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f60b416cbc0ca92827d6079803fe0352f73395e27d1e5faab3cee89bac5abc92 | from sympy.core.basic import Basic
from sympy.functions import adjoint, conjugate
from sympy.matrices.expressions.matexpr import MatrixExpr
class Transpose(MatrixExpr):
"""
The transpose of a matrix expression.
This is a symbolic object that simply stores its argument without
evaluating it. To actually compute the transpose, use the ``transpose()``
function, or the ``.T`` attribute of matrices.
Examples
========
>>> from sympy.matrices import MatrixSymbol, Transpose
>>> from sympy.functions import transpose
>>> A = MatrixSymbol('A', 3, 5)
>>> B = MatrixSymbol('B', 5, 3)
>>> Transpose(A)
A.T
>>> A.T == transpose(A) == Transpose(A)
True
>>> Transpose(A*B)
(A*B).T
>>> transpose(A*B)
B.T*A.T
"""
is_Transpose = True
def doit(self, **hints):
arg = self.arg
if hints.get('deep', True) and isinstance(arg, Basic):
arg = arg.doit(**hints)
_eval_transpose = getattr(arg, '_eval_transpose', None)
if _eval_transpose is not None:
result = _eval_transpose()
return result if result is not None else Transpose(arg)
else:
return Transpose(arg)
@property
def arg(self):
return self.args[0]
@property
def shape(self):
return self.arg.shape[::-1]
def _entry(self, i, j, expand=False, **kwargs):
return self.arg._entry(j, i, expand=expand, **kwargs)
def _eval_adjoint(self):
return conjugate(self.arg)
def _eval_conjugate(self):
return adjoint(self.arg)
def _eval_transpose(self):
return self.arg
def _eval_trace(self):
from .trace import Trace
return Trace(self.arg) # Trace(X.T) => Trace(X)
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import det
return det(self.arg)
def _eval_derivative(self, x):
# x is a scalar:
return self.arg._eval_derivative(x)
def _eval_derivative_matrix_lines(self, x):
lines = self.args[0]._eval_derivative_matrix_lines(x)
return [i.transpose() for i in lines]
def transpose(expr):
"""Matrix transpose"""
return Transpose(expr).doit(deep=False)
from sympy.assumptions.ask import ask, Q
from sympy.assumptions.refine import handlers_dict
def refine_Transpose(expr, assumptions):
"""
>>> from sympy import MatrixSymbol, Q, assuming, refine
>>> X = MatrixSymbol('X', 2, 2)
>>> X.T
X.T
>>> with assuming(Q.symmetric(X)):
... print(refine(X.T))
X
"""
if ask(Q.symmetric(expr), assumptions):
return expr.arg
return expr
handlers_dict['Transpose'] = refine_Transpose
|
d937ac1e46b084d501e8282b04fb2d54382e08e0095de53dc8fb3d22c0df8177 | from sympy.assumptions.ask import ask, Q
from sympy.assumptions.refine import handlers_dict
from sympy.core import Basic, sympify, S
from sympy.core.mul import mul, Mul
from sympy.core.numbers import Number, Integer
from sympy.core.symbol import Dummy
from sympy.functions import adjoint
from sympy.strategies import (rm_id, unpack, typed, flatten, exhaust,
do_one, new)
from sympy.matrices.common import ShapeError, NonInvertibleMatrixError
from sympy.matrices.matrices import MatrixBase
from .inverse import Inverse
from .matexpr import MatrixExpr
from .matpow import MatPow
from .transpose import transpose
from .permutation import PermutationMatrix
from .special import ZeroMatrix, Identity, GenericIdentity, OneMatrix
# XXX: MatMul should perhaps not subclass directly from Mul
class MatMul(MatrixExpr, Mul):
"""
A product of matrix expressions
Examples
========
>>> from sympy import MatMul, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 4)
>>> B = MatrixSymbol('B', 4, 3)
>>> C = MatrixSymbol('C', 3, 6)
>>> MatMul(A, B, C)
A*B*C
"""
is_MatMul = True
identity = GenericIdentity()
def __new__(cls, *args, evaluate=False, check=True, _sympify=True):
if not args:
return cls.identity
# This must be removed aggressively in the constructor to avoid
# TypeErrors from GenericIdentity().shape
args = list(filter(lambda i: cls.identity != i, args))
if _sympify:
args = list(map(sympify, args))
obj = Basic.__new__(cls, *args)
factor, matrices = obj.as_coeff_matrices()
if check:
validate(*matrices)
if not matrices:
# Should it be
#
# return Basic.__neq__(cls, factor, GenericIdentity()) ?
return factor
if evaluate:
return canonicalize(obj)
return obj
@property
def shape(self):
matrices = [arg for arg in self.args if arg.is_Matrix]
return (matrices[0].rows, matrices[-1].cols)
def could_extract_minus_sign(self):
return self.args[0].could_extract_minus_sign()
def _entry(self, i, j, expand=True, **kwargs):
# Avoid cyclic imports
from sympy.concrete.summations import Sum
from sympy.matrices.immutable import ImmutableMatrix
coeff, matrices = self.as_coeff_matrices()
if len(matrices) == 1: # situation like 2*X, matmul is just X
return coeff * matrices[0][i, j]
indices = [None]*(len(matrices) + 1)
ind_ranges = [None]*(len(matrices) - 1)
indices[0] = i
indices[-1] = j
def f():
counter = 1
while True:
yield Dummy("i_%i" % counter)
counter += 1
dummy_generator = kwargs.get("dummy_generator", f())
for i in range(1, len(matrices)):
indices[i] = next(dummy_generator)
for i, arg in enumerate(matrices[:-1]):
ind_ranges[i] = arg.shape[1] - 1
matrices = [arg._entry(indices[i], indices[i+1], dummy_generator=dummy_generator) for i, arg in enumerate(matrices)]
expr_in_sum = Mul.fromiter(matrices)
if any(v.has(ImmutableMatrix) for v in matrices):
expand = True
result = coeff*Sum(
expr_in_sum,
*zip(indices[1:-1], [0]*len(ind_ranges), ind_ranges)
)
# Don't waste time in result.doit() if the sum bounds are symbolic
if not any(isinstance(v, (Integer, int)) for v in ind_ranges):
expand = False
return result.doit() if expand else result
def as_coeff_matrices(self):
scalars = [x for x in self.args if not x.is_Matrix]
matrices = [x for x in self.args if x.is_Matrix]
coeff = Mul(*scalars)
if coeff.is_commutative is False:
raise NotImplementedError("noncommutative scalars in MatMul are not supported.")
return coeff, matrices
def as_coeff_mmul(self):
coeff, matrices = self.as_coeff_matrices()
return coeff, MatMul(*matrices)
def _eval_transpose(self):
"""Transposition of matrix multiplication.
Notes
=====
The following rules are applied.
Transposition for matrix multiplied with another matrix:
`\\left(A B\\right)^{T} = B^{T} A^{T}`
Transposition for matrix multiplied with scalar:
`\\left(c A\\right)^{T} = c A^{T}`
References
==========
.. [1] https://en.wikipedia.org/wiki/Transpose
"""
coeff, matrices = self.as_coeff_matrices()
return MatMul(
coeff, *[transpose(arg) for arg in matrices[::-1]]).doit()
def _eval_adjoint(self):
return MatMul(*[adjoint(arg) for arg in self.args[::-1]]).doit()
def _eval_trace(self):
factor, mmul = self.as_coeff_mmul()
if factor != 1:
from .trace import trace
return factor * trace(mmul.doit())
else:
raise NotImplementedError("Can't simplify any further")
def _eval_determinant(self):
from sympy.matrices.expressions.determinant import Determinant
factor, matrices = self.as_coeff_matrices()
square_matrices = only_squares(*matrices)
return factor**self.rows * Mul(*list(map(Determinant, square_matrices)))
def _eval_inverse(self):
try:
return MatMul(*[
arg.inverse() if isinstance(arg, MatrixExpr) else arg**-1
for arg in self.args[::-1]]).doit()
except ShapeError:
return Inverse(self)
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
# treat scalar*MatrixSymbol or scalar*MatPow separately
expr = canonicalize(MatMul(*args))
return expr
# Needed for partial compatibility with Mul
def args_cnc(self, **kwargs):
coeff_c = [x for x in self.args if x.is_commutative]
coeff_nc = [x for x in self.args if not x.is_commutative]
return [coeff_c, coeff_nc]
def _eval_derivative_matrix_lines(self, x):
from .transpose import Transpose
with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
lines = []
for ind in with_x_ind:
left_args = self.args[:ind]
right_args = self.args[ind+1:]
if right_args:
right_mat = MatMul.fromiter(right_args)
else:
right_mat = Identity(self.shape[1])
if left_args:
left_rev = MatMul.fromiter([Transpose(i).doit() if i.is_Matrix else i for i in reversed(left_args)])
else:
left_rev = Identity(self.shape[0])
d = self.args[ind]._eval_derivative_matrix_lines(x)
for i in d:
i.append_first(left_rev)
i.append_second(right_mat)
lines.append(i)
return lines
mul.register_handlerclass((Mul, MatMul), MatMul)
def validate(*matrices):
""" Checks for valid shapes for args of MatMul """
for i in range(len(matrices)-1):
A, B = matrices[i:i+2]
if A.cols != B.rows:
raise ShapeError("Matrices %s and %s are not aligned"%(A, B))
# Rules
def newmul(*args):
if args[0] == 1:
args = args[1:]
return new(MatMul, *args)
def any_zeros(mul):
if any(arg.is_zero or (arg.is_Matrix and arg.is_ZeroMatrix)
for arg in mul.args):
matrices = [arg for arg in mul.args if arg.is_Matrix]
return ZeroMatrix(matrices[0].rows, matrices[-1].cols)
return mul
def merge_explicit(matmul):
""" Merge explicit MatrixBase arguments
>>> from sympy import MatrixSymbol, Matrix, MatMul, pprint
>>> from sympy.matrices.expressions.matmul import merge_explicit
>>> A = MatrixSymbol('A', 2, 2)
>>> B = Matrix([[1, 1], [1, 1]])
>>> C = Matrix([[1, 2], [3, 4]])
>>> X = MatMul(A, B, C)
>>> pprint(X)
[1 1] [1 2]
A*[ ]*[ ]
[1 1] [3 4]
>>> pprint(merge_explicit(X))
[4 6]
A*[ ]
[4 6]
>>> X = MatMul(B, A, C)
>>> pprint(X)
[1 1] [1 2]
[ ]*A*[ ]
[1 1] [3 4]
>>> pprint(merge_explicit(X))
[1 1] [1 2]
[ ]*A*[ ]
[1 1] [3 4]
"""
if not any(isinstance(arg, MatrixBase) for arg in matmul.args):
return matmul
newargs = []
last = matmul.args[0]
for arg in matmul.args[1:]:
if isinstance(arg, (MatrixBase, Number)) and isinstance(last, (MatrixBase, Number)):
last = last * arg
else:
newargs.append(last)
last = arg
newargs.append(last)
return MatMul(*newargs)
def remove_ids(mul):
""" Remove Identities from a MatMul
This is a modified version of sympy.strategies.rm_id.
This is necesssary because MatMul may contain both MatrixExprs and Exprs
as args.
See Also
========
sympy.strategies.rm_id
"""
# Separate Exprs from MatrixExprs in args
factor, mmul = mul.as_coeff_mmul()
# Apply standard rm_id for MatMuls
result = rm_id(lambda x: x.is_Identity is True)(mmul)
if result != mmul:
return newmul(factor, *result.args) # Recombine and return
else:
return mul
def factor_in_front(mul):
factor, matrices = mul.as_coeff_matrices()
if factor != 1:
return newmul(factor, *matrices)
return mul
def combine_powers(mul):
r"""Combine consecutive powers with the same base into one, e.g.
$$A \times A^2 \Rightarrow A^3$$
This also cancels out the possible matrix inverses using the
knowledgebase of :class:`~.Inverse`, e.g.,
$$ Y \times X \times X^{-1} \Rightarrow Y $$
"""
factor, args = mul.as_coeff_matrices()
new_args = [args[0]]
for B in args[1:]:
A = new_args[-1]
if A.is_square == False or B.is_square == False:
new_args.append(B)
continue
if isinstance(A, MatPow):
A_base, A_exp = A.args
else:
A_base, A_exp = A, S.One
if isinstance(B, MatPow):
B_base, B_exp = B.args
else:
B_base, B_exp = B, S.One
if A_base == B_base:
new_exp = A_exp + B_exp
new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
continue
elif not isinstance(B_base, MatrixBase):
try:
B_base_inv = B_base.inverse()
except NonInvertibleMatrixError:
B_base_inv = None
if B_base_inv is not None and A_base == B_base_inv:
new_exp = A_exp - B_exp
new_args[-1] = MatPow(A_base, new_exp).doit(deep=False)
continue
new_args.append(B)
return newmul(factor, *new_args)
def combine_permutations(mul):
"""Refine products of permutation matrices as the products of cycles.
"""
args = mul.args
l = len(args)
if l < 2:
return mul
result = [args[0]]
for i in range(1, l):
A = result[-1]
B = args[i]
if isinstance(A, PermutationMatrix) and \
isinstance(B, PermutationMatrix):
cycle_1 = A.args[0]
cycle_2 = B.args[0]
result[-1] = PermutationMatrix(cycle_1 * cycle_2)
else:
result.append(B)
return MatMul(*result)
def combine_one_matrices(mul):
"""
Combine products of OneMatrix
e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4)
"""
factor, args = mul.as_coeff_matrices()
new_args = [args[0]]
for B in args[1:]:
A = new_args[-1]
if not isinstance(A, OneMatrix) or not isinstance(B, OneMatrix):
new_args.append(B)
continue
new_args.pop()
new_args.append(OneMatrix(A.shape[0], B.shape[1]))
factor *= A.shape[1]
return newmul(factor, *new_args)
def distribute_monom(mul):
"""
Simplify MatMul expressions but distributing
rational term to MatMul.
e.g. 2*(A+B) -> 2*A + 2*B
"""
args = mul.args
if len(args) == 2:
from .matadd import MatAdd
if args[0].is_MatAdd and args[1].is_Rational:
return MatAdd(*[MatMul(mat, args[1]).doit() for mat in args[0].args])
if args[1].is_MatAdd and args[0].is_Rational:
return MatAdd(*[MatMul(args[0], mat).doit() for mat in args[1].args])
return mul
rules = (
distribute_monom, any_zeros, remove_ids, combine_one_matrices, combine_powers, unpack, rm_id(lambda x: x == 1),
merge_explicit, factor_in_front, flatten, combine_permutations)
canonicalize = exhaust(typed({MatMul: do_one(*rules)}))
def only_squares(*matrices):
"""factor matrices only if they are square"""
if matrices[0].rows != matrices[-1].cols:
raise RuntimeError("Invalid matrices being multiplied")
out = []
start = 0
for i, M in enumerate(matrices):
if M.cols == matrices[start].rows:
out.append(MatMul(*matrices[start:i+1]).doit())
start = i+1
return out
def refine_MatMul(expr, assumptions):
"""
>>> from sympy import MatrixSymbol, Q, assuming, refine
>>> X = MatrixSymbol('X', 2, 2)
>>> expr = X * X.T
>>> print(expr)
X*X.T
>>> with assuming(Q.orthogonal(X)):
... print(refine(expr))
I
"""
newargs = []
exprargs = []
for args in expr.args:
if args.is_Matrix:
exprargs.append(args)
else:
newargs.append(args)
last = exprargs[0]
for arg in exprargs[1:]:
if arg == last.T and ask(Q.orthogonal(arg), assumptions):
last = Identity(arg.shape[0])
elif arg == last.conjugate() and ask(Q.unitary(arg), assumptions):
last = Identity(arg.shape[0])
else:
newargs.append(last)
last = arg
newargs.append(last)
return MatMul(*newargs)
handlers_dict['MatMul'] = refine_MatMul
|
97478e4c44c1a46ed91221f5a7028c184c2d15012fe715f91c5523ed93caeaca | from sympy.matrices.common import NonSquareMatrixError
from .matexpr import MatrixExpr
from .special import Identity
from sympy.core import S
from sympy.core.expr import ExprBuilder
from sympy.core.cache import cacheit
from sympy.core.power import Pow
from sympy.core.sympify import _sympify
from sympy.matrices import MatrixBase
class MatPow(MatrixExpr):
def __new__(cls, base, exp, evaluate=False, **options):
base = _sympify(base)
if not base.is_Matrix:
raise TypeError("MatPow base should be a matrix")
if not base.is_square:
raise NonSquareMatrixError("Power of non-square matrix %s" % base)
exp = _sympify(exp)
obj = super().__new__(cls, base, exp)
if evaluate:
obj = obj.doit(deep=False)
return obj
@property
def base(self):
return self.args[0]
@property
def exp(self):
return self.args[1]
@property
def shape(self):
return self.base.shape
@cacheit
def _get_explicit_matrix(self):
return self.base.as_explicit()**self.exp
def _entry(self, i, j, **kwargs):
from sympy.matrices.expressions import MatMul
A = self.doit()
if isinstance(A, MatPow):
