs(y, z) for r in roots]
assert set(post) == \
set(roots_cubic(Poly(p.subs(y, z), x)))
# /!\ if p is not made an expression, this is *very* slow
assert all(p.subs({y: z, x: i}).n(2, chop=True) == 0 for i in post)
def test_issue_8285():
roots = (Poly(4*x**8 - 1, x)*Poly(x**2 + 1)).all_roots()
assert _check(roots)
f = Poly(x**4 + 5*x**2 + 6, x)
ro = [rootof(f, i) for i in range(4)]
roots = Poly(x**4 + 5*x**2 + 6, x).all_roots()
assert roots == ro
assert _check(roots)
# more than 2 complex roots from which to identify the
# imaginary ones
roots = Poly(2*x**8 - 1).all_roots()
assert _check(roots)
assert len(Poly(2*x**10 - 1).all_roots()) == 10 # doesn't fail
def test_issue_8289():
roots = (Poly(x**2 + 2)*Poly(x**4 + 2)).all_roots()
assert _check(roots)
roots = Poly(x**6 + 3*x**3 + 2, x).all_roots()
assert _check(roots)
roots = Poly(x**6 - x + 1).all_roots()
assert _check(roots)
# all imaginary roots with multiplicity of 2
roots = Poly(x**4 + 4*x**2 + 4, x).all_roots()
assert _check(roots)
def test_issue_14291():
assert Poly(((x - 1)**2 + 1)*((x - 1)**2 + 2)*(x - 1)
).all_roots() == [1, 1 - I, 1 + I, 1 - sqrt(2)*I, 1 + sqrt(2)*I]
p = x**4 + 10*x**2 + 1
ans = [rootof(p, i) for i in range(4)]
assert Poly(p).all_roots() == ans
_check(ans)
def test_issue_13340():
eq = Poly(y**3 + exp(x)*y + x, y, domain='EX')
roots_d = roots(eq)
assert len(roots_d) == 3
def test_issue_14522():
eq = Poly(x**4 + x**3*(16 + 32*I) + x**2*(-285 + 386*I) + x*(-2824 - 448*I) - 2058 - 6053*I, x)
roots_eq = roots(eq)
assert all(eq(r) == 0 for r in roots_eq)
def test_issue_15076():
sol = roots_quartic(Poly(t**4 - 6*t**2 + t/x - 3, t))
assert sol[0].has(x)
def test_issue_16589():
eq = Poly(x**4 - 8*sqrt(2)*x**3 + 4*x**3 - 64*sqrt(2)*x**2 + 1024*x, x)
roots_eq = roots(eq)
assert 0 in roots_eq
def test_roots_cubic():
assert roots_cubic(Poly(2*x**3, x)) == [0, 0, 0]
assert roots_cubic(Poly(x**3 - 3*x**2 + 3*x - 1, x)) == [1, 1, 1]
# valid for arbitrary y (issue 21263)
r = root(y, 3)
assert roots_cubic(Poly(x**3 - y, x)) == [r,
r*(-S.Half + sqrt(3)*I/2),
r*(-S.Half - sqrt(3)*I/2)]
# simpler form when y is negative
assert roots_cubic(Poly(x**3 - -1, x)) == \
[-1, S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cubic(Poly(2*x**3 - 3*x**2 - 3*x - 1, x))[0] == \
S.Half + 3**Rational(1, 3)/2 + 3**Rational(2, 3)/2
eq = -x**3 + 2*x**2 + 3*x - 2
assert roots(eq, trig=True, multiple=True) == \
roots_cubic(Poly(eq, x), trig=True) == [
Rational(2, 3) + 2*sqrt(13)*cos(acos(8*sqrt(13)/169)/3)/3,
-2*sqrt(13)*sin(-acos(8*sqrt(13)/169)/3 + pi/6)/3 + Rational(2, 3),
-2*sqrt(13)*cos(-acos(8*sqrt(13)/169)/3 + pi/3)/3 + Rational(2, 3),
]
def test_roots_quartic():
assert roots_quartic(Poly(x**4, x)) == [0, 0, 0, 0]
assert roots_quartic(Poly(x**4 + x**3, x)) in [
[-1, 0, 0, 0],
[0, -1, 0, 0],
[0, 0, -1, 0],
[0, 0, 0, -1]
]
assert roots_quartic(Poly(x**4 - x**3, x)) in [
[1, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1]
]
lhs = roots_quartic(Poly(x**4 + x, x))
rhs = [S.Half + I*sqrt(3)/2, S.Half - I*sqrt(3)/2, S.Zero, -S.One]
assert sorted(lhs, key=hash) == sorted(rhs, key=hash)
# test of all branches of roots quartic
for i, (a, b, c, d) in enumerate([(1, 2, 3, 0),
(3, -7, -9, 9),
(1, 2, 3, 4),
(1, 2, 3, 4),
(-7, -3, 3, -6),
(-3, 5, -6, -4),
(6, -5, -10, -3)]):
if i == 2:
c = -a*(a**2/S(8) - b/S(2))
elif i == 3:
d = a*(a*(a**2*Rational(3, 256) - b/S(16)) + c/S(4))
eq = x**4 + a*x**3 + b*x**2 + c*x + d
ans = roots_quartic(Poly(eq, x))
assert all(eq.subs(x, ai).n(chop=True) == 0 for ai in ans)
# not all symbolic quartics are unresolvable
eq = Poly(q*x + q/4 + x**4 + x**3 + 2*x**2 - Rational(1, 3), x)
sol = roots_quartic(eq)
assert all(verify_numerically(eq.subs(x, i), 0) for i in sol)
z = symbols('z', negative=True)
eq = x**4 + 2*x**3 + 3*x**2 + x*(z + 11) + 5
zans = roots_quartic(Poly(eq, x))
assert all([verify_numerically(eq.subs(((x, i), (z, -1))), 0) for i in zans])
# but some are (see also issue 4989)
# it's ok if the solution is not Piecewise, but the tests below should pass
eq = Poly(y*x**4 + x**3 - x + z, x)
ans = roots_quartic(eq)
assert all(type(i) == Piecewise for i in ans)
reps = (
dict(y=Rational(-1, 3), z=Rational(-1, 4)), # 4 real
dict(y=Rational(-1, 3), z=Rational(-1, 2)), # 2 real
dict(y=Rational(-1, 3), z=-2)) # 0 real
for rep in reps:
sol = roots_quartic(Poly(eq.subs(rep), x))
assert all([verify_numerically(w.subs(rep) - s, 0) for w, s in zip(ans, sol)])
def test_issue_21287():
assert not any(isinstance(i, Piecewise) for i in roots_quartic(
Poly(x**4 - x**2*(3 + 5*I) + 2*x*(-1 + I) - 1 + 3*I, x)))
def test_roots_cyclotomic():
assert roots_cyclotomic(cyclotomic_poly(1, x, polys=True)) == [1]
assert roots_cyclotomic(cyclotomic_poly(2, x, polys=True)) == [-1]
assert roots_cyclotomic(cyclotomic_poly(
3, x, polys=True)) == [Rational(-1, 2) - I*sqrt(3)/2, Rational(-1, 2) + I*sqrt(3)/2]
assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True)) == [-I, I]
assert roots_cyclotomic(cyclotomic_poly(
6, x, polys=True)) == [S.Half - I*sqrt(3)/2, S.Half + I*sqrt(3)/2]
assert roots_cyclotomic(cyclotomic_poly(7, x, polys=True)) == [
-cos(pi/7) - I*sin(pi/7),
-cos(pi/7) + I*sin(pi/7),
-cos(pi*Rational(3, 7)) - I*sin(pi*Rational(3, 7)),
-cos(pi*Rational(3, 7)) + I*sin(pi*Rational(3, 7)),
cos(pi*Rational(2, 7)) - I*sin(pi*Rational(2, 7)),
cos(pi*Rational(2, 7)) + I*sin(pi*Rational(2, 7)),
]
assert roots_cyclotomic(cyclotomic_poly(8, x, polys=True)) == [
-sqrt(2)/2 - I*sqrt(2)/2,
-sqrt(2)/2 + I*sqrt(2)/2,
sqrt(2)/2 - I*sqrt(2)/2,
sqrt(2)/2 + I*sqrt(2)/2,
]
assert roots_cyclotomic(cyclotomic_poly(12, x, polys=True)) == [
-sqrt(3)/2 - I/2,
-sqrt(3)/2 + I/2,
sqrt(3)/2 - I/2,
sqrt(3)/2 + I/2,
]
assert roots_cyclotomic(
cyclotomic_poly(1, x, polys=True), factor=True) == [1]
assert roots_cyclotomic(
cyclotomic_poly(2, x, polys=True), factor=True) == [-1]
assert roots_cyclotomic(cyclotomic_poly(3, x, polys=True), factor=True) == \
[-root(-1, 3), -1 + root(-1, 3)]
assert roots_cyclotomic(cyclotomic_poly(4, x, polys=True), factor=True) == \
[-I, I]
assert roots_cyclotomic(cyclotomic_poly(5, x, polys=True), factor=True) == \
[-root(-1, 5), -root(-1, 5)**3, root(-1, 5)**2, -1 - root(-1, 5)**2 + root(-1, 5) + root(-1, 5)**3]
assert roots_cyclotomic(cyclotomic_poly(6, x, polys=True), factor=True) == \
[1 - root(-1, 3), root(-1, 3)]
def test_roots_binomial():
assert roots_binomial(Poly(5*x, x)) == [0]
assert roots_binomial(Poly(5*x**4, x)) == [0, 0, 0, 0]
assert roots_binomial(Poly(5*x + 2, x)) == [Rational(-2, 5)]
A = 10**Rational(3, 4)/10
assert roots_binomial(Poly(5*x**4 + 2, x)) == \
[-A - A*I, -A + A*I, A - A*I, A + A*I]
_check(roots_binomial(Poly(x**8 - 2)))
a1 = Symbol('a1', nonnegative=True)
b1 = Symbol('b1', nonnegative=True)
r0 = roots_quadratic(Poly(a1*x**2 + b1, x))
r1 = roots_binomial(Poly(a1*x**2 + b1, x))
assert powsimp(r0[0]) == powsimp(r1[0])
assert powsimp(r0[1]) == powsimp(r1[1])
for a, b, s, n in product((1, 2), (1, 2), (-1, 1), (2, 3, 4, 5)):
if a == b and a != 1: # a == b == 1 is sufficient
continue
p = Poly(a*x**n + s*b)
ans = roots_binomial(p)
assert ans == _nsort(ans)
# issue 8813
assert roots(Poly(2*x**3 - 16*y**3, x)) == {
2*y*(Rational(-1, 2) - sqrt(3)*I/2): 1,
2*y: 1,
2*y*(Rational(-1, 2) + sqrt(3)*I/2): 1}
def test_roots_preprocessing():
f = a*y*x**2 + y - b
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1
assert poly == Poly(a*y*x**2 + y - b, x)
f = c**3*x**3 + c**2*x**2 + c*x + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + x**2 + x + a, x)
f = c**3*x**3 + c**2*x**2 + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + x**2 + a, x)
f = c**3*x**3 + c*x + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + x + a, x)
f = c**3*x**3 + a
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 1/c
assert poly == Poly(x**3 + a, x)
E, F, J, L = symbols("E,F,J,L")
f = -21601054687500000000*E**8*J**8/L**16 + \
508232812500000000*F*x*E**7*J**7/L**14 - \
4269543750000000*E**6*F**2*J**6*x**2/L**12 + \
16194716250000*E**5*F**3*J**5*x**3/L**10 - \
27633173750*E**4*F**4*J**4*x**4/L**8 + \
14840215*E**3*F**5*J**3*x**5/L**6 + \
54794*E**2*F**6*J**2*x**6/(5*L**4) - \
1153*E*J*F**7*x**7/(80*L**2) + \
633*F**8*x**8/160000
coeff, poly = preprocess_roots(Poly(f, x))
assert coeff == 20*E*J/(F*L**2)
assert poly == 633*x**8 - 115300*x**7 + 4383520*x**6 + 296804300*x**5 - 27633173750*x**4 + \
809735812500*x**3 - 10673859375000*x**2 + 63529101562500*x - 135006591796875
f = Poly(-y**2 + x**2*exp(x), y, domain=ZZ[x, exp(x)])
g = Poly(-y**2 + exp(x), y, domain=ZZ[exp(x)])
assert preprocess_roots(f) == (x, g)
def test_roots0():
assert roots(1, x) == {}
assert roots(x, x) == {S.Zero: 1}
assert roots(x**9, x) == {S.Zero: 9}
assert roots(((x - 2)*(x + 3)*(x - 4)).expand(), x) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(2*x + 1, x) == {Rational(-1, 2): 1}
assert roots((2*x + 1)**2, x) == {Rational(-1, 2): 2}
assert roots((2*x + 1)**5, x) == {Rational(-1, 2): 5}
assert roots((2*x + 1)**10, x) == {Rational(-1, 2): 10}
assert roots(x**4 - 1, x) == {I: 1, S.One: 1, -S.One: 1, -I: 1}
assert roots((x**4 - 1)**2, x) == {I: 2, S.One: 2, -S.One: 2, -I: 2}
assert roots(((2*x - 3)**2).expand(), x) == {Rational( 3, 2): 2}
assert roots(((2*x + 3)**2).expand(), x) == {Rational(-3, 2): 2}
assert roots(((2*x - 3)**3).expand(), x) == {Rational( 3, 2): 3}
assert roots(((2*x + 3)**3).expand(), x) == {Rational(-3, 2): 3}
assert roots(((2*x - 3)**5).expand(), x) == {Rational( 3, 2): 5}
assert roots(((2*x + 3)**5).expand(), x) == {Rational(-3, 2): 5}
assert roots(((a*x - b)**5).expand(), x) == { b/a: 5}
assert roots(((a*x + b)**5).expand(), x) == {-b/a: 5}
assert roots(x**2 + (-a - 1)*x + a, x) == {a: 1, S.One: 1}
assert roots(x**4 - 2*x**2 + 1, x) == {S.One: 2, S.NegativeOne: 2}
assert roots(x**6 - 4*x**4 + 4*x**3 - x**2, x) == \
{S.One: 2, -1 - sqrt(2): 1, S.Zero: 2, -1 + sqrt(2): 1}
assert roots(x**8 - 1, x) == {
sqrt(2)/2 + I*sqrt(2)/2: 1,
sqrt(2)/2 - I*sqrt(2)/2: 1,
-sqrt(2)/2 + I*sqrt(2)/2: 1,
-sqrt(2)/2 - I*sqrt(2)/2: 1,
S.One: 1, -S.One: 1, I: 1, -I: 1
}
f = -2016*x**2 - 5616*x**3 - 2056*x**4 + 3324*x**5 + 2176*x**6 - \
224*x**7 - 384*x**8 - 64*x**9
assert roots(f) == {S.Zero: 2, -S(2): 2, S(2): 1, Rational(-7, 2): 1,
Rational(-3, 2): 1, Rational(-1, 2): 1, Rational(3, 2): 1}
assert roots((a + b + c)*x - (a + b + c + d), x) == {(a + b + c + d)/(a + b + c): 1}
assert roots(x**3 + x**2 - x + 1, x, cubics=False) == {}
assert roots(((x - 2)*(
x + 3)*(x - 4)).expand(), x, cubics=False) == {-S(3): 1, S(2): 1, S(4): 1}
assert roots(((x - 2)*(x + 3)*(x - 4)*(x - 5)).expand(), x, cubics=False) == \
{-S(3): 1, S(2): 1, S(4): 1, S(5): 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x) == {-S(2): 1, -2*I: 1, 2*I: 1}
assert roots(x**3 + 2*x**2 + 4*x + 8, x, cubics=True) == \
{-2*I: 1, 2*I: 1, -S(2): 1}
assert roots((x**2 - x)*(x**3 + 2*x**2 + 4*x + 8), x ) == \
{S.One: 1, S.Zero: 1, -S(2): 1, -2*I: 1, 2*I: 1}
r1_2, r1_3 = S.Half, Rational(1, 3)
x0 = (3*sqrt(33) + 19)**r1_3
x1 = 4/x0/3
x2 = x0/3
x3 = sqrt(3)*I/2
x4 = x3 - r1_2
x5 = -x3 - r1_2
assert roots(x**3 + x**2 - x + 1, x, cubics=True) == {
-x1 - x2 - r1_3: 1,
-x1/x4 - x2*x4 - r1_3: 1,
-x1/x5 - x2*x5 - r1_3: 1,
}
f = (x**2 + 2*x + 3).subs(x, 2*x**2 + 3*x).subs(x, 5*x - 4)
r13_20, r1_20 = [ Rational(*r)
for r in ((13, 20), (1, 20)) ]
s2 = sqrt(2)
assert roots(f, x) == {
r13_20 + r1_20*sqrt(1 - 8*I*s2): 1,
r13_20 - r1_20*sqrt(1 - 8*I*s2): 1,
r13_20 + r1_20*sqrt(1 + 8*I*s2): 1,
r13_20 - r1_20*sqrt(1 + 8*I*s2): 1,
}
f = x**4 + x**3 + x**2 + x + 1
r1_4, r1_8, r5_8 = [ Rational(*r) for r in ((1, 4), (1, 8), (5, 8)) ]
assert roots(f, x) == {
-r1_4 + r1_4*5**r1_2 + I*(r5_8 + r1_8*5**r1_2)**r1_2: 1,
-r1_4 + r1_4*5**r1_2 - I*(r5_8 + r1_8*5**r1_2)**r1_2: 1,
-r1_4 - r1_4*5**r1_2 + I*(r5_8 - r1_8*5**r1_2)**r1_2: 1,
-r1_4 - r1_4*5**r1_2 - I*(r5_8 - r1_8*5**r1_2)**r1_2: 1,
}
f = z**3 + (-2 - y)*z**2 + (1 + 2*y - 2*x**2)*z - y + 2*x**2
assert roots(f, z) == {
S.One: 1,
S.Half + S.Half*y + S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1,
S.Half + S.Half*y - S.Half*sqrt(1 - 2*y + y**2 + 8*x**2): 1,
}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=False) == {}
assert roots(a*b*c*x**3 + 2*x**2 + 4*x + 8, x, cubics=True) != {}
assert roots(x**4 - 1, x, filter='Z') == {S.One: 1, -S.One: 1}
assert roots(x**4 - 1, x, filter='I') == {I: 1, -I: 1}
assert roots((x - 1)*(x + 1), x) == {S.One: 1, -S.One: 1}
assert roots(
(x - 1)*(x + 1), x, predicate=lambda r: r.is_positive) == {S.One: 1}
assert roots(x**4 - 1, x, filter='Z', multiple=True) == [-S.One, S.One]
assert roots(x**4 - 1, x, filter='I', multiple=True) == [I, -I]
ar, br = symbols('a, b', real=True)
p = x**2*(ar-br)**2 + 2*x*(br-ar) + 1
assert roots(p, x, filter='R') == {1/(ar - br): 2}
assert roots(x**3, x, multiple=True) == [S.Zero, S.Zero, S.Zero]
assert roots(1234, x, multiple=True) == []
f = x**6 - x**5 + x**4 - x**3 + x**2 - x + 1
assert roots(f) == {
-I*sin(pi/7) + cos(pi/7): 1,
-I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1,
-I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1,
I*sin(pi/7) + cos(pi/7): 1,
I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 1,
I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 1,
}
g = ((x**2 + 1)*f**2).expand()
assert roots(g) == {
-I*sin(pi/7) + cos(pi/7): 2,
-I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2,
-I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2,
I*sin(pi/7) + cos(pi/7): 2,
I*sin(pi*Rational(2, 7)) - cos(pi*Rational(2, 7)): 2,
I*sin(pi*Rational(3, 7)) + cos(pi*Rational(3, 7)): 2,
-I: 1, I: 1,
}
r = roots(x**3 + 40*x + 64)
real_root = [rx for rx in r if rx.is_real][0]
cr = 108 + 6*sqrt(1074)
assert real_root == -2*root(cr, 3)/3 + 20/root(cr, 3)
eq = Poly((7 + 5*sqrt(2))*x**3 + (-6 - 4*sqrt(2))*x**2 + (-sqrt(2) - 1)*x + 2, x, domain='EX')
assert roots(eq) == {-1 + sqrt(2): 1, -2 + 2*sqrt(2): 1, -sqrt(2) + 1: 1}
eq = Poly(41*x**5 + 29*sqrt(2)*x**5 - 153*x**4 - 108*sqrt(2)*x**4 +
175*x**3 + 125*sqrt(2)*x**3 - 45*x**2 - 30*sqrt(2)*x**2 - 26*sqrt(2)*x -
26*x + 24, x, domain='EX')
assert roots(eq) == {-sqrt(2) + 1: 1, -2 + 2*sqrt(2): 1, -1 + sqrt(2): 1,
-4 + 4*sqrt(2): 1, -3 + 3*sqrt(2): 1}
eq = Poly(x**3 - 2*x**2 + 6*sqrt(2)*x**2 - 8*sqrt(2)*x + 23*x - 14 +
14*sqrt(2), x, domain='EX')
assert roots(eq) == {-2*sqrt(2) + 2: 1, -2*sqrt(2) + 1: 1, -2*sqrt(2) - 1: 1}
assert roots(Poly((x + sqrt(2))**3 - 7, x, domain='EX')) == \
{-sqrt(2) + root(7, 3)*(-S.Half - sqrt(3)*I/2): 1,
-sqrt(2) + root(7, 3)*(-S.Half + sqrt(3)*I/2): 1,
-sqrt(2) + root(7, 3): 1}
def test_roots_slow():
"""Just test that calculating these roots does not hang. """
a, b, c, d, x = symbols("a,b,c,d,x")
f1 = x**2*c + (a/b) + x*c*d - a
f2 = x**2*(a + b*(c - d)*a) + x*a*b*c/(b*d - d) + (a*d - c/d)
assert list(roots(f1, x).values()) == [1, 1]
assert list(roots(f2, x).values()) == [1, 1]
(zz, yy, xx, zy, zx, yx, k) = symbols("zz,yy,xx,zy,zx,yx,k")
e1 = (zz - k)*(yy - k)*(xx - k) + zy*yx*zx + zx - zy - yx
e2 = (zz - k)*yx*yx + zx*(yy - k)*zx + zy*zy*(xx - k)
assert list(roots(e1 - e2, k).values()) == [1, 1, 1]
f = x**3 + 2*x**2 + 8
R = list(roots(f).keys())
assert not any(i for i in [f.subs(x, ri).n(chop=True) for ri in R])
def test_roots_inexact():
R1 = roots(x**2 + x + 1, x, multiple=True)
R2 = roots(x**2 + x + 1.0, x, multiple=True)
for r1, r2 in zip(R1, R2):
assert abs(r1 - r2) < 1e-12
f = x**4 + 3.0*sqrt(2.0)*x**3 - (78.0 + 24.0*sqrt(3.0))*x**2 \
+ 144.0*(2*sqrt(3.0) + 9.0)
R1 = roots(f, multiple=True)
R2 = (-12.7530479110482, -3.85012393732929,
4.89897948556636, 7.46155167569183)
for r1, r2 in zip(R1, R2):
assert abs(r1 - r2) < 1e-10
def test_roots_preprocessed():
E, F, J, L = symbols("E,F,J,L")
f = -21601054687500000000*E**8*J**8/L**16 + \
508232812500000000*F*x*E**7*J**7/L**14 - \
4269543750000000*E**6*F**2*J**6*x**2/L**12 + \
16194716250000*E**5*F**3*J**5*x**3/L**10 - \
27633173750*E**4*F**4*J**4*x**4/L**8 + \
14840215*E**3*F**5*J**3*x**5/L**6 + \
54794*E**2*F**6*J**2*x**6/(5*L**4) - \
1153*E*J*F**7*x**7/(80*L**2) + \
633*F**8*x**8/160000
assert roots(f, x) == {}
R1 = roots(f.evalf(), x, multiple=True)
R2 = [-1304.88375606366, 97.1168816800648, 186.946430171876, 245.526792947065,
503.441004174773, 791.549343830097, 1273.16678129348, 1850.10650616851]
w = Wild('w')
p = w*E*J/(F*L**2)
assert len(R1) == len(R2)
for r1, r2 in zip(R1, R2):
match = r1.match(p)
assert match is not None and abs(match[w] - r2) < 1e-10
def test_roots_strict():
assert roots(x**2 - 2*x + 1, strict=False) == {1: 2}
assert roots(x**2 - 2*x + 1, strict=True) == {1: 2}
assert roots(x**6 - 2*x**5 - x**2 + 3*x - 2, strict=False) == {2: 1}
raises(UnsolvableFactorError, lambda: roots(x**6 - 2*x**5 - x**2 + 3*x - 2, strict=True))
def test_roots_mixed():
f = -1936 - 5056*x - 7592*x**2 + 2704*x**3 - 49*x**4
_re, _im = intervals(f, all=True)
_nroots = nroots(f)
_sroots = roots(f, multiple=True)
_re = [ Interval(a, b) for (a, b), _ in _re ]
_im = [ Interval(re(a), re(b))*Interval(im(a), im(b)) for (a, b),
_ in _im ]
_intervals = _re + _im
_sroots = [ r.evalf() for r in _sroots ]
_nroots = sorted(_nroots, key=lambda x: x.sort_key())
_sroots = sorted(_sroots, key=lambda x: x.sort_key())
for _roots in (_nroots, _sroots):
for i, r in zip(_intervals, _roots):
if r.is_real:
assert r in i
else:
assert (re(r), im(r)) in i
def test_root_factors():
assert root_factors(Poly(1, x)) == [Poly(1, x)]
assert root_factors(Poly(x, x)) == [Poly(x, x)]
assert root_factors(x**2 - 1, x) == [x + 1, x - 1]
assert root_factors(x**2 - y, x) == [x - sqrt(y), x + sqrt(y)]
assert root_factors((x**4 - 1)**2) == \
[x + 1, x + 1, x - 1, x - 1, x - I, x - I, x + I, x + I]
assert root_factors(Poly(x**4 - 1, x), filter='Z') == \
[Poly(x + 1, x), Poly(x - 1, x), Poly(x**2 + 1, x)]
assert root_factors(8*x**2 + 12*x**4 + 6*x**6 + x**8, x, filter='Q') == \
[x, x, x**6 + 6*x**4 + 12*x**2 + 8]
@slow
def test_nroots1():
n = 64
p = legendre_poly(n, x, polys=True)
raises(mpmath.mp.NoConvergence, lambda: p.nroots(n=3, maxsteps=5))
roots = p.nroots(n=3)
# The order of roots matters. They are ordered from smallest to the
# largest.
assert [str(r) for r in roots] == \
['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961',
'-0.946', '-0.930', '-0.911', '-0.889', '-0.866', '-0.841',
'-0.813', '-0.784', '-0.753', '-0.720', '-0.685', '-0.649',
'-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402',
'-0.357', '-0.311', '-0.265', '-0.217', '-0.170', '-0.121',
'-0.0730', '-0.0243', '0.0243', '0.0730', '0.121', '0.170',
'0.217', '0.265', '0.311', '0.357', '0.402', '0.446', '0.489',
'0.531', '0.572', '0.611', '0.649', '0.685', '0.720', '0.753',
'0.784', '0.813', '0.841', '0.866', '0.889', '0.911', '0.930',
'0.946', '0.961', '0.973', '0.983', '0.991', '0.996', '0.999']
def test_nroots2():
p = Poly(x**5 + 3*x + 1, x)
roots = p.nroots(n=3)
# The order of roots matters. The roots are ordered by their real
# components (if they agree, then by their imaginary components),
# with real roots appearing first.
assert [str(r) for r in roots] == \
['-0.332', '-0.839 - 0.944*I', '-0.839 + 0.944*I',
'1.01 - 0.937*I', '1.01 + 0.937*I']
roots = p.nroots(n=5)
assert [str(r) for r in roots] == \
['-0.33199', '-0.83907 - 0.94385*I', '-0.83907 + 0.94385*I',
'1.0051 - 0.93726*I', '1.0051 + 0.93726*I']
def test_roots_composite():
assert len(roots(Poly(y**3 + y**2*sqrt(x) + y + x, y, composite=True))) == 3
def test_issue_19113():
eq = cos(x)**3 - cos(x) + 1
raises(PolynomialError, lambda: roots(eq))
def test_issue_17454():
assert roots([1, -3*(-4 - 4*I)**2/8 + 12*I, 0], multiple=True) == [0, 0]
def test_issue_20913():
assert Poly(x + 9671406556917067856609794, x).real_roots() == [-9671406556917067856609794]
assert Poly(x**3 + 4, x).real_roots() == [-2**(S(2)/3)]
def test_issue_22768():
e = Rational(1, 3)
r = (-1/a)**e*(a + 1)**(5*e)
assert roots(Poly(a*x**3 + (a + 1)**5, x)) == {
r: 1,
-r*(1 + sqrt(3)*I)/2: 1,
r*(-1 + sqrt(3)*I)/2: 1}
|
7eeb92fcdbc5170eef3fb284b647eeaec3ad8099a72164aff4f119bf9ea8bce1 | #
# sympy.polys.matrices.linsolve module
#
# This module defines the _linsolve function which is the internal workhorse
# used by linsolve. This computes the solution of a system of linear equations
# using the SDM sparse matrix implementation in sympy.polys.matrices.sdm. This
# is a replacement for solve_lin_sys in sympy.polys.solvers which is
# inefficient for large sparse systems due to the use of a PolyRing with many
# generators:
#
# https://github.com/sympy/sympy/issues/20857
#
# The implementation of _linsolve here handles:
#
# - Extracting the coefficients from the Expr/Eq input equations.
# - Constructing a domain and converting the coefficients to
# that domain.
# - Using the SDM.rref, SDM.nullspace etc methods to generate the full
# solution working with arithmetic only in the domain of the coefficients.
#
# The routines here are particularly designed to be efficient for large sparse
# systems of linear equations although as well as dense systems. It is
# possible that for some small dense systems solve_lin_sys which uses the
# dense matrix implementation DDM will be more efficient. With smaller systems
# though the bulk of the time is spent just preprocessing the inputs and the
# relative time spent in rref is too small to be noticeable.
#
from collections import defaultdict
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.polys.constructor import construct_domain
from sympy.polys.solvers import PolyNonlinearError
from .sdm import (
SDM,
sdm_irref,
sdm_particular_from_rref,
sdm_nullspace_from_rref
)
from sympy.utilities.misc import filldedent
def _linsolve(eqs, syms):
"""Solve a linear system of equations.
Examples
========
Solve a linear system with a unique solution:
>>> from sympy import symbols, Eq
>>> from sympy.polys.matrices.linsolve import _linsolve
>>> x, y = symbols('x, y')
>>> eqs = [Eq(x + y, 1), Eq(x - y, 2)]
>>> _linsolve(eqs, [x, y])
{x: 3/2, y: -1/2}
In the case of underdetermined systems the solution will be expressed in
terms of the unknown symbols that are unconstrained:
>>> _linsolve([Eq(x + y, 0)], [x, y])
{x: -y, y: y}
"""
# Number of unknowns (columns in the non-augmented matrix)
nsyms = len(syms)
# Convert to sparse augmented matrix (len(eqs) x (nsyms+1))
eqsdict, const = _linear_eq_to_dict(eqs, syms)
Aaug = sympy_dict_to_dm(eqsdict, const, syms)
K = Aaug.domain
# sdm_irref has issues with float matrices. This uses the ddm_rref()
# function. When sdm_rref() can handle float matrices reasonably this
# should be removed...
if K.is_RealField or K.is_ComplexField:
Aaug = Aaug.to_ddm().rref()[0].to_sdm()
# Compute reduced-row echelon form (RREF)
Arref, pivots, nzcols = sdm_irref(Aaug)
# No solution:
if pivots and pivots[-1] == nsyms:
return None
# Particular solution for non-homogeneous system:
P = sdm_particular_from_rref(Arref, nsyms+1, pivots)
# Nullspace - general solution to homogeneous system
# Note: using nsyms not nsyms+1 to ignore last column
V, nonpivots = sdm_nullspace_from_rref(Arref, K.one, nsyms, pivots, nzcols)
# Collect together terms from particular and nullspace:
sol = defaultdict(list)
for i, v in P.items():
sol[syms[i]].append(K.to_sympy(v))
for npi, Vi in zip(nonpivots, V):
sym = syms[npi]
for i, v in Vi.items():
sol[syms[i]].append(sym * K.to_sympy(v))
# Use a single call to Add for each term:
sol = {s: Add(*terms) for s, terms in sol.items()}
# Fill in the zeros:
zero = S.Zero
for s in set(syms) - set(sol):
sol[s] = zero
# All done!
return sol
def sympy_dict_to_dm(eqs_coeffs, eqs_rhs, syms):
"""Convert a system of dict equations to a sparse augmented matrix"""
elems = set(eqs_rhs).union(*(e.values() for e in eqs_coeffs))
K, elems_K = construct_domain(elems, field=True, extension=True)
elem_map = dict(zip(elems, elems_K))
neqs = len(eqs_coeffs)
nsyms = len(syms)
sym2index = dict(zip(syms, range(nsyms)))
eqsdict = []
for eq, rhs in zip(eqs_coeffs, eqs_rhs):
eqdict = {sym2index[s]: elem_map[c] for s, c in eq.items()}
if rhs:
eqdict[nsyms] = -elem_map[rhs]
if eqdict:
eqsdict.append(eqdict)
sdm_aug = SDM(enumerate(eqsdict), (neqs, nsyms + 1), K)
return sdm_aug
def _linear_eq_to_dict(eqs, syms):
"""Convert a system Expr/Eq equations into dict form, returning
the coefficient dictionaries and a list of syms-independent terms
from each expression in ``eqs```.
Examples
========
>>> from sympy.polys.matrices.linsolve import _linear_eq_to_dict
>>> from sympy.abc import x
>>> _linear_eq_to_dict([2*x + 3], {x})
([{x: 2}], [3])
"""
coeffs = []
ind = []
symset = set(syms)
for i, e in enumerate(eqs):
if e.is_Equality:
coeff, terms = _lin_eq2dict(e.lhs, symset)
cR, tR = _lin_eq2dict(e.rhs, symset)
# there were no nonlinear errors so now
# cancellation is allowed
coeff -= cR
for k, v in tR.items():
if k in terms:
terms[k] -= v
else:
terms[k] = -v
# don't store coefficients of 0, however
terms = {k: v for k, v in terms.items() if v}
c, d = coeff, terms
else:
c, d = _lin_eq2dict(e, symset)
coeffs.append(d)
ind.append(c)
return coeffs, ind
def _lin_eq2dict(a, symset):
"""return (c, d) where c is the sym-independent part of ``a`` and
``d`` is an efficiently calculated dictionary mapping symbols to
their coefficients. A PolyNonlinearError is raised if non-linearity
is detected.
The values in the dictionary will be non-zero.
Examples
========
>>> from sympy.polys.matrices.linsolve import _lin_eq2dict
>>> from sympy.abc import x, y
>>> _lin_eq2dict(x + 2*y + 3, {x, y})
(3, {x: 1, y: 2})
"""
if a in symset:
return S.Zero, {a: S.One}
elif a.is_Add:
terms_list = defaultdict(list)
coeff_list = []
for ai in a.args:
ci, ti = _lin_eq2dict(ai, symset)
coeff_list.append(ci)
for mij, cij in ti.items():
terms_list[mij].append(cij)
coeff = Add(*coeff_list)
terms = {sym: Add(*coeffs) for sym, coeffs in terms_list.items()}
return coeff, terms
elif a.is_Mul:
terms = terms_coeff = None
coeff_list = []
for ai in a.args:
ci, ti = _lin_eq2dict(ai, symset)
if not ti:
coeff_list.append(ci)
elif terms is None:
terms = ti
terms_coeff = ci
else:
# since ti is not null and we already have
# a term, this is a cross term
raise PolyNonlinearError(filldedent('''
nonlinear cross-term: %s''' % a))
coeff = Mul._from_args(coeff_list)
if terms is None:
return coeff, {}
else:
terms = {sym: coeff * c for sym, c in terms.items()}
return coeff * terms_coeff, terms
elif not a.has_xfree(symset):
return a, {}
else:
raise PolyNonlinearError('nonlinear term: %s' % a)
|
1b52c20e407704352e0f342e5319f88c28ea5126c2c0c5ff95f9155c32defa4d | #
# test_linsolve.py
#
# Test the internal implementation of linsolve.
#
from sympy.testing.pytest import raises
from sympy.core.numbers import I
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.abc import x, y, z
from sympy.polys.matrices.linsolve import _linsolve
from sympy.polys.solvers import PolyNonlinearError
def test__linsolve():
assert _linsolve([], [x]) == {x:x}
assert _linsolve([S.Zero], [x]) == {x:x}
assert _linsolve([x-1,x-2], [x]) is None
assert _linsolve([x-1], [x]) == {x:1}
assert _linsolve([x-1, y], [x, y]) == {x:1, y:S.Zero}
assert _linsolve([2*I], [x]) is None
raises(PolyNonlinearError, lambda: _linsolve([x*(1 + x)], [x]))
def test__linsolve_float():
# This should give the exact answer:
eqs = [
y - x,
y - 0.0216 * x
]
sol = {x:0.0, y:0.0}
assert _linsolve(eqs, (x, y)) == sol
# Other cases should be close to eps
def all_close(sol1, sol2, eps=1e-15):
close = lambda a, b: abs(a - b) < eps
assert sol1.keys() == sol2.keys()
return all(close(sol1[s], sol2[s]) for s in sol1)
eqs = [
0.8*x + 0.8*z + 0.2,
0.9*x + 0.7*y + 0.2*z + 0.9,
0.7*x + 0.2*y + 0.2*z + 0.5
]
sol_exact = {x:-29/42, y:-11/21, z:37/84}
sol_linsolve = _linsolve(eqs, [x,y,z])
assert all_close(sol_exact, sol_linsolve)
eqs = [
0.9*x + 0.3*y + 0.4*z + 0.6,
0.6*x + 0.9*y + 0.1*z + 0.7,
0.4*x + 0.6*y + 0.9*z + 0.5
]
sol_exact = {x:-88/175, y:-46/105, z:-1/25}
sol_linsolve = _linsolve(eqs, [x,y,z])
assert all_close(sol_exact, sol_linsolve)
eqs = [
0.4*x + 0.3*y + 0.6*z + 0.7,
0.4*x + 0.3*y + 0.9*z + 0.9,
0.7*x + 0.9*y,
]
sol_exact = {x:-9/5, y:7/5, z:-2/3}
sol_linsolve = _linsolve(eqs, [x,y,z])
assert all_close(sol_exact, sol_linsolve)
eqs = [
x*(0.7 + 0.6*I) + y*(0.4 + 0.7*I) + z*(0.9 + 0.1*I) + 0.5,
0.2*I*x + 0.2*I*y + z*(0.9 + 0.2*I) + 0.1,
x*(0.9 + 0.7*I) + y*(0.9 + 0.7*I) + z*(0.9 + 0.4*I) + 0.4,
]
sol_exact = {
x:-6157/7995 - 411/5330*I,
y:8519/15990 + 1784/7995*I,
z:-34/533 + 107/1599*I,
}
sol_linsolve = _linsolve(eqs, [x,y,z])
assert all_close(sol_exact, sol_linsolve)
# XXX: This system for x and y over RR(z) is problematic.
#
# eqs = [
# x*(0.2*z + 0.9) + y*(0.5*z + 0.8) + 0.6,
# 0.1*x*z + y*(0.1*z + 0.6) + 0.9,
# ]
#
# linsolve(eqs, [x, y])
# The solution for x comes out as
#
# -3.9e-5*z**2 - 3.6e-5*z - 8.67361737988404e-20
# x = ----------------------------------------------
# 3.0e-6*z**3 - 1.3e-5*z**2 - 5.4e-5*z
#
# The 8e-20 in the numerator should be zero which would allow z to cancel
# from top and bottom. It should be possible to avoid this somehow because
# the inverse of the matrix only has a quadratic factor (the determinant)
# in the denominator.
def test__linsolve_deprecated():
raises(PolyNonlinearError, lambda:
_linsolve([Eq(x**2, x**2 + y)], [x, y]))
raises(PolyNonlinearError, lambda:
_linsolve([(x + y)**2 - x**2], [x]))
raises(PolyNonlinearError, lambda:
_linsolve([Eq((x + y)**2, x**2)], [x]))
|
bfd97c131f0ad8cb0f1b1a803e1a4482ee96028eb04baaaf7562ca40cc601380 | from sympy.core.function import Derivative
from sympy.vector.vector import Vector
from sympy.vector.coordsysrect import CoordSys3D
from sympy.simplify import simplify
from sympy.core.symbol import symbols
from sympy.core import S
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.vector.vector import Dot
from sympy.vector.operators import curl, divergence, gradient, Gradient, Divergence, Cross
from sympy.vector.deloperator import Del
from sympy.vector.functions import (is_conservative, is_solenoidal,
scalar_potential, directional_derivative,
laplacian, scalar_potential_difference)
from sympy.testing.pytest import raises
C = CoordSys3D('C')
i, j, k = C.base_vectors()
x, y, z = C.base_scalars()
delop = Del()
a, b, c, q = symbols('a b c q')
def test_del_operator():
# Tests for curl
assert delop ^ Vector.zero == Vector.zero
assert ((delop ^ Vector.zero).doit() == Vector.zero ==
curl(Vector.zero))
assert delop.cross(Vector.zero) == delop ^ Vector.zero
assert (delop ^ i).doit() == Vector.zero
assert delop.cross(2*y**2*j, doit=True) == Vector.zero
assert delop.cross(2*y**2*j) == delop ^ 2*y**2*j
v = x*y*z * (i + j + k)
assert ((delop ^ v).doit() ==
(-x*y + x*z)*i + (x*y - y*z)*j + (-x*z + y*z)*k ==
curl(v))
assert delop ^ v == delop.cross(v)
assert (delop.cross(2*x**2*j) ==
(Derivative(0, C.y) - Derivative(2*C.x**2, C.z))*C.i +
(-Derivative(0, C.x) + Derivative(0, C.z))*C.j +
(-Derivative(0, C.y) + Derivative(2*C.x**2, C.x))*C.k)
assert (delop.cross(2*x**2*j, doit=True) == 4*x*k ==
curl(2*x**2*j))
#Tests for divergence
assert delop & Vector.zero is S.Zero == divergence(Vector.zero)
assert (delop & Vector.zero).doit() is S.Zero
assert delop.dot(Vector.zero) == delop & Vector.zero
assert (delop & i).doit() is S.Zero
assert (delop & x**2*i).doit() == 2*x == divergence(x**2*i)
assert (delop.dot(v, doit=True) == x*y + y*z + z*x ==
divergence(v))
assert delop & v == delop.dot(v)
assert delop.dot(1/(x*y*z) * (i + j + k), doit=True) == \
- 1 / (x*y*z**2) - 1 / (x*y**2*z) - 1 / (x**2*y*z)
v = x*i + y*j + z*k
assert (delop & v == Derivative(C.x, C.x) +
Derivative(C.y, C.y) + Derivative(C.z, C.z))
assert delop.dot(v, doit=True) == 3 == divergence(v)
assert delop & v == delop.dot(v)
assert simplify((delop & v).doit()) == 3
#Tests for gradient
assert (delop.gradient(0, doit=True) == Vector.zero ==
gradient(0))
assert delop.gradient(0) == delop(0)
assert (delop(S.Zero)).doit() == Vector.zero
assert (delop(x) == (Derivative(C.x, C.x))*C.i +
(Derivative(C.x, C.y))*C.j + (Derivative(C.x, C.z))*C.k)
assert (delop(x)).doit() == i == gradient(x)
assert (delop(x*y*z) ==
(Derivative(C.x*C.y*C.z, C.x))*C.i +
(Derivative(C.x*C.y*C.z, C.y))*C.j +
(Derivative(C.x*C.y*C.z, C.z))*C.k)
assert (delop.gradient(x*y*z, doit=True) ==
y*z*i + z*x*j + x*y*k ==
gradient(x*y*z))
assert delop(x*y*z) == delop.gradient(x*y*z)
assert (delop(2*x**2)).doit() == 4*x*i
assert ((delop(a*sin(y) / x)).doit() ==
-a*sin(y)/x**2 * i + a*cos(y)/x * j)
#Tests for directional derivative
assert (Vector.zero & delop)(a) is S.Zero
assert ((Vector.zero & delop)(a)).doit() is S.Zero
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
assert ((v & delop)(S.Zero)).doit() is S.Zero
assert ((i & delop)(x)).doit() == 1
assert ((j & delop)(y)).doit() == 1
assert ((k & delop)(z)).doit() == 1
assert ((i & delop)(x*y*z)).doit() == y*z
assert ((v & delop)(x)).doit() == x
assert ((v & delop)(x*y*z)).doit() == 3*x*y*z
assert (v & delop)(x + y + z) == C.x + C.y + C.z
assert ((v & delop)(x + y + z)).doit() == x + y + z
assert ((v & delop)(v)).doit() == v
assert ((i & delop)(v)).doit() == i
assert ((j & delop)(v)).doit() == j
assert ((k & delop)(v)).doit() == k
assert ((v & delop)(Vector.zero)).doit() == Vector.zero
# Tests for laplacian on scalar fields
assert laplacian(x*y*z) is S.Zero
assert laplacian(x**2) == S(2)
assert laplacian(x**2*y**2*z**2) == \
2*y**2*z**2 + 2*x**2*z**2 + 2*x**2*y**2
A = CoordSys3D('A', transformation="spherical", variable_names=["r", "theta", "phi"])
B = CoordSys3D('B', transformation='cylindrical', variable_names=["r", "theta", "z"])
assert laplacian(A.r + A.theta + A.phi) == 2/A.r + cos(A.theta)/(A.r**2*sin(A.theta))
assert laplacian(B.r + B.theta + B.z) == 1/B.r
# Tests for laplacian on vector fields
assert laplacian(x*y*z*(i + j + k)) == Vector.zero
assert laplacian(x*y**2*z*(i + j + k)) == \
2*x*z*i + 2*x*z*j + 2*x*z*k
def test_product_rules():
"""
Tests the six product rules defined with respect to the Del
operator
References
==========
.. [1] https://en.wikipedia.org/wiki/Del
"""
#Define the scalar and vector functions
f = 2*x*y*z
g = x*y + y*z + z*x
u = x**2*i + 4*j - y**2*z*k
v = 4*i + x*y*z*k
# First product rule
lhs = delop(f * g, doit=True)
rhs = (f * delop(g) + g * delop(f)).doit()
assert simplify(lhs) == simplify(rhs)
# Second product rule
lhs = delop(u & v).doit()
rhs = ((u ^ (delop ^ v)) + (v ^ (delop ^ u)) + \
((u & delop)(v)) + ((v & delop)(u))).doit()
assert simplify(lhs) == simplify(rhs)
# Third product rule
lhs = (delop & (f*v)).doit()
rhs = ((f * (delop & v)) + (v & (delop(f)))).doit()
assert simplify(lhs) == simplify(rhs)
# Fourth product rule
lhs = (delop & (u ^ v)).doit()
rhs = ((v & (delop ^ u)) - (u & (delop ^ v))).doit()
assert simplify(lhs) == simplify(rhs)
# Fifth product rule
lhs = (delop ^ (f * v)).doit()
rhs = (((delop(f)) ^ v) + (f * (delop ^ v))).doit()
assert simplify(lhs) == simplify(rhs)
# Sixth product rule
lhs = (delop ^ (u ^ v)).doit()
rhs = (u * (delop & v) - v * (delop & u) +
(v & delop)(u) - (u & delop)(v)).doit()
assert simplify(lhs) == simplify(rhs)
P = C.orient_new_axis('P', q, C.k) # type: ignore
scalar_field = 2*x**2*y*z
grad_field = gradient(scalar_field)
vector_field = y**2*i + 3*x*j + 5*y*z*k
curl_field = curl(vector_field)
def test_conservative():
assert is_conservative(Vector.zero) is True
assert is_conservative(i) is True
assert is_conservative(2 * i + 3 * j + 4 * k) is True
assert (is_conservative(y*z*i + x*z*j + x*y*k) is
True)
assert is_conservative(x * j) is False
assert is_conservative(grad_field) is True
assert is_conservative(curl_field) is False
assert (is_conservative(4*x*y*z*i + 2*x**2*z*j) is
False)
assert is_conservative(z*P.i + P.x*k) is True
def test_solenoidal():
assert is_solenoidal(Vector.zero) is True
assert is_solenoidal(i) is True
assert is_solenoidal(2 * i + 3 * j + 4 * k) is True
assert (is_solenoidal(y*z*i + x*z*j + x*y*k) is
True)
assert is_solenoidal(y * j) is False
assert is_solenoidal(grad_field) is False
assert is_solenoidal(curl_field) is True
assert is_solenoidal((-2*y + 3)*k) is True
assert is_solenoidal(cos(q)*i + sin(q)*j + cos(q)*P.k) is True
assert is_solenoidal(z*P.i + P.x*k) is True
def test_directional_derivative():
assert directional_derivative(C.x*C.y*C.z, 3*C.i + 4*C.j + C.k) == C.x*C.y + 4*C.x*C.z + 3*C.y*C.z
assert directional_derivative(5*C.x**2*C.z, 3*C.i + 4*C.j + C.k) == 5*C.x**2 + 30*C.x*C.z
assert directional_derivative(5*C.x**2*C.z, 4*C.j) is S.Zero
D = CoordSys3D("D", "spherical", variable_names=["r", "theta", "phi"],
vector_names=["e_r", "e_theta", "e_phi"])
r, theta, phi = D.base_scalars()
e_r, e_theta, e_phi = D.base_vectors()
assert directional_derivative(r**2*e_r, e_r) == 2*r*e_r
assert directional_derivative(5*r**2*phi, 3*e_r + 4*e_theta + e_phi) == 5*r**2 + 30*r*phi
def test_scalar_potential():
assert scalar_potential(Vector.zero, C) == 0
assert scalar_potential(i, C) == x
assert scalar_potential(j, C) == y
assert scalar_potential(k, C) == z
assert scalar_potential(y*z*i + x*z*j + x*y*k, C) == x*y*z
assert scalar_potential(grad_field, C) == scalar_field
assert scalar_potential(z*P.i + P.x*k, C) == x*z*cos(q) + y*z*sin(q)
assert scalar_potential(z*P.i + P.x*k, P) == P.x*P.z
raises(ValueError, lambda: scalar_potential(x*j, C))
def test_scalar_potential_difference():
point1 = C.origin.locate_new('P1', 1*i + 2*j + 3*k)
point2 = C.origin.locate_new('P2', 4*i + 5*j + 6*k)
genericpointC = C.origin.locate_new('RP', x*i + y*j + z*k)
genericpointP = P.origin.locate_new('PP', P.x*P.i + P.y*P.j + P.z*P.k)
assert scalar_potential_difference(S.Zero, C, point1, point2) == 0
assert (scalar_potential_difference(scalar_field, C, C.origin,
genericpointC) ==
scalar_field)
assert (scalar_potential_difference(grad_field, C, C.origin,
genericpointC) ==
scalar_field)
assert scalar_potential_difference(grad_field, C, point1, point2) == 948
assert (scalar_potential_difference(y*z*i + x*z*j +
x*y*k, C, point1,
genericpointC) ==
x*y*z - 6)
potential_diff_P = (2*P.z*(P.x*sin(q) + P.y*cos(q))*
(P.x*cos(q) - P.y*sin(q))**2)
assert (scalar_potential_difference(grad_field, P, P.origin,
genericpointP).simplify() ==
potential_diff_P.simplify())
def test_differential_operators_curvilinear_system():
A = CoordSys3D('A', transformation="spherical", variable_names=["r", "theta", "phi"])
B = CoordSys3D('B', transformation='cylindrical', variable_names=["r", "theta", "z"])
# Test for spherical coordinate system and gradient
assert gradient(3*A.r + 4*A.theta) == 3*A.i + 4/A.r*A.j
assert gradient(3*A.r*A.phi + 4*A.theta) == 3*A.phi*A.i + 4/A.r*A.j + (3/sin(A.theta))*A.k
assert gradient(0*A.r + 0*A.theta+0*A.phi) == Vector.zero
assert gradient(A.r*A.theta*A.phi) == A.theta*A.phi*A.i + A.phi*A.j + (A.theta/sin(A.theta))*A.k
# Test for spherical coordinate system and divergence
assert divergence(A.r * A.i + A.theta * A.j + A.phi * A.k) == \
(sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 3 + 1/(sin(A.theta)*A.r)
assert divergence(3*A.r*A.phi*A.i + A.theta*A.j + A.r*A.theta*A.phi*A.k) == \
(sin(A.theta)*A.r + cos(A.theta)*A.r*A.theta)/(sin(A.theta)*A.r**2) + 9*A.phi + A.theta/sin(A.theta)
assert divergence(Vector.zero) == 0
assert divergence(0*A.i + 0*A.j + 0*A.k) == 0
# Test for spherical coordinate system and curl
assert curl(A.r*A.i + A.theta*A.j + A.phi*A.k) == \
(cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + A.theta/A.r*A.k
assert curl(A.r*A.j + A.phi*A.k) == (cos(A.theta)*A.phi/(sin(A.theta)*A.r))*A.i + (-A.phi/A.r)*A.j + 2*A.k
# Test for cylindrical coordinate system and gradient
assert gradient(0*B.r + 0*B.theta+0*B.z) == Vector.zero
assert gradient(B.r*B.theta*B.z) == B.theta*B.z*B.i + B.z*B.j + B.r*B.theta*B.k
assert gradient(3*B.r) == 3*B.i
assert gradient(2*B.theta) == 2/B.r * B.j
assert gradient(4*B.z) == 4*B.k
# Test for cylindrical coordinate system and divergence
assert divergence(B.r*B.i + B.theta*B.j + B.z*B.k) == 3 + 1/B.r
assert divergence(B.r*B.j + B.z*B.k) == 1
# Test for cylindrical coordinate system and curl
assert curl(B.r*B.j + B.z*B.k) == 2*B.k
assert curl(3*B.i + 2/B.r*B.j + 4*B.k) == Vector.zero
def test_mixed_coordinates():
# gradient
a = CoordSys3D('a')
b = CoordSys3D('b')
c = CoordSys3D('c')
assert gradient(a.x*b.y) == b.y*a.i + a.x*b.j
assert gradient(3*cos(q)*a.x*b.x+a.y*(a.x+(cos(q)+b.x))) ==\
(a.y + 3*b.x*cos(q))*a.i + (a.x + b.x + cos(q))*a.j + (3*a.x*cos(q) + a.y)*b.i
# Some tests need further work:
# assert gradient(a.x*(cos(a.x+b.x))) == (cos(a.x + b.x))*a.i + a.x*Gradient(cos(a.x + b.x))
# assert gradient(cos(a.x + b.x)*cos(a.x + b.z)) == Gradient(cos(a.x + b.x)*cos(a.x + b.z))
assert gradient(a.x**b.y) == Gradient(a.x**b.y)
# assert gradient(cos(a.x+b.y)*a.z) == None
assert gradient(cos(a.x*b.y)) == Gradient(cos(a.x*b.y))
assert gradient(3*cos(q)*a.x*b.x*a.z*a.y+ b.y*b.z + cos(a.x+a.y)*b.z) == \
(3*a.y*a.z*b.x*cos(q) - b.z*sin(a.x + a.y))*a.i + \
(3*a.x*a.z*b.x*cos(q) - b.z*sin(a.x + a.y))*a.j + (3*a.x*a.y*b.x*cos(q))*a.k + \
(3*a.x*a.y*a.z*cos(q))*b.i + b.z*b.j + (b.y + cos(a.x + a.y))*b.k
# divergence
assert divergence(a.i*a.x+a.j*a.y+a.z*a.k + b.i*b.x+b.j*b.y+b.z*b.k + c.i*c.x+c.j*c.y+c.z*c.k) == S(9)
# assert divergence(3*a.i*a.x*cos(a.x+b.z) + a.j*b.x*c.z) == None
assert divergence(3*a.i*a.x*a.z + b.j*b.x*c.z + 3*a.j*a.z*a.y) == \
6*a.z + b.x*Dot(b.j, c.k)
assert divergence(3*cos(q)*a.x*b.x*b.i*c.x) == \
3*a.x*b.x*cos(q)*Dot(b.i, c.i) + 3*a.x*c.x*cos(q) + 3*b.x*c.x*cos(q)*Dot(b.i, a.i)
assert divergence(a.x*b.x*c.x*Cross(a.x*a.i, a.y*b.j)) ==\
a.x*b.x*c.x*Divergence(Cross(a.x*a.i, a.y*b.j)) + \
b.x*c.x*Dot(Cross(a.x*a.i, a.y*b.j), a.i) + \
a.x*c.x*Dot(Cross(a.x*a.i, a.y*b.j), b.i) + \
a.x*b.x*Dot(Cross(a.x*a.i, a.y*b.j), c.i)
assert divergence(a.x*b.x*c.x*(a.x*a.i + b.x*b.i)) == \
4*a.x*b.x*c.x +\
a.x**2*c.x*Dot(a.i, b.i) +\
a.x**2*b.x*Dot(a.i, c.i) +\
b.x**2*c.x*Dot(b.i, a.i) +\
a.x*b.x**2*Dot(b.i, c.i)
|
a160a5a92d4b6747558d45b36b78ff182ce0bf6b9a58e45b2fbbce8b19496011 | from sympy.core.numbers import (Float, Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, cos, sin)
from sympy.sets import EmptySet
from sympy.simplify.simplify import simplify
from sympy.functions.elementary.trigonometric import tan
from sympy.geometry import (Circle, GeometryError, Line, Point, Ray,
Segment, Triangle, intersection, Point3D, Line3D, Ray3D, Segment3D,
Point2D, Line2D)
from sympy.geometry.line import Undecidable
from sympy.geometry.polygon import _asa as asa
from sympy.utilities.iterables import cartes
from sympy.testing.pytest import raises, warns, warns_deprecated_sympy
x = Symbol('x', real=True)
y = Symbol('y', real=True)
z = Symbol('z', real=True)
k = Symbol('k', real=True)
x1 = Symbol('x1', real=True)
y1 = Symbol('y1', real=True)
t = Symbol('t', real=True)
a, b = symbols('a,b', real=True)
m = symbols('m', real=True)
def test_object_from_equation():
from sympy.abc import x, y, a, b
assert Line(3*x + y + 18) == Line2D(Point2D(0, -18), Point2D(1, -21))
assert Line(3*x + 5 * y + 1) == Line2D(
Point2D(0, Rational(-1, 5)), Point2D(1, Rational(-4, 5)))
assert Line(3*a + b + 18, x="a", y="b") == Line2D(
Point2D(0, -18), Point2D(1, -21))
assert Line(3*x + y) == Line2D(Point2D(0, 0), Point2D(1, -3))
assert Line(x + y) == Line2D(Point2D(0, 0), Point2D(1, -1))
assert Line(Eq(3*a + b, -18), x="a", y=b) == Line2D(
Point2D(0, -18), Point2D(1, -21))
# issue 22361
assert Line(x - 1) == Line2D(Point2D(1, 0), Point2D(1, 1))
assert Line(2*x - 2, y=x) == Line2D(Point2D(0, 1), Point2D(1, 1))
assert Line(y) == Line2D(Point2D(0, 0), Point2D(1, 0))
assert Line(2*y, x=y) == Line2D(Point2D(0, 0), Point2D(0, 1))
assert Line(y, x=y) == Line2D(Point2D(0, 0), Point2D(0, 1))
raises(ValueError, lambda: Line(x / y))
raises(ValueError, lambda: Line(a / b, x='a', y='b'))
raises(ValueError, lambda: Line(y / x))
raises(ValueError, lambda: Line(b / a, x='a', y='b'))
raises(ValueError, lambda: Line((x + 1)**2 + y))
def feq(a, b):
"""Test if two floating point values are 'equal'."""
t_float = Float("1.0E-10")
return -t_float < a - b < t_float
def test_angle_between():
a = Point(1, 2, 3, 4)
b = a.orthogonal_direction
o = a.origin
assert feq(Line.angle_between(Line(Point(0, 0), Point(1, 1)),
Line(Point(0, 0), Point(5, 0))).evalf(), pi.evalf() / 4)
assert Line(a, o).angle_between(Line(b, o)) == pi / 2
z = Point3D(0, 0, 0)
assert Line3D.angle_between(Line3D(z, Point3D(1, 1, 1)),
Line3D(z, Point3D(5, 0, 0))) == acos(sqrt(3) / 3)
# direction of points is used to determine angle
assert Line3D.angle_between(Line3D(z, Point3D(1, 1, 1)),
Line3D(Point3D(5, 0, 0), z)) == acos(-sqrt(3) / 3)
def test_closing_angle():
a = Ray((0, 0), angle=0)
b = Ray((1, 2), angle=pi/2)
assert a.closing_angle(b) == -pi/2
assert b.closing_angle(a) == pi/2
assert a.closing_angle(a) == 0
def test_smallest_angle():
a = Line(Point(1, 1), Point(1, 2))
b = Line(Point(1, 1),Point(2, 3))
assert a.smallest_angle_between(b) == acos(2*sqrt(5)/5)
def test_svg():
a = Line(Point(1, 1),Point(1, 2))
assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 1.00000000000000,1.00000000000000 L 1.00000000000000,2.00000000000000" marker-start="url(#markerReverseArrow)" marker-end="url(#markerArrow)"/>'
a = Segment(Point(1, 0),Point(1, 1))
assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 1.00000000000000,0 L 1.00000000000000,1.00000000000000" />'
a = Ray(Point(2, 3), Point(3, 5))
assert a._svg() == '<path fill-rule="evenodd" fill="#66cc99" stroke="#555555" stroke-width="2.0" opacity="0.6" d="M 2.00000000000000,3.00000000000000 L 3.00000000000000,5.00000000000000" marker-start="url(#markerCircle)" marker-end="url(#markerArrow)"/>'
def test_arbitrary_point():
l1 = Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
l2 = Line(Point(x1, x1), Point(y1, y1))
assert l2.arbitrary_point() in l2
assert Ray((1, 1), angle=pi / 4).arbitrary_point() == \
Point(t + 1, t + 1)
assert Segment((1, 1), (2, 3)).arbitrary_point() == Point(1 + t, 1 + 2 * t)
assert l1.perpendicular_segment(l1.arbitrary_point()) == l1.arbitrary_point()
assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]).arbitrary_point() == \
Point3D(t + 1, 2 * t + 1, 3 * t + 1)
assert Segment3D(Point3D(0, 0, 0), Point3D(1, 1, 1)).midpoint == \
Point3D(S.Half, S.Half, S.Half)
assert Segment3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1)).length == sqrt(3) * sqrt((x1 - y1) ** 2)
assert Segment3D((1, 1, 1), (2, 3, 4)).arbitrary_point() == \
Point3D(t + 1, 2 * t + 1, 3 * t + 1)
raises(ValueError, (lambda: Line((x, 1), (2, 3)).arbitrary_point(x)))
def test_are_concurrent_2d():
l1 = Line(Point(0, 0), Point(1, 1))
l2 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert Line.are_concurrent(l1) is False
assert Line.are_concurrent(l1, l2)
assert Line.are_concurrent(l1, l1, l1, l2)
assert Line.are_concurrent(l1, l2, Line(Point(5, x1), Point(Rational(-3, 5), x1)))
assert Line.are_concurrent(l1, Line(Point(0, 0), Point(-x1, x1)), l2) is False
def test_are_concurrent_3d():
p1 = Point3D(0, 0, 0)
l1 = Line(p1, Point3D(1, 1, 1))
parallel_1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
parallel_2 = Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))
assert Line3D.are_concurrent(l1) is False
assert Line3D.are_concurrent(l1, Line(Point3D(x1, x1, x1), Point3D(y1, y1, y1))) is False
assert Line3D.are_concurrent(l1, Line3D(p1, Point3D(x1, x1, x1)),
Line(Point3D(x1, x1, x1), Point3D(x1, 1 + x1, 1))) is True
assert Line3D.are_concurrent(parallel_1, parallel_2) is False
def test_arguments():
"""Functions accepting `Point` objects in `geometry`
should also accept tuples, lists, and generators and
automatically convert them to points."""
from sympy.utilities.iterables import subsets
singles2d = ((1, 2), [1, 3], Point(1, 5))
doubles2d = subsets(singles2d, 2)
l2d = Line(Point2D(1, 2), Point2D(2, 3))
singles3d = ((1, 2, 3), [1, 2, 4], Point(1, 2, 6))
doubles3d = subsets(singles3d, 2)
l3d = Line(Point3D(1, 2, 3), Point3D(1, 1, 2))
singles4d = ((1, 2, 3, 4), [1, 2, 3, 5], Point(1, 2, 3, 7))
doubles4d = subsets(singles4d, 2)
l4d = Line(Point(1, 2, 3, 4), Point(2, 2, 2, 2))
# test 2D
test_single = ['contains', 'distance', 'equals', 'parallel_line', 'perpendicular_line', 'perpendicular_segment',
'projection', 'intersection']
for p in doubles2d:
Line2D(*p)
for func in test_single:
for p in singles2d:
getattr(l2d, func)(p)
# test 3D
for p in doubles3d:
Line3D(*p)
for func in test_single:
for p in singles3d:
getattr(l3d, func)(p)
# test 4D
for p in doubles4d:
Line(*p)
for func in test_single:
for p in singles4d:
getattr(l4d, func)(p)
def test_basic_properties_2d():
p1 = Point(0, 0)
p2 = Point(1, 1)
p10 = Point(2000, 2000)
p_r3 = Ray(p1, p2).random_point()
p_r4 = Ray(p2, p1).random_point()
l1 = Line(p1, p2)
l3 = Line(Point(x1, x1), Point(x1, 1 + x1))
l4 = Line(p1, Point(1, 0))
r1 = Ray(p1, Point(0, 1))
r2 = Ray(Point(0, 1), p1)
s1 = Segment(p1, p10)
p_s1 = s1.random_point()
assert Line((1, 1), slope=1) == Line((1, 1), (2, 2))
assert Line((1, 1), slope=oo) == Line((1, 1), (1, 2))
assert Line((1, 1), slope=oo).bounds == (1, 1, 1, 2)
assert Line((1, 1), slope=-oo) == Line((1, 1), (1, 2))
assert Line(p1, p2).scale(2, 1) == Line(p1, Point(2, 1))
assert Line(p1, p2) == Line(p1, p2)
assert Line(p1, p2) != Line(p2, p1)
assert l1 != Line(Point(x1, x1), Point(y1, y1))
assert l1 != l3
assert Line(p1, p10) != Line(p10, p1)
assert Line(p1, p10) != p1
assert p1 in l1 # is p1 on the line l1?