# We still have a MatPow, make an explicit MatMul out of it.
if A.exp.is_Integer and A.exp.is_positive:
A = MatMul(*[A.base for k in range(A.exp)])
elif not self._is_shape_symbolic():
return A._get_explicit_matrix()[i, j]
else:
# Leave the expression unevaluated:
from sympy.matrices.expressions.matexpr import MatrixElement
return MatrixElement(self, i, j)
return A[i, j]
def doit(self, **kwargs):
if kwargs.get('deep', True):
base, exp = [arg.doit(**kwargs) for arg in self.args]
else:
base, exp = self.args
# combine all powers, e.g. (A ** 2) ** 3 -> A ** 6
while isinstance(base, MatPow):
exp *= base.args[1]
base = base.args[0]
if isinstance(base, MatrixBase):
# Delegate
return base ** exp
# Handle simple cases so that _eval_power() in MatrixExpr sub-classes can ignore them
if exp == S.One:
return base
if exp == S.Zero:
return Identity(base.rows)
if exp == S.NegativeOne:
from sympy.matrices.expressions import Inverse
return Inverse(base).doit(**kwargs)
eval_power = getattr(base, '_eval_power', None)
if eval_power is not None:
return eval_power(exp)
return MatPow(base, exp)
def _eval_transpose(self):
base, exp = self.args
return MatPow(base.T, exp)
def _eval_derivative(self, x):
return Pow._eval_derivative(self, x)
def _eval_derivative_matrix_lines(self, x):
from sympy.tensor.array.expressions.array_expressions import ArrayContraction
from ...tensor.array.expressions.array_expressions import ArrayTensorProduct
from .matmul import MatMul
from .inverse import Inverse
exp = self.exp
if self.base.shape == (1, 1) and not exp.has(x):
lr = self.base._eval_derivative_matrix_lines(x)
for i in lr:
subexpr = ExprBuilder(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
Identity(1),
i._lines[0],
exp*self.base**(exp-1),
i._lines[1],
Identity(1),
]
),
(0, 3, 4), (5, 7, 8)
],
validator=ArrayContraction._validate
)
i._first_pointer_parent = subexpr.args[0].args
i._first_pointer_index = 0
i._second_pointer_parent = subexpr.args[0].args
i._second_pointer_index = 4
i._lines = [subexpr]
return lr
if (exp > 0) == True:
newexpr = MatMul.fromiter([self.base for i in range(exp)])
elif (exp == -1) == True:
return Inverse(self.base)._eval_derivative_matrix_lines(x)
elif (exp < 0) == True:
newexpr = MatMul.fromiter([Inverse(self.base) for i in range(-exp)])
elif (exp == 0) == True:
return self.doit()._eval_derivative_matrix_lines(x)
else:
raise NotImplementedError("cannot evaluate %s derived by %s" % (self, x))
return newexpr._eval_derivative_matrix_lines(x)
def _eval_inverse(self):
return MatPow(self.base, -self.exp)
|
0a12d4c0f4ded47cffeda2d96039d83ac39449b19c1aaadd3389c8d02981ceca | from sympy.assumptions.ask import ask, Q
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.sympify import _sympify
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.common import NonInvertibleMatrixError
from .matexpr import MatrixExpr
class ZeroMatrix(MatrixExpr):
"""The Matrix Zero 0 - additive identity
Examples
========
>>> from sympy import MatrixSymbol, ZeroMatrix
>>> A = MatrixSymbol('A', 3, 5)
>>> Z = ZeroMatrix(3, 5)
>>> A + Z
A
>>> Z*A.T
0
"""
is_ZeroMatrix = True
def __new__(cls, m, n):
m, n = _sympify(m), _sympify(n)
cls._check_dim(m)
cls._check_dim(n)
return super().__new__(cls, m, n)
@property
def shape(self):
return (self.args[0], self.args[1])
def _eval_power(self, exp):
# exp = -1, 0, 1 are already handled at this stage
if (exp < 0) == True:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
return self
def _eval_transpose(self):
return ZeroMatrix(self.cols, self.rows)
def _eval_trace(self):
return S.Zero
def _eval_determinant(self):
return S.Zero
def _eval_inverse(self):
raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
return S.Zero
class GenericZeroMatrix(ZeroMatrix):
"""
A zero matrix without a specified shape
This exists primarily so MatAdd() with no arguments can return something
meaningful.
"""
def __new__(cls):
# super(ZeroMatrix, cls) instead of super(GenericZeroMatrix, cls)
# because ZeroMatrix.__new__ doesn't have the same signature
return super(ZeroMatrix, cls).__new__(cls)
@property
def rows(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
@property
def cols(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
@property
def shape(self):
raise TypeError("GenericZeroMatrix does not have a specified shape")
# Avoid Matrix.__eq__ which might call .shape
def __eq__(self, other):
return isinstance(other, GenericZeroMatrix)
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return super().__hash__()
class Identity(MatrixExpr):
"""The Matrix Identity I - multiplicative identity
Examples
========
>>> from sympy.matrices import Identity, MatrixSymbol
>>> A = MatrixSymbol('A', 3, 5)
>>> I = Identity(3)
>>> I*A
A
"""
is_Identity = True
def __new__(cls, n):
n = _sympify(n)
cls._check_dim(n)
return super().__new__(cls, n)
@property
def rows(self):
return self.args[0]
@property
def cols(self):
return self.args[0]
@property
def shape(self):
return (self.args[0], self.args[0])
@property
def is_square(self):
return True
def _eval_transpose(self):
return self
def _eval_trace(self):
return self.rows
def _eval_inverse(self):
return self
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
eq = Eq(i, j)
if eq is S.true:
return S.One
elif eq is S.false:
return S.Zero
return KroneckerDelta(i, j, (0, self.cols-1))
def _eval_determinant(self):
return S.One
def _eval_power(self, exp):
return self
class GenericIdentity(Identity):
"""
An identity matrix without a specified shape
This exists primarily so MatMul() with no arguments can return something
meaningful.
"""
def __new__(cls):
# super(Identity, cls) instead of super(GenericIdentity, cls) because
# Identity.__new__ doesn't have the same signature
return super(Identity, cls).__new__(cls)
@property
def rows(self):
raise TypeError("GenericIdentity does not have a specified shape")
@property
def cols(self):
raise TypeError("GenericIdentity does not have a specified shape")
@property
def shape(self):
raise TypeError("GenericIdentity does not have a specified shape")
# Avoid Matrix.__eq__ which might call .shape
def __eq__(self, other):
return isinstance(other, GenericIdentity)
def __ne__(self, other):
return not (self == other)
def __hash__(self):
return super().__hash__()
class OneMatrix(MatrixExpr):
"""
Matrix whose all entries are ones.
"""
def __new__(cls, m, n, evaluate=False):
m, n = _sympify(m), _sympify(n)
cls._check_dim(m)
cls._check_dim(n)
if evaluate:
condition = Eq(m, 1) & Eq(n, 1)
if condition == True:
return Identity(1)
obj = super().__new__(cls, m, n)
return obj
@property
def shape(self):
return self._args
@property
def is_Identity(self):
return self._is_1x1() == True
def as_explicit(self):
from sympy.matrices.immutable import ImmutableDenseMatrix
return ImmutableDenseMatrix.ones(*self.shape)
def doit(self, **hints):
args = self.args
if hints.get('deep', True):
args = [a.doit(**hints) for a in args]
return self.func(*args, evaluate=True)
def _eval_power(self, exp):
# exp = -1, 0, 1 are already handled at this stage
if self._is_1x1() == True:
return Identity(1)
if (exp < 0) == True:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible")
if ask(Q.integer(exp)):
return self.shape[0] ** (exp - 1) * OneMatrix(*self.shape)
return super()._eval_power(exp)
def _eval_transpose(self):
return OneMatrix(self.cols, self.rows)
def _eval_trace(self):
return S.One*self.rows
def _is_1x1(self):
"""Returns true if the matrix is known to be 1x1"""
shape = self.shape
return Eq(shape[0], 1) & Eq(shape[1], 1)
def _eval_determinant(self):
condition = self._is_1x1()
if condition == True:
return S.One
elif condition == False:
return S.Zero
else:
from sympy.matrices.expressions.determinant import Determinant
return Determinant(self)
def _eval_inverse(self):
condition = self._is_1x1()
if condition == True:
return Identity(1)
elif condition == False:
raise NonInvertibleMatrixError("Matrix det == 0; not invertible.")
else:
from .inverse import Inverse
return Inverse(self)
def conjugate(self):
return self
def _entry(self, i, j, **kwargs):
return S.One
|
d955e247df1a2ae8c4fea709d3dfbdfb051cc96748e400890b5a400405d50b19 | from typing import Tuple as tTuple
from functools import wraps
from sympy.core import S, Integer, Basic, Mul, Add
from sympy.core.assumptions import check_assumptions
from sympy.core.decorators import call_highest_priority
from sympy.core.expr import Expr, ExprBuilder
from sympy.core.logic import FuzzyBool
from sympy.core.symbol import Str, Dummy, symbols, Symbol
from sympy.core.sympify import SympifyError, _sympify
from sympy.external.gmpy import SYMPY_INTS
from sympy.functions import conjugate, adjoint
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.common import NonSquareMatrixError
from sympy.matrices.matrices import MatrixKind, MatrixBase
from sympy.multipledispatch import dispatch
from sympy.simplify import simplify
from sympy.utilities.misc import filldedent
def _sympifyit(arg, retval=None):
# This version of _sympifyit sympifies MutableMatrix objects
def deco(func):
@wraps(func)
def __sympifyit_wrapper(a, b):
try:
b = _sympify(b)
return func(a, b)
except SympifyError:
return retval
return __sympifyit_wrapper
return deco
class MatrixExpr(Expr):
"""Superclass for Matrix Expressions
MatrixExprs represent abstract matrices, linear transformations represented
within a particular basis.
Examples
========
>>> from sympy import MatrixSymbol
>>> A = MatrixSymbol('A', 3, 3)
>>> y = MatrixSymbol('y', 3, 1)
>>> x = (A.T*A).I * A * y
See Also
========
MatrixSymbol, MatAdd, MatMul, Transpose, Inverse
"""
# Should not be considered iterable by the
# sympy.utilities.iterables.iterable function. Subclass that actually are
# iterable (i.e., explicit matrices) should set this to True.
_iterable = False
_op_priority = 11.0
is_Matrix = True # type: bool
is_MatrixExpr = True # type: bool
is_Identity = None # type: FuzzyBool
is_Inverse = False
is_Transpose = False
is_ZeroMatrix = False
is_MatAdd = False
is_MatMul = False
is_commutative = False
is_number = False
is_symbol = False
is_scalar = False
kind: MatrixKind = MatrixKind()
def __new__(cls, *args, **kwargs):
args = map(_sympify, args)
return Basic.__new__(cls, *args, **kwargs)
# The following is adapted from the core Expr object
@property
def shape(self) -> tTuple[Expr, Expr]:
raise NotImplementedError
@property
def _add_handler(self):
return MatAdd
@property
def _mul_handler(self):
return MatMul
def __neg__(self):
return MatMul(S.NegativeOne, self).doit()
def __abs__(self):
raise NotImplementedError
@_sympifyit('other', NotImplemented)
@call_highest_priority('__radd__')
def __add__(self, other):
return MatAdd(self, other, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__add__')
def __radd__(self, other):
return MatAdd(other, self, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rsub__')
def __sub__(self, other):
return MatAdd(self, -other, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__sub__')
def __rsub__(self, other):
return MatAdd(other, -self, check=True).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rmul__')
def __mul__(self, other):
return MatMul(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rmul__')
def __matmul__(self, other):
return MatMul(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__mul__')
def __rmul__(self, other):
return MatMul(other, self).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__mul__')
def __rmatmul__(self, other):
return MatMul(other, self).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rpow__')
def __pow__(self, other):
return MatPow(self, other).doit()
@_sympifyit('other', NotImplemented)
@call_highest_priority('__pow__')
def __rpow__(self, other):
raise NotImplementedError("Matrix Power not defined")
@_sympifyit('other', NotImplemented)
@call_highest_priority('__rtruediv__')
def __truediv__(self, other):
return self * other**S.NegativeOne
@_sympifyit('other', NotImplemented)
@call_highest_priority('__truediv__')
def __rtruediv__(self, other):
raise NotImplementedError()
#return MatMul(other, Pow(self, S.NegativeOne))
@property
def rows(self):
return self.shape[0]
@property
def cols(self):
return self.shape[1]
@property
def is_square(self):
return self.rows == self.cols
def _eval_conjugate(self):
from sympy.matrices.expressions.adjoint import Adjoint
return Adjoint(Transpose(self))
def as_real_imag(self, deep=True, **hints):
real = S.Half * (self + self._eval_conjugate())
im = (self - self._eval_conjugate())/(2*S.ImaginaryUnit)
return (real, im)
def _eval_inverse(self):
return Inverse(self)
def _eval_determinant(self):
return Determinant(self)
def _eval_transpose(self):
return Transpose(self)
def _eval_power(self, exp):
"""
Override this in sub-classes to implement simplification of powers. The cases where the exponent
is -1, 0, 1 are already covered in MatPow.doit(), so implementations can exclude these cases.
"""
return MatPow(self, exp)
def _eval_simplify(self, **kwargs):
if self.is_Atom:
return self
else:
return self.func(*[simplify(x, **kwargs) for x in self.args])
def _eval_adjoint(self):
from sympy.matrices.expressions.adjoint import Adjoint
return Adjoint(self)
def _eval_derivative_n_times(self, x, n):
return Basic._eval_derivative_n_times(self, x, n)
def _eval_derivative(self, x):
# `x` is a scalar:
if self.has(x):
# See if there are other methods using it:
return super()._eval_derivative(x)
else:
return ZeroMatrix(*self.shape)
@classmethod
def _check_dim(cls, dim):
"""Helper function to check invalid matrix dimensions"""
ok = check_assumptions(dim, integer=True, nonnegative=True)
if ok is False:
raise ValueError(
"The dimension specification {} should be "
"a nonnegative integer.".format(dim))
def _entry(self, i, j, **kwargs):
raise NotImplementedError(
"Indexing not implemented for %s" % self.__class__.__name__)
def adjoint(self):
return adjoint(self)
def as_coeff_Mul(self, rational=False):
"""Efficiently extract the coefficient of a product. """
return S.One, self
def conjugate(self):
return conjugate(self)
def transpose(self):
from sympy.matrices.expressions.transpose import transpose
return transpose(self)
@property
def T(self):
'''Matrix transposition'''
return self.transpose()
def inverse(self):
if not self.is_square:
raise NonSquareMatrixError('Inverse of non-square matrix')
return self._eval_inverse()
def inv(self):
return self.inverse()
def det(self):
from sympy.matrices.expressions.determinant import det
return det(self)
@property
def I(self):
return self.inverse()
def valid_index(self, i, j):
def is_valid(idx):
return isinstance(idx, (int, Integer, Symbol, Expr))
return (is_valid(i) and is_valid(j) and
(self.rows is None or
(0 <= i) != False and (i < self.rows) != False) and
(0 <= j) != False and (j < self.cols) != False)
def __getitem__(self, key):
if not isinstance(key, tuple) and isinstance(key, slice):
from sympy.matrices.expressions.slice import MatrixSlice
return MatrixSlice(self, key, (0, None, 1))
if isinstance(key, tuple) and len(key) == 2:
i, j = key
if isinstance(i, slice) or isinstance(j, slice):
from sympy.matrices.expressions.slice import MatrixSlice
return MatrixSlice(self, i, j)
i, j = _sympify(i), _sympify(j)
if self.valid_index(i, j) != False:
return self._entry(i, j)
else:
raise IndexError("Invalid indices (%s, %s)" % (i, j))
elif isinstance(key, (SYMPY_INTS, Integer)):
# row-wise decomposition of matrix
rows, cols = self.shape
# allow single indexing if number of columns is known
if not isinstance(cols, Integer):
raise IndexError(filldedent('''
Single indexing is only supported when the number
of columns is known.'''))
key = _sympify(key)
i = key // cols
j = key % cols
if self.valid_index(i, j) != False:
return self._entry(i, j)
else:
raise IndexError("Invalid index %s" % key)
elif isinstance(key, (Symbol, Expr)):
raise IndexError(filldedent('''
Only integers may be used when addressing the matrix
with a single index.'''))
raise IndexError("Invalid index, wanted %s[i,j]" % self)
def _is_shape_symbolic(self) -> bool:
return (not isinstance(self.rows, (SYMPY_INTS, Integer))
or not isinstance(self.cols, (SYMPY_INTS, Integer)))
def as_explicit(self):
"""
Returns a dense Matrix with elements represented explicitly
Returns an object of type ImmutableDenseMatrix.
Examples
========
>>> from sympy import Identity
>>> I = Identity(3)
>>> I
I
>>> I.as_explicit()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
See Also
========
as_mutable: returns mutable Matrix type
"""
if self._is_shape_symbolic():
raise ValueError(
'Matrix with symbolic shape '
'cannot be represented explicitly.')
from sympy.matrices.immutable import ImmutableDenseMatrix
return ImmutableDenseMatrix([[self[i, j]
for j in range(self.cols)]
for i in range(self.rows)])
def as_mutable(self):
"""
Returns a dense, mutable matrix with elements represented explicitly
Examples
========
>>> from sympy import Identity
>>> I = Identity(3)
>>> I
I
>>> I.shape
(3, 3)
>>> I.as_mutable()
Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]])
See Also
========
as_explicit: returns ImmutableDenseMatrix
"""
return self.as_explicit().as_mutable()
def __array__(self):
from numpy import empty
a = empty(self.shape, dtype=object)
for i in range(self.rows):
for j in range(self.cols):
a[i, j] = self[i, j]
return a
def equals(self, other):
"""
Test elementwise equality between matrices, potentially of different
types
>>> from sympy import Identity, eye
>>> Identity(3).equals(eye(3))
True
"""
return self.as_explicit().equals(other)
def canonicalize(self):
return self
def as_coeff_mmul(self):
return 1, MatMul(self)
@staticmethod
def from_index_summation(expr, first_index=None, last_index=None, dimensions=None):
r"""
Parse expression of matrices with explicitly summed indices into a
matrix expression without indices, if possible.
This transformation expressed in mathematical notation:
`\sum_{j=0}^{N-1} A_{i,j} B_{j,k} \Longrightarrow \mathbf{A}\cdot \mathbf{B}`
Optional parameter ``first_index``: specify which free index to use as
the index starting the expression.
Examples
========
>>> from sympy import MatrixSymbol, MatrixExpr, Sum
>>> from sympy.abc import i, j, k, l, N
>>> A = MatrixSymbol("A", N, N)
>>> B = MatrixSymbol("B", N, N)
>>> expr = Sum(A[i, j]*B[j, k], (j, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
A*B
Transposition is detected:
>>> expr = Sum(A[j, i]*B[j, k], (j, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
A.T*B
Detect the trace:
>>> expr = Sum(A[i, i], (i, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
Trace(A)
More complicated expressions:
>>> expr = Sum(A[i, j]*B[k, j]*A[l, k], (j, 0, N-1), (k, 0, N-1))
>>> MatrixExpr.from_index_summation(expr)
A*B.T*A.T
"""
from sympy.tensor.array.expressions.conv_indexed_to_array import convert_indexed_to_array
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
first_indices = []
if first_index is not None:
first_indices.append(first_index)
if last_index is not None:
first_indices.append(last_index)
arr = convert_indexed_to_array(expr, first_indices=first_indices)
return convert_array_to_matrix(arr)
def applyfunc(self, func):
from .applyfunc import ElementwiseApplyFunction
return ElementwiseApplyFunction(func, self)
@dispatch(MatrixExpr, Expr)
def _eval_is_eq(lhs, rhs): # noqa:F811
return False
@dispatch(MatrixExpr, MatrixExpr) # type: ignore
def _eval_is_eq(lhs, rhs): # noqa:F811
if lhs.shape != rhs.shape:
return False
if (lhs - rhs).is_ZeroMatrix:
return True
def get_postprocessor(cls):
def _postprocessor(expr):
# To avoid circular imports, we can't have MatMul/MatAdd on the top level
mat_class = {Mul: MatMul, Add: MatAdd}[cls]
nonmatrices = []
matrices = []
for term in expr.args:
if isinstance(term, MatrixExpr):
matrices.append(term)
else:
nonmatrices.append(term)
if not matrices:
return cls._from_args(nonmatrices)
if nonmatrices:
if cls == Mul:
for i in range(len(matrices)):
if not matrices[i].is_MatrixExpr:
# If one of the matrices explicit, absorb the scalar into it
# (doit will combine all explicit matrices into one, so it
# doesn't matter which)
matrices[i] = matrices[i].__mul__(cls._from_args(nonmatrices))
nonmatrices = []
break
else:
# Maintain the ability to create Add(scalar, matrix) without
# raising an exception. That way different algorithms can
# replace matrix expressions with non-commutative symbols to
# manipulate them like non-commutative scalars.
return cls._from_args(nonmatrices + [mat_class(*matrices).doit(deep=False)])
if mat_class == MatAdd:
return mat_class(*matrices).doit(deep=False)
return mat_class(cls._from_args(nonmatrices), *matrices).doit(deep=False)
return _postprocessor
Basic._constructor_postprocessor_mapping[MatrixExpr] = {
"Mul": [get_postprocessor(Mul)],
"Add": [get_postprocessor(Add)],
}
def _matrix_derivative(expr, x, old_algorithm=False):
if isinstance(expr, MatrixBase) or isinstance(x, MatrixBase):
# Do not use array expressions for explicit matrices:
old_algorithm = True
if old_algorithm:
return _matrix_derivative_old_algorithm(expr, x)
from sympy.tensor.array.expressions.conv_matrix_to_array import convert_matrix_to_array
from sympy.tensor.array.expressions.arrayexpr_derivatives import array_derive
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
array_expr = convert_matrix_to_array(expr)
diff_array_expr = array_derive(array_expr, x)
diff_matrix_expr = convert_array_to_matrix(diff_array_expr)
return diff_matrix_expr
def _matrix_derivative_old_algorithm(expr, x):
from sympy.tensor.array.array_derivatives import ArrayDerivative
lines = expr._eval_derivative_matrix_lines(x)
parts = [i.build() for i in lines]
from sympy.tensor.array.expressions.conv_array_to_matrix import convert_array_to_matrix
parts = [[convert_array_to_matrix(j) for j in i] for i in parts]
def _get_shape(elem):
if isinstance(elem, MatrixExpr):
return elem.shape
return 1, 1
def get_rank(parts):
return sum([j not in (1, None) for i in parts for j in _get_shape(i)])
ranks = [get_rank(i) for i in parts]
rank = ranks[0]
def contract_one_dims(parts):
if len(parts) == 1:
return parts[0]
else:
p1, p2 = parts[:2]
if p2.is_Matrix:
p2 = p2.T
if p1 == Identity(1):
pbase = p2
elif p2 == Identity(1):
pbase = p1
else:
pbase = p1*p2
if len(parts) == 2:
return pbase
else: # len(parts) > 2
if pbase.is_Matrix:
raise ValueError("")
return pbase*Mul.fromiter(parts[2:])
if rank <= 2:
return Add.fromiter([contract_one_dims(i) for i in parts])
return ArrayDerivative(expr, x)
class MatrixElement(Expr):
parent = property(lambda self: self.args[0])
i = property(lambda self: self.args[1])
j = property(lambda self: self.args[2])
_diff_wrt = True
is_symbol = True
is_commutative = True
def __new__(cls, name, n, m):
n, m = map(_sympify, (n, m))
from sympy.matrices.matrices import MatrixBase
if isinstance(name, (MatrixBase,)):
if n.is_Integer and m.is_Integer:
return name[n, m]
if isinstance(name, str):
name = Symbol(name)
else:
name = _sympify(name)
if not isinstance(name.kind, MatrixKind):
raise TypeError("First argument of MatrixElement should be a matrix")
obj = Expr.__new__(cls, name, n, m)
return obj
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
return args[0][args[1], args[2]]
@property
def indices(self):
return self.args[1:]
def _eval_derivative(self, v):
if not isinstance(v, MatrixElement):
from sympy.matrices.matrices import MatrixBase
if isinstance(self.parent, MatrixBase):
return self.parent.diff(v)[self.i, self.j]
return S.Zero
M = self.args[0]
m, n = self.parent.shape
if M == v.args[0]:
return KroneckerDelta(self.args[1], v.args[1], (0, m-1)) * \
KroneckerDelta(self.args[2], v.args[2], (0, n-1))
if isinstance(M, Inverse):
from sympy.concrete.summations import Sum
i, j = self.args[1:]
i1, i2 = symbols("z1, z2", cls=Dummy)
Y = M.args[0]
r1, r2 = Y.shape
return -Sum(M[i, i1]*Y[i1, i2].diff(v)*M[i2, j], (i1, 0, r1-1), (i2, 0, r2-1))
if self.has(v.args[0]):
return None
return S.Zero
class MatrixSymbol(MatrixExpr):
"""Symbolic representation of a Matrix object
Creates a SymPy Symbol to represent a Matrix. This matrix has a shape and
can be included in Matrix Expressions
Examples
========
>>> from sympy import MatrixSymbol, Identity
>>> A = MatrixSymbol('A', 3, 4) # A 3 by 4 Matrix
>>> B = MatrixSymbol('B', 4, 3) # A 4 by 3 Matrix
>>> A.shape
(3, 4)
>>> 2*A*B + Identity(3)
I + 2*A*B
"""
is_commutative = False
is_symbol = True
_diff_wrt = True
def __new__(cls, name, n, m):
n, m = _sympify(n), _sympify(m)
cls._check_dim(m)
cls._check_dim(n)
if isinstance(name, str):
name = Str(name)
obj = Basic.__new__(cls, name, n, m)
return obj
@property
def shape(self):
return self.args[1], self.args[2]
@property
def name(self):
return self.args[0].name
def _entry(self, i, j, **kwargs):
return MatrixElement(self, i, j)
@property
def free_symbols(self):
return {self}
def _eval_simplify(self, **kwargs):
return self
def _eval_derivative(self, x):
# x is a scalar:
return ZeroMatrix(self.shape[0], self.shape[1])
def _eval_derivative_matrix_lines(self, x):
if self != x:
first = ZeroMatrix(x.shape[0], self.shape[0]) if self.shape[0] != 1 else S.Zero
second = ZeroMatrix(x.shape[1], self.shape[1]) if self.shape[1] != 1 else S.Zero
return [_LeftRightArgs(
[first, second],
)]
else:
first = Identity(self.shape[0]) if self.shape[0] != 1 else S.One
second = Identity(self.shape[1]) if self.shape[1] != 1 else S.One
return [_LeftRightArgs(
[first, second],
)]
def matrix_symbols(expr):
return [sym for sym in expr.free_symbols if sym.is_Matrix]
class _LeftRightArgs:
r"""
Helper class to compute matrix derivatives.
The logic: when an expression is derived by a matrix `X_{mn}`, two lines of
matrix multiplications are created: the one contracted to `m` (first line),
and the one contracted to `n` (second line).
Transposition flips the side by which new matrices are connected to the
lines.
The trace connects the end of the two lines.
"""
def __init__(self, lines, higher=S.One):
self._lines = [i for i in lines]
self._first_pointer_parent = self._lines
self._first_pointer_index = 0
self._first_line_index = 0
self._second_pointer_parent = self._lines
self._second_pointer_index = 1
self._second_line_index = 1
self.higher = higher
@property
def first_pointer(self):
return self._first_pointer_parent[self._first_pointer_index]
@first_pointer.setter
def first_pointer(self, value):
self._first_pointer_parent[self._first_pointer_index] = value
@property
def second_pointer(self):
return self._second_pointer_parent[self._second_pointer_index]
@second_pointer.setter
def second_pointer(self, value):
self._second_pointer_parent[self._second_pointer_index] = value
def __repr__(self):
built = [self._build(i) for i in self._lines]
return "_LeftRightArgs(lines=%s, higher=%s)" % (
built,
self.higher,
)
def transpose(self):
self._first_pointer_parent, self._second_pointer_parent = self._second_pointer_parent, self._first_pointer_parent
self._first_pointer_index, self._second_pointer_index = self._second_pointer_index, self._first_pointer_index
self._first_line_index, self._second_line_index = self._second_line_index, self._first_line_index
return self
@staticmethod
def _build(expr):
if isinstance(expr, ExprBuilder):
return expr.build()
if isinstance(expr, list):
if len(expr) == 1:
return expr[0]
else:
return expr[0](*[_LeftRightArgs._build(i) for i in expr[1]])
else:
return expr
def build(self):
data = [self._build(i) for i in self._lines]
if self.higher != 1:
data += [self._build(self.higher)]
data = [i for i in data]
return data
def matrix_form(self):
if self.first != 1 and self.higher != 1:
raise ValueError("higher dimensional array cannot be represented")
def _get_shape(elem):
if isinstance(elem, MatrixExpr):
return elem.shape
return (None, None)
if _get_shape(self.first)[1] != _get_shape(self.second)[1]:
# Remove one-dimensional identity matrices:
# (this is needed by `a.diff(a)` where `a` is a vector)
if _get_shape(self.second) == (1, 1):
return self.first*self.second[0, 0]
if _get_shape(self.first) == (1, 1):
return self.first[1, 1]*self.second.T
raise ValueError("incompatible shapes")
if self.first != 1:
return self.first*self.second.T
else:
return self.higher
def rank(self):
"""
Number of dimensions different from trivial (warning: not related to
matrix rank).
"""
rank = 0
if self.first != 1:
rank += sum([i != 1 for i in self.first.shape])
if self.second != 1:
rank += sum([i != 1 for i in self.second.shape])
if self.higher != 1:
rank += 2
return rank
def _multiply_pointer(self, pointer, other):
from ...tensor.array.expressions.array_expressions import ArrayTensorProduct
from ...tensor.array.expressions.array_expressions import ArrayContraction
subexpr = ExprBuilder(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
pointer,
other
]
),
(1, 2)
],
validator=ArrayContraction._validate
)
return subexpr
def append_first(self, other):
self.first_pointer *= other
def append_second(self, other):
self.second_pointer *= other
def _make_matrix(x):
from sympy.matrices.immutable import ImmutableDenseMatrix
if isinstance(x, MatrixExpr):
return x
return ImmutableDenseMatrix([[x]])
from .matmul import MatMul
from .matadd import MatAdd
from .matpow import MatPow
from .transpose import Transpose
from .inverse import Inverse
from .special import ZeroMatrix, Identity
from .determinant import Determinant
|
76f69fb6007d8e0911ffc113ca1cf254bb5fd5926eff0402f996e94d83176461 | from sympy.core.sympify import _sympify
from sympy.matrices.expressions import MatrixExpr
from sympy.core.numbers import I
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
class DFT(MatrixExpr):
r"""
Returns a discrete Fourier transform matrix. The matrix is scaled
with :math:`\frac{1}{\sqrt{n}}` so that it is unitary.
Parameters
==========
n : integer or Symbol
Size of the transform.
Examples
========
>>> from sympy.abc import n
>>> from sympy.matrices.expressions.fourier import DFT
>>> DFT(3)
DFT(3)
>>> DFT(3).as_explicit()
Matrix([
[sqrt(3)/3, sqrt(3)/3, sqrt(3)/3],
[sqrt(3)/3, sqrt(3)*exp(-2*I*pi/3)/3, sqrt(3)*exp(2*I*pi/3)/3],
[sqrt(3)/3, sqrt(3)*exp(2*I*pi/3)/3, sqrt(3)*exp(-2*I*pi/3)/3]])
>>> DFT(n).shape
(n, n)
References
==========
.. [1] https://en.wikipedia.org/wiki/DFT_matrix
"""
def __new__(cls, n):
n = _sympify(n)
cls._check_dim(n)
obj = super().__new__(cls, n)
return obj
n = property(lambda self: self.args[0]) # type: ignore
shape = property(lambda self: (self.n, self.n)) # type: ignore
def _entry(self, i, j, **kwargs):
w = exp(-2*S.Pi*I/self.n)
return w**(i*j) / sqrt(self.n)
def _eval_inverse(self):
return IDFT(self.n)
class IDFT(DFT):
r"""
Returns an inverse discrete Fourier transform matrix. The matrix is scaled
with :math:`\frac{1}{\sqrt{n}}` so that it is unitary.
Parameters
==========
n : integer or Symbol
Size of the transform
Examples
========
>>> from sympy.matrices.expressions.fourier import DFT, IDFT
>>> IDFT(3)
IDFT(3)
>>> IDFT(4)*DFT(4)
I
See Also
========
DFT
"""
def _entry(self, i, j, **kwargs):
w = exp(-2*S.Pi*I/self.n)
return w**(-i*j) / sqrt(self.n)
def _eval_inverse(self):
return DFT(self.n)
|
4bbaf18e4c269f0a6929ae173fd338a7b8ae77599634d17376bcd8d5ad31ec82 | """Implementation of the Kronecker product"""
from sympy.core import Mul, prod, sympify
from sympy.functions import adjoint
from sympy.matrices.common import ShapeError
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.matrices.expressions.transpose import transpose
from sympy.matrices.expressions.special import Identity
from sympy.matrices.matrices import MatrixBase
from sympy.strategies import (
canon, condition, distribute, do_one, exhaust, flatten, typed, unpack)
from sympy.strategies.traverse import bottom_up
from sympy.utilities import sift
from .matadd import MatAdd
from .matmul import MatMul
from .matpow import MatPow
def kronecker_product(*matrices):
"""
The Kronecker product of two or more arguments.
This computes the explicit Kronecker product for subclasses of
``MatrixBase`` i.e. explicit matrices. Otherwise, a symbolic
``KroneckerProduct`` object is returned.
Examples
========
For ``MatrixSymbol`` arguments a ``KroneckerProduct`` object is returned.
Elements of this matrix can be obtained by indexing, or for MatrixSymbols
with known dimension the explicit matrix can be obtained with
``.as_explicit()``
>>> from sympy.matrices import kronecker_product, MatrixSymbol
>>> A = MatrixSymbol('A', 2, 2)
>>> B = MatrixSymbol('B', 2, 2)
>>> kronecker_product(A)
A
>>> kronecker_product(A, B)
KroneckerProduct(A, B)
>>> kronecker_product(A, B)[0, 1]
A[0, 0]*B[0, 1]
>>> kronecker_product(A, B).as_explicit()
Matrix([
[A[0, 0]*B[0, 0], A[0, 0]*B[0, 1], A[0, 1]*B[0, 0], A[0, 1]*B[0, 1]],
[A[0, 0]*B[1, 0], A[0, 0]*B[1, 1], A[0, 1]*B[1, 0], A[0, 1]*B[1, 1]],
[A[1, 0]*B[0, 0], A[1, 0]*B[0, 1], A[1, 1]*B[0, 0], A[1, 1]*B[0, 1]],
[A[1, 0]*B[1, 0], A[1, 0]*B[1, 1], A[1, 1]*B[1, 0], A[1, 1]*B[1, 1]]])
For explicit matrices the Kronecker product is returned as a Matrix
>>> from sympy.matrices import Matrix, kronecker_product
>>> sigma_x = Matrix([
... [0, 1],
... [1, 0]])
...
>>> Isigma_y = Matrix([
... [0, 1],
... [-1, 0]])
...
>>> kronecker_product(sigma_x, Isigma_y)
Matrix([
[ 0, 0, 0, 1],
[ 0, 0, -1, 0],
[ 0, 1, 0, 0],
[-1, 0, 0, 0]])
See Also
========
KroneckerProduct
"""
if not matrices:
raise TypeError("Empty Kronecker product is undefined")
validate(*matrices)
if len(matrices) == 1:
return matrices[0]
else:
return KroneckerProduct(*matrices).doit()
class KroneckerProduct(MatrixExpr):
"""
The Kronecker product of two or more arguments.
The Kronecker product is a non-commutative product of matrices.
Given two matrices of dimension (m, n) and (s, t) it produces a matrix
of dimension (m s, n t).
This is a symbolic object that simply stores its argument without
evaluating it. To actually compute the product, use the function
``kronecker_product()`` or call the ``.doit()`` or ``.as_explicit()``
methods.