assert p1 not in l3
assert s1 in Line(p1, p10)
assert Ray(Point(0, 0), Point(0, 1)) in Ray(Point(0, 0), Point(0, 2))
assert Ray(Point(0, 0), Point(0, 2)) in Ray(Point(0, 0), Point(0, 1))
assert Ray(Point(0, 0), Point(0, 2)).xdirection == S.Zero
assert Ray(Point(0, 0), Point(1, 2)).xdirection == S.Infinity
assert Ray(Point(0, 0), Point(-1, 2)).xdirection == S.NegativeInfinity
assert Ray(Point(0, 0), Point(2, 0)).ydirection == S.Zero
assert Ray(Point(0, 0), Point(2, 2)).ydirection == S.Infinity
assert Ray(Point(0, 0), Point(2, -2)).ydirection == S.NegativeInfinity
assert (r1 in s1) is False
assert Segment(p1, p2) in s1
assert Ray(Point(x1, x1), Point(x1, 1 + x1)) != Ray(p1, Point(-1, 5))
assert Segment(p1, p2).midpoint == Point(S.Half, S.Half)
assert Segment(p1, Point(-x1, x1)).length == sqrt(2 * (x1 ** 2))
assert l1.slope == 1
assert l3.slope is oo
assert l4.slope == 0
assert Line(p1, Point(0, 1)).slope is oo
assert Line(r1.source, r1.random_point()).slope == r1.slope
assert Line(r2.source, r2.random_point()).slope == r2.slope
assert Segment(Point(0, -1), Segment(p1, Point(0, 1)).random_point()).slope == Segment(p1, Point(0, 1)).slope
assert l4.coefficients == (0, 1, 0)
assert Line((-x, x), (-x + 1, x - 1)).coefficients == (1, 1, 0)
assert Line(p1, Point(0, 1)).coefficients == (1, 0, 0)
# issue 7963
r = Ray((0, 0), angle=x)
assert r.subs(x, 3 * pi / 4) == Ray((0, 0), (-1, 1))
assert r.subs(x, 5 * pi / 4) == Ray((0, 0), (-1, -1))
assert r.subs(x, -pi / 4) == Ray((0, 0), (1, -1))
assert r.subs(x, pi / 2) == Ray((0, 0), (0, 1))
assert r.subs(x, -pi / 2) == Ray((0, 0), (0, -1))
for ind in range(0, 5):
assert l3.random_point() in l3
assert p_r3.x >= p1.x and p_r3.y >= p1.y
assert p_r4.x <= p2.x and p_r4.y <= p2.y
assert p1.x <= p_s1.x <= p10.x and p1.y <= p_s1.y <= p10.y
assert hash(s1) != hash(Segment(p10, p1))
assert s1.plot_interval() == [t, 0, 1]
assert Line(p1, p10).plot_interval() == [t, -5, 5]
assert Ray((0, 0), angle=pi / 4).plot_interval() == [t, 0, 10]
def test_basic_properties_3d():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(x1, x1, x1)
p5 = Point3D(x1, 1 + x1, 1)
l1 = Line3D(p1, p2)
l3 = Line3D(p3, p5)
r1 = Ray3D(p1, Point3D(-1, 5, 0))
r3 = Ray3D(p1, p2)
s1 = Segment3D(p1, p2)
assert Line3D((1, 1, 1), direction_ratio=[2, 3, 4]) == Line3D(Point3D(1, 1, 1), Point3D(3, 4, 5))
assert Line3D((1, 1, 1), direction_ratio=[1, 5, 7]) == Line3D(Point3D(1, 1, 1), Point3D(2, 6, 8))
assert Line3D((1, 1, 1), direction_ratio=[1, 2, 3]) == Line3D(Point3D(1, 1, 1), Point3D(2, 3, 4))
assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).direction_cosine == [1, 0, 0]
assert Line3D(Line3D(p1, Point3D(0, 1, 0))) == Line3D(p1, Point3D(0, 1, 0))
assert Ray3D(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))) == Ray3D(p1, Point3D(1, 0, 0))
assert Line3D(p1, p2) != Line3D(p2, p1)
assert l1 != l3
assert l1 != Line3D(p3, Point3D(y1, y1, y1))
assert r3 != r1
assert Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1)) in Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2))
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)) in Ray3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).xdirection == S.Infinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).ydirection == S.Infinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 2)).zdirection == S.Infinity
assert Ray3D(Point3D(0, 0, 0), Point3D(-2, 2, 2)).xdirection == S.NegativeInfinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, -2, 2)).ydirection == S.NegativeInfinity
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, -2)).zdirection == S.NegativeInfinity
assert Ray3D(Point3D(0, 0, 0), Point3D(0, 2, 2)).xdirection == S.Zero
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 0, 2)).ydirection == S.Zero
assert Ray3D(Point3D(0, 0, 0), Point3D(2, 2, 0)).zdirection == S.Zero
assert p1 in l1
assert p1 not in l3
assert l1.direction_ratio == [1, 1, 1]
assert s1.midpoint == Point3D(S.Half, S.Half, S.Half)
# Test zdirection
assert Ray3D(p1, Point3D(0, 0, -1)).zdirection is S.NegativeInfinity
def test_contains():
p1 = Point(0, 0)
r = Ray(p1, Point(4, 4))
r1 = Ray3D(p1, Point3D(0, 0, -1))
r2 = Ray3D(p1, Point3D(0, 1, 0))
r3 = Ray3D(p1, Point3D(0, 0, 1))
l = Line(Point(0, 1), Point(3, 4))
# Segment contains
assert Point(0, (a + b) / 2) in Segment((0, a), (0, b))
assert Point((a + b) / 2, 0) in Segment((a, 0), (b, 0))
assert Point3D(0, 1, 0) in Segment3D((0, 1, 0), (0, 1, 0))
assert Point3D(1, 0, 0) in Segment3D((1, 0, 0), (1, 0, 0))
assert Segment3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).contains([]) is True
assert Segment3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).contains(
Segment3D(Point3D(2, 2, 2), Point3D(3, 2, 2))) is False
# Line contains
assert l.contains(Point(0, 1)) is True
assert l.contains((0, 1)) is True
assert l.contains((0, 0)) is False
# Ray contains
assert r.contains(p1) is True
assert r.contains((1, 1)) is True
assert r.contains((1, 3)) is False
assert r.contains(Segment((1, 1), (2, 2))) is True
assert r.contains(Segment((1, 2), (2, 5))) is False
assert r.contains(Ray((2, 2), (3, 3))) is True
assert r.contains(Ray((2, 2), (3, 5))) is False
assert r1.contains(Segment3D(p1, Point3D(0, 0, -10))) is True
assert r1.contains(Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))) is False
assert r2.contains(Point3D(0, 0, 0)) is True
assert r3.contains(Point3D(0, 0, 0)) is True
assert Ray3D(Point3D(1, 1, 1), Point3D(1, 0, 0)).contains([]) is False
assert Line3D((0, 0, 0), (x, y, z)).contains((2 * x, 2 * y, 2 * z))
with warns(UserWarning, test_stacklevel=False):
assert Line3D(p1, Point3D(0, 1, 0)).contains(Point(1.0, 1.0)) is False
with warns(UserWarning, test_stacklevel=False):
assert r3.contains(Point(1.0, 1.0)) is False
def test_contains_nonreal_symbols():
u, v, w, z = symbols('u, v, w, z')
l = Segment(Point(u, w), Point(v, z))
p = Point(u*Rational(2, 3) + v/3, w*Rational(2, 3) + z/3)
assert l.contains(p)
def test_distance_2d():
p1 = Point(0, 0)
p2 = Point(1, 1)
half = S.Half
s1 = Segment(Point(0, 0), Point(1, 1))
s2 = Segment(Point(half, half), Point(1, 0))
r = Ray(p1, p2)
assert s1.distance(Point(0, 0)) == 0
assert s1.distance((0, 0)) == 0
assert s2.distance(Point(0, 0)) == 2 ** half / 2
assert s2.distance(Point(Rational(3) / 2, Rational(3) / 2)) == 2 ** half
assert Line(p1, p2).distance(Point(-1, 1)) == sqrt(2)
assert Line(p1, p2).distance(Point(1, -1)) == sqrt(2)
assert Line(p1, p2).distance(Point(2, 2)) == 0
assert Line(p1, p2).distance((-1, 1)) == sqrt(2)
assert Line((0, 0), (0, 1)).distance(p1) == 0
assert Line((0, 0), (0, 1)).distance(p2) == 1
assert Line((0, 0), (1, 0)).distance(p1) == 0
assert Line((0, 0), (1, 0)).distance(p2) == 1
assert r.distance(Point(-1, -1)) == sqrt(2)
assert r.distance(Point(1, 1)) == 0
assert r.distance(Point(-1, 1)) == sqrt(2)
assert Ray((1, 1), (2, 2)).distance(Point(1.5, 3)) == 3 * sqrt(2) / 4
assert r.distance((1, 1)) == 0
def test_dimension_normalization():
with warns(UserWarning, test_stacklevel=False):
assert Ray((1, 1), (2, 1, 2)) == Ray((1, 1, 0), (2, 1, 2))
def test_distance_3d():
p1, p2 = Point3D(0, 0, 0), Point3D(1, 1, 1)
p3 = Point3D(Rational(3) / 2, Rational(3) / 2, Rational(3) / 2)
s1 = Segment3D(Point3D(0, 0, 0), Point3D(1, 1, 1))
s2 = Segment3D(Point3D(S.Half, S.Half, S.Half), Point3D(1, 0, 1))
r = Ray3D(p1, p2)
assert s1.distance(p1) == 0
assert s2.distance(p1) == sqrt(3) / 2
assert s2.distance(p3) == 2 * sqrt(6) / 3
assert s1.distance((0, 0, 0)) == 0
assert s2.distance((0, 0, 0)) == sqrt(3) / 2
assert s1.distance(p1) == 0
assert s2.distance(p1) == sqrt(3) / 2
assert s2.distance(p3) == 2 * sqrt(6) / 3
assert s1.distance((0, 0, 0)) == 0
assert s2.distance((0, 0, 0)) == sqrt(3) / 2
# Line to point
assert Line3D(p1, p2).distance(Point3D(-1, 1, 1)) == 2 * sqrt(6) / 3
assert Line3D(p1, p2).distance(Point3D(1, -1, 1)) == 2 * sqrt(6) / 3
assert Line3D(p1, p2).distance(Point3D(2, 2, 2)) == 0
assert Line3D(p1, p2).distance((2, 2, 2)) == 0
assert Line3D(p1, p2).distance((1, -1, 1)) == 2 * sqrt(6) / 3
assert Line3D((0, 0, 0), (0, 1, 0)).distance(p1) == 0
assert Line3D((0, 0, 0), (0, 1, 0)).distance(p2) == sqrt(2)
assert Line3D((0, 0, 0), (1, 0, 0)).distance(p1) == 0
assert Line3D((0, 0, 0), (1, 0, 0)).distance(p2) == sqrt(2)
# Ray to point
assert r.distance(Point3D(-1, -1, -1)) == sqrt(3)
assert r.distance(Point3D(1, 1, 1)) == 0
assert r.distance((-1, -1, -1)) == sqrt(3)
assert r.distance((1, 1, 1)) == 0
assert Ray3D((0, 0, 0), (1, 1, 2)).distance((-1, -1, 2)) == 4 * sqrt(3) / 3
assert Ray3D((1, 1, 1), (2, 2, 2)).distance(Point3D(1.5, -3, -1)) == Rational(9) / 2
assert Ray3D((1, 1, 1), (2, 2, 2)).distance(Point3D(1.5, 3, 1)) == sqrt(78) / 6
def test_equals():
p1 = Point(0, 0)
p2 = Point(1, 1)
l1 = Line(p1, p2)
l2 = Line((0, 5), slope=m)
l3 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert l1.perpendicular_line(p1.args).equals(Line(Point(0, 0), Point(1, -1)))
assert l1.perpendicular_line(p1).equals(Line(Point(0, 0), Point(1, -1)))
assert Line(Point(x1, x1), Point(y1, y1)).parallel_line(Point(-x1, x1)). \
equals(Line(Point(-x1, x1), Point(-y1, 2 * x1 - y1)))
assert l3.parallel_line(p1.args).equals(Line(Point(0, 0), Point(0, -1)))
assert l3.parallel_line(p1).equals(Line(Point(0, 0), Point(0, -1)))
assert (l2.distance(Point(2, 3)) - 2 * abs(m + 1) / sqrt(m ** 2 + 1)).equals(0)
assert Line3D(p1, Point3D(0, 1, 0)).equals(Point(1.0, 1.0)) is False
assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).equals(Line3D(Point3D(-5, 0, 0), Point3D(-1, 0, 0))) is True
assert Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)).equals(Line3D(p1, Point3D(0, 1, 0))) is False
assert Ray3D(p1, Point3D(0, 0, -1)).equals(Point(1.0, 1.0)) is False
assert Ray3D(p1, Point3D(0, 0, -1)).equals(Ray3D(p1, Point3D(0, 0, -1))) is True
assert Line3D((0, 0), (t, t)).perpendicular_line(Point(0, 1, 0)).equals(
Line3D(Point3D(0, 1, 0), Point3D(S.Half, S.Half, 0)))
assert Line3D((0, 0), (t, t)).perpendicular_segment(Point(0, 1, 0)).equals(Segment3D((0, 1), (S.Half, S.Half)))
assert Line3D(p1, Point3D(0, 1, 0)).equals(Point(1.0, 1.0)) is False
def test_equation():
p1 = Point(0, 0)
p2 = Point(1, 1)
l1 = Line(p1, p2)
l3 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert simplify(l1.equation()) in (x - y, y - x)
assert simplify(l3.equation()) in (x - x1, x1 - x)
assert Line(p1, Point(1, 0)).equation(x=x, y=y) == y
assert Line(p1, Point(0, 1)).equation() == x
assert Line(Point(2, 0), Point(2, 1)).equation() == x - 2
assert Line(p2, Point(2, 1)).equation() == y - 1
assert Line3D(Point(x1, x1, x1), Point(y1, y1, y1)
).equation() == (-x + y, -x + z)
assert Line3D(Point(1, 2, 3), Point(2, 3, 4)
).equation() == (-x + y - 1, -x + z - 2)
assert Line3D(Point(1, 2, 3), Point(1, 3, 4)
).equation() == (x - 1, -y + z - 1)
assert Line3D(Point(1, 2, 3), Point(2, 2, 4)
).equation() == (y - 2, -x + z - 2)
assert Line3D(Point(1, 2, 3), Point(2, 3, 3)
).equation() == (-x + y - 1, z - 3)
assert Line3D(Point(1, 2, 3), Point(1, 2, 4)
).equation() == (x - 1, y - 2)
assert Line3D(Point(1, 2, 3), Point(1, 3, 3)
).equation() == (x - 1, z - 3)
assert Line3D(Point(1, 2, 3), Point(2, 2, 3)
).equation() == (y - 2, z - 3)
with warns_deprecated_sympy():
assert Line3D(Point(1, 2, 3), Point(2, 2, 3)
).equation(k='k') == (y - 2, z - 3)
def test_intersection_2d():
p1 = Point(0, 0)
p2 = Point(1, 1)
p3 = Point(x1, x1)
p4 = Point(y1, y1)
l1 = Line(p1, p2)
l3 = Line(Point(0, 0), Point(3, 4))
r1 = Ray(Point(1, 1), Point(2, 2))
r2 = Ray(Point(0, 0), Point(3, 4))
r4 = Ray(p1, p2)
r6 = Ray(Point(0, 1), Point(1, 2))
r7 = Ray(Point(0.5, 0.5), Point(1, 1))
s1 = Segment(p1, p2)
s2 = Segment(Point(0.25, 0.25), Point(0.5, 0.5))
s3 = Segment(Point(0, 0), Point(3, 4))
assert intersection(l1, p1) == [p1]
assert intersection(l1, Point(x1, 1 + x1)) == []
assert intersection(l1, Line(p3, p4)) in [[l1], [Line(p3, p4)]]
assert intersection(l1, l1.parallel_line(Point(x1, 1 + x1))) == []
assert intersection(l3, l3) == [l3]
assert intersection(l3, r2) == [r2]
assert intersection(l3, s3) == [s3]
assert intersection(s3, l3) == [s3]
assert intersection(Segment(Point(-10, 10), Point(10, 10)), Segment(Point(-5, -5), Point(-5, 5))) == []
assert intersection(r2, l3) == [r2]
assert intersection(r1, Ray(Point(2, 2), Point(0, 0))) == [Segment(Point(1, 1), Point(2, 2))]
assert intersection(r1, Ray(Point(1, 1), Point(-1, -1))) == [Point(1, 1)]
assert intersection(r1, Segment(Point(0, 0), Point(2, 2))) == [Segment(Point(1, 1), Point(2, 2))]
assert r4.intersection(s2) == [s2]
assert r4.intersection(Segment(Point(2, 3), Point(3, 4))) == []
assert r4.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
assert r4.intersection(Ray(p2, p1)) == [s1]
assert Ray(p2, p1).intersection(r6) == []
assert r4.intersection(r7) == r7.intersection(r4) == [r7]
assert Ray3D((0, 0), (3, 0)).intersection(Ray3D((1, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
assert Ray3D((1, 0), (3, 0)).intersection(Ray3D((0, 0), (3, 0))) == [Ray3D((1, 0), (3, 0))]
assert Ray(Point(0, 0), Point(0, 4)).intersection(Ray(Point(0, 1), Point(0, -1))) == \
[Segment(Point(0, 0), Point(0, 1))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((1, 0), (2, 0))) == [Segment3D((1, 0), (2, 0))]
assert Segment3D((1, 0), (2, 0)).intersection(
Segment3D((0, 0), (3, 0))) == [Segment3D((1, 0), (2, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((3, 0), (4, 0))) == [Point3D((3, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((2, 0), (5, 0))) == [Segment3D((2, 0), (3, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((-2, 0), (1, 0))) == [Segment3D((0, 0), (1, 0))]
assert Segment3D((0, 0), (3, 0)).intersection(
Segment3D((-2, 0), (0, 0))) == [Point3D(0, 0)]
assert s1.intersection(Segment(Point(1, 1), Point(2, 2))) == [Point(1, 1)]
assert s1.intersection(Segment(Point(0.5, 0.5), Point(1.5, 1.5))) == [Segment(Point(0.5, 0.5), p2)]
assert s1.intersection(Segment(Point(4, 4), Point(5, 5))) == []
assert s1.intersection(Segment(Point(-1, -1), p1)) == [p1]
assert s1.intersection(Segment(Point(-1, -1), Point(0.5, 0.5))) == [Segment(p1, Point(0.5, 0.5))]
assert s1.intersection(Line(Point(1, 0), Point(2, 1))) == []
assert s1.intersection(s2) == [s2]
assert s2.intersection(s1) == [s2]
assert asa(120, 8, 52) == \
Triangle(
Point(0, 0),
Point(8, 0),
Point(-4 * cos(19 * pi / 90) / sin(2 * pi / 45),
4 * sqrt(3) * cos(19 * pi / 90) / sin(2 * pi / 45)))
assert Line((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
assert Line((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Ray((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (1, 1)).intersection(Segment((1, 0), (1, 2))) == [Point(1, 1)]
assert Ray((0, 0), (10, 10)).contains(Segment((1, 1), (2, 2))) is True
assert Segment((1, 1), (2, 2)) in Line((0, 0), (10, 10))
assert s1.intersection(Ray((1, 1), (4, 4))) == [Point(1, 1)]
# This test is disabled because it hangs after rref changes which simplify
# intermediate results and return a different representation from when the
# test was written.
# # 16628 - this should be fast
# p0 = Point2D(Rational(249, 5), Rational(497999, 10000))
# p1 = Point2D((-58977084786*sqrt(405639795226) + 2030690077184193 +
# 20112207807*sqrt(630547164901) + 99600*sqrt(255775022850776494562626))
# /(2000*sqrt(255775022850776494562626) + 1991998000*sqrt(405639795226)
# + 1991998000*sqrt(630547164901) + 1622561172902000),
# (-498000*sqrt(255775022850776494562626) - 995999*sqrt(630547164901) +
# 90004251917891999 +
# 496005510002*sqrt(405639795226))/(10000*sqrt(255775022850776494562626)
# + 9959990000*sqrt(405639795226) + 9959990000*sqrt(630547164901) +
# 8112805864510000))
# p2 = Point2D(Rational(497, 10), Rational(-497, 10))
# p3 = Point2D(Rational(-497, 10), Rational(-497, 10))
# l = Line(p0, p1)
# s = Segment(p2, p3)
# n = (-52673223862*sqrt(405639795226) - 15764156209307469 -
# 9803028531*sqrt(630547164901) +
# 33200*sqrt(255775022850776494562626))
# d = sqrt(405639795226) + 315274080450 + 498000*sqrt(
# 630547164901) + sqrt(255775022850776494562626)
# assert intersection(l, s) == [
# Point2D(n/d*Rational(3, 2000), Rational(-497, 10))]
def test_line_intersection():
# see also test_issue_11238 in test_matrices.py
x0 = tan(pi*Rational(13, 45))
x1 = sqrt(3)
x2 = x0**2
x, y = [8*x0/(x0 + x1), (24*x0 - 8*x1*x2)/(x2 - 3)]
assert Line(Point(0, 0), Point(1, -sqrt(3))).contains(Point(x, y)) is True
def test_intersection_3d():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
l1 = Line3D(p1, p2)
l2 = Line3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
r1 = Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
r2 = Ray3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
s1 = Segment3D(Point3D(0, 0, 0), Point3D(3, 4, 0))
assert intersection(l1, p1) == [p1]
assert intersection(l1, Point3D(x1, 1 + x1, 1)) == []
assert intersection(l1, l1.parallel_line(p1)) == [Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1))]
assert intersection(l2, r2) == [r2]
assert intersection(l2, s1) == [s1]
assert intersection(r2, l2) == [r2]
assert intersection(r1, Ray3D(Point3D(1, 1, 1), Point3D(-1, -1, -1))) == [Point3D(1, 1, 1)]
assert intersection(r1, Segment3D(Point3D(0, 0, 0), Point3D(2, 2, 2))) == [
Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
assert intersection(Ray3D(Point3D(1, 0, 0), Point3D(-1, 0, 0)), Ray3D(Point3D(0, 1, 0), Point3D(0, -1, 0))) \
== [Point3D(0, 0, 0)]
assert intersection(r1, Ray3D(Point3D(2, 2, 2), Point3D(0, 0, 0))) == \
[Segment3D(Point3D(1, 1, 1), Point3D(2, 2, 2))]
assert intersection(s1, r2) == [s1]
assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).intersection(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) == \
[Point3D(2, 2, 1)]
assert Line3D((0, 1, 2), (0, 2, 3)).intersection(Line3D((0, 1, 2), (0, 1, 1))) == [Point3D(0, 1, 2)]
assert Line3D((0, 0), (t, t)).intersection(Line3D((0, 1), (t, t))) == \
[Point3D(t, t)]
assert Ray3D(Point3D(0, 0, 0), Point3D(0, 4, 0)).intersection(Ray3D(Point3D(0, 1, 1), Point3D(0, -1, 1))) == []
def test_is_parallel():
p1 = Point3D(0, 0, 0)
p2 = Point3D(1, 1, 1)
p3 = Point3D(x1, x1, x1)
l2 = Line(Point(x1, x1), Point(y1, y1))
l2_1 = Line(Point(x1, x1), Point(x1, 1 + x1))
assert Line.is_parallel(Line(Point(0, 0), Point(1, 1)), l2)
assert Line.is_parallel(l2, Line(Point(x1, x1), Point(x1, 1 + x1))) is False
assert Line.is_parallel(l2, l2.parallel_line(Point(-x1, x1)))
assert Line.is_parallel(l2_1, l2_1.parallel_line(Point(0, 0)))
assert Line3D(p1, p2).is_parallel(Line3D(p1, p2)) # same as in 2D
assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).is_parallel(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) is False
assert Line3D(p1, p2).parallel_line(p3) == Line3D(Point3D(x1, x1, x1),
Point3D(x1 + 1, x1 + 1, x1 + 1))
assert Line3D(p1, p2).parallel_line(p3.args) == \
Line3D(Point3D(x1, x1, x1), Point3D(x1 + 1, x1 + 1, x1 + 1))
assert Line3D(Point3D(4, 0, 1), Point3D(0, 4, 1)).is_parallel(Line3D(Point3D(0, 0, 1), Point3D(4, 4, 1))) is False
def test_is_perpendicular():
p1 = Point(0, 0)
p2 = Point(1, 1)
l1 = Line(p1, p2)
l2 = Line(Point(x1, x1), Point(y1, y1))
l1_1 = Line(p1, Point(-x1, x1))
# 2D
assert Line.is_perpendicular(l1, l1_1)
assert Line.is_perpendicular(l1, l2) is False
p = l1.random_point()
assert l1.perpendicular_segment(p) == p
# 3D
assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)),
Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))) is True
assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0)),
Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))) is False
assert Line3D.is_perpendicular(Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)),
Line3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1))) is False
def test_is_similar():
p1 = Point(2000, 2000)
p2 = p1.scale(2, 2)
r1 = Ray3D(Point3D(1, 1, 1), Point3D(1, 0, 0))
r2 = Ray(Point(0, 0), Point(0, 1))
s1 = Segment(Point(0, 0), p1)
assert s1.is_similar(Segment(p1, p2))
assert s1.is_similar(r2) is False
assert r1.is_similar(Line3D(Point3D(1, 1, 1), Point3D(1, 0, 0))) is True
assert r1.is_similar(Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))) is False
def test_length():
s2 = Segment3D(Point3D(x1, x1, x1), Point3D(y1, y1, y1))
assert Line(Point(0, 0), Point(1, 1)).length is oo
assert s2.length == sqrt(3) * sqrt((x1 - y1) ** 2)
assert Line3D(Point3D(0, 0, 0), Point3D(1, 1, 1)).length is oo
def test_projection():
p1 = Point(0, 0)
p2 = Point3D(0, 0, 0)
p3 = Point(-x1, x1)
l1 = Line(p1, Point(1, 1))
l2 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
l3 = Line3D(p2, Point3D(1, 1, 1))
r1 = Ray(Point(1, 1), Point(2, 2))
s1 = Segment(Point2D(0, 0), Point2D(0, 1))
s2 = Segment(Point2D(1, 0), Point2D(2, 1/2))
assert Line(Point(x1, x1), Point(y1, y1)).projection(Point(y1, y1)) == Point(y1, y1)
assert Line(Point(x1, x1), Point(x1, 1 + x1)).projection(Point(1, 1)) == Point(x1, 1)
assert Segment(Point(-2, 2), Point(0, 4)).projection(r1) == Segment(Point(-1, 3), Point(0, 4))
assert Segment(Point(0, 4), Point(-2, 2)).projection(r1) == Segment(Point(0, 4), Point(-1, 3))
assert s2.projection(s1) == EmptySet
assert l1.projection(p3) == p1
assert l1.projection(Ray(p1, Point(-1, 5))) == Ray(Point(0, 0), Point(2, 2))
assert l1.projection(Ray(p1, Point(-1, 1))) == p1
assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1)
assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Ray(Point(1, 1), Point(-1, -1))) == Point(1, 1)
assert r1.projection(Ray(Point(0, 4), Point(-1, -5))) == Segment(Point(1, 1), Point(2, 2))
assert r1.projection(Segment(Point(-1, 5), Point(-5, -10))) == Segment(Point(1, 1), Point(2, 2))
assert l3.projection(Ray3D(p2, Point3D(-1, 5, 0))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(4, 3), Rational(4, 3), Rational(4, 3)))
assert l3.projection(Ray3D(p2, Point3D(-1, 1, 1))) == Ray3D(Point3D(0, 0, 0), Point3D(Rational(1, 3), Rational(1, 3), Rational(1, 3)))
assert l2.projection(Point3D(5, 5, 0)) == Point3D(5, 0)
assert l2.projection(Line3D(Point3D(0, 1, 0), Point3D(1, 1, 0))).equals(l2)
def test_perpendicular_line():
# 3d - requires a particular orthogonal to be selected
p1, p2, p3 = Point(0, 0, 0), Point(2, 3, 4), Point(-2, 2, 0)
l = Line(p1, p2)
p = l.perpendicular_line(p3)
assert p.p1 == p3
assert p.p2 in l
# 2d - does not require special selection
p1, p2, p3 = Point(0, 0), Point(2, 3), Point(-2, 2)
l = Line(p1, p2)
p = l.perpendicular_line(p3)
assert p.p1 == p3
# p is directed from l to p3
assert p.direction.unit == (p3 - l.projection(p3)).unit
def test_perpendicular_bisector():
s1 = Segment(Point(0, 0), Point(1, 1))
aline = Line(Point(S.Half, S.Half), Point(Rational(3, 2), Rational(-1, 2)))
on_line = Segment(Point(S.Half, S.Half), Point(Rational(3, 2), Rational(-1, 2))).midpoint
assert s1.perpendicular_bisector().equals(aline)
assert s1.perpendicular_bisector(on_line).equals(Segment(s1.midpoint, on_line))
assert s1.perpendicular_bisector(on_line + (1, 0)).equals(aline)
def test_raises():
d, e = symbols('a,b', real=True)
s = Segment((d, 0), (e, 0))
raises(TypeError, lambda: Line((1, 1), 1))
raises(ValueError, lambda: Line(Point(0, 0), Point(0, 0)))
raises(Undecidable, lambda: Point(2 * d, 0) in s)
raises(ValueError, lambda: Ray3D(Point(1.0, 1.0)))
raises(ValueError, lambda: Line3D(Point3D(0, 0, 0), Point3D(0, 0, 0)))
raises(TypeError, lambda: Line3D((1, 1), 1))
raises(ValueError, lambda: Line3D(Point3D(0, 0, 0)))
raises(TypeError, lambda: Ray((1, 1), 1))
raises(GeometryError, lambda: Line(Point(0, 0), Point(1, 0))
.projection(Circle(Point(0, 0), 1)))
def test_ray_generation():
assert Ray((1, 1), angle=pi / 4) == Ray((1, 1), (2, 2))
assert Ray((1, 1), angle=pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=-pi / 2) == Ray((1, 1), (1, 0))
assert Ray((1, 1), angle=-3 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=5.0 * pi / 2) == Ray((1, 1), (1, 2))
assert Ray((1, 1), angle=pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=3.0 * pi) == Ray((1, 1), (0, 1))
assert Ray((1, 1), angle=4.0 * pi) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=0) == Ray((1, 1), (2, 1))
assert Ray((1, 1), angle=4.05 * pi) == Ray(Point(1, 1),
Point(2, -sqrt(5) * sqrt(2 * sqrt(5) + 10) / 4 - sqrt(
2 * sqrt(5) + 10) / 4 + 2 + sqrt(5)))
assert Ray((1, 1), angle=4.02 * pi) == Ray(Point(1, 1),
Point(2, 1 + tan(4.02 * pi)))
assert Ray((1, 1), angle=5) == Ray((1, 1), (2, 1 + tan(5)))
assert Ray3D((1, 1, 1), direction_ratio=[4, 4, 4]) == Ray3D(Point3D(1, 1, 1), Point3D(5, 5, 5))
assert Ray3D((1, 1, 1), direction_ratio=[1, 2, 3]) == Ray3D(Point3D(1, 1, 1), Point3D(2, 3, 4))
assert Ray3D((1, 1, 1), direction_ratio=[1, 1, 1]) == Ray3D(Point3D(1, 1, 1), Point3D(2, 2, 2))
def test_issue_7814():
circle = Circle(Point(x, 0), y)
line = Line(Point(k, z), slope=0)
_s = sqrt((y - z)*(y + z))
assert line.intersection(circle) == [Point2D(x + _s, z), Point2D(x - _s, z)]
def test_issue_2941():
def _check():
for f, g in cartes(*[(Line, Ray, Segment)] * 2):
l1 = f(a, b)
l2 = g(c, d)
assert l1.intersection(l2) == l2.intersection(l1)
# intersect at end point
c, d = (-2, -2), (-2, 0)
a, b = (0, 0), (1, 1)
_check()
# midline intersection
c, d = (-2, -3), (-2, 0)
_check()
def test_parameter_value():
t = Symbol('t')
p1, p2 = Point(0, 1), Point(5, 6)
l = Line(p1, p2)
assert l.parameter_value((5, 6), t) == {t: 1}
raises(ValueError, lambda: l.parameter_value((0, 0), t))
def test_bisectors():
r1 = Line3D(Point3D(0, 0, 0), Point3D(1, 0, 0))
r2 = Line3D(Point3D(0, 0, 0), Point3D(0, 1, 0))
bisections = r1.bisectors(r2)
assert bisections == [Line3D(Point3D(0, 0, 0), Point3D(1, 1, 0)),
Line3D(Point3D(0, 0, 0), Point3D(1, -1, 0))]
ans = [Line3D(Point3D(0, 0, 0), Point3D(1, 0, 1)),
Line3D(Point3D(0, 0, 0), Point3D(-1, 0, 1))]
l1 = (0, 0, 0), (0, 0, 1)
l2 = (0, 0), (1, 0)
for a, b in cartes((Line, Segment, Ray), repeat=2):
assert a(*l1).bisectors(b(*l2)) == ans
def test_issue_8615():
a = Line3D(Point3D(6, 5, 0), Point3D(6, -6, 0))
b = Line3D(Point3D(6, -1, 19/10), Point3D(6, -1, 0))
assert a.intersection(b) == [Point3D(6, -1, 0)]
def test_issue_12598():
r1 = Ray(Point(0, 1), Point(0.98, 0.79).n(2))
r2 = Ray(Point(0, 0), Point(0.71, 0.71).n(2))
assert str(r1.intersection(r2)[0]) == 'Point2D(0.82, 0.82)'
l1 = Line((0, 0), (1, 1))
l2 = Segment((-1, 1), (0, -1)).n(2)
assert str(l1.intersection(l2)[0]) == 'Point2D(-0.33, -0.33)'
l2 = Segment((-1, 1), (-1/2, 1/2)).n(2)
assert not l1.intersection(l2)
|
91dd11539b0639c7e9ea383081732dbd3d81ea41b5f8cdfe88638b9d425f64ec | from sympy.core.numbers import (Rational, oo)
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.geometry.ellipse import (Circle, Ellipse)
from sympy.geometry.line import (Line, Ray2D, Segment2D)
from sympy.geometry.parabola import Parabola
from sympy.geometry.point import (Point, Point2D)
from sympy.testing.pytest import raises
from sympy.abc import x, y
def test_parabola_geom():
a, b = symbols('a b')
p1 = Point(0, 0)
p2 = Point(3, 7)
p3 = Point(0, 4)
p4 = Point(6, 0)
p5 = Point(a, a)
d1 = Line(Point(4, 0), Point(4, 9))
d2 = Line(Point(7, 6), Point(3, 6))
d3 = Line(Point(4, 0), slope=oo)
d4 = Line(Point(7, 6), slope=0)
d5 = Line(Point(b, a), slope=oo)
d6 = Line(Point(a, b), slope=0)
half = S.Half
pa1 = Parabola(None, d2)
pa2 = Parabola(directrix=d1)
pa3 = Parabola(p1, d1)
pa4 = Parabola(p2, d2)
pa5 = Parabola(p2, d4)
pa6 = Parabola(p3, d2)
pa7 = Parabola(p2, d1)
pa8 = Parabola(p4, d1)
pa9 = Parabola(p4, d3)
pa10 = Parabola(p5, d5)
pa11 = Parabola(p5, d6)
d = Line(Point(3, 7), Point(2, 9))
pa12 = Parabola(Point(7, 8), d)
pa12r = Parabola(Point(7, 8).reflect(d), d)
raises(ValueError, lambda:
Parabola(Point(7, 8, 9), Line(Point(6, 7), Point(7, 7))))
raises(ValueError, lambda:
Parabola(Point(0, 2), Line(Point(7, 2), Point(6, 2))))
raises(ValueError, lambda: Parabola(Point(7, 8), Point(3, 8)))
# Basic Stuff
assert pa1.focus == Point(0, 0)
assert pa1.ambient_dimension == S(2)
assert pa2 == pa3
assert pa4 != pa7
assert pa6 != pa7
assert pa6.focus == Point2D(0, 4)
assert pa6.focal_length == 1
assert pa6.p_parameter == -1
assert pa6.vertex == Point2D(0, 5)
assert pa6.eccentricity == 1
assert pa7.focus == Point2D(3, 7)
assert pa7.focal_length == half
assert pa7.p_parameter == -half
assert pa7.vertex == Point2D(7*half, 7)
assert pa4.focal_length == half
assert pa4.p_parameter == half
assert pa4.vertex == Point2D(3, 13*half)
assert pa8.focal_length == 1
assert pa8.p_parameter == 1
assert pa8.vertex == Point2D(5, 0)
assert pa4.focal_length == pa5.focal_length
assert pa4.p_parameter == pa5.p_parameter
assert pa4.vertex == pa5.vertex
assert pa4.equation() == pa5.equation()
assert pa8.focal_length == pa9.focal_length
assert pa8.p_parameter == pa9.p_parameter
assert pa8.vertex == pa9.vertex
assert pa8.equation() == pa9.equation()
assert pa10.focal_length == pa11.focal_length == sqrt((a - b) ** 2) / 2 # if a, b real == abs(a - b)/2
assert pa11.vertex == Point(*pa10.vertex[::-1]) == Point(a,
a - sqrt((a - b)**2)*sign(a - b)/2) # change axis x->y, y->x on pa10
aos = pa12.axis_of_symmetry
assert aos == Line(Point(7, 8), Point(5, 7))
assert pa12.directrix == Line(Point(3, 7), Point(2, 9))
assert pa12.directrix.angle_between(aos) == S.Pi/2
assert pa12.eccentricity == 1
assert pa12.equation(x, y) == (x - 7)**2 + (y - 8)**2 - (-2*x - y + 13)**2/5
assert pa12.focal_length == 9*sqrt(5)/10
assert pa12.focus == Point(7, 8)
assert pa12.p_parameter == 9*sqrt(5)/10
assert pa12.vertex == Point2D(S(26)/5, S(71)/10)
assert pa12r.focal_length == 9*sqrt(5)/10
assert pa12r.focus == Point(-S(1)/5, S(22)/5)
assert pa12r.p_parameter == -9*sqrt(5)/10
assert pa12r.vertex == Point(S(8)/5, S(53)/10)
def test_parabola_intersection():
l1 = Line(Point(1, -2), Point(-1,-2))
l2 = Line(Point(1, 2), Point(-1,2))
l3 = Line(Point(1, 0), Point(-1,0))
p1 = Point(0,0)
p2 = Point(0, -2)
p3 = Point(120, -12)
parabola1 = Parabola(p1, l1)
# parabola with parabola
assert parabola1.intersection(parabola1) == [parabola1]
assert parabola1.intersection(Parabola(p1, l2)) == [Point2D(-2, 0), Point2D(2, 0)]
assert parabola1.intersection(Parabola(p2, l3)) == [Point2D(0, -1)]
assert parabola1.intersection(Parabola(Point(16, 0), l1)) == [Point2D(8, 15)]
assert parabola1.intersection(Parabola(Point(0, 16), l1)) == [Point2D(-6, 8), Point2D(6, 8)]
assert parabola1.intersection(Parabola(p3, l3)) == []
# parabola with point
assert parabola1.intersection(p1) == []
assert parabola1.intersection(Point2D(0, -1)) == [Point2D(0, -1)]
assert parabola1.intersection(Point2D(4, 3)) == [Point2D(4, 3)]
# parabola with line
assert parabola1.intersection(Line(Point2D(-7, 3), Point(12, 3))) == [Point2D(-4, 3), Point2D(4, 3)]
assert parabola1.intersection(Line(Point(-4, -1), Point(4, -1))) == [Point(0, -1)]
assert parabola1.intersection(Line(Point(2, 0), Point(0, -2))) == [Point2D(2, 0)]
raises(TypeError, lambda: parabola1.intersection(Line(Point(0, 0, 0), Point(1, 1, 1))))
# parabola with segment
assert parabola1.intersection(Segment2D((-4, -5), (4, 3))) == [Point2D(0, -1), Point2D(4, 3)]
assert parabola1.intersection(Segment2D((0, -5), (0, 6))) == [Point2D(0, -1)]
assert parabola1.intersection(Segment2D((-12, -65), (14, -68))) == []
# parabola with ray
assert parabola1.intersection(Ray2D((-4, -5), (4, 3))) == [Point2D(0, -1), Point2D(4, 3)]
assert parabola1.intersection(Ray2D((0, 7), (1, 14))) == [Point2D(14 + 2*sqrt(57), 105 + 14*sqrt(57))]
assert parabola1.intersection(Ray2D((0, 7), (0, 14))) == []
# parabola with ellipse/circle
assert parabola1.intersection(Circle(p1, 2)) == [Point2D(-2, 0), Point2D(2, 0)]
assert parabola1.intersection(Circle(p2, 1)) == [Point2D(0, -1)]
assert parabola1.intersection(Ellipse(p2, 2, 1)) == [Point2D(0, -1)]
assert parabola1.intersection(Ellipse(Point(0, 19), 5, 7)) == []
assert parabola1.intersection(Ellipse((0, 3), 12, 4)) == [
Point2D(0, -1),
Point2D(-4*sqrt(17)/3, Rational(59, 9)),
Point2D(4*sqrt(17)/3, Rational(59, 9))]
# parabola with unsupported type
raises(TypeError, lambda: parabola1.intersection(2))
|
fb0351ec6c2f2475663d26714ae02ea17d7f14993cdced2d7038b4f6a618e899 | from sympy.parsing.sym_expr import SymPyExpression
from sympy.testing.pytest import raises
from sympy.external import import_module
lfortran = import_module('lfortran')
cin = import_module('clang.cindex', import_kwargs = {'fromlist': ['cindex']})
if lfortran and cin:
from sympy.codegen.ast import (Variable, IntBaseType, FloatBaseType, String,
Declaration, FloatType)
from sympy.core import Integer, Float
from sympy.core.symbol import Symbol
expr1 = SymPyExpression()
src = """\
integer :: a, b, c, d
real :: p, q, r, s
"""
def test_c_parse():
src1 = """\
int a, b = 4;
float c, d = 2.4;
"""
expr1.convert_to_expr(src1, 'c')
ls = expr1.return_expr()
assert ls[0] == Declaration(
Variable(
Symbol('a'),
type=IntBaseType(String('intc'))
)
)
assert ls[1] == Declaration(
Variable(
Symbol('b'),
type=IntBaseType(String('intc')),
value=Integer(4)
)
)
assert ls[2] == Declaration(
Variable(
Symbol('c'),
type=FloatType(
String('float32'),
nbits=Integer(32),
nmant=Integer(23),
nexp=Integer(8)
)
)
)
assert ls[3] == Declaration(
Variable(
Symbol('d'),
type=FloatType(
String('float32'),
nbits=Integer(32),
nmant=Integer(23),
nexp=Integer(8)
),
value=Float('2.3999999999999999', precision=53)
)
)
def test_fortran_parse():
expr = SymPyExpression(src, 'f')
ls = expr.return_expr()
assert ls[0] == Declaration(
Variable(
Symbol('a'),
type=IntBaseType(String('integer')),
value=Integer(0)
)
)
assert ls[1] == Declaration(
Variable(
Symbol('b'),
type=IntBaseType(String('integer')),
value=Integer(0)
)
)
assert ls[2] == Declaration(
Variable(
Symbol('c'),
type=IntBaseType(String('integer')),
value=Integer(0)
)
)
assert ls[3] == Declaration(
Variable(
Symbol('d'),
type=IntBaseType(String('integer')),
value=Integer(0)
)
)
assert ls[4] == Declaration(
Variable(
Symbol('p'),
type=FloatBaseType(String('real')),
value=Float('0.0', precision=53)
)
)
assert ls[5] == Declaration(
Variable(
Symbol('q'),
type=FloatBaseType(String('real')),
value=Float('0.0', precision=53)
)
)
assert ls[6] == Declaration(
Variable(
Symbol('r'),
type=FloatBaseType(String('real')),
value=Float('0.0', precision=53)
)
)
assert ls[7] == Declaration(
Variable(
Symbol('s'),
type=FloatBaseType(String('real')),
value=Float('0.0', precision=53)
)
)
def test_convert_py():
src1 = (
src +
"""\
a = b + c
s = p * q / r
"""
)
expr1.convert_to_expr(src1, 'f')
exp_py = expr1.convert_to_python()
assert exp_py == [
'a = 0',
'b = 0',
'c = 0',
'd = 0',
'p = 0.0',
'q = 0.0',
'r = 0.0',
's = 0.0',
'a = b + c',
's = p*q/r'
]
def test_convert_fort():
src1 = (
src +
"""\
a = b + c
s = p * q / r
"""
)
expr1.convert_to_expr(src1, 'f')
exp_fort = expr1.convert_to_fortran()
assert exp_fort == [
' integer*4 a',
' integer*4 b',
' integer*4 c',
' integer*4 d',
' real*8 p',
' real*8 q',
' real*8 r',
' real*8 s',
' a = b + c',
' s = p*q/r'
]
def test_convert_c():
src1 = (
src +
"""\
a = b + c
s = p * q / r
"""
)
expr1.convert_to_expr(src1, 'f')
exp_c = expr1.convert_to_c()
assert exp_c == [
'int a = 0',
'int b = 0',
'int c = 0',
'int d = 0',
'double p = 0.0',
'double q = 0.0',
'double r = 0.0',
'double s = 0.0',
'a = b + c;',
's = p*q/r;'
]
def test_exceptions():
src = 'int a;'
raises(ValueError, lambda: SymPyExpression(src))
raises(ValueError, lambda: SymPyExpression(mode = 'c'))
raises(NotImplementedError, lambda: SymPyExpression(src, mode = 'd'))
elif not lfortran and not cin:
def test_raise():
raises(ImportError, lambda: SymPyExpression('int a;', 'c'))
raises(ImportError, lambda: SymPyExpression('integer :: a', 'f'))
|
9cfccb29c00925aa1981d2e8ab5fc25e1896285047fbea78b7a76ae08dec57d0 | import sympy
from sympy.parsing.sympy_parser import (
parse_expr,
standard_transformations,
convert_xor,
implicit_multiplication_application,
implicit_multiplication,
implicit_application,
function_exponentiation,
split_symbols,
split_symbols_custom,
_token_splittable
)
from sympy.testing.pytest import raises
def test_implicit_multiplication():
cases = {
'5x': '5*x',
'abc': 'a*b*c',
'3sin(x)': '3*sin(x)',
'(x+1)(x+2)': '(x+1)*(x+2)',
'(5 x**2)sin(x)': '(5*x**2)*sin(x)',
'2 sin(x) cos(x)': '2*sin(x)*cos(x)',
'pi x': 'pi*x',
'x pi': 'x*pi',
'E x': 'E*x',
'EulerGamma y': 'EulerGamma*y',
'E pi': 'E*pi',
'pi (x + 2)': 'pi*(x+2)',
'(x + 2) pi': '(x+2)*pi',
'pi sin(x)': 'pi*sin(x)',
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (split_symbols,
implicit_multiplication)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal)
application = ['sin x', 'cos 2*x', 'sin cos x']
for case in application:
raises(SyntaxError,
lambda: parse_expr(case, transformations=transformations2))
raises(TypeError,
lambda: parse_expr('sin**2(x)', transformations=transformations2))
def test_implicit_application():
cases = {
'factorial': 'factorial',
'sin x': 'sin(x)',
'tan y**3': 'tan(y**3)',
'cos 2*x': 'cos(2*x)',
'(cot)': 'cot',
'sin cos tan x': 'sin(cos(tan(x)))'
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (implicit_application,)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal), (implicit, normal)
multiplication = ['x y', 'x sin x', '2x']
for case in multiplication:
raises(SyntaxError,
lambda: parse_expr(case, transformations=transformations2))
raises(TypeError,
lambda: parse_expr('sin**2(x)', transformations=transformations2))
def test_function_exponentiation():
cases = {
'sin**2(x)': 'sin(x)**2',
'exp^y(z)': 'exp(z)^y',
'sin**2(E^(x))': 'sin(E^(x))**2'
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (function_exponentiation,)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal)
other_implicit = ['x y', 'x sin x', '2x', 'sin x',
'cos 2*x', 'sin cos x']
for case in other_implicit:
raises(SyntaxError,
lambda: parse_expr(case, transformations=transformations2))
assert parse_expr('x**2', local_dict={ 'x': sympy.Symbol('x') },
transformations=transformations2) == parse_expr('x**2')
def test_symbol_splitting():
# By default Greek letter names should not be split (lambda is a keyword
# so skip it)
transformations = standard_transformations + (split_symbols,)
greek_letters = ('alpha', 'beta', 'gamma', 'delta', 'epsilon', 'zeta',
'eta', 'theta', 'iota', 'kappa', 'mu', 'nu', 'xi',
'omicron', 'pi', 'rho', 'sigma', 'tau', 'upsilon',
'phi', 'chi', 'psi', 'omega')
for letter in greek_letters:
assert(parse_expr(letter, transformations=transformations) ==
parse_expr(letter))
# Make sure symbol splitting resolves names
transformations += (implicit_multiplication,)
local_dict = { 'e': sympy.E }
cases = {
'xe': 'E*x',
'Iy': 'I*y',
'ee': 'E*E',
}
for case, expected in cases.items():
assert(parse_expr(case, local_dict=local_dict,
transformations=transformations) ==
parse_expr(expected))
# Make sure custom splitting works
def can_split(symbol):
if symbol not in ('unsplittable', 'names'):
return _token_splittable(symbol)
return False
transformations = standard_transformations
transformations += (split_symbols_custom(can_split),
implicit_multiplication)
assert(parse_expr('unsplittable', transformations=transformations) ==
parse_expr('unsplittable'))
assert(parse_expr('names', transformations=transformations) ==
parse_expr('names'))
assert(parse_expr('xy', transformations=transformations) ==
parse_expr('x*y'))
for letter in greek_letters:
assert(parse_expr(letter, transformations=transformations) ==
parse_expr(letter))
def test_all_implicit_steps():
cases = {
'2x': '2*x', # implicit multiplication
'x y': 'x*y',
'xy': 'x*y',
'sin x': 'sin(x)', # add parentheses
'2sin x': '2*sin(x)',
'x y z': 'x*y*z',
'sin(2 * 3x)': 'sin(2 * 3 * x)',
'sin(x) (1 + cos(x))': 'sin(x) * (1 + cos(x))',
'(x + 2) sin(x)': '(x + 2) * sin(x)',
'(x + 2) sin x': '(x + 2) * sin(x)',
'sin(sin x)': 'sin(sin(x))',
'sin x!': 'sin(factorial(x))',
'sin x!!': 'sin(factorial2(x))',
'factorial': 'factorial', # don't apply a bare function
'x sin x': 'x * sin(x)', # both application and multiplication
'xy sin x': 'x * y * sin(x)',
'(x+2)(x+3)': '(x + 2) * (x+3)',
'x**2 + 2xy + y**2': 'x**2 + 2 * x * y + y**2', # split the xy
'pi': 'pi', # don't mess with constants
'None': 'None',
'ln sin x': 'ln(sin(x))', # multiple implicit function applications
'factorial': 'factorial', # don't add parentheses
'sin x**2': 'sin(x**2)', # implicit application to an exponential
'alpha': 'Symbol("alpha")', # don't split Greek letters/subscripts
'x_2': 'Symbol("x_2")',
'sin^2 x**2': 'sin(x**2)**2', # function raised to a power
'sin**3(x)': 'sin(x)**3',
'(factorial)': 'factorial',
'tan 3x': 'tan(3*x)',
'sin^2(3*E^(x))': 'sin(3*E**(x))**2',
'sin**2(E^(3x))': 'sin(E**(3*x))**2',
'sin^2 (3x*E^(x))': 'sin(3*x*E^x)**2',
'pi sin x': 'pi*sin(x)',
}
transformations = standard_transformations + (convert_xor,)
transformations2 = transformations + (implicit_multiplication_application,)
for case in cases:
implicit = parse_expr(case, transformations=transformations2)
normal = parse_expr(cases[case], transformations=transformations)
assert(implicit == normal)
def test_no_methods_implicit_multiplication():
# Issue 21020
u = sympy.Symbol('u')
transformations = standard_transformations + \
(implicit_multiplication,)
expr = parse_expr('x.is_polynomial(x)', transformations=transformations)
assert expr == True
expr = parse_expr('(exp(x) / (1 + exp(2x))).subs(exp(x), u)',
transformations=transformations)
assert expr == u/(u**2 + 1)
|
4b42198dddbcf4f2ff35b97207abea44744828378d11b5d371c7efae5244ccd5 | # coding=utf-8
from abc import ABC, abstractmethod
from sympy.core.numbers import pi
from sympy.physics.mechanics.body import Body
from sympy.physics.vector import Vector, dynamicsymbols, cross
from sympy.physics.vector.frame import ReferenceFrame
import warnings
__all__ = ['Joint', 'PinJoint', 'PrismaticJoint']
class Joint(ABC):
"""Abstract base class for all specific joints.
Explanation
===========
A joint subtracts degrees of freedom from a body. This is the base class
for all specific joints and holds all common methods acting as an interface
for all joints. Custom joint can be created by inheriting Joint class and
defining all abstract functions.
The abstract methods are:
- ``_generate_coordinates``
- ``_generate_speeds``
- ``_orient_frames``
- ``_set_angular_velocity``
- ``_set_linar_velocity``
Parameters
==========
name : string
A unique name for the joint.
parent : Body
The parent body of joint.
child : Body
The child body of joint.
coordinates: List of dynamicsymbols, optional
Generalized coordinates of the joint.
speeds : List of dynamicsymbols, optional
Generalized speeds of joint.
parent_joint_pos : Vector, optional
Vector from the parent body's mass center to the point where the parent
and child are connected. The default value is the zero vector.
child_joint_pos : Vector, optional
Vector from the child body's mass center to the point where the parent
and child are connected. The default value is the zero vector.
parent_axis : Vector, optional
Axis fixed in the parent body which aligns with an axis fixed in the
child body. The default is x axis in parent's reference frame.
child_axis : Vector, optional
Axis fixed in the child body which aligns with an axis fixed in the
parent body. The default is x axis in child's reference frame.
Attributes
==========
name : string
The joint's name.
parent : Body
The joint's parent body.
child : Body
The joint's child body.
coordinates : list
List of the joint's generalized coordinates.
speeds : list
List of the joint's generalized speeds.
parent_point : Point
The point fixed in the parent body that represents the joint.
child_point : Point
The point fixed in the child body that represents the joint.
parent_axis : Vector
The axis fixed in the parent frame that represents the joint.
child_axis : Vector
The axis fixed in the child frame that represents the joint.
kdes : list
Kinematical differential equations of the joint.
Notes
=====
The direction cosine matrix between the child and parent is formed using a
simple rotation about an axis that is normal to both ``child_axis`` and
``parent_axis``. In general, the normal axis is formed by crossing the
``child_axis`` into the ``parent_axis`` except if the child and parent axes
are in exactly opposite directions. In that case the rotation vector is chosen
using the rules in the following table where ``C`` is the child reference
frame and ``P`` is the parent reference frame:
.. list-table::
:header-rows: 1
* - ``child_axis``
- ``parent_axis``
- ``rotation_axis``
* - ``-C.x``
- ``P.x``
- ``P.z``
* - ``-C.y``
- ``P.y``
- ``P.x``
* - ``-C.z``
- ``P.z``
- ``P.y``
* - ``-C.x-C.y``
- ``P.x+P.y``
- ``P.z``
* - ``-C.y-C.z``
- ``P.y+P.z``
- ``P.x``
* - ``-C.x-C.z``
- ``P.x+P.z``
- ``P.y``
* - ``-C.x-C.y-C.z``
- ``P.x+P.y+P.z``
- ``(P.x+P.y+P.z) × P.x``
"""
def __init__(self, name, parent, child, coordinates=None, speeds=None,
parent_joint_pos=None, child_joint_pos=None, parent_axis=None,
child_axis=None):
if not isinstance(name, str):
raise TypeError('Supply a valid name.')
self._name = name
if not isinstance(parent, Body):
raise TypeError('Parent must be an instance of Body.')
self._parent = parent
if not isinstance(child, Body):
raise TypeError('Parent must be an instance of Body.')
self._child = child
self._coordinates = self._generate_coordinates(coordinates)
self._speeds = self._generate_speeds(speeds)
self._kdes = self._generate_kdes()
self._parent_axis = self._axis(parent, parent_axis)
self._child_axis = self._axis(child, child_axis)
self._parent_point = self._locate_joint_pos(parent, parent_joint_pos)
self._child_point = self._locate_joint_pos(child, child_joint_pos)
self._orient_frames()
self._set_angular_velocity()
self._set_linear_velocity()
def __str__(self):
return self.name
def __repr__(self):
return self.__str__()
@property
def name(self):
return self._name
@property
def parent(self):
"""Parent body of Joint."""
return self._parent
@property
def child(self):
"""Child body of Joint."""
return self._child
@property
def coordinates(self):
"""List generalized coordinates of the joint."""
return self._coordinates
@property
def speeds(self):
"""List generalized coordinates of the joint.."""
return self._speeds
@property
def kdes(self):
"""Kinematical differential equations of the joint."""
return self._kdes
@property
def parent_axis(self):
"""The axis of parent frame."""
return self._parent_axis
@property
def child_axis(self):
"""The axis of child frame."""
return self._child_axis
@property
def parent_point(self):
"""The joint's point where parent body is connected to the joint."""
return self._parent_point
@property
def child_point(self):
"""The joint's point where child body is connected to the joint."""
return self._child_point
@abstractmethod
def _generate_coordinates(self, coordinates):
"""Generate list generalized coordinates of the joint."""
pass
@abstractmethod
def _generate_speeds(self, speeds):
"""Generate list generalized speeds of the joint."""
pass
@abstractmethod
def _orient_frames(self):
"""Orient frames as per the joint."""
pass
@abstractmethod
def _set_angular_velocity(self):
pass
@abstractmethod
def _set_linear_velocity(self):
pass
def _generate_kdes(self):
kdes = []
t = dynamicsymbols._t
for i in range(len(self.coordinates)):
kdes.append(-self.coordinates[i].diff(t) + self.speeds[i])
return kdes
def _axis(self, body, ax):
if ax is None:
ax = body.frame.x
return ax
if not isinstance(ax, Vector):
raise TypeError("Axis must be of type Vector.")
if not ax.dt(body.frame) == 0:
msg = ('Axis cannot be time-varying when viewed from the '
'associated body.')
raise ValueError(msg)
return ax
def _locate_joint_pos(self, body, joint_pos):
if joint_pos is None:
joint_pos = Vector(0)
if not isinstance(joint_pos, Vector):
raise ValueError('Joint Position must be supplied as Vector.')
if not joint_pos.dt(body.frame) == 0:
msg = ('Position Vector cannot be time-varying when viewed from '
'the associated body.')
raise ValueError(msg)
point_name = self._name + '_' + body.name + '_joint'
return body.masscenter.locatenew(point_name, joint_pos)
def _alignment_rotation(self, parent, child):
# Returns the axis and angle between two axis(vectors).
angle = parent.angle_between(child)
axis = cross(child, parent).normalize()
return angle, axis
def _generate_vector(self):
parent_frame = self.parent.frame
components = self.parent_axis.to_matrix(parent_frame)
x, y, z = components[0], components[1], components[2]
if x != 0:
if y!=0:
if z!=0:
return cross(self.parent_axis,
parent_frame.x)
if z!=0:
return parent_frame.y
return parent_frame.z
if x == 0:
if y!=0:
if z!=0:
return parent_frame.x
return parent_frame.x
return parent_frame.y
def _set_orientation(self):
#Helper method for `orient_axis()`
self.child.frame.orient_axis(self.parent.frame, self.parent_axis, 0)
angle, axis = self._alignment_rotation(self.parent_axis,
self.child_axis)
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=UserWarning)
if axis != Vector(0) or angle == pi:
if angle == pi:
axis = self._generate_vector()
int_frame = ReferenceFrame('int_frame')
int_frame.orient_axis(self.child.frame, self.child_axis, 0)
int_frame.orient_axis(self.parent.frame, axis, angle)
return int_frame
return self.parent.frame
class PinJoint(Joint):
"""Pin (Revolute) Joint.
.. image:: PinJoint.svg
Explanation
===========
A pin joint is defined such that the joint rotation axis is fixed in both
the child and parent and the location of the joint is relative to the mass
center of each body. The child rotates an angle, θ, from the parent about
the rotation axis and has a simple angular speed, ω, relative to the
parent. The direction cosine matrix between the child and parent is formed
using a simple rotation about an axis that is normal to both ``child_axis``
and ``parent_axis``, see the Notes section for a detailed explanation of
this.
Parameters
==========
name : string
A unique name for the joint.
parent : Body
The parent body of joint.
child : Body
The child body of joint.
coordinates: dynamicsymbol, optional
Generalized coordinates of the joint.
speeds : dynamicsymbol, optional
Generalized speeds of joint.
parent_joint_pos : Vector, optional
Vector from the parent body's mass center to the point where the parent
and child are connected. The default value is the zero vector.
child_joint_pos : Vector, optional
Vector from the child body's mass center to the point where the parent
and child are connected. The default value is the zero vector.
parent_axis : Vector, optional
Axis fixed in the parent body which aligns with an axis fixed in the
child body. The default is x axis in parent's reference frame.
child_axis : Vector, optional
Axis fixed in the child body which aligns with an axis fixed in the
parent body. The default is x axis in child's reference frame.
Attributes
==========
name : string
The joint's name.
parent : Body
The joint's parent body.
child : Body
The joint's child body.
coordinates : list
List of the joint's generalized coordinates.
speeds : list
List of the joint's generalized speeds.
parent_point : Point
The point fixed in the parent body that represents the joint.
child_point : Point
The point fixed in the child body that represents the joint.
parent_axis : Vector
The axis fixed in the parent frame that represents the joint.
child_axis : Vector
The axis fixed in the child frame that represents the joint.
kdes : list
Kinematical differential equations of the joint.
Examples
=========
A single pin joint is created from two bodies and has the following basic
attributes:
>>> from sympy.physics.mechanics import Body, PinJoint
>>> parent = Body('P')
>>> parent
P
>>> child = Body('C')
>>> child
C
>>> joint = PinJoint('PC', parent, child)
>>> joint
PinJoint: PC parent: P child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
PC_P_joint
>>> joint.child_point
PC_C_joint
>>> joint.parent_axis
P_frame.x
>>> joint.child_axis
C_frame.x
>>> joint.coordinates
[theta_PC(t)]
>>> joint.speeds
[omega_PC(t)]
>>> joint.child.frame.ang_vel_in(joint.parent.frame)
omega_PC(t)*P_frame.x
>>> joint.child.frame.dcm(joint.parent.frame)
Matrix([
[1, 0, 0],
[0, cos(theta_PC(t)), sin(theta_PC(t))],
[0, -sin(theta_PC(t)), cos(theta_PC(t))]])
>>> joint.child_point.pos_from(joint.parent_point)
0
To further demonstrate the use of the pin joint, the kinematics of simple
double pendulum that rotates about the Z axis of each connected body can be
created as follows.
>>> from sympy import symbols, trigsimp
>>> from sympy.physics.mechanics import Body, PinJoint
>>> l1, l2 = symbols('l1 l2')
First create bodies to represent the fixed ceiling and one to represent
each pendulum bob.
>>> ceiling = Body('C')
>>> upper_bob = Body('U')
>>> lower_bob = Body('L')
The first joint will connect the upper bob to the ceiling by a distance of
``l1`` and the joint axis will be about the Z axis for each body.
>>> ceiling_joint = PinJoint('P1', ceiling, upper_bob,
... child_joint_pos=-l1*upper_bob.frame.x,
... parent_axis=ceiling.frame.z,
... child_axis=upper_bob.frame.z)
The second joint will connect the lower bob to the upper bob by a distance
of ``l2`` and the joint axis will also be about the Z axis for each body.
>>> pendulum_joint = PinJoint('P2', upper_bob, lower_bob,
... child_joint_pos=-l2*lower_bob.frame.x,
... parent_axis=upper_bob.frame.z,
... child_axis=lower_bob.frame.z)
Once the joints are established the kinematics of the connected bodies can
be accessed. First the direction cosine matrices of pendulum link relative
to the ceiling are found:
>>> upper_bob.frame.dcm(ceiling.frame)
Matrix([
[ cos(theta_P1(t)), sin(theta_P1(t)), 0],
[-sin(theta_P1(t)), cos(theta_P1(t)), 0],
[ 0, 0, 1]])
>>> trigsimp(lower_bob.frame.dcm(ceiling.frame))
Matrix([
[ cos(theta_P1(t) + theta_P2(t)), sin(theta_P1(t) + theta_P2(t)), 0],
[-sin(theta_P1(t) + theta_P2(t)), cos(theta_P1(t) + theta_P2(t)), 0],
[ 0, 0, 1]])
The position of the lower bob's masscenter is found with:
>>> lower_bob.masscenter.pos_from(ceiling.masscenter)
l1*U_frame.x + l2*L_frame.x
The angular velocities of the two pendulum links can be computed with
respect to the ceiling.
>>> upper_bob.frame.ang_vel_in(ceiling.frame)
omega_P1(t)*C_frame.z
>>> lower_bob.frame.ang_vel_in(ceiling.frame)
omega_P1(t)*C_frame.z + omega_P2(t)*U_frame.z
And finally, the linear velocities of the two pendulum bobs can be computed
with respect to the ceiling.
>>> upper_bob.masscenter.vel(ceiling.frame)
l1*omega_P1(t)*U_frame.y
>>> lower_bob.masscenter.vel(ceiling.frame)
l1*omega_P1(t)*U_frame.y + l2*(omega_P1(t) + omega_P2(t))*L_frame.y
"""
def __init__(self, name, parent, child, coordinates=None, speeds=None,
parent_joint_pos=None, child_joint_pos=None, parent_axis=None,
child_axis=None):
super().__init__(name, parent, child, coordinates, speeds,
parent_joint_pos, child_joint_pos, parent_axis,
child_axis)
def __str__(self):
return (f'PinJoint: {self.name} parent: {self.parent} '
f'child: {self.child}')
def _generate_coordinates(self, coordinate):
coordinates = []
if coordinate is None:
theta = dynamicsymbols('theta' + '_' + self._name)
coordinate = theta
coordinates.append(coordinate)
return coordinates
def _generate_speeds(self, speed):
speeds = []
if speed is None:
omega = dynamicsymbols('omega' + '_' + self._name)
speed = omega
speeds.append(speed)
return speeds
def _orient_frames(self):
frame = self._set_orientation()
self.child.frame.orient_axis(frame, self.parent_axis,
self.coordinates[0])
def _set_angular_velocity(self):
self.child.frame.set_ang_vel(self.parent.frame, self.speeds[0] *
self.parent_axis.normalize())
def _set_linear_velocity(self):
self.parent_point.set_vel(self.parent.frame, 0)
self.child_point.set_vel(self.child.frame, 0)
self.child_point.set_pos(self.parent_point, 0)
self.child.masscenter.v2pt_theory(self.parent_point,
self.parent.frame, self.child.frame)
class PrismaticJoint(Joint):
"""Prismatic (Sliding) Joint.
.. image:: PrismaticJoint.svg
Explanation
===========
It is defined such that the child body translates with respect to the parent
body along the body fixed parent axis. The location of the joint is defined
by two points in each body which coincides when the generalized coordinate is zero. The direction cosine matrix between
the child and parent is formed using a simple rotation about an axis that is normal to
both ``child_axis`` and ``parent_axis``, see the Notes section for a detailed explanation of
this.
Parameters
==========
name : string
A unique name for the joint.
parent : Body
The parent body of joint.
child : Body
The child body of joint.
coordinates: dynamicsymbol, optional
Generalized coordinates of the joint.
speeds : dynamicsymbol, optional
Generalized speeds of joint.
parent_joint_pos : Vector, optional
Vector from the parent body's mass center to the point where the parent
and child are connected. The default value is the zero vector.
child_joint_pos : Vector, optional
Vector from the child body's mass center to the point where the parent
and child are connected. The default value is the zero vector.
parent_axis : Vector, optional
Axis fixed in the parent body which aligns with an axis fixed in the
child body. The default is x axis in parent's reference frame.
child_axis : Vector, optional
Axis fixed in the child body which aligns with an axis fixed in the
parent body. The default is x axis in child's reference frame.
Attributes
==========
name : string
The joint's name.
parent : Body
The joint's parent body.
child : Body
The joint's child body.
coordinates : list
List of the joint's generalized coordinates.
speeds : list
List of the joint's generalized speeds.
parent_point : Point
The point fixed in the parent body that represents the joint.
child_point : Point
The point fixed in the child body that represents the joint.
parent_axis : Vector
The axis fixed in the parent frame that represents the joint.
child_axis : Vector
The axis fixed in the child frame that represents the joint.
kdes : list
Kinematical differential equations of the joint.
Examples
=========
A single prismatic joint is created from two bodies and has the following basic
attributes:
>>> from sympy.physics.mechanics import Body, PrismaticJoint
>>> parent = Body('P')
>>> parent
P
>>> child = Body('C')
>>> child
C
>>> joint = PrismaticJoint('PC', parent, child)
>>> joint
PrismaticJoint: PC parent: P child: C
>>> joint.name
'PC'
>>> joint.parent
P
>>> joint.child
C
>>> joint.parent_point
PC_P_joint
>>> joint.child_point
PC_C_joint
>>> joint.parent_axis
P_frame.x
>>> joint.child_axis
C_frame.x
>>> joint.coordinates
[x_PC(t)]
>>> joint.speeds
[v_PC(t)]
>>> joint.child.frame.ang_vel_in(joint.parent.frame)
0
>>> joint.child.frame.dcm(joint.parent.frame)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> joint.child_point.pos_from(joint.parent_point)
x_PC(t)*P_frame.x
To further demonstrate the use of the prismatic joint, the kinematics of
two masses sliding, one moving relative to a fixed body and the other relative to the
moving body. about the X axis of each connected body can be created as follows.
>>> from sympy.physics.mechanics import PrismaticJoint, Body
First create bodies to represent the fixed ceiling and one to represent
a particle.
>>> wall = Body('W')
>>> Part1 = Body('P1')
>>> Part2 = Body('P2')
The first joint will connect the particle to the ceiling and the
joint axis will be about the X axis for each body.
>>> J1 = PrismaticJoint('J1', wall, Part1)
The second joint will connect the second particle to the first particle
and the joint axis will also be about the X axis for each body.
>>> J2 = PrismaticJoint('J2', Part1, Part2)
Once the joint is established the kinematics of the connected bodies can
be accessed. First the direction cosine matrices of Part relative
to the ceiling are found:
>>> Part1.dcm(wall)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
>>> Part2.dcm(wall)
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
The position of the particles' masscenter is found with:
>>> Part1.masscenter.pos_from(wall.masscenter)
x_J1(t)*W_frame.x
>>> Part2.masscenter.pos_from(wall.masscenter)
x_J1(t)*W_frame.x + x_J2(t)*P1_frame.x
The angular velocities of the two particle links can be computed with
respect to the ceiling.
>>> Part1.ang_vel_in(wall)
0
>>> Part2.ang_vel_in(wall)
0
And finally, the linear velocities of the two particles can be computed
with respect to the ceiling.
>>> Part1.masscenter_vel(wall)
v_J1(t)*W_frame.x
>>> Part2.masscenter.vel(wall.frame)
v_J1(t)*W_frame.x + Derivative(x_J2(t), t)*P1_frame.x
"""
def __init__(self, name, parent, child, coordinates=None, speeds=None, parent_joint_pos=None,
child_joint_pos=None, parent_axis=None, child_axis=None):
super().__init__(name, parent, child, coordinates, speeds, parent_joint_pos,
child_joint_pos, parent_axis, child_axis)
def __str__(self):
return (f'PrismaticJoint: {self.name} parent: {self.parent} '
f'child: {self.child}')
def _generate_coordinates(self, coordinate):
coordinates = []
if coordinate is None:
x = dynamicsymbols('x' + '_' + self._name)
coordinate = x
coordinates.append(coordinate)
return coordinates
def _generate_speeds(self, speed):
speeds = []
if speed is None:
y = dynamicsymbols('v' + '_' + self._name)
speed = y
speeds.append(speed)
return speeds
def _orient_frames(self):
frame = self._set_orientation()
self.child.frame.orient_axis(frame, self.parent_axis, 0)
def _set_angular_velocity(self):
self.child.frame.set_ang_vel(self.parent.frame, 0)
def _set_linear_velocity(self):
self.parent_point.set_vel(self.parent.frame, 0)
self.child_point.set_vel(self.child.frame, 0)
self.child_point.set_pos(self.parent_point, self.coordinates[0] * self.parent_axis.normalize())
self.child_point.set_vel(self.parent.frame, self.speeds[0] * self.parent_axis.normalize())
self.child.masscenter.set_vel(self.parent.frame, self.speeds[0] * self.parent_axis.normalize())
|
e204cdabf353e9681c7978d749f25cf3dcfe0765d315556082d5fb88f7498a67 | """
Module to handle gamma matrices expressed as tensor objects.
Examples
========
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex
>>> from sympy.tensor.tensor import tensor_indices
>>> i = tensor_indices('i', LorentzIndex)
>>> G(i)
GammaMatrix(i)
Note that there is already an instance of GammaMatrixHead in four dimensions:
GammaMatrix, which is simply declare as
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix
>>> from sympy.tensor.tensor import tensor_indices
>>> i = tensor_indices('i', LorentzIndex)
>>> GammaMatrix(i)
GammaMatrix(i)
To access the metric tensor
>>> LorentzIndex.metric
metric(LorentzIndex,LorentzIndex)
"""
from sympy.core.mul import Mul
from sympy.core.singleton import S
from sympy.matrices.dense import eye
from sympy.matrices.expressions.trace import trace
from sympy.tensor.tensor import TensorIndexType, TensorIndex,\
TensMul, TensAdd, tensor_mul, Tensor, TensorHead, TensorSymmetry
# DiracSpinorIndex = TensorIndexType('DiracSpinorIndex', dim=4, dummy_name="S")
LorentzIndex = TensorIndexType('LorentzIndex', dim=4, dummy_name="L")
GammaMatrix = TensorHead("GammaMatrix", [LorentzIndex],
TensorSymmetry.no_symmetry(1), comm=None)
def extract_type_tens(expression, component):
"""
Extract from a ``TensExpr`` all tensors with `component`.
Returns two tensor expressions:
* the first contains all ``Tensor`` of having `component`.
* the second contains all remaining.
"""
if isinstance(expression, Tensor):
sp = [expression]
elif isinstance(expression, TensMul):
sp = expression.args
else:
raise ValueError('wrong type')
# Collect all gamma matrices of the same dimension
new_expr = S.One
residual_expr = S.One
for i in sp:
if isinstance(i, Tensor) and i.component == component:
new_expr *= i
else:
residual_expr *= i
return new_expr, residual_expr
def simplify_gamma_expression(expression):
extracted_expr, residual_expr = extract_type_tens(expression, GammaMatrix)
res_expr = _simplify_single_line(extracted_expr)
return res_expr * residual_expr
def simplify_gpgp(ex, sort=True):
"""
simplify products ``G(i)*p(-i)*G(j)*p(-j) -> p(i)*p(-i)``
Examples
========
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
LorentzIndex, simplify_gpgp
>>> from sympy.tensor.tensor import tensor_indices, tensor_heads
>>> p, q = tensor_heads('p, q', [LorentzIndex])
>>> i0,i1,i2,i3,i4,i5 = tensor_indices('i0:6', LorentzIndex)
>>> ps = p(i0)*G(-i0)
>>> qs = q(i0)*G(-i0)
>>> simplify_gpgp(ps*qs*qs)
GammaMatrix(-L_0)*p(L_0)*q(L_1)*q(-L_1)
"""
def _simplify_gpgp(ex):
components = ex.components
a = []
comp_map = []
for i, comp in enumerate(components):
comp_map.extend([i]*comp.rank)
dum = [(i[0], i[1], comp_map[i[0]], comp_map[i[1]]) for i in ex.dum]
for i in range(len(components)):
if components[i] != GammaMatrix:
continue
for dx in dum:
if dx[2] == i:
p_pos1 = dx[3]
elif dx[3] == i:
p_pos1 = dx[2]
else:
continue
comp1 = components[p_pos1]
if comp1.comm == 0 and comp1.rank == 1:
a.append((i, p_pos1))
if not a:
return ex
elim = set()
tv = []
hit = True
coeff = S.One
ta = None
while hit:
hit = False
for i, ai in enumerate(a[:-1]):
if ai[0] in elim:
continue
if ai[0] != a[i + 1][0] - 1:
continue
if components[ai[1]] != components[a[i + 1][1]]:
continue
elim.add(ai[0])
elim.add(ai[1])
elim.add(a[i + 1][0])
elim.add(a[i + 1][1])
if not ta:
ta = ex.split()
mu = TensorIndex('mu', LorentzIndex)
hit = True
if i == 0:
coeff = ex.coeff
tx = components[ai[1]](mu)*components[ai[1]](-mu)
if len(a) == 2:
tx *= 4 # eye(4)
tv.append(tx)
break
if tv:
a = [x for j, x in enumerate(ta) if j not in elim]
a.extend(tv)
t = tensor_mul(*a)*coeff
# t = t.replace(lambda x: x.is_Matrix, lambda x: 1)
return t
else:
return ex
if sort:
ex = ex.sorted_components()
# this would be better off with pattern matching
while 1:
t = _simplify_gpgp(ex)
if t != ex:
ex = t
else:
return t
def gamma_trace(t):
"""
trace of a single line of gamma matrices
Examples
========
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
gamma_trace, LorentzIndex
>>> from sympy.tensor.tensor import tensor_indices, tensor_heads
>>> p, q = tensor_heads('p, q', [LorentzIndex])
>>> i0,i1,i2,i3,i4,i5 = tensor_indices('i0:6', LorentzIndex)
>>> ps = p(i0)*G(-i0)
>>> qs = q(i0)*G(-i0)
>>> gamma_trace(G(i0)*G(i1))
4*metric(i0, i1)
>>> gamma_trace(ps*ps) - 4*p(i0)*p(-i0)
0
>>> gamma_trace(ps*qs + ps*ps) - 4*p(i0)*p(-i0) - 4*p(i0)*q(-i0)
0
"""
if isinstance(t, TensAdd):
res = TensAdd(*[gamma_trace(x) for x in t.args])
return res
t = _simplify_single_line(t)
res = _trace_single_line(t)
return res
def _simplify_single_line(expression):
"""
Simplify single-line product of gamma matrices.