>>> from sympy.matrices import KroneckerProduct, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> isinstance(KroneckerProduct(A, B), KroneckerProduct)
True
"""
is_KroneckerProduct = True
def __new__(cls, *args, check=True):
args = list(map(sympify, args))
if all(a.is_Identity for a in args):
ret = Identity(prod(a.rows for a in args))
if all(isinstance(a, MatrixBase) for a in args):
return ret.as_explicit()
else:
return ret
if check:
validate(*args)
return super().__new__(cls, *args)
@property
def shape(self):
rows, cols = self.args[0].shape
for mat in self.args[1:]:
rows *= mat.rows
cols *= mat.cols
return (rows, cols)
def _entry(self, i, j, **kwargs):
result = 1
for mat in reversed(self.args):
i, m = divmod(i, mat.rows)
j, n = divmod(j, mat.cols)
result *= mat[m, n]
return result
def _eval_adjoint(self):
return KroneckerProduct(*list(map(adjoint, self.args))).doit()
def _eval_conjugate(self):
return KroneckerProduct(*[a.conjugate() for a in self.args]).doit()
def _eval_transpose(self):
return KroneckerProduct(*list(map(transpose, self.args))).doit()
def _eval_trace(self):
from .trace import trace
return prod(trace(a) for a in self.args)
def _eval_determinant(self):
from .determinant import det, Determinant
if not all(a.is_square for a in self.args):
return Determinant(self)
m = self.rows
return prod(det(a)**(m/a.rows) for a in self.args)
def _eval_inverse(self):
try:
return KroneckerProduct(*[a.inverse() for a in self.args])
except ShapeError:
from sympy.matrices.expressions.inverse import Inverse
return Inverse(self)
def structurally_equal(self, other):
'''Determine whether two matrices have the same Kronecker product structure
Examples
========
>>> from sympy import KroneckerProduct, MatrixSymbol, symbols
>>> m, n = symbols(r'm, n', integer=True)
>>> A = MatrixSymbol('A', m, m)
>>> B = MatrixSymbol('B', n, n)
>>> C = MatrixSymbol('C', m, m)
>>> D = MatrixSymbol('D', n, n)
>>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(C, D))
True
>>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(D, C))
False
>>> KroneckerProduct(A, B).structurally_equal(C)
False
'''
# Inspired by BlockMatrix
return (isinstance(other, KroneckerProduct)
and self.shape == other.shape
and len(self.args) == len(other.args)
and all(a.shape == b.shape for (a, b) in zip(self.args, other.args)))
def has_matching_shape(self, other):
'''Determine whether two matrices have the appropriate structure to bring matrix
multiplication inside the KroneckerProdut
Examples
========
>>> from sympy import KroneckerProduct, MatrixSymbol, symbols
>>> m, n = symbols(r'm, n', integer=True)
>>> A = MatrixSymbol('A', m, n)
>>> B = MatrixSymbol('B', n, m)
>>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(B, A))
True
>>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(A, B))
False
>>> KroneckerProduct(A, B).has_matching_shape(A)
False
'''
return (isinstance(other, KroneckerProduct)
and self.cols == other.rows
and len(self.args) == len(other.args)
and all(a.cols == b.rows for (a, b) in zip(self.args, other.args)))
def _eval_expand_kroneckerproduct(self, **hints):
return flatten(canon(typed({KroneckerProduct: distribute(KroneckerProduct, MatAdd)}))(self))
def _kronecker_add(self, other):
if self.structurally_equal(other):
return self.__class__(*[a + b for (a, b) in zip(self.args, other.args)])
else:
return self + other
def _kronecker_mul(self, other):
if self.has_matching_shape(other):
return self.__class__(*[a*b for (a, b) in zip(self.args, other.args)])
else:
return self * other
def doit(self, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) for arg in self.args]
else:
args = self.args
return canonicalize(KroneckerProduct(*args))
def validate(*args):
if not all(arg.is_Matrix for arg in args):
raise TypeError("Mix of Matrix and Scalar symbols")
# rules
def extract_commutative(kron):
c_part = []
nc_part = []
for arg in kron.args:
c, nc = arg.args_cnc()
c_part.extend(c)
nc_part.append(Mul._from_args(nc))
c_part = Mul(*c_part)
if c_part != 1:
return c_part*KroneckerProduct(*nc_part)
return kron
def matrix_kronecker_product(*matrices):
"""Compute the Kronecker product of a sequence of SymPy Matrices.
This is the standard Kronecker product of matrices [1].
Parameters
==========
matrices : tuple of MatrixBase instances
The matrices to take the Kronecker product of.
Returns
=======
matrix : MatrixBase
The Kronecker product matrix.
Examples
========
>>> from sympy import Matrix
>>> from sympy.matrices.expressions.kronecker import (
... matrix_kronecker_product)
>>> m1 = Matrix([[1,2],[3,4]])
>>> m2 = Matrix([[1,0],[0,1]])
>>> matrix_kronecker_product(m1, m2)
Matrix([
[1, 0, 2, 0],
[0, 1, 0, 2],
[3, 0, 4, 0],
[0, 3, 0, 4]])
>>> matrix_kronecker_product(m2, m1)
Matrix([
[1, 2, 0, 0],
[3, 4, 0, 0],
[0, 0, 1, 2],
[0, 0, 3, 4]])
References
==========
.. [1] https://en.wikipedia.org/wiki/Kronecker_product
"""
# Make sure we have a sequence of Matrices
if not all(isinstance(m, MatrixBase) for m in matrices):
raise TypeError(
'Sequence of Matrices expected, got: %s' % repr(matrices)
)
# Pull out the first element in the product.
matrix_expansion = matrices[-1]
# Do the kronecker product working from right to left.
for mat in reversed(matrices[:-1]):
rows = mat.rows
cols = mat.cols
# Go through each row appending kronecker product to.
# running matrix_expansion.
for i in range(rows):
start = matrix_expansion*mat[i*cols]
# Go through each column joining each item
for j in range(cols - 1):
start = start.row_join(
matrix_expansion*mat[i*cols + j + 1]
)
# If this is the first element, make it the start of the
# new row.
if i == 0:
next = start
else:
next = next.col_join(start)
matrix_expansion = next
MatrixClass = max(matrices, key=lambda M: M._class_priority).__class__
if isinstance(matrix_expansion, MatrixClass):
return matrix_expansion
else:
return MatrixClass(matrix_expansion)
def explicit_kronecker_product(kron):
# Make sure we have a sequence of Matrices
if not all(isinstance(m, MatrixBase) for m in kron.args):
return kron
return matrix_kronecker_product(*kron.args)
rules = (unpack,
explicit_kronecker_product,
flatten,
extract_commutative)
canonicalize = exhaust(condition(lambda x: isinstance(x, KroneckerProduct),
do_one(*rules)))
def _kronecker_dims_key(expr):
if isinstance(expr, KroneckerProduct):
return tuple(a.shape for a in expr.args)
else:
return (0,)
def kronecker_mat_add(expr):
from functools import reduce
args = sift(expr.args, _kronecker_dims_key)
nonkrons = args.pop((0,), None)
if not args:
return expr
krons = [reduce(lambda x, y: x._kronecker_add(y), group)
for group in args.values()]
if not nonkrons:
return MatAdd(*krons)
else:
return MatAdd(*krons) + nonkrons
def kronecker_mat_mul(expr):
# modified from block matrix code
factor, matrices = expr.as_coeff_matrices()
i = 0
while i < len(matrices) - 1:
A, B = matrices[i:i+2]
if isinstance(A, KroneckerProduct) and isinstance(B, KroneckerProduct):
matrices[i] = A._kronecker_mul(B)
matrices.pop(i+1)
else:
i += 1
return factor*MatMul(*matrices)
def kronecker_mat_pow(expr):
if isinstance(expr.base, KroneckerProduct) and all(a.is_square for a in expr.base.args):
return KroneckerProduct(*[MatPow(a, expr.exp) for a in expr.base.args])
else:
return expr
def combine_kronecker(expr):
"""Combine KronekeckerProduct with expression.
If possible write operations on KroneckerProducts of compatible shapes
as a single KroneckerProduct.
Examples
========
>>> from sympy.matrices.expressions import MatrixSymbol, KroneckerProduct, combine_kronecker
>>> from sympy import symbols
>>> m, n = symbols(r'm, n', integer=True)
>>> A = MatrixSymbol('A', m, n)
>>> B = MatrixSymbol('B', n, m)
>>> combine_kronecker(KroneckerProduct(A, B)*KroneckerProduct(B, A))
KroneckerProduct(A*B, B*A)
>>> combine_kronecker(KroneckerProduct(A, B)+KroneckerProduct(B.T, A.T))
KroneckerProduct(A + B.T, B + A.T)
>>> C = MatrixSymbol('C', n, n)
>>> D = MatrixSymbol('D', m, m)
>>> combine_kronecker(KroneckerProduct(C, D)**m)
KroneckerProduct(C**m, D**m)
"""
def haskron(expr):
return isinstance(expr, MatrixExpr) and expr.has(KroneckerProduct)
rule = exhaust(
bottom_up(exhaust(condition(haskron, typed(
{MatAdd: kronecker_mat_add,
MatMul: kronecker_mat_mul,
MatPow: kronecker_mat_pow})))))
result = rule(expr)
doit = getattr(result, 'doit', None)
if doit is not None:
return doit()
else:
return result
|
36d20501fa6823f8e11257e427a582c5ce2bf6258df1504e4772cbc4a13326c1 | from sympy.core.sympify import _sympify
from sympy.matrices.expressions import MatrixExpr
from sympy.core import S, Eq, Ge
from sympy.core.mul import Mul
from sympy.functions.special.tensor_functions import KroneckerDelta
class DiagonalMatrix(MatrixExpr):
"""DiagonalMatrix(M) will create a matrix expression that
behaves as though all off-diagonal elements,
`M[i, j]` where `i != j`, are zero.
Examples
========
>>> from sympy import MatrixSymbol, DiagonalMatrix, Symbol
>>> n = Symbol('n', integer=True)
>>> m = Symbol('m', integer=True)
>>> D = DiagonalMatrix(MatrixSymbol('x', 2, 3))
>>> D[1, 2]
0
>>> D[1, 1]
x[1, 1]
The length of the diagonal -- the lesser of the two dimensions of `M` --
is accessed through the `diagonal_length` property:
>>> D.diagonal_length
2
>>> DiagonalMatrix(MatrixSymbol('x', n + 1, n)).diagonal_length
n
When one of the dimensions is symbolic the other will be treated as
though it is smaller:
>>> tall = DiagonalMatrix(MatrixSymbol('x', n, 3))
>>> tall.diagonal_length
3
>>> tall[10, 1]
0
When the size of the diagonal is not known, a value of None will
be returned:
>>> DiagonalMatrix(MatrixSymbol('x', n, m)).diagonal_length is None
True
"""
arg = property(lambda self: self.args[0])
shape = property(lambda self: self.arg.shape) # type:ignore
@property
def diagonal_length(self):
r, c = self.shape
if r.is_Integer and c.is_Integer:
m = min(r, c)
elif r.is_Integer and not c.is_Integer:
m = r
elif c.is_Integer and not r.is_Integer:
m = c
elif r == c:
m = r
else:
try:
m = min(r, c)
except TypeError:
m = None
return m
def _entry(self, i, j, **kwargs):
if self.diagonal_length is not None:
if Ge(i, self.diagonal_length) is S.true:
return S.Zero
elif Ge(j, self.diagonal_length) is S.true:
return S.Zero
eq = Eq(i, j)
if eq is S.true:
return self.arg[i, i]
elif eq is S.false:
return S.Zero
return self.arg[i, j]*KroneckerDelta(i, j)
class DiagonalOf(MatrixExpr):
"""DiagonalOf(M) will create a matrix expression that
is equivalent to the diagonal of `M`, represented as
a single column matrix.
Examples
========
>>> from sympy import MatrixSymbol, DiagonalOf, Symbol
>>> n = Symbol('n', integer=True)
>>> m = Symbol('m', integer=True)
>>> x = MatrixSymbol('x', 2, 3)
>>> diag = DiagonalOf(x)
>>> diag.shape
(2, 1)
The diagonal can be addressed like a matrix or vector and will
return the corresponding element of the original matrix:
>>> diag[1, 0] == diag[1] == x[1, 1]
True
The length of the diagonal -- the lesser of the two dimensions of `M` --
is accessed through the `diagonal_length` property:
>>> diag.diagonal_length
2
>>> DiagonalOf(MatrixSymbol('x', n + 1, n)).diagonal_length
n
When only one of the dimensions is symbolic the other will be
treated as though it is smaller:
>>> dtall = DiagonalOf(MatrixSymbol('x', n, 3))
>>> dtall.diagonal_length
3
When the size of the diagonal is not known, a value of None will
be returned:
>>> DiagonalOf(MatrixSymbol('x', n, m)).diagonal_length is None
True
"""
arg = property(lambda self: self.args[0])
@property
def shape(self):
r, c = self.arg.shape
if r.is_Integer and c.is_Integer:
m = min(r, c)
elif r.is_Integer and not c.is_Integer:
m = r
elif c.is_Integer and not r.is_Integer:
m = c
elif r == c:
m = r
else:
try:
m = min(r, c)
except TypeError:
m = None
return m, S.One
@property
def diagonal_length(self):
return self.shape[0]
def _entry(self, i, j, **kwargs):
return self.arg._entry(i, i, **kwargs)
class DiagMatrix(MatrixExpr):
"""
Turn a vector into a diagonal matrix.
"""
def __new__(cls, vector):
vector = _sympify(vector)
obj = MatrixExpr.__new__(cls, vector)
shape = vector.shape
dim = shape[1] if shape[0] == 1 else shape[0]
if vector.shape[0] != 1:
obj._iscolumn = True
else:
obj._iscolumn = False
obj._shape = (dim, dim)
obj._vector = vector
return obj
@property
def shape(self):
return self._shape
def _entry(self, i, j, **kwargs):
if self._iscolumn:
result = self._vector._entry(i, 0, **kwargs)
else:
result = self._vector._entry(0, j, **kwargs)
if i != j:
result *= KroneckerDelta(i, j)
return result
def _eval_transpose(self):
return self
def as_explicit(self):
from sympy.matrices.dense import diag
return diag(*list(self._vector.as_explicit()))
def doit(self, **hints):
from sympy.assumptions import ask, Q
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.transpose import Transpose
from sympy.matrices.dense import eye
from sympy.matrices.matrices import MatrixBase
vector = self._vector
# This accounts for shape (1, 1) and identity matrices, among others:
if ask(Q.diagonal(vector)):
return vector
if isinstance(vector, MatrixBase):
ret = eye(max(vector.shape))
for i in range(ret.shape[0]):
ret[i, i] = vector[i]
return type(vector)(ret)
if vector.is_MatMul:
matrices = [arg for arg in vector.args if arg.is_Matrix]
scalars = [arg for arg in vector.args if arg not in matrices]
if scalars:
return Mul.fromiter(scalars)*DiagMatrix(MatMul.fromiter(matrices).doit()).doit()
if isinstance(vector, Transpose):
vector = vector.arg
return DiagMatrix(vector)
def diagonalize_vector(vector):
return DiagMatrix(vector).doit()
|
5a21b64a1d1b2d7bbb9d9bfc6d28dae27d7598ef8c96507553693a2235eeb052 | from sympy.core.basic import Basic
from sympy.core.expr import Expr, ExprBuilder
from sympy.core.singleton import S
from sympy.core.sorting import default_sort_key
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.matrices.matrices import MatrixBase
from sympy.matrices.common import NonSquareMatrixError
class Trace(Expr):
"""Matrix Trace
Represents the trace of a matrix expression.
Examples
========
>>> from sympy import MatrixSymbol, Trace, eye
>>> A = MatrixSymbol('A', 3, 3)
>>> Trace(A)
Trace(A)
>>> Trace(eye(3))
Trace(Matrix([
[1, 0, 0],
[0, 1, 0],
[0, 0, 1]]))
>>> Trace(eye(3)).simplify()
3
"""
is_Trace = True
is_commutative = True
def __new__(cls, mat):
mat = sympify(mat)
if not mat.is_Matrix:
raise TypeError("input to Trace, %s, is not a matrix" % str(mat))
if not mat.is_square:
raise NonSquareMatrixError("Trace of a non-square matrix")
return Basic.__new__(cls, mat)
def _eval_transpose(self):
return self
def _eval_derivative(self, v):
from sympy.concrete.summations import Sum
from .matexpr import MatrixElement
if isinstance(v, MatrixElement):
return self.rewrite(Sum).diff(v)
expr = self.doit()
if isinstance(expr, Trace):
# Avoid looping infinitely:
raise NotImplementedError
return expr._eval_derivative(v)
def _eval_derivative_matrix_lines(self, x):
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct, ArrayContraction
r = self.args[0]._eval_derivative_matrix_lines(x)
for lr in r:
if lr.higher == 1:
lr.higher = ExprBuilder(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
lr._lines[0],
lr._lines[1],
]
),
(1, 3),
],
validator=ArrayContraction._validate
)
else:
# This is not a matrix line:
lr.higher = ExprBuilder(
ArrayContraction,
[
ExprBuilder(
ArrayTensorProduct,
[
lr._lines[0],
lr._lines[1],
lr.higher,
]
),
(1, 3), (0, 2)
]
)
lr._lines = [S.One, S.One]
lr._first_pointer_parent = lr._lines
lr._second_pointer_parent = lr._lines
lr._first_pointer_index = 0
lr._second_pointer_index = 1
return r
@property
def arg(self):
return self.args[0]
def doit(self, **kwargs):
if kwargs.get('deep', True):
arg = self.arg.doit(**kwargs)
try:
return arg._eval_trace()
except (AttributeError, NotImplementedError):
return Trace(arg)
else:
# _eval_trace would go too deep here
if isinstance(self.arg, MatrixBase):
return trace(self.arg)
else:
return Trace(self.arg)
def as_explicit(self):
return Trace(self.arg.as_explicit()).doit()
def _normalize(self):
# Normalization of trace of matrix products. Use transposition and
# cyclic properties of traces to make sure the arguments of the matrix
# product are sorted and the first argument is not a trasposition.
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.transpose import Transpose
trace_arg = self.arg
if isinstance(trace_arg, MatMul):
def get_arg_key(x):
a = trace_arg.args[x]
if isinstance(a, Transpose):
a = a.arg
return default_sort_key(a)
indmin = min(range(len(trace_arg.args)), key=get_arg_key)
if isinstance(trace_arg.args[indmin], Transpose):
trace_arg = Transpose(trace_arg).doit()
indmin = min(range(len(trace_arg.args)), key=lambda x: default_sort_key(trace_arg.args[x]))
trace_arg = MatMul.fromiter(trace_arg.args[indmin:] + trace_arg.args[:indmin])
return Trace(trace_arg)
return self
def _eval_rewrite_as_Sum(self, expr, **kwargs):
from sympy.concrete.summations import Sum
i = Dummy('i')
return Sum(self.arg[i, i], (i, 0, self.arg.rows-1)).doit()
def trace(expr):
"""Trace of a Matrix. Sum of the diagonal elements.
Examples
========
>>> from sympy import trace, Symbol, MatrixSymbol, eye
>>> n = Symbol('n')
>>> X = MatrixSymbol('X', n, n) # A square matrix
>>> trace(2*X)
2*Trace(X)
>>> trace(eye(3))
3
"""
return Trace(expr).doit()
|
f5b524849c8edd129c1e2769d144f56cbc7ca674b9682067f2fe89620da6ea16 | from sympy.core import Mul, sympify
from sympy.core.add import Add
from sympy.core.expr import ExprBuilder
from sympy.core.sorting import default_sort_key
from sympy.matrices.common import ShapeError
from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.matrices.expressions.special import ZeroMatrix, OneMatrix
from sympy.strategies import (
unpack, flatten, condition, exhaust, rm_id, sort
)
def hadamard_product(*matrices):
"""
Return the elementwise (aka Hadamard) product of matrices.
Examples
========
>>> from sympy.matrices import hadamard_product, MatrixSymbol
>>> A = MatrixSymbol('A', 2, 3)
>>> B = MatrixSymbol('B', 2, 3)
>>> hadamard_product(A)
A
>>> hadamard_product(A, B)
HadamardProduct(A, B)
>>> hadamard_product(A, B)[0, 1]
A[0, 1]*B[0, 1]
"""
if not matrices:
raise TypeError("Empty Hadamard product is undefined")
validate(*matrices)
if len(matrices) == 1:
return matrices[0]
else:
matrices = [i for i in matrices if not i.is_Identity]
return HadamardProduct(*matrices).doit()
class HadamardProduct(MatrixExpr):
"""
Elementwise product of matrix expressions
Examples
========
Hadamard product for matrix symbols:
>>> from sympy.matrices import hadamard_product, HadamardProduct, MatrixSymbol
>>> A = MatrixSymbol('A', 5, 5)
>>> B = MatrixSymbol('B', 5, 5)
>>> isinstance(hadamard_product(A, B), HadamardProduct)
True
Notes
=====
This is a symbolic object that simply stores its argument without
evaluating it. To actually compute the product, use the function
``hadamard_product()`` or ``HadamardProduct.doit``
"""
is_HadamardProduct = True
def __new__(cls, *args, evaluate=False, check=True):
args = list(map(sympify, args))
if check:
validate(*args)
obj = super().__new__(cls, *args)
if evaluate:
obj = obj.doit(deep=False)
return obj
@property
def shape(self):
return self.args[0].shape
def _entry(self, i, j, **kwargs):
return Mul(*[arg._entry(i, j, **kwargs) for arg in self.args])
def _eval_transpose(self):
from sympy.matrices.expressions.transpose import transpose
return HadamardProduct(*list(map(transpose, self.args)))
def doit(self, **ignored):
expr = self.func(*[i.doit(**ignored) for i in self.args])
# Check for explicit matrices:
from sympy.matrices.matrices import MatrixBase
from sympy.matrices.immutable import ImmutableMatrix
explicit = [i for i in expr.args if isinstance(i, MatrixBase)]
if explicit:
remainder = [i for i in expr.args if i not in explicit]
expl_mat = ImmutableMatrix([
Mul.fromiter(i) for i in zip(*explicit)
]).reshape(*self.shape)
expr = HadamardProduct(*([expl_mat] + remainder))
return canonicalize(expr)
def _eval_derivative(self, x):
terms = []
args = list(self.args)
for i in range(len(args)):
factors = args[:i] + [args[i].diff(x)] + args[i+1:]
terms.append(hadamard_product(*factors))
return Add.fromiter(terms)
def _eval_derivative_matrix_lines(self, x):
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.matrices.expressions.matexpr import _make_matrix
with_x_ind = [i for i, arg in enumerate(self.args) if arg.has(x)]
lines = []
for ind in with_x_ind:
left_args = self.args[:ind]
right_args = self.args[ind+1:]
d = self.args[ind]._eval_derivative_matrix_lines(x)
hadam = hadamard_product(*(right_args + left_args))
diagonal = [(0, 2), (3, 4)]
diagonal = [e for j, e in enumerate(diagonal) if self.shape[j] != 1]
for i in d:
l1 = i._lines[i._first_line_index]
l2 = i._lines[i._second_line_index]
subexpr = ExprBuilder(
ArrayDiagonal,
[
ExprBuilder(
ArrayTensorProduct,
[
ExprBuilder(_make_matrix, [l1]),
hadam,
ExprBuilder(_make_matrix, [l2]),
]
),
*diagonal],
)
i._first_pointer_parent = subexpr.args[0].args[0].args
i._first_pointer_index = 0
i._second_pointer_parent = subexpr.args[0].args[2].args
i._second_pointer_index = 0
i._lines = [subexpr]
lines.append(i)
return lines
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))
# TODO Implement algorithm for rewriting Hadamard product as diagonal matrix
# if matmul identy matrix is multiplied.
def canonicalize(x):
"""Canonicalize the Hadamard product ``x`` with mathematical properties.
Examples
========
>>> from sympy.matrices.expressions import MatrixSymbol, HadamardProduct
>>> from sympy.matrices.expressions import OneMatrix, ZeroMatrix
>>> from sympy.matrices.expressions.hadamard import canonicalize
>>> from sympy import init_printing
>>> init_printing(use_unicode=False)
>>> A = MatrixSymbol('A', 2, 2)
>>> B = MatrixSymbol('B', 2, 2)
>>> C = MatrixSymbol('C', 2, 2)
Hadamard product associativity:
>>> X = HadamardProduct(A, HadamardProduct(B, C))
>>> X
A.*(B.*C)
>>> canonicalize(X)
A.*B.*C
Hadamard product commutativity:
>>> X = HadamardProduct(A, B)
>>> Y = HadamardProduct(B, A)
>>> X
A.*B
>>> Y
B.*A
>>> canonicalize(X)
A.*B
>>> canonicalize(Y)
A.*B
Hadamard product identity:
>>> X = HadamardProduct(A, OneMatrix(2, 2))
>>> X
A.*1
>>> canonicalize(X)
A
Absorbing element of Hadamard product:
>>> X = HadamardProduct(A, ZeroMatrix(2, 2))
>>> X
A.*0
>>> canonicalize(X)
0
Rewriting to Hadamard Power
>>> X = HadamardProduct(A, A, A)
>>> X
A.*A.*A
>>> canonicalize(X)
.3
A
Notes
=====
As the Hadamard product is associative, nested products can be flattened.
The Hadamard product is commutative so that factors can be sorted for
canonical form.
A matrix of only ones is an identity for Hadamard product,
so every matrices of only ones can be removed.
Any zero matrix will make the whole product a zero matrix.
Duplicate elements can be collected and rewritten as HadamardPower
References
==========
.. [1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
"""
# Associativity
rule = condition(
lambda x: isinstance(x, HadamardProduct),
flatten
)
fun = exhaust(rule)
x = fun(x)
# Identity
fun = condition(
lambda x: isinstance(x, HadamardProduct),
rm_id(lambda x: isinstance(x, OneMatrix))
)
x = fun(x)
# Absorbing by Zero Matrix
def absorb(x):
if any(isinstance(c, ZeroMatrix) for c in x.args):
return ZeroMatrix(*x.shape)
else:
return x
fun = condition(
lambda x: isinstance(x, HadamardProduct),
absorb
)
x = fun(x)
# Rewriting with HadamardPower
if isinstance(x, HadamardProduct):
from collections import Counter
tally = Counter(x.args)
new_arg = []
for base, exp in tally.items():
if exp == 1:
new_arg.append(base)
else:
new_arg.append(HadamardPower(base, exp))
x = HadamardProduct(*new_arg)
# Commutativity
fun = condition(
lambda x: isinstance(x, HadamardProduct),
sort(default_sort_key)
)
x = fun(x)
# Unpacking
x = unpack(x)
return x
def hadamard_power(base, exp):
base = sympify(base)
exp = sympify(exp)
if exp == 1:
return base
if not base.is_Matrix:
return base**exp
if exp.is_Matrix:
raise ValueError("cannot raise expression to a matrix")
return HadamardPower(base, exp)
class HadamardPower(MatrixExpr):
r"""
Elementwise power of matrix expressions
Parameters
==========
base : scalar or matrix
exp : scalar or matrix
Notes
=====
There are four definitions for the hadamard power which can be used.
Let's consider `A, B` as `(m, n)` matrices, and `a, b` as scalars.
Matrix raised to a scalar exponent:
.. math::
A^{\circ b} = \begin{bmatrix}
A_{0, 0}^b & A_{0, 1}^b & \cdots & A_{0, n-1}^b \\
A_{1, 0}^b & A_{1, 1}^b & \cdots & A_{1, n-1}^b \\
\vdots & \vdots & \ddots & \vdots \\
A_{m-1, 0}^b & A_{m-1, 1}^b & \cdots & A_{m-1, n-1}^b
\end{bmatrix}
Scalar raised to a matrix exponent:
.. math::
a^{\circ B} = \begin{bmatrix}
a^{B_{0, 0}} & a^{B_{0, 1}} & \cdots & a^{B_{0, n-1}} \\
a^{B_{1, 0}} & a^{B_{1, 1}} & \cdots & a^{B_{1, n-1}} \\
\vdots & \vdots & \ddots & \vdots \\
a^{B_{m-1, 0}} & a^{B_{m-1, 1}} & \cdots & a^{B_{m-1, n-1}}
\end{bmatrix}
Matrix raised to a matrix exponent:
.. math::
A^{\circ B} = \begin{bmatrix}
A_{0, 0}^{B_{0, 0}} & A_{0, 1}^{B_{0, 1}} &
\cdots & A_{0, n-1}^{B_{0, n-1}} \\
A_{1, 0}^{B_{1, 0}} & A_{1, 1}^{B_{1, 1}} &
\cdots & A_{1, n-1}^{B_{1, n-1}} \\
\vdots & \vdots &
\ddots & \vdots \\
A_{m-1, 0}^{B_{m-1, 0}} & A_{m-1, 1}^{B_{m-1, 1}} &
\cdots & A_{m-1, n-1}^{B_{m-1, n-1}}
\end{bmatrix}
Scalar raised to a scalar exponent:
.. math::
a^{\circ b} = a^b
"""
def __new__(cls, base, exp):
base = sympify(base)
exp = sympify(exp)
if base.is_scalar and exp.is_scalar:
return base ** exp
if base.is_Matrix and exp.is_Matrix and base.shape != exp.shape:
raise ValueError(
'The shape of the base {} and '
'the shape of the exponent {} do not match.'
.format(base.shape, exp.shape)
)
obj = super().__new__(cls, base, exp)
return obj
@property
def base(self):
return self._args[0]
@property
def exp(self):
return self._args[1]
@property
def shape(self):
if self.base.is_Matrix:
return self.base.shape
return self.exp.shape
def _entry(self, i, j, **kwargs):
base = self.base
exp = self.exp
if base.is_Matrix:
a = base._entry(i, j, **kwargs)
elif base.is_scalar:
a = base
else:
raise ValueError(
'The base {} must be a scalar or a matrix.'.format(base))
if exp.is_Matrix:
b = exp._entry(i, j, **kwargs)
elif exp.is_scalar:
b = exp
else:
raise ValueError(
'The exponent {} must be a scalar or a matrix.'.format(exp))
return a ** b
def _eval_transpose(self):
from sympy.matrices.expressions.transpose import transpose
return HadamardPower(transpose(self.base), self.exp)
def _eval_derivative(self, x):
from sympy.functions.elementary.exponential import log
dexp = self.exp.diff(x)
logbase = self.base.applyfunc(log)
dlbase = logbase.diff(x)
return hadamard_product(
dexp*logbase + self.exp*dlbase,
self
)
def _eval_derivative_matrix_lines(self, x):
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.tensor.array.expressions.array_expressions import ArrayDiagonal
from sympy.matrices.expressions.matexpr import _make_matrix
lr = self.base._eval_derivative_matrix_lines(x)
for i in lr:
diagonal = [(1, 2), (3, 4)]
diagonal = [e for j, e in enumerate(diagonal) if self.base.shape[j] != 1]
l1 = i._lines[i._first_line_index]
l2 = i._lines[i._second_line_index]
subexpr = ExprBuilder(
ArrayDiagonal,
[
ExprBuilder(
ArrayTensorProduct,
[
ExprBuilder(_make_matrix, [l1]),
self.exp*hadamard_power(self.base, self.exp-1),
ExprBuilder(_make_matrix, [l2]),
]
),
*diagonal],
validator=ArrayDiagonal._validate
)
i._first_pointer_parent = subexpr.args[0].args[0].args
i._first_pointer_index = 0
i._first_line_index = 0
i._second_pointer_parent = subexpr.args[0].args[2].args
i._second_pointer_index = 0
i._second_line_index = 0
i._lines = [subexpr]
return lr
|
0e0261b29ad66c54b2dede932949d770b117fed0decabf74c2c0ab31a9a0ad9f | from sympy.assumptions.ask import (Q, ask)
from sympy.core import Basic, Add, Mul, S
from sympy.core.sympify import _sympify
from sympy.functions.elementary.complexes import re, im
from sympy.strategies import typed, exhaust, condition, do_one, unpack
from sympy.strategies.traverse import bottom_up
from sympy.utilities.iterables import is_sequence, sift
from sympy.utilities.misc import filldedent
from sympy.matrices import Matrix, ShapeError
from sympy.matrices.common import NonInvertibleMatrixError
from sympy.matrices.expressions.determinant import det, Determinant
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.matexpr import MatrixExpr, MatrixElement
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions.slice import MatrixSlice
from sympy.matrices.expressions.special import ZeroMatrix, Identity
from sympy.matrices.expressions.trace import trace
from sympy.matrices.expressions.transpose import Transpose, transpose
class BlockMatrix(MatrixExpr):
"""A BlockMatrix is a Matrix comprised of other matrices.
The submatrices are stored in a SymPy Matrix object but accessed as part of
a Matrix Expression
>>> from sympy import (MatrixSymbol, BlockMatrix, symbols,
... Identity, ZeroMatrix, block_collapse)
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m, m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])
>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])
>>> print(block_collapse(C*B))
Matrix([[X, Z + Z*Y]])
Some matrices might be comprised of rows of blocks with
the matrices in each row having the same height and the
rows all having the same total number of columns but
not having the same number of columns for each matrix
in each row. In this case, the matrix is not a block
matrix and should be instantiated by Matrix.
>>> from sympy import ones, Matrix
>>> dat = [
... [ones(3,2), ones(3,3)*2],
... [ones(2,3)*3, ones(2,2)*4]]
...
>>> BlockMatrix(dat)
Traceback (most recent call last):
...
ValueError:
Although this matrix is comprised of blocks, the blocks do not fill
the matrix in a size-symmetric fashion. To create a full matrix from
these arguments, pass them directly to Matrix.