Examples
========
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
LorentzIndex, _simplify_single_line
>>> from sympy.tensor.tensor import tensor_indices, TensorHead
>>> p = TensorHead('p', [LorentzIndex])
>>> i0,i1 = tensor_indices('i0:2', LorentzIndex)
>>> _simplify_single_line(G(i0)*G(i1)*p(-i1)*G(-i0)) + 2*G(i0)*p(-i0)
0
"""
t1, t2 = extract_type_tens(expression, GammaMatrix)
if t1 != 1:
t1 = kahane_simplify(t1)
res = t1*t2
return res
def _trace_single_line(t):
"""
Evaluate the trace of a single gamma matrix line inside a ``TensExpr``.
Notes
=====
If there are ``DiracSpinorIndex.auto_left`` and ``DiracSpinorIndex.auto_right``
indices trace over them; otherwise traces are not implied (explain)
Examples
========
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, \
LorentzIndex, _trace_single_line
>>> from sympy.tensor.tensor import tensor_indices, TensorHead
>>> p = TensorHead('p', [LorentzIndex])
>>> i0,i1,i2,i3,i4,i5 = tensor_indices('i0:6', LorentzIndex)
>>> _trace_single_line(G(i0)*G(i1))
4*metric(i0, i1)
>>> _trace_single_line(G(i0)*p(-i0)*G(i1)*p(-i1)) - 4*p(i0)*p(-i0)
0
"""
def _trace_single_line1(t):
t = t.sorted_components()
components = t.components
ncomps = len(components)
g = LorentzIndex.metric
# gamma matirices are in a[i:j]
hit = 0
for i in range(ncomps):
if components[i] == GammaMatrix:
hit = 1
break
for j in range(i + hit, ncomps):
if components[j] != GammaMatrix:
break
else:
j = ncomps
numG = j - i
if numG == 0:
tcoeff = t.coeff
return t.nocoeff if tcoeff else t
if numG % 2 == 1:
return TensMul.from_data(S.Zero, [], [], [])
elif numG > 4:
# find the open matrix indices and connect them:
a = t.split()
ind1 = a[i].get_indices()[0]
ind2 = a[i + 1].get_indices()[0]
aa = a[:i] + a[i + 2:]
t1 = tensor_mul(*aa)*g(ind1, ind2)
t1 = t1.contract_metric(g)
args = [t1]
sign = 1
for k in range(i + 2, j):
sign = -sign
ind2 = a[k].get_indices()[0]
aa = a[:i] + a[i + 1:k] + a[k + 1:]
t2 = sign*tensor_mul(*aa)*g(ind1, ind2)
t2 = t2.contract_metric(g)
t2 = simplify_gpgp(t2, False)
args.append(t2)
t3 = TensAdd(*args)
t3 = _trace_single_line(t3)
return t3
else:
a = t.split()
t1 = _gamma_trace1(*a[i:j])
a2 = a[:i] + a[j:]
t2 = tensor_mul(*a2)
t3 = t1*t2
if not t3:
return t3
t3 = t3.contract_metric(g)
return t3
t = t.expand()
if isinstance(t, TensAdd):
a = [_trace_single_line1(x)*x.coeff for x in t.args]
return TensAdd(*a)
elif isinstance(t, (Tensor, TensMul)):
r = t.coeff*_trace_single_line1(t)
return r
else:
return trace(t)
def _gamma_trace1(*a):
gctr = 4 # FIXME specific for d=4
g = LorentzIndex.metric
if not a:
return gctr
n = len(a)
if n%2 == 1:
#return TensMul.from_data(S.Zero, [], [], [])
return S.Zero
if n == 2:
ind0 = a[0].get_indices()[0]
ind1 = a[1].get_indices()[0]
return gctr*g(ind0, ind1)
if n == 4:
ind0 = a[0].get_indices()[0]
ind1 = a[1].get_indices()[0]
ind2 = a[2].get_indices()[0]
ind3 = a[3].get_indices()[0]
return gctr*(g(ind0, ind1)*g(ind2, ind3) - \
g(ind0, ind2)*g(ind1, ind3) + g(ind0, ind3)*g(ind1, ind2))
def kahane_simplify(expression):
r"""
This function cancels contracted elements in a product of four
dimensional gamma matrices, resulting in an expression equal to the given
one, without the contracted gamma matrices.
Parameters
==========
`expression` the tensor expression containing the gamma matrices to simplify.
Notes
=====
If spinor indices are given, the matrices must be given in
the order given in the product.
Algorithm
=========
The idea behind the algorithm is to use some well-known identities,
i.e., for contractions enclosing an even number of `\gamma` matrices
`\gamma^\mu \gamma_{a_1} \cdots \gamma_{a_{2N}} \gamma_\mu = 2 (\gamma_{a_{2N}} \gamma_{a_1} \cdots \gamma_{a_{2N-1}} + \gamma_{a_{2N-1}} \cdots \gamma_{a_1} \gamma_{a_{2N}} )`
for an odd number of `\gamma` matrices
`\gamma^\mu \gamma_{a_1} \cdots \gamma_{a_{2N+1}} \gamma_\mu = -2 \gamma_{a_{2N+1}} \gamma_{a_{2N}} \cdots \gamma_{a_{1}}`
Instead of repeatedly applying these identities to cancel out all contracted indices,
it is possible to recognize the links that would result from such an operation,
the problem is thus reduced to a simple rearrangement of free gamma matrices.
Examples
========
When using, always remember that the original expression coefficient
has to be handled separately
>>> from sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex
>>> from sympy.physics.hep.gamma_matrices import kahane_simplify
>>> from sympy.tensor.tensor import tensor_indices
>>> i0, i1, i2 = tensor_indices('i0:3', LorentzIndex)
>>> ta = G(i0)*G(-i0)
>>> kahane_simplify(ta)
Matrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 4, 0],
[0, 0, 0, 4]])
>>> tb = G(i0)*G(i1)*G(-i0)
>>> kahane_simplify(tb)
-2*GammaMatrix(i1)
>>> t = G(i0)*G(-i0)
>>> kahane_simplify(t)
Matrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 4, 0],
[0, 0, 0, 4]])
>>> t = G(i0)*G(-i0)
>>> kahane_simplify(t)
Matrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 4, 0],
[0, 0, 0, 4]])
If there are no contractions, the same expression is returned
>>> tc = G(i0)*G(i1)
>>> kahane_simplify(tc)
GammaMatrix(i0)*GammaMatrix(i1)
References
==========
[1] Algorithm for Reducing Contracted Products of gamma Matrices,
Joseph Kahane, Journal of Mathematical Physics, Vol. 9, No. 10, October 1968.
"""
if isinstance(expression, Mul):
return expression
if isinstance(expression, TensAdd):
return TensAdd(*[kahane_simplify(arg) for arg in expression.args])
if isinstance(expression, Tensor):
return expression
assert isinstance(expression, TensMul)
gammas = expression.args
for gamma in gammas:
assert gamma.component == GammaMatrix
free = expression.free
# spinor_free = [_ for _ in expression.free_in_args if _[1] != 0]
# if len(spinor_free) == 2:
# spinor_free.sort(key=lambda x: x[2])
# assert spinor_free[0][1] == 1 and spinor_free[-1][1] == 2
# assert spinor_free[0][2] == 0
# elif spinor_free:
# raise ValueError('spinor indices do not match')
dum = []
for dum_pair in expression.dum:
if expression.index_types[dum_pair[0]] == LorentzIndex:
dum.append((dum_pair[0], dum_pair[1]))
dum = sorted(dum)
if len(dum) == 0: # or GammaMatrixHead:
# no contractions in `expression`, just return it.
return expression
# find the `first_dum_pos`, i.e. the position of the first contracted
# gamma matrix, Kahane's algorithm as described in his paper requires the
# gamma matrix expression to start with a contracted gamma matrix, this is
# a workaround which ignores possible initial free indices, and re-adds
# them later.
first_dum_pos = min(map(min, dum))
# for p1, p2, a1, a2 in expression.dum_in_args:
# if p1 != 0 or p2 != 0:
# # only Lorentz indices, skip Dirac indices:
# continue
# first_dum_pos = min(p1, p2)
# break
total_number = len(free) + len(dum)*2
number_of_contractions = len(dum)
free_pos = [None]*total_number
for i in free:
free_pos[i[1]] = i[0]
# `index_is_free` is a list of booleans, to identify index position
# and whether that index is free or dummy.
index_is_free = [False]*total_number
for i, indx in enumerate(free):
index_is_free[indx[1]] = True
# `links` is a dictionary containing the graph described in Kahane's paper,
# to every key correspond one or two values, representing the linked indices.
# All values in `links` are integers, negative numbers are used in the case
# where it is necessary to insert gamma matrices between free indices, in
# order to make Kahane's algorithm work (see paper).
links = {i: [] for i in range(first_dum_pos, total_number)}
# `cum_sign` is a step variable to mark the sign of every index, see paper.
cum_sign = -1
# `cum_sign_list` keeps storage for all `cum_sign` (every index).
cum_sign_list = [None]*total_number
block_free_count = 0
# multiply `resulting_coeff` by the coefficient parameter, the rest
# of the algorithm ignores a scalar coefficient.
resulting_coeff = S.One
# initialize a list of lists of indices. The outer list will contain all
# additive tensor expressions, while the inner list will contain the
# free indices (rearranged according to the algorithm).
resulting_indices = [[]]
# start to count the `connected_components`, which together with the number
# of contractions, determines a -1 or +1 factor to be multiplied.
connected_components = 1
# First loop: here we fill `cum_sign_list`, and draw the links
# among consecutive indices (they are stored in `links`). Links among
# non-consecutive indices will be drawn later.
for i, is_free in enumerate(index_is_free):
# if `expression` starts with free indices, they are ignored here;
# they are later added as they are to the beginning of all
# `resulting_indices` list of lists of indices.
if i < first_dum_pos:
continue
if is_free:
block_free_count += 1
# if previous index was free as well, draw an arch in `links`.
if block_free_count > 1:
links[i - 1].append(i)
links[i].append(i - 1)
else:
# Change the sign of the index (`cum_sign`) if the number of free
# indices preceding it is even.
cum_sign *= 1 if (block_free_count % 2) else -1
if block_free_count == 0 and i != first_dum_pos:
# check if there are two consecutive dummy indices:
# in this case create virtual indices with negative position,
# these "virtual" indices represent the insertion of two
# gamma^0 matrices to separate consecutive dummy indices, as
# Kahane's algorithm requires dummy indices to be separated by
# free indices. The product of two gamma^0 matrices is unity,
# so the new expression being examined is the same as the
# original one.
if cum_sign == -1:
links[-1-i] = [-1-i+1]
links[-1-i+1] = [-1-i]
if (i - cum_sign) in links:
if i != first_dum_pos:
links[i].append(i - cum_sign)
if block_free_count != 0:
if i - cum_sign < len(index_is_free):
if index_is_free[i - cum_sign]:
links[i - cum_sign].append(i)
block_free_count = 0
cum_sign_list[i] = cum_sign
# The previous loop has only created links between consecutive free indices,
# it is necessary to properly create links among dummy (contracted) indices,
# according to the rules described in Kahane's paper. There is only one exception
# to Kahane's rules: the negative indices, which handle the case of some
# consecutive free indices (Kahane's paper just describes dummy indices
# separated by free indices, hinting that free indices can be added without
# altering the expression result).
for i in dum:
# get the positions of the two contracted indices:
pos1 = i[0]
pos2 = i[1]
# create Kahane's upper links, i.e. the upper arcs between dummy
# (i.e. contracted) indices:
links[pos1].append(pos2)
links[pos2].append(pos1)
# create Kahane's lower links, this corresponds to the arcs below
# the line described in the paper:
# first we move `pos1` and `pos2` according to the sign of the indices:
linkpos1 = pos1 + cum_sign_list[pos1]
linkpos2 = pos2 + cum_sign_list[pos2]
# otherwise, perform some checks before creating the lower arcs:
# make sure we are not exceeding the total number of indices:
if linkpos1 >= total_number:
continue
if linkpos2 >= total_number:
continue
# make sure we are not below the first dummy index in `expression`:
if linkpos1 < first_dum_pos:
continue
if linkpos2 < first_dum_pos:
continue
# check if the previous loop created "virtual" indices between dummy
# indices, in such a case relink `linkpos1` and `linkpos2`:
if (-1-linkpos1) in links:
linkpos1 = -1-linkpos1
if (-1-linkpos2) in links:
linkpos2 = -1-linkpos2
# move only if not next to free index:
if linkpos1 >= 0 and not index_is_free[linkpos1]:
linkpos1 = pos1
if linkpos2 >=0 and not index_is_free[linkpos2]:
linkpos2 = pos2
# create the lower arcs:
if linkpos2 not in links[linkpos1]:
links[linkpos1].append(linkpos2)
if linkpos1 not in links[linkpos2]:
links[linkpos2].append(linkpos1)
# This loop starts from the `first_dum_pos` index (first dummy index)
# walks through the graph deleting the visited indices from `links`,
# it adds a gamma matrix for every free index in encounters, while it
# completely ignores dummy indices and virtual indices.
pointer = first_dum_pos
previous_pointer = 0
while True:
if pointer in links:
next_ones = links.pop(pointer)
else:
break
if previous_pointer in next_ones:
next_ones.remove(previous_pointer)
previous_pointer = pointer
if next_ones:
pointer = next_ones[0]
else:
break
if pointer == previous_pointer:
break
if pointer >=0 and free_pos[pointer] is not None:
for ri in resulting_indices:
ri.append(free_pos[pointer])
# The following loop removes the remaining connected components in `links`.
# If there are free indices inside a connected component, it gives a
# contribution to the resulting expression given by the factor
# `gamma_a gamma_b ... gamma_z + gamma_z ... gamma_b gamma_a`, in Kahanes's
# paper represented as {gamma_a, gamma_b, ... , gamma_z},
# virtual indices are ignored. The variable `connected_components` is
# increased by one for every connected component this loop encounters.
# If the connected component has virtual and dummy indices only
# (no free indices), it contributes to `resulting_indices` by a factor of two.
# The multiplication by two is a result of the
# factor {gamma^0, gamma^0} = 2 I, as it appears in Kahane's paper.
# Note: curly brackets are meant as in the paper, as a generalized
# multi-element anticommutator!
while links:
connected_components += 1
pointer = min(links.keys())
previous_pointer = pointer
# the inner loop erases the visited indices from `links`, and it adds
# all free indices to `prepend_indices` list, virtual indices are
# ignored.
prepend_indices = []
while True:
if pointer in links:
next_ones = links.pop(pointer)
else:
break
if previous_pointer in next_ones:
if len(next_ones) > 1:
next_ones.remove(previous_pointer)
previous_pointer = pointer
if next_ones:
pointer = next_ones[0]
if pointer >= first_dum_pos and free_pos[pointer] is not None:
prepend_indices.insert(0, free_pos[pointer])
# if `prepend_indices` is void, it means there are no free indices
# in the loop (and it can be shown that there must be a virtual index),
# loops of virtual indices only contribute by a factor of two:
if len(prepend_indices) == 0:
resulting_coeff *= 2
# otherwise, add the free indices in `prepend_indices` to
# the `resulting_indices`:
else:
expr1 = prepend_indices
expr2 = list(reversed(prepend_indices))
resulting_indices = [expri + ri for ri in resulting_indices for expri in (expr1, expr2)]
# sign correction, as described in Kahane's paper:
resulting_coeff *= -1 if (number_of_contractions - connected_components + 1) % 2 else 1
# power of two factor, as described in Kahane's paper:
resulting_coeff *= 2**(number_of_contractions)
# If `first_dum_pos` is not zero, it means that there are trailing free gamma
# matrices in front of `expression`, so multiply by them:
resulting_indices = list( free_pos[0:first_dum_pos] + ri for ri in resulting_indices )
resulting_expr = S.Zero
for i in resulting_indices:
temp_expr = S.One
for j in i:
temp_expr *= GammaMatrix(j)
resulting_expr += temp_expr
t = resulting_coeff * resulting_expr
t1 = None
if isinstance(t, TensAdd):
t1 = t.args[0]
elif isinstance(t, TensMul):
t1 = t
if t1:
pass
else:
t = eye(4)*t
return t
|
a93ee68eebb6dfa37875b4eec69fef009f8333cf0e3d83662d33f41459a51b10 | """
This module can be used to solve 2D beam bending problems with
singularity functions in mechanics.
"""
from sympy.core import S, Symbol, diff, symbols
from sympy.core.add import Add
from sympy.core.expr import Expr
from sympy.core.function import (Derivative, Function)
from sympy.core.mul import Mul
from sympy.core.relational import Eq
from sympy.core.sympify import sympify
from sympy.solvers import linsolve
from sympy.solvers.ode.ode import dsolve
from sympy.solvers.solvers import solve
from sympy.printing import sstr
from sympy.functions import SingularityFunction, Piecewise, factorial
from sympy.integrals import integrate
from sympy.series import limit
from sympy.plotting import plot, PlotGrid
from sympy.geometry.entity import GeometryEntity
from sympy.external import import_module
from sympy.sets.sets import Interval
from sympy.utilities.lambdify import lambdify
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.iterables import iterable
numpy = import_module('numpy', import_kwargs={'fromlist':['arange']})
class Beam:
"""
A Beam is a structural element that is capable of withstanding load
primarily by resisting against bending. Beams are characterized by
their cross sectional profile(Second moment of area), their length
and their material.
.. note::
A consistent sign convention must be used while solving a beam
bending problem; the results will
automatically follow the chosen sign convention. However, the
chosen sign convention must respect the rule that, on the positive
side of beam's axis (in respect to current section), a loading force
giving positive shear yields a negative moment, as below (the
curved arrow shows the positive moment and rotation):
.. image:: allowed-sign-conventions.png
Examples
========
There is a beam of length 4 meters. A constant distributed load of 6 N/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. The deflection of the beam at the end is restricted.
Using the sign convention of downwards forces being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols, Piecewise
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(4, E, I)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(6, 2, 0)
>>> b.apply_load(R2, 4, -1)
>>> b.bc_deflection = [(0, 0), (4, 0)]
>>> b.boundary_conditions
{'deflection': [(0, 0), (4, 0)], 'slope': []}
>>> b.load
R1*SingularityFunction(x, 0, -1) + R2*SingularityFunction(x, 4, -1) + 6*SingularityFunction(x, 2, 0)
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.load
-3*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 2, 0) - 9*SingularityFunction(x, 4, -1)
>>> b.shear_force()
3*SingularityFunction(x, 0, 0) - 6*SingularityFunction(x, 2, 1) + 9*SingularityFunction(x, 4, 0)
>>> b.bending_moment()
3*SingularityFunction(x, 0, 1) - 3*SingularityFunction(x, 2, 2) + 9*SingularityFunction(x, 4, 1)
>>> b.slope()
(-3*SingularityFunction(x, 0, 2)/2 + SingularityFunction(x, 2, 3) - 9*SingularityFunction(x, 4, 2)/2 + 7)/(E*I)
>>> b.deflection()
(7*x - SingularityFunction(x, 0, 3)/2 + SingularityFunction(x, 2, 4)/4 - 3*SingularityFunction(x, 4, 3)/2)/(E*I)
>>> b.deflection().rewrite(Piecewise)
(7*x - Piecewise((x**3, x > 0), (0, True))/2
- 3*Piecewise(((x - 4)**3, x > 4), (0, True))/2
+ Piecewise(((x - 2)**4, x > 2), (0, True))/4)/(E*I)
"""
def __init__(self, length, elastic_modulus, second_moment, area=Symbol('A'), variable=Symbol('x'), base_char='C'):
"""Initializes the class.
Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material. It can
also be a continuous function of position along the beam.
second_moment : Sympifyable or Geometry object
Describes the cross-section of the beam via a SymPy expression
representing the Beam's second moment of area. It is a geometrical
property of an area which reflects how its points are distributed
with respect to its neutral axis. It can also be a continuous
function of position along the beam. Alternatively ``second_moment``
can be a shape object such as a ``Polygon`` from the geometry module
representing the shape of the cross-section of the beam. In such cases,
it is assumed that the x-axis of the shape object is aligned with the
bending axis of the beam. The second moment of area will be computed
from the shape object internally.
area : Symbol/float
Represents the cross-section area of beam
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to ``Symbol('x')``.
base_char : String, optional
A String that will be used as base character to generate sequential
symbols for integration constants in cases where boundary conditions
are not sufficient to solve them.
"""
self.length = length
self.elastic_modulus = elastic_modulus
if isinstance(second_moment, GeometryEntity):
self.cross_section = second_moment
else:
self.cross_section = None
self.second_moment = second_moment
self.variable = variable
self._base_char = base_char
self._boundary_conditions = {'deflection': [], 'slope': []}
self._load = 0
self.area = area
self._applied_supports = []
self._support_as_loads = []
self._applied_loads = []
self._reaction_loads = {}
self._ild_reactions = {}
self._ild_shear = 0
self._ild_moment = 0
# _original_load is a copy of _load equations with unsubstituted reaction
# forces. It is used for calculating reaction forces in case of I.L.D.
self._original_load = 0
self._composite_type = None
self._hinge_position = None
def __str__(self):
shape_description = self._cross_section if self._cross_section else self._second_moment
str_sol = 'Beam({}, {}, {})'.format(sstr(self._length), sstr(self._elastic_modulus), sstr(shape_description))
return str_sol
@property
def reaction_loads(self):
""" Returns the reaction forces in a dictionary."""
return self._reaction_loads
@property
def ild_shear(self):
""" Returns the I.L.D. shear equation."""
return self._ild_shear
@property
def ild_reactions(self):
""" Returns the I.L.D. reaction forces in a dictionary."""
return self._ild_reactions
@property
def ild_moment(self):
""" Returns the I.L.D. moment equation."""
return self._ild_moment
@property
def length(self):
"""Length of the Beam."""
return self._length
@length.setter
def length(self, l):
self._length = sympify(l)
@property
def area(self):
"""Cross-sectional area of the Beam. """
return self._area
@area.setter
def area(self, a):
self._area = sympify(a)
@property
def variable(self):
"""
A symbol that can be used as a variable along the length of the beam
while representing load distribution, shear force curve, bending
moment, slope curve and the deflection curve. By default, it is set
to ``Symbol('x')``, but this property is mutable.
Examples
========
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I, A = symbols('E, I, A')
>>> x, y, z = symbols('x, y, z')
>>> b = Beam(4, E, I)
>>> b.variable
x
>>> b.variable = y
>>> b.variable
y
>>> b = Beam(4, E, I, A, z)
>>> b.variable
z
"""
return self._variable
@variable.setter
def variable(self, v):
if isinstance(v, Symbol):
self._variable = v
else:
raise TypeError("""The variable should be a Symbol object.""")
@property
def elastic_modulus(self):
"""Young's Modulus of the Beam. """
return self._elastic_modulus
@elastic_modulus.setter
def elastic_modulus(self, e):
self._elastic_modulus = sympify(e)
@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment
@second_moment.setter
def second_moment(self, i):
self._cross_section = None
if isinstance(i, GeometryEntity):
raise ValueError("To update cross-section geometry use `cross_section` attribute")
else:
self._second_moment = sympify(i)
@property
def cross_section(self):
"""Cross-section of the beam"""
return self._cross_section
@cross_section.setter
def cross_section(self, s):
if s:
self._second_moment = s.second_moment_of_area()[0]
self._cross_section = s
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has three keywords namely moment, slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains location and value of a boundary condition in the format
(location, value).
Examples
========
There is a beam of length 4 meters. The bending moment at 0 should be 4
and at 4 it should be 0. The slope of the beam should be 1 at 0. The
deflection should be 2 at 0.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.bc_deflection = [(0, 2)]
>>> b.bc_slope = [(0, 1)]
>>> b.boundary_conditions
{'deflection': [(0, 2)], 'slope': [(0, 1)]}
Here the deflection of the beam should be ``2`` at ``0``.
Similarly, the slope of the beam should be ``1`` at ``0``.
"""
return self._boundary_conditions
@property
def bc_slope(self):
return self._boundary_conditions['slope']
@bc_slope.setter
def bc_slope(self, s_bcs):
self._boundary_conditions['slope'] = s_bcs
@property
def bc_deflection(self):
return self._boundary_conditions['deflection']
@bc_deflection.setter
def bc_deflection(self, d_bcs):
self._boundary_conditions['deflection'] = d_bcs
def join(self, beam, via="fixed"):
"""
This method joins two beams to make a new composite beam system.
Passed Beam class instance is attached to the right end of calling
object. This method can be used to form beams having Discontinuous
values of Elastic modulus or Second moment.
Parameters
==========
beam : Beam class object
The Beam object which would be connected to the right of calling
object.
via : String
States the way two Beam object would get connected
- For axially fixed Beams, via="fixed"
- For Beams connected via hinge, via="hinge"
Examples
========
There is a cantilever beam of length 4 meters. For first 2 meters
its moment of inertia is `1.5*I` and `I` for the other end.
A pointload of magnitude 4 N is applied from the top at its free end.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b1 = Beam(2, E, 1.5*I)
>>> b2 = Beam(2, E, I)
>>> b = b1.join(b2, "fixed")
>>> b.apply_load(20, 4, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 0, -2)
>>> b.bc_slope = [(0, 0)]
>>> b.bc_deflection = [(0, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.load
80*SingularityFunction(x, 0, -2) - 20*SingularityFunction(x, 0, -1) + 20*SingularityFunction(x, 4, -1)
>>> b.slope()
(-((-80*SingularityFunction(x, 0, 1) + 10*SingularityFunction(x, 0, 2) - 10*SingularityFunction(x, 4, 2))/I + 120/I)/E + 80.0/(E*I))*SingularityFunction(x, 2, 0)
- 0.666666666666667*(-80*SingularityFunction(x, 0, 1) + 10*SingularityFunction(x, 0, 2) - 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 0, 0)/(E*I)
+ 0.666666666666667*(-80*SingularityFunction(x, 0, 1) + 10*SingularityFunction(x, 0, 2) - 10*SingularityFunction(x, 4, 2))*SingularityFunction(x, 2, 0)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
new_length = self.length + beam.length
if self.second_moment != beam.second_moment:
new_second_moment = Piecewise((self.second_moment, x<=self.length),
(beam.second_moment, x<=new_length))
else:
new_second_moment = self.second_moment
if via == "fixed":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "fixed"
return new_beam
if via == "hinge":
new_beam = Beam(new_length, E, new_second_moment, x)
new_beam._composite_type = "hinge"
new_beam._hinge_position = self.length
return new_beam
def apply_support(self, loc, type="fixed"):
"""
This method applies support to a particular beam object.
Parameters
==========
loc : Sympifyable
Location of point at which support is applied.
type : String
Determines type of Beam support applied. To apply support structure
with
- zero degree of freedom, type = "fixed"
- one degree of freedom, type = "pin"
- two degrees of freedom, type = "roller"
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(30, E, I)
>>> b.apply_support(10, 'roller')
>>> b.apply_support(30, 'roller')
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(120, 30, -2)
>>> R_10, R_30 = symbols('R_10, R_30')
>>> b.solve_for_reaction_loads(R_10, R_30)
>>> b.load
-8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
>>> b.slope()
(-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)
"""
loc = sympify(loc)
self._applied_supports.append((loc, type))
if type in ("pin", "roller"):
reaction_load = Symbol('R_'+str(loc))
self.apply_load(reaction_load, loc, -1)
self.bc_deflection.append((loc, 0))
else:
reaction_load = Symbol('R_'+str(loc))
reaction_moment = Symbol('M_'+str(loc))
self.apply_load(reaction_load, loc, -1)
self.apply_load(reaction_moment, loc, -2)
self.bc_deflection.append((loc, 0))
self.bc_slope.append((loc, 0))
self._support_as_loads.append((reaction_moment, loc, -2, None))
self._support_as_loads.append((reaction_load, loc, -1, None))
def apply_load(self, value, start, order, end=None):
"""
This method adds up the loads given to a particular beam object.
Parameters
==========
value : Sympifyable
The value inserted should have the units [Force/(Distance**(n+1)]
where n is the order of applied load.
Units for applied loads:
- For moments, unit = kN*m
- For point loads, unit = kN
- For constant distributed load, unit = kN/m
- For ramp loads, unit = kN/m/m
- For parabolic ramp loads, unit = kN/m/m/m
- ... so on.
start : Sympifyable
The starting point of the applied load. For point moments and
point forces this is the location of application.
order : Integer
The order of the applied load.
- For moments, order = -2
- For point loads, order =-1
- For constant distributed load, order = 0
- For ramp loads, order = 1
- For parabolic ramp loads, order = 2
- ... so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point
within the length of the beam.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
self._applied_loads.append((value, start, order, end))
self._load += value*SingularityFunction(x, start, order)
self._original_load += value*SingularityFunction(x, start, order)
if end:
# load has an end point within the length of the beam.
self._handle_end(x, value, start, order, end, type="apply")
def remove_load(self, value, start, order, end=None):
"""
This method removes a particular load present on the beam object.
Returns a ValueError if the load passed as an argument is not
present on the beam.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
start : Sympifyable
The starting point of the applied load. For point moments and
point forces this is the location of application.
order : Integer
The order of the applied load.
- For moments, order= -2
- For point loads, order=-1
- For constant distributed load, order=0
- For ramp loads, order=1
- For parabolic ramp loads, order=2
- ... so on.
end : Sympifyable, optional
An optional argument that can be used if the load has an end point
within the length of the beam.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 2 meters to 3 meters
away from the starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 2, 2, end=3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) + 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
>>> b.remove_load(-2, 2, 2, end = 3)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if (value, start, order, end) in self._applied_loads:
self._load -= value*SingularityFunction(x, start, order)
self._original_load -= value*SingularityFunction(x, start, order)
self._applied_loads.remove((value, start, order, end))
else:
msg = "No such load distribution exists on the beam object."
raise ValueError(msg)
if end:
# load has an end point within the length of the beam.
self._handle_end(x, value, start, order, end, type="remove")
def _handle_end(self, x, value, start, order, end, type):
"""
This functions handles the optional `end` value in the
`apply_load` and `remove_load` functions. When the value
of end is not NULL, this function will be executed.
"""
if order.is_negative:
msg = ("If 'end' is provided the 'order' of the load cannot "
"be negative, i.e. 'end' is only valid for distributed "
"loads.")
raise ValueError(msg)
# NOTE : A Taylor series can be used to define the summation of
# singularity functions that subtract from the load past the end
# point such that it evaluates to zero past 'end'.
f = value*x**order
if type == "apply":
# iterating for "apply_load" method
for i in range(0, order + 1):
self._load -= (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i)/factorial(i))
self._original_load -= (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i)/factorial(i))
elif type == "remove":
# iterating for "remove_load" method
for i in range(0, order + 1):
self._load += (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i)/factorial(i))
self._original_load += (f.diff(x, i).subs(x, end - start) *
SingularityFunction(x, end, i)/factorial(i))
@property
def load(self):
"""
Returns a Singularity Function expression which represents
the load distribution curve of the Beam object.
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A point load of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point and a parabolic ramp load of magnitude
2 N/m is applied below the beam starting from 3 meters away from the
starting point of the beam.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(-2, 3, 2)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1) - 2*SingularityFunction(x, 3, 2)
"""
return self._load
@property
def applied_loads(self):
"""
Returns a list of all loads applied on the beam object.
Each load in the list is a tuple of form (value, start, order, end).
Examples
========
There is a beam of length 4 meters. A moment of magnitude 3 Nm is
applied in the clockwise direction at the starting point of the beam.
A pointload of magnitude 4 N is applied from the top of the beam at
2 meters from the starting point. Another pointload of magnitude 5 N
is applied at same position.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(4, E, I)
>>> b.apply_load(-3, 0, -2)
>>> b.apply_load(4, 2, -1)
>>> b.apply_load(5, 2, -1)
>>> b.load
-3*SingularityFunction(x, 0, -2) + 9*SingularityFunction(x, 2, -1)
>>> b.applied_loads
[(-3, 0, -2, None), (4, 2, -1, None), (5, 2, -1, None)]
"""
return self._applied_loads
def _solve_hinge_beams(self, *reactions):
"""Method to find integration constants and reactional variables in a
composite beam connected via hinge.
This method resolves the composite Beam into its sub-beams and then
equations of shear force, bending moment, slope and deflection are
evaluated for both of them separately. These equations are then solved
for unknown reactions and integration constants using the boundary
conditions applied on the Beam. Equal deflection of both sub-beams
at the hinge joint gives us another equation to solve the system.
Examples
========
A combined beam, with constant fkexural rigidity E*I, is formed by joining
a Beam of length 2*l to the right of another Beam of length l. The whole beam
is fixed at both of its both end. A point load of magnitude P is also applied
from the top at a distance of 2*l from starting point.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> l=symbols('l', positive=True)
>>> b1=Beam(l, E, I)
>>> b2=Beam(2*l, E, I)
>>> b=b1.join(b2,"hinge")
>>> M1, A1, M2, A2, P = symbols('M1 A1 M2 A2 P')
>>> b.apply_load(A1,0,-1)
>>> b.apply_load(M1,0,-2)
>>> b.apply_load(P,2*l,-1)
>>> b.apply_load(A2,3*l,-1)
>>> b.apply_load(M2,3*l,-2)
>>> b.bc_slope=[(0,0), (3*l, 0)]
>>> b.bc_deflection=[(0,0), (3*l, 0)]
>>> b.solve_for_reaction_loads(M1, A1, M2, A2)
>>> b.reaction_loads
{A1: -5*P/18, A2: -13*P/18, M1: 5*P*l/18, M2: -4*P*l/9}
>>> b.slope()
(5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 1)/18 - 5*P*SingularityFunction(x, 0, 2)/36 + 5*P*SingularityFunction(x, l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
+ (P*l**2/18 - 4*P*l*SingularityFunction(-l + x, 2*l, 1)/9 - 5*P*SingularityFunction(-l + x, 0, 2)/36 + P*SingularityFunction(-l + x, l, 2)/2
- 13*P*SingularityFunction(-l + x, 2*l, 2)/36)*SingularityFunction(x, l, 0)/(E*I)
>>> b.deflection()
(5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, 0, 0)/(E*I)
- (5*P*l*SingularityFunction(x, 0, 2)/36 - 5*P*SingularityFunction(x, 0, 3)/108 + 5*P*SingularityFunction(x, l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
+ (5*P*l**3/54 + P*l**2*(-l + x)/18 - 2*P*l*SingularityFunction(-l + x, 2*l, 2)/9 - 5*P*SingularityFunction(-l + x, 0, 3)/108 + P*SingularityFunction(-l + x, l, 3)/6
- 13*P*SingularityFunction(-l + x, 2*l, 3)/108)*SingularityFunction(x, l, 0)/(E*I)
"""
x = self.variable
l = self._hinge_position
E = self._elastic_modulus
I = self._second_moment
if isinstance(I, Piecewise):
I1 = I.args[0][0]
I2 = I.args[1][0]
else:
I1 = I2 = I
load_1 = 0 # Load equation on first segment of composite beam
load_2 = 0 # Load equation on second segment of composite beam
# Distributing load on both segments
for load in self.applied_loads:
if load[1] < l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
if load[2] == 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2])
elif load[2] > 0:
load_1 -= load[0]*SingularityFunction(x, load[3], load[2]) + load[0]*SingularityFunction(x, load[3], 0)
elif load[1] == l:
load_1 += load[0]*SingularityFunction(x, load[1], load[2])
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
elif load[1] > l:
load_2 += load[0]*SingularityFunction(x, load[1] - l, load[2])
if load[2] == 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2])
elif load[2] > 0:
load_2 -= load[0]*SingularityFunction(x, load[3] - l, load[2]) + load[0]*SingularityFunction(x, load[3] - l, 0)
h = Symbol('h') # Force due to hinge
load_1 += h*SingularityFunction(x, l, -1)
load_2 -= h*SingularityFunction(x, 0, -1)
eq = []
shear_1 = integrate(load_1, x)
shear_curve_1 = limit(shear_1, x, l)
eq.append(shear_curve_1)
bending_1 = integrate(shear_1, x)
moment_curve_1 = limit(bending_1, x, l)
eq.append(moment_curve_1)
shear_2 = integrate(load_2, x)
shear_curve_2 = limit(shear_2, x, self.length - l)
eq.append(shear_curve_2)
bending_2 = integrate(shear_2, x)
moment_curve_2 = limit(bending_2, x, self.length - l)
eq.append(moment_curve_2)
C1 = Symbol('C1')
C2 = Symbol('C2')
C3 = Symbol('C3')
C4 = Symbol('C4')
slope_1 = S.One/(E*I1)*(integrate(bending_1, x) + C1)
def_1 = S.One/(E*I1)*(integrate((E*I)*slope_1, x) + C1*x + C2)
slope_2 = S.One/(E*I2)*(integrate(integrate(integrate(load_2, x), x), x) + C3)
def_2 = S.One/(E*I2)*(integrate((E*I)*slope_2, x) + C4)
for position, value in self.bc_slope:
if position<l:
eq.append(slope_1.subs(x, position) - value)
else:
eq.append(slope_2.subs(x, position - l) - value)
for position, value in self.bc_deflection:
if position<l:
eq.append(def_1.subs(x, position) - value)
else:
eq.append(def_2.subs(x, position - l) - value)
eq.append(def_1.subs(x, l) - def_2.subs(x, 0)) # Deflection of both the segments at hinge would be equal
constants = list(linsolve(eq, C1, C2, C3, C4, h, *reactions))
reaction_values = list(constants[0])[5:]
self._reaction_loads = dict(zip(reactions, reaction_values))
self._load = self._load.subs(self._reaction_loads)
# Substituting constants and reactional load and moments with their corresponding values
slope_1 = slope_1.subs({C1: constants[0][0], h:constants[0][4]}).subs(self._reaction_loads)
def_1 = def_1.subs({C1: constants[0][0], C2: constants[0][1], h:constants[0][4]}).subs(self._reaction_loads)
slope_2 = slope_2.subs({x: x-l, C3: constants[0][2], h:constants[0][4]}).subs(self._reaction_loads)
def_2 = def_2.subs({x: x-l,C3: constants[0][2], C4: constants[0][3], h:constants[0][4]}).subs(self._reaction_loads)
self._hinge_beam_slope = slope_1*SingularityFunction(x, 0, 0) - slope_1*SingularityFunction(x, l, 0) + slope_2*SingularityFunction(x, l, 0)
self._hinge_beam_deflection = def_1*SingularityFunction(x, 0, 0) - def_1*SingularityFunction(x, l, 0) + def_2*SingularityFunction(x, l, 0)
def solve_for_reaction_loads(self, *reactions):
"""
Solves for the reaction forces.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1) # Reaction force at x = 10
>>> b.apply_load(R2, 30, -1) # Reaction force at x = 30
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.load
R1*SingularityFunction(x, 10, -1) + R2*SingularityFunction(x, 30, -1)
- 8*SingularityFunction(x, 0, -1) + 120*SingularityFunction(x, 30, -2)
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.reaction_loads
{R1: 6, R2: 2}
>>> b.load
-8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1)
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
"""
if self._composite_type == "hinge":
return self._solve_hinge_beams(*reactions)
x = self.variable
l = self.length
C3 = Symbol('C3')
C4 = Symbol('C4')
shear_curve = limit(self.shear_force(), x, l)
moment_curve = limit(self.bending_moment(), x, l)
slope_eqs = []
deflection_eqs = []
slope_curve = integrate(self.bending_moment(), x) + C3
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
slope_eqs.append(eqs)
deflection_curve = integrate(slope_curve, x) + C4
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
deflection_eqs.append(eqs)
solution = list((linsolve([shear_curve, moment_curve] + slope_eqs
+ deflection_eqs, (C3, C4) + reactions).args)[0])
solution = solution[2:]
self._reaction_loads = dict(zip(reactions, solution))
self._load = self._load.subs(self._reaction_loads)
def shear_force(self):
"""
Returns a Singularity Function expression which represents
the shear force curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.shear_force()
8*SingularityFunction(x, 0, 0) - 6*SingularityFunction(x, 10, 0) - 120*SingularityFunction(x, 30, -1) - 2*SingularityFunction(x, 30, 0)
"""
x = self.variable
return -integrate(self.load, x)
def max_shear_force(self):
"""Returns maximum Shear force and its coordinate
in the Beam object."""
shear_curve = self.shear_force()
x = self.variable
terms = shear_curve.args
singularity = [] # Points at which shear function changes
for term in terms:
if isinstance(term, Mul):
term = term.args[-1] # SingularityFunction in the term
singularity.append(term.args[1])
singularity.sort()
singularity = list(set(singularity))
intervals = [] # List of Intervals with discrete value of shear force
shear_values = [] # List of values of shear force in each interval
for i, s in enumerate(singularity):
if s == 0:
continue
try:
shear_slope = Piecewise((float("nan"), x<=singularity[i-1]),(self._load.rewrite(Piecewise), x<s), (float("nan"), True))
points = solve(shear_slope, x)
val = []
for point in points:
val.append(abs(shear_curve.subs(x, point)))
points.extend([singularity[i-1], s])
val += [abs(limit(shear_curve, x, singularity[i-1], '+')), abs(limit(shear_curve, x, s, '-'))]
max_shear = max(val)
shear_values.append(max_shear)
intervals.append(points[val.index(max_shear)])
# If shear force in a particular Interval has zero or constant
# slope, then above block gives NotImplementedError as
# solve can't represent Interval solutions.
except NotImplementedError:
initial_shear = limit(shear_curve, x, singularity[i-1], '+')
final_shear = limit(shear_curve, x, s, '-')
# If shear_curve has a constant slope(it is a line).
if shear_curve.subs(x, (singularity[i-1] + s)/2) == (initial_shear + final_shear)/2 and initial_shear != final_shear:
shear_values.extend([initial_shear, final_shear])
intervals.extend([singularity[i-1], s])
else: # shear_curve has same value in whole Interval
shear_values.append(final_shear)
intervals.append(Interval(singularity[i-1], s))
shear_values = list(map(abs, shear_values))
maximum_shear = max(shear_values)
point = intervals[shear_values.index(maximum_shear)]
return (point, maximum_shear)
def bending_moment(self):
"""
Returns a Singularity Function expression which represents
the bending moment curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.bending_moment()
8*SingularityFunction(x, 0, 1) - 6*SingularityFunction(x, 10, 1) - 120*SingularityFunction(x, 30, 0) - 2*SingularityFunction(x, 30, 1)
"""
x = self.variable
return integrate(self.shear_force(), x)
def max_bmoment(self):
"""Returns maximum Shear force and its coordinate
in the Beam object."""
bending_curve = self.bending_moment()
x = self.variable
terms = bending_curve.args
singularity = [] # Points at which bending moment changes
for term in terms:
if isinstance(term, Mul):
term = term.args[-1] # SingularityFunction in the term
singularity.append(term.args[1])
singularity.sort()
singularity = list(set(singularity))
intervals = [] # List of Intervals with discrete value of bending moment
moment_values = [] # List of values of bending moment in each interval
for i, s in enumerate(singularity):
if s == 0:
continue
try:
moment_slope = Piecewise((float("nan"), x<=singularity[i-1]),(self.shear_force().rewrite(Piecewise), x<s), (float("nan"), True))
points = solve(moment_slope, x)
val = []
for point in points:
val.append(abs(bending_curve.subs(x, point)))
points.extend([singularity[i-1], s])
val += [abs(limit(bending_curve, x, singularity[i-1], '+')), abs(limit(bending_curve, x, s, '-'))]
max_moment = max(val)
moment_values.append(max_moment)
intervals.append(points[val.index(max_moment)])
# If bending moment in a particular Interval has zero or constant
# slope, then above block gives NotImplementedError as solve
# can't represent Interval solutions.
except NotImplementedError:
initial_moment = limit(bending_curve, x, singularity[i-1], '+')
final_moment = limit(bending_curve, x, s, '-')
# If bending_curve has a constant slope(it is a line).
if bending_curve.subs(x, (singularity[i-1] + s)/2) == (initial_moment + final_moment)/2 and initial_moment != final_moment:
moment_values.extend([initial_moment, final_moment])
intervals.extend([singularity[i-1], s])
else: # bending_curve has same value in whole Interval
moment_values.append(final_moment)
intervals.append(Interval(singularity[i-1], s))
moment_values = list(map(abs, moment_values))
maximum_moment = max(moment_values)
point = intervals[moment_values.index(maximum_moment)]
return (point, maximum_moment)
def point_cflexure(self):
"""
Returns a Set of point(s) with zero bending moment and
where bending moment curve of the beam object changes
its sign from negative to positive or vice versa.
Examples
========
There is is 10 meter long overhanging beam. There are
two simple supports below the beam. One at the start
and another one at a distance of 6 meters from the start.
Point loads of magnitude 10KN and 20KN are applied at
2 meters and 4 meters from start respectively. A Uniformly
distribute load of magnitude of magnitude 3KN/m is also
applied on top starting from 6 meters away from starting
point till end.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> b = Beam(10, E, I)
>>> b.apply_load(-4, 0, -1)
>>> b.apply_load(-46, 6, -1)
>>> b.apply_load(10, 2, -1)
>>> b.apply_load(20, 4, -1)
>>> b.apply_load(3, 6, 0)
>>> b.point_cflexure()
[10/3]
"""
# To restrict the range within length of the Beam
moment_curve = Piecewise((float("nan"), self.variable<=0),
(self.bending_moment(), self.variable<self.length),
(float("nan"), True))
points = solve(moment_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
return points
def slope(self):
"""
Returns a Singularity Function expression which represents
the slope the elastic curve of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.slope()
(-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + 4000/3)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment
if self._composite_type == "hinge":
return self._hinge_beam_slope
if not self._boundary_conditions['slope']:
return diff(self.deflection(), x)
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
slope = 0
prev_slope = 0
prev_end = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = -S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
if i != len(args) - 1:
slope += (prev_slope + slope_value)*SingularityFunction(x, prev_end, 0) - \
(prev_slope + slope_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
slope += (prev_slope + slope_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
return slope
C3 = Symbol('C3')
slope_curve = -integrate(S.One/(E*I)*self.bending_moment(), x) + C3
bc_eqs = []
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, C3))
slope_curve = slope_curve.subs({C3: constants[0][0]})
return slope_curve
def deflection(self):
"""
Returns a Singularity Function expression which represents
the elastic curve or deflection of the Beam object.
Examples
========
There is a beam of length 30 meters. A moment of magnitude 120 Nm is
applied in the clockwise direction at the end of the beam. A pointload
of magnitude 8 N is applied from the top of the beam at the starting
point. There are two simple supports below the beam. One at the end
and another one at a distance of 10 meters from the start. The
deflection is restricted at both the supports.
Using the sign convention of upward forces and clockwise moment
being positive.
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> E, I = symbols('E, I')
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(30, E, I)
>>> b.apply_load(-8, 0, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(120, 30, -2)
>>> b.bc_deflection = [(10, 0), (30, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.deflection()
(4000*x/3 - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3)
+ 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000)/(E*I)
"""
x = self.variable
E = self.elastic_modulus
I = self.second_moment
if self._composite_type == "hinge":
return self._hinge_beam_deflection
if not self._boundary_conditions['deflection'] and not self._boundary_conditions['slope']:
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = -S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
base_char = self._base_char
constants = symbols(base_char + '3:5')
return S.One/(E*I)*integrate(-integrate(self.bending_moment(), x), x) + constants[0]*x + constants[1]
elif not self._boundary_conditions['deflection']:
base_char = self._base_char
constant = symbols(base_char + '4')
return integrate(self.slope(), x) + constant
elif not self._boundary_conditions['slope'] and self._boundary_conditions['deflection']:
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = -S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
base_char = self._base_char
C3, C4 = symbols(base_char + '3:5') # Integration constants
slope_curve = -integrate(self.bending_moment(), x) + C3
deflection_curve = integrate(slope_curve, x) + C4
bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, (C3, C4)))
deflection_curve = deflection_curve.subs({C3: constants[0][0], C4: constants[0][1]})
return S.One/(E*I)*deflection_curve
if isinstance(I, Piecewise) and self._composite_type == "fixed":
args = I.args
prev_slope = 0
prev_def = 0
prev_end = 0
deflection = 0
for i in range(len(args)):
if i != 0:
prev_end = args[i-1][1].args[1]
slope_value = S.One/E*integrate(self.bending_moment()/args[i][0], (x, prev_end, x))
recent_segment_slope = prev_slope + slope_value
deflection_value = integrate(recent_segment_slope, (x, prev_end, x))
if i != len(args) - 1:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0) \
- (prev_def + deflection_value)*SingularityFunction(x, args[i][1].args[1], 0)
else:
deflection += (prev_def + deflection_value)*SingularityFunction(x, prev_end, 0)
prev_slope = slope_value.subs(x, args[i][1].args[1])
prev_def = deflection_value.subs(x, args[i][1].args[1])
return deflection
C4 = Symbol('C4')
deflection_curve = integrate(self.slope(), x) + C4
bc_eqs = []
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
bc_eqs.append(eqs)
constants = list(linsolve(bc_eqs, C4))
deflection_curve = deflection_curve.subs({C4: constants[0][0]})
return deflection_curve
def max_deflection(self):
"""
Returns point of max deflection and its corresponding deflection value
in a Beam object.
"""
# To restrict the range within length of the Beam
slope_curve = Piecewise((float("nan"), self.variable<=0),
(self.slope(), self.variable<self.length),
(float("nan"), True))
points = solve(slope_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
deflection_curve = self.deflection()
deflections = [deflection_curve.subs(self.variable, x) for x in points]
deflections = list(map(abs, deflections))
if len(deflections) != 0:
max_def = max(deflections)
return (points[deflections.index(max_def)], max_def)
else:
return None
def shear_stress(self):
"""
Returns an expression representing the Shear Stress
curve of the Beam object.
"""
return self.shear_force()/self._area
def plot_shear_stress(self, subs=None):
"""
Returns a plot of shear stress present in the beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters and area of cross section 2 square
meters. A constant distributed load of 10 KN/m is applied from half of
the beam till the end. There are two simple supports below the beam,
one at the starting point and another at the ending point of the beam.
A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6), 2)
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_shear_stress()
Plot object containing:
[0]: cartesian line: 6875*SingularityFunction(x, 0, 0) - 2500*SingularityFunction(x, 2, 0)
- 5000*SingularityFunction(x, 4, 1) + 15625*SingularityFunction(x, 8, 0)
+ 5000*SingularityFunction(x, 8, 1) for x over (0.0, 8.0)
"""
shear_stress = self.shear_stress()
x = self.variable
length = self.length
if subs is None:
subs = {}
for sym in shear_stress.atoms(Symbol):
if sym != x and sym not in subs:
raise ValueError('value of %s was not passed.' %sym)
if length in subs:
length = subs[length]
# Returns Plot of Shear Stress
return plot (shear_stress.subs(subs), (x, 0, length),
title='Shear Stress', xlabel=r'$\mathrm{x}$', ylabel=r'$\tau$',
line_color='r')
def plot_shear_force(self, subs=None):
"""
Returns a plot for Shear force present in the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_shear_force()
Plot object containing:
[0]: cartesian line: 13750*SingularityFunction(x, 0, 0) - 5000*SingularityFunction(x, 2, 0)
- 10000*SingularityFunction(x, 4, 1) + 31250*SingularityFunction(x, 8, 0)
+ 10000*SingularityFunction(x, 8, 1) for x over (0.0, 8.0)
"""
shear_force = self.shear_force()
if subs is None:
subs = {}
for sym in shear_force.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(shear_force.subs(subs), (self.variable, 0, length), title='Shear Force',
xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{V}$', line_color='g')
def plot_bending_moment(self, subs=None):
"""
Returns a plot for Bending moment present in the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_bending_moment()
Plot object containing:
[0]: cartesian line: 13750*SingularityFunction(x, 0, 1) - 5000*SingularityFunction(x, 2, 1)
- 5000*SingularityFunction(x, 4, 2) + 31250*SingularityFunction(x, 8, 1)
+ 5000*SingularityFunction(x, 8, 2) for x over (0.0, 8.0)
"""
bending_moment = self.bending_moment()
if subs is None:
subs = {}
for sym in bending_moment.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(bending_moment.subs(subs), (self.variable, 0, length), title='Bending Moment',
xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{M}$', line_color='b')
def plot_slope(self, subs=None):
"""
Returns a plot for slope of deflection curve of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_slope()
Plot object containing:
[0]: cartesian line: -8.59375e-5*SingularityFunction(x, 0, 2) + 3.125e-5*SingularityFunction(x, 2, 2)
+ 2.08333333333333e-5*SingularityFunction(x, 4, 3) - 0.0001953125*SingularityFunction(x, 8, 2)
- 2.08333333333333e-5*SingularityFunction(x, 8, 3) + 0.00138541666666667 for x over (0.0, 8.0)
"""
slope = self.slope()
if subs is None:
subs = {}
for sym in slope.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(slope.subs(subs), (self.variable, 0, length), title='Slope',
xlabel=r'$\mathrm{x}$', ylabel=r'$\theta$', line_color='m')
def plot_deflection(self, subs=None):
"""
Returns a plot for deflection curve of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> b.plot_deflection()
Plot object containing:
[0]: cartesian line: 0.00138541666666667*x - 2.86458333333333e-5*SingularityFunction(x, 0, 3)
+ 1.04166666666667e-5*SingularityFunction(x, 2, 3) + 5.20833333333333e-6*SingularityFunction(x, 4, 4)
- 6.51041666666667e-5*SingularityFunction(x, 8, 3) - 5.20833333333333e-6*SingularityFunction(x, 8, 4)
for x over (0.0, 8.0)
"""
deflection = self.deflection()
if subs is None:
subs = {}
for sym in deflection.atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(deflection.subs(subs), (self.variable, 0, length),
title='Deflection', xlabel=r'$\mathrm{x}$', ylabel=r'$\delta$',
line_color='r')
def plot_loading_results(self, subs=None):
"""
Returns a subplot of Shear Force, Bending Moment,
Slope and Deflection of the Beam object.
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 8 meters. A constant distributed load of 10 KN/m
is applied from half of the beam till the end. There are two simple supports
below the beam, one at the starting point and another at the ending point
of the beam. A pointload of magnitude 5 KN is also applied from top of the
beam, at a distance of 4 meters from the starting point.
Take E = 200 GPa and I = 400*(10**-6) meter**4.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> b = Beam(8, 200*(10**9), 400*(10**-6))
>>> b.apply_load(5000, 2, -1)
>>> b.apply_load(R1, 0, -1)
>>> b.apply_load(R2, 8, -1)
>>> b.apply_load(10000, 4, 0, end=8)
>>> b.bc_deflection = [(0, 0), (8, 0)]
>>> b.solve_for_reaction_loads(R1, R2)
>>> axes = b.plot_loading_results()
"""
length = self.length
variable = self.variable
if subs is None:
subs = {}
for sym in self.deflection().atoms(Symbol):
if sym == self.variable:
continue
if sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if length in subs:
length = subs[length]
ax1 = plot(self.shear_force().subs(subs), (variable, 0, length),
title="Shear Force", xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{V}$',
line_color='g', show=False)
ax2 = plot(self.bending_moment().subs(subs), (variable, 0, length),
title="Bending Moment", xlabel=r'$\mathrm{x}$', ylabel=r'$\mathrm{M}$',
line_color='b', show=False)
ax3 = plot(self.slope().subs(subs), (variable, 0, length),
title="Slope", xlabel=r'$\mathrm{x}$', ylabel=r'$\theta$',
line_color='m', show=False)
ax4 = plot(self.deflection().subs(subs), (variable, 0, length),
title="Deflection", xlabel=r'$\mathrm{x}$', ylabel=r'$\delta$',
line_color='r', show=False)
return PlotGrid(4, 1, ax1, ax2, ax3, ax4)
def _solve_for_ild_equations(self):
"""
Helper function for I.L.D. It takes the unsubstituted
copy of the load equation and uses it to calculate shear force and bending
moment equations.
"""
x = self.variable
shear_force = -integrate(self._original_load, x)
bending_moment = integrate(shear_force, x)
return shear_force, bending_moment
def solve_for_ild_reactions(self, value, *reactions):
"""
Determines the Influence Line Diagram equations for reaction
forces under the effect of a moving load.
Parameters
==========
value : Integer
Magnitude of moving load
reactions :
The reaction forces applied on the beam.
Examples
========
There is a beam of length 10 meters. There are two simple supports
below the beam, one at the starting point and another at the ending
point of the beam. Calculate the I.L.D. equations for reaction forces
under the effect of a moving load of magnitude 1kN.
.. image:: ildreaction.png
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> E, I = symbols('E, I')
>>> R_0, R_10 = symbols('R_0, R_10')
>>> b = Beam(10, E, I)
>>> b.apply_support(0, 'roller')
>>> b.apply_support(10, 'roller')
>>> b.solve_for_ild_reactions(1,R_0,R_10)
>>> b.ild_reactions
{R_0: x/10 - 1, R_10: -x/10}
"""
shear_force, bending_moment = self._solve_for_ild_equations()
x = self.variable
l = self.length
C3 = Symbol('C3')
C4 = Symbol('C4')
shear_curve = limit(shear_force, x, l) - value
moment_curve = limit(bending_moment, x, l) - value*(l-x)
slope_eqs = []
deflection_eqs = []
slope_curve = integrate(bending_moment, x) + C3
for position, value in self._boundary_conditions['slope']:
eqs = slope_curve.subs(x, position) - value
slope_eqs.append(eqs)
deflection_curve = integrate(slope_curve, x) + C4
for position, value in self._boundary_conditions['deflection']:
eqs = deflection_curve.subs(x, position) - value
deflection_eqs.append(eqs)
solution = list((linsolve([shear_curve, moment_curve] + slope_eqs
+ deflection_eqs, (C3, C4) + reactions).args)[0])
solution = solution[2:]
# Determining the equations and solving them.
self._ild_reactions = dict(zip(reactions, solution))
def plot_ild_reactions(self, subs=None):
"""
Plots the Influence Line Diagram of Reaction Forces
under the effect of a moving load. This function
should be called after calling solve_for_ild_reactions().
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 10 meters. A point load of magnitude 5KN
is also applied from top of the beam, at a distance of 4 meters
from the starting point. There are two simple supports below the
beam, located at the starting point and at a distance of 7 meters
from the starting point. Plot the I.L.D. equations for reactions
at both support points under the effect of a moving load
of magnitude 1kN.
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> E, I = symbols('E, I')
>>> R_0, R_7 = symbols('R_0, R_7')
>>> b = Beam(10, E, I)
>>> b.apply_support(0, 'roller')
>>> b.apply_support(7, 'roller')
>>> b.apply_load(5,4,-1)
>>> b.solve_for_ild_reactions(1,R_0,R_7)
>>> b.ild_reactions
{R_0: x/7 - 22/7, R_7: -x/7 - 20/7}
>>> b.plot_ild_reactions()
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: x/7 - 22/7 for x over (0.0, 10.0)
Plot[1]:Plot object containing:
[0]: cartesian line: -x/7 - 20/7 for x over (0.0, 10.0)
"""
if not self._ild_reactions:
raise ValueError("I.L.D. reaction equations not found. Please use solve_for_ild_reactions() to generate the I.L.D. reaction equations.")
x = self.variable
ildplots = []
if subs is None:
subs = {}
for reaction in self._ild_reactions:
for sym in self._ild_reactions[reaction].atoms(Symbol):
if sym != x and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
for sym in self._length.atoms(Symbol):
if sym != x and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
for reaction in self._ild_reactions:
ildplots.append(plot(self._ild_reactions[reaction].subs(subs),
(x, 0, self._length.subs(subs)), title='I.L.D. for Reactions',
xlabel=x, ylabel=reaction, line_color='blue', show=False))
return PlotGrid(len(ildplots), 1, *ildplots)
def solve_for_ild_shear(self, distance, value, *reactions):
"""
Determines the Influence Line Diagram equations for shear at a
specified point under the effect of a moving load.
Parameters
==========
distance : Integer
Distance of the point from the start of the beam
for which equations are to be determined
value : Integer
Magnitude of moving load
reactions :
The reaction forces applied on the beam.
Examples
========
There is a beam of length 12 meters. There are two simple supports
below the beam, one at the starting point and another at a distance
of 8 meters. Calculate the I.L.D. equations for Shear at a distance
of 4 meters under the effect of a moving load of magnitude 1kN.
.. image:: ildshear.png
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> E, I = symbols('E, I')
>>> R_0, R_8 = symbols('R_0, R_8')
>>> b = Beam(12, E, I)
>>> b.apply_support(0, 'roller')
>>> b.apply_support(8, 'roller')
>>> b.solve_for_ild_reactions(1, R_0, R_8)
>>> b.solve_for_ild_shear(4, 1, R_0, R_8)
>>> b.ild_shear
Piecewise((x/8, x < 4), (x/8 - 1, x > 4))
"""
x = self.variable
l = self.length
shear_force, _ = self._solve_for_ild_equations()
shear_curve1 = value - limit(shear_force, x, distance)
shear_curve2 = (limit(shear_force, x, l) - limit(shear_force, x, distance)) - value
for reaction in reactions:
shear_curve1 = shear_curve1.subs(reaction,self._ild_reactions[reaction])
shear_curve2 = shear_curve2.subs(reaction,self._ild_reactions[reaction])
shear_eq = Piecewise((shear_curve1, x < distance), (shear_curve2, x > distance))
self._ild_shear = shear_eq
def plot_ild_shear(self,subs=None):
"""
Plots the Influence Line Diagram for Shear under the effect
of a moving load. This function should be called after
calling solve_for_ild_shear().
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 12 meters. There are two simple supports
below the beam, one at the starting point and another at a distance
of 8 meters. Plot the I.L.D. for Shear at a distance
of 4 meters under the effect of a moving load of magnitude 1kN.
.. image:: ildshear.png
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> E, I = symbols('E, I')
>>> R_0, R_8 = symbols('R_0, R_8')
>>> b = Beam(12, E, I)
>>> b.apply_support(0, 'roller')
>>> b.apply_support(8, 'roller')
>>> b.solve_for_ild_reactions(1, R_0, R_8)
>>> b.solve_for_ild_shear(4, 1, R_0, R_8)
>>> b.ild_shear
Piecewise((x/8, x < 4), (x/8 - 1, x > 4))
>>> b.plot_ild_shear()
Plot object containing:
[0]: cartesian line: Piecewise((x/8, x < 4), (x/8 - 1, x > 4)) for x over (0.0, 12.0)
"""
if not self._ild_shear:
raise ValueError("I.L.D. shear equation not found. Please use solve_for_ild_shear() to generate the I.L.D. shear equations.")
x = self.variable
l = self._length
if subs is None:
subs = {}
for sym in self._ild_shear.atoms(Symbol):
if sym != x and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
for sym in self._length.atoms(Symbol):
if sym != x and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
return plot(self._ild_shear.subs(subs), (x, 0, l), title='I.L.D. for Shear',
xlabel=r'$\mathrm{X}$', ylabel=r'$\mathrm{V}$', line_color='blue',show=True)
def solve_for_ild_moment(self, distance, value, *reactions):
"""
Determines the Influence Line Diagram equations for moment at a
specified point under the effect of a moving load.
Parameters
==========
distance : Integer
Distance of the point from the start of the beam
for which equations are to be determined
value : Integer
Magnitude of moving load
reactions :
The reaction forces applied on the beam.
Examples
========
There is a beam of length 12 meters. There are two simple supports
below the beam, one at the starting point and another at a distance
of 8 meters. Calculate the I.L.D. equations for Moment at a distance
of 4 meters under the effect of a moving load of magnitude 1kN.
.. image:: ildshear.png
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> E, I = symbols('E, I')
>>> R_0, R_8 = symbols('R_0, R_8')
>>> b = Beam(12, E, I)
>>> b.apply_support(0, 'roller')
>>> b.apply_support(8, 'roller')
>>> b.solve_for_ild_reactions(1, R_0, R_8)
>>> b.solve_for_ild_moment(4, 1, R_0, R_8)
>>> b.ild_moment
Piecewise((-x/2, x < 4), (x/2 - 4, x > 4))
"""
x = self.variable
l = self.length
_, moment = self._solve_for_ild_equations()
moment_curve1 = value*(distance-x) - limit(moment, x, distance)
moment_curve2= (limit(moment, x, l)-limit(moment, x, distance))-value*(l-x)
for reaction in reactions:
moment_curve1 = moment_curve1.subs(reaction, self._ild_reactions[reaction])
moment_curve2 = moment_curve2.subs(reaction, self._ild_reactions[reaction])
moment_eq = Piecewise((moment_curve1, x < distance), (moment_curve2, x > distance))
self._ild_moment = moment_eq
def plot_ild_moment(self,subs=None):
"""
Plots the Influence Line Diagram for Moment under the effect
of a moving load. This function should be called after
calling solve_for_ild_moment().
Parameters
==========
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 12 meters. There are two simple supports
below the beam, one at the starting point and another at a distance
of 8 meters. Plot the I.L.D. for Moment at a distance
of 4 meters under the effect of a moving load of magnitude 1kN.
.. image:: ildshear.png
Using the sign convention of downwards forces being positive.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy import symbols
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> E, I = symbols('E, I')
>>> R_0, R_8 = symbols('R_0, R_8')
>>> b = Beam(12, E, I)
>>> b.apply_support(0, 'roller')
>>> b.apply_support(8, 'roller')
>>> b.solve_for_ild_reactions(1, R_0, R_8)
>>> b.solve_for_ild_moment(4, 1, R_0, R_8)
>>> b.ild_moment
Piecewise((-x/2, x < 4), (x/2 - 4, x > 4))
>>> b.plot_ild_moment()
Plot object containing:
[0]: cartesian line: Piecewise((-x/2, x < 4), (x/2 - 4, x > 4)) for x over (0.0, 12.0)
"""
if not self._ild_moment:
raise ValueError("I.L.D. moment equation not found. Please use solve_for_ild_moment() to generate the I.L.D. moment equations.")
x = self.variable
if subs is None:
subs = {}
for sym in self._ild_moment.atoms(Symbol):
if sym != x and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
for sym in self._length.atoms(Symbol):
if sym != x and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
return plot(self._ild_moment.subs(subs), (x, 0, self._length), title='I.L.D. for Moment',
xlabel=r'$\mathrm{X}$', ylabel=r'$\mathrm{M}$', line_color='blue', show=True)
@doctest_depends_on(modules=('numpy',))
def draw(self, pictorial=True):
"""
Returns a plot object representing the beam diagram of the beam.
.. note::
The user must be careful while entering load values.
The draw function assumes a sign convention which is used
for plotting loads.
Given a right handed coordinate system with XYZ coordinates,
the beam's length is assumed to be along the positive X axis.
The draw function recognizes positve loads(with n>-2) as loads
acting along negative Y direction and positve moments acting
along positive Z direction.
Parameters
==========
pictorial: Boolean (default=True)
Setting ``pictorial=True`` would simply create a pictorial (scaled) view
of the beam diagram not with the exact dimensions.
Although setting ``pictorial=False`` would create a beam diagram with
the exact dimensions on the plot
Examples
========
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam
>>> from sympy import symbols
>>> R1, R2 = symbols('R1, R2')
>>> E, I = symbols('E, I')
>>> b = Beam(50, 20, 30)
>>> b.apply_load(10, 2, -1)
>>> b.apply_load(R1, 10, -1)
>>> b.apply_load(R2, 30, -1)
>>> b.apply_load(90, 5, 0, 23)
>>> b.apply_load(10, 30, 1, 50)
>>> b.apply_support(50, "pin")
>>> b.apply_support(0, "fixed")
>>> b.apply_support(20, "roller")
>>> p = b.draw()
>>> p
Plot object containing:
[0]: cartesian line: 25*SingularityFunction(x, 5, 0) - 25*SingularityFunction(x, 23, 0)
+ SingularityFunction(x, 30, 1) - 20*SingularityFunction(x, 50, 0)
- SingularityFunction(x, 50, 1) + 5 for x over (0.0, 50.0)
[1]: cartesian line: 5 for x over (0.0, 50.0)
>>> p.show()
"""
if not numpy:
raise ImportError("To use this function numpy module is required")
x = self.variable
# checking whether length is an expression in terms of any Symbol.
if isinstance(self.length, Expr):
l = list(self.length.atoms(Symbol))
# assigning every Symbol a default value of 10
l = {i:10 for i in l}
length = self.length.subs(l)
else:
l = {}
length = self.length
height = length/10
rectangles = []
rectangles.append({'xy':(0, 0), 'width':length, 'height': height, 'facecolor':"brown"})
annotations, markers, load_eq,load_eq1, fill = self._draw_load(pictorial, length, l)
support_markers, support_rectangles = self._draw_supports(length, l)
rectangles += support_rectangles
markers += support_markers
sing_plot = plot(height + load_eq, height + load_eq1, (x, 0, length),
xlim=(-height, length + height), ylim=(-length, 1.25*length), annotations=annotations,
markers=markers, rectangles=rectangles, line_color='brown', fill=fill, axis=False, show=False)
return sing_plot
def _draw_load(self, pictorial, length, l):
loads = list(set(self.applied_loads) - set(self._support_as_loads))
height = length/10
x = self.variable
annotations = []
markers = []
load_args = []
scaled_load = 0
load_args1 = []
scaled_load1 = 0
load_eq = 0 # For positive valued higher order loads
load_eq1 = 0 # For negative valued higher order loads
fill = None
plus = 0 # For positive valued higher order loads
minus = 0 # For negative valued higher order loads
for load in loads:
# check if the position of load is in terms of the beam length.
if l:
pos = load[1].subs(l)
else:
pos = load[1]
# point loads
if load[2] == -1:
if isinstance(load[0], Symbol) or load[0].is_negative:
annotations.append({'text':'', 'xy':(pos, 0), 'xytext':(pos, height - 4*height), 'arrowprops':dict(width= 1.5, headlength=5, headwidth=5, facecolor='black')})
else:
annotations.append({'text':'', 'xy':(pos, height), 'xytext':(pos, height*4), 'arrowprops':dict(width= 1.5, headlength=4, headwidth=4, facecolor='black')})
# moment loads
elif load[2] == -2:
if load[0].is_negative:
markers.append({'args':[[pos], [height/2]], 'marker': r'$\circlearrowright$', 'markersize':15})
else:
markers.append({'args':[[pos], [height/2]], 'marker': r'$\circlearrowleft$', 'markersize':15})
# higher order loads
elif load[2] >= 0:
# `fill` will be assigned only when higher order loads are present
value, start, order, end = load
# Positive loads have their seperate equations
if(value>0):
plus = 1
# if pictorial is True we remake the load equation again with
# some constant magnitude values.
if pictorial:
value = 10**(1-order) if order > 0 else length/2
scaled_load += value*SingularityFunction(x, start, order)
if end:
f2 = 10**(1-order)*x**order if order > 0 else length/2*x**order
for i in range(0, order + 1):
scaled_load -= (f2.diff(x, i).subs(x, end - start)*
SingularityFunction(x, end, i)/factorial(i))
if pictorial:
if isinstance(scaled_load, Add):
load_args = scaled_load.args
else:
# when the load equation consists of only a single term
load_args = (scaled_load,)
load_eq = [i.subs(l) for i in load_args]
else:
if isinstance(self.load, Add):
load_args = self.load.args
else:
load_args = (self.load,)
load_eq = [i.subs(l) for i in load_args if list(i.atoms(SingularityFunction))[0].args[2] >= 0]
load_eq = Add(*load_eq)
# filling higher order loads with colour
expr = height + load_eq.rewrite(Piecewise)
y1 = lambdify(x, expr, 'numpy')
# For loads with negative value
else:
minus = 1
# if pictorial is True we remake the load equation again with
# some constant magnitude values.
if pictorial:
value = 10**(1-order) if order > 0 else length/2
scaled_load1 += value*SingularityFunction(x, start, order)
if end:
f2 = 10**(1-order)*x**order if order > 0 else length/2*x**order
for i in range(0, order + 1):
scaled_load1 -= (f2.diff(x, i).subs(x, end - start)*
SingularityFunction(x, end, i)/factorial(i))
if pictorial:
if isinstance(scaled_load1, Add):
load_args1 = scaled_load1.args
else:
# when the load equation consists of only a single term
load_args1 = (scaled_load1,)
load_eq1 = [i.subs(l) for i in load_args1]
else:
if isinstance(self.load, Add):
load_args1 = self.load.args1
else:
load_args1 = (self.load,)
load_eq1 = [i.subs(l) for i in load_args if list(i.atoms(SingularityFunction))[0].args[2] >= 0]
load_eq1 = -Add(*load_eq1)-height
# filling higher order loads with colour
expr = height + load_eq1.rewrite(Piecewise)
y1_ = lambdify(x, expr, 'numpy')
y = numpy.arange(0, float(length), 0.001)
y2 = float(height)
if(plus == 1 and minus == 1):
fill = {'x': y, 'y1': y1(y), 'y2': y1_(y), 'color':'darkkhaki'}
elif(plus == 1):
fill = {'x': y, 'y1': y1(y), 'y2': y2, 'color':'darkkhaki'}
else:
fill = {'x': y, 'y1': y1_(y), 'y2': y2, 'color':'darkkhaki'}
return annotations, markers, load_eq, load_eq1, fill
def _draw_supports(self, length, l):
height = float(length/10)
support_markers = []
support_rectangles = []
for support in self._applied_supports:
if l:
pos = support[0].subs(l)
else:
pos = support[0]
if support[1] == "pin":
support_markers.append({'args':[pos, [0]], 'marker':6, 'markersize':13, 'color':"black"})
elif support[1] == "roller":
support_markers.append({'args':[pos, [-height/2.5]], 'marker':'o', 'markersize':11, 'color':"black"})
elif support[1] == "fixed":
if pos == 0:
support_rectangles.append({'xy':(0, -3*height), 'width':-length/20, 'height':6*height + height, 'fill':False, 'hatch':'/////'})
else:
support_rectangles.append({'xy':(length, -3*height), 'width':length/20, 'height': 6*height + height, 'fill':False, 'hatch':'/////'})
return support_markers, support_rectangles
class Beam3D(Beam):
"""
This class handles loads applied in any direction of a 3D space along
with unequal values of Second moment along different axes.
.. note::
A consistent sign convention must be used while solving a beam
bending problem; the results will
automatically follow the chosen sign convention.
This class assumes that any kind of distributed load/moment is
applied through out the span of a beam.
Examples
========
There is a beam of l meters long. A constant distributed load of magnitude q
is applied along y-axis from start till the end of beam. A constant distributed
moment of magnitude m is also applied along z-axis from start till the end of beam.
Beam is fixed at both of its end. So, deflection of the beam at the both ends
is restricted.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols, simplify, collect, factor
>>> l, E, G, I, A = symbols('l, E, G, I, A')
>>> b = Beam3D(l, E, G, I, A)
>>> x, q, m = symbols('x, q, m')
>>> b.apply_load(q, 0, 0, dir="y")
>>> b.apply_moment_load(m, 0, -1, dir="z")
>>> b.shear_force()
[0, -q*x, 0]
>>> b.bending_moment()
[0, 0, -m*x + q*x**2/2]
>>> b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])]
>>> b.solve_slope_deflection()
>>> factor(b.slope())
[0, 0, x*(-l + x)*(-A*G*l**3*q + 2*A*G*l**2*q*x - 12*E*I*l*q
- 72*E*I*m + 24*E*I*q*x)/(12*E*I*(A*G*l**2 + 12*E*I))]
>>> dx, dy, dz = b.deflection()
>>> dy = collect(simplify(dy), x)
>>> dx == dz == 0
True
>>> dy == (x*(12*E*I*l*(A*G*l**2*q - 2*A*G*l*m + 12*E*I*q)
... + x*(A*G*l*(3*l*(A*G*l**2*q - 2*A*G*l*m + 12*E*I*q) + x*(-2*A*G*l**2*q + 4*A*G*l*m - 24*E*I*q))
... + A*G*(A*G*l**2 + 12*E*I)*(-2*l**2*q + 6*l*m - 4*m*x + q*x**2)
... - 12*E*I*q*(A*G*l**2 + 12*E*I)))/(24*A*E*G*I*(A*G*l**2 + 12*E*I)))
True
References
==========
.. [1] http://homes.civil.aau.dk/jc/FemteSemester/Beams3D.pdf
"""
def __init__(self, length, elastic_modulus, shear_modulus, second_moment,
area, variable=Symbol('x')):
"""Initializes the class.
Parameters
==========
length : Sympifyable
A Symbol or value representing the Beam's length.
elastic_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of Elasticity.
It is a measure of the stiffness of the Beam material.
shear_modulus : Sympifyable
A SymPy expression representing the Beam's Modulus of rigidity.
It is a measure of rigidity of the Beam material.
second_moment : Sympifyable or list
A list of two elements having SymPy expression representing the
Beam's Second moment of area. First value represent Second moment
across y-axis and second across z-axis.
Single SymPy expression can be passed if both values are same
area : Sympifyable
A SymPy expression representing the Beam's cross-sectional area
in a plane prependicular to length of the Beam.
variable : Symbol, optional
A Symbol object that will be used as the variable along the beam
while representing the load, shear, moment, slope and deflection
curve. By default, it is set to ``Symbol('x')``.
"""
super().__init__(length, elastic_modulus, second_moment, variable)
self.shear_modulus = shear_modulus
self.area = area
self._load_vector = [0, 0, 0]
self._moment_load_vector = [0, 0, 0]
self._torsion_moment = {}
self._load_Singularity = [0, 0, 0]
self._slope = [0, 0, 0]
self._deflection = [0, 0, 0]
self._angular_deflection = 0
@property
def shear_modulus(self):
"""Young's Modulus of the Beam. """
return self._shear_modulus
@shear_modulus.setter
def shear_modulus(self, e):
self._shear_modulus = sympify(e)
@property
def second_moment(self):
"""Second moment of area of the Beam. """
return self._second_moment
@second_moment.setter
def second_moment(self, i):
if isinstance(i, list):
i = [sympify(x) for x in i]
self._second_moment = i
else:
self._second_moment = sympify(i)
@property
def area(self):
"""Cross-sectional area of the Beam. """
return self._area
@area.setter
def area(self, a):
self._area = sympify(a)
@property
def load_vector(self):
"""
Returns a three element list representing the load vector.
"""
return self._load_vector
@property
def moment_load_vector(self):
"""
Returns a three element list representing moment loads on Beam.
"""
return self._moment_load_vector
@property
def boundary_conditions(self):
"""
Returns a dictionary of boundary conditions applied on the beam.
The dictionary has two keywords namely slope and deflection.
The value of each keyword is a list of tuple, where each tuple
contains location and value of a boundary condition in the format
(location, value). Further each value is a list corresponding to
slope or deflection(s) values along three axes at that location.
Examples
========
There is a beam of length 4 meters. The slope at 0 should be 4 along
the x-axis and 0 along others. At the other end of beam, deflection
along all the three axes should be zero.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(30, E, G, I, A, x)
>>> b.bc_slope = [(0, (4, 0, 0))]
>>> b.bc_deflection = [(4, [0, 0, 0])]
>>> b.boundary_conditions
{'deflection': [(4, [0, 0, 0])], 'slope': [(0, (4, 0, 0))]}
Here the deflection of the beam should be ``0`` along all the three axes at ``4``.
Similarly, the slope of the beam should be ``4`` along x-axis and ``0``
along y and z axis at ``0``.
"""
return self._boundary_conditions
def polar_moment(self):
"""
Returns the polar moment of area of the beam
about the X axis with respect to the centroid.
Examples
========
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A = symbols('l, E, G, I, A')
>>> b = Beam3D(l, E, G, I, A)
>>> b.polar_moment()
2*I
>>> I1 = [9, 15]
>>> b = Beam3D(l, E, G, I1, A)
>>> b.polar_moment()
24
"""
if not iterable(self.second_moment):
return 2*self.second_moment
return sum(self.second_moment)
def apply_load(self, value, start, order, dir="y"):
"""
This method adds up the force load to a particular beam object.
Parameters
==========
value : Sympifyable
The magnitude of an applied load.
dir : String
Axis along which load is applied.
order : Integer
The order of the applied load.
- For point loads, order=-1
- For constant distributed load, order=0
- For ramp loads, order=1
- For parabolic ramp loads, order=2
- ... so on.
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if dir == "x":
if not order == -1:
self._load_vector[0] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
elif dir == "y":
if not order == -1:
self._load_vector[1] += value
self._load_Singularity[1] += value*SingularityFunction(x, start, order)
else:
if not order == -1:
self._load_vector[2] += value
self._load_Singularity[2] += value*SingularityFunction(x, start, order)
def apply_moment_load(self, value, start, order, dir="y"):
"""
This method adds up the moment loads to a particular beam object.
Parameters
==========
value : Sympifyable
The magnitude of an applied moment.
dir : String
Axis along which moment is applied.
order : Integer
The order of the applied load.
- For point moments, order=-2
- For constant distributed moment, order=-1
- For ramp moments, order=0
- For parabolic ramp moments, order=1
- ... so on.
"""
x = self.variable
value = sympify(value)
start = sympify(start)
order = sympify(order)
if dir == "x":
if not order == -2:
self._moment_load_vector[0] += value
else:
if start in list(self._torsion_moment):
self._torsion_moment[start] += value
else:
self._torsion_moment[start] = value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
elif dir == "y":
if not order == -2:
self._moment_load_vector[1] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
else:
if not order == -2:
self._moment_load_vector[2] += value
self._load_Singularity[0] += value*SingularityFunction(x, start, order)
def apply_support(self, loc, type="fixed"):
if type in ("pin", "roller"):
reaction_load = Symbol('R_'+str(loc))
self._reaction_loads[reaction_load] = reaction_load
self.bc_deflection.append((loc, [0, 0, 0]))
else:
reaction_load = Symbol('R_'+str(loc))
reaction_moment = Symbol('M_'+str(loc))
self._reaction_loads[reaction_load] = [reaction_load, reaction_moment]
self.bc_deflection.append((loc, [0, 0, 0]))
self.bc_slope.append((loc, [0, 0, 0]))
def solve_for_reaction_loads(self, *reaction):
"""
Solves for the reaction forces.
Examples
========
There is a beam of length 30 meters. It it supported by rollers at
of its end. A constant distributed load of magnitude 8 N is applied
from start till its end along y-axis. Another linear load having
slope equal to 9 is applied along z-axis.
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(30, E, G, I, A, x)
>>> b.apply_load(8, start=0, order=0, dir="y")
>>> b.apply_load(9*x, start=0, order=0, dir="z")
>>> b.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="y")
>>> b.apply_load(R2, start=30, order=-1, dir="y")
>>> b.apply_load(R3, start=0, order=-1, dir="z")
>>> b.apply_load(R4, start=30, order=-1, dir="z")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.reaction_loads
{R1: -120, R2: -120, R3: -1350, R4: -2700}
"""
x = self.variable
l = self.length
q = self._load_Singularity
shear_curves = [integrate(load, x) for load in q]
moment_curves = [integrate(shear, x) for shear in shear_curves]
for i in range(3):
react = [r for r in reaction if (shear_curves[i].has(r) or moment_curves[i].has(r))]
if len(react) == 0:
continue
shear_curve = limit(shear_curves[i], x, l)
moment_curve = limit(moment_curves[i], x, l)
sol = list((linsolve([shear_curve, moment_curve], react).args)[0])
sol_dict = dict(zip(react, sol))
reaction_loads = self._reaction_loads
# Check if any of the evaluated rection exists in another direction
# and if it exists then it should have same value.
for key in sol_dict:
if key in reaction_loads and sol_dict[key] != reaction_loads[key]:
raise ValueError("Ambiguous solution for %s in different directions." % key)
self._reaction_loads.update(sol_dict)
def shear_force(self):
"""
Returns a list of three expressions which represents the shear force
curve of the Beam object along all three axes.
"""
x = self.variable
q = self._load_vector
return [integrate(-q[0], x), integrate(-q[1], x), integrate(-q[2], x)]
def axial_force(self):
"""
Returns expression of Axial shear force present inside the Beam object.
"""
return self.shear_force()[0]
def shear_stress(self):
"""
Returns a list of three expressions which represents the shear stress
curve of the Beam object along all three axes.
"""
return [self.shear_force()[0]/self._area, self.shear_force()[1]/self._area, self.shear_force()[2]/self._area]
def axial_stress(self):
"""
Returns expression of Axial stress present inside the Beam object.
"""
return self.axial_force()/self._area
def bending_moment(self):
"""
Returns a list of three expressions which represents the bending moment
curve of the Beam object along all three axes.
"""
x = self.variable
m = self._moment_load_vector
shear = self.shear_force()
return [integrate(-m[0], x), integrate(-m[1] + shear[2], x),
integrate(-m[2] - shear[1], x) ]
def torsional_moment(self):
"""
Returns expression of Torsional moment present inside the Beam object.
"""
return self.bending_moment()[0]
def solve_for_torsion(self):
"""
Solves for the angular deflection due to the torsional effects of
moments being applied in the x-direction i.e. out of or into the beam.
Here, a positive torque means the direction of the torque is positive
i.e. out of the beam along the beam-axis. Likewise, a negative torque
signifies a torque into the beam cross-section.
Examples
========
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, E, G, I, A, x)
>>> b.apply_moment_load(4, 4, -2, dir='x')
>>> b.apply_moment_load(4, 8, -2, dir='x')
>>> b.apply_moment_load(4, 8, -2, dir='x')
>>> b.solve_for_torsion()
>>> b.angular_deflection().subs(x, 3)
18/(G*I)
"""
x = self.variable
sum_moments = 0
for point in list(self._torsion_moment):
sum_moments += self._torsion_moment[point]
list(self._torsion_moment).sort()
pointsList = list(self._torsion_moment)
torque_diagram = Piecewise((sum_moments, x<=pointsList[0]), (0, x>=pointsList[0]))
for i in range(len(pointsList))[1:]:
sum_moments -= self._torsion_moment[pointsList[i-1]]
torque_diagram += Piecewise((0, x<=pointsList[i-1]), (sum_moments, x<=pointsList[i]), (0, x>=pointsList[i]))
integrated_torque_diagram = integrate(torque_diagram)
self._angular_deflection = integrated_torque_diagram/(self.shear_modulus*self.polar_moment())
def solve_slope_deflection(self):
x = self.variable
l = self.length
E = self.elastic_modulus
G = self.shear_modulus
I = self.second_moment
if isinstance(I, list):
I_y, I_z = I[0], I[1]
else:
I_y = I_z = I
A = self._area
load = self._load_vector
moment = self._moment_load_vector
defl = Function('defl')
theta = Function('theta')
# Finding deflection along x-axis(and corresponding slope value by differentiating it)
# Equation used: Derivative(E*A*Derivative(def_x(x), x), x) + load_x = 0
eq = Derivative(E*A*Derivative(defl(x), x), x) + load[0]
def_x = dsolve(Eq(eq, 0), defl(x)).args[1]
# Solving constants originated from dsolve
C1 = Symbol('C1')
C2 = Symbol('C2')
constants = list((linsolve([def_x.subs(x, 0), def_x.subs(x, l)], C1, C2).args)[0])
def_x = def_x.subs({C1:constants[0], C2:constants[1]})
slope_x = def_x.diff(x)
self._deflection[0] = def_x
self._slope[0] = slope_x
# Finding deflection along y-axis and slope across z-axis. System of equation involved:
# 1: Derivative(E*I_z*Derivative(theta_z(x), x), x) + G*A*(Derivative(defl_y(x), x) - theta_z(x)) + moment_z = 0
# 2: Derivative(G*A*(Derivative(defl_y(x), x) - theta_z(x)), x) + load_y = 0
C_i = Symbol('C_i')
# Substitute value of `G*A*(Derivative(defl_y(x), x) - theta_z(x))` from (2) in (1)
eq1 = Derivative(E*I_z*Derivative(theta(x), x), x) + (integrate(-load[1], x) + C_i) + moment[2]
slope_z = dsolve(Eq(eq1, 0)).args[1]
# Solve for constants originated from using dsolve on eq1
constants = list((linsolve([slope_z.subs(x, 0), slope_z.subs(x, l)], C1, C2).args)[0])
slope_z = slope_z.subs({C1:constants[0], C2:constants[1]})
# Put value of slope obtained back in (2) to solve for `C_i` and find deflection across y-axis
eq2 = G*A*(Derivative(defl(x), x)) + load[1]*x - C_i - G*A*slope_z
def_y = dsolve(Eq(eq2, 0), defl(x)).args[1]
# Solve for constants originated from using dsolve on eq2
constants = list((linsolve([def_y.subs(x, 0), def_y.subs(x, l)], C1, C_i).args)[0])
self._deflection[1] = def_y.subs({C1:constants[0], C_i:constants[1]})
self._slope[2] = slope_z.subs(C_i, constants[1])
# Finding deflection along z-axis and slope across y-axis. System of equation involved:
# 1: Derivative(E*I_y*Derivative(theta_y(x), x), x) - G*A*(Derivative(defl_z(x), x) + theta_y(x)) + moment_y = 0
# 2: Derivative(G*A*(Derivative(defl_z(x), x) + theta_y(x)), x) + load_z = 0
# Substitute value of `G*A*(Derivative(defl_y(x), x) + theta_z(x))` from (2) in (1)
eq1 = Derivative(E*I_y*Derivative(theta(x), x), x) + (integrate(load[2], x) - C_i) + moment[1]
slope_y = dsolve(Eq(eq1, 0)).args[1]
# Solve for constants originated from using dsolve on eq1
constants = list((linsolve([slope_y.subs(x, 0), slope_y.subs(x, l)], C1, C2).args)[0])
slope_y = slope_y.subs({C1:constants[0], C2:constants[1]})
# Put value of slope obtained back in (2) to solve for `C_i` and find deflection across z-axis
eq2 = G*A*(Derivative(defl(x), x)) + load[2]*x - C_i + G*A*slope_y
def_z = dsolve(Eq(eq2,0)).args[1]
# Solve for constants originated from using dsolve on eq2
constants = list((linsolve([def_z.subs(x, 0), def_z.subs(x, l)], C1, C_i).args)[0])
self._deflection[2] = def_z.subs({C1:constants[0], C_i:constants[1]})
self._slope[1] = slope_y.subs(C_i, constants[1])
def slope(self):
"""
Returns a three element list representing slope of deflection curve
along all the three axes.
"""
return self._slope
def deflection(self):
"""
Returns a three element list representing deflection curve along all
the three axes.
"""
return self._deflection
def angular_deflection(self):
"""
Returns a function in x depicting how the angular deflection, due to moments
in the x-axis on the beam, varies with x.
"""
return self._angular_deflection
def _plot_shear_force(self, dir, subs=None):
shear_force = self.shear_force()
if dir == 'x':
dir_num = 0
color = 'r'
elif dir == 'y':
dir_num = 1
color = 'g'
elif dir == 'z':
dir_num = 2
color = 'b'
if subs is None:
subs = {}
for sym in shear_force[dir_num].atoms(Symbol):
if sym != self.variable and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(shear_force[dir_num].subs(subs), (self.variable, 0, length), show = False, title='Shear Force along %c direction'%dir,
xlabel=r'$\mathrm{X}$', ylabel=r'$\mathrm{V(%c)}$'%dir, line_color=color)
def plot_shear_force(self, dir="all", subs=None):
"""
Returns a plot for Shear force along all three directions
present in the Beam object.
Parameters
==========
dir : string (default : "all")
Direction along which shear force plot is required.
If no direction is specified, all plots are displayed.
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, E, G, I, A, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.plot_shear_force()
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: 0 for x over (0.0, 20.0)
Plot[1]:Plot object containing:
[0]: cartesian line: -6*x**2 for x over (0.0, 20.0)
Plot[2]:Plot object containing:
[0]: cartesian line: -15*x for x over (0.0, 20.0)
"""
dir = dir.lower()
# For shear force along x direction
if dir == "x":
Px = self._plot_shear_force('x', subs)
return Px.show()
# For shear force along y direction
elif dir == "y":
Py = self._plot_shear_force('y', subs)
return Py.show()
# For shear force along z direction
elif dir == "z":
Pz = self._plot_shear_force('z', subs)
return Pz.show()
# For shear force along all direction
else:
Px = self._plot_shear_force('x', subs)
Py = self._plot_shear_force('y', subs)
Pz = self._plot_shear_force('z', subs)
return PlotGrid(3, 1, Px, Py, Pz)
def _plot_bending_moment(self, dir, subs=None):
bending_moment = self.bending_moment()
if dir == 'x':
dir_num = 0
color = 'g'
elif dir == 'y':
dir_num = 1
color = 'c'
elif dir == 'z':
dir_num = 2
color = 'm'
if subs is None:
subs = {}
for sym in bending_moment[dir_num].atoms(Symbol):
if sym != self.variable and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(bending_moment[dir_num].subs(subs), (self.variable, 0, length), show = False, title='Bending Moment along %c direction'%dir,
xlabel=r'$\mathrm{X}$', ylabel=r'$\mathrm{M(%c)}$'%dir, line_color=color)
def plot_bending_moment(self, dir="all", subs=None):
"""
Returns a plot for bending moment along all three directions
present in the Beam object.
Parameters
==========
dir : string (default : "all")
Direction along which bending moment plot is required.
If no direction is specified, all plots are displayed.
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, E, G, I, A, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.plot_bending_moment()
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: 0 for x over (0.0, 20.0)
Plot[1]:Plot object containing:
[0]: cartesian line: -15*x**2/2 for x over (0.0, 20.0)
Plot[2]:Plot object containing:
[0]: cartesian line: 2*x**3 for x over (0.0, 20.0)
"""
dir = dir.lower()
# For bending moment along x direction
if dir == "x":
Px = self._plot_bending_moment('x', subs)
return Px.show()
# For bending moment along y direction
elif dir == "y":
Py = self._plot_bending_moment('y', subs)
return Py.show()
# For bending moment along z direction
elif dir == "z":
Pz = self._plot_bending_moment('z', subs)
return Pz.show()
# For bending moment along all direction
else:
Px = self._plot_bending_moment('x', subs)
Py = self._plot_bending_moment('y', subs)
Pz = self._plot_bending_moment('z', subs)
return PlotGrid(3, 1, Px, Py, Pz)
def _plot_slope(self, dir, subs=None):
slope = self.slope()
if dir == 'x':
dir_num = 0
color = 'b'
elif dir == 'y':
dir_num = 1
color = 'm'
elif dir == 'z':
dir_num = 2
color = 'g'
if subs is None:
subs = {}
for sym in slope[dir_num].atoms(Symbol):
if sym != self.variable and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(slope[dir_num].subs(subs), (self.variable, 0, length), show = False, title='Slope along %c direction'%dir,
xlabel=r'$\mathrm{X}$', ylabel=r'$\mathrm{\theta(%c)}$'%dir, line_color=color)
def plot_slope(self, dir="all", subs=None):
"""
Returns a plot for Slope along all three directions
present in the Beam object.
Parameters
==========
dir : string (default : "all")
Direction along which Slope plot is required.
If no direction is specified, all plots are displayed.
subs : dictionary
Python dictionary containing Symbols as keys and their
corresponding values.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, 40, 21, 100, 25, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.solve_slope_deflection()
>>> b.plot_slope()
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: 0 for x over (0.0, 20.0)
Plot[1]:Plot object containing:
[0]: cartesian line: -x**3/1600 + 3*x**2/160 - x/8 for x over (0.0, 20.0)
Plot[2]:Plot object containing:
[0]: cartesian line: x**4/8000 - 19*x**2/172 + 52*x/43 for x over (0.0, 20.0)
"""
dir = dir.lower()
# For Slope along x direction
if dir == "x":
Px = self._plot_slope('x', subs)
return Px.show()
# For Slope along y direction
elif dir == "y":
Py = self._plot_slope('y', subs)
return Py.show()
# For Slope along z direction
elif dir == "z":
Pz = self._plot_slope('z', subs)
return Pz.show()
# For Slope along all direction
else:
Px = self._plot_slope('x', subs)
Py = self._plot_slope('y', subs)
Pz = self._plot_slope('z', subs)
return PlotGrid(3, 1, Px, Py, Pz)
def _plot_deflection(self, dir, subs=None):
deflection = self.deflection()
if dir == 'x':
dir_num = 0
color = 'm'
elif dir == 'y':
dir_num = 1
color = 'r'
elif dir == 'z':
dir_num = 2
color = 'c'
if subs is None:
subs = {}
for sym in deflection[dir_num].atoms(Symbol):
if sym != self.variable and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(deflection[dir_num].subs(subs), (self.variable, 0, length), show = False, title='Deflection along %c direction'%dir,
xlabel=r'$\mathrm{X}$', ylabel=r'$\mathrm{\delta(%c)}$'%dir, line_color=color)
def plot_deflection(self, dir="all", subs=None):
"""
Returns a plot for Deflection along all three directions
present in the Beam object.
Parameters
==========
dir : string (default : "all")
Direction along which deflection plot is required.
If no direction is specified, all plots are displayed.
subs : dictionary
Python dictionary containing Symbols as keys and their
corresponding values.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, 40, 21, 100, 25, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.solve_slope_deflection()
>>> b.plot_deflection()
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: 0 for x over (0.0, 20.0)
Plot[1]:Plot object containing:
[0]: cartesian line: x**5/40000 - 4013*x**3/90300 + 26*x**2/43 + 1520*x/903 for x over (0.0, 20.0)
Plot[2]:Plot object containing:
[0]: cartesian line: x**4/6400 - x**3/160 + 27*x**2/560 + 2*x/7 for x over (0.0, 20.0)
"""
dir = dir.lower()
# For deflection along x direction
if dir == "x":
Px = self._plot_deflection('x', subs)
return Px.show()
# For deflection along y direction
elif dir == "y":
Py = self._plot_deflection('y', subs)
return Py.show()
# For deflection along z direction
elif dir == "z":
Pz = self._plot_deflection('z', subs)
return Pz.show()
# For deflection along all direction
else:
Px = self._plot_deflection('x', subs)
Py = self._plot_deflection('y', subs)
Pz = self._plot_deflection('z', subs)
return PlotGrid(3, 1, Px, Py, Pz)
def plot_loading_results(self, dir='x', subs=None):
"""
Returns a subplot of Shear Force, Bending Moment,
Slope and Deflection of the Beam object along the direction specified.
Parameters
==========
dir : string (default : "x")
Direction along which plots are required.
If no direction is specified, plots along x-axis are displayed.
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, E, G, I, A, x)
>>> subs = {E:40, G:21, I:100, A:25}
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.solve_slope_deflection()
>>> b.plot_loading_results('y',subs)
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: -6*x**2 for x over (0.0, 20.0)
Plot[1]:Plot object containing:
[0]: cartesian line: -15*x**2/2 for x over (0.0, 20.0)
Plot[2]:Plot object containing:
[0]: cartesian line: -x**3/1600 + 3*x**2/160 - x/8 for x over (0.0, 20.0)
Plot[3]:Plot object containing:
[0]: cartesian line: x**5/40000 - 4013*x**3/90300 + 26*x**2/43 + 1520*x/903 for x over (0.0, 20.0)
"""
dir = dir.lower()
if subs is None:
subs = {}
ax1 = self._plot_shear_force(dir, subs)
ax2 = self._plot_bending_moment(dir, subs)
ax3 = self._plot_slope(dir, subs)
ax4 = self._plot_deflection(dir, subs)
return PlotGrid(4, 1, ax1, ax2, ax3, ax4)
def _plot_shear_stress(self, dir, subs=None):
shear_stress = self.shear_stress()
if dir == 'x':
dir_num = 0
color = 'r'
elif dir == 'y':
dir_num = 1
color = 'g'
elif dir == 'z':
dir_num = 2
color = 'b'
if subs is None:
subs = {}
for sym in shear_stress[dir_num].atoms(Symbol):
if sym != self.variable and sym not in subs:
raise ValueError('Value of %s was not passed.' %sym)
if self.length in subs:
length = subs[self.length]
else:
length = self.length
return plot(shear_stress[dir_num].subs(subs), (self.variable, 0, length), show = False, title='Shear stress along %c direction'%dir,
xlabel=r'$\mathrm{X}$', ylabel=r'$\tau(%c)$'%dir, line_color=color)
def plot_shear_stress(self, dir="all", subs=None):
"""
Returns a plot for Shear Stress along all three directions
present in the Beam object.
Parameters
==========
dir : string (default : "all")
Direction along which shear stress plot is required.
If no direction is specified, all plots are displayed.
subs : dictionary
Python dictionary containing Symbols as key and their
corresponding values.
Examples
========
There is a beam of length 20 meters and area of cross section 2 square
meters. It it supported by rollers at of its end. A linear load having
slope equal to 12 is applied along y-axis. A constant distributed load
of magnitude 15 N is applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, E, G, I, 2, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.plot_shear_stress()
PlotGrid object containing:
Plot[0]:Plot object containing:
[0]: cartesian line: 0 for x over (0.0, 20.0)
Plot[1]:Plot object containing:
[0]: cartesian line: -3*x**2 for x over (0.0, 20.0)
Plot[2]:Plot object containing:
[0]: cartesian line: -15*x/2 for x over (0.0, 20.0)
"""
dir = dir.lower()
# For shear stress along x direction
if dir == "x":
Px = self._plot_shear_stress('x', subs)
return Px.show()
# For shear stress along y direction
elif dir == "y":
Py = self._plot_shear_stress('y', subs)
return Py.show()
# For shear stress along z direction
elif dir == "z":
Pz = self._plot_shear_stress('z', subs)
return Pz.show()
# For shear stress along all direction
else:
Px = self._plot_shear_stress('x', subs)
Py = self._plot_shear_stress('y', subs)
Pz = self._plot_shear_stress('z', subs)
return PlotGrid(3, 1, Px, Py, Pz)
def _max_shear_force(self, dir):
"""
Helper function for max_shear_force().
"""
dir = dir.lower()
if dir == 'x':
dir_num = 0
elif dir == 'y':
dir_num = 1
elif dir == 'z':
dir_num = 2
if not self.shear_force()[dir_num]:
return (0,0)
# To restrict the range within length of the Beam
load_curve = Piecewise((float("nan"), self.variable<=0),
(self._load_vector[dir_num], self.variable<self.length),
(float("nan"), True))
points = solve(load_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
points.append(0)
points.append(self.length)
shear_curve = self.shear_force()[dir_num]
shear_values = [shear_curve.subs(self.variable, x) for x in points]
shear_values = list(map(abs, shear_values))
max_shear = max(shear_values)
return (points[shear_values.index(max_shear)], max_shear)
def max_shear_force(self):
"""
Returns point of max shear force and its corresponding shear value
along all directions in a Beam object as a list.
solve_for_reaction_loads() must be called before using this function.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, 40, 21, 100, 25, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.max_shear_force()
[(0, 0), (20, 2400), (20, 300)]
"""
max_shear = []
max_shear.append(self._max_shear_force('x'))
max_shear.append(self._max_shear_force('y'))
max_shear.append(self._max_shear_force('z'))
return max_shear
def _max_bending_moment(self, dir):
"""
Helper function for max_bending_moment().
"""
dir = dir.lower()
if dir == 'x':
dir_num = 0
elif dir == 'y':
dir_num = 1
elif dir == 'z':
dir_num = 2
if not self.bending_moment()[dir_num]:
return (0,0)
# To restrict the range within length of the Beam
shear_curve = Piecewise((float("nan"), self.variable<=0),
(self.shear_force()[dir_num], self.variable<self.length),
(float("nan"), True))
points = solve(shear_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
points.append(0)
points.append(self.length)
bending_moment_curve = self.bending_moment()[dir_num]
bending_moments = [bending_moment_curve.subs(self.variable, x) for x in points]
bending_moments = list(map(abs, bending_moments))
max_bending_moment = max(bending_moments)
return (points[bending_moments.index(max_bending_moment)], max_bending_moment)
def max_bending_moment(self):
"""
Returns point of max bending moment and its corresponding bending moment value
along all directions in a Beam object as a list.
solve_for_reaction_loads() must be called before using this function.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, 40, 21, 100, 25, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.max_bending_moment()
[(0, 0), (20, 3000), (20, 16000)]
"""
max_bmoment = []
max_bmoment.append(self._max_bending_moment('x'))
max_bmoment.append(self._max_bending_moment('y'))
max_bmoment.append(self._max_bending_moment('z'))
return max_bmoment
max_bmoment = max_bending_moment
def _max_deflection(self, dir):
"""
Helper function for max_Deflection()
"""
dir = dir.lower()
if dir == 'x':
dir_num = 0
elif dir == 'y':
dir_num = 1
elif dir == 'z':
dir_num = 2
if not self.deflection()[dir_num]:
return (0,0)
# To restrict the range within length of the Beam
slope_curve = Piecewise((float("nan"), self.variable<=0),
(self.slope()[dir_num], self.variable<self.length),
(float("nan"), True))
points = solve(slope_curve.rewrite(Piecewise), self.variable,
domain=S.Reals)
points.append(0)
points.append(self._length)
deflection_curve = self.deflection()[dir_num]
deflections = [deflection_curve.subs(self.variable, x) for x in points]
deflections = list(map(abs, deflections))
max_def = max(deflections)
return (points[deflections.index(max_def)], max_def)
def max_deflection(self):
"""
Returns point of max deflection and its corresponding deflection value
along all directions in a Beam object as a list.
solve_for_reaction_loads() and solve_slope_deflection() must be called
before using this function.
Examples
========
There is a beam of length 20 meters. It it supported by rollers
at of its end. A linear load having slope equal to 12 is applied
along y-axis. A constant distributed load of magnitude 15 N is
applied from start till its end along z-axis.
.. plot::
:context: close-figs
:format: doctest
:include-source: True
>>> from sympy.physics.continuum_mechanics.beam import Beam3D
>>> from sympy import symbols
>>> l, E, G, I, A, x = symbols('l, E, G, I, A, x')
>>> b = Beam3D(20, 40, 21, 100, 25, x)
>>> b.apply_load(15, start=0, order=0, dir="z")
>>> b.apply_load(12*x, start=0, order=0, dir="y")
>>> b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
>>> R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
>>> b.apply_load(R1, start=0, order=-1, dir="z")
>>> b.apply_load(R2, start=20, order=-1, dir="z")
>>> b.apply_load(R3, start=0, order=-1, dir="y")
>>> b.apply_load(R4, start=20, order=-1, dir="y")
>>> b.solve_for_reaction_loads(R1, R2, R3, R4)
>>> b.solve_slope_deflection()
>>> b.max_deflection()
[(0, 0), (10, 495/14), (-10 + 10*sqrt(10793)/43, (10 - 10*sqrt(10793)/43)**3/160 - 20/7 + (10 - 10*sqrt(10793)/43)**4/6400 + 20*sqrt(10793)/301 + 27*(10 - 10*sqrt(10793)/43)**2/560)]
"""
max_def = []
max_def.append(self._max_deflection('x'))
max_def.append(self._max_deflection('y'))
max_def.append(self._max_deflection('z'))
return max_def
|
ca5726541dc97ba16473b756e5fdab76a3c5da8e45959f8b8dcff88b8dd19cd6 | """
This module can be used to solve problems related
to 2D Trusses.
"""
from cmath import inf
from sympy.core.add import Add
from sympy.core.mul import Mul
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy import Matrix, pi
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.matrices.dense import zeros
import math
class Truss:
"""
A Truss is an assembly of members such as beams,
connected by nodes, that create a rigid structure.
In engineering, a truss is a structure that
consists of two-force members only.
Trusses are extremely important in engineering applications
and can be seen in numerous real-world applications like bridges.
Examples
========
There is a Truss consisting of four nodes and five
members connecting the nodes. A force P acts
downward on the node D and there also exist pinned
and roller joints on the nodes A and B respectively.
.. image:: truss_example.png
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node("node_1", 0, 0)
>>> t.add_node("node_2", 6, 0)
>>> t.add_node("node_3", 2, 2)
>>> t.add_node("node_4", 2, 0)
>>> t.add_member("member_1", "node_1", "node_4")
>>> t.add_member("member_2", "node_2", "node_4")
>>> t.add_member("member_3", "node_1", "node_3")
>>> t.add_member("member_4", "node_2", "node_3")
>>> t.add_member("member_5", "node_3", "node_4")
>>> t.apply_load("node_4", magnitude=10, direction=270)
>>> t.apply_support("node_1", type="fixed")
>>> t.apply_support("node_2", type="roller")
"""
def __init__(self):
"""
Initializes the class
"""
self._nodes = []
self._members = {}
self._loads = {}
self._supports = {}
self._node_labels = []
self._node_positions = []
self._node_position_x = []
self._node_position_y = []
self._nodes_occupied = {}
self._reaction_loads = {}
self._internal_forces = {}
self._node_coordinates = {}
@property
def nodes(self):
"""
Returns the nodes of the truss along with their positions.
"""
return self._nodes
@property
def node_labels(self):
"""
Returns the node labels of the truss.
"""
return self._node_labels
@property
def node_positions(self):
"""
Returns the positions of the nodes of the truss.
"""
return self._node_positions
@property
def members(self):
"""
Returns the members of the truss along with the start and end points.
"""
return self._members
@property
def member_labels(self):
"""
Returns the members of the truss along with the start and end points.
"""
return self._member_labels
@property
def supports(self):
"""
Returns the nodes with provided supports along with the kind of support provided i.e.
pinned or roller.
"""
return self._supports
@property
def loads(self):
"""
Returns the loads acting on the truss.
"""
return self._loads
@property
def reaction_loads(self):
"""
Returns the reaction forces for all supports which are all initialized to 0.
"""
return self._reaction_loads
@property
def internal_forces(self):
"""
Returns the internal forces for all members which are all initialized to 0.
"""
return self._internal_forces
def add_node(self, label, x, y):
"""
This method adds a node to the truss along with its name/label and its location.
Parameters
==========
label: String or a Symbol
The label for a node. It is the only way to identify a particular node.
x: Sympifyable
The x-coordinate of the position of the node.
y: Sympifyable
The y-coordinate of the position of the node.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.nodes
[('A', 0, 0)]
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
"""
x = sympify(x)
y = sympify(y)
if label in self._node_labels:
raise ValueError("Node needs to have a unique label")
elif x in self._node_position_x and y in self._node_position_y and self._node_position_x.index(x)==self._node_position_y.index(y):
raise ValueError("A node already exists at the given position")
else :
self._nodes.append((label, x, y))
self._node_labels.append(label)
self._node_positions.append((x, y))
self._node_position_x.append(x)
self._node_position_y.append(y)
self._node_coordinates[label] = [x, y]
def remove_node(self, label):
"""
This method removes a node from the truss.
Parameters
==========
label: String or Symbol
The label of the node to be removed.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.nodes
[('A', 0, 0)]
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
>>> t.remove_node('A')
>>> t.nodes
[('B', 3, 0)]
"""
for i in range(len(self.nodes)):
if self._node_labels[i] == label:
x = self._node_position_x[i]
y = self._node_position_y[i]
if label not in self._node_labels:
raise ValueError("No such node exists in the truss")
else:
members_duplicate = self._members.copy()
for member in members_duplicate:
if label == self._members[member][0] or label == self._members[member][1]:
raise ValueError("The node given has members already attached to it")
self._nodes.remove((label, x, y))
self._node_labels.remove(label)
self._node_positions.remove((x, y))
self._node_position_x.remove(x)
self._node_position_y.remove(y)
if label in list(self._loads):
self._loads.pop(label)
if label in list(self._supports):
self._supports.pop(label)
self._node_coordinates.pop(label)
def add_member(self, label, start, end):
"""
This method adds a member between any two nodes in the given truss.
Parameters
==========
label: String or Symbol
The label for a member. It is the only way to identify a particular member.
start: String or Symbol
The label of the starting point/node of the member.
end: String or Symbol
The label of the ending point/node of the member.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.add_node('C', 2, 2)
>>> t.add_member('AB', 'A', 'B')
>>> t.members
{'AB': ['A', 'B']}
"""
if start not in self._node_labels or end not in self._node_labels or start==end:
raise ValueError("The start and end points of the member must be unique nodes")
elif label in list(self._members):
raise ValueError("A member with the same label already exists for the truss")
elif self._nodes_occupied.get(tuple([start, end])):
raise ValueError("A member already exists between the two nodes")
else:
self._members[label] = [start, end]
self._nodes_occupied[start, end] = True
self._nodes_occupied[end, start] = True
self._internal_forces[label] = 0
def remove_member(self, label):
"""
This method removes a member from the given truss.
Parameters
==========
label: String or Symbol
The label for the member to be removed.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.add_node('C', 2, 2)
>>> t.add_member('AB', 'A', 'B')
>>> t.add_member('AC', 'A', 'C')
>>> t.add_member('BC', 'B', 'C')
>>> t.members
{'AB': ['A', 'B'], 'AC': ['A', 'C'], 'BC': ['B', 'C']}
>>> t.remove_member('AC')
>>> t.members
{'AB': ['A', 'B'], 'BC': ['B', 'C']}
"""
if label not in list(self._members):
raise ValueError("No such member exists in the Truss")
else:
self._nodes_occupied.pop(tuple([self._members[label][0], self._members[label][1]]))
self._nodes_occupied.pop(tuple([self._members[label][1], self._members[label][0]]))
self._members.pop(label)
self._internal_forces.pop(label)
def change_node_label(self, label, new_label):
"""
This method changes the label of a node.
Parameters
==========
label: String or Symbol
The label of the node for which the label has
to be changed.
new_label: String or Symbol
The new label of the node.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
>>> t.change_node_label('A', 'C')
>>> t.nodes
[('C', 0, 0), ('B', 3, 0)]
"""
if label not in self._node_labels:
raise ValueError("No such node exists for the Truss")
elif new_label in self._node_labels:
raise ValueError("A node with the given label already exists")
else:
for node in self._nodes:
if node[0] == label:
self._nodes[self._nodes.index((label, node[1], node[2]))] = (new_label, node[1], node[2])
self._node_labels[self._node_labels.index(node[0])] = new_label
self._node_coordinates[new_label] = self._node_coordinates[label]
self._node_coordinates.pop(label)
if node[0] in list(self._supports):
self._supports[new_label] = self._supports[node[0]]
self._supports.pop(node[0])
if new_label in list(self._supports):
if self._supports[new_label] == 'pinned':
if 'R_'+str(label)+'_x' in list(self._reaction_loads) and 'R_'+str(label)+'_y' in list(self._reaction_loads):
self._reaction_loads['R_'+str(new_label)+'_x'] = self._reaction_loads['R_'+str(label)+'_x']
self._reaction_loads['R_'+str(new_label)+'_y'] = self._reaction_loads['R_'+str(label)+'_y']
self._reaction_loads.pop('R_'+str(label)+'_x')
self._reaction_loads.pop('R_'+str(label)+'_y')
self._loads[new_label] = self._loads[label]
for load in self._loads[new_label]:
if load[1] == 90:
load[0] -= Symbol('R_'+str(label)+'_y')
if load[0] == 0:
self._loads[label].remove(load)
break
for load in self._loads[new_label]:
if load[1] == 0:
load[0] -= Symbol('R_'+str(label)+'_x')
if load[0] == 0:
self._loads[label].remove(load)
break
self.apply_load(new_label, Symbol('R_'+str(new_label)+'_x'), 0)
self.apply_load(new_label, Symbol('R_'+str(new_label)+'_y'), 90)
self._loads.pop(label)
elif self._supports[new_label] == 'roller':
self._loads[new_label] = self._loads[label]
for load in self._loads[label]:
if load[1] == 90:
load[0] -= Symbol('R_'+str(label)+'_y')
if load[0] == 0:
self._loads[label].remove(load)
break
self.apply_load(new_label, Symbol('R_'+str(new_label)+'_y'), 90)
self._loads.pop(label)
else:
if label in list(self._loads):
self._loads[new_label] = self._loads[label]
self._loads.pop(label)
for member in list(self._members):
if self._members[member][0] == node[0]:
self._members[member][0] = new_label
self._nodes_occupied[(new_label, self._members[member][1])] = True
self._nodes_occupied[(self._members[member][1], new_label)] = True
self._nodes_occupied.pop(tuple([label, self._members[member][1]]))
self._nodes_occupied.pop(tuple([self._members[member][1], label]))
elif self._members[member][1] == node[0]:
self._members[member][1] = new_label
self._nodes_occupied[(self._members[member][0], new_label)] = True
self._nodes_occupied[(new_label, self._members[member][0])] = True
self._nodes_occupied.pop(tuple([self._members[member][0], label]))
self._nodes_occupied.pop(tuple([label, self._members[member][0]]))
def change_member_label(self, label, new_label):
"""
This method changes the label of a member.
Parameters
==========
label: String or Symbol
The label of the member for which the label has
to be changed.
new_label: String or Symbol
The new label of the member.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.nodes
[('A', 0, 0), ('B', 3, 0)]
>>> t.change_node_label('A', 'C')
>>> t.nodes
[('C', 0, 0), ('B', 3, 0)]
>>> t.add_member('BC', 'B', 'C')
>>> t.members
{'BC': ['B', 'C']}
>>> t.change_member_label('BC', 'BC_new')
>>> t.members
{'BC_new': ['B', 'C']}
"""
if label not in list(self._members):
raise ValueError("No such member exists for the Truss")
else:
members_duplicate = list(self._members).copy()
for member in members_duplicate:
if member == label:
self._members[new_label] = [self._members[member][0], self._members[member][1]]
self._members.pop(label)
self._internal_forces[new_label] = self._internal_forces[label]
self._internal_forces.pop(label)
def apply_load(self, location, magnitude, direction):
"""
This method applies an external load at a particular node
Parameters
==========
location: String or Symbol
Label of the Node at which load is applied.
magnitude: Sympifyable
Magnitude of the load applied. It must always be positive and any changes in
the direction of the load are not reflected here.
direction: Sympifyable
The angle, in degrees, that the load vector makes with the horizontal
in the counter-clockwise direction. It takes the values 0 to 360,
inclusive.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> from sympy import symbols
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> P = symbols('P')
>>> t.apply_load('A', P, 90)
>>> t.apply_load('A', P/2, 45)
>>> t.apply_load('A', P/4, 90)
>>> t.loads
{'A': [[P, 90], [P/2, 45], [P/4, 90]]}
"""
magnitude = sympify(magnitude)
direction = sympify(direction)
if location not in self.node_labels:
raise ValueError("Load must be applied at a known node")
else:
if location in list(self._loads):
self._loads[location].append([magnitude, direction])
else:
self._loads[location] = [[magnitude, direction]]
def remove_load(self, location, magnitude, direction):
"""
This method removes an already
present external load at a particular node
Parameters
==========
location: String or Symbol
Label of the Node at which load is applied and is to be removed.
magnitude: Sympifyable
Magnitude of the load applied.
direction: Sympifyable
The angle, in degrees, that the load vector makes with the horizontal
in the counter-clockwise direction. It takes the values 0 to 360,
inclusive.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> from sympy import symbols
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> P = symbols('P')
>>> t.apply_load('A', P, 90)
>>> t.apply_load('A', P/2, 45)
>>> t.apply_load('A', P/4, 90)
>>> t.loads
{'A': [[P, 90], [P/2, 45], [P/4, 90]]}
>>> t.remove_load('A', P/4, 90)
>>> t.loads
{'A': [[P, 90], [P/2, 45]]}
"""
magnitude = sympify(magnitude)
direction = sympify(direction)
if location not in self.node_labels:
raise ValueError("Load must be removed from a known node")
else:
if [magnitude, direction] not in self._loads[location]:
raise ValueError("No load of this magnitude and direction has been applied at this node")
else:
self._loads[location].remove([magnitude, direction])
if self._loads[location] == []:
self._loads.pop(location)
def apply_support(self, location, type):
"""
This method adds a pinned or roller support at a particular node
Parameters
==========
location: String or Symbol
Label of the Node at which support is added.
type: String
Type of the support being provided at the node.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.apply_support('A', 'pinned')
>>> t.supports
{'A': 'pinned'}
"""
if location not in self._node_labels:
raise ValueError("Support must be added on a known node")
else:
if location not in list(self._supports):
if type == 'pinned':
self.apply_load(location, Symbol('R_'+str(location)+'_x'), 0)
self.apply_load(location, Symbol('R_'+str(location)+'_y'), 90)
elif type == 'roller':
self.apply_load(location, Symbol('R_'+str(location)+'_y'), 90)
elif self._supports[location] == 'pinned':
if type == 'roller':
self.remove_load(location, Symbol('R_'+str(location)+'_x'), 0)
elif self._supports[location] == 'roller':
if type == 'pinned':
self.apply_load(location, Symbol('R_'+str(location)+'_x'), 0)
self._supports[location] = type
def remove_support(self, location):
"""
This method removes support from a particular node
Parameters
==========
location: String or Symbol
Label of the Node at which support is to be removed.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node('A', 0, 0)
>>> t.add_node('B', 3, 0)
>>> t.apply_support('A', 'pinned')
>>> t.supports
{'A': 'pinned'}
>>> t.remove_support('A')
>>> t.supports
{}
"""
if location not in self._node_labels:
raise ValueError("No such node exists in the Truss")
elif location not in list(self._supports):
raise ValueError("No support has been added to the given node")
else:
if self._supports[location] == 'pinned':
self.remove_load(location, Symbol('R_'+str(location)+'_x'), 0)
self.remove_load(location, Symbol('R_'+str(location)+'_y'), 90)
elif self._supports[location] == 'roller':
self.remove_load(location, Symbol('R_'+str(location)+'_y'), 90)
self._supports.pop(location)
def solve(self):
"""
This method solves for all reaction forces of all supports and all internal forces
of all the members in the truss, provided the Truss is solvable.
A Truss is solvable if the following condition is met,
2n >= r + m
Where n is the number of nodes, r is the number of reaction forces, where each pinned
support has 2 reaction forces and each roller has 1, and m is the number of members.
The given condition is derived from the fact that a system of equations is solvable
only when the number of variables is lesser than or equal to the number of equations.
Equilibrium Equations in x and y directions give two equations per node giving 2n number
equations. However, the truss needs to be stable as well and may be unstable if 2n > r + m.
The number of variables is simply the sum of the number of reaction forces and member
forces.
.. note::
The sign convention for the internal forces present in a member revolves around whether each
force is compressive or tensile. While forming equations for each node, internal force due
to a member on the node is assumed to be away from the node i.e. each force is assumed to
be compressive by default. Hence, a positive value for an internal force implies the
presence of compressive force in the member and a negative value implies a tensile force.
Examples
========
>>> from sympy.physics.continuum_mechanics.truss import Truss
>>> t = Truss()
>>> t.add_node("node_1", 0, 0)
>>> t.add_node("node_2", 6, 0)
>>> t.add_node("node_3", 2, 2)
>>> t.add_node("node_4", 2, 0)
>>> t.add_member("member_1", "node_1", "node_4")
>>> t.add_member("member_2", "node_2", "node_4")
>>> t.add_member("member_3", "node_1", "node_3")
>>> t.add_member("member_4", "node_2", "node_3")
>>> t.add_member("member_5", "node_3", "node_4")
>>> t.apply_load("node_4", magnitude=10, direction=270)
>>> t.apply_support("node_1", type="pinned")
>>> t.apply_support("node_2", type="roller")
>>> t.solve()
>>> t.reaction_loads
{'R_node_1_x': 0, 'R_node_1_y': 6.66666666666667, 'R_node_2_y': 3.33333333333333}
>>> t.internal_forces
{'member_1': 6.66666666666666, 'member_2': 6.66666666666667, 'member_3': -6.66666666666667*sqrt(2), 'member_4': -3.33333333333333*sqrt(5), 'member_5': 10.0}
"""
count_reaction_loads = 0
for node in self._nodes:
if node[0] in list(self._supports):
if self._supports[node[0]]=='pinned':
count_reaction_loads += 2
elif self._supports[node[0]]=='roller':
count_reaction_loads += 1
if 2*len(self._nodes) != len(self._members) + count_reaction_loads:
raise ValueError("The given truss cannot be solved")
coefficients_matrix = [[0 for i in range(2*len(self._nodes))] for j in range(2*len(self._nodes))]
load_matrix = zeros(2*len(self.nodes), 1)
load_matrix_row = 0
for node in self._nodes:
if node[0] in list(self._loads):
for load in self._loads[node[0]]:
if load[0]!=Symbol('R_'+str(node[0])+'_x') and load[0]!=Symbol('R_'+str(node[0])+'_y'):
load_matrix[load_matrix_row] -= load[0]*math.cos(pi*load[1]/180)
load_matrix[load_matrix_row + 1] -= load[0]*math.sin(pi*load[1]/180)
load_matrix_row += 2
cols = 0
row = 0
for node in self._nodes:
if node[0] in list(self._supports):
if self._supports[node[0]]=='pinned':
coefficients_matrix[row][cols] += 1
coefficients_matrix[row+1][cols+1] += 1
cols += 2
elif self._supports[node[0]]=='roller':
coefficients_matrix[row+1][cols] += 1
cols += 1
row += 2
for member in list(self._members):
start = self._members[member][0]
end = self._members[member][1]
length = sqrt((self._node_coordinates[start][0]-self._node_coordinates[end][0])**2 + (self._node_coordinates[start][1]-self._node_coordinates[end][1])**2)
start_index = self._node_labels.index(start)
end_index = self._node_labels.index(end)
horizontal_component_start = (self._node_coordinates[end][0]-self._node_coordinates[start][0])/length
vertical_component_start = (self._node_coordinates[end][1]-self._node_coordinates[start][1])/length
horizontal_component_end = (self._node_coordinates[start][0]-self._node_coordinates[end][0])/length
vertical_component_end = (self._node_coordinates[start][1]-self._node_coordinates[end][1])/length
coefficients_matrix[start_index*2][cols] += horizontal_component_start
coefficients_matrix[start_index*2+1][cols] += vertical_component_start
coefficients_matrix[end_index*2][cols] += horizontal_component_end
coefficients_matrix[end_index*2+1][cols] += vertical_component_end
cols += 1
forces_matrix = (Matrix(coefficients_matrix)**-1)*load_matrix
self._reaction_loads = {}
i = 0
min_load = inf
for node in self._nodes:
if node[0] in list(self._loads):
for load in self._loads[node[0]]:
if type(load[0]) not in [Symbol, Mul, Add]:
min_load = min(min_load, load[0])
for j in range(len(forces_matrix)):
if type(forces_matrix[j]) not in [Symbol, Mul, Add]:
if abs(forces_matrix[j]/min_load) <1E-10:
forces_matrix[j] = 0
for node in self._nodes:
if node[0] in list(self._supports):
if self._supports[node[0]]=='pinned':
self._reaction_loads['R_'+str(node[0])+'_x'] = forces_matrix[i]
self._reaction_loads['R_'+str(node[0])+'_y'] = forces_matrix[i+1]
i += 2
elif self._supports[node[0]]=='roller':
self._reaction_loads['R_'+str(node[0])+'_y'] = forces_matrix[i]
i += 1
for member in list(self._members):
self._internal_forces[member] = forces_matrix[i]
i += 1
return
|
88aa880a3d1b3fb10ad2c4ae605b0f74f9034db1c66ba27f2cc3d9033bf58f8a | from sympy.matrices.dense import eye, Matrix
from sympy.tensor.tensor import tensor_indices, TensorHead, tensor_heads, \
TensExpr, canon_bp
from sympy.physics.hep.gamma_matrices import GammaMatrix as G, LorentzIndex, \
kahane_simplify, gamma_trace, _simplify_single_line, simplify_gamma_expression
from sympy import Symbol
def _is_tensor_eq(arg1, arg2):
arg1 = canon_bp(arg1)
arg2 = canon_bp(arg2)
if isinstance(arg1, TensExpr):
return arg1.equals(arg2)
elif isinstance(arg2, TensExpr):
return arg2.equals(arg1)
return arg1 == arg2
def execute_gamma_simplify_tests_for_function(tfunc, D):
"""
Perform tests to check if sfunc is able to simplify gamma matrix expressions.
Parameters
==========
`sfunc` a function to simplify a `TIDS`, shall return the simplified `TIDS`.
`D` the number of dimension (in most cases `D=4`).
"""
mu, nu, rho, sigma = tensor_indices("mu, nu, rho, sigma", LorentzIndex)
a1, a2, a3, a4, a5, a6 = tensor_indices("a1:7", LorentzIndex)
mu11, mu12, mu21, mu31, mu32, mu41, mu51, mu52 = tensor_indices("mu11, mu12, mu21, mu31, mu32, mu41, mu51, mu52", LorentzIndex)
mu61, mu71, mu72 = tensor_indices("mu61, mu71, mu72", LorentzIndex)
m0, m1, m2, m3, m4, m5, m6 = tensor_indices("m0:7", LorentzIndex)
def g(xx, yy):
return (G(xx)*G(yy) + G(yy)*G(xx))/2
# Some examples taken from Kahane's paper, 4 dim only:
if D == 4:
t = (G(a1)*G(mu11)*G(a2)*G(mu21)*G(-a1)*G(mu31)*G(-a2))
assert _is_tensor_eq(tfunc(t), -4*G(mu11)*G(mu31)*G(mu21) - 4*G(mu31)*G(mu11)*G(mu21))
t = (G(a1)*G(mu11)*G(mu12)*\
G(a2)*G(mu21)*\
G(a3)*G(mu31)*G(mu32)*\
G(a4)*G(mu41)*\
G(-a2)*G(mu51)*G(mu52)*\
G(-a1)*G(mu61)*\
G(-a3)*G(mu71)*G(mu72)*\
G(-a4))
assert _is_tensor_eq(tfunc(t), \
16*G(mu31)*G(mu32)*G(mu72)*G(mu71)*G(mu11)*G(mu52)*G(mu51)*G(mu12)*G(mu61)*G(mu21)*G(mu41) + 16*G(mu31)*G(mu32)*G(mu72)*G(mu71)*G(mu12)*G(mu51)*G(mu52)*G(mu11)*G(mu61)*G(mu21)*G(mu41) + 16*G(mu71)*G(mu72)*G(mu32)*G(mu31)*G(mu11)*G(mu52)*G(mu51)*G(mu12)*G(mu61)*G(mu21)*G(mu41) + 16*G(mu71)*G(mu72)*G(mu32)*G(mu31)*G(mu12)*G(mu51)*G(mu52)*G(mu11)*G(mu61)*G(mu21)*G(mu41))
# Fully Lorentz-contracted expressions, these return scalars:
def add_delta(ne):
return ne * eye(4) # DiracSpinorIndex.delta(DiracSpinorIndex.auto_left, -DiracSpinorIndex.auto_right)
t = (G(mu)*G(-mu))
ts = add_delta(D)
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu)*G(nu)*G(-mu)*G(-nu))
ts = add_delta(2*D - D**2) # -8
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu)*G(nu)*G(-nu)*G(-mu))
ts = add_delta(D**2) # 16
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu)*G(nu)*G(-rho)*G(-nu)*G(-mu)*G(rho))
ts = add_delta(4*D - 4*D**2 + D**3) # 16
assert _is_tensor_eq(tfunc(t), ts)
t = (G(mu)*G(nu)*G(rho)*G(-rho)*G(-nu)*G(-mu))
ts = add_delta(D**3) # 64
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1)*G(a2)*G(a3)*G(a4)*G(-a3)*G(-a1)*G(-a2)*G(-a4))
ts = add_delta(-8*D + 16*D**2 - 8*D**3 + D**4) # -32
assert _is_tensor_eq(tfunc(t), ts)
t = (G(-mu)*G(-nu)*G(-rho)*G(-sigma)*G(nu)*G(mu)*G(sigma)*G(rho))
ts = add_delta(-16*D + 24*D**2 - 8*D**3 + D**4) # 64
assert _is_tensor_eq(tfunc(t), ts)
t = (G(-mu)*G(nu)*G(-rho)*G(sigma)*G(rho)*G(-nu)*G(mu)*G(-sigma))
ts = add_delta(8*D - 12*D**2 + 6*D**3 - D**4) # -32
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1)*G(a2)*G(a3)*G(a4)*G(a5)*G(-a3)*G(-a2)*G(-a1)*G(-a5)*G(-a4))
ts = add_delta(64*D - 112*D**2 + 60*D**3 - 12*D**4 + D**5) # 256
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1)*G(a2)*G(a3)*G(a4)*G(a5)*G(-a3)*G(-a1)*G(-a2)*G(-a4)*G(-a5))
ts = add_delta(64*D - 120*D**2 + 72*D**3 - 16*D**4 + D**5) # -128
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1)*G(a2)*G(a3)*G(a4)*G(a5)*G(a6)*G(-a3)*G(-a2)*G(-a1)*G(-a6)*G(-a5)*G(-a4))
ts = add_delta(416*D - 816*D**2 + 528*D**3 - 144*D**4 + 18*D**5 - D**6) # -128
assert _is_tensor_eq(tfunc(t), ts)
t = (G(a1)*G(a2)*G(a3)*G(a4)*G(a5)*G(a6)*G(-a2)*G(-a3)*G(-a1)*G(-a6)*G(-a4)*G(-a5))
ts = add_delta(416*D - 848*D**2 + 584*D**3 - 172*D**4 + 22*D**5 - D**6) # -128
assert _is_tensor_eq(tfunc(t), ts)
# Expressions with free indices:
t = (G(mu)*G(nu)*G(rho)*G(sigma)*G(-mu))
assert _is_tensor_eq(tfunc(t), (-2*G(sigma)*G(rho)*G(nu) + (4-D)*G(nu)*G(rho)*G(sigma)))
t = (G(mu)*G(nu)*G(-mu))
assert _is_tensor_eq(tfunc(t), (2-D)*G(nu))
t = (G(mu)*G(nu)*G(rho)*G(-mu))
assert _is_tensor_eq(tfunc(t), 2*G(nu)*G(rho) + 2*G(rho)*G(nu) - (4-D)*G(nu)*G(rho))
t = 2*G(m2)*G(m0)*G(m1)*G(-m0)*G(-m1)
st = tfunc(t)
assert _is_tensor_eq(st, (D*(-2*D + 4))*G(m2))
t = G(m2)*G(m0)*G(m1)*G(-m0)*G(-m2)
st = tfunc(t)
assert _is_tensor_eq(st, ((-D + 2)**2)*G(m1))
t = G(m0)*G(m1)*G(m2)*G(m3)*G(-m1)
st = tfunc(t)
assert _is_tensor_eq(st, (D - 4)*G(m0)*G(m2)*G(m3) + 4*G(m0)*g(m2, m3))
t = G(m0)*G(m1)*G(m2)*G(m3)*G(-m1)*G(-m0)
st = tfunc(t)
assert _is_tensor_eq(st, ((D - 4)**2)*G(m2)*G(m3) + (8*D - 16)*g(m2, m3))
t = G(m2)*G(m0)*G(m1)*G(-m2)*G(-m0)
st = tfunc(t)
assert _is_tensor_eq(st, ((-D + 2)*(D - 4) + 4)*G(m1))
t = G(m3)*G(m1)*G(m0)*G(m2)*G(-m3)*G(-m0)*G(-m2)
st = tfunc(t)
assert _is_tensor_eq(st, (-4*D + (-D + 2)**2*(D - 4) + 8)*G(m1))
t = 2*G(m0)*G(m1)*G(m2)*G(m3)*G(-m0)
st = tfunc(t)
assert _is_tensor_eq(st, ((-2*D + 8)*G(m1)*G(m2)*G(m3) - 4*G(m3)*G(m2)*G(m1)))
t = G(m5)*G(m0)*G(m1)*G(m4)*G(m2)*G(-m4)*G(m3)*G(-m0)
st = tfunc(t)
assert _is_tensor_eq(st, (((-D + 2)*(-D + 4))*G(m5)*G(m1)*G(m2)*G(m3) + (2*D - 4)*G(m5)*G(m3)*G(m2)*G(m1)))
t = -G(m0)*G(m1)*G(m2)*G(m3)*G(-m0)*G(m4)
st = tfunc(t)
assert _is_tensor_eq(st, ((D - 4)*G(m1)*G(m2)*G(m3)*G(m4) + 2*G(m3)*G(m2)*G(m1)*G(m4)))
t = G(-m5)*G(m0)*G(m1)*G(m2)*G(m3)*G(m4)*G(-m0)*G(m5)
st = tfunc(t)
result1 = ((-D + 4)**2 + 4)*G(m1)*G(m2)*G(m3)*G(m4) +\
(4*D - 16)*G(m3)*G(m2)*G(m1)*G(m4) + (4*D - 16)*G(m4)*G(m1)*G(m2)*G(m3)\
+ 4*G(m2)*G(m1)*G(m4)*G(m3) + 4*G(m3)*G(m4)*G(m1)*G(m2) +\
4*G(m4)*G(m3)*G(m2)*G(m1)
# Kahane's algorithm yields this result, which is equivalent to `result1`
# in four dimensions, but is not automatically recognized as equal:
result2 = 8*G(m1)*G(m2)*G(m3)*G(m4) + 8*G(m4)*G(m3)*G(m2)*G(m1)
if D == 4:
assert _is_tensor_eq(st, (result1)) or _is_tensor_eq(st, (result2))
else:
assert _is_tensor_eq(st, (result1))
# and a few very simple cases, with no contracted indices:
t = G(m0)
st = tfunc(t)
assert _is_tensor_eq(st, t)
t = -7*G(m0)
st = tfunc(t)
assert _is_tensor_eq(st, t)
t = 224*G(m0)*G(m1)*G(-m2)*G(m3)
st = tfunc(t)
assert _is_tensor_eq(st, t)
def test_kahane_algorithm():
# Wrap this function to convert to and from TIDS:
def tfunc(e):
return _simplify_single_line(e)
execute_gamma_simplify_tests_for_function(tfunc, D=4)
def test_kahane_simplify1():
i0,i1,i2,i3,i4,i5,i6,i7,i8,i9,i10,i11,i12,i13,i14,i15 = tensor_indices('i0:16', LorentzIndex)
mu, nu, rho, sigma = tensor_indices("mu, nu, rho, sigma", LorentzIndex)
D = 4
t = G(i0)*G(i1)
r = kahane_simplify(t)
assert r.equals(t)
t = G(i0)*G(i1)*G(-i0)
r = kahane_simplify(t)
assert r.equals(-2*G(i1))
t = G(i0)*G(i1)*G(-i0)
r = kahane_simplify(t)
assert r.equals(-2*G(i1))
t = G(i0)*G(i1)
r = kahane_simplify(t)
assert r.equals(t)
t = G(i0)*G(i1)
r = kahane_simplify(t)
assert r.equals(t)
t = G(i0)*G(-i0)
r = kahane_simplify(t)
assert r.equals(4*eye(4))
t = G(i0)*G(-i0)
r = kahane_simplify(t)
assert r.equals(4*eye(4))
t = G(i0)*G(-i0)
r = kahane_simplify(t)
assert r.equals(4*eye(4))
t = G(i0)*G(i1)*G(-i0)
r = kahane_simplify(t)
assert r.equals(-2*G(i1))
t = G(i0)*G(i1)*G(-i0)*G(-i1)
r = kahane_simplify(t)
assert r.equals((2*D - D**2)*eye(4))
t = G(i0)*G(i1)*G(-i0)*G(-i1)
r = kahane_simplify(t)
assert r.equals((2*D - D**2)*eye(4))
t = G(i0)*G(-i0)*G(i1)*G(-i1)
r = kahane_simplify(t)
assert r.equals(16*eye(4))
t = (G(mu)*G(nu)*G(-nu)*G(-mu))
r = kahane_simplify(t)
assert r.equals(D**2*eye(4))
t = (G(mu)*G(nu)*G(-nu)*G(-mu))
r = kahane_simplify(t)
assert r.equals(D**2*eye(4))
t = (G(mu)*G(nu)*G(-nu)*G(-mu))
r = kahane_simplify(t)
assert r.equals(D**2*eye(4))
t = (G(mu)*G(nu)*G(-rho)*G(-nu)*G(-mu)*G(rho))
r = kahane_simplify(t)
assert r.equals((4*D - 4*D**2 + D**3)*eye(4))
t = (G(-mu)*G(-nu)*G(-rho)*G(-sigma)*G(nu)*G(mu)*G(sigma)*G(rho))
r = kahane_simplify(t)
assert r.equals((-16*D + 24*D**2 - 8*D**3 + D**4)*eye(4))
t = (G(-mu)*G(nu)*G(-rho)*G(sigma)*G(rho)*G(-nu)*G(mu)*G(-sigma))
r = kahane_simplify(t)
assert r.equals((8*D - 12*D**2 + 6*D**3 - D**4)*eye(4))
# Expressions with free indices:
t = (G(mu)*G(nu)*G(rho)*G(sigma)*G(-mu))
r = kahane_simplify(t)
assert r.equals(-2*G(sigma)*G(rho)*G(nu))
t = (G(mu)*G(-mu)*G(rho)*G(sigma))
r = kahane_simplify(t)
assert r.equals(4*G(rho)*G(sigma))
t = (G(rho)*G(sigma)*G(mu)*G(-mu))
r = kahane_simplify(t)
assert r.equals(4*G(rho)*G(sigma))
def test_gamma_matrix_class():
i, j, k = tensor_indices('i,j,k', LorentzIndex)
# define another type of TensorHead to see if exprs are correctly handled:
A = TensorHead('A', [LorentzIndex])
t = A(k)*G(i)*G(-i)
ts = simplify_gamma_expression(t)
assert _is_tensor_eq(ts, Matrix([
[4, 0, 0, 0],
[0, 4, 0, 0],
[0, 0, 4, 0],
[0, 0, 0, 4]])*A(k))
t = G(i)*A(k)*G(j)
ts = simplify_gamma_expression(t)
assert _is_tensor_eq(ts, A(k)*G(i)*G(j))
execute_gamma_simplify_tests_for_function(simplify_gamma_expression, D=4)
def test_gamma_matrix_trace():
g = LorentzIndex.metric
m0, m1, m2, m3, m4, m5, m6 = tensor_indices('m0:7', LorentzIndex)
n0, n1, n2, n3, n4, n5 = tensor_indices('n0:6', LorentzIndex)
# working in D=4 dimensions
D = 4
# traces of odd number of gamma matrices are zero:
t = G(m0)
t1 = gamma_trace(t)
assert t1.equals(0)
t = G(m0)*G(m1)*G(m2)
t1 = gamma_trace(t)
assert t1.equals(0)
t = G(m0)*G(m1)*G(-m0)
t1 = gamma_trace(t)
assert t1.equals(0)
t = G(m0)*G(m1)*G(m2)*G(m3)*G(m4)
t1 = gamma_trace(t)
assert t1.equals(0)
# traces without internal contractions:
t = G(m0)*G(m1)
t1 = gamma_trace(t)
assert _is_tensor_eq(t1, 4*g(m0, m1))
t = G(m0)*G(m1)*G(m2)*G(m3)
t1 = gamma_trace(t)
t2 = -4*g(m0, m2)*g(m1, m3) + 4*g(m0, m1)*g(m2, m3) + 4*g(m0, m3)*g(m1, m2)
assert _is_tensor_eq(t1, t2)
t = G(m0)*G(m1)*G(m2)*G(m3)*G(m4)*G(m5)
t1 = gamma_trace(t)
t2 = t1*g(-m0, -m5)
t2 = t2.contract_metric(g)
assert _is_tensor_eq(t2, D*gamma_trace(G(m1)*G(m2)*G(m3)*G(m4)))
# traces of expressions with internal contractions:
t = G(m0)*G(-m0)
t1 = gamma_trace(t)
assert t1.equals(4*D)
t = G(m0)*G(m1)*G(-m0)*G(-m1)
t1 = gamma_trace(t)
assert t1.equals(8*D - 4*D**2)
t = G(m0)*G(m1)*G(m2)*G(m3)*G(m4)*G(-m0)
t1 = gamma_trace(t)
t2 = (-4*D)*g(m1, m3)*g(m2, m4) + (4*D)*g(m1, m2)*g(m3, m4) + \
(4*D)*g(m1, m4)*g(m2, m3)
assert _is_tensor_eq(t1, t2)
t = G(-m5)*G(m0)*G(m1)*G(m2)*G(m3)*G(m4)*G(-m0)*G(m5)
t1 = gamma_trace(t)
t2 = (32*D + 4*(-D + 4)**2 - 64)*(g(m1, m2)*g(m3, m4) - \
g(m1, m3)*g(m2, m4) + g(m1, m4)*g(m2, m3))
assert _is_tensor_eq(t1, t2)
t = G(m0)*G(m1)*G(-m0)*G(m3)
t1 = gamma_trace(t)
assert t1.equals((-4*D + 8)*g(m1, m3))
# p, q = S1('p,q')
# ps = p(m0)*G(-m0)
# qs = q(m0)*G(-m0)
# t = ps*qs*ps*qs
# t1 = gamma_trace(t)
# assert t1 == 8*p(m0)*q(-m0)*p(m1)*q(-m1) - 4*p(m0)*p(-m0)*q(m1)*q(-m1)
t = G(m0)*G(m1)*G(m2)*G(m3)*G(m4)*G(m5)*G(-m0)*G(-m1)*G(-m2)*G(-m3)*G(-m4)*G(-m5)
t1 = gamma_trace(t)
assert t1.equals(-4*D**6 + 120*D**5 - 1040*D**4 + 3360*D**3 - 4480*D**2 + 2048*D)
t = G(m0)*G(m1)*G(n1)*G(m2)*G(n2)*G(m3)*G(m4)*G(-n2)*G(-n1)*G(-m0)*G(-m1)*G(-m2)*G(-m3)*G(-m4)
t1 = gamma_trace(t)
tresu = -7168*D + 16768*D**2 - 14400*D**3 + 5920*D**4 - 1232*D**5 + 120*D**6 - 4*D**7
assert t1.equals(tresu)
# checked with Mathematica
# In[1]:= <<Tracer.m
# In[2]:= Spur[l];
# In[3]:= GammaTrace[l, {m0},{m1},{n1},{m2},{n2},{m3},{m4},{n3},{n4},{m0},{m1},{m2},{m3},{m4}]
t = G(m0)*G(m1)*G(n1)*G(m2)*G(n2)*G(m3)*G(m4)*G(n3)*G(n4)*G(-m0)*G(-m1)*G(-m2)*G(-m3)*G(-m4)
t1 = gamma_trace(t)
# t1 = t1.expand_coeff()
c1 = -4*D**5 + 120*D**4 - 1200*D**3 + 5280*D**2 - 10560*D + 7808
c2 = -4*D**5 + 88*D**4 - 560*D**3 + 1440*D**2 - 1600*D + 640
assert _is_tensor_eq(t1, c1*g(n1, n4)*g(n2, n3) + c2*g(n1, n2)*g(n3, n4) + \
(-c1)*g(n1, n3)*g(n2, n4))
p, q = tensor_heads('p,q', [LorentzIndex])
ps = p(m0)*G(-m0)
qs = q(m0)*G(-m0)
p2 = p(m0)*p(-m0)
q2 = q(m0)*q(-m0)
pq = p(m0)*q(-m0)
t = ps*qs*ps*qs
r = gamma_trace(t)
assert _is_tensor_eq(r, 8*pq*pq - 4*p2*q2)
t = ps*qs*ps*qs*ps*qs
r = gamma_trace(t)
assert _is_tensor_eq(r, -12*p2*pq*q2 + 16*pq*pq*pq)
t = ps*qs*ps*qs*ps*qs*ps*qs
r = gamma_trace(t)
assert _is_tensor_eq(r, -32*pq*pq*p2*q2 + 32*pq*pq*pq*pq + 4*p2*p2*q2*q2)
t = 4*p(m1)*p(m0)*p(-m0)*q(-m1)*q(m2)*q(-m2)
assert _is_tensor_eq(gamma_trace(t), t)
t = ps*ps*ps*ps*ps*ps*ps*ps
r = gamma_trace(t)
assert r.equals(4*p2*p2*p2*p2)
def test_bug_13636():
"""Test issue 13636 regarding handling traces of sums of products
of GammaMatrix mixed with other factors."""