>>> Matrix(dat)
Matrix([
[1, 1, 2, 2, 2],
[1, 1, 2, 2, 2],
[1, 1, 2, 2, 2],
[3, 3, 3, 4, 4],
[3, 3, 3, 4, 4]])
See Also
========
sympy.matrices.matrices.MatrixBase.irregular
"""
def __new__(cls, *args, **kwargs):
from sympy.matrices.immutable import ImmutableDenseMatrix
isMat = lambda i: getattr(i, 'is_Matrix', False)
if len(args) != 1 or \
not is_sequence(args[0]) or \
len({isMat(r) for r in args[0]}) != 1:
raise ValueError(filldedent('''
expecting a sequence of 1 or more rows
containing Matrices.'''))
rows = args[0] if args else []
if not isMat(rows):
if rows and isMat(rows[0]):
rows = [rows] # rows is not list of lists or []
# regularity check
# same number of matrices in each row
blocky = ok = len({len(r) for r in rows}) == 1
if ok:
# same number of rows for each matrix in a row
for r in rows:
ok = len({i.rows for i in r}) == 1
if not ok:
break
blocky = ok
if ok:
# same number of cols for each matrix in each col
for c in range(len(rows[0])):
ok = len({rows[i][c].cols
for i in range(len(rows))}) == 1
if not ok:
break
if not ok:
# same total cols in each row
ok = len({
sum([i.cols for i in r]) for r in rows}) == 1
if blocky and ok:
raise ValueError(filldedent('''
Although this matrix is comprised of blocks,
the blocks do not fill the matrix in a
size-symmetric fashion. To create a full matrix
from these arguments, pass them directly to
Matrix.'''))
raise ValueError(filldedent('''
When there are not the same number of rows in each
row's matrices or there are not the same number of
total columns in each row, the matrix is not a
block matrix. If this matrix is known to consist of
blocks fully filling a 2-D space then see
Matrix.irregular.'''))
mat = ImmutableDenseMatrix(rows, evaluate=False)
obj = Basic.__new__(cls, mat)
return obj
@property
def shape(self):
numrows = numcols = 0
M = self.blocks
for i in range(M.shape[0]):
numrows += M[i, 0].shape[0]
for i in range(M.shape[1]):
numcols += M[0, i].shape[1]
return (numrows, numcols)
@property
def blockshape(self):
return self.blocks.shape
@property
def blocks(self):
return self.args[0]
@property
def rowblocksizes(self):
return [self.blocks[i, 0].rows for i in range(self.blockshape[0])]
@property
def colblocksizes(self):
return [self.blocks[0, i].cols for i in range(self.blockshape[1])]
def structurally_equal(self, other):
return (isinstance(other, BlockMatrix)
and self.shape == other.shape
and self.blockshape == other.blockshape
and self.rowblocksizes == other.rowblocksizes
and self.colblocksizes == other.colblocksizes)
def _blockmul(self, other):
if (isinstance(other, BlockMatrix) and
self.colblocksizes == other.rowblocksizes):
return BlockMatrix(self.blocks*other.blocks)
return self * other
def _blockadd(self, other):
if (isinstance(other, BlockMatrix)
and self.structurally_equal(other)):
return BlockMatrix(self.blocks + other.blocks)
return self + other
def _eval_transpose(self):
# Flip all the individual matrices
matrices = [transpose(matrix) for matrix in self.blocks]
# Make a copy
M = Matrix(self.blockshape[0], self.blockshape[1], matrices)
# Transpose the block structure
M = M.transpose()
return BlockMatrix(M)
def _eval_trace(self):
if self.rowblocksizes == self.colblocksizes:
return Add(*[trace(self.blocks[i, i])
for i in range(self.blockshape[0])])
raise NotImplementedError(
"Can't perform trace of irregular blockshape")
def _eval_determinant(self):
if self.blockshape == (1, 1):
return det(self.blocks[0, 0])
if self.blockshape == (2, 2):
[[A, B],
[C, D]] = self.blocks.tolist()
if ask(Q.invertible(A)):
return det(A)*det(D - C*A.I*B)
elif ask(Q.invertible(D)):
return det(D)*det(A - B*D.I*C)
return Determinant(self)
def as_real_imag(self):
real_matrices = [re(matrix) for matrix in self.blocks]
real_matrices = Matrix(self.blockshape[0], self.blockshape[1], real_matrices)
im_matrices = [im(matrix) for matrix in self.blocks]
im_matrices = Matrix(self.blockshape[0], self.blockshape[1], im_matrices)
return (real_matrices, im_matrices)
def transpose(self):
"""Return transpose of matrix.
Examples
========
>>> from sympy import MatrixSymbol, BlockMatrix, ZeroMatrix
>>> from sympy.abc import m, n
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m, m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m,n), Y]])
>>> B.transpose()
Matrix([
[X.T, 0],
[Z.T, Y.T]])
>>> _.transpose()
Matrix([
[X, Z],
[0, Y]])
"""
return self._eval_transpose()
def schur(self, mat = 'A', generalized = False):
"""Return the Schur Complement of the 2x2 BlockMatrix
Parameters
==========
mat : String, optional
The matrix with respect to which the
Schur Complement is calculated. 'A' is
used by default
generalized : bool, optional
If True, returns the generalized Schur
Component which uses Moore-Penrose Inverse
Examples
========
>>> from sympy import symbols, MatrixSymbol, BlockMatrix
>>> m, n = symbols('m n')
>>> 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]])
The default Schur Complement is evaluated with "A"
>>> X.schur()
-C*A**(-1)*B + D
>>> X.schur('D')
A - B*D**(-1)*C
Schur complement with non-invertible matrices is not
defined. Instead, the generalized Schur complement can
be calculated which uses the Moore-Penrose Inverse. To
achieve this, `generalized` must be set to `True`
>>> X.schur('B', generalized=True)
C - D*(B.T*B)**(-1)*B.T*A
>>> X.schur('C', generalized=True)
-A*(C.T*C)**(-1)*C.T*D + B
Returns
=======
M : Matrix
The Schur Complement Matrix
Raises
======
ShapeError
If the block matrix is not a 2x2 matrix
NonInvertibleMatrixError
If given matrix is non-invertible
References
==========
.. [1] Wikipedia Article on Schur Component : https://en.wikipedia.org/wiki/Schur_complement
See Also
========
sympy.matrices.matrices.MatrixBase.pinv
"""
if self.blockshape == (2, 2):
[[A, B],
[C, D]] = self.blocks.tolist()
d={'A' : A, 'B' : B, 'C' : C, 'D' : D}
try:
inv = (d[mat].T*d[mat]).inv()*d[mat].T if generalized else d[mat].inv()
if mat == 'A':
return D - C * inv * B
elif mat == 'B':
return C - D * inv * A
elif mat == 'C':
return B - A * inv * D
elif mat == 'D':
return A - B * inv * C
#For matrices where no sub-matrix is square
return self
except NonInvertibleMatrixError:
raise NonInvertibleMatrixError('The given matrix is not invertible. Please set generalized=True \
to compute the generalized Schur Complement which uses Moore-Penrose Inverse')
else:
raise ShapeError('Schur Complement can only be calculated for 2x2 block matrices')
def LDUdecomposition(self):
"""Returns the Block LDU decomposition of
a 2x2 Block Matrix
Returns
=======
(L, D, U) : Matrices
L : Lower Diagonal Matrix
D : Diagonal Matrix
U : Upper Diagonal Matrix
Examples
========
>>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
>>> m, n = symbols('m n')
>>> 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]])
>>> L, D, U = X.LDUdecomposition()
>>> block_collapse(L*D*U)
Matrix([
[A, B],
[C, D]])
Raises
======
ShapeError
If the block matrix is not a 2x2 matrix
NonInvertibleMatrixError
If the matrix "A" is non-invertible
See Also
========
sympy.matrices.expressions.blockmatrix.BlockMatrix.UDLdecomposition
sympy.matrices.expressions.blockmatrix.BlockMatrix.LUdecomposition
"""
if self.blockshape == (2,2):
[[A, B],
[C, D]] = self.blocks.tolist()
try:
AI = A.I
except NonInvertibleMatrixError:
raise NonInvertibleMatrixError('Block LDU decomposition cannot be calculated when\
"A" is singular')
Ip = Identity(B.shape[0])
Iq = Identity(B.shape[1])
Z = ZeroMatrix(*B.shape)
L = BlockMatrix([[Ip, Z], [C*AI, Iq]])
D = BlockDiagMatrix(A, self.schur())
U = BlockMatrix([[Ip, AI*B],[Z.T, Iq]])
return L, D, U
else:
raise ShapeError("Block LDU decomposition is supported only for 2x2 block matrices")
def UDLdecomposition(self):
"""Returns the Block UDL decomposition of
a 2x2 Block Matrix
Returns
=======
(U, D, L) : Matrices
U : Upper Diagonal Matrix
D : Diagonal Matrix
L : Lower Diagonal Matrix
Examples
========
>>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
>>> m, n = symbols('m n')
>>> 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]])
>>> U, D, L = X.UDLdecomposition()
>>> block_collapse(U*D*L)
Matrix([
[A, B],
[C, D]])
Raises
======
ShapeError
If the block matrix is not a 2x2 matrix
NonInvertibleMatrixError
If the matrix "D" is non-invertible
See Also
========
sympy.matrices.expressions.blockmatrix.BlockMatrix.LDUdecomposition
sympy.matrices.expressions.blockmatrix.BlockMatrix.LUdecomposition
"""
if self.blockshape == (2,2):
[[A, B],
[C, D]] = self.blocks.tolist()
try:
DI = D.I
except NonInvertibleMatrixError:
raise NonInvertibleMatrixError('Block UDL decomposition cannot be calculated when\
"D" is singular')
Ip = Identity(A.shape[0])
Iq = Identity(B.shape[1])
Z = ZeroMatrix(*B.shape)
U = BlockMatrix([[Ip, B*DI], [Z.T, Iq]])
D = BlockDiagMatrix(self.schur('D'), D)
L = BlockMatrix([[Ip, Z],[DI*C, Iq]])
return U, D, L
else:
raise ShapeError("Block UDL decomposition is supported only for 2x2 block matrices")
def LUdecomposition(self):
"""Returns the Block LU decomposition of
a 2x2 Block Matrix
Returns
=======
(L, U) : Matrices
L : Lower Diagonal Matrix
U : Upper Diagonal Matrix
Examples
========
>>> from sympy import symbols, MatrixSymbol, BlockMatrix, block_collapse
>>> m, n = symbols('m n')
>>> 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]])
>>> L, U = X.LUdecomposition()
>>> block_collapse(L*U)
Matrix([
[A, B],
[C, D]])
Raises
======
ShapeError
If the block matrix is not a 2x2 matrix
NonInvertibleMatrixError
If the matrix "A" is non-invertible
See Also
========
sympy.matrices.expressions.blockmatrix.BlockMatrix.UDLdecomposition
sympy.matrices.expressions.blockmatrix.BlockMatrix.LDUdecomposition
"""
if self.blockshape == (2,2):
[[A, B],
[C, D]] = self.blocks.tolist()
try:
A = A**0.5
AI = A.I
except NonInvertibleMatrixError:
raise NonInvertibleMatrixError('Block LU decomposition cannot be calculated when\
"A" is singular')
Z = ZeroMatrix(*B.shape)
Q = self.schur()**0.5
L = BlockMatrix([[A, Z], [C*AI, Q]])
U = BlockMatrix([[A, AI*B],[Z.T, Q]])
return L, U
else:
raise ShapeError("Block LU decomposition is supported only for 2x2 block matrices")
def _entry(self, i, j, **kwargs):
# Find row entry
orig_i, orig_j = i, j
for row_block, numrows in enumerate(self.rowblocksizes):
cmp = i < numrows
if cmp == True:
break
elif cmp == False:
i -= numrows
elif row_block < self.blockshape[0] - 1:
# Can't tell which block and it's not the last one, return unevaluated
return MatrixElement(self, orig_i, orig_j)
for col_block, numcols in enumerate(self.colblocksizes):
cmp = j < numcols
if cmp == True:
break
elif cmp == False:
j -= numcols
elif col_block < self.blockshape[1] - 1:
return MatrixElement(self, orig_i, orig_j)
return self.blocks[row_block, col_block][i, j]
@property
def is_Identity(self):
if self.blockshape[0] != self.blockshape[1]:
return False
for i in range(self.blockshape[0]):
for j in range(self.blockshape[1]):
if i==j and not self.blocks[i, j].is_Identity:
return False
if i!=j and not self.blocks[i, j].is_ZeroMatrix:
return False
return True
@property
def is_structurally_symmetric(self):
return self.rowblocksizes == self.colblocksizes
def equals(self, other):
if self == other:
return True
if (isinstance(other, BlockMatrix) and self.blocks == other.blocks):
return True
return super().equals(other)
class BlockDiagMatrix(BlockMatrix):
"""A sparse matrix with block matrices along its diagonals
Examples
========
>>> from sympy import MatrixSymbol, BlockDiagMatrix, symbols
>>> n, m, l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m, m)
>>> BlockDiagMatrix(X, Y)
Matrix([
[X, 0],
[0, Y]])
Notes
=====
If you want to get the individual diagonal blocks, use
:meth:`get_diag_blocks`.
See Also
========
sympy.matrices.dense.diag
"""
def __new__(cls, *mats):
return Basic.__new__(BlockDiagMatrix, *[_sympify(m) for m in mats])
@property
def diag(self):
return self.args
@property
def blocks(self):
from sympy.matrices.immutable import ImmutableDenseMatrix
mats = self.args
data = [[mats[i] if i == j else ZeroMatrix(mats[i].rows, mats[j].cols)
for j in range(len(mats))]
for i in range(len(mats))]
return ImmutableDenseMatrix(data, evaluate=False)
@property
def shape(self):
return (sum(block.rows for block in self.args),
sum(block.cols for block in self.args))
@property
def blockshape(self):
n = len(self.args)
return (n, n)
@property
def rowblocksizes(self):
return [block.rows for block in self.args]
@property
def colblocksizes(self):
return [block.cols for block in self.args]
def _all_square_blocks(self):
"""Returns true if all blocks are square"""
return all(mat.is_square for mat in self.args)
def _eval_determinant(self):
if self._all_square_blocks():
return Mul(*[det(mat) for mat in self.args])
# At least one block is non-square. Since the entire matrix must be square we know there must
# be at least two blocks in this matrix, in which case the entire matrix is necessarily rank-deficient
return S.Zero
def _eval_inverse(self, expand='ignored'):
if self._all_square_blocks():
return BlockDiagMatrix(*[mat.inverse() for mat in self.args])
# See comment in _eval_determinant()
raise NonInvertibleMatrixError('Matrix det == 0; not invertible.')
def _eval_transpose(self):
return BlockDiagMatrix(*[mat.transpose() for mat in self.args])
def _blockmul(self, other):
if (isinstance(other, BlockDiagMatrix) and
self.colblocksizes == other.rowblocksizes):
return BlockDiagMatrix(*[a*b for a, b in zip(self.args, other.args)])
else:
return BlockMatrix._blockmul(self, other)
def _blockadd(self, other):
if (isinstance(other, BlockDiagMatrix) and
self.blockshape == other.blockshape and
self.rowblocksizes == other.rowblocksizes and
self.colblocksizes == other.colblocksizes):
return BlockDiagMatrix(*[a + b for a, b in zip(self.args, other.args)])
else:
return BlockMatrix._blockadd(self, other)
def get_diag_blocks(self):
"""Return the list of diagonal blocks of the matrix.
Examples
========
>>> from sympy.matrices import BlockDiagMatrix, Matrix
>>> A = Matrix([[1, 2], [3, 4]])
>>> B = Matrix([[5, 6], [7, 8]])
>>> M = BlockDiagMatrix(A, B)
How to get diagonal blocks from the block diagonal matrix:
>>> diag_blocks = M.get_diag_blocks()
>>> diag_blocks[0]
Matrix([
[1, 2],
[3, 4]])
>>> diag_blocks[1]
Matrix([
[5, 6],
[7, 8]])
"""
return self.args
def block_collapse(expr):
"""Evaluates a block matrix expression
>>> from sympy import MatrixSymbol, BlockMatrix, symbols, Identity, ZeroMatrix, block_collapse
>>> n,m,l = symbols('n m l')
>>> X = MatrixSymbol('X', n, n)
>>> Y = MatrixSymbol('Y', m, m)
>>> Z = MatrixSymbol('Z', n, m)
>>> B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])
>>> print(B)
Matrix([
[X, Z],
[0, Y]])
>>> C = BlockMatrix([[Identity(n), Z]])
>>> print(C)
Matrix([[I, Z]])
>>> print(block_collapse(C*B))
Matrix([[X, Z + Z*Y]])
"""
from sympy.strategies.util import expr_fns
hasbm = lambda expr: isinstance(expr, MatrixExpr) and expr.has(BlockMatrix)
conditioned_rl = condition(
hasbm,
typed(
{MatAdd: do_one(bc_matadd, bc_block_plus_ident),
MatMul: do_one(bc_matmul, bc_dist),
MatPow: bc_matmul,
Transpose: bc_transpose,
Inverse: bc_inverse,
BlockMatrix: do_one(bc_unpack, deblock)}
)
)
rule = exhaust(
bottom_up(
exhaust(conditioned_rl),
fns=expr_fns
)
)
result = rule(expr)
doit = getattr(result, 'doit', None)
if doit is not None:
return doit()
else:
return result
def bc_unpack(expr):
if expr.blockshape == (1, 1):
return expr.blocks[0, 0]
return expr
def bc_matadd(expr):
args = sift(expr.args, lambda M: isinstance(M, BlockMatrix))
blocks = args[True]
if not blocks:
return expr
nonblocks = args[False]
block = blocks[0]
for b in blocks[1:]:
block = block._blockadd(b)
if nonblocks:
return MatAdd(*nonblocks) + block
else:
return block
def bc_block_plus_ident(expr):
idents = [arg for arg in expr.args if arg.is_Identity]
if not idents:
return expr
blocks = [arg for arg in expr.args if isinstance(arg, BlockMatrix)]
if (blocks and all(b.structurally_equal(blocks[0]) for b in blocks)
and blocks[0].is_structurally_symmetric):
block_id = BlockDiagMatrix(*[Identity(k)
for k in blocks[0].rowblocksizes])
rest = [arg for arg in expr.args if not arg.is_Identity and not isinstance(arg, BlockMatrix)]
return MatAdd(block_id * len(idents), *blocks, *rest).doit()
return expr
def bc_dist(expr):
""" Turn a*[X, Y] into [a*X, a*Y] """
factor, mat = expr.as_coeff_mmul()
if factor == 1:
return expr
unpacked = unpack(mat)
if isinstance(unpacked, BlockDiagMatrix):
B = unpacked.diag
new_B = [factor * mat for mat in B]
return BlockDiagMatrix(*new_B)
elif isinstance(unpacked, BlockMatrix):
B = unpacked.blocks
new_B = [
[factor * B[i, j] for j in range(B.cols)] for i in range(B.rows)]
return BlockMatrix(new_B)
return unpacked
def bc_matmul(expr):
if isinstance(expr, MatPow):
if expr.args[1].is_Integer:
factor, matrices = (1, [expr.args[0]]*expr.args[1])
else:
return expr
else:
factor, matrices = expr.as_coeff_matrices()
i = 0
while (i+1 < len(matrices)):
A, B = matrices[i:i+2]
if isinstance(A, BlockMatrix) and isinstance(B, BlockMatrix):
matrices[i] = A._blockmul(B)
matrices.pop(i+1)
elif isinstance(A, BlockMatrix):
matrices[i] = A._blockmul(BlockMatrix([[B]]))
matrices.pop(i+1)
elif isinstance(B, BlockMatrix):
matrices[i] = BlockMatrix([[A]])._blockmul(B)
matrices.pop(i+1)
else:
i+=1
return MatMul(factor, *matrices).doit()
def bc_transpose(expr):
collapse = block_collapse(expr.arg)
return collapse._eval_transpose()
def bc_inverse(expr):
if isinstance(expr.arg, BlockDiagMatrix):
return expr.inverse()
expr2 = blockinverse_1x1(expr)
if expr != expr2:
return expr2
return blockinverse_2x2(Inverse(reblock_2x2(expr.arg)))
def blockinverse_1x1(expr):
if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (1, 1):
mat = Matrix([[expr.arg.blocks[0].inverse()]])
return BlockMatrix(mat)
return expr
def blockinverse_2x2(expr):
if isinstance(expr.arg, BlockMatrix) and expr.arg.blockshape == (2, 2):
# See: Inverses of 2x2 Block Matrices, Tzon-Tzer Lu and Sheng-Hua Shiou
[[A, B],
[C, D]] = expr.arg.blocks.tolist()
formula = _choose_2x2_inversion_formula(A, B, C, D)
if formula != None:
MI = expr.arg.schur(formula).I
if formula == 'A':
AI = A.I
return BlockMatrix([[AI + AI * B * MI * C * AI, -AI * B * MI], [-MI * C * AI, MI]])
if formula == 'B':
BI = B.I
return BlockMatrix([[-MI * D * BI, MI], [BI + BI * A * MI * D * BI, -BI * A * MI]])
if formula == 'C':
CI = C.I
return BlockMatrix([[-CI * D * MI, CI + CI * D * MI * A * CI], [MI, -MI * A * CI]])
if formula == 'D':
DI = D.I
return BlockMatrix([[MI, -MI * B * DI], [-DI * C * MI, DI + DI * C * MI * B * DI]])
return expr
def _choose_2x2_inversion_formula(A, B, C, D):
"""
Assuming [[A, B], [C, D]] would form a valid square block matrix, find
which of the classical 2x2 block matrix inversion formulas would be
best suited.