pi, ki, pf = tensor_heads("pi, ki, pf", [LorentzIndex])
i0, i1, i2, i3, i4 = tensor_indices("i0:5", LorentzIndex)
x = Symbol("x")
pis = pi(i2) * G(-i2)
kis = ki(i3) * G(-i3)
pfs = pf(i4) * G(-i4)
a = pfs * G(i0) * kis * G(i1) * pis * G(-i1) * kis * G(-i0)
b = pfs * G(i0) * kis * G(i1) * pis * x * G(-i0) * pi(-i1)
ta = gamma_trace(a)
tb = gamma_trace(b)
t_a_plus_b = gamma_trace(a + b)
assert ta == 4 * (
-4 * ki(i0) * ki(-i0) * pf(i1) * pi(-i1)
+ 8 * ki(i0) * ki(i1) * pf(-i0) * pi(-i1)
)
assert tb == -8 * x * ki(i0) * pf(-i0) * pi(i1) * pi(-i1)
assert t_a_plus_b == ta + tb
|
22e6d86733a443d74173247facb03d171524159ff1d415f5e8c294bf225a3a41 | from sympy.core.function import expand
from sympy.core.numbers import (Rational, pi)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.sets.sets import Interval
from sympy.simplify.simplify import simplify
from sympy.physics.continuum_mechanics.beam import Beam
from sympy.functions import SingularityFunction, Piecewise, meijerg, Abs, log
from sympy.testing.pytest import raises
from sympy.physics.units import meter, newton, kilo, giga, milli
from sympy.physics.continuum_mechanics.beam import Beam3D
from sympy.geometry import Circle, Polygon, Point2D, Triangle
from sympy.core.sympify import sympify
x = Symbol('x')
y = Symbol('y')
R1, R2 = symbols('R1, R2')
def test_Beam():
E = Symbol('E')
E_1 = Symbol('E_1')
I = Symbol('I')
I_1 = Symbol('I_1')
A = Symbol('A')
b = Beam(1, E, I)
assert b.length == 1
assert b.elastic_modulus == E
assert b.second_moment == I
assert b.variable == x
# Test the length setter
b.length = 4
assert b.length == 4
# Test the E setter
b.elastic_modulus = E_1
assert b.elastic_modulus == E_1
# Test the I setter
b.second_moment = I_1
assert b.second_moment is I_1
# Test the variable setter
b.variable = y
assert b.variable is y
# Test for all boundary conditions.
b.bc_deflection = [(0, 2)]
b.bc_slope = [(0, 1)]
assert b.boundary_conditions == {'deflection': [(0, 2)], 'slope': [(0, 1)]}
# Test for slope boundary condition method
b.bc_slope.extend([(4, 3), (5, 0)])
s_bcs = b.bc_slope
assert s_bcs == [(0, 1), (4, 3), (5, 0)]
# Test for deflection boundary condition method
b.bc_deflection.extend([(4, 3), (5, 0)])
d_bcs = b.bc_deflection
assert d_bcs == [(0, 2), (4, 3), (5, 0)]
# Test for updated boundary conditions
bcs_new = b.boundary_conditions
assert bcs_new == {
'deflection': [(0, 2), (4, 3), (5, 0)],
'slope': [(0, 1), (4, 3), (5, 0)]}
b1 = Beam(30, E, I)
b1.apply_load(-8, 0, -1)
b1.apply_load(R1, 10, -1)
b1.apply_load(R2, 30, -1)
b1.apply_load(120, 30, -2)
b1.bc_deflection = [(10, 0), (30, 0)]
b1.solve_for_reaction_loads(R1, R2)
# Test for finding reaction forces
p = b1.reaction_loads
q = {R1: 6, R2: 2}
assert p == q
# Test for load distribution function.
p = b1.load
q = -8*SingularityFunction(x, 0, -1) + 6*SingularityFunction(x, 10, -1) \
+ 120*SingularityFunction(x, 30, -2) + 2*SingularityFunction(x, 30, -1)
assert p == q
# Test for shear force distribution function
p = b1.shear_force()
q = 8*SingularityFunction(x, 0, 0) - 6*SingularityFunction(x, 10, 0) \
- 120*SingularityFunction(x, 30, -1) - 2*SingularityFunction(x, 30, 0)
assert p == q
# Test for shear stress distribution function
p = b1.shear_stress()
q = (8*SingularityFunction(x, 0, 0) - 6*SingularityFunction(x, 10, 0) \
- 120*SingularityFunction(x, 30, -1) \
- 2*SingularityFunction(x, 30, 0))/A
assert p==q
# Test for bending moment distribution function
p = b1.bending_moment()
q = 8*SingularityFunction(x, 0, 1) - 6*SingularityFunction(x, 10, 1) \
- 120*SingularityFunction(x, 30, 0) - 2*SingularityFunction(x, 30, 1)
assert p == q
# Test for slope distribution function
p = b1.slope()
q = -4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2) \
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) \
+ Rational(4000, 3)
assert p == q/(E*I)
# Test for deflection distribution function
p = b1.deflection()
q = x*Rational(4000, 3) - 4*SingularityFunction(x, 0, 3)/3 \
+ SingularityFunction(x, 10, 3) + 60*SingularityFunction(x, 30, 2) \
+ SingularityFunction(x, 30, 3)/3 - 12000
assert p == q/(E*I)
# Test using symbols
l = Symbol('l')
w0 = Symbol('w0')
w2 = Symbol('w2')
a1 = Symbol('a1')
c = Symbol('c')
c1 = Symbol('c1')
d = Symbol('d')
e = Symbol('e')
f = Symbol('f')
b2 = Beam(l, E, I)
b2.apply_load(w0, a1, 1)
b2.apply_load(w2, c1, -1)
b2.bc_deflection = [(c, d)]
b2.bc_slope = [(e, f)]
# Test for load distribution function.
p = b2.load
q = w0*SingularityFunction(x, a1, 1) + w2*SingularityFunction(x, c1, -1)
assert p == q
# Test for shear force distribution function
p = b2.shear_force()
q = -w0*SingularityFunction(x, a1, 2)/2 \
- w2*SingularityFunction(x, c1, 0)
assert p == q
# Test for shear stress distribution function
p = b2.shear_stress()
q = (-w0*SingularityFunction(x, a1, 2)/2 \
- w2*SingularityFunction(x, c1, 0))/A
assert p == q
# Test for bending moment distribution function
p = b2.bending_moment()
q = -w0*SingularityFunction(x, a1, 3)/6 - w2*SingularityFunction(x, c1, 1)
assert p == q
# Test for slope distribution function
p = b2.slope()
q = (w0*SingularityFunction(x, a1, 4)/24 + w2*SingularityFunction(x, c1, 2)/2)/(E*I) + (E*I*f - w0*SingularityFunction(e, a1, 4)/24 - w2*SingularityFunction(e, c1, 2)/2)/(E*I)
assert expand(p) == expand(q)
# Test for deflection distribution function
p = b2.deflection()
q = x*(E*I*f - w0*SingularityFunction(e, a1, 4)/24 \
- w2*SingularityFunction(e, c1, 2)/2)/(E*I) \
+ (w0*SingularityFunction(x, a1, 5)/120 \
+ w2*SingularityFunction(x, c1, 3)/6)/(E*I) \
+ (E*I*(-c*f + d) + c*w0*SingularityFunction(e, a1, 4)/24 \
+ c*w2*SingularityFunction(e, c1, 2)/2 \
- w0*SingularityFunction(c, a1, 5)/120 \
- w2*SingularityFunction(c, c1, 3)/6)/(E*I)
assert simplify(p - q) == 0
b3 = Beam(9, E, I, 2)
b3.apply_load(value=-2, start=2, order=2, end=3)
b3.bc_slope.append((0, 2))
C3 = symbols('C3')
C4 = symbols('C4')
p = b3.load
q = -2*SingularityFunction(x, 2, 2) + 2*SingularityFunction(x, 3, 0) \
+ 4*SingularityFunction(x, 3, 1) + 2*SingularityFunction(x, 3, 2)
assert p == q
p = b3.shear_force()
q = 2*SingularityFunction(x, 2, 3)/3 - 2*SingularityFunction(x, 3, 1) \
- 2*SingularityFunction(x, 3, 2) - 2*SingularityFunction(x, 3, 3)/3
assert p == q
p = b3.shear_stress()
q = SingularityFunction(x, 2, 3)/3 - 1*SingularityFunction(x, 3, 1) \
- 1*SingularityFunction(x, 3, 2) - 1*SingularityFunction(x, 3, 3)/3
assert p == q
p = b3.slope()
q = 2 - (SingularityFunction(x, 2, 5)/30 - SingularityFunction(x, 3, 3)/3 \
- SingularityFunction(x, 3, 4)/6 - SingularityFunction(x, 3, 5)/30)/(E*I)
assert p == q
p = b3.deflection()
q = 2*x - (SingularityFunction(x, 2, 6)/180 \
- SingularityFunction(x, 3, 4)/12 - SingularityFunction(x, 3, 5)/30 \
- SingularityFunction(x, 3, 6)/180)/(E*I)
assert p == q + C4
b4 = Beam(4, E, I, 3)
b4.apply_load(-3, 0, 0, end=3)
p = b4.load
q = -3*SingularityFunction(x, 0, 0) + 3*SingularityFunction(x, 3, 0)
assert p == q
p = b4.shear_force()
q = 3*SingularityFunction(x, 0, 1) \
- 3*SingularityFunction(x, 3, 1)
assert p == q
p = b4.shear_stress()
q = SingularityFunction(x, 0, 1) - SingularityFunction(x, 3, 1)
assert p == q
p = b4.slope()
q = -3*SingularityFunction(x, 0, 3)/6 + 3*SingularityFunction(x, 3, 3)/6
assert p == q/(E*I) + C3
p = b4.deflection()
q = -3*SingularityFunction(x, 0, 4)/24 + 3*SingularityFunction(x, 3, 4)/24
assert p == q/(E*I) + C3*x + C4
# can't use end with point loads
raises(ValueError, lambda: b4.apply_load(-3, 0, -1, end=3))
with raises(TypeError):
b4.variable = 1
def test_insufficient_bconditions():
# Test cases when required number of boundary conditions
# are not provided to solve the integration constants.
L = symbols('L', positive=True)
E, I, P, a3, a4 = symbols('E I P a3 a4')
b = Beam(L, E, I, base_char='a')
b.apply_load(R2, L, -1)
b.apply_load(R1, 0, -1)
b.apply_load(-P, L/2, -1)
b.solve_for_reaction_loads(R1, R2)
p = b.slope()
q = P*SingularityFunction(x, 0, 2)/4 - P*SingularityFunction(x, L/2, 2)/2 + P*SingularityFunction(x, L, 2)/4
assert p == q/(E*I) + a3
p = b.deflection()
q = P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
assert p == q/(E*I) + a3*x + a4
b.bc_deflection = [(0, 0)]
p = b.deflection()
q = a3*x + P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
assert p == q/(E*I)
b.bc_deflection = [(0, 0), (L, 0)]
p = b.deflection()
q = -L**2*P*x/16 + P*SingularityFunction(x, 0, 3)/12 - P*SingularityFunction(x, L/2, 3)/6 + P*SingularityFunction(x, L, 3)/12
assert p == q/(E*I)
def test_statically_indeterminate():
E = Symbol('E')
I = Symbol('I')
M1, M2 = symbols('M1, M2')
F = Symbol('F')
l = Symbol('l', positive=True)
b5 = Beam(l, E, I)
b5.bc_deflection = [(0, 0),(l, 0)]
b5.bc_slope = [(0, 0),(l, 0)]
b5.apply_load(R1, 0, -1)
b5.apply_load(M1, 0, -2)
b5.apply_load(R2, l, -1)
b5.apply_load(M2, l, -2)
b5.apply_load(-F, l/2, -1)
b5.solve_for_reaction_loads(R1, R2, M1, M2)
p = b5.reaction_loads
q = {R1: F/2, R2: F/2, M1: -F*l/8, M2: F*l/8}
assert p == q
def test_beam_units():
E = Symbol('E')
I = Symbol('I')
R1, R2 = symbols('R1, R2')
kN = kilo*newton
gN = giga*newton
b = Beam(8*meter, 200*gN/meter**2, 400*1000000*(milli*meter)**4)
b.apply_load(5*kN, 2*meter, -1)
b.apply_load(R1, 0*meter, -1)
b.apply_load(R2, 8*meter, -1)
b.apply_load(10*kN/meter, 4*meter, 0, end=8*meter)
b.bc_deflection = [(0*meter, 0*meter), (8*meter, 0*meter)]
b.solve_for_reaction_loads(R1, R2)
assert b.reaction_loads == {R1: -13750*newton, R2: -31250*newton}
b = Beam(3*meter, E*newton/meter**2, I*meter**4)
b.apply_load(8*kN, 1*meter, -1)
b.apply_load(R1, 0*meter, -1)
b.apply_load(R2, 3*meter, -1)
b.apply_load(12*kN*meter, 2*meter, -2)
b.bc_deflection = [(0*meter, 0*meter), (3*meter, 0*meter)]
b.solve_for_reaction_loads(R1, R2)
assert b.reaction_loads == {R1: newton*Rational(-28000, 3), R2: newton*Rational(4000, 3)}
assert b.deflection().subs(x, 1*meter) == 62000*meter/(9*E*I)
def test_variable_moment():
E = Symbol('E')
I = Symbol('I')
b = Beam(4, E, 2*(4 - x))
b.apply_load(20, 4, -1)
R, M = symbols('R, M')
b.apply_load(R, 0, -1)
b.apply_load(M, 0, -2)
b.bc_deflection = [(0, 0)]
b.bc_slope = [(0, 0)]
b.solve_for_reaction_loads(R, M)
assert b.slope().expand() == ((10*x*SingularityFunction(x, 0, 0)
- 10*(x - 4)*SingularityFunction(x, 4, 0))/E).expand()
assert b.deflection().expand() == ((5*x**2*SingularityFunction(x, 0, 0)
- 10*Piecewise((0, Abs(x)/4 < 1), (16*meijerg(((3, 1), ()), ((), (2, 0)), x/4), True))
+ 40*SingularityFunction(x, 4, 1))/E).expand()
b = Beam(4, E - x, I)
b.apply_load(20, 4, -1)
R, M = symbols('R, M')
b.apply_load(R, 0, -1)
b.apply_load(M, 0, -2)
b.bc_deflection = [(0, 0)]
b.bc_slope = [(0, 0)]
b.solve_for_reaction_loads(R, M)
assert b.slope().expand() == ((-80*(-log(-E) + log(-E + x))*SingularityFunction(x, 0, 0)
+ 80*(-log(-E + 4) + log(-E + x))*SingularityFunction(x, 4, 0) + 20*(-E*log(-E)
+ E*log(-E + x) + x)*SingularityFunction(x, 0, 0) - 20*(-E*log(-E + 4) + E*log(-E + x)
+ x - 4)*SingularityFunction(x, 4, 0))/I).expand()
def test_composite_beam():
E = Symbol('E')
I = Symbol('I')
b1 = Beam(2, E, 1.5*I)
b2 = Beam(2, E, I)
b = b1.join(b2, "fixed")
b.apply_load(-20, 0, -1)
b.apply_load(80, 0, -2)
b.apply_load(20, 4, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0)]
assert b.length == 4
assert b.second_moment == Piecewise((1.5*I, x <= 2), (I, x <= 4))
assert b.slope().subs(x, 4) == 120.0/(E*I)
assert b.slope().subs(x, 2) == 80.0/(E*I)
assert int(b.deflection().subs(x, 4).args[0]) == -302 # Coefficient of 1/(E*I)
l = symbols('l', positive=True)
R1, M1, R2, R3, P = symbols('R1 M1 R2 R3 P')
b1 = Beam(2*l, E, I)
b2 = Beam(2*l, E, I)
b = b1.join(b2,"hinge")
b.apply_load(M1, 0, -2)
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(R3, 4*l, -1)
b.apply_load(P, 3*l, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0), (l, 0), (4*l, 0)]
b.solve_for_reaction_loads(M1, R1, R2, R3)
assert b.reaction_loads == {R3: -P/2, R2: P*Rational(-5, 4), M1: -P*l/4, R1: P*Rational(3, 4)}
assert b.slope().subs(x, 3*l) == -7*P*l**2/(48*E*I)
assert b.deflection().subs(x, 2*l) == 7*P*l**3/(24*E*I)
assert b.deflection().subs(x, 3*l) == 5*P*l**3/(16*E*I)
# When beams having same second moment are joined.
b1 = Beam(2, 500, 10)
b2 = Beam(2, 500, 10)
b = b1.join(b2, "fixed")
b.apply_load(M1, 0, -2)
b.apply_load(R1, 0, -1)
b.apply_load(R2, 1, -1)
b.apply_load(R3, 4, -1)
b.apply_load(10, 3, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0), (1, 0), (4, 0)]
b.solve_for_reaction_loads(M1, R1, R2, R3)
assert b.slope() == -2*SingularityFunction(x, 0, 1)/5625 + SingularityFunction(x, 0, 2)/1875\
- 133*SingularityFunction(x, 1, 2)/135000 + SingularityFunction(x, 3, 2)/1000\
- 37*SingularityFunction(x, 4, 2)/67500
assert b.deflection() == -SingularityFunction(x, 0, 2)/5625 + SingularityFunction(x, 0, 3)/5625\
- 133*SingularityFunction(x, 1, 3)/405000 + SingularityFunction(x, 3, 3)/3000\
- 37*SingularityFunction(x, 4, 3)/202500
def test_point_cflexure():
E = Symbol('E')
I = Symbol('I')
b = Beam(10, E, I)
b.apply_load(-4, 0, -1)
b.apply_load(-46, 6, -1)
b.apply_load(10, 2, -1)
b.apply_load(20, 4, -1)
b.apply_load(3, 6, 0)
assert b.point_cflexure() == [Rational(10, 3)]
def test_remove_load():
E = Symbol('E')
I = Symbol('I')
b = Beam(4, E, I)
try:
b.remove_load(2, 1, -1)
# As no load is applied on beam, ValueError should be returned.
except ValueError:
assert True
else:
assert False
b.apply_load(-3, 0, -2)
b.apply_load(4, 2, -1)
b.apply_load(-2, 2, 2, end = 3)
b.remove_load(-2, 2, 2, end = 3)
assert b.load == -3*SingularityFunction(x, 0, -2) + 4*SingularityFunction(x, 2, -1)
assert b.applied_loads == [(-3, 0, -2, None), (4, 2, -1, None)]
try:
b.remove_load(1, 2, -1)
# As load of this magnitude was never applied at
# this position, method should return a ValueError.
except ValueError:
assert True
else:
assert False
b.remove_load(-3, 0, -2)
b.remove_load(4, 2, -1)
assert b.load == 0
assert b.applied_loads == []
def test_apply_support():
E = Symbol('E')
I = Symbol('I')
b = Beam(4, E, I)
b.apply_support(0, "cantilever")
b.apply_load(20, 4, -1)
M_0, R_0 = symbols('M_0, R_0')
b.solve_for_reaction_loads(R_0, M_0)
assert simplify(b.slope()) == simplify((80*SingularityFunction(x, 0, 1) - 10*SingularityFunction(x, 0, 2)
+ 10*SingularityFunction(x, 4, 2))/(E*I))
assert simplify(b.deflection()) == simplify((40*SingularityFunction(x, 0, 2) - 10*SingularityFunction(x, 0, 3)/3
+ 10*SingularityFunction(x, 4, 3)/3)/(E*I))
b = Beam(30, E, I)
b.apply_support(10, "pin")
b.apply_support(30, "roller")
b.apply_load(-8, 0, -1)
b.apply_load(120, 30, -2)
R_10, R_30 = symbols('R_10, R_30')
b.solve_for_reaction_loads(R_10, R_30)
assert b.slope() == (-4*SingularityFunction(x, 0, 2) + 3*SingularityFunction(x, 10, 2)
+ 120*SingularityFunction(x, 30, 1) + SingularityFunction(x, 30, 2) + Rational(4000, 3))/(E*I)
assert b.deflection() == (x*Rational(4000, 3) - 4*SingularityFunction(x, 0, 3)/3 + SingularityFunction(x, 10, 3)
+ 60*SingularityFunction(x, 30, 2) + SingularityFunction(x, 30, 3)/3 - 12000)/(E*I)
P = Symbol('P', positive=True)
L = Symbol('L', positive=True)
b = Beam(L, E, I)
b.apply_support(0, type='fixed')
b.apply_support(L, type='fixed')
b.apply_load(-P, L/2, -1)
R_0, R_L, M_0, M_L = symbols('R_0, R_L, M_0, M_L')
b.solve_for_reaction_loads(R_0, R_L, M_0, M_L)
assert b.reaction_loads == {R_0: P/2, R_L: P/2, M_0: -L*P/8, M_L: L*P/8}
def test_max_shear_force():
E = Symbol('E')
I = Symbol('I')
b = Beam(3, E, I)
R, M = symbols('R, M')
b.apply_load(R, 0, -1)
b.apply_load(M, 0, -2)
b.apply_load(2, 3, -1)
b.apply_load(4, 2, -1)
b.apply_load(2, 2, 0, end=3)
b.solve_for_reaction_loads(R, M)
assert b.max_shear_force() == (Interval(0, 2), 8)
l = symbols('l', positive=True)
P = Symbol('P')
b = Beam(l, E, I)
R1, R2 = symbols('R1, R2')
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(P, 0, 0, end=l)
b.solve_for_reaction_loads(R1, R2)
assert b.max_shear_force() == (0, l*Abs(P)/2)
def test_max_bmoment():
E = Symbol('E')
I = Symbol('I')
l, P = symbols('l, P', positive=True)
b = Beam(l, E, I)
R1, R2 = symbols('R1, R2')
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(P, l/2, -1)
b.solve_for_reaction_loads(R1, R2)
b.reaction_loads
assert b.max_bmoment() == (l/2, P*l/4)
b = Beam(l, E, I)
R1, R2 = symbols('R1, R2')
b.apply_load(R1, 0, -1)
b.apply_load(R2, l, -1)
b.apply_load(P, 0, 0, end=l)
b.solve_for_reaction_loads(R1, R2)
assert b.max_bmoment() == (l/2, P*l**2/8)
def test_max_deflection():
E, I, l, F = symbols('E, I, l, F', positive=True)
b = Beam(l, E, I)
b.bc_deflection = [(0, 0),(l, 0)]
b.bc_slope = [(0, 0),(l, 0)]
b.apply_load(F/2, 0, -1)
b.apply_load(-F*l/8, 0, -2)
b.apply_load(F/2, l, -1)
b.apply_load(F*l/8, l, -2)
b.apply_load(-F, l/2, -1)
assert b.max_deflection() == (l/2, F*l**3/(192*E*I))
def test_Beam3D():
l, E, G, I, A = symbols('l, E, G, I, A')
R1, R2, R3, R4 = symbols('R1, R2, R3, R4')
b = Beam3D(l, E, G, I, A)
m, q = symbols('m, q')
b.apply_load(q, 0, 0, dir="y")
b.apply_moment_load(m, 0, 0, dir="z")
b.bc_slope = [(0, [0, 0, 0]), (l, [0, 0, 0])]
b.bc_deflection = [(0, [0, 0, 0]), (l, [0, 0, 0])]
b.solve_slope_deflection()
assert b.polar_moment() == 2*I
assert b.shear_force() == [0, -q*x, 0]
assert b.shear_stress() == [0, -q*x/A, 0]
assert b.axial_stress() == 0
assert b.bending_moment() == [0, 0, -m*x + q*x**2/2]
expected_deflection = (x*(A*G*q*x**3/4 + A*G*x**2*(-l*(A*G*l*(l*q - 2*m) +
12*E*I*q)/(A*G*l**2 + 12*E*I)/2 - m) + 3*E*I*l*(A*G*l*(l*q - 2*m) +
12*E*I*q)/(A*G*l**2 + 12*E*I) + x*(-A*G*l**2*q/2 +
3*A*G*l**2*(A*G*l*(l*q - 2*m) + 12*E*I*q)/(A*G*l**2 + 12*E*I)/4 +
A*G*l*m*Rational(3, 2) - 3*E*I*q))/(6*A*E*G*I))
dx, dy, dz = b.deflection()
assert dx == dz == 0
assert simplify(dy - expected_deflection) == 0
b2 = Beam3D(30, E, G, I, A, x)
b2.apply_load(50, start=0, order=0, dir="y")
b2.bc_deflection = [(0, [0, 0, 0]), (30, [0, 0, 0])]
b2.apply_load(R1, start=0, order=-1, dir="y")
b2.apply_load(R2, start=30, order=-1, dir="y")
b2.solve_for_reaction_loads(R1, R2)
assert b2.reaction_loads == {R1: -750, R2: -750}
b2.solve_slope_deflection()
assert b2.slope() == [0, 0, 25*x**3/(3*E*I) - 375*x**2/(E*I) + 3750*x/(E*I)]
expected_deflection = 25*x**4/(12*E*I) - 125*x**3/(E*I) + 1875*x**2/(E*I) - \
25*x**2/(A*G) + 750*x/(A*G)
dx, dy, dz = b2.deflection()
assert dx == dz == 0
assert dy == expected_deflection
# Test for solve_for_reaction_loads
b3 = Beam3D(30, E, G, I, A, x)
b3.apply_load(8, start=0, order=0, dir="y")
b3.apply_load(9*x, start=0, order=0, dir="z")
b3.apply_load(R1, start=0, order=-1, dir="y")
b3.apply_load(R2, start=30, order=-1, dir="y")
b3.apply_load(R3, start=0, order=-1, dir="z")
b3.apply_load(R4, start=30, order=-1, dir="z")
b3.solve_for_reaction_loads(R1, R2, R3, R4)
assert b3.reaction_loads == {R1: -120, R2: -120, R3: -1350, R4: -2700}
def test_polar_moment_Beam3D():
l, E, G, A, I1, I2 = symbols('l, E, G, A, I1, I2')
I = [I1, I2]
b = Beam3D(l, E, G, I, A)
assert b.polar_moment() == I1 + I2
def test_parabolic_loads():
E, I, L = symbols('E, I, L', positive=True, real=True)
R, M, P = symbols('R, M, P', real=True)
# cantilever beam fixed at x=0 and parabolic distributed loading across
# length of beam
beam = Beam(L, E, I)
beam.bc_deflection.append((0, 0))
beam.bc_slope.append((0, 0))
beam.apply_load(R, 0, -1)
beam.apply_load(M, 0, -2)
# parabolic load
beam.apply_load(1, 0, 2)
beam.solve_for_reaction_loads(R, M)
assert beam.reaction_loads[R] == -L**3/3
# cantilever beam fixed at x=0 and parabolic distributed loading across
# first half of beam
beam = Beam(2*L, E, I)
beam.bc_deflection.append((0, 0))
beam.bc_slope.append((0, 0))
beam.apply_load(R, 0, -1)
beam.apply_load(M, 0, -2)
# parabolic load from x=0 to x=L
beam.apply_load(1, 0, 2, end=L)
beam.solve_for_reaction_loads(R, M)
# result should be the same as the prior example
assert beam.reaction_loads[R] == -L**3/3
# check constant load
beam = Beam(2*L, E, I)
beam.apply_load(P, 0, 0, end=L)
loading = beam.load.xreplace({L: 10, E: 20, I: 30, P: 40})
assert loading.xreplace({x: 5}) == 40
assert loading.xreplace({x: 15}) == 0
# check ramp load
beam = Beam(2*L, E, I)
beam.apply_load(P, 0, 1, end=L)
assert beam.load == (P*SingularityFunction(x, 0, 1) -
P*SingularityFunction(x, L, 1) -
P*L*SingularityFunction(x, L, 0))
# check higher order load: x**8 load from x=0 to x=L
beam = Beam(2*L, E, I)
beam.apply_load(P, 0, 8, end=L)
loading = beam.load.xreplace({L: 10, E: 20, I: 30, P: 40})
assert loading.xreplace({x: 5}) == 40*5**8
assert loading.xreplace({x: 15}) == 0
def test_cross_section():
I = Symbol('I')
l = Symbol('l')
E = Symbol('E')
C3, C4 = symbols('C3, C4')
a, c, g, h, r, n = symbols('a, c, g, h, r, n')
# test for second_moment and cross_section setter
b0 = Beam(l, E, I)
assert b0.second_moment == I
assert b0.cross_section == None
b0.cross_section = Circle((0, 0), 5)
assert b0.second_moment == pi*Rational(625, 4)
assert b0.cross_section == Circle((0, 0), 5)
b0.second_moment = 2*n - 6
assert b0.second_moment == 2*n-6
assert b0.cross_section == None
with raises(ValueError):
b0.second_moment = Circle((0, 0), 5)
# beam with a circular cross-section
b1 = Beam(50, E, Circle((0, 0), r))
assert b1.cross_section == Circle((0, 0), r)
assert b1.second_moment == pi*r*Abs(r)**3/4
b1.apply_load(-10, 0, -1)
b1.apply_load(R1, 5, -1)
b1.apply_load(R2, 50, -1)
b1.apply_load(90, 45, -2)
b1.solve_for_reaction_loads(R1, R2)
assert b1.load == (-10*SingularityFunction(x, 0, -1) + 82*SingularityFunction(x, 5, -1)/S(9)
+ 90*SingularityFunction(x, 45, -2) + 8*SingularityFunction(x, 50, -1)/9)
assert b1.bending_moment() == (10*SingularityFunction(x, 0, 1) - 82*SingularityFunction(x, 5, 1)/9
- 90*SingularityFunction(x, 45, 0) - 8*SingularityFunction(x, 50, 1)/9)
q = (-5*SingularityFunction(x, 0, 2) + 41*SingularityFunction(x, 5, 2)/S(9)
+ 90*SingularityFunction(x, 45, 1) + 4*SingularityFunction(x, 50, 2)/S(9))/(pi*E*r*Abs(r)**3)
assert b1.slope() == C3 + 4*q
q = (-5*SingularityFunction(x, 0, 3)/3 + 41*SingularityFunction(x, 5, 3)/27 + 45*SingularityFunction(x, 45, 2)
+ 4*SingularityFunction(x, 50, 3)/27)/(pi*E*r*Abs(r)**3)
assert b1.deflection() == C3*x + C4 + 4*q
# beam with a recatangular cross-section
b2 = Beam(20, E, Polygon((0, 0), (a, 0), (a, c), (0, c)))
assert b2.cross_section == Polygon((0, 0), (a, 0), (a, c), (0, c))
assert b2.second_moment == a*c**3/12
# beam with a triangular cross-section
b3 = Beam(15, E, Triangle((0, 0), (g, 0), (g/2, h)))
assert b3.cross_section == Triangle(Point2D(0, 0), Point2D(g, 0), Point2D(g/2, h))
assert b3.second_moment == g*h**3/36
# composite beam
b = b2.join(b3, "fixed")
b.apply_load(-30, 0, -1)
b.apply_load(65, 0, -2)
b.apply_load(40, 0, -1)
b.bc_slope = [(0, 0)]
b.bc_deflection = [(0, 0)]
assert b.second_moment == Piecewise((a*c**3/12, x <= 20), (g*h**3/36, x <= 35))
assert b.cross_section == None
assert b.length == 35
assert b.slope().subs(x, 7) == 8400/(E*a*c**3)
assert b.slope().subs(x, 25) == 52200/(E*g*h**3) + 39600/(E*a*c**3)
assert b.deflection().subs(x, 30) == -537000/(E*g*h**3) - 712000/(E*a*c**3)
def test_max_shear_force_Beam3D():
x = symbols('x')
b = Beam3D(20, 40, 21, 100, 25)
b.apply_load(15, start=0, order=0, dir="z")
b.apply_load(12*x, start=0, order=0, dir="y")
b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
assert b.max_shear_force() == [(0, 0), (20, 2400), (20, 300)]
def test_max_bending_moment_Beam3D():
x = symbols('x')
b = Beam3D(20, 40, 21, 100, 25)
b.apply_load(15, start=0, order=0, dir="z")
b.apply_load(12*x, start=0, order=0, dir="y")
b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
assert b.max_bmoment() == [(0, 0), (20, 3000), (20, 16000)]
def test_max_deflection_Beam3D():
x = symbols('x')
b = Beam3D(20, 40, 21, 100, 25)
b.apply_load(15, start=0, order=0, dir="z")
b.apply_load(12*x, start=0, order=0, dir="y")
b.bc_deflection = [(0, [0, 0, 0]), (20, [0, 0, 0])]
b.solve_slope_deflection()
c = sympify("495/14")
p = sympify("-10 + 10*sqrt(10793)/43")
q = sympify("(10 - 10*sqrt(10793)/43)**3/160 - 20/7 + (10 - 10*sqrt(10793)/43)**4/6400 + 20*sqrt(10793)/301 + 27*(10 - 10*sqrt(10793)/43)**2/560")
assert b.max_deflection() == [(0, 0), (10, c), (p, q)]
def test_torsion_Beam3D():
x = symbols('x')
b = Beam3D(20, 40, 21, 100, 25)
b.apply_moment_load(15, 5, -2, dir='x')
b.apply_moment_load(25, 10, -2, dir='x')
b.apply_moment_load(-5, 20, -2, dir='x')
b.solve_for_torsion()
assert b.angular_deflection().subs(x, 3) == sympify("1/40")
assert b.angular_deflection().subs(x, 9) == sympify("17/280")
assert b.angular_deflection().subs(x, 12) == sympify("53/840")
assert b.angular_deflection().subs(x, 17) == sympify("2/35")
assert b.angular_deflection().subs(x, 20) == sympify("3/56")
|
68e8a3f84624a45396d0c8dfdb1c75a61b6fe8bfb24a1b03f0a6f1ad08a9dec4 | from sympy.core.symbol import Symbol, symbols
from sympy.physics.continuum_mechanics.truss import Truss
def test_truss():
A = Symbol('A')
B = Symbol('B')
C = Symbol('C')
AB, BC, AC = symbols('AB, BC, AC')
P = Symbol('P')
t = Truss()
assert t.nodes == []
assert t.node_labels == []
assert t.node_positions == []
assert t.members == {}
assert t.loads == {}
assert t.supports == {}
assert t.reaction_loads == {}
assert t.internal_forces == {}
# testing the add_node method
t.add_node(A, 0, 0)
t.add_node(B, 2, 2)
t.add_node(C, 3, 0)
assert t.nodes == [(A, 0, 0), (B, 2, 2), (C, 3, 0)]
assert t.node_labels == [A, B, C]
assert t.node_positions == [(0, 0), (2, 2), (3, 0)]
assert t.loads == {}
assert t.supports == {}
assert t.reaction_loads == {}
# testing the remove_node method
t.remove_node(C)
assert t.nodes == [(A, 0, 0), (B, 2, 2)]
assert t.node_labels == [A, B]
assert t.node_positions == [(0, 0), (2, 2)]
assert t.loads == {}
assert t.supports == {}
t.add_node(C, 3, 0)
# testing the add_member method
t.add_member(AB, A, B)
t.add_member(BC, B, C)
t.add_member(AC, A, C)
assert t.members == {AB: [A, B], BC: [B, C], AC: [A, C]}
assert t.internal_forces == {AB: 0, BC: 0, AC: 0}
# testing the remove_member method
t.remove_member(BC)
assert t.members == {AB: [A, B], AC: [A, C]}
assert t.internal_forces == {AB: 0, AC: 0}
t.add_member(BC, B, C)
D, CD = symbols('D, CD')
# testing the change_label methods
t.change_node_label(B, D)
assert t.nodes == [(A, 0, 0), (D, 2, 2), (C, 3, 0)]
assert t.node_labels == [A, D, C]
assert t.loads == {}
assert t.supports == {}
assert t.members == {AB: [A, D], BC: [D, C], AC: [A, C]}
t.change_member_label(BC, CD)
assert t.members == {AB: [A, D], CD: [D, C], AC: [A, C]}
assert t.internal_forces == {AB: 0, CD: 0, AC: 0}
# testing the apply_load method
t.apply_load(A, P, 90)
t.apply_load(A, P/4, 90)
t.apply_load(A, 2*P,45)
t.apply_load(D, P/2, 90)
assert t.loads == {A: [[P, 90], [P/4, 90], [2*P, 45]], D: [[P/2, 90]]}
assert t.loads[A] == [[P, 90], [P/4, 90], [2*P, 45]]
# testing the remove_load method
t.remove_load(A, P/4, 90)
assert t.loads == {A: [[P, 90], [2*P, 45]], D: [[P/2, 90]]}
assert t.loads[A] == [[P, 90], [2*P, 45]]
# testing the apply_support method
t.apply_support(A, "pinned")
t.apply_support(D, "roller")
assert t.supports == {A: 'pinned', D: 'roller'}
assert t.reaction_loads == {}
assert t.loads == {A: [[P, 90], [2*P, 45], [Symbol('R_A_x'), 0], [Symbol('R_A_y'), 90]], D: [[P/2, 90], [Symbol('R_D_y'), 90]]}
# testing the remove_support method
t.remove_support(A)
assert t.supports == {D: 'roller'}
assert t.reaction_loads == {}
assert t.loads == {A: [[P, 90], [2*P, 45]], D: [[P/2, 90], [Symbol('R_D_y'), 90]]}
t.apply_support(A, "pinned")
# testing the solve method
t.solve()
assert t.reaction_loads['R_A_x']/P - (-1.4142135623731) < 1e-10
assert t.reaction_loads['R_A_y']/P - (-2.41421356237309) < 1e-10
assert t.reaction_loads['R_D_y']/P - (-0.5) < 1e-10
assert t.internal_forces[AB]/P < 1e-10
assert t.internal_forces[CD] < 1e-10
assert t.internal_forces[AC] < 1e-10
# assert t.internal_forces == {AB: 3.06161699786838e-17*P, CD: 0, AC: 0}
|
b831f6877796fb8f90a99f696fbdb268cc1c6a1f478adc7a6785099f60c2b200 | import functools
from typing import List
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.singleton import S
from sympy.core.sympify import _sympify
from sympy.tensor.array.mutable_ndim_array import MutableNDimArray
from sympy.tensor.array.ndim_array import NDimArray, ImmutableNDimArray, ArrayKind
from sympy.utilities.iterables import flatten
class DenseNDimArray(NDimArray):
_array: List[Basic]
def __new__(self, *args, **kwargs):
return ImmutableDenseNDimArray(*args, **kwargs)
@property
def kind(self) -> ArrayKind:
return ArrayKind._union(self._array)
def __getitem__(self, index):
"""
Allows to get items from N-dim array.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray([0, 1, 2, 3], (2, 2))
>>> a
[[0, 1], [2, 3]]
>>> a[0, 0]
0
>>> a[1, 1]
3
>>> a[0]
[0, 1]
>>> a[1]
[2, 3]
Symbolic index:
>>> from sympy.abc import i, j
>>> a[i, j]
[[0, 1], [2, 3]][i, j]
Replace `i` and `j` to get element `(1, 1)`:
>>> a[i, j].subs({i: 1, j: 1})
3
"""
syindex = self._check_symbolic_index(index)
if syindex is not None:
return syindex
index = self._check_index_for_getitem(index)
if isinstance(index, tuple) and any(isinstance(i, slice) for i in index):
sl_factors, eindices = self._get_slice_data_for_array_access(index)
array = [self._array[self._parse_index(i)] for i in eindices]
nshape = [len(el) for i, el in enumerate(sl_factors) if isinstance(index[i], slice)]
return type(self)(array, nshape)
else:
index = self._parse_index(index)
return self._array[index]
@classmethod
def zeros(cls, *shape):
list_length = functools.reduce(lambda x, y: x*y, shape, S.One)
return cls._new(([0]*list_length,), shape)
def tomatrix(self):
"""
Converts MutableDenseNDimArray to Matrix. Can convert only 2-dim array, else will raise error.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray([1 for i in range(9)], (3, 3))
>>> b = a.tomatrix()
>>> b
Matrix([
[1, 1, 1],
[1, 1, 1],
[1, 1, 1]])
"""
from sympy.matrices import Matrix
if self.rank() != 2:
raise ValueError('Dimensions must be of size of 2')
return Matrix(self.shape[0], self.shape[1], self._array)
def reshape(self, *newshape):
"""
Returns MutableDenseNDimArray instance with new shape. Elements number
must be suitable to new shape. The only argument of method sets
new shape.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray([1, 2, 3, 4, 5, 6], (2, 3))
>>> a.shape
(2, 3)
>>> a
[[1, 2, 3], [4, 5, 6]]
>>> b = a.reshape(3, 2)
>>> b.shape
(3, 2)
>>> b
[[1, 2], [3, 4], [5, 6]]
"""
new_total_size = functools.reduce(lambda x,y: x*y, newshape)
if new_total_size != self._loop_size:
raise ValueError('Expecting reshape size to %d but got prod(%s) = %d' % (
self._loop_size, str(newshape), new_total_size))
# there is no `.func` as this class does not subtype `Basic`:
return type(self)(self._array, newshape)
class ImmutableDenseNDimArray(DenseNDimArray, ImmutableNDimArray): # type: ignore
"""
"""
def __new__(cls, iterable, shape=None, **kwargs):
return cls._new(iterable, shape, **kwargs)
@classmethod
def _new(cls, iterable, shape, **kwargs):
shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs)
shape = Tuple(*map(_sympify, shape))
cls._check_special_bounds(flat_list, shape)
flat_list = flatten(flat_list)
flat_list = Tuple(*flat_list)
self = Basic.__new__(cls, flat_list, shape, **kwargs)
self._shape = shape
self._array = list(flat_list)
self._rank = len(shape)
self._loop_size = functools.reduce(lambda x,y: x*y, shape, 1)
return self
def __setitem__(self, index, value):
raise TypeError('immutable N-dim array')
def as_mutable(self):
return MutableDenseNDimArray(self)
def _eval_simplify(self, **kwargs):
from sympy.simplify.simplify import simplify
return self.applyfunc(simplify)
class MutableDenseNDimArray(DenseNDimArray, MutableNDimArray):
def __new__(cls, iterable=None, shape=None, **kwargs):
return cls._new(iterable, shape, **kwargs)
@classmethod
def _new(cls, iterable, shape, **kwargs):
shape, flat_list = cls._handle_ndarray_creation_inputs(iterable, shape, **kwargs)
flat_list = flatten(flat_list)
self = object.__new__(cls)
self._shape = shape
self._array = list(flat_list)
self._rank = len(shape)
self._loop_size = functools.reduce(lambda x,y: x*y, shape) if shape else len(flat_list)
return self
def __setitem__(self, index, value):
"""Allows to set items to MutableDenseNDimArray.
Examples
========
>>> from sympy import MutableDenseNDimArray
>>> a = MutableDenseNDimArray.zeros(2, 2)
>>> a[0,0] = 1
>>> a[1,1] = 1
>>> a
[[1, 0], [0, 1]]
"""
if isinstance(index, tuple) and any(isinstance(i, slice) for i in index):
value, eindices, slice_offsets = self._get_slice_data_for_array_assignment(index, value)
for i in eindices:
other_i = [ind - j for ind, j in zip(i, slice_offsets) if j is not None]
self._array[self._parse_index(i)] = value[other_i]
else:
index = self._parse_index(index)
self._setter_iterable_check(value)
value = _sympify(value)
self._array[index] = value
def as_immutable(self):
return ImmutableDenseNDimArray(self)
@property
def free_symbols(self):
return {i for j in self._array for i in j.free_symbols}
|
b0ec2ef94dbcd3df0b834895ffd45f631bc9f3001de5193afef022bd676473d4 | from sympy import sin, cos
from sympy.testing.pytest import raises
from sympy.tensor.toperators import PartialDerivative
from sympy.tensor.tensor import (TensorIndexType,
tensor_indices,
TensorHead, tensor_heads)
from sympy.core.numbers import Rational
from sympy.core.symbol import symbols
from sympy.matrices.dense import diag
from sympy.tensor.array import Array
from sympy.core.random import randint
L = TensorIndexType("L")
i, j, k, m, m1, m2, m3, m4 = tensor_indices("i j k m m1 m2 m3 m4", L)
i0 = tensor_indices("i0", L)
L_0, L_1 = tensor_indices("L_0 L_1", L)
A, B, C, D = tensor_heads("A B C D", [L])
H = TensorHead("H", [L, L])
def test_invalid_partial_derivative_valence():
raises(ValueError, lambda: PartialDerivative(C(j), D(-j)))
raises(ValueError, lambda: PartialDerivative(C(-j), D(j)))
def test_tensor_partial_deriv():
# Test flatten:
expr = PartialDerivative(PartialDerivative(A(i), A(j)), A(k))
assert expr == PartialDerivative(A(i), A(j), A(k))
assert expr.expr == A(i)
assert expr.variables == (A(j), A(k))
assert expr.get_indices() == [i, -j, -k]
assert expr.get_free_indices() == [i, -j, -k]
expr = PartialDerivative(PartialDerivative(A(i), A(j)), A(i))
assert expr.expr == A(L_0)
assert expr.variables == (A(j), A(L_0))
expr1 = PartialDerivative(A(i), A(j))
assert expr1.expr == A(i)
assert expr1.variables == (A(j),)
expr2 = A(i)*PartialDerivative(H(k, -i), A(j))
assert expr2.get_indices() == [L_0, k, -L_0, -j]
expr2b = A(i)*PartialDerivative(H(k, -i), A(-j))
assert expr2b.get_indices() == [L_0, k, -L_0, j]
expr3 = A(i)*PartialDerivative(B(k)*C(-i) + 3*H(k, -i), A(j))
assert expr3.get_indices() == [L_0, k, -L_0, -j]
expr4 = (A(i) + B(i))*PartialDerivative(C(j), D(j))
assert expr4.get_indices() == [i, L_0, -L_0]
expr4b = (A(i) + B(i))*PartialDerivative(C(-j), D(-j))
assert expr4b.get_indices() == [i, -L_0, L_0]
expr5 = (A(i) + B(i))*PartialDerivative(C(-i), D(j))
assert expr5.get_indices() == [L_0, -L_0, -j]
def test_replace_arrays_partial_derivative():
x, y, z, t = symbols("x y z t")
expr = PartialDerivative(A(i), B(j))
repl = expr.replace_with_arrays({A(i): [sin(x)*cos(y), x**3*y**2], B(i): [x, y]})
assert repl == Array([[cos(x)*cos(y), -sin(x)*sin(y)], [3*x**2*y**2, 2*x**3*y]])
repl = expr.replace_with_arrays({A(i): [sin(x)*cos(y), x**3*y**2], B(i): [x, y]}, [-j, i])
assert repl == Array([[cos(x)*cos(y), 3*x**2*y**2], [-sin(x)*sin(y), 2*x**3*y]])
# d(A^i)/d(A_j) = d(g^ik A_k)/d(A_j) = g^ik delta_jk
expr = PartialDerivative(A(i), A(-j))
assert expr.get_free_indices() == [i, j]
assert expr.get_indices() == [i, j]
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, [i, j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, [i, j]) == Array([[1, 0], [0, -1]])
assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, 1)}, [i, j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, -1)}, [i, j]) == Array([[1, 0], [0, -1]])
expr = PartialDerivative(A(i), A(j))
assert expr.get_free_indices() == [i, -j]
assert expr.get_indices() == [i, -j]
assert expr.replace_with_arrays({A(i): [x, y]}, [i, -j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, [i, -j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, [i, -j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, 1)}, [i, -j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, -1)}, [i, -j]) == Array([[1, 0], [0, 1]])
expr = PartialDerivative(A(-i), A(-j))
assert expr.get_free_indices() == [-i, j]
assert expr.get_indices() == [-i, j]
assert expr.replace_with_arrays({A(-i): [x, y]}, [-i, j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, 1)}, [-i, j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(-i): [x, y], L: diag(1, -1)}, [-i, j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, [-i, j]) == Array([[1, 0], [0, 1]])
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, [-i, j]) == Array([[1, 0], [0, 1]])
expr = PartialDerivative(A(i), A(i))
assert expr.get_free_indices() == []
assert expr.get_indices() == [L_0, -L_0]
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 2
expr = PartialDerivative(A(-i), A(-i))
assert expr.get_free_indices() == []
assert expr.get_indices() == [-L_0, L_0]
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, 1)}, []) == 2
assert expr.replace_with_arrays({A(i): [x, y], L: diag(1, -1)}, []) == 2
expr = PartialDerivative(H(i, j) + H(j, i), A(i))
assert expr.get_indices() == [L_0, j, -L_0]
assert expr.get_free_indices() == [j]
expr = PartialDerivative(H(i, j) + H(j, i), A(k))*B(-i)
assert expr.get_indices() == [L_0, j, -k, -L_0]
assert expr.get_free_indices() == [j, -k]
expr = PartialDerivative(A(i)*(H(-i, j) + H(j, -i)), A(j))
assert expr.get_indices() == [L_0, -L_0, L_1, -L_1]
assert expr.get_free_indices() == []
expr = A(j)*A(-j) + expr
assert expr.get_indices() == [L_0, -L_0, L_1, -L_1]
assert expr.get_free_indices() == []
expr = A(i)*(B(j)*PartialDerivative(C(-j), D(i)) + C(j)*PartialDerivative(D(-j), B(i)))
assert expr.get_indices() == [L_0, L_1, -L_1, -L_0]
assert expr.get_free_indices() == []
expr = A(i)*PartialDerivative(C(-j), D(i))
assert expr.get_indices() == [L_0, -j, -L_0]
assert expr.get_free_indices() == [-j]
def test_expand_partial_derivative_sum_rule():
tau = symbols("tau")
# check sum rule for D(tensor, symbol)
expr1aa = PartialDerivative(A(i), tau)
assert expr1aa._expand_partial_derivative() == PartialDerivative(A(i), tau)
expr1ab = PartialDerivative(A(i) + B(i), tau)
assert (expr1ab._expand_partial_derivative() ==
PartialDerivative(A(i), tau) +
PartialDerivative(B(i), tau))
expr1ac = PartialDerivative(A(i) + B(i) + C(i), tau)
assert (expr1ac._expand_partial_derivative() ==
PartialDerivative(A(i), tau) +
PartialDerivative(B(i), tau) +
PartialDerivative(C(i), tau))
# check sum rule for D(tensor, D(j))
expr1ba = PartialDerivative(A(i), D(j))
assert expr1ba._expand_partial_derivative() ==\
PartialDerivative(A(i), D(j))
expr1bb = PartialDerivative(A(i) + B(i), D(j))
assert (expr1bb._expand_partial_derivative() ==
PartialDerivative(A(i), D(j)) +
PartialDerivative(B(i), D(j)))
expr1bc = PartialDerivative(A(i) + B(i) + C(i), D(j))
assert expr1bc._expand_partial_derivative() ==\
PartialDerivative(A(i), D(j))\
+ PartialDerivative(B(i), D(j))\
+ PartialDerivative(C(i), D(j))
# check sum rule for D(tensor, H(j, k))
expr1ca = PartialDerivative(A(i), H(j, k))
assert expr1ca._expand_partial_derivative() ==\
PartialDerivative(A(i), H(j, k))
expr1cb = PartialDerivative(A(i) + B(i), H(j, k))
assert (expr1cb._expand_partial_derivative() ==
PartialDerivative(A(i), H(j, k))
+ PartialDerivative(B(i), H(j, k)))
expr1cc = PartialDerivative(A(i) + B(i) + C(i), H(j, k))
assert (expr1cc._expand_partial_derivative() ==
PartialDerivative(A(i), H(j, k))
+ PartialDerivative(B(i), H(j, k))
+ PartialDerivative(C(i), H(j, k)))
# check sum rule for D(D(tensor, D(j)), H(k, m))
expr1da = PartialDerivative(A(i), (D(j), H(k, m)))
assert expr1da._expand_partial_derivative() ==\
PartialDerivative(A(i), (D(j), H(k, m)))
expr1db = PartialDerivative(A(i) + B(i), (D(j), H(k, m)))
assert expr1db._expand_partial_derivative() ==\
PartialDerivative(A(i), (D(j), H(k, m)))\
+ PartialDerivative(B(i), (D(j), H(k, m)))
expr1dc = PartialDerivative(A(i) + B(i) + C(i), (D(j), H(k, m)))
assert expr1dc._expand_partial_derivative() ==\
PartialDerivative(A(i), (D(j), H(k, m)))\
+ PartialDerivative(B(i), (D(j), H(k, m)))\
+ PartialDerivative(C(i), (D(j), H(k, m)))
def test_expand_partial_derivative_constant_factor_rule():
nneg = randint(0, 1000)
pos = randint(1, 1000)
neg = -randint(1, 1000)
c1 = Rational(nneg, pos)
c2 = Rational(neg, pos)
c3 = Rational(nneg, neg)
expr2a = PartialDerivative(nneg*A(i), D(j))
assert expr2a._expand_partial_derivative() ==\
nneg*PartialDerivative(A(i), D(j))
expr2b = PartialDerivative(neg*A(i), D(j))
assert expr2b._expand_partial_derivative() ==\
neg*PartialDerivative(A(i), D(j))
expr2ca = PartialDerivative(c1*A(i), D(j))
assert expr2ca._expand_partial_derivative() ==\
c1*PartialDerivative(A(i), D(j))
expr2cb = PartialDerivative(c2*A(i), D(j))
assert expr2cb._expand_partial_derivative() ==\
c2*PartialDerivative(A(i), D(j))
expr2cc = PartialDerivative(c3*A(i), D(j))
assert expr2cc._expand_partial_derivative() ==\
c3*PartialDerivative(A(i), D(j))
def test_expand_partial_derivative_full_linearity():
nneg = randint(0, 1000)
pos = randint(1, 1000)
neg = -randint(1, 1000)
c1 = Rational(nneg, pos)
c2 = Rational(neg, pos)
c3 = Rational(nneg, neg)
# check full linearity
p = PartialDerivative(42, D(j))
assert p and not p._expand_partial_derivative()
expr3a = PartialDerivative(nneg*A(i) + pos*B(i), D(j))
assert expr3a._expand_partial_derivative() ==\
nneg*PartialDerivative(A(i), D(j))\
+ pos*PartialDerivative(B(i), D(j))
expr3b = PartialDerivative(nneg*A(i) + neg*B(i), D(j))
assert expr3b._expand_partial_derivative() ==\
nneg*PartialDerivative(A(i), D(j))\
+ neg*PartialDerivative(B(i), D(j))
expr3c = PartialDerivative(neg*A(i) + pos*B(i), D(j))
assert expr3c._expand_partial_derivative() ==\
neg*PartialDerivative(A(i), D(j))\
+ pos*PartialDerivative(B(i), D(j))
expr3d = PartialDerivative(c1*A(i) + c2*B(i), D(j))
assert expr3d._expand_partial_derivative() ==\
c1*PartialDerivative(A(i), D(j))\
+ c2*PartialDerivative(B(i), D(j))
expr3e = PartialDerivative(c2*A(i) + c1*B(i), D(j))
assert expr3e._expand_partial_derivative() ==\
c2*PartialDerivative(A(i), D(j))\
+ c1*PartialDerivative(B(i), D(j))
expr3f = PartialDerivative(c2*A(i) + c3*B(i), D(j))
assert expr3f._expand_partial_derivative() ==\
c2*PartialDerivative(A(i), D(j))\
+ c3*PartialDerivative(B(i), D(j))
expr3g = PartialDerivative(c3*A(i) + c2*B(i), D(j))
assert expr3g._expand_partial_derivative() ==\
c3*PartialDerivative(A(i), D(j))\
+ c2*PartialDerivative(B(i), D(j))
expr3h = PartialDerivative(c3*A(i) + c1*B(i), D(j))
assert expr3h._expand_partial_derivative() ==\
c3*PartialDerivative(A(i), D(j))\
+ c1*PartialDerivative(B(i), D(j))
expr3i = PartialDerivative(c1*A(i) + c3*B(i), D(j))
assert expr3i._expand_partial_derivative() ==\
c1*PartialDerivative(A(i), D(j))\
+ c3*PartialDerivative(B(i), D(j))
def test_expand_partial_derivative_product_rule():
# check product rule
expr4a = PartialDerivative(A(i)*B(j), D(k))
assert expr4a._expand_partial_derivative() == \
PartialDerivative(A(i), D(k))*B(j)\
+ A(i)*PartialDerivative(B(j), D(k))
expr4b = PartialDerivative(A(i)*B(j)*C(k), D(m))
assert expr4b._expand_partial_derivative() ==\
PartialDerivative(A(i), D(m))*B(j)*C(k)\
+ A(i)*PartialDerivative(B(j), D(m))*C(k)\
+ A(i)*B(j)*PartialDerivative(C(k), D(m))
expr4c = PartialDerivative(A(i)*B(j), C(k), D(m))
assert expr4c._expand_partial_derivative() ==\
PartialDerivative(A(i), C(k), D(m))*B(j) \
+ PartialDerivative(A(i), C(k))*PartialDerivative(B(j), D(m))\
+ PartialDerivative(A(i), D(m))*PartialDerivative(B(j), C(k))\
+ A(i)*PartialDerivative(B(j), C(k), D(m))
def test_eval_partial_derivative_expr_by_symbol():
tau, alpha = symbols("tau alpha")
expr1 = PartialDerivative(tau**alpha, tau)
assert expr1._perform_derivative() == alpha * 1 / tau * tau ** alpha
expr2 = PartialDerivative(2*tau + 3*tau**4, tau)
assert expr2._perform_derivative() == 2 + 12 * tau ** 3
expr3 = PartialDerivative(2*tau + 3*tau**4, alpha)
assert expr3._perform_derivative() == 0
def test_eval_partial_derivative_single_tensors_by_scalar():
tau, mu = symbols("tau mu")
expr = PartialDerivative(tau**mu, tau)
assert expr._perform_derivative() == mu*tau**mu/tau
expr1a = PartialDerivative(A(i), tau)
assert expr1a._perform_derivative() == 0
expr1b = PartialDerivative(A(-i), tau)
assert expr1b._perform_derivative() == 0
expr2a = PartialDerivative(H(i, j), tau)
assert expr2a._perform_derivative() == 0
expr2b = PartialDerivative(H(i, -j), tau)
assert expr2b._perform_derivative() == 0
expr2c = PartialDerivative(H(-i, j), tau)
assert expr2c._perform_derivative() == 0
expr2d = PartialDerivative(H(-i, -j), tau)
assert expr2d._perform_derivative() == 0
def test_eval_partial_derivative_single_1st_rank_tensors_by_tensor():
expr1 = PartialDerivative(A(i), A(j))
assert expr1._perform_derivative() - L.delta(i, -j) == 0
expr2 = PartialDerivative(A(i), A(-j))
assert expr2._perform_derivative() - L.metric(i, L_0) * L.delta(-L_0, j) == 0
expr3 = PartialDerivative(A(-i), A(-j))
assert expr3._perform_derivative() - L.delta(-i, j) == 0
expr4 = PartialDerivative(A(-i), A(j))
assert expr4._perform_derivative() - L.metric(-i, -L_0) * L.delta(L_0, -j) == 0
expr5 = PartialDerivative(A(i), B(j))
expr6 = PartialDerivative(A(i), C(j))
expr7 = PartialDerivative(A(i), D(j))
expr8 = PartialDerivative(A(i), H(j, k))
assert expr5._perform_derivative() == 0
assert expr6._perform_derivative() == 0
assert expr7._perform_derivative() == 0
assert expr8._perform_derivative() == 0
expr9 = PartialDerivative(A(i), A(i))
assert expr9._perform_derivative() - L.delta(L_0, -L_0) == 0
expr10 = PartialDerivative(A(-i), A(-i))
assert expr10._perform_derivative() - L.delta(-L_0, L_0) == 0
def test_eval_partial_derivative_single_2nd_rank_tensors_by_tensor():
expr1 = PartialDerivative(H(i, j), H(m, m1))
assert expr1._perform_derivative() - L.delta(i, -m) * L.delta(j, -m1) == 0
expr2 = PartialDerivative(H(i, j), H(-m, m1))
assert expr2._perform_derivative() - L.metric(i, L_0) * L.delta(-L_0, m) * L.delta(j, -m1) == 0
expr3 = PartialDerivative(H(i, j), H(m, -m1))
assert expr3._perform_derivative() - L.delta(i, -m) * L.metric(j, L_0) * L.delta(-L_0, m1) == 0
expr4 = PartialDerivative(H(i, j), H(-m, -m1))
assert expr4._perform_derivative() - L.metric(i, L_0) * L.delta(-L_0, m) * L.metric(j, L_1) * L.delta(-L_1, m1) == 0
def test_eval_partial_derivative_divergence_type():
expr1a = PartialDerivative(A(i), A(i))
expr1b = PartialDerivative(A(i), A(k))
expr1c = PartialDerivative(L.delta(-i, k) * A(i), A(k))
assert (expr1a._perform_derivative()
- (L.delta(-i, k) * expr1b._perform_derivative())).contract_delta(L.delta) == 0
assert (expr1a._perform_derivative()
- expr1c._perform_derivative()).contract_delta(L.delta) == 0
expr2a = PartialDerivative(H(i, j), H(i, j))
expr2b = PartialDerivative(H(i, j), H(k, m))
expr2c = PartialDerivative(L.delta(-i, k) * L.delta(-j, m) * H(i, j), H(k, m))
assert (expr2a._perform_derivative()
- (L.delta(-i, k) * L.delta(-j, m) * expr2b._perform_derivative())).contract_delta(L.delta) == 0
assert (expr2a._perform_derivative()
- expr2c._perform_derivative()).contract_delta(L.delta) == 0
def test_eval_partial_derivative_expr1():
tau, alpha = symbols("tau alpha")
# this is only some special expression
# tested: vector derivative
# tested: scalar derivative
# tested: tensor derivative
base_expr1 = A(i)*H(-i, j) + A(i)*A(-i)*A(j) + tau**alpha*A(j)
tensor_derivative = PartialDerivative(base_expr1, H(k, m))._perform_derivative()
vector_derivative = PartialDerivative(base_expr1, A(k))._perform_derivative()
scalar_derivative = PartialDerivative(base_expr1, tau)._perform_derivative()
assert (tensor_derivative - A(L_0)*L.metric(-L_0, -L_1)*L.delta(L_1, -k)*L.delta(j, -m)) == 0
assert (vector_derivative - (tau**alpha*L.delta(j, -k) +
L.delta(L_0, -k)*A(-L_0)*A(j) +
A(L_0)*L.metric(-L_0, -L_1)*L.delta(L_1, -k)*A(j) +
A(L_0)*A(-L_0)*L.delta(j, -k) +
L.delta(L_0, -k)*H(-L_0, j))).expand() == 0
assert (vector_derivative.contract_metric(L.metric).contract_delta(L.delta) -
(tau**alpha*L.delta(j, -k) + A(L_0)*A(-L_0)*L.delta(j, -k) + H(-k, j) + 2*A(j)*A(-k))).expand() == 0
assert scalar_derivative - alpha*1/tau*tau**alpha*A(j) == 0
def test_eval_partial_derivative_mixed_scalar_tensor_expr2():
tau, alpha = symbols("tau alpha")
base_expr2 = A(i)*A(-i) + tau**2
vector_expression = PartialDerivative(base_expr2, A(k))._perform_derivative()
assert (vector_expression -
(L.delta(L_0, -k)*A(-L_0) + A(L_0)*L.metric(-L_0, -L_1)*L.delta(L_1, -k))).expand() == 0
scalar_expression = PartialDerivative(base_expr2, tau)._perform_derivative()
assert scalar_expression == 2*tau
|
204b51759a6bb32841c9d8ff1ca2edbb228813350f59345a5841dba0f47a1d2d | from sympy.assumptions.ask import Q
from sympy.assumptions.refine import refine
from sympy.core.numbers import oo
from sympy.core.relational import Equality, Eq, Ne
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, symbols)
from sympy.functions import Piecewise
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.sets.sets import (Interval, Union)
from sympy.simplify.simplify import simplify
from sympy.logic.boolalg import (
And, Boolean, Equivalent, ITE, Implies, Nand, Nor, Not, Or,
POSform, SOPform, Xor, Xnor, conjuncts, disjuncts,
distribute_or_over_and, distribute_and_over_or,
eliminate_implications, is_nnf, is_cnf, is_dnf, simplify_logic,
to_nnf, to_cnf, to_dnf, to_int_repr, bool_map, true, false,
BooleanAtom, is_literal, term_to_integer,
truth_table, as_Boolean, to_anf, is_anf, distribute_xor_over_and,
anf_coeffs, ANFform, bool_minterm, bool_maxterm, bool_monomial,
_check_pair, _convert_to_varsSOP, _convert_to_varsPOS, Exclusive,
gateinputcount)
from sympy.assumptions.cnf import CNF
from sympy.testing.pytest import raises, XFAIL, slow
from itertools import combinations, permutations, product
A, B, C, D = symbols('A:D')
a, b, c, d, e, w, x, y, z = symbols('a:e w:z')
def test_overloading():
"""Test that |, & are overloaded as expected"""
assert A & B == And(A, B)
assert A | B == Or(A, B)
assert (A & B) | C == Or(And(A, B), C)
assert A >> B == Implies(A, B)
assert A << B == Implies(B, A)
assert ~A == Not(A)
assert A ^ B == Xor(A, B)
def test_And():
assert And() is true
assert And(A) == A
assert And(True) is true
assert And(False) is false
assert And(True, True) is true
assert And(True, False) is false
assert And(False, False) is false
assert And(True, A) == A
assert And(False, A) is false
assert And(True, True, True) is true
assert And(True, True, A) == A
assert And(True, False, A) is false
assert And(1, A) == A
raises(TypeError, lambda: And(2, A))
raises(TypeError, lambda: And(A < 2, A))
assert And(A < 1, A >= 1) is false
e = A > 1
assert And(e, e.canonical) == e.canonical
g, l, ge, le = A > B, B < A, A >= B, B <= A
assert And(g, l, ge, le) == And(ge, g)
assert {And(*i) for i in permutations((l,g,le,ge))} == {And(ge, g)}
assert And(And(Eq(a, 0), Eq(b, 0)), And(Ne(a, 0), Eq(c, 0))) is false
def test_Or():
assert Or() is false
assert Or(A) == A
assert Or(True) is true
assert Or(False) is false
assert Or(True, True) is true
assert Or(True, False) is true
assert Or(False, False) is false
assert Or(True, A) is true
assert Or(False, A) == A
assert Or(True, False, False) is true
assert Or(True, False, A) is true
assert Or(False, False, A) == A
assert Or(1, A) is true
raises(TypeError, lambda: Or(2, A))
raises(TypeError, lambda: Or(A < 2, A))
assert Or(A < 1, A >= 1) is true
e = A > 1
assert Or(e, e.canonical) == e
g, l, ge, le = A > B, B < A, A >= B, B <= A
assert Or(g, l, ge, le) == Or(g, ge)
def test_Xor():
assert Xor() is false
assert Xor(A) == A
assert Xor(A, A) is false
assert Xor(True, A, A) is true
assert Xor(A, A, A, A, A) == A
assert Xor(True, False, False, A, B) == ~Xor(A, B)
assert Xor(True) is true
assert Xor(False) is false
assert Xor(True, True) is false
assert Xor(True, False) is true
assert Xor(False, False) is false
assert Xor(True, A) == ~A
assert Xor(False, A) == A
assert Xor(True, False, False) is true
assert Xor(True, False, A) == ~A
assert Xor(False, False, A) == A
assert isinstance(Xor(A, B), Xor)
assert Xor(A, B, Xor(C, D)) == Xor(A, B, C, D)
assert Xor(A, B, Xor(B, C)) == Xor(A, C)
assert Xor(A < 1, A >= 1, B) == Xor(0, 1, B) == Xor(1, 0, B)
e = A > 1
assert Xor(e, e.canonical) == Xor(0, 0) == Xor(1, 1)
def test_rewrite_as_And():
expr = x ^ y
assert expr.rewrite(And) == (x | y) & (~x | ~y)
def test_rewrite_as_Or():
expr = x ^ y
assert expr.rewrite(Or) == (x & ~y) | (y & ~x)
def test_rewrite_as_Nand():
expr = (y & z) | (z & ~w)
assert expr.rewrite(Nand) == ~(~(y & z) & ~(z & ~w))
def test_rewrite_as_Nor():
expr = z & (y | ~w)
assert expr.rewrite(Nor) == ~(~z | ~(y | ~w))
def test_Not():
raises(TypeError, lambda: Not(True, False))
assert Not(True) is false
assert Not(False) is true
assert Not(0) is true
assert Not(1) is false
assert Not(2) is false
def test_Nand():
assert Nand() is false
assert Nand(A) == ~A
assert Nand(True) is false
assert Nand(False) is true
assert Nand(True, True) is false
assert Nand(True, False) is true
assert Nand(False, False) is true
assert Nand(True, A) == ~A
assert Nand(False, A) is true
assert Nand(True, True, True) is false
assert Nand(True, True, A) == ~A
assert Nand(True, False, A) is true
def test_Nor():
assert Nor() is true
assert Nor(A) == ~A
assert Nor(True) is false
assert Nor(False) is true
assert Nor(True, True) is false
assert Nor(True, False) is false
assert Nor(False, False) is true
assert Nor(True, A) is false
assert Nor(False, A) == ~A
assert Nor(True, True, True) is false
assert Nor(True, True, A) is false
assert Nor(True, False, A) is false
def test_Xnor():
assert Xnor() is true
assert Xnor(A) == ~A
assert Xnor(A, A) is true
assert Xnor(True, A, A) is false
assert Xnor(A, A, A, A, A) == ~A
assert Xnor(True) is false
assert Xnor(False) is true
assert Xnor(True, True) is true
assert Xnor(True, False) is false
assert Xnor(False, False) is true
assert Xnor(True, A) == A
assert Xnor(False, A) == ~A
assert Xnor(True, False, False) is false
assert Xnor(True, False, A) == A
assert Xnor(False, False, A) == ~A
def test_Implies():
raises(ValueError, lambda: Implies(A, B, C))
assert Implies(True, True) is true
assert Implies(True, False) is false
assert Implies(False, True) is true
assert Implies(False, False) is true
assert Implies(0, A) is true
assert Implies(1, 1) is true
assert Implies(1, 0) is false
assert A >> B == B << A
assert (A < 1) >> (A >= 1) == (A >= 1)
assert (A < 1) >> (S.One > A) is true
assert A >> A is true
def test_Equivalent():
assert Equivalent(A, B) == Equivalent(B, A) == Equivalent(A, B, A)
assert Equivalent() is true
assert Equivalent(A, A) == Equivalent(A) is true
assert Equivalent(True, True) == Equivalent(False, False) is true
assert Equivalent(True, False) == Equivalent(False, True) is false
assert Equivalent(A, True) == A
assert Equivalent(A, False) == Not(A)
assert Equivalent(A, B, True) == A & B
assert Equivalent(A, B, False) == ~A & ~B
assert Equivalent(1, A) == A
assert Equivalent(0, A) == Not(A)
assert Equivalent(A, Equivalent(B, C)) != Equivalent(Equivalent(A, B), C)
assert Equivalent(A < 1, A >= 1) is false
assert Equivalent(A < 1, A >= 1, 0) is false
assert Equivalent(A < 1, A >= 1, 1) is false
assert Equivalent(A < 1, S.One > A) == Equivalent(1, 1) == Equivalent(0, 0)
assert Equivalent(Equality(A, B), Equality(B, A)) is true
def test_Exclusive():
assert Exclusive(False, False, False) is true
assert Exclusive(True, False, False) is true
assert Exclusive(True, True, False) is false
assert Exclusive(True, True, True) is false
def test_equals():
assert Not(Or(A, B)).equals(And(Not(A), Not(B))) is True
assert Equivalent(A, B).equals((A >> B) & (B >> A)) is True
assert ((A | ~B) & (~A | B)).equals((~A & ~B) | (A & B)) is True
assert (A >> B).equals(~A >> ~B) is False
assert (A >> (B >> A)).equals(A >> (C >> A)) is False
raises(NotImplementedError, lambda: (A & B).equals(A > B))
def test_simplification_boolalg():
"""
Test working of simplification methods.