Returns 'A', 'B', 'C', 'D' to represent the algorithm involving inversion
of the given argument or None if the matrix cannot be inverted using
any of those formulas.
"""
# Try to find a known invertible matrix. Note that the Schur complement
# is currently not being considered for this
A_inv = ask(Q.invertible(A))
if A_inv == True:
return 'A'
B_inv = ask(Q.invertible(B))
if B_inv == True:
return 'B'
C_inv = ask(Q.invertible(C))
if C_inv == True:
return 'C'
D_inv = ask(Q.invertible(D))
if D_inv == True:
return 'D'
# Otherwise try to find a matrix that isn't known to be non-invertible
if A_inv != False:
return 'A'
if B_inv != False:
return 'B'
if C_inv != False:
return 'C'
if D_inv != False:
return 'D'
return None
def deblock(B):
""" Flatten a BlockMatrix of BlockMatrices """
if not isinstance(B, BlockMatrix) or not B.blocks.has(BlockMatrix):
return B
wrap = lambda x: x if isinstance(x, BlockMatrix) else BlockMatrix([[x]])
bb = B.blocks.applyfunc(wrap) # everything is a block
try:
MM = Matrix(0, sum(bb[0, i].blocks.shape[1] for i in range(bb.shape[1])), [])
for row in range(0, bb.shape[0]):
M = Matrix(bb[row, 0].blocks)
for col in range(1, bb.shape[1]):
M = M.row_join(bb[row, col].blocks)
MM = MM.col_join(M)
return BlockMatrix(MM)
except ShapeError:
return B
def reblock_2x2(expr):
"""
Reblock a BlockMatrix so that it has 2x2 blocks of block matrices. If
possible in such a way that the matrix continues to be invertible using the
classical 2x2 block inversion formulas.
"""
if not isinstance(expr, BlockMatrix) or not all(d > 2 for d in expr.blockshape):
return expr
BM = BlockMatrix # for brevity's sake
rowblocks, colblocks = expr.blockshape
blocks = expr.blocks
for i in range(1, rowblocks):
for j in range(1, colblocks):
# try to split rows at i and cols at j
A = bc_unpack(BM(blocks[:i, :j]))
B = bc_unpack(BM(blocks[:i, j:]))
C = bc_unpack(BM(blocks[i:, :j]))
D = bc_unpack(BM(blocks[i:, j:]))
formula = _choose_2x2_inversion_formula(A, B, C, D)
if formula is not None:
return BlockMatrix([[A, B], [C, D]])
# else: nothing worked, just split upper left corner
return BM([[blocks[0, 0], BM(blocks[0, 1:])],
[BM(blocks[1:, 0]), BM(blocks[1:, 1:])]])
def bounds(sizes):
""" Convert sequence of numbers into pairs of low-high pairs
>>> from sympy.matrices.expressions.blockmatrix import bounds
>>> bounds((1, 10, 50))
[(0, 1), (1, 11), (11, 61)]
"""
low = 0
rv = []
for size in sizes:
rv.append((low, low + size))
low += size
return rv
def blockcut(expr, rowsizes, colsizes):
""" Cut a matrix expression into Blocks
>>> from sympy import ImmutableMatrix, blockcut
>>> M = ImmutableMatrix(4, 4, range(16))
>>> B = blockcut(M, (1, 3), (1, 3))
>>> type(B).__name__
'BlockMatrix'
>>> ImmutableMatrix(B.blocks[0, 1])
Matrix([[1, 2, 3]])
"""
rowbounds = bounds(rowsizes)
colbounds = bounds(colsizes)
return BlockMatrix([[MatrixSlice(expr, rowbound, colbound)
for colbound in colbounds]
for rowbound in rowbounds])
|
86283ae8549d24225fb36cddcef583a9c709da89540722125e9fc0b6b20cf428 | from sympy.core.assumptions import check_assumptions
from sympy.core.logic import fuzzy_and
from sympy.core.sympify import _sympify
from sympy.sets.sets import Set
from .matexpr import MatrixExpr
class MatrixSet(Set):
"""
MatrixSet represents the set of matrices with ``shape = (n, m)`` over the
given set.
Examples
========
>>> from sympy.matrices import MatrixSet, Matrix
>>> from sympy import S, I
>>> M = MatrixSet(2, 2, set=S.Reals)
>>> X = Matrix([[1, 2], [3, 4]])
>>> X in M
True
>>> X = Matrix([[1, 2], [I, 4]])
>>> X in M
False
"""
is_empty = False
def __new__(cls, n, m, set):
n, m, set = _sympify(n), _sympify(m), _sympify(set)
cls._check_dim(n)
cls._check_dim(m)
if not isinstance(set, Set):
raise TypeError("{} should be an instance of Set.".format(set))
return Set.__new__(cls, n, m, set)
@property
def shape(self):
return self.args[:2]
@property
def set(self):
return self.args[2]
def _contains(self, other):
if not isinstance(other, MatrixExpr):
raise TypeError("{} should be an instance of MatrixExpr.".format(other))
if other.shape != self.shape:
are_symbolic = any(_sympify(x).is_Symbol for x in other.shape + self.shape)
if are_symbolic:
return None
return False
return fuzzy_and(self.set.contains(x) for x in other)
@classmethod
def _check_dim(cls, dim):
"""Helper function to check invalid matrix dimensions"""
ok = check_assumptions(dim, integer=True, nonnegative=True)
if ok is False:
raise ValueError(
"The dimension specification {} should be "
"a nonnegative integer.".format(dim))
|
26594d39f65dc253343068387c5e775b5c9d417e88c2c1d1bb6939c8583aca14 | 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 as_real_imag(self):
return (re(Matrix(self)), im(Matrix(self)))
|
a7df8dbd3049eae140df01ad9be89e3bec479bcd074337bf8281d9ed0a326a6e | 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 import sift
# 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=False, _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:
if not any(isinstance(i, MatrixExpr) for i in args):
return Add.fromiter(args)
validate(*args)
if evaluate:
if not any(isinstance(i, MatrixExpr) for i in args):
return Add(*args, evaluate=True)
obj = canonicalize(obj)
return obj
@property
def shape(self):
return self.args[0].shape
def could_extract_minus_sign(self):
return _could_extract_minus_sign(self)
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, **kwargs):
deep = kwargs.get('deep', True)
if deep:
args = [arg.doit(**kwargs) 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)))
|
68d758d44cb854947e502be83915f6921d83980e908cea7ec1a10dbd683c5951 | from sympy.matrices.expressions.matexpr import MatrixExpr
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.functions.elementary.integers import floor
def normalize(i, parentsize):
if isinstance(i, slice):
i = (i.start, i.stop, i.step)
if not isinstance(i, (tuple, list, Tuple)):
if (i < 0) == True:
i += parentsize
i = (i, i+1, 1)
i = list(i)
if len(i) == 2:
i.append(1)
start, stop, step = i
start = start or 0
if stop is None:
stop = parentsize
if (start < 0) == True:
start += parentsize
if (stop < 0) == True:
stop += parentsize
step = step or 1
if ((stop - start) * step < 1) == True:
raise IndexError()
return (start, stop, step)
class MatrixSlice(MatrixExpr):
""" A MatrixSlice of a Matrix Expression
Examples
========
>>> from sympy import MatrixSlice, ImmutableMatrix
>>> M = ImmutableMatrix(4, 4, range(16))
>>> M
Matrix([
[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
>>> B = MatrixSlice(M, (0, 2), (2, 4))
>>> ImmutableMatrix(B)
Matrix([
[2, 3],
[6, 7]])
"""
parent = property(lambda self: self.args[0])
rowslice = property(lambda self: self.args[1])
colslice = property(lambda self: self.args[2])
def __new__(cls, parent, rowslice, colslice):
rowslice = normalize(rowslice, parent.shape[0])
colslice = normalize(colslice, parent.shape[1])
if not (len(rowslice) == len(colslice) == 3):
raise IndexError()
if ((0 > rowslice[0]) == True or
(parent.shape[0] < rowslice[1]) == True or
(0 > colslice[0]) == True or
(parent.shape[1] < colslice[1]) == True):
raise IndexError()
if isinstance(parent, MatrixSlice):
return mat_slice_of_slice(parent, rowslice, colslice)
return Basic.__new__(cls, parent, Tuple(*rowslice), Tuple(*colslice))
@property
def shape(self):
rows = self.rowslice[1] - self.rowslice[0]
rows = rows if self.rowslice[2] == 1 else floor(rows/self.rowslice[2])
cols = self.colslice[1] - self.colslice[0]
cols = cols if self.colslice[2] == 1 else floor(cols/self.colslice[2])
return rows, cols
def _entry(self, i, j, **kwargs):
return self.parent._entry(i*self.rowslice[2] + self.rowslice[0],
j*self.colslice[2] + self.colslice[0],
**kwargs)
@property
def on_diag(self):
return self.rowslice == self.colslice
def slice_of_slice(s, t):
start1, stop1, step1 = s
start2, stop2, step2 = t
start = start1 + start2*step1
step = step1 * step2
stop = start1 + step1*stop2
if stop > stop1:
raise IndexError()
return start, stop, step
def mat_slice_of_slice(parent, rowslice, colslice):
""" Collapse nested matrix slices
>>> from sympy import MatrixSymbol
>>> X = MatrixSymbol('X', 10, 10)
>>> X[:, 1:5][5:8, :]
X[5:8, 1:5]
>>> X[1:9:2, 2:6][1:3, 2]
X[3:7:2, 4:5]
"""
row = slice_of_slice(parent.rowslice, rowslice)
col = slice_of_slice(parent.colslice, colslice)
return MatrixSlice(parent.parent, row, col)
|
fa7d92bcdf071e064d98f9cee7956d28fe810f8a6863b6ce1c0aa7646b729148 | 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)
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
|
02c05bde1f16f321f25d675535bf1e3d8b3cb0247b3dc05518ac57867ba9c9bf | 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)
from sympy.matrices.common import NonInvertibleMatrixError
from sympy.matrices import (
Matrix, ImmutableMatrix, ImmutableSparseMatrix)
from sympy.core import Tuple, symbols, Expr
from sympy.functions import transpose
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
|
ded45b126662da06539a4066ee60165036d7e23c047c2f2d0b8918a0ebb36948 | from sympy.functions.elementary.miscellaneous import sqrt
from sympy.simplify.powsimp import powsimp
from sympy.testing.pytest import raises
from sympy.core.expr import unchanged
from sympy.core import symbols, S
from sympy.matrices import Identity, MatrixSymbol, ImmutableMatrix, ZeroMatrix, OneMatrix, Matrix
from sympy.matrices.common import NonSquareMatrixError
from sympy.matrices.expressions import MatPow, MatAdd, MatMul
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions.matexpr import MatrixElement
n, m, l, k = symbols('n m l k', integer=True)
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_entry_matrix():
X = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(X, 0)[0, 0] == 1
assert MatPow(X, 0)[0, 1] == 0
assert MatPow(X, 1)[0, 0] == 1
assert MatPow(X, 1)[0, 1] == 2
assert MatPow(X, 2)[0, 0] == 7
def test_entry_symbol():
from sympy.concrete import Sum
assert MatPow(C, 0)[0, 0] == 1
assert MatPow(C, 0)[0, 1] == 0
assert MatPow(C, 1)[0, 0] == C[0, 0]
assert isinstance(MatPow(C, 2)[0, 0], Sum)
assert isinstance(MatPow(C, n)[0, 0], MatrixElement)
def test_as_explicit_symbol():
X = MatrixSymbol('X', 2, 2)
assert MatPow(X, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).as_explicit() == X.as_explicit()
assert MatPow(X, 2).as_explicit() == (X.as_explicit())**2
assert MatPow(X, n).as_explicit() == ImmutableMatrix([
[(X ** n)[0, 0], (X ** n)[0, 1]],
[(X ** n)[1, 0], (X ** n)[1, 1]],
])
a = MatrixSymbol("a", 3, 1)
b = MatrixSymbol("b", 3, 1)
c = MatrixSymbol("c", 3, 1)
expr = (a.T*b)**S.Half
assert expr.as_explicit() == Matrix([[sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0])]])
expr = c*(a.T*b)**S.Half
m = sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0])
assert expr.as_explicit() == Matrix([[c[0, 0]*m], [c[1, 0]*m], [c[2, 0]*m]])
expr = (a*b.T)**S.Half
denom = sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0])
expected = (a*b.T).as_explicit()/denom
assert expr.as_explicit() == expected
expr = X**-1
det = X[0, 0]*X[1, 1] - X[1, 0]*X[0, 1]
expected = Matrix([[X[1, 1], -X[0, 1]], [-X[1, 0], X[0, 0]]])/det
assert expr.as_explicit() == expected
expr = X**m
assert expr.as_explicit() == X.as_explicit()**m
def test_as_explicit_matrix():
A = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(A, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(A, 1).as_explicit() == A
assert MatPow(A, 2).as_explicit() == A**2
assert MatPow(A, -1).as_explicit() == A.inv()
assert MatPow(A, -2).as_explicit() == (A.inv())**2
# less expensive than testing on a 2x2
A = ImmutableMatrix([4])
assert MatPow(A, S.Half).as_explicit() == A**S.Half
def test_doit_symbol():
assert MatPow(C, 0).doit() == Identity(n)
assert MatPow(C, 1).doit() == C
assert MatPow(C, -1).doit() == C.I
for r in [2, S.Half, S.Pi, n]:
assert MatPow(C, r).doit() == MatPow(C, r)
def test_doit_matrix():
X = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(X, 0).doit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).doit() == X
assert MatPow(X, 2).doit() == X**2
assert MatPow(X, -1).doit() == X.inv()
assert MatPow(X, -2).doit() == (X.inv())**2
# less expensive than testing on a 2x2
assert MatPow(ImmutableMatrix([4]), S.Half).doit() == ImmutableMatrix([2])
X = ImmutableMatrix([[0, 2], [0, 4]]) # det() == 0
raises(ValueError, lambda: MatPow(X,-1).doit())
raises(ValueError, lambda: MatPow(X,-2).doit())
def test_nonsquare():
A = MatrixSymbol('A', 2, 3)
B = ImmutableMatrix([[1, 2, 3], [4, 5, 6]])
for r in [-1, 0, 1, 2, S.Half, S.Pi, n]:
raises(NonSquareMatrixError, lambda: MatPow(A, r))
raises(NonSquareMatrixError, lambda: MatPow(B, r))
def test_doit_equals_pow(): #17179
X = ImmutableMatrix ([[1,0],[0,1]])
assert MatPow(X, n).doit() == X**n == X
def test_doit_nested_MatrixExpr():
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[2, 3], [4, 5]])
assert MatPow(MatMul(X, Y), 2).doit() == (X*Y)**2
assert MatPow(MatAdd(X, Y), 2).doit() == (X + Y)**2
def test_identity_power():
k = Identity(n)
assert MatPow(k, 4).doit() == k
assert MatPow(k, n).doit() == k
assert MatPow(k, -3).doit() == k
assert MatPow(k, 0).doit() == k
l = Identity(3)
assert MatPow(l, n).doit() == l
assert MatPow(l, -1).doit() == l
assert MatPow(l, 0).doit() == l
def test_zero_power():
z1 = ZeroMatrix(n, n)
assert MatPow(z1, 3).doit() == z1
raises(ValueError, lambda:MatPow(z1, -1).doit())
assert MatPow(z1, 0).doit() == Identity(n)
assert MatPow(z1, n).doit() == z1
raises(ValueError, lambda:MatPow(z1, -2).doit())
z2 = ZeroMatrix(4, 4)
assert MatPow(z2, n).doit() == z2
raises(ValueError, lambda:MatPow(z2, -3).doit())
assert MatPow(z2, 2).doit() == z2
assert MatPow(z2, 0).doit() == Identity(4)
raises(ValueError, lambda:MatPow(z2, -1).doit())
def test_OneMatrix_power():
o = OneMatrix(3, 3)
assert o ** 0 == Identity(3)
assert o ** 1 == o
assert o * o == o ** 2 == 3 * o
assert o * o * o == o ** 3 == 9 * o
o = OneMatrix(n, n)
assert o * o == o ** 2 == n * o