"""
set1 = [[0, 0, 1], [0, 1, 1], [1, 0, 0], [1, 1, 0]]
set2 = [[0, 0, 0], [0, 1, 0], [1, 0, 1], [1, 1, 1]]
assert SOPform([x, y, z], set1) == Or(And(Not(x), z), And(Not(z), x))
assert Not(SOPform([x, y, z], set2)) == \
Not(Or(And(Not(x), Not(z)), And(x, z)))
assert POSform([x, y, z], set1 + set2) is true
assert SOPform([x, y, z], set1 + set2) is true
assert SOPform([Dummy(), Dummy(), Dummy()], set1 + set2) is true
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [[0, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 1]]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, 3, 7, 11, 15]
dontcares = [0, 2, 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, [0, 0, 1, 1], 7, [1, 0, 1, 1],
[1, 1, 1, 1]]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [1, {y: 1, z: 1}]
dontcares = [0, [0, 0, 1, 0], 5]
assert (
SOPform([w, x, y, z], minterms, dontcares) ==
Or(And(y, z), And(Not(w), Not(x))))
assert POSform([w, x, y, z], minterms, dontcares) == And(Or(Not(w), y), z)
minterms = [{y: 1, z: 1}, 1]
dontcares = [[0, 0, 0, 0]]
minterms = [[0, 0, 0]]
raises(ValueError, lambda: SOPform([w, x, y, z], minterms))
raises(ValueError, lambda: POSform([w, x, y, z], minterms))
raises(TypeError, lambda: POSform([w, x, y, z], ["abcdefg"]))
# test simplification
ans = And(A, Or(B, C))
assert simplify_logic(A & (B | C)) == ans
assert simplify_logic((A & B) | (A & C)) == ans
assert simplify_logic(Implies(A, B)) == Or(Not(A), B)
assert simplify_logic(Equivalent(A, B)) == \
Or(And(A, B), And(Not(A), Not(B)))
assert simplify_logic(And(Equality(A, 2), C)) == And(Equality(A, 2), C)
assert simplify_logic(And(Equality(A, 2), A)) is S.false
assert simplify_logic(And(Equality(A, 2), A)) == And(Equality(A, 2), A)
assert simplify_logic(And(Equality(A, B), C)) == And(Equality(A, B), C)
assert simplify_logic(Or(And(Equality(A, 3), B), And(Equality(A, 3), C))) \
== And(Equality(A, 3), Or(B, C))
b = (~x & ~y & ~z) | (~x & ~y & z)
e = And(A, b)
assert simplify_logic(e) == A & ~x & ~y
raises(ValueError, lambda: simplify_logic(A & (B | C), form='blabla'))
assert simplify(Or(x <= y, And(x < y, z))) == (x <= y)
assert simplify(Or(x <= y, And(y > x, z))) == (x <= y)
assert simplify(Or(x >= y, And(y < x, z))) == (x >= y)
# Check that expressions with nine variables or more are not simplified
# (without the force-flag)
a, b, c, d, e, f, g, h, j = symbols('a b c d e f g h j')
expr = a & b & c & d & e & f & g & h & j | \
a & b & c & d & e & f & g & h & ~j
# This expression can be simplified to get rid of the j variables
assert simplify_logic(expr) == expr
# check input
ans = SOPform([x, y], [[1, 0]])
assert SOPform([x, y], [[1, 0]]) == ans
assert POSform([x, y], [[1, 0]]) == ans
raises(ValueError, lambda: SOPform([x], [[1]], [[1]]))
assert SOPform([x], [[1]], [[0]]) is true
assert SOPform([x], [[0]], [[1]]) is true
assert SOPform([x], [], []) is false
raises(ValueError, lambda: POSform([x], [[1]], [[1]]))
assert POSform([x], [[1]], [[0]]) is true
assert POSform([x], [[0]], [[1]]) is true
assert POSform([x], [], []) is false
# check working of simplify
assert simplify((A & B) | (A & C)) == And(A, Or(B, C))
assert simplify(And(x, Not(x))) == False
assert simplify(Or(x, Not(x))) == True
assert simplify(And(Eq(x, 0), Eq(x, y))) == And(Eq(x, 0), Eq(y, 0))
assert And(Eq(x - 1, 0), Eq(x, y)).simplify() == And(Eq(x, 1), Eq(y, 1))
assert And(Ne(x - 1, 0), Ne(x, y)).simplify() == And(Ne(x, 1), Ne(x, y))
assert And(Eq(x - 1, 0), Ne(x, y)).simplify() == And(Eq(x, 1), Ne(y, 1))
assert And(Eq(x - 1, 0), Eq(x, z + y), Eq(y + x, 0)).simplify(
) == And(Eq(x, 1), Eq(y, -1), Eq(z, 2))
assert And(Eq(x - 1, 0), Eq(x + 2, 3)).simplify() == Eq(x, 1)
assert And(Ne(x - 1, 0), Ne(x + 2, 3)).simplify() == Ne(x, 1)
assert And(Eq(x - 1, 0), Eq(x + 2, 2)).simplify() == False
assert And(Ne(x - 1, 0), Ne(x + 2, 2)).simplify(
) == And(Ne(x, 1), Ne(x, 0))
def test_bool_map():
"""
Test working of bool_map function.
"""
minterms = [[0, 0, 0, 1], [0, 0, 1, 1], [0, 1, 1, 1], [1, 0, 1, 1],
[1, 1, 1, 1]]
assert bool_map(Not(Not(a)), a) == (a, {a: a})
assert bool_map(SOPform([w, x, y, z], minterms),
POSform([w, x, y, z], minterms)) == \
(And(Or(Not(w), y), Or(Not(x), y), z), {x: x, w: w, z: z, y: y})
assert bool_map(SOPform([x, z, y], [[1, 0, 1]]),
SOPform([a, b, c], [[1, 0, 1]])) != False
function1 = SOPform([x, z, y], [[1, 0, 1], [0, 0, 1]])
function2 = SOPform([a, b, c], [[1, 0, 1], [1, 0, 0]])
assert bool_map(function1, function2) == \
(function1, {y: a, z: b})
assert bool_map(Xor(x, y), ~Xor(x, y)) == False
assert bool_map(And(x, y), Or(x, y)) is None
assert bool_map(And(x, y), And(x, y, z)) is None
# issue 16179
assert bool_map(Xor(x, y, z), ~Xor(x, y, z)) == False
assert bool_map(Xor(a, x, y, z), ~Xor(a, x, y, z)) == False
def test_bool_symbol():
"""Test that mixing symbols with boolean values
works as expected"""
assert And(A, True) == A
assert And(A, True, True) == A
assert And(A, False) is false
assert And(A, True, False) is false
assert Or(A, True) is true
assert Or(A, False) == A
def test_is_boolean():
assert isinstance(True, Boolean) is False
assert isinstance(true, Boolean) is True
assert 1 == True
assert 1 != true
assert (1 == true) is False
assert 0 == False
assert 0 != false
assert (0 == false) is False
assert true.is_Boolean is True
assert (A & B).is_Boolean
assert (A | B).is_Boolean
assert (~A).is_Boolean
assert (A ^ B).is_Boolean
assert A.is_Boolean != isinstance(A, Boolean)
assert isinstance(A, Boolean)
def test_subs():
assert (A & B).subs(A, True) == B
assert (A & B).subs(A, False) is false
assert (A & B).subs(B, True) == A
assert (A & B).subs(B, False) is false
assert (A & B).subs({A: True, B: True}) is true
assert (A | B).subs(A, True) is true
assert (A | B).subs(A, False) == B
assert (A | B).subs(B, True) is true
assert (A | B).subs(B, False) == A
assert (A | B).subs({A: True, B: True}) is true
"""
we test for axioms of boolean algebra
see https://en.wikipedia.org/wiki/Boolean_algebra_(structure)
"""
def test_commutative():
"""Test for commutativity of And and Or"""
A, B = map(Boolean, symbols('A,B'))
assert A & B == B & A
assert A | B == B | A
def test_and_associativity():
"""Test for associativity of And"""
assert (A & B) & C == A & (B & C)
def test_or_assicativity():
assert ((A | B) | C) == (A | (B | C))
def test_double_negation():
a = Boolean()
assert ~(~a) == a
# test methods
def test_eliminate_implications():
assert eliminate_implications(Implies(A, B, evaluate=False)) == (~A) | B
assert eliminate_implications(
A >> (C >> Not(B))) == Or(Or(Not(B), Not(C)), Not(A))
assert eliminate_implications(Equivalent(A, B, C, D)) == \
(~A | B) & (~B | C) & (~C | D) & (~D | A)
def test_conjuncts():
assert conjuncts(A & B & C) == {A, B, C}
assert conjuncts((A | B) & C) == {A | B, C}
assert conjuncts(A) == {A}
assert conjuncts(True) == {True}
assert conjuncts(False) == {False}
def test_disjuncts():
assert disjuncts(A | B | C) == {A, B, C}
assert disjuncts((A | B) & C) == {(A | B) & C}
assert disjuncts(A) == {A}
assert disjuncts(True) == {True}
assert disjuncts(False) == {False}
def test_distribute():
assert distribute_and_over_or(Or(And(A, B), C)) == And(Or(A, C), Or(B, C))
assert distribute_or_over_and(And(A, Or(B, C))) == Or(And(A, B), And(A, C))
assert distribute_xor_over_and(And(A, Xor(B, C))) == Xor(And(A, B), And(A, C))
def test_to_anf():
x, y, z = symbols('x,y,z')
assert to_anf(And(x, y)) == And(x, y)
assert to_anf(Or(x, y)) == Xor(x, y, And(x, y))
assert to_anf(Or(Implies(x, y), And(x, y), y)) == \
Xor(x, True, x & y, remove_true=False)
assert to_anf(Or(Nand(x, y), Nor(x, y), Xnor(x, y), Implies(x, y))) == True
assert to_anf(Or(x, Not(y), Nor(x,z), And(x, y), Nand(y, z))) == \
Xor(True, And(y, z), And(x, y, z), remove_true=False)
assert to_anf(Xor(x, y)) == Xor(x, y)
assert to_anf(Not(x)) == Xor(x, True, remove_true=False)
assert to_anf(Nand(x, y)) == Xor(True, And(x, y), remove_true=False)
assert to_anf(Nor(x, y)) == Xor(x, y, True, And(x, y), remove_true=False)
assert to_anf(Implies(x, y)) == Xor(x, True, And(x, y), remove_true=False)
assert to_anf(Equivalent(x, y)) == Xor(x, y, True, remove_true=False)
assert to_anf(Nand(x | y, x >> y), deep=False) == \
Xor(True, And(Or(x, y), Implies(x, y)), remove_true=False)
assert to_anf(Nor(x ^ y, x & y), deep=False) == \
Xor(True, Or(Xor(x, y), And(x, y)), remove_true=False)
def test_to_nnf():
assert to_nnf(true) is true
assert to_nnf(false) is false
assert to_nnf(A) == A
assert to_nnf(A | ~A | B) is true
assert to_nnf(A & ~A & B) is false
assert to_nnf(A >> B) == ~A | B
assert to_nnf(Equivalent(A, B, C)) == (~A | B) & (~B | C) & (~C | A)
assert to_nnf(A ^ B ^ C) == \
(A | B | C) & (~A | ~B | C) & (A | ~B | ~C) & (~A | B | ~C)
assert to_nnf(ITE(A, B, C)) == (~A | B) & (A | C)
assert to_nnf(Not(A | B | C)) == ~A & ~B & ~C
assert to_nnf(Not(A & B & C)) == ~A | ~B | ~C
assert to_nnf(Not(A >> B)) == A & ~B
assert to_nnf(Not(Equivalent(A, B, C))) == And(Or(A, B, C), Or(~A, ~B, ~C))
assert to_nnf(Not(A ^ B ^ C)) == \
(~A | B | C) & (A | ~B | C) & (A | B | ~C) & (~A | ~B | ~C)
assert to_nnf(Not(ITE(A, B, C))) == (~A | ~B) & (A | ~C)
assert to_nnf((A >> B) ^ (B >> A)) == (A & ~B) | (~A & B)
assert to_nnf((A >> B) ^ (B >> A), False) == \
(~A | ~B | A | B) & ((A & ~B) | (~A & B))
assert ITE(A, 1, 0).to_nnf() == A
assert ITE(A, 0, 1).to_nnf() == ~A
# although ITE can hold non-Boolean, it will complain if
# an attempt is made to convert the ITE to Boolean nnf
raises(TypeError, lambda: ITE(A < 1, [1], B).to_nnf())
def test_to_cnf():
assert to_cnf(~(B | C)) == And(Not(B), Not(C))
assert to_cnf((A & B) | C) == And(Or(A, C), Or(B, C))
assert to_cnf(A >> B) == (~A) | B
assert to_cnf(A >> (B & C)) == (~A | B) & (~A | C)
assert to_cnf(A & (B | C) | ~A & (B | C), True) == B | C
assert to_cnf(A & B) == And(A, B)
assert to_cnf(Equivalent(A, B)) == And(Or(A, Not(B)), Or(B, Not(A)))
assert to_cnf(Equivalent(A, B & C)) == \
(~A | B) & (~A | C) & (~B | ~C | A)
assert to_cnf(Equivalent(A, B | C), True) == \
And(Or(Not(B), A), Or(Not(C), A), Or(B, C, Not(A)))
assert to_cnf(A + 1) == A + 1
def test_issue_18904():
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, x15 = symbols('x1:16')
eq = (( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x8 & x9 ) |
( x1 & x2 & x3 & x4 & x5 & x6 & x7 & x10 & x9 ) |
( x1 & x11 & x3 & x12 & x5 & x13 & x14 & x15 & x9 ))
assert is_cnf(to_cnf(eq))
raises(ValueError, lambda: to_cnf(eq, simplify=True))
for f, t in zip((And, Or), (to_cnf, to_dnf)):
eq = f(x1, x2, x3, x4, x5, x6, x7, x8, x9)
raises(ValueError, lambda: to_cnf(eq, simplify=True))
assert t(eq, simplify=True, force=True) == eq
def test_issue_9949():
assert is_cnf(to_cnf((b > -5) | (a > 2) & (a < 4)))
def test_to_CNF():
assert CNF.CNF_to_cnf(CNF.to_CNF(~(B | C))) == to_cnf(~(B | C))
assert CNF.CNF_to_cnf(CNF.to_CNF((A & B) | C)) == to_cnf((A & B) | C)
assert CNF.CNF_to_cnf(CNF.to_CNF(A >> B)) == to_cnf(A >> B)
assert CNF.CNF_to_cnf(CNF.to_CNF(A >> (B & C))) == to_cnf(A >> (B & C))
assert CNF.CNF_to_cnf(CNF.to_CNF(A & (B | C) | ~A & (B | C))) == to_cnf(A & (B | C) | ~A & (B | C))
assert CNF.CNF_to_cnf(CNF.to_CNF(A & B)) == to_cnf(A & B)
def test_to_dnf():
assert to_dnf(~(B | C)) == And(Not(B), Not(C))
assert to_dnf(A & (B | C)) == Or(And(A, B), And(A, C))
assert to_dnf(A >> B) == (~A) | B
assert to_dnf(A >> (B & C)) == (~A) | (B & C)
assert to_dnf(A | B) == A | B
assert to_dnf(Equivalent(A, B), True) == \
Or(And(A, B), And(Not(A), Not(B)))
assert to_dnf(Equivalent(A, B & C), True) == \
Or(And(A, B, C), And(Not(A), Not(B)), And(Not(A), Not(C)))
assert to_dnf(A + 1) == A + 1
def test_to_int_repr():
x, y, z = map(Boolean, symbols('x,y,z'))
def sorted_recursive(arg):
try:
return sorted(sorted_recursive(x) for x in arg)
except TypeError: # arg is not a sequence
return arg
assert sorted_recursive(to_int_repr([x | y, z | x], [x, y, z])) == \
sorted_recursive([[1, 2], [1, 3]])
assert sorted_recursive(to_int_repr([x | y, z | ~x], [x, y, z])) == \
sorted_recursive([[1, 2], [3, -1]])
def test_is_anf():
x, y = symbols('x,y')
assert is_anf(true) is True
assert is_anf(false) is True
assert is_anf(x) is True
assert is_anf(And(x, y)) is True
assert is_anf(Xor(x, y, And(x, y))) is True
assert is_anf(Xor(x, y, Or(x, y))) is False
assert is_anf(Xor(Not(x), y)) is False
def test_is_nnf():
assert is_nnf(true) is True
assert is_nnf(A) is True
assert is_nnf(~A) is True
assert is_nnf(A & B) is True
assert is_nnf((A & B) | (~A & A) | (~B & B) | (~A & ~B), False) is True
assert is_nnf((A | B) & (~A | ~B)) is True
assert is_nnf(Not(Or(A, B))) is False
assert is_nnf(A ^ B) is False
assert is_nnf((A & B) | (~A & A) | (~B & B) | (~A & ~B), True) is False
def test_is_cnf():
assert is_cnf(x) is True
assert is_cnf(x | y | z) is True
assert is_cnf(x & y & z) is True
assert is_cnf((x | y) & z) is True
assert is_cnf((x & y) | z) is False
assert is_cnf(~(x & y) | z) is False
def test_is_dnf():
assert is_dnf(x) is True
assert is_dnf(x | y | z) is True
assert is_dnf(x & y & z) is True
assert is_dnf((x & y) | z) is True
assert is_dnf((x | y) & z) is False
assert is_dnf(~(x | y) & z) is False
def test_ITE():
A, B, C = symbols('A:C')
assert ITE(True, False, True) is false
assert ITE(True, True, False) is true
assert ITE(False, True, False) is false
assert ITE(False, False, True) is true
assert isinstance(ITE(A, B, C), ITE)
A = True
assert ITE(A, B, C) == B
A = False
assert ITE(A, B, C) == C
B = True
assert ITE(And(A, B), B, C) == C
assert ITE(Or(A, False), And(B, True), False) is false
assert ITE(x, A, B) == Not(x)
assert ITE(x, B, A) == x
assert ITE(1, x, y) == x
assert ITE(0, x, y) == y
raises(TypeError, lambda: ITE(2, x, y))
raises(TypeError, lambda: ITE(1, [], y))
raises(TypeError, lambda: ITE(1, (), y))
raises(TypeError, lambda: ITE(1, y, []))
assert ITE(1, 1, 1) is S.true
assert isinstance(ITE(1, 1, 1, evaluate=False), ITE)
raises(TypeError, lambda: ITE(x > 1, y, x))
assert ITE(Eq(x, True), y, x) == ITE(x, y, x)
assert ITE(Eq(x, False), y, x) == ITE(~x, y, x)
assert ITE(Ne(x, True), y, x) == ITE(~x, y, x)
assert ITE(Ne(x, False), y, x) == ITE(x, y, x)
assert ITE(Eq(S. true, x), y, x) == ITE(x, y, x)
assert ITE(Eq(S.false, x), y, x) == ITE(~x, y, x)
assert ITE(Ne(S.true, x), y, x) == ITE(~x, y, x)
assert ITE(Ne(S.false, x), y, x) == ITE(x, y, x)
# 0 and 1 in the context are not treated as True/False
# so the equality must always be False since dissimilar
# objects cannot be equal
assert ITE(Eq(x, 0), y, x) == x
assert ITE(Eq(x, 1), y, x) == x
assert ITE(Ne(x, 0), y, x) == y
assert ITE(Ne(x, 1), y, x) == y
assert ITE(Eq(x, 0), y, z).subs(x, 0) == y
assert ITE(Eq(x, 0), y, z).subs(x, 1) == z
raises(ValueError, lambda: ITE(x > 1, y, x, z))
def test_is_literal():
assert is_literal(True) is True
assert is_literal(False) is True
assert is_literal(A) is True
assert is_literal(~A) is True
assert is_literal(Or(A, B)) is False
assert is_literal(Q.zero(A)) is True
assert is_literal(Not(Q.zero(A))) is True
assert is_literal(Or(A, B)) is False
assert is_literal(And(Q.zero(A), Q.zero(B))) is False
assert is_literal(x < 3)
assert not is_literal(x + y < 3)
def test_operators():
# Mostly test __and__, __rand__, and so on
assert True & A == A & True == A
assert False & A == A & False == False
assert A & B == And(A, B)
assert True | A == A | True == True
assert False | A == A | False == A
assert A | B == Or(A, B)
assert ~A == Not(A)
assert True >> A == A << True == A
assert False >> A == A << False == True
assert A >> True == True << A == True
assert A >> False == False << A == ~A
assert A >> B == B << A == Implies(A, B)
assert True ^ A == A ^ True == ~A
assert False ^ A == A ^ False == A
assert A ^ B == Xor(A, B)
def test_true_false():
assert true is S.true
assert false is S.false
assert true is not True
assert false is not False
assert true
assert not false
assert true == True
assert false == False
assert not (true == False)
assert not (false == True)
assert not (true == false)
assert hash(true) == hash(True)
assert hash(false) == hash(False)
assert len({true, True}) == len({false, False}) == 1
assert isinstance(true, BooleanAtom)
assert isinstance(false, BooleanAtom)
# We don't want to subclass from bool, because bool subclasses from
# int. But operators like &, |, ^, <<, >>, and ~ act differently on 0 and
# 1 then we want them to on true and false. See the docstrings of the
# various And, Or, etc. functions for examples.
assert not isinstance(true, bool)
assert not isinstance(false, bool)
# Note: using 'is' comparison is important here. We want these to return
# true and false, not True and False
assert Not(true) is false
assert Not(True) is false
assert Not(false) is true
assert Not(False) is true
assert ~true is false
assert ~false is true
for T, F in product((True, true), (False, false)):
assert And(T, F) is false
assert And(F, T) is false
assert And(F, F) is false
assert And(T, T) is true
assert And(T, x) == x
assert And(F, x) is false
if not (T is True and F is False):
assert T & F is false
assert F & T is false
if F is not False:
assert F & F is false
if T is not True:
assert T & T is true
assert Or(T, F) is true
assert Or(F, T) is true
assert Or(F, F) is false
assert Or(T, T) is true
assert Or(T, x) is true
assert Or(F, x) == x
if not (T is True and F is False):
assert T | F is true
assert F | T is true
if F is not False:
assert F | F is false
if T is not True:
assert T | T is true
assert Xor(T, F) is true
assert Xor(F, T) is true
assert Xor(F, F) is false
assert Xor(T, T) is false
assert Xor(T, x) == ~x
assert Xor(F, x) == x
if not (T is True and F is False):
assert T ^ F is true
assert F ^ T is true
if F is not False:
assert F ^ F is false
if T is not True:
assert T ^ T is false
assert Nand(T, F) is true
assert Nand(F, T) is true
assert Nand(F, F) is true
assert Nand(T, T) is false
assert Nand(T, x) == ~x
assert Nand(F, x) is true
assert Nor(T, F) is false
assert Nor(F, T) is false
assert Nor(F, F) is true
assert Nor(T, T) is false
assert Nor(T, x) is false
assert Nor(F, x) == ~x
assert Implies(T, F) is false
assert Implies(F, T) is true
assert Implies(F, F) is true
assert Implies(T, T) is true
assert Implies(T, x) == x
assert Implies(F, x) is true
assert Implies(x, T) is true
assert Implies(x, F) == ~x
if not (T is True and F is False):
assert T >> F is false
assert F << T is false
assert F >> T is true
assert T << F is true
if F is not False:
assert F >> F is true
assert F << F is true
if T is not True:
assert T >> T is true
assert T << T is true
assert Equivalent(T, F) is false
assert Equivalent(F, T) is false
assert Equivalent(F, F) is true
assert Equivalent(T, T) is true
assert Equivalent(T, x) == x
assert Equivalent(F, x) == ~x
assert Equivalent(x, T) == x
assert Equivalent(x, F) == ~x
assert ITE(T, T, T) is true
assert ITE(T, T, F) is true
assert ITE(T, F, T) is false
assert ITE(T, F, F) is false
assert ITE(F, T, T) is true
assert ITE(F, T, F) is false
assert ITE(F, F, T) is true
assert ITE(F, F, F) is false
assert all(i.simplify(1, 2) is i for i in (S.true, S.false))
def test_bool_as_set():
assert ITE(y <= 0, False, y >= 1).as_set() == Interval(1, oo)
assert And(x <= 2, x >= -2).as_set() == Interval(-2, 2)
assert Or(x >= 2, x <= -2).as_set() == Interval(-oo, -2) + Interval(2, oo)
assert Not(x > 2).as_set() == Interval(-oo, 2)
# issue 10240
assert Not(And(x > 2, x < 3)).as_set() == \
Union(Interval(-oo, 2), Interval(3, oo))
assert true.as_set() == S.UniversalSet
assert false.as_set() is S.EmptySet
assert x.as_set() == S.UniversalSet
assert And(Or(x < 1, x > 3), x < 2).as_set() == Interval.open(-oo, 1)
assert And(x < 1, sin(x) < 3).as_set() == (x < 1).as_set()
raises(NotImplementedError, lambda: (sin(x) < 1).as_set())
# watch for object morph in as_set
assert Eq(-1, cos(2*x)**2/sin(2*x)**2).as_set() is S.EmptySet
@XFAIL
def test_multivariate_bool_as_set():
x, y = symbols('x,y')
assert And(x >= 0, y >= 0).as_set() == Interval(0, oo)*Interval(0, oo)
assert Or(x >= 0, y >= 0).as_set() == S.Reals*S.Reals - \
Interval(-oo, 0, True, True)*Interval(-oo, 0, True, True)
def test_all_or_nothing():
x = symbols('x', extended_real=True)
args = x >= -oo, x <= oo
v = And(*args)
if v.func is And:
assert len(v.args) == len(args) - args.count(S.true)
else:
assert v == True
v = Or(*args)
if v.func is Or:
assert len(v.args) == 2
else:
assert v == True
def test_canonical_atoms():
assert true.canonical == true
assert false.canonical == false
def test_negated_atoms():
assert true.negated == false
assert false.negated == true
def test_issue_8777():
assert And(x > 2, x < oo).as_set() == Interval(2, oo, left_open=True)
assert And(x >= 1, x < oo).as_set() == Interval(1, oo)
assert (x < oo).as_set() == Interval(-oo, oo)
assert (x > -oo).as_set() == Interval(-oo, oo)
def test_issue_8975():
assert Or(And(-oo < x, x <= -2), And(2 <= x, x < oo)).as_set() == \
Interval(-oo, -2) + Interval(2, oo)
def test_term_to_integer():
assert term_to_integer([1, 0, 1, 0, 0, 1, 0]) == 82
assert term_to_integer('0010101000111001') == 10809
def test_issue_21971():
a, b, c, d = symbols('a b c d')
f = a & b & c | a & c
assert f.subs(a & c, d) == b & d | d
assert f.subs(a & b & c, d) == a & c | d
f = (a | b | c) & (a | c)
assert f.subs(a | c, d) == (b | d) & d
assert f.subs(a | b | c, d) == (a | c) & d
f = (a ^ b ^ c) & (a ^ c)
assert f.subs(a ^ c, d) == (b ^ d) & d
assert f.subs(a ^ b ^ c, d) == (a ^ c) & d
def test_truth_table():
assert list(truth_table(And(x, y), [x, y], input=False)) == \
[False, False, False, True]
assert list(truth_table(x | y, [x, y], input=False)) == \
[False, True, True, True]
assert list(truth_table(x >> y, [x, y], input=False)) == \
[True, True, False, True]
assert list(truth_table(And(x, y), [x, y])) == \
[([0, 0], False), ([0, 1], False), ([1, 0], False), ([1, 1], True)]
def test_issue_8571():
for t in (S.true, S.false):
raises(TypeError, lambda: +t)
raises(TypeError, lambda: -t)
raises(TypeError, lambda: abs(t))
# use int(bool(t)) to get 0 or 1
raises(TypeError, lambda: int(t))
for o in [S.Zero, S.One, x]:
for _ in range(2):
raises(TypeError, lambda: o + t)
raises(TypeError, lambda: o - t)
raises(TypeError, lambda: o % t)
raises(TypeError, lambda: o*t)
raises(TypeError, lambda: o/t)
raises(TypeError, lambda: o**t)
o, t = t, o # do again in reversed order
def test_expand_relational():
n = symbols('n', negative=True)
p, q = symbols('p q', positive=True)
r = ((n + q*(-n/q + 1))/(q*(-n/q + 1)) < 0)
assert r is not S.false
assert r.expand() is S.false
assert (q > 0).expand() is S.true
def test_issue_12717():
assert S.true.is_Atom == True
assert S.false.is_Atom == True
def test_as_Boolean():
nz = symbols('nz', nonzero=True)
assert all(as_Boolean(i) is S.true for i in (True, S.true, 1, nz))
z = symbols('z', zero=True)
assert all(as_Boolean(i) is S.false for i in (False, S.false, 0, z))
assert all(as_Boolean(i) == i for i in (x, x < 0))
for i in (2, S(2), x + 1, []):
raises(TypeError, lambda: as_Boolean(i))
def test_binary_symbols():
assert ITE(x < 1, y, z).binary_symbols == {y, z}
for f in (Eq, Ne):
assert f(x, 1).binary_symbols == set()
assert f(x, True).binary_symbols == {x}
assert f(x, False).binary_symbols == {x}
assert S.true.binary_symbols == set()
assert S.false.binary_symbols == set()
assert x.binary_symbols == {x}
assert And(x, Eq(y, False), Eq(z, 1)).binary_symbols == {x, y}
assert Q.prime(x).binary_symbols == set()
assert Q.lt(x, 1).binary_symbols == set()
assert Q.is_true(x).binary_symbols == {x}
assert Q.eq(x, True).binary_symbols == {x}
assert Q.prime(x).binary_symbols == set()
def test_BooleanFunction_diff():
assert And(x, y).diff(x) == Piecewise((0, Eq(y, False)), (1, True))
def test_issue_14700():
A, B, C, D, E, F, G, H = symbols('A B C D E F G H')
q = ((B & D & H & ~F) | (B & H & ~C & ~D) | (B & H & ~C & ~F) |
(B & H & ~D & ~G) | (B & H & ~F & ~G) | (C & G & ~B & ~D) |
(C & G & ~D & ~H) | (C & G & ~F & ~H) | (D & F & H & ~B) |
(D & F & ~G & ~H) | (B & D & F & ~C & ~H) | (D & E & F & ~B & ~C) |
(D & F & ~A & ~B & ~C) | (D & F & ~A & ~C & ~H) |
(A & B & D & F & ~E & ~H))
soldnf = ((B & D & H & ~F) | (D & F & H & ~B) | (B & H & ~C & ~D) |
(B & H & ~D & ~G) | (C & G & ~B & ~D) | (C & G & ~D & ~H) |
(C & G & ~F & ~H) | (D & F & ~G & ~H) | (D & E & F & ~C & ~H) |
(D & F & ~A & ~C & ~H) | (A & B & D & F & ~E & ~H))
solcnf = ((B | C | D) & (B | D | G) & (C | D | H) & (C | F | H) &
(D | G | H) & (F | G | H) & (B | F | ~D | ~H) &
(~B | ~D | ~F | ~H) & (D | ~B | ~C | ~G | ~H) &
(A | H | ~C | ~D | ~F | ~G) & (H | ~C | ~D | ~E | ~F | ~G) &
(B | E | H | ~A | ~D | ~F | ~G))
assert simplify_logic(q, "dnf") == soldnf
assert simplify_logic(q, "cnf") == solcnf
minterms = [[0, 1, 0, 0], [0, 1, 0, 1], [0, 1, 1, 0], [0, 1, 1, 1],
[0, 0, 1, 1], [1, 0, 1, 1]]
dontcares = [[1, 0, 0, 0], [1, 0, 0, 1], [1, 1, 0, 0], [1, 1, 0, 1]]
assert SOPform([w, x, y, z], minterms) == (x & ~w) | (y & z & ~x)
# Should not be more complicated with don't cares
assert SOPform([w, x, y, z], minterms, dontcares) == \
(x & ~w) | (y & z & ~x)
def test_relational_simplification():
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
# Test all combinations or sign and order
assert Or(x >= y, x < y).simplify() == S.true
assert Or(x >= y, y > x).simplify() == S.true
assert Or(x >= y, -x > -y).simplify() == S.true
assert Or(x >= y, -y < -x).simplify() == S.true
assert Or(-x <= -y, x < y).simplify() == S.true
assert Or(-x <= -y, -x > -y).simplify() == S.true
assert Or(-x <= -y, y > x).simplify() == S.true
assert Or(-x <= -y, -y < -x).simplify() == S.true
assert Or(y <= x, x < y).simplify() == S.true
assert Or(y <= x, y > x).simplify() == S.true
assert Or(y <= x, -x > -y).simplify() == S.true
assert Or(y <= x, -y < -x).simplify() == S.true
assert Or(-y >= -x, x < y).simplify() == S.true
assert Or(-y >= -x, y > x).simplify() == S.true
assert Or(-y >= -x, -x > -y).simplify() == S.true
assert Or(-y >= -x, -y < -x).simplify() == S.true
assert Or(x < y, x >= y).simplify() == S.true
assert Or(y > x, x >= y).simplify() == S.true
assert Or(-x > -y, x >= y).simplify() == S.true
assert Or(-y < -x, x >= y).simplify() == S.true
assert Or(x < y, -x <= -y).simplify() == S.true
assert Or(-x > -y, -x <= -y).simplify() == S.true
assert Or(y > x, -x <= -y).simplify() == S.true
assert Or(-y < -x, -x <= -y).simplify() == S.true
assert Or(x < y, y <= x).simplify() == S.true
assert Or(y > x, y <= x).simplify() == S.true
assert Or(-x > -y, y <= x).simplify() == S.true
assert Or(-y < -x, y <= x).simplify() == S.true
assert Or(x < y, -y >= -x).simplify() == S.true
assert Or(y > x, -y >= -x).simplify() == S.true
assert Or(-x > -y, -y >= -x).simplify() == S.true
assert Or(-y < -x, -y >= -x).simplify() == S.true
# Some other tests
assert Or(x >= y, w < z, x <= y).simplify() == S.true
assert And(x >= y, x < y).simplify() == S.false
assert Or(x >= y, Eq(y, x)).simplify() == (x >= y)
assert And(x >= y, Eq(y, x)).simplify() == Eq(x, y)
assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) & (x >= 1) & (y >= 5) & (y > z))
assert Or(Eq(x, y), x >= y, w < y, z < y).simplify() == \
(x >= y) | (y > z) | (w < y)
assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify() == \
Eq(x, y) & (y > z) & (w < y)
# assert And(Eq(x, y), x >= y, w < y, y >= z, z < y).simplify(relational_minmax=True) == \
# And(Eq(x, y), y > Max(w, z))
# assert Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify(relational_minmax=True) == \
# (Eq(x, y) | (x >= 1) | (y > Min(2, z)))
assert And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y).simplify() == \
(Eq(x, y) & (x >= 1) & (y >= 5) & (y > z))
assert (Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)).simplify() == \
(Eq(x, y) & Eq(d, e) & (d >= e))
assert And(Eq(x, y), Eq(x, -y)).simplify() == And(Eq(x, 0), Eq(y, 0))
assert Xor(x >= y, x <= y).simplify() == Ne(x, y)
assert And(x > 1, x < -1, Eq(x, y)).simplify() == S.false
# From #16690
assert And(x >= y, Eq(y, 0)).simplify() == And(x >= 0, Eq(y, 0))
def test_issue_8373():
x = symbols('x', real=True)
assert Or(x < 1, x > -1).simplify() == S.true
assert Or(x < 1, x >= 1).simplify() == S.true
assert And(x < 1, x >= 1).simplify() == S.false
assert Or(x <= 1, x >= 1).simplify() == S.true
def test_issue_7950():
x = symbols('x', real=True)
assert And(Eq(x, 1), Eq(x, 2)).simplify() == S.false
@slow
def test_relational_simplification_numerically():
def test_simplification_numerically_function(original, simplified):
symb = original.free_symbols
n = len(symb)
valuelist = list(set(list(combinations(list(range(-(n-1), n))*n, n))))
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.subs(sublist)
simplifiedvalue = simplified.subs(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for {}"\
"".format(original, simplified, sublist)
w, x, y, z = symbols('w x y z', real=True)
d, e = symbols('d e', real=False)
expressions = (And(Eq(x, y), x >= y, w < y, y >= z, z < y),
And(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
Or(Eq(x, y), x >= 1, 2 < y, y >= 5, z < y),
And(x >= y, Eq(y, x)),
Or(And(Eq(x, y), x >= y, w < y, Or(y >= z, z < y)),
And(Eq(x, y), x >= 1, 2 < y, y >= -1, z < y)),
(Eq(x, y) & Eq(d, e) & (x >= y) & (d >= e)),
)
for expression in expressions:
test_simplification_numerically_function(expression,
expression.simplify())
def test_relational_simplification_patterns_numerically():
from sympy.core import Wild
from sympy.logic.boolalg import _simplify_patterns_and, \
_simplify_patterns_or, _simplify_patterns_xor
a = Wild('a')
b = Wild('b')
c = Wild('c')
symb = [a, b, c]
patternlists = [[And, _simplify_patterns_and()],
[Or, _simplify_patterns_or()],
[Xor, _simplify_patterns_xor()]]
valuelist = list(set(list(combinations(list(range(-2, 3))*3, 3))))
# Skip combinations of +/-2 and 0, except for all 0
valuelist = [v for v in valuelist if any([w % 2 for w in v]) or not any(v)]
for func, patternlist in patternlists:
for pattern in patternlist:
original = func(*pattern[0].args)
simplified = pattern[1]
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.xreplace(sublist)
simplifiedvalue = simplified.xreplace(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for"\
"{}".format(pattern[0], simplified, sublist)
def test_issue_16803():
n = symbols('n')
# No simplification done, but should not raise an exception
assert ((n > 3) | (n < 0) | ((n > 0) & (n < 3))).simplify() == \
(n > 3) | (n < 0) | ((n > 0) & (n < 3))
def test_issue_17530():
r = {x: oo, y: oo}
assert Or(x + y > 0, x - y < 0).subs(r)
assert not And(x + y < 0, x - y < 0).subs(r)
raises(TypeError, lambda: Or(x + y < 0, x - y < 0).subs(r))
raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r))
raises(TypeError, lambda: And(x + y > 0, x - y < 0).subs(r))
def test_anf_coeffs():
assert anf_coeffs([1, 0]) == [1, 1]
assert anf_coeffs([0, 0, 0, 1]) == [0, 0, 0, 1]
assert anf_coeffs([0, 1, 1, 1]) == [0, 1, 1, 1]
assert anf_coeffs([1, 1, 1, 0]) == [1, 0, 0, 1]
assert anf_coeffs([1, 0, 0, 0]) == [1, 1, 1, 1]
assert anf_coeffs([1, 0, 0, 1]) == [1, 1, 1, 0]
assert anf_coeffs([1, 1, 0, 1]) == [1, 0, 1, 1]
def test_ANFform():
x, y = symbols('x,y')
assert ANFform([x], [1, 1]) == True
assert ANFform([x], [0, 0]) == False
assert ANFform([x], [1, 0]) == Xor(x, True, remove_true=False)
assert ANFform([x, y], [1, 1, 1, 0]) == \
Xor(True, And(x, y), remove_true=False)
def test_bool_minterm():
x, y = symbols('x,y')
assert bool_minterm(3, [x, y]) == And(x, y)
assert bool_minterm([1, 0], [x, y]) == And(Not(y), x)
def test_bool_maxterm():
x, y = symbols('x,y')
assert bool_maxterm(2, [x, y]) == Or(Not(x), y)
assert bool_maxterm([0, 1], [x, y]) == Or(Not(y), x)
def test_bool_monomial():
x, y = symbols('x,y')
assert bool_monomial(1, [x, y]) == y
assert bool_monomial([1, 1], [x, y]) == And(x, y)
def test_check_pair():
assert _check_pair([0, 1, 0], [0, 1, 1]) == 2
assert _check_pair([0, 1, 0], [1, 1, 1]) == -1
def test_issue_19114():
expr = (B & C) | (A & ~C) | (~A & ~B)
# Expression is minimal, but there are multiple minimal forms possible
res1 = (A & B) | (C & ~A) | (~B & ~C)
result = to_dnf(expr, simplify=True)
assert result in (expr, res1)
def test_issue_20870():
result = SOPform([a, b, c, d], [1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15])
expected = ((d & ~b) | (a & b & c) | (a & ~c & ~d) |
(b & ~a & ~c) | (c & ~a & ~d))
assert result == expected
def test_convert_to_varsSOP():
assert _convert_to_varsSOP([0, 1, 0], [x, y, z]) == And(Not(x), y, Not(z))
assert _convert_to_varsSOP([3, 1, 0], [x, y, z]) == And(y, Not(z))
def test_convert_to_varsPOS():
assert _convert_to_varsPOS([0, 1, 0], [x, y, z]) == Or(x, Not(y), z)
assert _convert_to_varsPOS([3, 1, 0], [x, y, z]) == Or(Not(y), z)
def test_gateinputcount():
a, b, c, d, e = symbols('a:e')
assert gateinputcount(And(a, b)) == 2
assert gateinputcount(a | b & c & d ^ (e | a)) == 9
assert gateinputcount(And(a, True)) == 0
raises(TypeError, lambda: gateinputcount(a*b))
def test_refine():
# relational
assert not refine(x < 0, ~(x < 0))
assert refine(x < 0, (x < 0))
assert refine(x < 0, (0 > x)) is S.true
assert refine(x < 0, (y < 0)) == (x < 0)
assert not refine(x <= 0, ~(x <= 0))
assert refine(x <= 0, (x <= 0))
assert refine(x <= 0, (0 >= x)) is S.true
assert refine(x <= 0, (y <= 0)) == (x <= 0)
assert not refine(x > 0, ~(x > 0))
assert refine(x > 0, (x > 0))
assert refine(x > 0, (0 < x)) is S.true
assert refine(x > 0, (y > 0)) == (x > 0)
assert not refine(x >= 0, ~(x >= 0))
assert refine(x >= 0, (x >= 0))
assert refine(x >= 0, (0 <= x)) is S.true
assert refine(x >= 0, (y >= 0)) == (x >= 0)
assert not refine(Eq(x, 0), ~(Eq(x, 0)))
assert refine(Eq(x, 0), (Eq(x, 0)))
assert refine(Eq(x, 0), (Eq(0, x))) is S.true
assert refine(Eq(x, 0), (Eq(y, 0))) == Eq(x, 0)
assert not refine(Ne(x, 0), ~(Ne(x, 0)))
assert refine(Ne(x, 0), (Ne(0, x))) is S.true
assert refine(Ne(x, 0), (Ne(x, 0)))
assert refine(Ne(x, 0), (Ne(y, 0))) == (Ne(x, 0))
# boolean functions
assert refine(And(x > 0, y > 0), (x > 0)) == (y > 0)
assert refine(And(x > 0, y > 0), (x > 0) & (y > 0)) is S.true
# predicates
assert refine(Q.positive(x), Q.positive(x)) is S.true
assert refine(Q.positive(x), Q.negative(x)) is S.false
assert refine(Q.positive(x), Q.real(x)) == Q.positive(x)
def test_relational_threeterm_simplification_patterns_numerically():
from sympy.core import Wild
from sympy.logic.boolalg import _simplify_patterns_and3
a = Wild('a')
b = Wild('b')
c = Wild('c')
symb = [a, b, c]
patternlists = [[And, _simplify_patterns_and3()]]
valuelist = list(set(list(combinations(list(range(-2, 3))*3, 3))))
# Skip combinations of +/-2 and 0, except for all 0
valuelist = [v for v in valuelist if any([w % 2 for w in v]) or not any(v)]
for func, patternlist in patternlists:
for pattern in patternlist:
original = func(*pattern[0].args)
simplified = pattern[1]
for values in valuelist:
sublist = dict(zip(symb, values))
originalvalue = original.xreplace(sublist)
simplifiedvalue = simplified.xreplace(sublist)
assert originalvalue == simplifiedvalue, "Original: {}\nand"\
" simplified: {}\ndo not evaluate to the same value for"\
"{}".format(pattern[0], simplified, sublist)
|
4af8c620a65ac18ca8b736dbc29c29702edf57e9d944d943c1186ad402f6010d | from sympy.core.function import expand_mul
from sympy.core.numbers import I, Rational
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.complexes import Abs
from sympy.simplify.simplify import simplify
from sympy.matrices.matrices import NonSquareMatrixError
from sympy.matrices import Matrix, zeros, eye, SparseMatrix
from sympy.abc import x, y, z
from sympy.testing.pytest import raises, slow
from sympy.testing.matrices import allclose
def test_LUdecomp():
testmat = Matrix([[0, 2, 5, 3],
[3, 3, 7, 4],
[8, 4, 0, 2],
[-2, 6, 3, 4]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
testmat = Matrix([[6, -2, 7, 4],
[0, 3, 6, 7],
[1, -2, 7, 4],
[-9, 2, 6, 3]])
L, U, p = testmat.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
# non-square
testmat = Matrix([[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
[10, 11, 12]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4, 3)
# square and singular
testmat = Matrix([[1, 2, 3],
[2, 4, 6],
[4, 5, 6]])
L, U, p = testmat.LUdecomposition(rankcheck=False)
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - testmat == zeros(3)
M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
L, U, p = M.LUdecomposition()
assert L.is_lower
assert U.is_upper
assert (L*U).permute_rows(p, 'backward') - M == zeros(3)
mL = Matrix((
(1, 0, 0),
(2, 3, 0),
))
assert mL.is_lower is True
assert mL.is_upper is False
mU = Matrix((
(1, 2, 3),
(0, 4, 5),
))
assert mU.is_lower is False
assert mU.is_upper is True
# test FF LUdecomp
M = Matrix([[1, 3, 3],
[3, 2, 6],
[3, 2, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[1, 2, 3, 4],
[3, -1, 2, 3],
[3, 1, 3, -2],
[6, -1, 0, 2]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
M = Matrix([[0, 0, 1],
[2, 3, 0],
[3, 1, 4]])
P, L, Dee, U = M.LUdecompositionFF()
assert P*M == L*Dee.inv()*U
# issue 15794
M = Matrix(
[[1, 2, 3],
[4, 5, 6],
[7, 8, 9]]
)
raises(ValueError, lambda : M.LUdecomposition_Simple(rankcheck=True))
def test_singular_value_decompositionD():
A = Matrix([[1, 2], [2, 1]])
U, S, V = A.singular_value_decomposition()
assert U * S * V.T == A
assert U.T * U == eye(U.cols)
assert V.T * V == eye(V.cols)
B = Matrix([[1, 2]])
U, S, V = B.singular_value_decomposition()
assert U * S * V.T == B
assert U.T * U == eye(U.cols)
assert V.T * V == eye(V.cols)
C = Matrix([
[1, 0, 0, 0, 2],
[0, 0, 3, 0, 0],
[0, 0, 0, 0, 0],
[0, 2, 0, 0, 0],
])
U, S, V = C.singular_value_decomposition()
assert U * S * V.T == C
assert U.T * U == eye(U.cols)
assert V.T * V == eye(V.cols)
D = Matrix([[Rational(1, 3), sqrt(2)], [0, Rational(1, 4)]])
U, S, V = D.singular_value_decomposition()
assert simplify(U.T * U) == eye(U.cols)
assert simplify(V.T * V) == eye(V.cols)
assert simplify(U * S * V.T) == D
def test_QR():
A = Matrix([[1, 2], [2, 3]])
Q, S = A.QRdecomposition()
R = Rational
assert Q == Matrix([
[ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)],
[2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
assert S == Matrix([[5**R(1, 2), 8*5**R(-1, 2)], [0, (R(1)/5)**R(1, 2)]])
assert Q*S == A
assert Q.T * Q == eye(2)
A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[12, 0, -51], [6, 0, 167], [-4, 0, 24]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
x = Symbol('x')
A = Matrix([x])
Q, R = A.QRdecomposition()
assert Q == Matrix([x / Abs(x)])
assert R == Matrix([Abs(x)])
A = Matrix([[x, 0], [0, x]])
Q, R = A.QRdecomposition()
assert Q == x / Abs(x) * Matrix([[1, 0], [0, 1]])
assert R == Abs(x) * Matrix([[1, 0], [0, 1]])
def test_QR_non_square():
# Narrow (cols < rows) matrices
A = Matrix([[9, 0, 26], [12, 0, -7], [0, 4, 4], [0, -3, -3]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(2, 1, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Wide (cols > rows) matrices
A = Matrix([[1, 2, 3], [4, 5, 6]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 2, 3, 4], [1, 4, 9, 16], [1, 8, 27, 64]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix(1, 2, [1, 2])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_trivial():
# Rank deficient matrices
A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Zero rank matrices
A = Matrix([[0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]])
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
# Rank deficient matrices with zero norm from beginning columns
A = Matrix([[0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0], [2, 4, 6, 8]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 2, 3]]).T
Q, R = A.QRdecomposition()
assert Q.T * Q == eye(Q.cols)
assert R.is_upper
assert A == Q*R
def test_QR_float():
A = Matrix([[1, 1], [1, 1.01]])
Q, R = A.QRdecomposition()
assert allclose(Q * R, A)
assert allclose(Q * Q.T, Matrix.eye(2))
assert allclose(Q.T * Q, Matrix.eye(2))
A = Matrix([[1, 1], [1, 1.001]])
Q, R = A.QRdecomposition()
assert allclose(Q * R, A)
assert allclose(Q * Q.T, Matrix.eye(2))
assert allclose(Q.T * Q, Matrix.eye(2))
def test_LUdecomposition_Simple_iszerofunc():
# Test if callable passed to matrices.LUdecomposition_Simple() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
lu, p = Matrix([[1, 0], [0, 1]]).LUdecomposition_Simple(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_LUdecomposition_iszerofunc():
# Test if callable passed to matrices.LUdecomposition() as iszerofunc keyword argument is used inside
# matrices.LUdecomposition_Simple()
magic_string = "I got passed in!"
def goofyiszero(value):
raise ValueError(magic_string)
try:
l, u, p = Matrix([[1, 0], [0, 1]]).LUdecomposition(iszerofunc=goofyiszero)
except ValueError as err:
assert magic_string == err.args[0]
return
assert False
def test_LDLdecomposition():
raises(NonSquareMatrixError, lambda: Matrix((1, 2)).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).LDLdecomposition())
raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
A = Matrix(((1, 5), (5, 1)))
L, D = A.LDLdecomposition(hermitian=False)
assert L * D * L.T == A
A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert L * D * L.T == A
assert L.is_lower
assert L == Matrix([[1, 0, 0], [ Rational(3, 5), 1, 0], [Rational(-1, 5), Rational(1, 3), 1]])
assert D.is_diagonal()
assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
L, D = A.LDLdecomposition()
assert expand_mul(L * D * L.H) == A
assert L.expand() == Matrix([[1, 0, 0], [I/2, 1, 0], [S.Half - I/2, 0, 1]])
assert D.expand() == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
raises(NonSquareMatrixError, lambda: SparseMatrix((1, 2)).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((1, 2), (3, 4))).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((5 + I, 0), (0, 1))).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((1, 5), (5, 1))).LDLdecomposition())
raises(ValueError, lambda: SparseMatrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
A = SparseMatrix(((1, 5), (5, 1)))
L, D = A.LDLdecomposition(hermitian=False)
assert L * D * L.T == A
A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
L, D = A.LDLdecomposition()
assert L * D * L.T == A
assert L.is_lower
assert L == Matrix([[1, 0, 0], [ Rational(3, 5), 1, 0], [Rational(-1, 5), Rational(1, 3), 1]])
assert D.is_diagonal()
assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
A = SparseMatrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
L, D = A.LDLdecomposition()
assert expand_mul(L * D * L.H) == A
assert L == Matrix(((1, 0, 0), (I/2, 1, 0), (S.Half - I/2, 0, 1)))
assert D == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
def test_pinv_succeeds_with_rank_decomposition_method():
# Test rank decomposition method of pseudoinverse succeeding
As = [Matrix([
[61, 89, 55, 20, 71, 0],
[62, 96, 85, 85, 16, 0],
[69, 56, 17, 4, 54, 0],
[10, 54, 91, 41, 71, 0],
[ 7, 30, 10, 48, 90, 0],
[0,0,0,0,0,0]])]
for A in As:
A_pinv = A.pinv(method="RD")
AAp = A * A_pinv
ApA = A_pinv * A
assert simplify(AAp * A) == A
assert simplify(ApA * A_pinv) == A_pinv
assert AAp.H == AAp
assert ApA.H == ApA
def test_rank_decomposition():
a = Matrix(0, 0, [])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(1, 1, [5])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
a = Matrix([
[0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
c, f = a.rank_decomposition()
assert f.is_echelon
assert c.cols == f.rows == a.rank()
assert c * f == a
@slow
def test_upper_hessenberg_decomposition():
A = Matrix([
[1, 0, sqrt(3)],
[sqrt(2), Rational(1, 2), 2],
[1, Rational(1, 4), 3],
])
H, P = A.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert (simplify(P * H * P.H)) == A
B = Matrix([
[1, 2, 10],
[8, 2, 5],
[3, 12, 34],
])
H, P = B.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == B
C = Matrix([
[1, sqrt(2), 2, 3],
[0, 5, 3, 4],
[1, 1, 4, sqrt(5)],
[0, 2, 2, 3]
])
H, P = C.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == C
D = Matrix([
[1, 2, 3],
[-3, 5, 6],
[4, -8, 9],
])
H, P = D.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == D
E = Matrix([
[1, 0, 0, 0],
[0, 1, 0, 0],
[1, 1, 0, 1],
[1, 1, 1, 0]
])
H, P = E.upper_hessenberg_decomposition()
assert simplify(P * P.H) == eye(P.cols)
assert simplify(P.H * P) == eye(P.cols)
assert H.is_upper_hessenberg
assert simplify(P * H * P.H) == E
|
28a2245189caa1501e2226f5c04700c54d4b5ba40490a277717ca03e63bd3a3c | from sympy.core.add import Add
from sympy.core.expr import unchanged
from sympy.core.mul import Mul
from sympy.core.symbol import symbols
from sympy.core.relational import Eq
from sympy.concrete.summations import Sum
from sympy.functions.elementary.complexes import im, re
from sympy.functions.elementary.piecewise import Piecewise
from sympy.matrices.common import NonSquareMatrixError, ShapeError
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.special import (
ZeroMatrix, GenericZeroMatrix, Identity, GenericIdentity, OneMatrix)
from sympy.matrices.expressions.matmul import MatMul
from sympy.testing.pytest import raises
def test_zero_matrix_creation():
assert unchanged(ZeroMatrix, 2, 2)
assert unchanged(ZeroMatrix, 0, 0)
raises(ValueError, lambda: ZeroMatrix(-1, 2))
raises(ValueError, lambda: ZeroMatrix(2.0, 2))
raises(ValueError, lambda: ZeroMatrix(2j, 2))
raises(ValueError, lambda: ZeroMatrix(2, -1))
raises(ValueError, lambda: ZeroMatrix(2, 2.0))
raises(ValueError, lambda: ZeroMatrix(2, 2j))
n = symbols('n')
assert unchanged(ZeroMatrix, n, n)
n = symbols('n', integer=False)
raises(ValueError, lambda: ZeroMatrix(n, n))
n = symbols('n', negative=True)
raises(ValueError, lambda: ZeroMatrix(n, n))
def test_generic_zero_matrix():
z = GenericZeroMatrix()
n = symbols('n', integer=True)
A = MatrixSymbol("A", n, n)
assert z == z
assert z != A
assert A != z
assert z.is_ZeroMatrix
raises(TypeError, lambda: z.shape)
raises(TypeError, lambda: z.rows)
raises(TypeError, lambda: z.cols)
assert MatAdd() == z
assert MatAdd(z, A) == MatAdd(A)
# Make sure it is hashable
hash(z)
def test_identity_matrix_creation():
assert Identity(2)
assert Identity(0)
raises(ValueError, lambda: Identity(-1))
raises(ValueError, lambda: Identity(2.0))
raises(ValueError, lambda: Identity(2j))
n = symbols('n')
assert Identity(n)
n = symbols('n', integer=False)
raises(ValueError, lambda: Identity(n))
n = symbols('n', negative=True)
raises(ValueError, lambda: Identity(n))
def test_generic_identity():
I = GenericIdentity()
n = symbols('n', integer=True)
A = MatrixSymbol("A", n, n)
assert I == I
assert I != A
assert A != I
assert I.is_Identity
assert I**-1 == I
raises(TypeError, lambda: I.shape)
raises(TypeError, lambda: I.rows)
raises(TypeError, lambda: I.cols)
assert MatMul() == I
assert MatMul(I, A) == MatMul(A)
# Make sure it is hashable
hash(I)
def test_one_matrix_creation():
assert OneMatrix(2, 2)
assert OneMatrix(0, 0)
assert Eq(OneMatrix(1, 1), Identity(1))
raises(ValueError, lambda: OneMatrix(-1, 2))
raises(ValueError, lambda: OneMatrix(2.0, 2))
raises(ValueError, lambda: OneMatrix(2j, 2))
raises(ValueError, lambda: OneMatrix(2, -1))
raises(ValueError, lambda: OneMatrix(2, 2.0))
raises(ValueError, lambda: OneMatrix(2, 2j))
n = symbols('n')
assert OneMatrix(n, n)
n = symbols('n', integer=False)
raises(ValueError, lambda: OneMatrix(n, n))
n = symbols('n', negative=True)
raises(ValueError, lambda: OneMatrix(n, n))
def test_ZeroMatrix():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
Z = ZeroMatrix(n, m)
assert A + Z == A
assert A*Z.T == ZeroMatrix(n, n)
assert Z*A.T == ZeroMatrix(n, n)
assert A - A == ZeroMatrix(*A.shape)
assert Z
assert Z.transpose() == ZeroMatrix(m, n)
assert Z.conjugate() == Z
assert Z.adjoint() == ZeroMatrix(m, n)
assert re(Z) == Z
assert im(Z) == Z
assert ZeroMatrix(n, n)**0 == Identity(n)
with raises(NonSquareMatrixError):
Z**0
with raises(NonSquareMatrixError):
Z**1
with raises(NonSquareMatrixError):
Z**2
assert ZeroMatrix(3, 3).as_explicit() == ImmutableDenseMatrix.zeros(3, 3)
def test_ZeroMatrix_doit():
n = symbols('n', integer=True)
Znn = ZeroMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Znn.rows, Add)
assert Znn.doit() == ZeroMatrix(2*n, n)
assert isinstance(Znn.doit().rows, Mul)
def test_OneMatrix():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
a = MatrixSymbol('a', n, 1)
U = OneMatrix(n, m)
assert U.shape == (n, m)
assert isinstance(A + U, Add)
assert U.transpose() == OneMatrix(m, n)
assert U.conjugate() == U
assert U.adjoint() == OneMatrix(m, n)
assert re(U) == U
assert im(U) == ZeroMatrix(n, m)
assert OneMatrix(n, n) ** 0 == Identity(n)
with raises(NonSquareMatrixError):
U ** 0
with raises(NonSquareMatrixError):
U ** 1
with raises(NonSquareMatrixError):
U ** 2
with raises(ShapeError):
a + U
U = OneMatrix(n, n)
assert U[1, 2] == 1
U = OneMatrix(2, 3)
assert U.as_explicit() == ImmutableDenseMatrix.ones(2, 3)
def test_OneMatrix_doit():
n = symbols('n', integer=True)
Unn = OneMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Unn.rows, Add)
assert Unn.doit() == OneMatrix(2 * n, n)
assert isinstance(Unn.doit().rows, Mul)
def test_OneMatrix_mul():
n, m, k = symbols('n m k', integer=True)
w = MatrixSymbol('w', n, 1)
assert OneMatrix(n, m) * OneMatrix(m, k) == OneMatrix(n, k) * m
assert w * OneMatrix(1, 1) == w
assert OneMatrix(1, 1) * w.T == w.T
def test_Identity():
n, m = symbols('n m', integer=True)
A = MatrixSymbol('A', n, m)
i, j = symbols('i j')
In = Identity(n)
Im = Identity(m)
assert A*Im == A
assert In*A == A
assert In.transpose() == In
assert In.inverse() == In
assert In.conjugate() == In
assert In.adjoint() == In
assert re(In) == In
assert im(In) == ZeroMatrix(n, n)
assert In[i, j] != 0
assert Sum(In[i, j], (i, 0, n-1), (j, 0, n-1)).subs(n,3).doit() == 3
assert Sum(Sum(In[i, j], (i, 0, n-1)), (j, 0, n-1)).subs(n,3).doit() == 3
# If range exceeds the limit `(0, n-1)`, do not remove `Piecewise`:
expr = Sum(In[i, j], (i, 0, n-1))
assert expr.doit() == 1
expr = Sum(In[i, j], (i, 0, n-2))
assert expr.doit().dummy_eq(
Piecewise(
(1, (j >= 0) & (j <= n-2)),
(0, True)
)
)
expr = Sum(In[i, j], (i, 1, n-1))
assert expr.doit().dummy_eq(
Piecewise(
(1, (j >= 1) & (j <= n-1)),
(0, True)
)
)
assert Identity(3).as_explicit() == ImmutableDenseMatrix.eye(3)
def test_Identity_doit():
n = symbols('n', integer=True)
Inn = Identity(Add(n, n, evaluate=False))
assert isinstance(Inn.rows, Add)
assert Inn.doit() == Identity(2*n)
assert isinstance(Inn.doit().rows, Mul)
|
a31f1fd269712b8bf46b6d2985a1be13898e61ce6bb9af39fcf0f981f251a064 | from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.special import (Identity, OneMatrix, ZeroMatrix)
from sympy.core import symbols
from sympy.testing.pytest import raises, warns_deprecated_sympy
from sympy.matrices import ShapeError, MatrixSymbol
from sympy.matrices.expressions import (HadamardProduct, hadamard_product, HadamardPower, hadamard_power)
n, m, k = symbols('n,m,k')
Z = MatrixSymbol('Z', n, n)
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, k)
def test_HadamardProduct():
assert HadamardProduct(A, B, A).shape == A.shape
raises(ShapeError, lambda: HadamardProduct(A, B.T))
raises(TypeError, lambda: HadamardProduct(A, n))
raises(TypeError, lambda: HadamardProduct(A, 1))
assert HadamardProduct(A, 2*B, -A)[1, 1] == \
-2 * A[1, 1] * B[1, 1] * A[1, 1]
mix = HadamardProduct(Z*A, B)*C
assert mix.shape == (n, k)
assert set(HadamardProduct(A, B, A).T.args) == {A.T, A.T, B.T}
def test_HadamardProduct_isnt_commutative():
assert HadamardProduct(A, B) != HadamardProduct(B, A)
def test_mixed_indexing():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
Z = MatrixSymbol('Z', 2, 2)
assert (X*HadamardProduct(Y, Z))[0, 0] == \
X[0, 0]*Y[0, 0]*Z[0, 0] + X[0, 1]*Y[1, 0]*Z[1, 0]
def test_canonicalize():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
with warns_deprecated_sympy():
expr = HadamardProduct(X, check=False)
assert isinstance(expr, HadamardProduct)
expr2 = expr.doit() # unpack is called
assert isinstance(expr2, MatrixSymbol)
Z = ZeroMatrix(2, 2)
U = OneMatrix(2, 2)
assert HadamardProduct(Z, X).doit() == Z
assert HadamardProduct(U, X, X, U).doit() == HadamardPower(X, 2)
assert HadamardProduct(X, U, Y).doit() == HadamardProduct(X, Y)
assert HadamardProduct(X, Z, U, Y).doit() == Z
def test_hadamard():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
B = MatrixSymbol('B', m, n)
C = MatrixSymbol('C', m, p)
X = MatrixSymbol('X', m, m)
I = Identity(m)
with raises(TypeError):
hadamard_product()
assert hadamard_product(A) == A
assert isinstance(hadamard_product(A, B), HadamardProduct)
assert hadamard_product(A, B).doit() == hadamard_product(A, B)
with raises(ShapeError):
hadamard_product(A, C)
hadamard_product(A, I)
assert hadamard_product(X, I) == X
assert isinstance(hadamard_product(X, I), MatrixSymbol)
a = MatrixSymbol("a", k, 1)
expr = MatAdd(ZeroMatrix(k, 1), OneMatrix(k, 1))
expr = HadamardProduct(expr, a)
assert expr.doit() == a
def test_hadamard_product_with_explicit_mat():
A = MatrixSymbol("A", 3, 3).as_explicit()
B = MatrixSymbol("B", 3, 3).as_explicit()
X = MatrixSymbol("X", 3, 3)
expr = hadamard_product(A, B)
ret = Matrix([i*j for i, j in zip(A, B)]).reshape(3, 3)
assert expr == ret
expr = hadamard_product(A, X, B)
assert expr == HadamardProduct(ret, X)
def test_hadamard_power():
m, n, p = symbols('m, n, p', integer=True)
A = MatrixSymbol('A', m, n)
assert hadamard_power(A, 1) == A
assert isinstance(hadamard_power(A, 2), HadamardPower)
assert hadamard_power(A, n).T == hadamard_power(A.T, n)
assert hadamard_power(A, n)[0, 0] == A[0, 0]**n
assert hadamard_power(m, n) == m**n
raises(ValueError, lambda: hadamard_power(A, A))
def test_hadamard_power_explicit():
A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
a, b = symbols('a b')
assert HadamardPower(a, b) == a**b
assert HadamardPower(a, B).as_explicit() == \
Matrix([
[a**B[0, 0], a**B[0, 1]],
[a**B[1, 0], a**B[1, 1]]])
assert HadamardPower(A, b).as_explicit() == \
Matrix([
[A[0, 0]**b, A[0, 1]**b],
[A[1, 0]**b, A[1, 1]**b]])
assert HadamardPower(A, B).as_explicit() == \
Matrix([
[A[0, 0]**B[0, 0], A[0, 1]**B[0, 1]],
[A[1, 0]**B[1, 0], A[1, 1]**B[1, 1]]])
|
ac87514ea150cdbb1ad0d19ea735a67e82a101df3b684ab138d74e82e8d1ff40 | from sympy.core.function import Lambda, expand_complex
from sympy.core.mul import Mul
from sympy.core.numbers import ilcm
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, symbols)
from sympy.core.sorting import ordered
from sympy.functions.elementary.complexes import sign
from sympy.functions.elementary.integers import floor, ceiling
from sympy.sets.fancysets import ComplexRegion
from sympy.sets.sets import (FiniteSet, Intersection, Interval, Set, Union)
from sympy.multipledispatch import Dispatcher
from sympy.sets.conditionset import ConditionSet
from sympy.sets.fancysets import (Integers, Naturals, Reals, Range,
ImageSet, Rationals)
from sympy.sets.sets import EmptySet, UniversalSet, imageset, ProductSet
from sympy.simplify.radsimp import numer
intersection_sets = Dispatcher('intersection_sets')
@intersection_sets.register(ConditionSet, ConditionSet)
def _(a, b):
return None
@intersection_sets.register(ConditionSet, Set)
def _(a, b):
return ConditionSet(a.sym, a.condition, Intersection(a.base_set, b))
@intersection_sets.register(Naturals, Integers)
def _(a, b):
return a
@intersection_sets.register(Naturals, Naturals)
def _(a, b):
return a if a is S.Naturals else b
@intersection_sets.register(Interval, Naturals)
def _(a, b):
return intersection_sets(b, a)
@intersection_sets.register(ComplexRegion, Set)
def _(self, other):
if other.is_ComplexRegion:
# self in rectangular form
if (not self.polar) and (not other.polar):
return ComplexRegion(Intersection(self.sets, other.sets))
# self in polar form
elif self.polar and other.polar:
r1, theta1 = self.a_interval, self.b_interval
r2, theta2 = other.a_interval, other.b_interval
new_r_interval = Intersection(r1, r2)
new_theta_interval = Intersection(theta1, theta2)
# 0 and 2*Pi means the same
if ((2*S.Pi in theta1 and S.Zero in theta2) or
(2*S.Pi in theta2 and S.Zero in theta1)):
new_theta_interval = Union(new_theta_interval,
FiniteSet(0))
return ComplexRegion(new_r_interval*new_theta_interval,
polar=True)
if other.is_subset(S.Reals):
new_interval = []
x = symbols("x", cls=Dummy, real=True)
# self in rectangular form
if not self.polar:
for element in self.psets:
if S.Zero in element.args[1]:
new_interval.append(element.args[0])
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
# self in polar form
elif self.polar:
for element in self.psets:
if S.Zero in element.args[1]:
new_interval.append(element.args[0])
if S.Pi in element.args[1]:
new_interval.append(ImageSet(Lambda(x, -x), element.args[0]))
if S.Zero in element.args[0]:
new_interval.append(FiniteSet(0))
new_interval = Union(*new_interval)
return Intersection(new_interval, other)
@intersection_sets.register(Integers, Reals)
def _(a, b):
return a
@intersection_sets.register(Range, Interval)
def _(a, b):
# Check that there are no symbolic arguments
if not all(i.is_number for i in a.args + b.args[:2]):
return
# In case of null Range, return an EmptySet.
if a.size == 0:
return S.EmptySet
# trim down to self's size, and represent
# as a Range with step 1.
start = ceiling(max(b.inf, a.inf))
if start not in b:
start += 1
end = floor(min(b.sup, a.sup))
if end not in b:
end -= 1
return intersection_sets(a, Range(start, end + 1))
@intersection_sets.register(Range, Naturals)
def _(a, b):
return intersection_sets(a, Interval(b.inf, S.Infinity))
@intersection_sets.register(Range, Range)
def _(a, b):
# Check that there are no symbolic range arguments
if not all(all(v.is_number for v in r.args) for r in [a, b]):
return None
# non-overlap quick exits
if not b:
return S.EmptySet
if not a:
return S.EmptySet
if b.sup < a.inf:
return S.EmptySet
if b.inf > a.sup:
return S.EmptySet
# work with finite end at the start
r1 = a
if r1.start.is_infinite:
r1 = r1.reversed
r2 = b
if r2.start.is_infinite:
r2 = r2.reversed
# If both ends are infinite then it means that one Range is just the set
# of all integers (the step must be 1).
if r1.start.is_infinite:
return b
if r2.start.is_infinite:
return a
from sympy.solvers.diophantine.diophantine import diop_linear
# this equation represents the values of the Range;
# it's a linear equation
eq = lambda r, i: r.start + i*r.step
# we want to know when the two equations might
# have integer solutions so we use the diophantine
# solver
va, vb = diop_linear(eq(r1, Dummy('a')) - eq(r2, Dummy('b')))
# check for no solution
no_solution = va is None and vb is None
if no_solution:
return S.EmptySet
# there is a solution
# -------------------
# find the coincident point, c
a0 = va.as_coeff_Add()[0]
c = eq(r1, a0)
# find the first point, if possible, in each range
# since c may not be that point
def _first_finite_point(r1, c):
if c == r1.start:
return c
# st is the signed step we need to take to
# get from c to r1.start
st = sign(r1.start - c)*step
# use Range to calculate the first point:
# we want to get as close as possible to
# r1.start; the Range will not be null since
# it will at least contain c
s1 = Range(c, r1.start + st, st)[-1]
if s1 == r1.start:
pass
else:
# if we didn't hit r1.start then, if the
# sign of st didn't match the sign of r1.step
# we are off by one and s1 is not in r1
if sign(r1.step) != sign(st):
s1 -= st
if s1 not in r1:
return
return s1
# calculate the step size of the new Range
step = abs(ilcm(r1.step, r2.step))
s1 = _first_finite_point(r1, c)
if s1 is None:
return S.EmptySet
s2 = _first_finite_point(r2, c)
if s2 is None:
return S.EmptySet
# replace the corresponding start or stop in
# the original Ranges with these points; the
# result must have at least one point since
# we know that s1 and s2 are in the Ranges
def _updated_range(r, first):
st = sign(r.step)*step
if r.start.is_finite:
rv = Range(first, r.stop, st)
else:
rv = Range(r.start, first + st, st)
return rv
r1 = _updated_range(a, s1)
r2 = _updated_range(b, s2)
# work with them both in the increasing direction
if sign(r1.step) < 0:
r1 = r1.reversed
if sign(r2.step) < 0:
r2 = r2.reversed
# return clipped Range with positive step; it
# can't be empty at this point
start = max(r1.start, r2.start)
stop = min(r1.stop, r2.stop)
return Range(start, stop, step)
@intersection_sets.register(Range, Integers)
def _(a, b):
return a
@intersection_sets.register(Range, Rationals)
def _(a, b):
return a
@intersection_sets.register(ImageSet, Set)
def _(self, other):
from sympy.solvers.diophantine import diophantine
# Only handle the straight-forward univariate case
if (len(self.lamda.variables) > 1
or self.lamda.signature != self.lamda.variables):
return None
base_set = self.base_sets[0]
# Intersection between ImageSets with Integers as base set
# For {f(n) : n in Integers} & {g(m) : m in Integers} we solve the
# diophantine equations f(n)=g(m).
# If the solutions for n are {h(t) : t in Integers} then we return
# {f(h(t)) : t in integers}.
# If the solutions for n are {n_1, n_2, ..., n_k} then we return
# {f(n_i) : 1 <= i <= k}.
if base_set is S.Integers:
gm = None
if isinstance(other, ImageSet) and other.base_sets == (S.Integers,):
gm = other.lamda.expr
var = other.lamda.variables[0]
# Symbol of second ImageSet lambda must be distinct from first
m = Dummy('m')
gm = gm.subs(var, m)
elif other is S.Integers:
m = gm = Dummy('m')
if gm is not None:
fn = self.lamda.expr
n = self.lamda.variables[0]
try:
solns = list(diophantine(fn - gm, syms=(n, m), permute=True))
except (TypeError, NotImplementedError):
# TypeError if equation not polynomial with rational coeff.
# NotImplementedError if correct format but no solver.
return
# 3 cases are possible for solns:
# - empty set,
# - one or more parametric (infinite) solutions,
# - a finite number of (non-parametric) solution couples.
# Among those, there is one type of solution set that is
# not helpful here: multiple parametric solutions.
if len(solns) == 0:
return S.EmptySet
elif any(s.free_symbols for tupl in solns for s in tupl):
if len(solns) == 1:
soln, solm = solns[0]
(t,) = soln.free_symbols
expr = fn.subs(n, soln.subs(t, n)).expand()
return imageset(Lambda(n, expr), S.Integers)
else:
return
else:
return FiniteSet(*(fn.subs(n, s[0]) for s in solns))
if other == S.Reals:
from sympy.solvers.solvers import denoms, solve_linear
def _solution_union(exprs, sym):
# return a union of linear solutions to i in expr;
# if i cannot be solved, use a ConditionSet for solution
sols = []
for i in exprs:
x, xis = solve_linear(i, 0, [sym])
if x == sym:
sols.append(FiniteSet(xis))
else:
sols.append(ConditionSet(sym, Eq(i, 0)))
return Union(*sols)
f = self.lamda.expr
n = self.lamda.variables[0]
n_ = Dummy(n.name, real=True)
f_ = f.subs(n, n_)
re, im = f_.as_real_imag()
im = expand_complex(im)
re = re.subs(n_, n)
im = im.subs(n_, n)
ifree = im.free_symbols
lam = Lambda(n, re)
if im.is_zero:
# allow re-evaluation
# of self in this case to make
# the result canonical
pass
elif im.is_zero is False:
return S.EmptySet
elif ifree != {n}:
return None
else:
# univarite imaginary part in same variable;
# use numer instead of as_numer_denom to keep
# this as fast as possible while still handling
# simple cases
base_set &= _solution_union(
Mul.make_args(numer(im)), n)
# exclude values that make denominators 0
base_set -= _solution_union(denoms(f), n)
return imageset(lam, base_set)
elif isinstance(other, Interval):
from sympy.solvers.solveset import (invert_real, invert_complex,
solveset)
f = self.lamda.expr
n = self.lamda.variables[0]
new_inf, new_sup = None, None
new_lopen, new_ropen = other.left_open, other.right_open
if f.is_real:
inverter = invert_real
else:
inverter = invert_complex
g1, h1 = inverter(f, other.inf, n)
g2, h2 = inverter(f, other.sup, n)
if all(isinstance(i, FiniteSet) for i in (h1, h2)):
if g1 == n:
if len(h1) == 1:
new_inf = h1.args[0]
if g2 == n:
if len(h2) == 1:
new_sup = h2.args[0]
# TODO: Design a technique to handle multiple-inverse
# functions
# Any of the new boundary values cannot be determined
if any(i is None for i in (new_sup, new_inf)):
return
range_set = S.EmptySet
if all(i.is_real for i in (new_sup, new_inf)):
# this assumes continuity of underlying function
# however fixes the case when it is decreasing
if new_inf > new_sup:
new_inf, new_sup = new_sup, new_inf
new_interval = Interval(new_inf, new_sup, new_lopen, new_ropen)
range_set = base_set.intersect(new_interval)
else:
if other.is_subset(S.Reals):
solutions = solveset(f, n, S.Reals)
if not isinstance(range_set, (ImageSet, ConditionSet)):
range_set = solutions.intersect(other)
else:
return
if range_set is S.EmptySet:
return S.EmptySet
elif isinstance(range_set, Range) and range_set.size is not S.Infinity:
range_set = FiniteSet(*list(range_set))
if range_set is not None:
return imageset(Lambda(n, f), range_set)
return
else:
return
@intersection_sets.register(ProductSet, ProductSet)
def _(a, b):
if len(b.args) != len(a.args):
return S.EmptySet
return ProductSet(*(i.intersect(j) for i, j in zip(a.sets, b.sets)))
@intersection_sets.register(Interval, Interval)
def _(a, b):
# handle (-oo, oo)
infty = S.NegativeInfinity, S.Infinity
if a == Interval(*infty):
l, r = a.left, a.right
if l.is_real or l in infty or r.is_real or r in infty:
return b
# We can't intersect [0,3] with [x,6] -- we don't know if x>0 or x<0
if not a._is_comparable(b):
return None
empty = False
if a.start <= b.end and b.start <= a.end:
# Get topology right.
if a.start < b.start:
start = b.start
left_open = b.left_open
elif a.start > b.start:
start = a.start
left_open = a.left_open
else:
#this is to ensure that if Eq(a.start,b.start) but
#type(a.start) != type(b.start) the order of a and b
#does not matter for the result
start = list(ordered([a,b]))[0].start
left_open = a.left_open or b.left_open
if a.end < b.end:
end = a.end
right_open = a.right_open
elif a.end > b.end:
end = b.end
right_open = b.right_open
else:
end = list(ordered([a,b]))[0].end
right_open = a.right_open or b.right_open
if end - start == 0 and (left_open or right_open):
empty = True
else:
empty = True
if empty:
return S.EmptySet
return Interval(start, end, left_open, right_open)
@intersection_sets.register(EmptySet, Set)
def _(a, b):
return S.EmptySet
@intersection_sets.register(UniversalSet, Set)
def _(a, b):
return b
@intersection_sets.register(FiniteSet, FiniteSet)
def _(a, b):
return FiniteSet(*(a._elements & b._elements))
@intersection_sets.register(FiniteSet, Set)
def _(a, b):
try:
return FiniteSet(*[el for el in a if el in b])
except TypeError:
return None # could not evaluate `el in b` due to symbolic ranges.
@intersection_sets.register(Set, Set)
def _(a, b):
return None
@intersection_sets.register(Integers, Rationals)
def _(a, b):
return a
@intersection_sets.register(Naturals, Rationals)
def _(a, b):
return a
@intersection_sets.register(Rationals, Reals)
def _(a, b):
return a
def _intlike_interval(a, b):
try:
if b._inf is S.NegativeInfinity and b._sup is S.Infinity:
return a
s = Range(max(a.inf, ceiling(b.left)), floor(b.right) + 1)
return intersection_sets(s, b) # take out endpoints if open interval
except ValueError:
return None
@intersection_sets.register(Integers, Interval)
def _(a, b):
return _intlike_interval(a, b)
@intersection_sets.register(Naturals, Interval)
def _(a, b):
return _intlike_interval(a, b)
|
13b0aa2c8b2f6954540eabbb6ae64e59f475d0cfdb021cb90daeae77c4800995 | from sympy.sets.setexpr import SetExpr
from sympy.sets import Interval, FiniteSet, Intersection, ImageSet, Union
from sympy.core.expr import Expr
from sympy.core.function import Lambda
from sympy.core.numbers import (I, Rational, oo)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.trigonometric import cos
from sympy.sets.sets import Set
a, x = symbols("a, x")
_d = Dummy("d")
def test_setexpr():
se = SetExpr(Interval(0, 1))
assert isinstance(se.set, Set)
assert isinstance(se, Expr)
def test_scalar_funcs():
assert SetExpr(Interval(0, 1)).set == Interval(0, 1)
a, b = Symbol('a', real=True), Symbol('b', real=True)
a, b = 1, 2
# TODO: add support for more functions in the future:
for f in [exp, log]:
input_se = f(SetExpr(Interval(a, b)))
output = input_se.set
expected = Interval(Min(f(a), f(b)), Max(f(a), f(b)))
assert output == expected
def test_Add_Mul():
assert (SetExpr(Interval(0, 1)) + 1).set == Interval(1, 2)
assert (SetExpr(Interval(0, 1))*2).set == Interval(0, 2)
def test_Pow():
assert (SetExpr(Interval(0, 2))**2).set == Interval(0, 4)
def test_compound():
assert (exp(SetExpr(Interval(0, 1))*2 + 1)).set == \
Interval(exp(1), exp(3))
def test_Interval_Interval():
assert (SetExpr(Interval(1, 2)) + SetExpr(Interval(10, 20))).set == \
Interval(11, 22)
assert (SetExpr(Interval(1, 2))*SetExpr(Interval(10, 20))).set == \
Interval(10, 40)
def test_FiniteSet_FiniteSet():
assert (SetExpr(FiniteSet(1, 2, 3)) + SetExpr(FiniteSet(1, 2))).set == \
FiniteSet(2, 3, 4, 5)
assert (SetExpr(FiniteSet(1, 2, 3))*SetExpr(FiniteSet(1, 2))).set == \
FiniteSet(1, 2, 3, 4, 6)
def test_Interval_FiniteSet():
assert (SetExpr(FiniteSet(1, 2)) + SetExpr(Interval(0, 10))).set == \
Interval(1, 12)
def test_Many_Sets():
assert (SetExpr(Interval(0, 1)) +
SetExpr(Interval(2, 3)) +
SetExpr(FiniteSet(10, 11, 12))).set == Interval(12, 16)
def test_same_setexprs_are_not_identical():
a = SetExpr(FiniteSet(0, 1))
b = SetExpr(FiniteSet(0, 1))
assert (a + b).set == FiniteSet(0, 1, 2)
# Cannont detect the set being the same:
# assert (a + a).set == FiniteSet(0, 2)
def test_Interval_arithmetic():
i12cc = SetExpr(Interval(1, 2))
i12lo = SetExpr(Interval.Lopen(1, 2))
i12ro = SetExpr(Interval.Ropen(1, 2))
i12o = SetExpr(Interval.open(1, 2))
n23cc = SetExpr(Interval(-2, 3))
n23lo = SetExpr(Interval.Lopen(-2, 3))
n23ro = SetExpr(Interval.Ropen(-2, 3))
n23o = SetExpr(Interval.open(-2, 3))
n3n2cc = SetExpr(Interval(-3, -2))
assert i12cc + i12cc == SetExpr(Interval(2, 4))
assert i12cc - i12cc == SetExpr(Interval(-1, 1))
assert i12cc*i12cc == SetExpr(Interval(1, 4))
assert i12cc/i12cc == SetExpr(Interval(S.Half, 2))
assert i12cc**2 == SetExpr(Interval(1, 4))
assert i12cc**3 == SetExpr(Interval(1, 8))
assert i12lo + i12ro == SetExpr(Interval.open(2, 4))
assert i12lo - i12ro == SetExpr(Interval.Lopen(-1, 1))
assert i12lo*i12ro == SetExpr(Interval.open(1, 4))
assert i12lo/i12ro == SetExpr(Interval.Lopen(S.Half, 2))
assert i12lo + i12lo == SetExpr(Interval.Lopen(2, 4))
assert i12lo - i12lo == SetExpr(Interval.open(-1, 1))
assert i12lo*i12lo == SetExpr(Interval.Lopen(1, 4))
assert i12lo/i12lo == SetExpr(Interval.open(S.Half, 2))
assert i12lo + i12cc == SetExpr(Interval.Lopen(2, 4))
assert i12lo - i12cc == SetExpr(Interval.Lopen(-1, 1))
assert i12lo*i12cc == SetExpr(Interval.Lopen(1, 4))
assert i12lo/i12cc == SetExpr(Interval.Lopen(S.Half, 2))
assert i12lo + i12o == SetExpr(Interval.open(2, 4))
assert i12lo - i12o == SetExpr(Interval.open(-1, 1))
assert i12lo*i12o == SetExpr(Interval.open(1, 4))
assert i12lo/i12o == SetExpr(Interval.open(S.Half, 2))
assert i12lo**2 == SetExpr(Interval.Lopen(1, 4))
assert i12lo**3 == SetExpr(Interval.Lopen(1, 8))
assert i12ro + i12ro == SetExpr(Interval.Ropen(2, 4))
assert i12ro - i12ro == SetExpr(Interval.open(-1, 1))
assert i12ro*i12ro == SetExpr(Interval.Ropen(1, 4))
assert i12ro/i12ro == SetExpr(Interval.open(S.Half, 2))
assert i12ro + i12cc == SetExpr(Interval.Ropen(2, 4))
assert i12ro - i12cc == SetExpr(Interval.Ropen(-1, 1))
assert i12ro*i12cc == SetExpr(Interval.Ropen(1, 4))
assert i12ro/i12cc == SetExpr(Interval.Ropen(S.Half, 2))
assert i12ro + i12o == SetExpr(Interval.open(2, 4))
assert i12ro - i12o == SetExpr(Interval.open(-1, 1))
assert i12ro*i12o == SetExpr(Interval.open(1, 4))
assert i12ro/i12o == SetExpr(Interval.open(S.Half, 2))
assert i12ro**2 == SetExpr(Interval.Ropen(1, 4))
assert i12ro**3 == SetExpr(Interval.Ropen(1, 8))
assert i12o + i12lo == SetExpr(Interval.open(2, 4))
assert i12o - i12lo == SetExpr(Interval.open(-1, 1))
assert i12o*i12lo == SetExpr(Interval.open(1, 4))
assert i12o/i12lo == SetExpr(Interval.open(S.Half, 2))
assert i12o + i12ro == SetExpr(Interval.open(2, 4))
assert i12o - i12ro == SetExpr(Interval.open(-1, 1))
assert i12o*i12ro == SetExpr(Interval.open(1, 4))
assert i12o/i12ro == SetExpr(Interval.open(S.Half, 2))
assert i12o + i12cc == SetExpr(Interval.open(2, 4))
assert i12o - i12cc == SetExpr(Interval.open(-1, 1))
assert i12o*i12cc == SetExpr(Interval.open(1, 4))
assert i12o/i12cc == SetExpr(Interval.open(S.Half, 2))
assert i12o**2 == SetExpr(Interval.open(1, 4))
assert i12o**3 == SetExpr(Interval.open(1, 8))
assert n23cc + n23cc == SetExpr(Interval(-4, 6))
assert n23cc - n23cc == SetExpr(Interval(-5, 5))
assert n23cc*n23cc == SetExpr(Interval(-6, 9))
assert n23cc/n23cc == SetExpr(Interval.open(-oo, oo))
assert n23cc + n23ro == SetExpr(Interval.Ropen(-4, 6))
assert n23cc - n23ro == SetExpr(Interval.Lopen(-5, 5))
assert n23cc*n23ro == SetExpr(Interval.Ropen(-6, 9))
assert n23cc/n23ro == SetExpr(Interval.Lopen(-oo, oo))
assert n23cc + n23lo == SetExpr(Interval.Lopen(-4, 6))
assert n23cc - n23lo == SetExpr(Interval.Ropen(-5, 5))
assert n23cc*n23lo == SetExpr(Interval(-6, 9))
assert n23cc/n23lo == SetExpr(Interval.open(-oo, oo))
assert n23cc + n23o == SetExpr(Interval.open(-4, 6))
assert n23cc - n23o == SetExpr(Interval.open(-5, 5))
assert n23cc*n23o == SetExpr(Interval.open(-6, 9))
assert n23cc/n23o == SetExpr(Interval.open(-oo, oo))
assert n23cc**2 == SetExpr(Interval(0, 9))
assert n23cc**3 == SetExpr(Interval(-8, 27))
n32cc = SetExpr(Interval(-3, 2))
n32lo = SetExpr(Interval.Lopen(-3, 2))
n32ro = SetExpr(Interval.Ropen(-3, 2))
assert n32cc*n32lo == SetExpr(Interval.Ropen(-6, 9))
assert n32cc*n32cc == SetExpr(Interval(-6, 9))
assert n32lo*n32cc == SetExpr(Interval.Ropen(-6, 9))
assert n32cc*n32ro == SetExpr(Interval(-6, 9))
assert n32lo*n32ro == SetExpr(Interval.Ropen(-6, 9))
assert n32cc/n32lo == SetExpr(Interval.Ropen(-oo, oo))
assert i12cc/n32lo == SetExpr(Interval.Ropen(-oo, oo))
assert n3n2cc**2 == SetExpr(Interval(4, 9))
assert n3n2cc**3 == SetExpr(Interval(-27, -8))
assert n23cc + i12cc == SetExpr(Interval(-1, 5))
assert n23cc - i12cc == SetExpr(Interval(-4, 2))
assert n23cc*i12cc == SetExpr(Interval(-4, 6))
assert n23cc/i12cc == SetExpr(Interval(-2, 3))
def test_SetExpr_Intersection():
x, y, z, w = symbols("x y z w")
set1 = Interval(x, y)
set2 = Interval(w, z)
inter = Intersection(set1, set2)
se = SetExpr(inter)
assert exp(se).set == Intersection(
ImageSet(Lambda(x, exp(x)), set1),
ImageSet(Lambda(x, exp(x)), set2))
assert cos(se).set == ImageSet(Lambda(x, cos(x)), inter)
def test_SetExpr_Interval_div():
# TODO: some expressions cannot be calculated due to bugs (currently
# commented):
assert SetExpr(Interval(-3, -2))/SetExpr(Interval(-2, 1)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(2, 3))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-3, -2))/SetExpr(Interval(0, 4)) == SetExpr(Interval(-oo, Rational(-1, 2)))
assert SetExpr(Interval(2, 4))/SetExpr(Interval(-3, 0)) == SetExpr(Interval(-oo, Rational(-2, 3)))
assert SetExpr(Interval(2, 4))/SetExpr(Interval(0, 3)) == SetExpr(Interval(Rational(2, 3), oo))
# assert SetExpr(Interval(0, 1))/SetExpr(Interval(0, 1)) == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(-1, 0))/SetExpr(Interval(0, 1)) == SetExpr(Interval(-oo, 0))
assert SetExpr(Interval(-1, 2))/SetExpr(Interval(-2, 2)) == SetExpr(Interval(-oo, oo))
assert 1/SetExpr(Interval(-1, 2)) == SetExpr(Union(Interval(-oo, -1), Interval(S.Half, oo)))
assert 1/SetExpr(Interval(0, 2)) == SetExpr(Interval(S.Half, oo))
assert (-1)/SetExpr(Interval(0, 2)) == SetExpr(Interval(-oo, Rational(-1, 2)))
assert 1/SetExpr(Interval(-oo, 0)) == SetExpr(Interval.open(-oo, 0))
assert 1/SetExpr(Interval(-1, 0)) == SetExpr(Interval(-oo, -1))
# assert (-2)/SetExpr(Interval(-oo, 0)) == SetExpr(Interval(0, oo))
# assert 1/SetExpr(Interval(-oo, -1)) == SetExpr(Interval(-1, 0))
# assert SetExpr(Interval(1, 2))/a == Mul(SetExpr(Interval(1, 2)), 1/a, evaluate=False)
# assert SetExpr(Interval(1, 2))/0 == SetExpr(Interval(1, 2))*zoo
# assert SetExpr(Interval(1, oo))/oo == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(1, oo))/(-oo) == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, -1))/oo == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, -1))/(-oo) == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(-oo, oo))/oo == SetExpr(Interval(-oo, oo))
# assert SetExpr(Interval(-oo, oo))/(-oo) == SetExpr(Interval(-oo, oo))
# assert SetExpr(Interval(-1, oo))/oo == SetExpr(Interval(0, oo))
# assert SetExpr(Interval(-1, oo))/(-oo) == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, 1))/oo == SetExpr(Interval(-oo, 0))
# assert SetExpr(Interval(-oo, 1))/(-oo) == SetExpr(Interval(0, oo))
def test_SetExpr_Interval_pow():
assert SetExpr(Interval(0, 2))**2 == SetExpr(Interval(0, 4))
assert SetExpr(Interval(-1, 1))**2 == SetExpr(Interval(0, 1))
assert SetExpr(Interval(1, 2))**2 == SetExpr(Interval(1, 4))
assert SetExpr(Interval(-1, 2))**3 == SetExpr(Interval(-1, 8))
assert SetExpr(Interval(-1, 1))**0 == SetExpr(FiniteSet(1))
assert SetExpr(Interval(1, 2))**Rational(5, 2) == SetExpr(Interval(1, 4*sqrt(2)))
#assert SetExpr(Interval(-1, 2))**Rational(1, 3) == SetExpr(Interval(-1, 2**Rational(1, 3)))
#assert SetExpr(Interval(0, 2))**S.Half == SetExpr(Interval(0, sqrt(2)))
#assert SetExpr(Interval(-4, 2))**Rational(2, 3) == SetExpr(Interval(0, 2*2**Rational(1, 3)))
#assert SetExpr(Interval(-1, 5))**S.Half == SetExpr(Interval(0, sqrt(5)))
#assert SetExpr(Interval(-oo, 2))**S.Half == SetExpr(Interval(0, sqrt(2)))
#assert SetExpr(Interval(-2, 3))**(Rational(-1, 4)) == SetExpr(Interval(0, oo))
assert SetExpr(Interval(1, 5))**(-2) == SetExpr(Interval(Rational(1, 25), 1))
assert SetExpr(Interval(-1, 3))**(-2) == SetExpr(Interval(0, oo))
assert SetExpr(Interval(0, 2))**(-2) == SetExpr(Interval(Rational(1, 4), oo))
assert SetExpr(Interval(-1, 2))**(-3) == SetExpr(Union(Interval(-oo, -1), Interval(Rational(1, 8), oo)))
assert SetExpr(Interval(-3, -2))**(-3) == SetExpr(Interval(Rational(-1, 8), Rational(-1, 27)))
assert SetExpr(Interval(-3, -2))**(-2) == SetExpr(Interval(Rational(1, 9), Rational(1, 4)))
#assert SetExpr(Interval(0, oo))**S.Half == SetExpr(Interval(0, oo))
#assert SetExpr(Interval(-oo, -1))**Rational(1, 3) == SetExpr(Interval(-oo, -1))
#assert SetExpr(Interval(-2, 3))**(Rational(-1, 3)) == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-oo, 0))**(-2) == SetExpr(Interval.open(0, oo))
assert SetExpr(Interval(-2, 0))**(-2) == SetExpr(Interval(Rational(1, 4), oo))
assert SetExpr(Interval(Rational(1, 3), S.Half))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(0, S.Half))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(S.Half, 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(0, 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(2, 3))**oo == SetExpr(FiniteSet(oo))
assert SetExpr(Interval(1, 2))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(S.Half, 3))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(Rational(-1, 3), Rational(-1, 4)))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(-1, Rational(-1, 2)))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-3, -2))**oo == SetExpr(FiniteSet(-oo, oo))
assert SetExpr(Interval(-2, -1))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-2, Rational(-1, 2)))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(Rational(-1, 2), S.Half))**oo == SetExpr(FiniteSet(0))
assert SetExpr(Interval(Rational(-1, 2), 1))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(Rational(-2, 3), 2))**oo == SetExpr(Interval(0, oo))
assert SetExpr(Interval(-1, 1))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-1, S.Half))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-1, 2))**oo == SetExpr(Interval(-oo, oo))
assert SetExpr(Interval(-2, S.Half))**oo == SetExpr(Interval(-oo, oo))
assert (SetExpr(Interval(1, 2))**x).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**x), Interval(1, 2))))
assert SetExpr(Interval(2, 3))**(-oo) == SetExpr(FiniteSet(0))
assert SetExpr(Interval(0, 2))**(-oo) == SetExpr(Interval(0, oo))
assert (SetExpr(Interval(-1, 2))**(-oo)).dummy_eq(SetExpr(ImageSet(Lambda(_d, _d**(-oo)), Interval(-1, 2))))
def test_SetExpr_Integers():
assert SetExpr(S.Integers) + 1 == SetExpr(S.Integers)
assert (SetExpr(S.Integers) + I).dummy_eq(
SetExpr(ImageSet(Lambda(_d, _d + I), S.Integers)))
assert SetExpr(S.Integers)*(-1) == SetExpr(S.Integers)
assert (SetExpr(S.Integers)*2).dummy_eq(
SetExpr(ImageSet(Lambda(_d, 2*_d), S.Integers)))
assert (SetExpr(S.Integers)*I).dummy_eq(
SetExpr(ImageSet(Lambda(_d, I*_d), S.Integers)))
# issue #18050:
assert SetExpr(S.Integers)._eval_func(Lambda(x, I*x + 1)).dummy_eq(
SetExpr(ImageSet(Lambda(_d, I*_d + 1), S.Integers)))
# needs improvement:
assert (SetExpr(S.Integers)*I + 1).dummy_eq(
SetExpr(ImageSet(Lambda(x, x + 1),
ImageSet(Lambda(_d, _d*I), S.Integers))))
|
1ac45b40664afca006a7d68806ce932d694bcc3349bf392bce3cdd50532145bf |
from sympy.core.expr import unchanged
from sympy.sets.contains import Contains
from sympy.sets.fancysets import (ImageSet, Range, normalize_theta_set,
ComplexRegion)
from sympy.sets.sets import (FiniteSet, Interval, Union, imageset,
Intersection, ProductSet, SetKind)
from sympy.sets.conditionset import ConditionSet
from sympy.simplify.simplify import simplify
from sympy.core.basic import Basic
from sympy.core.containers import Tuple, TupleKind
from sympy.core.function import Lambda
from sympy.core.kind import NumberKind
from sympy.core.numbers import (I, Rational, oo, pi)
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol, symbols)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
from sympy.logic.boolalg import And
from sympy.matrices.dense import eye
from sympy.testing.pytest import XFAIL, raises
from sympy.abc import x, y, t, z
from sympy.core.mod import Mod
import itertools
def test_naturals():
N = S.Naturals
assert 5 in N
assert -5 not in N
assert 5.5 not in N
ni = iter(N)
a, b, c, d = next(ni), next(ni), next(ni), next(ni)
assert (a, b, c, d) == (1, 2, 3, 4)
assert isinstance(a, Basic)
assert N.intersect(Interval(-5, 5)) == Range(1, 6)
assert N.intersect(Interval(-5, 5, True, True)) == Range(1, 5)
assert N.boundary == N
assert N.is_open == False
assert N.is_closed == True
assert N.inf == 1
assert N.sup is oo
assert not N.contains(oo)
for s in (S.Naturals0, S.Naturals):
assert s.intersection(S.Reals) is s
assert s.is_subset(S.Reals)
assert N.as_relational(x) == And(Eq(floor(x), x), x >= 1, x < oo)
def test_naturals0():
N = S.Naturals0
assert 0 in N
assert -1 not in N
assert next(iter(N)) == 0
assert not N.contains(oo)
assert N.contains(sin(x)) == Contains(sin(x), N)
def test_integers():
Z = S.Integers
assert 5 in Z
assert -5 in Z
assert 5.5 not in Z
assert not Z.contains(oo)
assert not Z.contains(-oo)
zi = iter(Z)
a, b, c, d = next(zi), next(zi), next(zi), next(zi)
assert (a, b, c, d) == (0, 1, -1, 2)
assert isinstance(a, Basic)
assert Z.intersect(Interval(-5, 5)) == Range(-5, 6)
assert Z.intersect(Interval(-5, 5, True, True)) == Range(-4, 5)
assert Z.intersect(Interval(5, S.Infinity)) == Range(5, S.Infinity)
assert Z.intersect(Interval.Lopen(5, S.Infinity)) == Range(6, S.Infinity)
assert Z.inf is -oo
assert Z.sup is oo
assert Z.boundary == Z
assert Z.is_open == False
assert Z.is_closed == True
assert Z.as_relational(x) == And(Eq(floor(x), x), -oo < x, x < oo)
def test_ImageSet():
raises(ValueError, lambda: ImageSet(x, S.Integers))
assert ImageSet(Lambda(x, 1), S.Integers) == FiniteSet(1)
assert ImageSet(Lambda(x, y), S.Integers) == {y}
assert ImageSet(Lambda(x, 1), S.EmptySet) == S.EmptySet
empty = Intersection(FiniteSet(log(2)/pi), S.Integers)
assert unchanged(ImageSet, Lambda(x, 1), empty) # issue #17471
squares = ImageSet(Lambda(x, x**2), S.Naturals)
assert 4 in squares
assert 5 not in squares
assert FiniteSet(*range(10)).intersect(squares) == FiniteSet(1, 4, 9)
assert 16 not in squares.intersect(Interval(0, 10))
si = iter(squares)
a, b, c, d = next(si), next(si), next(si), next(si)
assert (a, b, c, d) == (1, 4, 9, 16)
harmonics = ImageSet(Lambda(x, 1/x), S.Naturals)
assert Rational(1, 5) in harmonics
assert Rational(.25) in harmonics
assert 0.25 not in harmonics
assert Rational(.3) not in harmonics
assert (1, 2) not in harmonics
assert harmonics.is_iterable
assert imageset(x, -x, Interval(0, 1)) == Interval(-1, 0)
assert ImageSet(Lambda(x, x**2), Interval(0, 2)).doit() == Interval(0, 4)
assert ImageSet(Lambda((x, y), 2*x), {4}, {3}).doit() == FiniteSet(8)
assert (ImageSet(Lambda((x, y), x+y), {1, 2, 3}, {10, 20, 30}).doit() ==
FiniteSet(11, 12, 13, 21, 22, 23, 31, 32, 33))
c = Interval(1, 3) * Interval(1, 3)
assert Tuple(2, 6) in ImageSet(Lambda(((x, y),), (x, 2*y)), c)
assert Tuple(2, S.Half) in ImageSet(Lambda(((x, y),), (x, 1/y)), c)
assert Tuple(2, -2) not in ImageSet(Lambda(((x, y),), (x, y**2)), c)
assert Tuple(2, -2) in ImageSet(Lambda(((x, y),), (x, -2)), c)
c3 = ProductSet(Interval(3, 7), Interval(8, 11), Interval(5, 9))
assert Tuple(8, 3, 9) in ImageSet(Lambda(((t, y, x),), (y, t, x)), c3)
assert Tuple(Rational(1, 8), 3, 9) in ImageSet(Lambda(((t, y, x),), (1/y, t, x)), c3)
assert 2/pi not in ImageSet(Lambda(((x, y),), 2/x), c)
assert 2/S(100) not in ImageSet(Lambda(((x, y),), 2/x), c)
assert Rational(2, 3) in ImageSet(Lambda(((x, y),), 2/x), c)
S1 = imageset(lambda x, y: x + y, S.Integers, S.Naturals)
assert S1.base_pset == ProductSet(S.Integers, S.Naturals)
assert S1.base_sets == (S.Integers, S.Naturals)
# Passing a set instead of a FiniteSet shouldn't raise
assert unchanged(ImageSet, Lambda(x, x**2), {1, 2, 3})
S2 = ImageSet(Lambda(((x, y),), x+y), {(1, 2), (3, 4)})
assert 3 in S2.doit()
# FIXME: This doesn't yet work:
#assert 3 in S2
assert S2._contains(3) is None
raises(TypeError, lambda: ImageSet(Lambda(x, x**2), 1))
def test_image_is_ImageSet():
assert isinstance(imageset(x, sqrt(sin(x)), Range(5)), ImageSet)
def test_halfcircle():
r, th = symbols('r, theta', real=True)
L = Lambda(((r, th),), (r*cos(th), r*sin(th)))
halfcircle = ImageSet(L, Interval(0, 1)*Interval(0, pi))
assert (1, 0) in halfcircle
assert (0, -1) not in halfcircle
assert (0, 0) in halfcircle
assert halfcircle._contains((r, 0)) is None
# This one doesn't work:
#assert (r, 2*pi) not in halfcircle
assert not halfcircle.is_iterable
def test_ImageSet_iterator_not_injective():
L = Lambda(x, x - x % 2) # produces 0, 2, 2, 4, 4, 6, 6, ...
evens = ImageSet(L, S.Naturals)
i = iter(evens)
# No repeats here
assert (next(i), next(i), next(i), next(i)) == (0, 2, 4, 6)
def test_inf_Range_len():
raises(ValueError, lambda: len(Range(0, oo, 2)))
assert Range(0, oo, 2).size is S.Infinity
assert Range(0, -oo, -2).size is S.Infinity
assert Range(oo, 0, -2).size is S.Infinity
assert Range(-oo, 0, 2).size is S.Infinity
def test_Range_set():
empty = Range(0)
assert Range(5) == Range(0, 5) == Range(0, 5, 1)
r = Range(10, 20, 2)
assert 12 in r
assert 8 not in r
assert 11 not in r
assert 30 not in r
assert list(Range(0, 5)) == list(range(5))
assert list(Range(5, 0, -1)) == list(range(5, 0, -1))
assert Range(5, 15).sup == 14
assert Range(5, 15).inf == 5
assert Range(15, 5, -1).sup == 15
assert Range(15, 5, -1).inf == 6
assert Range(10, 67, 10).sup == 60
assert Range(60, 7, -10).inf == 10
assert len(Range(10, 38, 10)) == 3
assert Range(0, 0, 5) == empty
assert Range(oo, oo, 1) == empty
assert Range(oo, 1, 1) == empty
assert Range(-oo, 1, -1) == empty
assert Range(1, oo, -1) == empty
assert Range(1, -oo, 1) == empty
assert Range(1, -4, oo) == empty
ip = symbols('ip', positive=True)
assert Range(0, ip, -1) == empty
assert Range(0, -ip, 1) == empty
assert Range(1, -4, -oo) == Range(1, 2)
assert Range(1, 4, oo) == Range(1, 2)
assert Range(-oo, oo).size == oo
assert Range(oo, -oo, -1).size == oo
raises(ValueError, lambda: Range(-oo, oo, 2))
raises(ValueError, lambda: Range(x, pi, y))
raises(ValueError, lambda: Range(x, y, 0))
assert 5 in Range(0, oo, 5)
assert -5 in Range(-oo, 0, 5)
assert oo not in Range(0, oo)
ni = symbols('ni', integer=False)
assert ni not in Range(oo)
u = symbols('u', integer=None)
assert Range(oo).contains(u) is not False
inf = symbols('inf', infinite=True)
assert inf not in Range(-oo, oo)
raises(ValueError, lambda: Range(0, oo, 2)[-1])
raises(ValueError, lambda: Range(0, -oo, -2)[-1])
assert Range(-oo, 1, 1)[-1] is S.Zero
assert Range(oo, 1, -1)[-1] == 2
assert inf not in Range(oo)
assert Range(1, 10, 1)[-1] == 9
assert all(i.is_Integer for i in Range(0, -1, 1))
it = iter(Range(-oo, 0, 2))
raises(TypeError, lambda: next(it))
assert empty.intersect(S.Integers) == empty
assert Range(-1, 10, 1).intersect(S.Complexes) == Range(-1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Reals) == Range(-1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Rationals) == Range(-1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Integers) == Range(-1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Naturals) == Range(1, 10, 1)
assert Range(-1, 10, 1).intersect(S.Naturals0) == Range(0, 10, 1)
# test slicing
assert Range(1, 10, 1)[5] == 6
assert Range(1, 12, 2)[5] == 11
assert Range(1, 10, 1)[-1] == 9
assert Range(1, 10, 3)[-1] == 7
raises(ValueError, lambda: Range(oo,0,-1)[1:3:0])
raises(ValueError, lambda: Range(oo,0,-1)[:1])
raises(ValueError, lambda: Range(1, oo)[-2])
raises(ValueError, lambda: Range(-oo, 1)[2])
raises(IndexError, lambda: Range(10)[-20])
raises(IndexError, lambda: Range(10)[20])
raises(ValueError, lambda: Range(2, -oo, -2)[2:2:0])
assert Range(2, -oo, -2)[2:2:2] == empty
assert Range(2, -oo, -2)[:2:2] == Range(2, -2, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[:2:2])
assert Range(-oo, 4, 2)[::-2] == Range(2, -oo, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[::2])
assert Range(oo, 2, -2)[::] == Range(oo, 2, -2)
assert Range(-oo, 4, 2)[:-2:-2] == Range(2, 0, -4)
assert Range(-oo, 4, 2)[:-2:2] == Range(-oo, 0, 4)
raises(ValueError, lambda: Range(-oo, 4, 2)[:0:-2])
raises(ValueError, lambda: Range(-oo, 4, 2)[:2:-2])
assert Range(-oo, 4, 2)[-2::-2] == Range(0, -oo, -4)
raises(ValueError, lambda: Range(-oo, 4, 2)[-2:0:-2])
raises(ValueError, lambda: Range(-oo, 4, 2)[0::2])
assert Range(oo, 2, -2)[0::] == Range(oo, 2, -2)
raises(ValueError, lambda: Range(-oo, 4, 2)[0:-2:2])
assert Range(oo, 2, -2)[0:-2:] == Range(oo, 6, -2)
raises(ValueError, lambda: Range(oo, 2, -2)[0:2:])
raises(ValueError, lambda: Range(-oo, 4, 2)[2::-1])
assert Range(-oo, 4, 2)[-2::2] == Range(0, 4, 4)
assert Range(oo, 0, -2)[-10:0:2] == empty
raises(ValueError, lambda: Range(oo, 0, -2)[0])
raises(ValueError, lambda: Range(oo, 0, -2)[-10:10:2])
raises(ValueError, lambda: Range(oo, 0, -2)[0::-2])
assert Range(oo, 0, -2)[0:-4:-2] == empty
assert Range(oo, 0, -2)[:0:2] == empty
raises(ValueError, lambda: Range(oo, 0, -2)[:1:-1])
# test empty Range
assert Range(x, x, y) == empty
assert empty.reversed == empty
assert 0 not in empty
assert list(empty) == []
assert len(empty) == 0
assert empty.size is S.Zero
assert empty.intersect(FiniteSet(0)) is S.EmptySet
assert bool(empty) is False
raises(IndexError, lambda: empty[0])
assert empty[:0] == empty
raises(NotImplementedError, lambda: empty.inf)
raises(NotImplementedError, lambda: empty.sup)
assert empty.as_relational(x) is S.false
AB = [None] + list(range(12))
for R in [
Range(1, 10),
Range(1, 10, 2),
]:
r = list(R)
for a, b, c in itertools.product(AB, AB, [-3, -1, None, 1, 3]):
for reverse in range(2):
r = list(reversed(r))
R = R.reversed
result = list(R[a:b:c])
ans = r[a:b:c]
txt = ('\n%s[%s:%s:%s] = %s -> %s' % (
R, a, b, c, result, ans))
check = ans == result
assert check, txt
assert Range(1, 10, 1).boundary == Range(1, 10, 1)
for r in (Range(1, 10, 2), Range(1, oo, 2)):
rev = r.reversed
assert r.inf == rev.inf and r.sup == rev.sup
assert r.step == -rev.step
builtin_range = range
raises(TypeError, lambda: Range(builtin_range(1)))
assert S(builtin_range(10)) == Range(10)
assert S(builtin_range(1000000000000)) == Range(1000000000000)
# test Range.as_relational
assert Range(1, 4).as_relational(x) == (x >= 1) & (x <= 3) & Eq(Mod(x, 1), 0)
assert Range(oo, 1, -2).as_relational(x) == (x >= 3) & (x < oo) & Eq(Mod(x + 1, -2), 0)
def test_Range_symbolic():
# symbolic Range
xr = Range(x, x + 4, 5)
sr = Range(x, y, t)
i = Symbol('i', integer=True)
ip = Symbol('i', integer=True, positive=True)
ipr = Range(ip)
inr = Range(0, -ip, -1)
ir = Range(i, i + 19, 2)
ir2 = Range(i, i*8, 3*i)
i = Symbol('i', integer=True)
inf = symbols('inf', infinite=True)
raises(ValueError, lambda: Range(inf))
raises(ValueError, lambda: Range(inf, 0, -1))
raises(ValueError, lambda: Range(inf, inf, 1))
raises(ValueError, lambda: Range(1, 1, inf))
# args
assert xr.args == (x, x + 5, 5)
assert sr.args == (x, y, t)
assert ir.args == (i, i + 20, 2)
assert ir2.args == (i, 10*i, 3*i)
# reversed
raises(ValueError, lambda: xr.reversed)
raises(ValueError, lambda: sr.reversed)
assert ipr.reversed.args == (ip - 1, -1, -1)
assert inr.reversed.args == (-ip + 1, 1, 1)
assert ir.reversed.args == (i + 18, i - 2, -2)
assert ir2.reversed.args == (7*i, -2*i, -3*i)
# contains
assert inf not in sr
assert inf not in ir
assert 0 in ipr
assert 0 in inr
raises(TypeError, lambda: 1 in ipr)
raises(TypeError, lambda: -1 in inr)
assert .1 not in sr
assert .1 not in ir
assert i + 1 not in ir
assert i + 2 in ir
raises(TypeError, lambda: x in xr) # XXX is this what contains is supposed to do?
raises(TypeError, lambda: 1 in sr) # XXX is this what contains is supposed to do?
# iter
raises(ValueError, lambda: next(iter(xr)))
raises(ValueError, lambda: next(iter(sr)))
assert next(iter(ir)) == i
assert next(iter(ir2)) == i
assert sr.intersect(S.Integers) == sr
assert sr.intersect(FiniteSet(x)) == Intersection({x}, sr)
raises(ValueError, lambda: sr[:2])
raises(ValueError, lambda: xr[0])
raises(ValueError, lambda: sr[0])
# len
assert len(ir) == ir.size == 10
assert len(ir2) == ir2.size == 3
raises(ValueError, lambda: len(xr))
raises(ValueError, lambda: xr.size)
raises(ValueError, lambda: len(sr))
raises(ValueError, lambda: sr.size)
# bool
assert bool(Range(0)) == False
assert bool(xr)
assert bool(ir)
assert bool(ipr)
assert bool(inr)
raises(ValueError, lambda: bool(sr))
raises(ValueError, lambda: bool(ir2))
# inf
raises(ValueError, lambda: xr.inf)
raises(ValueError, lambda: sr.inf)
assert ipr.inf == 0
assert inr.inf == -ip + 1
assert ir.inf == i
raises(ValueError, lambda: ir2.inf)
# sup
raises(ValueError, lambda: xr.sup)
raises(ValueError, lambda: sr.sup)
assert ipr.sup == ip - 1
assert inr.sup == 0
assert ir.inf == i
raises(ValueError, lambda: ir2.sup)
# getitem
raises(ValueError, lambda: xr[0])
raises(ValueError, lambda: sr[0])
raises(ValueError, lambda: sr[-1])
raises(ValueError, lambda: sr[:2])
assert ir[:2] == Range(i, i + 4, 2)
assert ir[0] == i
assert ir[-2] == i + 16
assert ir[-1] == i + 18
assert ir2[:2] == Range(i, 7*i, 3*i)
assert ir2[0] == i
assert ir2[-2] == 4*i
assert ir2[-1] == 7*i
raises(ValueError, lambda: Range(i)[-1])
assert ipr[0] == ipr.inf == 0
assert ipr[-1] == ipr.sup == ip - 1
assert inr[0] == inr.sup == 0
assert inr[-1] == inr.inf == -ip + 1
raises(ValueError, lambda: ipr[-2])
assert ir.inf == i
assert ir.sup == i + 18
raises(ValueError, lambda: Range(i).inf)
# as_relational
assert ir.as_relational(x) == ((x >= i) & (x <= i + 18) &
Eq(Mod(-i + x, 2), 0))
assert ir2.as_relational(x) == Eq(
Mod(-i + x, 3*i), 0) & (((x >= i) & (x <= 7*i) & (3*i >= 1)) |
((x <= i) & (x >= 7*i) & (3*i <= -1)))
assert Range(i, i + 1).as_relational(x) == Eq(x, i)
assert sr.as_relational(z) == Eq(
Mod(t, 1), 0) & Eq(Mod(x, 1), 0) & Eq(Mod(-x + z, t), 0
) & (((z >= x) & (z <= -t + y) & (t >= 1)) |
((z <= x) & (z >= -t + y) & (t <= -1)))
assert xr.as_relational(z) == Eq(z, x) & Eq(Mod(x, 1), 0)
# symbols can clash if user wants (but it must be integer)
assert xr.as_relational(x) == Eq(Mod(x, 1), 0)
# contains() for symbolic values (issue #18146)
e = Symbol('e', integer=True, even=True)
o = Symbol('o', integer=True, odd=True)
assert Range(5).contains(i) == And(i >= 0, i <= 4)
assert Range(1).contains(i) == Eq(i, 0)
assert Range(-oo, 5, 1).contains(i) == (i <= 4)
assert Range(-oo, oo).contains(i) == True
assert Range(0, 8, 2).contains(i) == Contains(i, Range(0, 8, 2))
assert Range(0, 8, 2).contains(e) == And(e >= 0, e <= 6)
assert Range(0, 8, 2).contains(2*i) == And(2*i >= 0, 2*i <= 6)
assert Range(0, 8, 2).contains(o) == False
assert Range(1, 9, 2).contains(e) == False
assert Range(1, 9, 2).contains(o) == And(o >= 1, o <= 7)
assert Range(8, 0, -2).contains(o) == False
assert Range(9, 1, -2).contains(o) == And(o >= 3, o <= 9)
assert Range(-oo, 8, 2).contains(i) == Contains(i, Range(-oo, 8, 2))
def test_range_range_intersection():
for a, b, r in [
(Range(0), Range(1), S.EmptySet),
(Range(3), Range(4, oo), S.EmptySet),
(Range(3), Range(-3, -1), S.EmptySet),
(Range(1, 3), Range(0, 3), Range(1, 3)),
(Range(1, 3), Range(1, 4), Range(1, 3)),
(Range(1, oo, 2), Range(2, oo, 2), S.EmptySet),
(Range(0, oo, 2), Range(oo), Range(0, oo, 2)),
(Range(0, oo, 2), Range(100), Range(0, 100, 2)),
(Range(2, oo, 2), Range(oo), Range(2, oo, 2)),
(Range(0, oo, 2), Range(5, 6), S.EmptySet),
(Range(2, 80, 1), Range(55, 71, 4), Range(55, 71, 4)),
(Range(0, 6, 3), Range(-oo, 5, 3), S.EmptySet),
(Range(0, oo, 2), Range(5, oo, 3), Range(8, oo, 6)),
(Range(4, 6, 2), Range(2, 16, 7), S.EmptySet),]:
assert a.intersect(b) == r
assert a.intersect(b.reversed) == r
assert a.reversed.intersect(b) == r
assert a.reversed.intersect(b.reversed) == r
a, b = b, a
assert a.intersect(b) == r
assert a.intersect(b.reversed) == r
assert a.reversed.intersect(b) == r
assert a.reversed.intersect(b.reversed) == r
def test_range_interval_intersection():
p = symbols('p', positive=True)
assert isinstance(Range(3).intersect(Interval(p, p + 2)), Intersection)
assert Range(4).intersect(Interval(0, 3)) == Range(4)
assert Range(4).intersect(Interval(-oo, oo)) == Range(4)
assert Range(4).intersect(Interval(1, oo)) == Range(1, 4)
assert Range(4).intersect(Interval(1.1, oo)) == Range(2, 4)
assert Range(4).intersect(Interval(0.1, 3)) == Range(1, 4)
assert Range(4).intersect(Interval(0.1, 3.1)) == Range(1, 4)
assert Range(4).intersect(Interval.open(0, 3)) == Range(1, 3)
assert Range(4).intersect(Interval.open(0.1, 0.5)) is S.EmptySet
assert Interval(-1, 5).intersect(S.Complexes) == Interval(-1, 5)
assert Interval(-1, 5).intersect(S.Reals) == Interval(-1, 5)
assert Interval(-1, 5).intersect(S.Integers) == Range(-1, 6)
assert Interval(-1, 5).intersect(S.Naturals) == Range(1, 6)
assert Interval(-1, 5).intersect(S.Naturals0) == Range(0, 6)
# Null Range intersections
assert Range(0).intersect(Interval(0.2, 0.8)) is S.EmptySet
assert Range(0).intersect(Interval(-oo, oo)) is S.EmptySet
def test_range_is_finite_set():
assert Range(-100, 100).is_finite_set is True
assert Range(2, oo).is_finite_set is False
assert Range(-oo, 50).is_finite_set is False
assert Range(-oo, oo).is_finite_set is False
assert Range(oo, -oo).is_finite_set is True
assert Range(0, 0).is_finite_set is True
assert Range(oo, oo).is_finite_set is True
assert Range(-oo, -oo).is_finite_set is True
n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
assert Range(n, n + 49).is_finite_set is True
assert Range(n, 0).is_finite_set is True
assert Range(-3, n + 7).is_finite_set is True
assert Range(n, m).is_finite_set is True
assert Range(n + m, m - n).is_finite_set is True
assert Range(n, n + m + n).is_finite_set is True
assert Range(n, oo).is_finite_set is False
assert Range(-oo, n).is_finite_set is False
assert Range(n, -oo).is_finite_set is True
assert Range(oo, n).is_finite_set is True
def test_Range_is_iterable():
assert Range(-100, 100).is_iterable is True
assert Range(2, oo).is_iterable is False
assert Range(-oo, 50).is_iterable is False
assert Range(-oo, oo).is_iterable is False
assert Range(oo, -oo).is_iterable is True
assert Range(0, 0).is_iterable is True
assert Range(oo, oo).is_iterable is True
assert Range(-oo, -oo).is_iterable is True
n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
p = Symbol('p', integer=True, positive=True)
assert Range(n, n + 49).is_iterable is True
assert Range(n, 0).is_iterable is False
assert Range(-3, n + 7).is_iterable is False
assert Range(-3, p + 7).is_iterable is False # Should work with better __iter__
assert Range(n, m).is_iterable is False
assert Range(n + m, m - n).is_iterable is False
assert Range(n, n + m + n).is_iterable is False
assert Range(n, oo).is_iterable is False
assert Range(-oo, n).is_iterable is False
x = Symbol('x')
assert Range(x, x + 49).is_iterable is False
assert Range(x, 0).is_iterable is False
assert Range(-3, x + 7).is_iterable is False
assert Range(x, m).is_iterable is False
assert Range(x + m, m - x).is_iterable is False
assert Range(x, x + m + x).is_iterable is False
assert Range(x, oo).is_iterable is False
assert Range(-oo, x).is_iterable is False
def test_Integers_eval_imageset():
ans = ImageSet(Lambda(x, 2*x + Rational(3, 7)), S.Integers)
im = imageset(Lambda(x, -2*x + Rational(3, 7)), S.Integers)
assert im == ans
im = imageset(Lambda(x, -2*x - Rational(11, 7)), S.Integers)
assert im == ans
y = Symbol('y')
L = imageset(x, 2*x + y, S.Integers)
assert y + 4 in L
a, b, c = 0.092, 0.433, 0.341
assert a in imageset(x, a + c*x, S.Integers)
assert b in imageset(x, b + c*x, S.Integers)
_x = symbols('x', negative=True)
eq = _x**2 - _x + 1
assert imageset(_x, eq, S.Integers).lamda.expr == _x**2 + _x + 1
eq = 3*_x - 1
assert imageset(_x, eq, S.Integers).lamda.expr == 3*_x + 2
assert imageset(x, (x, 1/x), S.Integers) == \
ImageSet(Lambda(x, (x, 1/x)), S.Integers)
def test_Range_eval_imageset():
a, b, c = symbols('a b c')
assert imageset(x, a*(x + b) + c, Range(3)) == \
imageset(x, a*x + a*b + c, Range(3))
eq = (x + 1)**2
assert imageset(x, eq, Range(3)).lamda.expr == eq
eq = a*(x + b) + c
r = Range(3, -3, -2)
imset = imageset(x, eq, r)
assert imset.lamda.expr != eq
assert list(imset) == [eq.subs(x, i).expand() for i in list(r)]
def test_fun():
assert (FiniteSet(*ImageSet(Lambda(x, sin(pi*x/4)),
Range(-10, 11))) == FiniteSet(-1, -sqrt(2)/2, 0, sqrt(2)/2, 1))
def test_Range_is_empty():
i = Symbol('i', integer=True)
n = Symbol('n', negative=True, integer=True)
p = Symbol('p', positive=True, integer=True)
assert Range(0).is_empty
assert not Range(1).is_empty
assert Range(1, 0).is_empty
assert not Range(-1, 0).is_empty
assert Range(i).is_empty is None
assert Range(n).is_empty
assert Range(p).is_empty is False
assert Range(n, 0).is_empty is False
assert Range(n, p).is_empty is False
assert Range(p, n).is_empty
assert Range(n, -1).is_empty is None
assert Range(p, n, -1).is_empty is False
def test_Reals():
assert 5 in S.Reals
assert S.Pi in S.Reals
assert -sqrt(2) in S.Reals
assert (2, 5) not in S.Reals
assert sqrt(-1) not in S.Reals
assert S.Reals == Interval(-oo, oo)
assert S.Reals != Interval(0, oo)
assert S.Reals.is_subset(Interval(-oo, oo))
assert S.Reals.intersect(Range(-oo, oo)) == Range(-oo, oo)
assert S.ComplexInfinity not in S.Reals
assert S.NaN not in S.Reals
assert x + S.ComplexInfinity not in S.Reals
def test_Complex():
assert 5 in S.Complexes
assert 5 + 4*I in S.Complexes
assert S.Pi in S.Complexes
assert -sqrt(2) in S.Complexes
assert -I in S.Complexes
assert sqrt(-1) in S.Complexes
assert S.Complexes.intersect(S.Reals) == S.Reals
assert S.Complexes.union(S.Reals) == S.Complexes
assert S.Complexes == ComplexRegion(S.Reals*S.Reals)
assert (S.Complexes == ComplexRegion(Interval(1, 2)*Interval(3, 4))) == False
assert str(S.Complexes) == "Complexes"
assert repr(S.Complexes) == "Complexes"
def take(n, iterable):
"Return first n items of the iterable as a list"
return list(itertools.islice(iterable, n))
def test_intersections():
assert S.Integers.intersect(S.Reals) == S.Integers
assert 5 in S.Integers.intersect(S.Reals)
assert 5 in S.Integers.intersect(S.Reals)
assert -5 not in S.Naturals.intersect(S.Reals)
assert 5.5 not in S.Integers.intersect(S.Reals)
assert 5 in S.Integers.intersect(Interval(3, oo))
assert -5 in S.Integers.intersect(Interval(-oo, 3))
assert all(x.is_Integer
for x in take(10, S.Integers.intersect(Interval(3, oo)) ))
def test_infinitely_indexed_set_1():
from sympy.abc import n, m
assert imageset(Lambda(n, n), S.Integers) == imageset(Lambda(m, m), S.Integers)
assert imageset(Lambda(n, 2*n), S.Integers).intersect(
imageset(Lambda(m, 2*m + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(n, 2*n), S.Integers).intersect(
imageset(Lambda(n, 2*n + 1), S.Integers)) is S.EmptySet
assert imageset(Lambda(m, 2*m), S.Integers).intersect(
imageset(Lambda(n, 3*n), S.Integers)).dummy_eq(
ImageSet(Lambda(t, 6*t), S.Integers))
assert imageset(x, x/2 + Rational(1, 3), S.Integers).intersect(S.Integers) is S.EmptySet
assert imageset(x, x/2 + S.Half, S.Integers).intersect(S.Integers) is S.Integers
# https://github.com/sympy/sympy/issues/17355
S53 = ImageSet(Lambda(n, 5*n + 3), S.Integers)
assert S53.intersect(S.Integers) == S53
def test_infinitely_indexed_set_2():
from sympy.abc import n
a = Symbol('a', integer=True)
assert imageset(Lambda(n, n), S.Integers) == \
imageset(Lambda(n, n + a), S.Integers)
assert imageset(Lambda(n, n + pi), S.Integers) == \
imageset(Lambda(n, n + a + pi), S.Integers)
assert imageset(Lambda(n, n), S.Integers) == \
imageset(Lambda(n, -n + a), S.Integers)
assert imageset(Lambda(n, -6*n), S.Integers) == \
ImageSet(Lambda(n, 6*n), S.Integers)
assert imageset(Lambda(n, 2*n + pi), S.Integers) == \
ImageSet(Lambda(n, 2*n + pi - 2), S.Integers)
def test_imageset_intersect_real():
from sympy.abc import n
assert imageset(Lambda(n, n + (n - 1)*(n + 1)*I), S.Integers).intersect(S.Reals) == FiniteSet(-1, 1)
im = (n - 1)*(n + S.Half)
assert imageset(Lambda(n, n + im*I), S.Integers
).intersect(S.Reals) == FiniteSet(1)
assert imageset(Lambda(n, n + im*(n + 1)*I), S.Naturals0
).intersect(S.Reals) == FiniteSet(1)
assert imageset(Lambda(n, n/2 + im.expand()*I), S.Integers
).intersect(S.Reals) == ImageSet(Lambda(x, x/2), ConditionSet(
n, Eq(n**2 - n/2 - S(1)/2, 0), S.Integers))
assert imageset(Lambda(n, n/(1/n - 1) + im*(n + 1)*I), S.Integers
).intersect(S.Reals) == FiniteSet(S.Half)
assert imageset(Lambda(n, n/(n - 6) +
(n - 3)*(n + 1)*I/(2*n + 2)), S.Integers).intersect(
S.Reals) == FiniteSet(-1)
assert imageset(Lambda(n, n/(n**2 - 9) +
(n - 3)*(n + 1)*I/(2*n + 2)), S.Integers).intersect(
S.Reals) is S.EmptySet
s = ImageSet(
Lambda(n, -I*(I*(2*pi*n - pi/4) + log(Abs(sqrt(-I))))),
S.Integers)
# s is unevaluated, but after intersection the result
# should be canonical
assert s.intersect(S.Reals) == imageset(
Lambda(n, 2*n*pi - pi/4), S.Integers) == ImageSet(
Lambda(n, 2*pi*n + pi*Rational(7, 4)), S.Integers)
def test_imageset_intersect_interval():
from sympy.abc import n
f1 = ImageSet(Lambda(n, n*pi), S.Integers)
f2 = ImageSet(Lambda(n, 2*n), Interval(0, pi))
f3 = ImageSet(Lambda(n, 2*n*pi + pi/2), S.Integers)
# complex expressions
f4 = ImageSet(Lambda(n, n*I*pi), S.Integers)
f5 = ImageSet(Lambda(n, 2*I*n*pi + pi/2), S.Integers)
# non-linear expressions
f6 = ImageSet(Lambda(n, log(n)), S.Integers)
f7 = ImageSet(Lambda(n, n**2), S.Integers)
f8 = ImageSet(Lambda(n, Abs(n)), S.Integers)
f9 = ImageSet(Lambda(n, exp(n)), S.Naturals0)
assert f1.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f1.intersect(Interval(0, 2*pi, False, True)) == FiniteSet(0, pi)
assert f2.intersect(Interval(1, 2)) == Interval(1, 2)
assert f3.intersect(Interval(-1, 1)) == S.EmptySet
assert f3.intersect(Interval(-5, 5)) == FiniteSet(pi*Rational(-3, 2), pi/2)
assert f4.intersect(Interval(-1, 1)) == FiniteSet(0)
assert f4.intersect(Interval(1, 2)) == S.EmptySet
assert f5.intersect(Interval(0, 1)) == S.EmptySet
assert f6.intersect(Interval(0, 1)) == FiniteSet(S.Zero, log(2))
assert f7.intersect(Interval(0, 10)) == Intersection(f7, Interval(0, 10))
assert f8.intersect(Interval(0, 2)) == Intersection(f8, Interval(0, 2))
assert f9.intersect(Interval(1, 2)) == Intersection(f9, Interval(1, 2))
def test_imageset_intersect_diophantine():
from sympy.abc import m, n
# Check that same lambda variable for both ImageSets is handled correctly
img1 = ImageSet(Lambda(n, 2*n + 1), S.Integers)
img2 = ImageSet(Lambda(n, 4*n + 1), S.Integers)
assert img1.intersect(img2) == img2
# Empty solution set returned by diophantine:
assert ImageSet(Lambda(n, 2*n), S.Integers).intersect(
ImageSet(Lambda(n, 2*n + 1), S.Integers)) == S.EmptySet
# Check intersection with S.Integers:
assert ImageSet(Lambda(n, 9/n + 20*n/3), S.Integers).intersect(
S.Integers) == FiniteSet(-61, -23, 23, 61)
# Single solution (2, 3) for diophantine solution:
assert ImageSet(Lambda(n, (n - 2)**2), S.Integers).intersect(
ImageSet(Lambda(n, -(n - 3)**2), S.Integers)) == FiniteSet(0)
# Single parametric solution for diophantine solution:
assert ImageSet(Lambda(n, n**2 + 5), S.Integers).intersect(
ImageSet(Lambda(m, 2*m), S.Integers)).dummy_eq(ImageSet(
Lambda(n, 4*n**2 + 4*n + 6), S.Integers))
# 4 non-parametric solution couples for dioph. equation:
assert ImageSet(Lambda(n, n**2 - 9), S.Integers).intersect(
ImageSet(Lambda(m, -m**2), S.Integers)) == FiniteSet(-9, 0)
# Double parametric solution for diophantine solution:
assert ImageSet(Lambda(m, m**2 + 40), S.Integers).intersect(
ImageSet(Lambda(n, 41*n), S.Integers)).dummy_eq(Intersection(
ImageSet(Lambda(m, m**2 + 40), S.Integers),
ImageSet(Lambda(n, 41*n), S.Integers)))
# Check that diophantine returns *all* (8) solutions (permute=True)
assert ImageSet(Lambda(n, n**4 - 2**4), S.Integers).intersect(
ImageSet(Lambda(m, -m**4 + 3**4), S.Integers)) == FiniteSet(0, 65)
assert ImageSet(Lambda(n, pi/12 + n*5*pi/12), S.Integers).intersect(
ImageSet(Lambda(n, 7*pi/12 + n*11*pi/12), S.Integers)).dummy_eq(ImageSet(
Lambda(n, 55*pi*n/12 + 17*pi/4), S.Integers))
# TypeError raised by diophantine (#18081)
assert ImageSet(Lambda(n, n*log(2)), S.Integers).intersection(
S.Integers).dummy_eq(Intersection(ImageSet(
Lambda(n, n*log(2)), S.Integers), S.Integers))
# NotImplementedError raised by diophantine (no solver for cubic_thue)
assert ImageSet(Lambda(n, n**3 + 1), S.Integers).intersect(
ImageSet(Lambda(n, n**3), S.Integers)).dummy_eq(Intersection(
ImageSet(Lambda(n, n**3 + 1), S.Integers),
ImageSet(Lambda(n, n**3), S.Integers)))
def test_infinitely_indexed_set_3():
from sympy.abc import n, m
assert imageset(Lambda(m, 2*pi*m), S.Integers).intersect(
imageset(Lambda(n, 3*pi*n), S.Integers)).dummy_eq(
ImageSet(Lambda(t, 6*pi*t), S.Integers))
assert imageset(Lambda(n, 2*n + 1), S.Integers) == \
imageset(Lambda(n, 2*n - 1), S.Integers)
assert imageset(Lambda(n, 3*n + 2), S.Integers) == \
imageset(Lambda(n, 3*n - 1), S.Integers)
def test_ImageSet_simplification():
from sympy.abc import n, m
assert imageset(Lambda(n, n), S.Integers) == S.Integers
assert imageset(Lambda(n, sin(n)),
imageset(Lambda(m, tan(m)), S.Integers)) == \
imageset(Lambda(m, sin(tan(m))), S.Integers)
assert imageset(n, 1 + 2*n, S.Naturals) == Range(3, oo, 2)
assert imageset(n, 1 + 2*n, S.Naturals0) == Range(1, oo, 2)
assert imageset(n, 1 - 2*n, S.Naturals) == Range(-1, -oo, -2)
def test_ImageSet_contains():
assert (2, S.Half) in imageset(x, (x, 1/x), S.Integers)
assert imageset(x, x + I*3, S.Integers).intersection(S.Reals) is S.EmptySet
i = Dummy(integer=True)
q = imageset(x, x + I*y, S.Integers).intersection(S.Reals)
assert q.subs(y, I*i).intersection(S.Integers) is S.Integers
q = imageset(x, x + I*y/x, S.Integers).intersection(S.Reals)
assert q.subs(y, 0) is S.Integers
assert q.subs(y, I*i*x).intersection(S.Integers) is S.Integers
z = cos(1)**2 + sin(1)**2 - 1
q = imageset(x, x + I*z, S.Integers).intersection(S.Reals)
assert q is not S.EmptySet
def test_ComplexRegion_contains():
r = Symbol('r', real=True)
# contains in ComplexRegion
a = Interval(2, 3)
b = Interval(4, 6)
c = Interval(7, 9)
c1 = ComplexRegion(a*b)
c2 = ComplexRegion(Union(a*b, c*a))
assert 2.5 + 4.5*I in c1
assert 2 + 4*I in c1
assert 3 + 4*I in c1
assert 8 + 2.5*I in c2
assert 2.5 + 6.1*I not in c1
assert 4.5 + 3.2*I not in c1
assert c1.contains(x) == Contains(x, c1, evaluate=False)
assert c1.contains(r) == False
assert c2.contains(x) == Contains(x, c2, evaluate=False)
assert c2.contains(r) == False
r1 = Interval(0, 1)
theta1 = Interval(0, 2*S.Pi)
c3 = ComplexRegion(r1*theta1, polar=True)
assert (0.5 + I*6/10) in c3
assert (S.Half + I*6/10) in c3
assert (S.Half + .6*I) in c3
assert (0.5 + .6*I) in c3
assert I in c3
assert 1 in c3
assert 0 in c3
assert 1 + I not in c3
assert 1 - I not in c3
assert c3.contains(x) == Contains(x, c3, evaluate=False)
assert c3.contains(r + 2*I) == Contains(
r + 2*I, c3, evaluate=False) # is in fact False
assert c3.contains(1/(1 + r**2)) == Contains(
1/(1 + r**2), c3, evaluate=False) # is in fact True
r2 = Interval(0, 3)
theta2 = Interval(pi, 2*pi, left_open=True)
c4 = ComplexRegion(r2*theta2, polar=True)
assert c4.contains(0) == True
assert c4.contains(2 + I) == False
assert c4.contains(-2 + I) == False
assert c4.contains(-2 - I) == True
assert c4.contains(2 - I) == True
assert c4.contains(-2) == False
assert c4.contains(2) == True
assert c4.contains(x) == Contains(x, c4, evaluate=False)
assert c4.contains(3/(1 + r**2)) == Contains(
3/(1 + r**2), c4, evaluate=False) # is in fact True
raises(ValueError, lambda: ComplexRegion(r1*theta1, polar=2))
def test_symbolic_Range():
n = Symbol('n')
raises(ValueError, lambda: Range(n)[0])
raises(IndexError, lambda: Range(n, n)[0])
raises(ValueError, lambda: Range(n, n+1)[0])
raises(ValueError, lambda: Range(n).size)
n = Symbol('n', integer=True)
raises(ValueError, lambda: Range(n)[0])
raises(IndexError, lambda: Range(n, n)[0])
assert Range(n, n+1)[0] == n
raises(ValueError, lambda: Range(n).size)
assert Range(n, n+1).size == 1
n = Symbol('n', integer=True, nonnegative=True)
raises(ValueError, lambda: Range(n)[0])
raises(IndexError, lambda: Range(n, n)[0])
assert Range(n+1)[0] == 0
assert Range(n, n+1)[0] == n
assert Range(n).size == n
assert Range(n+1).size == n+1
assert Range(n, n+1).size == 1
n = Symbol('n', integer=True, positive=True)
assert Range(n)[0] == 0
assert Range(n, n+1)[0] == n
assert Range(n).size == n
assert Range(n, n+1).size == 1
m = Symbol('m', integer=True, positive=True)
assert Range(n, n+m)[0] == n
assert Range(n, n+m).size == m
assert Range(n, n+1).size == 1
assert Range(n, n+m, 2).size == floor(m/2)
m = Symbol('m', integer=True, positive=True, even=True)
assert Range(n, n+m, 2).size == m/2
def test_issue_18400():
n = Symbol('n', integer=True)
raises(ValueError, lambda: imageset(lambda x: x*2, Range(n)))
n = Symbol('n', integer=True, positive=True)
# No exception
assert imageset(lambda x: x*2, Range(n)) == imageset(lambda x: x*2, Range(n))
def test_ComplexRegion_intersect():
# Polar form
X_axis = ComplexRegion(Interval(0, oo)*FiniteSet(0, S.Pi), polar=True)
unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
upper_half_unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True)
upper_half_disk = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi), polar=True)
lower_half_disk = ComplexRegion(Interval(0, oo)*Interval(S.Pi, 2*S.Pi), polar=True)
right_half_disk = ComplexRegion(Interval(0, oo)*Interval(-S.Pi/2, S.Pi/2), polar=True)
first_quad_disk = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi/2), polar=True)
assert upper_half_disk.intersect(unit_disk) == upper_half_unit_disk
assert right_half_disk.intersect(first_quad_disk) == first_quad_disk
assert upper_half_disk.intersect(right_half_disk) == first_quad_disk
assert upper_half_disk.intersect(lower_half_disk) == X_axis
c1 = ComplexRegion(Interval(0, 4)*Interval(0, 2*S.Pi), polar=True)
assert c1.intersect(Interval(1, 5)) == Interval(1, 4)
assert c1.intersect(Interval(4, 9)) == FiniteSet(4)
assert c1.intersect(Interval(5, 12)) is S.EmptySet
# Rectangular form
X_axis = ComplexRegion(Interval(-oo, oo)*FiniteSet(0))
unit_square = ComplexRegion(Interval(-1, 1)*Interval(-1, 1))
upper_half_unit_square = ComplexRegion(Interval(-1, 1)*Interval(0, 1))
upper_half_plane = ComplexRegion(Interval(-oo, oo)*Interval(0, oo))
lower_half_plane = ComplexRegion(Interval(-oo, oo)*Interval(-oo, 0))
right_half_plane = ComplexRegion(Interval(0, oo)*Interval(-oo, oo))
first_quad_plane = ComplexRegion(Interval(0, oo)*Interval(0, oo))
assert upper_half_plane.intersect(unit_square) == upper_half_unit_square
assert right_half_plane.intersect(first_quad_plane) == first_quad_plane
assert upper_half_plane.intersect(right_half_plane) == first_quad_plane
assert upper_half_plane.intersect(lower_half_plane) == X_axis
c1 = ComplexRegion(Interval(-5, 5)*Interval(-10, 10))
assert c1.intersect(Interval(2, 7)) == Interval(2, 5)
assert c1.intersect(Interval(5, 7)) == FiniteSet(5)
assert c1.intersect(Interval(6, 9)) is S.EmptySet
# unevaluated object
C1 = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
C2 = ComplexRegion(Interval(-1, 1)*Interval(-1, 1))
assert C1.intersect(C2) == Intersection(C1, C2, evaluate=False)
def test_ComplexRegion_union():
# Polar form
c1 = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
c2 = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True)
c3 = ComplexRegion(Interval(0, oo)*Interval(0, S.Pi), polar=True)
c4 = ComplexRegion(Interval(0, oo)*Interval(S.Pi, 2*S.Pi), polar=True)
p1 = Union(Interval(0, 1)*Interval(0, 2*S.Pi), Interval(0, 1)*Interval(0, S.Pi))
p2 = Union(Interval(0, oo)*Interval(0, S.Pi), Interval(0, oo)*Interval(S.Pi, 2*S.Pi))
assert c1.union(c2) == ComplexRegion(p1, polar=True)
assert c3.union(c4) == ComplexRegion(p2, polar=True)
# Rectangular form
c5 = ComplexRegion(Interval(2, 5)*Interval(6, 9))
c6 = ComplexRegion(Interval(4, 6)*Interval(10, 12))
c7 = ComplexRegion(Interval(0, 10)*Interval(-10, 0))
c8 = ComplexRegion(Interval(12, 16)*Interval(14, 20))
p3 = Union(Interval(2, 5)*Interval(6, 9), Interval(4, 6)*Interval(10, 12))
p4 = Union(Interval(0, 10)*Interval(-10, 0), Interval(12, 16)*Interval(14, 20))
assert c5.union(c6) == ComplexRegion(p3)
assert c7.union(c8) == ComplexRegion(p4)
assert c1.union(Interval(2, 4)) == Union(c1, Interval(2, 4), evaluate=False)
assert c5.union(Interval(2, 4)) == Union(c5, ComplexRegion.from_real(Interval(2, 4)))
def test_ComplexRegion_from_real():
c1 = ComplexRegion(Interval(0, 1) * Interval(0, 2 * S.Pi), polar=True)
raises(ValueError, lambda: c1.from_real(c1))
assert c1.from_real(Interval(-1, 1)) == ComplexRegion(Interval(-1, 1) * FiniteSet(0), False)
def test_ComplexRegion_measure():
a, b = Interval(2, 5), Interval(4, 8)
theta1, theta2 = Interval(0, 2*S.Pi), Interval(0, S.Pi)
c1 = ComplexRegion(a*b)
c2 = ComplexRegion(Union(a*theta1, b*theta2), polar=True)
assert c1.measure == 12
assert c2.measure == 9*pi
def test_normalize_theta_set():
# Interval
assert normalize_theta_set(Interval(pi, 2*pi)) == \
Union(FiniteSet(0), Interval.Ropen(pi, 2*pi))
assert normalize_theta_set(Interval(pi*Rational(9, 2), 5*pi)) == Interval(pi/2, pi)
assert normalize_theta_set(Interval(pi*Rational(-3, 2), pi/2)) == Interval.Ropen(0, 2*pi)
assert normalize_theta_set(Interval.open(pi*Rational(-3, 2), pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval.open(pi*Rational(-7, 2), pi*Rational(-3, 2))) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi/2, 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.open(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval(-4*pi, 3*pi)) == Interval.Ropen(0, 2*pi)
assert normalize_theta_set(Interval(pi*Rational(-3, 2), -pi/2)) == Interval(pi/2, pi*Rational(3, 2))
assert normalize_theta_set(Interval.open(0, 2*pi)) == Interval.open(0, 2*pi)
assert normalize_theta_set(Interval.Ropen(-pi/2, pi/2)) == \
Union(Interval.Ropen(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.Lopen(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.open(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval(-pi/2, pi/2)) == \
Union(Interval(0, pi/2), Interval.Ropen(pi*Rational(3, 2), 2*pi))
assert normalize_theta_set(Interval.open(4*pi, pi*Rational(9, 2))) == Interval.open(0, pi/2)
assert normalize_theta_set(Interval.Lopen(4*pi, pi*Rational(9, 2))) == Interval.Lopen(0, pi/2)
assert normalize_theta_set(Interval.Ropen(4*pi, pi*Rational(9, 2))) == Interval.Ropen(0, pi/2)
assert normalize_theta_set(Interval.open(3*pi, 5*pi)) == \
Union(Interval.Ropen(0, pi), Interval.open(pi, 2*pi))
# FiniteSet
assert normalize_theta_set(FiniteSet(0, pi, 3*pi)) == FiniteSet(0, pi)
assert normalize_theta_set(FiniteSet(0, pi/2, pi, 2*pi)) == FiniteSet(0, pi/2, pi)
assert normalize_theta_set(FiniteSet(0, -pi/2, -pi, -2*pi)) == FiniteSet(0, pi, pi*Rational(3, 2))
assert normalize_theta_set(FiniteSet(pi*Rational(-3, 2), pi/2)) == \
FiniteSet(pi/2)
assert normalize_theta_set(FiniteSet(2*pi)) == FiniteSet(0)
# Unions
assert normalize_theta_set(Union(Interval(0, pi/3), Interval(pi/2, pi))) == \
Union(Interval(0, pi/3), Interval(pi/2, pi))
assert normalize_theta_set(Union(Interval(0, pi), Interval(2*pi, pi*Rational(7, 3)))) == \
Interval(0, pi)
# ValueError for non-real sets
raises(ValueError, lambda: normalize_theta_set(S.Complexes))
# NotImplementedError for subset of reals
raises(NotImplementedError, lambda: normalize_theta_set(Interval(0, 1)))
# NotImplementedError without pi as coefficient
raises(NotImplementedError, lambda: normalize_theta_set(Interval(1, 2*pi)))
raises(NotImplementedError, lambda: normalize_theta_set(Interval(2*pi, 10)))
raises(NotImplementedError, lambda: normalize_theta_set(FiniteSet(0, 3, 3*pi)))
def test_ComplexRegion_FiniteSet():
x, y, z, a, b, c = symbols('x y z a b c')
# Issue #9669
assert ComplexRegion(FiniteSet(a, b, c)*FiniteSet(x, y, z)) == \
FiniteSet(a + I*x, a + I*y, a + I*z, b + I*x, b + I*y,
b + I*z, c + I*x, c + I*y, c + I*z)
assert ComplexRegion(FiniteSet(2)*FiniteSet(3)) == FiniteSet(2 + 3*I)
def test_union_RealSubSet():
assert (S.Complexes).union(Interval(1, 2)) == S.Complexes
assert (S.Complexes).union(S.Integers) == S.Complexes
def test_SetKind_fancySet():
G = lambda *args: ImageSet(Lambda(x, x ** 2), *args)
assert G(Interval(1, 4)).kind is SetKind(NumberKind)
assert G(FiniteSet(1, 4)).kind is SetKind(NumberKind)
assert S.Rationals.kind is SetKind(NumberKind)
assert S.Naturals.kind is SetKind(NumberKind)
assert S.Integers.kind is SetKind(NumberKind)
assert Range(3).kind is SetKind(NumberKind)
a = Interval(2, 3)
b = Interval(4, 6)
c1 = ComplexRegion(a*b)
assert c1.kind is SetKind(TupleKind(NumberKind, NumberKind))
def test_issue_9980():
c1 = ComplexRegion(Interval(1, 2)*Interval(2, 3))
c2 = ComplexRegion(Interval(1, 5)*Interval(1, 3))
R = Union(c1, c2)
assert simplify(R) == ComplexRegion(Union(Interval(1, 2)*Interval(2, 3), \
Interval(1, 5)*Interval(1, 3)), False)
assert c1.func(*c1.args) == c1
assert R.func(*R.args) == R
def test_issue_11732():
interval12 = Interval(1, 2)
finiteset1234 = FiniteSet(1, 2, 3, 4)
pointComplex = Tuple(1, 5)
assert (interval12 in S.Naturals) == False
assert (interval12 in S.Naturals0) == False
assert (interval12 in S.Integers) == False
assert (interval12 in S.Complexes) == False
assert (finiteset1234 in S.Naturals) == False
assert (finiteset1234 in S.Naturals0) == False
assert (finiteset1234 in S.Integers) == False
assert (finiteset1234 in S.Complexes) == False
assert (pointComplex in S.Naturals) == False
assert (pointComplex in S.Naturals0) == False
assert (pointComplex in S.Integers) == False
assert (pointComplex in S.Complexes) == True
def test_issue_11730():
unit = Interval(0, 1)
square = ComplexRegion(unit ** 2)
assert Union(S.Complexes, FiniteSet(oo)) != S.Complexes
assert Union(S.Complexes, FiniteSet(eye(4))) != S.Complexes
assert Union(unit, square) == square
assert Intersection(S.Reals, square) == unit
def test_issue_11938():
unit = Interval(0, 1)
ival = Interval(1, 2)
cr1 = ComplexRegion(ival * unit)
assert Intersection(cr1, S.Reals) == ival
assert Intersection(cr1, unit) == FiniteSet(1)
arg1 = Interval(0, S.Pi)
arg2 = FiniteSet(S.Pi)
arg3 = Interval(S.Pi / 4, 3 * S.Pi / 4)
cp1 = ComplexRegion(unit * arg1, polar=True)
cp2 = ComplexRegion(unit * arg2, polar=True)
cp3 = ComplexRegion(unit * arg3, polar=True)
assert Intersection(cp1, S.Reals) == Interval(-1, 1)
assert Intersection(cp2, S.Reals) == Interval(-1, 0)
assert Intersection(cp3, S.Reals) == FiniteSet(0)
def test_issue_11914():
a, b = Interval(0, 1), Interval(0, pi)
c, d = Interval(2, 3), Interval(pi, 3 * pi / 2)
cp1 = ComplexRegion(a * b, polar=True)
cp2 = ComplexRegion(c * d, polar=True)
assert -3 in cp1.union(cp2)
assert -3 in cp2.union(cp1)
assert -5 not in cp1.union(cp2)
def test_issue_9543():
assert ImageSet(Lambda(x, x**2), S.Naturals).is_subset(S.Reals)
def test_issue_16871():
assert ImageSet(Lambda(x, x), FiniteSet(1)) == {1}
assert ImageSet(Lambda(x, x - 3), S.Integers
).intersection(S.Integers) is S.Integers
@XFAIL
def test_issue_16871b():
assert ImageSet(Lambda(x, x - 3), S.Integers).is_subset(S.Integers)
def test_issue_18050():
assert imageset(Lambda(x, I*x + 1), S.Integers
) == ImageSet(Lambda(x, I*x + 1), S.Integers)
assert imageset(Lambda(x, 3*I*x + 4 + 8*I), S.Integers
) == ImageSet(Lambda(x, 3*I*x + 4 + 2*I), S.Integers)
# no 'Mod' for next 2 tests:
assert imageset(Lambda(x, 2*x + 3*I), S.Integers
) == ImageSet(Lambda(x, 2*x + 3*I), S.Integers)
r = Symbol('r', positive=True)
assert imageset(Lambda(x, r*x + 10), S.Integers
) == ImageSet(Lambda(x, r*x + 10), S.Integers)
# reduce real part:
assert imageset(Lambda(x, 3*x + 8 + 5*I), S.Integers
) == ImageSet(Lambda(x, 3*x + 2 + 5*I), S.Integers)
def test_Rationals():
assert S.Integers.is_subset(S.Rationals)
assert S.Naturals.is_subset(S.Rationals)
assert S.Naturals0.is_subset(S.Rationals)
assert S.Rationals.is_subset(S.Reals)
assert S.Rationals.inf is -oo
assert S.Rationals.sup is oo
it = iter(S.Rationals)
assert [next(it) for i in range(12)] == [
0, 1, -1, S.Half, 2, Rational(-1, 2), -2,
Rational(1, 3), 3, Rational(-1, 3), -3, Rational(2, 3)]
assert Basic() not in S.Rationals
assert S.Half in S.Rationals
assert S.Rationals.contains(0.5) == Contains(0.5, S.Rationals, evaluate=False)
assert 2 in S.Rationals
r = symbols('r', rational=True)
assert r in S.Rationals
raises(TypeError, lambda: x in S.Rationals)
# issue #18134:
assert S.Rationals.boundary == S.Reals
assert S.Rationals.closure == S.Reals
assert S.Rationals.is_open == False
assert S.Rationals.is_closed == False
def test_NZQRC_unions():
# check that all trivial number set unions are simplified:
nbrsets = (S.Naturals, S.Naturals0, S.Integers, S.Rationals,
S.Reals, S.Complexes)
unions = (Union(a, b) for a in nbrsets for b in nbrsets)
assert all(u.is_Union is False for u in unions)
def test_imageset_intersection():
n = Dummy()
s = ImageSet(Lambda(n, -I*(I*(2*pi*n - pi/4) +
log(Abs(sqrt(-I))))), S.Integers)
assert s.intersect(S.Reals) == ImageSet(
Lambda(n, 2*pi*n + pi*Rational(7, 4)), S.Integers)
def test_issue_17858():
assert 1 in Range(-oo, oo)
assert 0 in Range(oo, -oo, -1)
assert oo not in Range(-oo, oo)
assert -oo not in Range(-oo, oo)
def test_issue_17859():
r = Range(-oo,oo)
raises(ValueError,lambda: r[::2])
raises(ValueError, lambda: r[::-2])
r = Range(oo,-oo,-1)
raises(ValueError,lambda: r[::2])
raises(ValueError, lambda: r[::-2])
|
00b7daf7265506c27ff752dccd43937a3dd0fd80f82a86d3891971c6000614f1 | #!/usr/bin/env python
""" Update the ``ask_generated.py`` file.
This must be run each time ``known_facts()`` in ``assumptions.facts`` module
is changed.
This must be run each time ``_generate_assumption_rules()`` in
``sympy.core.assumptions`` module is changed.
Should be run from sympy root directory.
$ python bin/ask_update.py
"""
# hook in-tree SymPy into Python path, if possible
import os
import sys
import pprint
isympy_path = os.path.abspath(__file__)
isympy_dir = os.path.dirname(isympy_path)
sympy_top = os.path.split(isympy_dir)[0]
sympy_dir = os.path.join(sympy_top, 'sympy')
if os.path.isdir(sympy_dir):
sys.path.insert(0, sympy_top)
from sympy.core.assumptions import _generate_assumption_rules
from sympy.assumptions.cnf import CNF, Literal
from sympy.assumptions.facts import (get_known_facts,
generate_known_facts_dict, get_known_facts_keys)
from sympy.core import Symbol
from textwrap import dedent, wrap
def generate_code():
LINE = ",\n "
HANG = ' '*8
code_string = dedent('''\
"""
Do NOT manually edit this file.
Instead, run ./bin/ask_update.py.
"""
from sympy.assumptions.ask import Q
from sympy.assumptions.cnf import Literal
from sympy.core.cache import cacheit
@cacheit
def get_all_known_facts():
"""
Known facts between unary predicates as CNF clauses.
"""
return {
%s
}
@cacheit
def get_known_facts_dict():
"""
Logical relations between unary predicates as dictionary.
Each key is a predicate, and item is two groups of predicates.
First group contains the predicates which are implied by the key, and
second group contains the predicates which are rejected by the key.
"""
return {
%s
}
''')
x = Symbol('x')
fact = get_known_facts(x)
# Generate CNF of facts between known unary predicates
cnf = CNF.to_CNF(fact)
p = LINE.join(sorted([
'frozenset(('
+ ', '.join(str(Literal(lit.arg.function, lit.is_Not))
for lit in sorted(clause, key=str))
+ '))' for clause in cnf.clauses]))
# Generate dictionary of facts between known unary predicates
keys = [pred(x) for pred in get_known_facts_keys()]
mapping = generate_known_facts_dict(keys, fact)
items = sorted(mapping.items(), key=str)
keys = [str(i[0]) for i in items]
values = ['(set(%s), set(%s))' % (sorted(i[1][0], key=str),
sorted(i[1][1], key=str))
for i in items]
m = LINE.join(['\n'.join(
wrap("{}: {}".format(k, v),
subsequent_indent=HANG,
break_long_words=False))
for k, v in zip(keys, values)]) + ','
return code_string % (p, m)
with open('sympy/assumptions/ask_generated.py', 'w') as f:
code = generate_code()
f.write(code)
with open('sympy/core/assumptions_generated.py', 'w') as f:
representation = _generate_assumption_rules()._to_python()
code_string = dedent('''\
"""
Do NOT manually edit this file.
Instead, run ./bin/ask_update.py.
"""
%s
''')
code = code_string % (representation,)
f.write(code)
|
626da83146229253da4db782ac0b78d6eb9dcfbaf78d21f3a25d4ef26d3c97ef | """
SymPy is a Python library for symbolic mathematics. It aims to become a
full-featured computer algebra system (CAS) while keeping the code as simple
as possible in order to be comprehensible and easily extensible. SymPy is
written entirely in Python. It depends on mpmath, and other external libraries
may be optionally for things like plotting support.
See the webpage for more information and documentation:
https://sympy.org
"""
import sys
if sys.version_info < (3, 8):
raise ImportError("Python version 3.8 or above is required for SymPy.")
del sys
try:
import mpmath
except ImportError:
raise ImportError("SymPy now depends on mpmath as an external library. "
"See https://docs.sympy.org/latest/install.html#mpmath for more information.")
del mpmath
from sympy.release import __version__
from sympy.core.cache import lazy_function
if 'dev' in __version__:
def enable_warnings():
import warnings
warnings.filterwarnings('default', '.*', DeprecationWarning, module='sympy.*')
del warnings
enable_warnings()
del enable_warnings
def __sympy_debug():
# helper function so we don't import os globally
import os
debug_str = os.getenv('SYMPY_DEBUG', 'False')
if debug_str in ('True', 'False'):
return eval(debug_str)
else:
raise RuntimeError("unrecognized value for SYMPY_DEBUG: %s" %
debug_str)
SYMPY_DEBUG = __sympy_debug() # type: bool
from .core import (sympify, SympifyError, cacheit, Basic, Atom,
preorder_traversal, S, Expr, AtomicExpr, UnevaluatedExpr, Symbol,
Wild, Dummy, symbols, var, Number, Float, Rational, Integer,
NumberSymbol, RealNumber, igcd, ilcm, seterr, E, I, nan, oo, pi, zoo,
AlgebraicNumber, comp, mod_inverse, Pow, integer_nthroot, integer_log,
Mul, prod, Add, Mod, Rel, Eq, Ne, Lt, Le, Gt, Ge, Equality,
GreaterThan, LessThan, Unequality, StrictGreaterThan, StrictLessThan,
vectorize, Lambda, WildFunction, Derivative, diff, FunctionClass,
Function, Subs, expand, PoleError, count_ops, expand_mul, expand_log,
expand_func, expand_trig, expand_complex, expand_multinomial, nfloat,
expand_power_base, expand_power_exp, arity, PrecisionExhausted, N,
evalf, Tuple, Dict, gcd_terms, factor_terms, factor_nc, evaluate,
Catalan, EulerGamma, GoldenRatio, TribonacciConstant, bottom_up, use,
postorder_traversal, default_sort_key, ordered)
from .logic import (to_cnf, to_dnf, to_nnf, And, Or, Not, Xor, Nand, Nor,
Implies, Equivalent, ITE, POSform, SOPform, simplify_logic, bool_map,
true, false, satisfiable)
from .assumptions import (AppliedPredicate, Predicate, AssumptionsContext,
assuming, Q, ask, register_handler, remove_handler, refine)
from .polys import (Poly, PurePoly, poly_from_expr, parallel_poly_from_expr,
degree, total_degree, degree_list, LC, LM, LT, pdiv, prem, pquo,
pexquo, div, rem, quo, exquo, half_gcdex, gcdex, invert,
subresultants, resultant, discriminant, cofactors, gcd_list, gcd,
lcm_list, lcm, terms_gcd, trunc, monic, content, primitive, compose,
decompose, sturm, gff_list, gff, sqf_norm, sqf_part, sqf_list, sqf,
factor_list, factor, intervals, refine_root, count_roots, real_roots,
nroots, ground_roots, nth_power_roots_poly, cancel, reduced, groebner,
is_zero_dimensional, GroebnerBasis, poly, symmetrize, horner,
interpolate, rational_interpolate, viete, together,
BasePolynomialError, ExactQuotientFailed, PolynomialDivisionFailed,
OperationNotSupported, HeuristicGCDFailed, HomomorphismFailed,
IsomorphismFailed, ExtraneousFactors, EvaluationFailed,
RefinementFailed, CoercionFailed, NotInvertible, NotReversible,
NotAlgebraic, DomainError, PolynomialError, UnificationFailed,
GeneratorsError, GeneratorsNeeded, ComputationFailed,
UnivariatePolynomialError, MultivariatePolynomialError,
PolificationFailed, OptionError, FlagError, minpoly,
minimal_polynomial, primitive_element, field_isomorphism,
to_number_field, isolate, round_two, prime_decomp, prime_valuation,
itermonomials, Monomial, lex, grlex,
grevlex, ilex, igrlex, igrevlex, CRootOf, rootof, RootOf,
ComplexRootOf, RootSum, roots, Domain, FiniteField, IntegerRing,
RationalField, RealField, ComplexField, PythonFiniteField,
GMPYFiniteField, PythonIntegerRing, GMPYIntegerRing, PythonRational,
GMPYRationalField, AlgebraicField, PolynomialRing, FractionField,
ExpressionDomain, FF_python, FF_gmpy, ZZ_python, ZZ_gmpy, QQ_python,
QQ_gmpy, GF, FF, ZZ, QQ, ZZ_I, QQ_I, RR, CC, EX, EXRAW,
construct_domain, swinnerton_dyer_poly, cyclotomic_poly,
symmetric_poly, random_poly, interpolating_poly, jacobi_poly,
chebyshevt_poly, chebyshevu_poly, hermite_poly, hermite_prob_poly,
legendre_poly, laguerre_poly, apart, apart_list, assemble_partfrac_list,
Options, ring, xring, vring, sring, field, xfield, vfield, sfield)
from .series import (Order, O, limit, Limit, gruntz, series, approximants,
residue, EmptySequence, SeqPer, SeqFormula, sequence, SeqAdd, SeqMul,
fourier_series, fps, difference_delta, limit_seq)
from .functions import (factorial, factorial2, rf, ff, binomial,
RisingFactorial, FallingFactorial, subfactorial, carmichael,
fibonacci, lucas, motzkin, tribonacci, harmonic, bernoulli, bell, euler,
catalan, genocchi, andre, partition, sqrt, root, Min, Max, Id,
real_root, Rem, cbrt, re, im, sign, Abs, conjugate, arg, polar_lift,
periodic_argument, unbranched_argument, principal_branch, transpose,
adjoint, polarify, unpolarify, sin, cos, tan, sec, csc, cot, sinc,
asin, acos, atan, asec, acsc, acot, atan2, exp_polar, exp, ln, log,
LambertW, sinh, cosh, tanh, coth, sech, csch, asinh, acosh, atanh,
acoth, asech, acsch, floor, ceiling, frac, Piecewise, piecewise_fold,
piecewise_exclusive, erf, erfc, erfi, erf2, erfinv, erfcinv, erf2inv,
Ei, expint, E1, li, Li, Si, Ci, Shi, Chi, fresnels, fresnelc, gamma,
lowergamma, uppergamma, polygamma, loggamma, digamma, trigamma,
multigamma, dirichlet_eta, zeta, lerchphi, polylog, stieltjes, Eijk,
LeviCivita, KroneckerDelta, SingularityFunction, DiracDelta, Heaviside,
bspline_basis, bspline_basis_set, interpolating_spline, besselj,
bessely, besseli, besselk, hankel1, hankel2, jn, yn, jn_zeros, hn1,
hn2, airyai, airybi, airyaiprime, airybiprime, marcumq, hyper,
meijerg, appellf1, legendre, assoc_legendre, hermite, hermite_prob,
chebyshevt, chebyshevu, chebyshevu_root, chebyshevt_root, laguerre,
assoc_laguerre, gegenbauer, jacobi, jacobi_normalized, Ynm, Ynm_c,
Znm, elliptic_k, elliptic_f, elliptic_e, elliptic_pi, beta, mathieus,
mathieuc, mathieusprime, mathieucprime, riemann_xi, betainc, betainc_regularized)
from .ntheory import (nextprime, prevprime, prime, primepi, primerange,
randprime, Sieve, sieve, primorial, cycle_length, composite,
compositepi, isprime, divisors, proper_divisors, factorint,
multiplicity, perfect_power, pollard_pm1, pollard_rho, primefactors,
totient, trailing, divisor_count, proper_divisor_count, divisor_sigma,
factorrat, reduced_totient, primenu, primeomega,
mersenne_prime_exponent, is_perfect, is_mersenne_prime, is_abundant,
is_deficient, is_amicable, abundance, npartitions, is_primitive_root,
is_quad_residue, legendre_symbol, jacobi_symbol, n_order, sqrt_mod,
quadratic_residues, primitive_root, nthroot_mod, is_nthpow_residue,
sqrt_mod_iter, mobius, discrete_log, quadratic_congruence,
binomial_coefficients, binomial_coefficients_list,
multinomial_coefficients, continued_fraction_periodic,
continued_fraction_iterator, continued_fraction_reduce,
continued_fraction_convergents, continued_fraction, egyptian_fraction)
from .concrete import product, Product, summation, Sum
from .discrete import (fft, ifft, ntt, intt, fwht, ifwht, mobius_transform,
inverse_mobius_transform, convolution, covering_product,
intersecting_product)
from .simplify import (simplify, hypersimp, hypersimilar, logcombine,
separatevars, posify, besselsimp, kroneckersimp, signsimp,
nsimplify, FU, fu, sqrtdenest, cse, epath, EPath, hyperexpand,
collect, rcollect, radsimp, collect_const, fraction, numer, denom,
trigsimp, exptrigsimp, powsimp, powdenest, combsimp, gammasimp,
ratsimp, ratsimpmodprime)
from .sets import (Set, Interval, Union, EmptySet, FiniteSet, ProductSet,
Intersection, DisjointUnion, imageset, Complement, SymmetricDifference, ImageSet,
Range, ComplexRegion, Complexes, Reals, Contains, ConditionSet, Ordinal,
OmegaPower, ord0, PowerSet, Naturals, Naturals0, UniversalSet,
Integers, Rationals)
from .solvers import (solve, solve_linear_system, solve_linear_system_LU,
solve_undetermined_coeffs, nsolve, solve_linear, checksol, det_quick,
inv_quick, check_assumptions, failing_assumptions, diophantine,
rsolve, rsolve_poly, rsolve_ratio, rsolve_hyper, checkodesol,
classify_ode, dsolve, homogeneous_order, solve_poly_system,
solve_triangulated, pde_separate, pde_separate_add, pde_separate_mul,
pdsolve, classify_pde, checkpdesol, ode_order, reduce_inequalities,
reduce_abs_inequality, reduce_abs_inequalities, solve_poly_inequality,
solve_rational_inequalities, solve_univariate_inequality, decompogen,
solveset, linsolve, linear_eq_to_matrix, nonlinsolve, substitution)
from .matrices import (ShapeError, NonSquareMatrixError, GramSchmidt,
casoratian, diag, eye, hessian, jordan_cell, list2numpy, matrix2numpy,
matrix_multiply_elementwise, ones, randMatrix, rot_axis1, rot_axis2,
rot_axis3, symarray, wronskian, zeros, MutableDenseMatrix,
DeferredVector, MatrixBase, Matrix, MutableMatrix,
MutableSparseMatrix, banded, ImmutableDenseMatrix,
ImmutableSparseMatrix, ImmutableMatrix, SparseMatrix, MatrixSlice,
BlockDiagMatrix, BlockMatrix, FunctionMatrix, Identity, Inverse,
MatAdd, MatMul, MatPow, MatrixExpr, MatrixSymbol, Trace, Transpose,
ZeroMatrix, OneMatrix, blockcut, block_collapse, matrix_symbols,
Adjoint, hadamard_product, HadamardProduct, HadamardPower,
Determinant, det, diagonalize_vector, DiagMatrix, DiagonalMatrix,
DiagonalOf, trace, DotProduct, kronecker_product, KroneckerProduct,
PermutationMatrix, MatrixPermute, Permanent, per)
from .geometry import (Point, Point2D, Point3D, Line, Ray, Segment, Line2D,
Segment2D, Ray2D, Line3D, Segment3D, Ray3D, Plane, Ellipse, Circle,
Polygon, RegularPolygon, Triangle, rad, deg, are_similar, centroid,
convex_hull, idiff, intersection, closest_points, farthest_points,
GeometryError, Curve, Parabola)
from .utilities import (flatten, group, take, subsets, variations,
numbered_symbols, cartes, capture, dict_merge, prefixes, postfixes,
sift, topological_sort, unflatten, has_dups, has_variety, reshape,
rotations, filldedent, lambdify, source,
threaded, xthreaded, public, memoize_property, timed)
from .integrals import (integrate, Integral, line_integrate, mellin_transform,
inverse_mellin_transform, MellinTransform, InverseMellinTransform,
laplace_transform, inverse_laplace_transform, LaplaceTransform,
InverseLaplaceTransform, fourier_transform, inverse_fourier_transform,
FourierTransform, InverseFourierTransform, sine_transform,
inverse_sine_transform, SineTransform, InverseSineTransform,
cosine_transform, inverse_cosine_transform, CosineTransform,
InverseCosineTransform, hankel_transform, inverse_hankel_transform,
HankelTransform, InverseHankelTransform, singularityintegrate)
from .tensor import (IndexedBase, Idx, Indexed, get_contraction_structure,
get_indices, shape, MutableDenseNDimArray, ImmutableDenseNDimArray,
MutableSparseNDimArray, ImmutableSparseNDimArray, NDimArray,
tensorproduct, tensorcontraction, tensordiagonal, derive_by_array,
permutedims, Array, DenseNDimArray, SparseNDimArray)
from .parsing import parse_expr
from .calculus import (euler_equations, singularities, is_increasing,
is_strictly_increasing, is_decreasing, is_strictly_decreasing,
is_monotonic, finite_diff_weights, apply_finite_diff,
differentiate_finite, periodicity, not_empty_in, AccumBounds,
is_convex, stationary_points, minimum, maximum)
from .algebras import Quaternion
from .printing import (pager_print, pretty, pretty_print, pprint,
pprint_use_unicode, pprint_try_use_unicode, latex, print_latex,
multiline_latex, mathml, print_mathml, python, print_python, pycode,
ccode, print_ccode, smtlib_code, glsl_code, print_glsl, cxxcode, fcode,
print_fcode, rcode, print_rcode, jscode, print_jscode, julia_code,
mathematica_code, octave_code, rust_code, print_gtk, preview, srepr,
print_tree, StrPrinter, sstr, sstrrepr, TableForm, dotprint,
maple_code, print_maple_code)
test = lazy_function('sympy.testing.runtests', 'test')
doctest = lazy_function('sympy.testing.runtests', 'doctest')
# This module causes conflicts with other modules:
# from .stats import *
# Adds about .04-.05 seconds of import time
# from combinatorics import *
# This module is slow to import:
#from physics import units
from .plotting import plot, textplot, plot_backends, plot_implicit, plot_parametric
from .interactive import init_session, init_printing, interactive_traversal
evalf._create_evalf_table()
__all__ = [
'__version__',
# sympy.core
'sympify', 'SympifyError', 'cacheit', 'Basic', 'Atom',
'preorder_traversal', 'S', 'Expr', 'AtomicExpr', 'UnevaluatedExpr',
'Symbol', 'Wild', 'Dummy', 'symbols', 'var', 'Number', 'Float',
'Rational', 'Integer', 'NumberSymbol', 'RealNumber', 'igcd', 'ilcm',
'seterr', 'E', 'I', 'nan', 'oo', 'pi', 'zoo', 'AlgebraicNumber', 'comp',
'mod_inverse', 'Pow', 'integer_nthroot', 'integer_log', 'Mul', 'prod',
'Add', 'Mod', 'Rel', 'Eq', 'Ne', 'Lt', 'Le', 'Gt', 'Ge', 'Equality',
'GreaterThan', 'LessThan', 'Unequality', 'StrictGreaterThan',
'StrictLessThan', 'vectorize', 'Lambda', 'WildFunction', 'Derivative',
'diff', 'FunctionClass', 'Function', 'Subs', 'expand', 'PoleError',
'count_ops', 'expand_mul', 'expand_log', 'expand_func', 'expand_trig',
'expand_complex', 'expand_multinomial', 'nfloat', 'expand_power_base',
'expand_power_exp', 'arity', 'PrecisionExhausted', 'N', 'evalf', 'Tuple',
'Dict', 'gcd_terms', 'factor_terms', 'factor_nc', 'evaluate', 'Catalan',
'EulerGamma', 'GoldenRatio', 'TribonacciConstant', 'bottom_up', 'use',
'postorder_traversal', 'default_sort_key', 'ordered',
# sympy.logic
'to_cnf', 'to_dnf', 'to_nnf', 'And', 'Or', 'Not', 'Xor', 'Nand', 'Nor',
'Implies', 'Equivalent', 'ITE', 'POSform', 'SOPform', 'simplify_logic',
'bool_map', 'true', 'false', 'satisfiable',
# sympy.assumptions
'AppliedPredicate', 'Predicate', 'AssumptionsContext', 'assuming', 'Q',
'ask', 'register_handler', 'remove_handler', 'refine',
# sympy.polys
'Poly', 'PurePoly', 'poly_from_expr', 'parallel_poly_from_expr', 'degree',
'total_degree', 'degree_list', 'LC', 'LM', 'LT', 'pdiv', 'prem', 'pquo',
'pexquo', 'div', 'rem', 'quo', 'exquo', 'half_gcdex', 'gcdex', 'invert',
'subresultants', 'resultant', 'discriminant', 'cofactors', 'gcd_list',
'gcd', 'lcm_list', 'lcm', 'terms_gcd', 'trunc', 'monic', 'content',
'primitive', 'compose', 'decompose', 'sturm', 'gff_list', 'gff',
'sqf_norm', 'sqf_part', 'sqf_list', 'sqf', 'factor_list', 'factor',
'intervals', 'refine_root', 'count_roots', 'real_roots', 'nroots',
'ground_roots', 'nth_power_roots_poly', 'cancel', 'reduced', 'groebner',
'is_zero_dimensional', 'GroebnerBasis', 'poly', 'symmetrize', 'horner',
'interpolate', 'rational_interpolate', 'viete', 'together',
'BasePolynomialError', 'ExactQuotientFailed', 'PolynomialDivisionFailed',
'OperationNotSupported', 'HeuristicGCDFailed', 'HomomorphismFailed',
'IsomorphismFailed', 'ExtraneousFactors', 'EvaluationFailed',
'RefinementFailed', 'CoercionFailed', 'NotInvertible', 'NotReversible',
'NotAlgebraic', 'DomainError', 'PolynomialError', 'UnificationFailed',
'GeneratorsError', 'GeneratorsNeeded', 'ComputationFailed',
'UnivariatePolynomialError', 'MultivariatePolynomialError',
'PolificationFailed', 'OptionError', 'FlagError', 'minpoly',
'minimal_polynomial', 'primitive_element', 'field_isomorphism',
'to_number_field', 'isolate', 'round_two', 'prime_decomp',
'prime_valuation', 'itermonomials', 'Monomial', 'lex', 'grlex',
'grevlex', 'ilex', 'igrlex', 'igrevlex', 'CRootOf', 'rootof', 'RootOf',
'ComplexRootOf', 'RootSum', 'roots', 'Domain', 'FiniteField',
'IntegerRing', 'RationalField', 'RealField', 'ComplexField',
'PythonFiniteField', 'GMPYFiniteField', 'PythonIntegerRing',
'GMPYIntegerRing', 'PythonRational', 'GMPYRationalField',
'AlgebraicField', 'PolynomialRing', 'FractionField', 'ExpressionDomain',
'FF_python', 'FF_gmpy', 'ZZ_python', 'ZZ_gmpy', 'QQ_python', 'QQ_gmpy',
'GF', 'FF', 'ZZ', 'QQ', 'ZZ_I', 'QQ_I', 'RR', 'CC', 'EX', 'EXRAW',
'construct_domain', 'swinnerton_dyer_poly', 'cyclotomic_poly',
'symmetric_poly', 'random_poly', 'interpolating_poly', 'jacobi_poly',
'chebyshevt_poly', 'chebyshevu_poly', 'hermite_poly', 'hermite_prob_poly',
'legendre_poly', 'laguerre_poly', 'apart', 'apart_list', 'assemble_partfrac_list',
'Options', 'ring', 'xring', 'vring', 'sring', 'field', 'xfield', 'vfield',
'sfield',
# sympy.series
'Order', 'O', 'limit', 'Limit', 'gruntz', 'series', 'approximants',
'residue', 'EmptySequence', 'SeqPer', 'SeqFormula', 'sequence', 'SeqAdd',
'SeqMul', 'fourier_series', 'fps', 'difference_delta', 'limit_seq',
# sympy.functions
'factorial', 'factorial2', 'rf', 'ff', 'binomial', 'RisingFactorial',
'FallingFactorial', 'subfactorial', 'carmichael', 'fibonacci', 'lucas',
'motzkin', 'tribonacci', 'harmonic', 'bernoulli', 'bell', 'euler', 'catalan',
'genocchi', 'andre', 'partition', 'sqrt', 'root', 'Min', 'Max', 'Id', 'real_root',
'Rem', 'cbrt', 're', 'im', 'sign', 'Abs', 'conjugate', 'arg', 'polar_lift',
'periodic_argument', 'unbranched_argument', 'principal_branch',
'transpose', 'adjoint', 'polarify', 'unpolarify', 'sin', 'cos', 'tan',
'sec', 'csc', 'cot', 'sinc', 'asin', 'acos', 'atan', 'asec', 'acsc',
'acot', 'atan2', 'exp_polar', 'exp', 'ln', 'log', 'LambertW', 'sinh',
'cosh', 'tanh', 'coth', 'sech', 'csch', 'asinh', 'acosh', 'atanh',
'acoth', 'asech', 'acsch', 'floor', 'ceiling', 'frac', 'Piecewise',
'piecewise_fold', 'piecewise_exclusive', 'erf', 'erfc', 'erfi', 'erf2',
'erfinv', 'erfcinv', 'erf2inv', 'Ei', 'expint', 'E1', 'li', 'Li', 'Si',
'Ci', 'Shi', 'Chi', 'fresnels', 'fresnelc', 'gamma', 'lowergamma',
'uppergamma', 'polygamma', 'loggamma', 'digamma', 'trigamma', 'multigamma',
'dirichlet_eta', 'zeta', 'lerchphi', 'polylog', 'stieltjes', 'Eijk', 'LeviCivita',
'KroneckerDelta', 'SingularityFunction', 'DiracDelta', 'Heaviside',
'bspline_basis', 'bspline_basis_set', 'interpolating_spline', 'besselj',
'bessely', 'besseli', 'besselk', 'hankel1', 'hankel2', 'jn', 'yn',
'jn_zeros', 'hn1', 'hn2', 'airyai', 'airybi', 'airyaiprime',
'airybiprime', 'marcumq', 'hyper', 'meijerg', 'appellf1', 'legendre',
'assoc_legendre', 'hermite', 'hermite_prob', 'chebyshevt', 'chebyshevu',
'chebyshevu_root', 'chebyshevt_root', 'laguerre', 'assoc_laguerre',
'gegenbauer', 'jacobi', 'jacobi_normalized', 'Ynm', 'Ynm_c', 'Znm',
'elliptic_k', 'elliptic_f', 'elliptic_e', 'elliptic_pi', 'beta',
'mathieus', 'mathieuc', 'mathieusprime', 'mathieucprime', 'riemann_xi','betainc',
'betainc_regularized',
# sympy.ntheory
'nextprime', 'prevprime', 'prime', 'primepi', 'primerange', 'randprime',
'Sieve', 'sieve', 'primorial', 'cycle_length', 'composite', 'compositepi',
'isprime', 'divisors', 'proper_divisors', 'factorint', 'multiplicity',
'perfect_power', 'pollard_pm1', 'pollard_rho', 'primefactors', 'totient',
'trailing', 'divisor_count', 'proper_divisor_count', 'divisor_sigma',
'factorrat', 'reduced_totient', 'primenu', 'primeomega',
'mersenne_prime_exponent', 'is_perfect', 'is_mersenne_prime',
'is_abundant', 'is_deficient', 'is_amicable', 'abundance', 'npartitions',
'is_primitive_root', 'is_quad_residue', 'legendre_symbol',
'jacobi_symbol', 'n_order', 'sqrt_mod', 'quadratic_residues',
'primitive_root', 'nthroot_mod', 'is_nthpow_residue', 'sqrt_mod_iter',
'mobius', 'discrete_log', 'quadratic_congruence', 'binomial_coefficients',
'binomial_coefficients_list', 'multinomial_coefficients',
'continued_fraction_periodic', 'continued_fraction_iterator',
'continued_fraction_reduce', 'continued_fraction_convergents',
'continued_fraction', 'egyptian_fraction',
# sympy.concrete
'product', 'Product', 'summation', 'Sum',
# sympy.discrete
'fft', 'ifft', 'ntt', 'intt', 'fwht', 'ifwht', 'mobius_transform',
'inverse_mobius_transform', 'convolution', 'covering_product',
'intersecting_product',
# sympy.simplify
'simplify', 'hypersimp', 'hypersimilar', 'logcombine', 'separatevars',
'posify', 'besselsimp', 'kroneckersimp', 'signsimp',
'nsimplify', 'FU', 'fu', 'sqrtdenest', 'cse', 'epath', 'EPath',
'hyperexpand', 'collect', 'rcollect', 'radsimp', 'collect_const',
'fraction', 'numer', 'denom', 'trigsimp', 'exptrigsimp', 'powsimp',
'powdenest', 'combsimp', 'gammasimp', 'ratsimp', 'ratsimpmodprime',
# sympy.sets
'Set', 'Interval', 'Union', 'EmptySet', 'FiniteSet', 'ProductSet',
'Intersection', 'imageset', 'DisjointUnion', 'Complement', 'SymmetricDifference',
'ImageSet', 'Range', 'ComplexRegion', 'Reals', 'Contains', 'ConditionSet',
'Ordinal', 'OmegaPower', 'ord0', 'PowerSet', 'Naturals',
'Naturals0', 'UniversalSet', 'Integers', 'Rationals', 'Complexes',
# sympy.solvers
'solve', 'solve_linear_system', 'solve_linear_system_LU',
'solve_undetermined_coeffs', 'nsolve', 'solve_linear', 'checksol',
'det_quick', 'inv_quick', 'check_assumptions', 'failing_assumptions',
'diophantine', 'rsolve', 'rsolve_poly', 'rsolve_ratio', 'rsolve_hyper',
'checkodesol', 'classify_ode', 'dsolve', 'homogeneous_order',
'solve_poly_system', 'solve_triangulated', 'pde_separate',
'pde_separate_add', 'pde_separate_mul', 'pdsolve', 'classify_pde',
'checkpdesol', 'ode_order', 'reduce_inequalities',
'reduce_abs_inequality', 'reduce_abs_inequalities',
'solve_poly_inequality', 'solve_rational_inequalities',
'solve_univariate_inequality', 'decompogen', 'solveset', 'linsolve',
'linear_eq_to_matrix', 'nonlinsolve', 'substitution',
# sympy.matrices
'ShapeError', 'NonSquareMatrixError', 'GramSchmidt', 'casoratian', 'diag',
'eye', 'hessian', 'jordan_cell', 'list2numpy', 'matrix2numpy',
'matrix_multiply_elementwise', 'ones', 'randMatrix', 'rot_axis1',
'rot_axis2', 'rot_axis3', 'symarray', 'wronskian', 'zeros',
'MutableDenseMatrix', 'DeferredVector', 'MatrixBase', 'Matrix',
'MutableMatrix', 'MutableSparseMatrix', 'banded', 'ImmutableDenseMatrix',
'ImmutableSparseMatrix', 'ImmutableMatrix', 'SparseMatrix', 'MatrixSlice',
'BlockDiagMatrix', 'BlockMatrix', 'FunctionMatrix', 'Identity', 'Inverse',
'MatAdd', 'MatMul', 'MatPow', 'MatrixExpr', 'MatrixSymbol', 'Trace',
'Transpose', 'ZeroMatrix', 'OneMatrix', 'blockcut', 'block_collapse',
'matrix_symbols', 'Adjoint', 'hadamard_product', 'HadamardProduct',
'HadamardPower', 'Determinant', 'det', 'diagonalize_vector', 'DiagMatrix',
'DiagonalMatrix', 'DiagonalOf', 'trace', 'DotProduct',
'kronecker_product', 'KroneckerProduct', 'PermutationMatrix',
'MatrixPermute', 'Permanent', 'per',
# sympy.geometry
'Point', 'Point2D', 'Point3D', 'Line', 'Ray', 'Segment', 'Line2D',
'Segment2D', 'Ray2D', 'Line3D', 'Segment3D', 'Ray3D', 'Plane', 'Ellipse',
'Circle', 'Polygon', 'RegularPolygon', 'Triangle', 'rad', 'deg',
'are_similar', 'centroid', 'convex_hull', 'idiff', 'intersection',
'closest_points', 'farthest_points', 'GeometryError', 'Curve', 'Parabola',
# sympy.utilities
'flatten', 'group', 'take', 'subsets', 'variations', 'numbered_symbols',
'cartes', 'capture', 'dict_merge', 'prefixes', 'postfixes', 'sift',
'topological_sort', 'unflatten', 'has_dups', 'has_variety', 'reshape',
'rotations', 'filldedent', 'lambdify', 'source', 'threaded', 'xthreaded',
'public', 'memoize_property', 'timed',
# sympy.integrals
'integrate', 'Integral', 'line_integrate', 'mellin_transform',
'inverse_mellin_transform', 'MellinTransform', 'InverseMellinTransform',
'laplace_transform', 'inverse_laplace_transform', 'LaplaceTransform',
'InverseLaplaceTransform', 'fourier_transform',
'inverse_fourier_transform', 'FourierTransform',
'InverseFourierTransform', 'sine_transform', 'inverse_sine_transform',
'SineTransform', 'InverseSineTransform', 'cosine_transform',
'inverse_cosine_transform', 'CosineTransform', 'InverseCosineTransform',
'hankel_transform', 'inverse_hankel_transform', 'HankelTransform',
'InverseHankelTransform', 'singularityintegrate',
# sympy.tensor
'IndexedBase', 'Idx', 'Indexed', 'get_contraction_structure',
'get_indices', 'shape', 'MutableDenseNDimArray', 'ImmutableDenseNDimArray',
'MutableSparseNDimArray', 'ImmutableSparseNDimArray', 'NDimArray',
'tensorproduct', 'tensorcontraction', 'tensordiagonal', 'derive_by_array',
'permutedims', 'Array', 'DenseNDimArray', 'SparseNDimArray',
# sympy.parsing
'parse_expr',
# sympy.calculus
'euler_equations', 'singularities', 'is_increasing',
'is_strictly_increasing', 'is_decreasing', 'is_strictly_decreasing',
'is_monotonic', 'finite_diff_weights', 'apply_finite_diff',
'differentiate_finite', 'periodicity', 'not_empty_in',
'AccumBounds', 'is_convex', 'stationary_points', 'minimum', 'maximum',
# sympy.algebras
'Quaternion',
# sympy.printing
'pager_print', 'pretty', 'pretty_print', 'pprint', 'pprint_use_unicode',
'pprint_try_use_unicode', 'latex', 'print_latex', 'multiline_latex',
'mathml', 'print_mathml', 'python', 'print_python', 'pycode', 'ccode',
'print_ccode', 'smtlib_code', 'glsl_code', 'print_glsl', 'cxxcode', 'fcode',
'print_fcode', 'rcode', 'print_rcode', 'jscode', 'print_jscode',
'julia_code', 'mathematica_code', 'octave_code', 'rust_code', 'print_gtk',
'preview', 'srepr', 'print_tree', 'StrPrinter', 'sstr', 'sstrrepr',
'TableForm', 'dotprint', 'maple_code', 'print_maple_code',
# sympy.plotting
'plot', 'textplot', 'plot_backends', 'plot_implicit', 'plot_parametric',
# sympy.interactive
'init_session', 'init_printing', 'interactive_traversal',
# sympy.testing
'test', 'doctest',
]
#===========================================================================#
# #
# XXX: The names below were importable before SymPy 1.6 using #
# #
# from sympy import * #
# #
# This happened implicitly because there was no __all__ defined in this #
# __init__.py file. Not every package is imported. The list matches what #
# would have been imported before. It is possible that these packages will #
# not be imported by a star-import from sympy in future. #
# #
#===========================================================================#
__all__.extend((
'algebras',
'assumptions',
'calculus',
'concrete',
'discrete',
'external',
'functions',
'geometry',
'interactive',
'multipledispatch',
'ntheory',
'parsing',
'plotting',
'polys',
'printing',
'release',
'strategies',
'tensor',
'utilities',
))
|
4a9078c0c58560d49ba4890e426fd53e19faf7b1c512f1db27de9da4c54f3d29 | #
# SymPy documentation build configuration file, created by
# sphinx-quickstart.py on Sat Mar 22 19:34:32 2008.
#
# This file is execfile()d with the current directory set to its containing dir.
#
# The contents of this file are pickled, so don't put values in the namespace
# that aren't pickleable (module imports are okay, they're removed automatically).
#
# All configuration values have a default value; values that are commented out
# serve to show the default value.
import sys
import inspect
import os
import subprocess
from datetime import datetime
# Make sure we import sympy from git
sys.path.insert(0, os.path.abspath('../..'))
import sympy
# If your extensions are in another directory, add it here.
sys.path = ['ext'] + sys.path
# General configuration
# ---------------------
# Add any Sphinx extension module names here, as strings. They can be extensions
# coming with Sphinx (named 'sphinx.addons.*') or your custom ones.
extensions = ['sphinx.ext.autodoc', 'sphinx.ext.linkcode',
'sphinx_math_dollar', 'sphinx.ext.mathjax', 'numpydoc',
'sphinx_reredirects', 'sphinx_copybutton',
'sphinx.ext.graphviz', 'matplotlib.sphinxext.plot_directive',
'myst_parser', 'convert-svg-to-pdf', 'sphinx.ext.intersphinx'
]
# Add redirects here. This should be done whenever a page that is in the
# existing release docs is moved somewhere else so that the URLs don't break.
# The format is
# "page/path/without/extension": "../relative_path_with.html"
# Note that the html path is relative to the redirected page. Always test the
# redirect manually (they aren't tested automatically). See
# https://documatt.gitlab.io/sphinx-reredirects/usage.html
redirects = {
"guides/getting_started/install": "../../install.html",
"documentation-style-guide": "contributing/documentation-style-guide.html",
"gotchas": "explanation/gotchas.html",
"special_topics/classification": "../explanation/classification.html",
"special_topics/finite_diff_derivatives": "../explanation/finite_diff_derivatives.html",
"special_topics/intro": "../explanation/index.html",
"special_topics/index": "../explanation/index.html",
"modules/index": "../reference/index.html",
"modules/physics/index": "../../reference/public/physics/index.html",
"guides/contributing/index": "../../contributing/index.html",
"guides/contributing/dev-setup": "../../contributing/dev-setup.html",
"guides/contributing/dependencies": "../../contributing/dependencies.html",
"guides/contributing/build-docs": "../../contributing/build-docs.html",
"guides/contributing/debug": "../../contributing/debug.html",
"guides/contributing/docstring": "../../contributing/docstring.html",
"guides/documentation-style-guide": "../../contributing/contributing/documentation-style-guide.html",
"guides/make-a-contribution": "../../contributing/make-a-contribution.html",
"guides/contributing/deprecations": "../../contributing/deprecations.html",
"tutorial/preliminaries": "../tutorials/intro-tutorial/preliminaries.html",
"tutorial/intro": "../tutorials/intro-tutorial/intro.html",
"tutorial/gotchas": "../tutorials/intro-tutorial/gotchas.html",
"tutorial/features": "../tutorials/intro-tutorial/features.html",
"tutorial/next": "../tutorials/intro-tutorial/next.html",
"tutorial/basic_operations": "../tutorials/intro-tutorial/basic_operations.html",
"tutorial/printing": "../tutorials/intro-tutorial/printing.html",
"tutorial/simplification": "../tutorials/intro-tutorial/simplification.html",
"tutorial/calculus": "../tutorials/intro-tutorial/calculus.html",
"tutorial/solvers": "../tutorials/intro-tutorial/solvers.html",
"tutorial/matrices": "../tutorials/intro-tutorial/matrices.html",
"tutorial/manipulation": "../tutorials/intro-tutorial/manipulation.html",
}
html_baseurl = "https://docs.sympy.org/latest/"
# Configure Sphinx copybutton (see https://sphinx-copybutton.readthedocs.io/en/latest/use.html)
copybutton_prompt_text = r">>> |\.\.\. |\$ |In \[\d*\]: | {2,5}\.\.\.: | {5,8}: "
copybutton_prompt_is_regexp = True
# Use this to use pngmath instead
#extensions = ['sphinx.ext.autodoc', 'sphinx.ext.viewcode', 'sphinx.ext.pngmath', ]
# Enable warnings for all bad cross references. These are turned into errors
# with the -W flag in the Makefile.
nitpicky = True
nitpick_ignore = [
('py:class', 'sympy.logic.boolalg.Boolean')
]
# To stop docstrings inheritance.
autodoc_inherit_docstrings = False
# See https://www.sympy.org/sphinx-math-dollar/
mathjax3_config = {
"tex": {
"inlineMath": [['\\(', '\\)']],
"displayMath": [["\\[", "\\]"]],
}
}
# Myst configuration (for .md files). See
# https://myst-parser.readthedocs.io/en/latest/syntax/optional.html
myst_enable_extensions = ["dollarmath", "linkify"]
myst_heading_anchors = 6
# myst_update_mathjax = False
# Add any paths that contain templates here, relative to this directory.
templates_path = ['_templates']
# The suffix of source filenames.
source_suffix = '.rst'
# The master toctree document.
master_doc = 'index'
suppress_warnings = ['ref.citation', 'ref.footnote']
# General substitutions.
project = 'SymPy'
copyright = '{} SymPy Development Team'.format(datetime.utcnow().year)
# The default replacements for |version| and |release|, also used in various
# other places throughout the built documents.
#
# The short X.Y version.
version = sympy.__version__
# The full version, including alpha/beta/rc tags.
release = version
# There are two options for replacing |today|: either, you set today to some
# non-false value, then it is used:
#today = ''
# Else, today_fmt is used as the format for a strftime call.
today_fmt = '%B %d, %Y'
# List of documents that shouldn't be included in the build.
#unused_docs = []
# If true, '()' will be appended to :func: etc. cross-reference text.
#add_function_parentheses = True
# If true, the current module name will be prepended to all description
# unit titles (such as .. function::).
#add_module_names = True
# If true, sectionauthor and moduleauthor directives will be shown in the
# output. They are ignored by default.
#show_authors = False
# The name of the Pygments (syntax highlighting) style to use.
sys.path.append(os.path.abspath("./_pygments"))
pygments_style = 'styles.SphinxHighContrastStyle'
pygments_dark_style = 'styles.NativeHighContrastStyle'
# Don't show the source code hyperlinks when using matplotlib plot directive.
plot_html_show_source_link = False
# Options for HTML output
# -----------------------
# The style sheet to use for HTML and HTML Help pages. A file of that name
# must exist either in Sphinx' static/ path, or in one of the custom paths
# given in html_static_path.
# html_style = 'default.css'
# Add any paths that contain custom static files (such as style sheets) here,
# relative to this directory. They are copied after the builtin static files,
# so a file named "default.css" will overwrite the builtin "default.css".
html_static_path = ['_static']
# If not '', a 'Last updated on:' timestamp is inserted at every page bottom,
# using the given strftime format.
html_last_updated_fmt = '%b %d, %Y'
# was classic
# html_theme = "classic"
html_theme = "furo"
# Adjust the sidebar so that the entire sidebar is scrollable
html_sidebars = {
"**": [
"sidebar/scroll-start.html",
"sidebar/brand.html",
"sidebar/search.html",
"sidebar/navigation.html",
"sidebar/versions.html",
"sidebar/scroll-end.html",
],
}
common_theme_variables = {
# Main "SymPy green" colors. Many things uses these colors.
"color-brand-primary": "#52833A",
"color-brand-content": "#307748",
# The left sidebar.
"color-sidebar-background": "#3B5526",
"color-sidebar-background-border": "var(--color-background-primary)",
"color-sidebar-link-text": "#FFFFFF",
"color-sidebar-brand-text": "var(--color-sidebar-link-text--top-level)",
"color-sidebar-link-text--top-level": "#FFFFFF",
"color-sidebar-item-background--hover": "var(--color-brand-primary)",
"color-sidebar-item-expander-background--hover": "var(--color-brand-primary)",
"color-link-underline--hover": "var(--color-link)",
"color-api-keyword": "#000000bd",
"color-api-name": "var(--color-brand-content)",
"color-api-pre-name": "var(--color-brand-content)",
"api-font-size": "var(--font-size--normal)",
"color-foreground-secondary": "#53555B",
# TODO: Add the other types of admonitions here if anyone uses them.
"color-admonition-title-background--seealso": "#CCCCCC",
"color-admonition-title--seealso": "black",
"color-admonition-title-background--note": "#CCCCCC",
"color-admonition-title--note": "black",
"color-admonition-title-background--warning": "var(--color-problematic)",
"color-admonition-title--warning": "white",
"admonition-font-size": "var(--font-size--normal)",
"admonition-title-font-size": "var(--font-size--normal)",
# Note: this doesn't work. If we want to change this, we have to set
# it as the .highlight background in custom.css.
"color-code-background": "hsl(80deg 100% 95%)",
"code-font-size": "var(--font-size--small)",
"font-stack--monospace": 'DejaVu Sans Mono,"SFMono-Regular",Menlo,Consolas,Monaco,Liberation Mono,Lucida Console,monospace;'
}
html_theme_options = {
"light_css_variables": common_theme_variables,
# The dark variables automatically inherit values from the light variables
"dark_css_variables": {
**common_theme_variables,
"color-brand-primary": "#33CB33",
"color-brand-content": "#1DBD1D",
"color-api-keyword": "#FFFFFFbd",
"color-api-overall": "#FFFFFF90",
"color-api-paren": "#FFFFFF90",
"color-sidebar-item-background--hover": "#52833A",
"color-sidebar-item-expander-background--hover": "#52833A",
# This is the color of the text in the right sidebar
"color-foreground-secondary": "#9DA1AC",
"color-admonition-title-background--seealso": "#555555",
"color-admonition-title-background--note": "#555555",
"color-problematic": "#B30000",
},
# See https://pradyunsg.me/furo/customisation/footer/
"footer_icons": [
{
"name": "GitHub",
"url": "https://github.com/sympy/sympy",
"html": """
<svg stroke="currentColor" fill="currentColor" stroke-width="0" viewBox="0 0 16 16">
<path fill-rule="evenodd" d="M8 0C3.58 0 0 3.58 0 8c0 3.54 2.29 6.53 5.47 7.59.4.07.55-.17.55-.38 0-.19-.01-.82-.01-1.49-2.01.37-2.53-.49-2.69-.94-.09-.23-.48-.94-.82-1.13-.28-.15-.68-.52-.01-.53.63-.01 1.08.58 1.23.82.72 1.21 1.87.87 2.33.66.07-.52.28-.87.51-1.07-1.78-.2-3.64-.89-3.64-3.95 0-.87.31-1.59.82-2.15-.08-.2-.36-1.02.08-2.12 0 0 .67-.21 2.2.82.64-.18 1.32-.27 2-.27.68 0 1.36.09 2 .27 1.53-1.04 2.2-.82 2.2-.82.44 1.1.16 1.92.08 2.12.51.56.82 1.27.82 2.15 0 3.07-1.87 3.75-3.65 3.95.29.25.54.73.54 1.48 0 1.07-.01 1.93-.01 2.2 0 .21.15.46.55.38A8.013 8.013 0 0 0 16 8c0-4.42-3.58-8-8-8z"></path>
</svg>
""",
"class": "",
},
],
}
# Add a header for PR preview builds. See the Circle CI configuration.
if os.environ.get("CIRCLECI") == "true":
PR_NUMBER = os.environ.get('CIRCLE_PR_NUMBER')
SHA1 = os.environ.get('CIRCLE_SHA1')
html_theme_options['announcement'] = f"""This is a preview build from
SymPy pull request <a href="https://github.com/sympy/sympy/pull/{PR_NUMBER}">
#{PR_NUMBER}</a>. It was built against <a
href="https://github.com/sympy/sympy/pull/{PR_NUMBER}/commits/{SHA1}">{SHA1[:7]}</a>.
If you aren't looking for a PR preview, go to <a
href="https://docs.sympy.org/">the main SymPy documentation</a>. """
# custom.css contains changes that aren't possible with the above because they
# aren't specified in the Furo theme as CSS variables
html_css_files = ['custom.css']
# html_js_files = []
# If true, SmartyPants will be used to convert quotes and dashes to
# typographically correct entities.
#html_use_smartypants = True
# Content template for the index page.
#html_index = ''
# Custom sidebar templates, maps document names to template names.
#html_sidebars = {}
# Additional templates that should be rendered to pages, maps page names to
# template names.
#html_additional_pages = {}
# If false, no module index is generated.
#html_use_modindex = True
html_domain_indices = ['py-modindex']
# If true, the reST sources are included in the HTML build as _sources/<name>.
# html_copy_source = True
# Output file base name for HTML help builder.
htmlhelp_basename = 'SymPydoc'
language = 'en'
# Options for LaTeX output
# ------------------------
# The paper size ('letter' or 'a4').
#latex_paper_size = 'letter'
# The font size ('10pt', '11pt' or '12pt').
#latex_font_size = '10pt'
# Grouping the document tree into LaTeX files. List of tuples
# (source start file, target name, title, author, document class [howto/manual], toctree_only).
# toctree_only is set to True so that the start file document itself is not included in the
# output, only the documents referenced by it via TOC trees. The extra stuff in the master
# document is intended to show up in the HTML, but doesn't really belong in the LaTeX output.
latex_documents = [('index', 'sympy-%s.tex' % release, 'SymPy Documentation',
'SymPy Development Team', 'manual', True)]
# Additional stuff for the LaTeX preamble.
# Tweaked to work with XeTeX.
latex_elements = {
'babel': '',
'fontenc': r'''
% Define version of \LaTeX that is usable in math mode
\let\OldLaTeX\LaTeX
\renewcommand{\LaTeX}{\text{\OldLaTeX}}
\usepackage{bm}
\usepackage{amssymb}
\usepackage{fontspec}
\usepackage[english]{babel}
\defaultfontfeatures{Mapping=tex-text}
\setmainfont{DejaVu Serif}
\setsansfont{DejaVu Sans}
\setmonofont{DejaVu Sans Mono}
''',
'fontpkg': '',
'inputenc': '',
'utf8extra': '',
'preamble': r'''
'''
}
# SymPy logo on title page
html_logo = '_static/sympylogo.png'
latex_logo = '_static/sympylogo_big.png'
html_favicon = '../_build/logo/sympy-notailtext-favicon.ico'
# Documents to append as an appendix to all manuals.
#latex_appendices = []
# Show page numbers next to internal references
latex_show_pagerefs = True
# We use False otherwise the module index gets generated twice.
latex_use_modindex = False
default_role = 'math'
pngmath_divpng_args = ['-gamma 1.5', '-D 110']
# Note, this is ignored by the mathjax extension
# Any \newcommand should be defined in the file
pngmath_latex_preamble = '\\usepackage{amsmath}\n' \
'\\usepackage{bm}\n' \
'\\usepackage{amsfonts}\n' \
'\\usepackage{amssymb}\n' \
'\\setlength{\\parindent}{0pt}\n'
texinfo_documents = [
(master_doc, 'sympy', 'SymPy Documentation', 'SymPy Development Team',
'SymPy', 'Computer algebra system (CAS) in Python', 'Programming', 1),
]
# Use svg for graphviz
graphviz_output_format = 'svg'
# Enable links to other packages
intersphinx_mapping = {
'matplotlib': ('https://matplotlib.org/stable/', None),
'mpmath': ('https://mpmath.org/doc/current/', None),
"scipy": ("https://docs.scipy.org/doc/scipy/", None),
"numpy": ("https://numpy.org/doc/stable/", None),
}
# Require :external: to reference intersphinx. Prevents accidentally linking
# to something from matplotlib.
intersphinx_disabled_reftypes = ['*']
# Requried for linkcode extension.
# Get commit hash from the external file.
commit_hash_filepath = '../commit_hash.txt'
commit_hash = None
if os.path.isfile(commit_hash_filepath):
with open(commit_hash_filepath) as f:
commit_hash = f.readline()
# Get commit hash from the external file.
if not commit_hash:
try:
commit_hash = subprocess.check_output(['git', 'rev-parse', 'HEAD'])
commit_hash = commit_hash.decode('ascii')
commit_hash = commit_hash.rstrip()
except:
import warnings
warnings.warn(
"Failed to get the git commit hash as the command " \
"'git rev-parse HEAD' is not working. The commit hash will be " \
"assumed as the SymPy master, but the lines may be misleading " \
"or nonexistent as it is not the correct branch the doc is " \
"built with. Check your installation of 'git' if you want to " \
"resolve this warning.")
commit_hash = 'master'
fork = 'sympy'
blobpath = \
"https://github.com/{}/sympy/blob/{}/sympy/".format(fork, commit_hash)
def linkcode_resolve(domain, info):
"""Determine the URL corresponding to Python object."""
if domain != 'py':
return
modname = info['module']
fullname = info['fullname']
submod = sys.modules.get(modname)
if submod is None:
return
obj = submod
for part in fullname.split('.'):
try:
obj = getattr(obj, part)
except Exception:
return
# strip decorators, which would resolve to the source of the decorator
# possibly an upstream bug in getsourcefile, bpo-1764286
try:
unwrap = inspect.unwrap
except AttributeError:
pass
else:
obj = unwrap(obj)
try:
fn = inspect.getsourcefile(obj)
except Exception:
fn = None
if not fn:
return
try:
source, lineno = inspect.getsourcelines(obj)
except Exception:
lineno = None
if lineno:
linespec = "#L%d-L%d" % (lineno, lineno + len(source) - 1)
else:
linespec = ""
fn = os.path.relpath(fn, start=os.path.dirname(sympy.__file__))
return blobpath + fn + linespec
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Subsets and Splits