# powsimp necessary as n ** (n - 2) * n does not produce n ** (n - 1)
assert powsimp(o ** (n - 1) * o) == o ** n == n ** (n - 1) * o
def test_transpose_power():
from sympy.matrices.expressions.transpose import Transpose as TP
assert (C*D).T**5 == ((C*D)**5).T == (D.T * C.T)**5
assert ((C*D).T**5).T == (C*D)**5
assert (C.T.I.T)**7 == C**-7
assert (C.T**l).T**k == C**(l*k)
assert ((E.T * A.T)**5).T == (A*E)**5
assert ((A*E).T**5).T**7 == (A*E)**35
assert TP(TP(C**2 * D**3)**5).doit() == (C**2 * D**3)**5
assert ((D*C)**-5).T**-5 == ((D*C)**25).T
assert (((D*C)**l).T**k).T == (D*C)**(l*k)
def test_Inverse():
assert Inverse(MatPow(C, 0)).doit() == Identity(n)
assert Inverse(MatPow(C, 1)).doit() == Inverse(C)
assert Inverse(MatPow(C, 2)).doit() == MatPow(C, -2)
assert Inverse(MatPow(C, -1)).doit() == C
assert MatPow(Inverse(C), 0).doit() == Identity(n)
assert MatPow(Inverse(C), 1).doit() == Inverse(C)
assert MatPow(Inverse(C), 2).doit() == MatPow(C, -2)
assert MatPow(Inverse(C), -1).doit() == C
def test_combine_powers():
assert (C ** 1) ** 1 == C
assert (C ** 2) ** 3 == MatPow(C, 6)
assert (C ** -2) ** -3 == MatPow(C, 6)
assert (C ** -1) ** -1 == C
assert (((C ** 2) ** 3) ** 4) ** 5 == MatPow(C, 120)
assert (C ** n) ** n == C ** (n ** 2)
def test_unchanged():
assert unchanged(MatPow, C, 0)
assert unchanged(MatPow, C, 1)
assert unchanged(MatPow, Inverse(C), -1)
assert unchanged(Inverse, MatPow(C, -1), -1)
assert unchanged(MatPow, MatPow(C, -1), -1)
assert unchanged(MatPow, MatPow(C, 1), 1)
def test_no_exponentiation():
# if this passes, Pow.as_numer_denom should recognize
# MatAdd as exponent
raises(NotImplementedError, lambda: 3**(-2*C))
|
f770aa8e3da4ddaa11ddac8c24b74951ae61d6ef3b6d14714f723c078150b2f5 | from sympy.combinatorics import Permutation
from sympy.core.expr import unchanged
from sympy.matrices import Matrix
from sympy.matrices.expressions import \
MatMul, BlockDiagMatrix, Determinant, Inverse
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.special import ZeroMatrix, OneMatrix, Identity
from sympy.matrices.expressions.permutation import \
MatrixPermute, PermutationMatrix
from sympy.testing.pytest import raises
from sympy.core.symbol import Symbol
def test_PermutationMatrix_basic():
p = Permutation([1, 0])
assert unchanged(PermutationMatrix, p)
raises(ValueError, lambda: PermutationMatrix((0, 1, 2)))
assert PermutationMatrix(p).as_explicit() == Matrix([[0, 1], [1, 0]])
assert isinstance(PermutationMatrix(p)*MatrixSymbol('A', 2, 2), MatMul)
def test_PermutationMatrix_matmul():
p = Permutation([1, 2, 0])
P = PermutationMatrix(p)
M = Matrix([[0, 1, 2], [3, 4, 5], [6, 7, 8]])
assert (P*M).as_explicit() == P.as_explicit()*M
assert (M*P).as_explicit() == M*P.as_explicit()
P1 = PermutationMatrix(Permutation([1, 2, 0]))
P2 = PermutationMatrix(Permutation([2, 1, 0]))
P3 = PermutationMatrix(Permutation([1, 0, 2]))
assert P1*P2 == P3
def test_PermutationMatrix_matpow():
p1 = Permutation([1, 2, 0])
P1 = PermutationMatrix(p1)
p2 = Permutation([2, 0, 1])
P2 = PermutationMatrix(p2)
assert P1**2 == P2
assert P1**3 == Identity(3)
def test_PermutationMatrix_identity():
p = Permutation([0, 1])
assert PermutationMatrix(p).is_Identity
p = Permutation([1, 0])
assert not PermutationMatrix(p).is_Identity
def test_PermutationMatrix_determinant():
P = PermutationMatrix(Permutation([0, 1, 2]))
assert Determinant(P).doit() == 1
P = PermutationMatrix(Permutation([0, 2, 1]))
assert Determinant(P).doit() == -1
P = PermutationMatrix(Permutation([2, 0, 1]))
assert Determinant(P).doit() == 1
def test_PermutationMatrix_inverse():
P = PermutationMatrix(Permutation(0, 1, 2))
assert Inverse(P).doit() == PermutationMatrix(Permutation(0, 2, 1))
def test_PermutationMatrix_rewrite_BlockDiagMatrix():
P = PermutationMatrix(Permutation([0, 1, 2, 3, 4, 5]))
P0 = PermutationMatrix(Permutation([0]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P0, P0, P0, P0, P0)
P = PermutationMatrix(Permutation([0, 1, 3, 2, 4, 5]))
P10 = PermutationMatrix(Permutation(0, 1))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P0, P10, P0, P0)
P = PermutationMatrix(Permutation([1, 0, 3, 2, 5, 4]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P10, P10, P10)
P = PermutationMatrix(Permutation([0, 4, 3, 2, 1, 5]))
P3210 = PermutationMatrix(Permutation([3, 2, 1, 0]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P3210, P0)
P = PermutationMatrix(Permutation([0, 4, 2, 3, 1, 5]))
P3120 = PermutationMatrix(Permutation([3, 1, 2, 0]))
assert P.rewrite(BlockDiagMatrix) == \
BlockDiagMatrix(P0, P3120, P0)
P = PermutationMatrix(Permutation(0, 3)(1, 4)(2, 5))
assert P.rewrite(BlockDiagMatrix) == BlockDiagMatrix(P)
def test_MartrixPermute_basic():
p = Permutation(0, 1)
P = PermutationMatrix(p)
A = MatrixSymbol('A', 2, 2)
raises(ValueError, lambda: MatrixPermute(Symbol('x'), p))
raises(ValueError, lambda: MatrixPermute(A, Symbol('x')))
assert MatrixPermute(A, P) == MatrixPermute(A, p)
raises(ValueError, lambda: MatrixPermute(A, p, 2))
pp = Permutation(0, 1, size=3)
assert MatrixPermute(A, pp) == MatrixPermute(A, p)
pp = Permutation(0, 1, 2)
raises(ValueError, lambda: MatrixPermute(A, pp))
def test_MatrixPermute_shape():
p = Permutation(0, 1)
A = MatrixSymbol('A', 2, 3)
assert MatrixPermute(A, p).shape == (2, 3)
def test_MatrixPermute_explicit():
p = Permutation(0, 1, 2)
A = MatrixSymbol('A', 3, 3)
AA = A.as_explicit()
assert MatrixPermute(A, p, 0).as_explicit() == \
AA.permute(p, orientation='rows')
assert MatrixPermute(A, p, 1).as_explicit() == \
AA.permute(p, orientation='cols')
def test_MatrixPermute_rewrite_MatMul():
p = Permutation(0, 1, 2)
A = MatrixSymbol('A', 3, 3)
assert MatrixPermute(A, p, 0).rewrite(MatMul).as_explicit() == \
MatrixPermute(A, p, 0).as_explicit()
assert MatrixPermute(A, p, 1).rewrite(MatMul).as_explicit() == \
MatrixPermute(A, p, 1).as_explicit()
def test_MatrixPermute_doit():
p = Permutation(0, 1, 2)
A = MatrixSymbol('A', 3, 3)
assert MatrixPermute(A, p).doit() == MatrixPermute(A, p)
p = Permutation(0, size=3)
A = MatrixSymbol('A', 3, 3)
assert MatrixPermute(A, p).doit().as_explicit() == \
MatrixPermute(A, p).as_explicit()
p = Permutation(0, 1, 2)
A = Identity(3)
assert MatrixPermute(A, p, 0).doit().as_explicit() == \
MatrixPermute(A, p, 0).as_explicit()
assert MatrixPermute(A, p, 1).doit().as_explicit() == \
MatrixPermute(A, p, 1).as_explicit()
A = ZeroMatrix(3, 3)
assert MatrixPermute(A, p).doit() == A
A = OneMatrix(3, 3)
assert MatrixPermute(A, p).doit() == A
A = MatrixSymbol('A', 4, 4)
p1 = Permutation(0, 1, 2, 3)
p2 = Permutation(0, 2, 3, 1)
expr = MatrixPermute(MatrixPermute(A, p1, 0), p2, 0)
assert expr.as_explicit() == expr.doit().as_explicit()
expr = MatrixPermute(MatrixPermute(A, p1, 1), p2, 1)
assert expr.as_explicit() == expr.doit().as_explicit()
|
5b8390602cdde3b07b647af3f4934c63bf05f8598d3e68a616c0e1de05aaff2d | from sympy.core import S, symbols
from sympy.matrices import eye, ones, Matrix, ShapeError
from sympy.matrices.expressions import (
Identity, MatrixExpr, MatrixSymbol, Determinant,
det, per, ZeroMatrix, Transpose,
Permanent
)
from sympy.matrices.expressions.special import OneMatrix
from sympy.testing.pytest import raises
from sympy.assumptions.ask import Q
from sympy.assumptions.refine import refine
n = symbols('n', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
C = MatrixSymbol('C', 3, 4)
def test_det():
assert isinstance(Determinant(A), Determinant)
assert not isinstance(Determinant(A), MatrixExpr)
raises(ShapeError, lambda: Determinant(C))
assert det(eye(3)) == 1
assert det(Matrix(3, 3, [1, 3, 2, 4, 1, 3, 2, 5, 2])) == 17
A / det(A) # Make sure this is possible
raises(TypeError, lambda: Determinant(S.One))
assert Determinant(A).arg is A
def test_eval_determinant():
assert det(Identity(n)) == 1
assert det(ZeroMatrix(n, n)) == 0
assert det(OneMatrix(n, n)) == Determinant(OneMatrix(n, n))
assert det(OneMatrix(1, 1)) == 1
assert det(OneMatrix(2, 2)) == 0
assert det(Transpose(A)) == det(A)
def test_refine():
assert refine(det(A), Q.orthogonal(A)) == 1
assert refine(det(A), Q.singular(A)) == 0
assert refine(det(A), Q.unit_triangular(A)) == 1
assert refine(det(A), Q.normal(A)) == det(A)
def test_commutative():
det_a = Determinant(A)
det_b = Determinant(B)
assert det_a.is_commutative
assert det_b.is_commutative
assert det_a * det_b == det_b * det_a
def test_permanent():
assert isinstance(Permanent(A), Permanent)
assert not isinstance(Permanent(A), MatrixExpr)
assert isinstance(Permanent(C), Permanent)
assert Permanent(ones(3, 3)).doit() == 6
C / per(C)
assert per(Matrix(3, 3, [1, 3, 2, 4, 1, 3, 2, 5, 2])) == 103
raises(TypeError, lambda: Permanent(S.One))
assert Permanent(A).arg is A
|
3f7c0f3d1cb02a7ac0a695fa82cb200faeafd1121ca25c6a0e409a28618f55c3 | from sympy.functions import adjoint, conjugate, transpose
from sympy.matrices.expressions import MatrixSymbol, Adjoint, trace, Transpose
from sympy.matrices import eye, Matrix
from sympy.assumptions.ask import Q
from sympy.assumptions.refine import refine
from sympy.core.singleton import S
from sympy.core.symbol import symbols
n, m, l, k, p = symbols('n m l k p', integer=True)
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', n, n)
def test_transpose():
Sq = MatrixSymbol('Sq', n, n)
assert transpose(A) == Transpose(A)
assert Transpose(A).shape == (m, n)
assert Transpose(A*B).shape == (l, n)
assert transpose(Transpose(A)) == A
assert isinstance(Transpose(Transpose(A)), Transpose)
assert adjoint(Transpose(A)) == Adjoint(Transpose(A))
assert conjugate(Transpose(A)) == Adjoint(A)
assert Transpose(eye(3)).doit() == eye(3)
assert Transpose(S(5)).doit() == S(5)
assert Transpose(Matrix([[1, 2], [3, 4]])).doit() == Matrix([[1, 3], [2, 4]])
assert transpose(trace(Sq)) == trace(Sq)
assert trace(Transpose(Sq)) == trace(Sq)
assert Transpose(Sq)[0, 1] == Sq[1, 0]
assert Transpose(A*B).doit() == Transpose(B) * Transpose(A)
def test_transpose_MatAdd_MatMul():
# Issue 16807
from sympy.functions.elementary.trigonometric import cos
x = symbols('x')
M = MatrixSymbol('M', 3, 3)
N = MatrixSymbol('N', 3, 3)
assert (N + (cos(x) * M)).T == cos(x)*M.T + N.T
def test_refine():
assert refine(C.T, Q.symmetric(C)) == C
def test_transpose1x1():
m = MatrixSymbol('m', 1, 1)
assert m == refine(m.T)
assert m == refine(m.T.T)
def test_issue_9817():
from sympy.matrices.expressions import Identity
v = MatrixSymbol('v', 3, 1)
A = MatrixSymbol('A', 3, 3)
x = Matrix([i + 1 for i in range(3)])
X = Identity(3)
quadratic = v.T * A * v
subbed = quadratic.xreplace({v:x, A:X})
assert subbed.as_explicit() == Matrix([[14]])
|
bd4bc868bef697649ca60b04daa979d3d2c5a0f50ef35ec782df54b2c50d71d8 | """
Some examples have been taken from:
http://www.math.uwaterloo.ca/~hwolkowi//matrixcookbook.pdf
"""
from sympy.combinatorics import Permutation
from sympy.concrete.summations import Sum
from sympy.core.numbers import Rational
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.expressions.determinant import Determinant
from sympy.matrices.expressions.diagonal import DiagMatrix
from sympy.matrices.expressions.hadamard import (HadamardPower, HadamardProduct, hadamard_product)
from sympy.matrices.expressions.inverse import Inverse
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.special import OneMatrix
from sympy.matrices.expressions.trace import Trace
from sympy.matrices.expressions.matadd import MatAdd
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.special import (Identity, ZeroMatrix)
from sympy.tensor.array.array_derivatives import ArrayDerivative
from sympy.matrices.expressions import hadamard_power
from sympy.tensor.array.expressions.array_expressions import ArrayAdd, ArrayTensorProduct, PermuteDims
k = symbols("k")
i, j = symbols("i j")
m, n = symbols("m n")
X = MatrixSymbol("X", k, k)
x = MatrixSymbol("x", k, 1)
y = MatrixSymbol("y", k, 1)
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
D = MatrixSymbol("D", k, k)
a = MatrixSymbol("a", k, 1)
b = MatrixSymbol("b", k, 1)
c = MatrixSymbol("c", k, 1)
d = MatrixSymbol("d", k, 1)
KDelta = lambda i, j: KroneckerDelta(i, j, (0, k-1))
def _check_derivative_with_explicit_matrix(expr, x, diffexpr, dim=2):
# TODO: this is commented because it slows down the tests.
return
expr = expr.xreplace({k: dim})
x = x.xreplace({k: dim})
diffexpr = diffexpr.xreplace({k: dim})
expr = expr.as_explicit()
x = x.as_explicit()
diffexpr = diffexpr.as_explicit()
assert expr.diff(x).reshape(*diffexpr.shape).tomatrix() == diffexpr
def test_matrix_derivative_by_scalar():
assert A.diff(i) == ZeroMatrix(k, k)
assert (A*(X + B)*c).diff(i) == ZeroMatrix(k, 1)
assert x.diff(i) == ZeroMatrix(k, 1)
assert (x.T*y).diff(i) == ZeroMatrix(1, 1)
assert (x*x.T).diff(i) == ZeroMatrix(k, k)
assert (x + y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, 2).diff(i) == ZeroMatrix(k, 1)
assert hadamard_power(x, i).diff(i).dummy_eq(
HadamardProduct(x.applyfunc(log), HadamardPower(x, i)))
assert hadamard_product(x, y).diff(i) == ZeroMatrix(k, 1)
assert hadamard_product(i*OneMatrix(k, 1), x, y).diff(i) == hadamard_product(x, y)
assert (i*x).diff(i) == x
assert (sin(i)*A*B*x).diff(i) == cos(i)*A*B*x
assert x.applyfunc(sin).diff(i) == ZeroMatrix(k, 1)
assert Trace(i**2*X).diff(i) == 2*i*Trace(X)
mu = symbols("mu")
expr = (2*mu*x)
assert expr.diff(x) == 2*mu*Identity(k)
def test_matrix_derivative_non_matrix_result():
# This is a 4-dimensional array:
I = Identity(k)
AdA = PermuteDims(ArrayTensorProduct(I, I), Permutation(3)(1, 2))
assert A.diff(A) == AdA
assert A.T.diff(A) == PermuteDims(ArrayTensorProduct(I, I), Permutation(3)(1, 2, 3))
assert (2*A).diff(A) == PermuteDims(ArrayTensorProduct(2*I, I), Permutation(3)(1, 2))
assert MatAdd(A, A).diff(A) == ArrayAdd(AdA, AdA)
assert (A + B).diff(A) == AdA
def test_matrix_derivative_trivial_cases():
# Cookbook example 33:
# TODO: find a way to represent a four-dimensional zero-array:
assert X.diff(A) == ArrayDerivative(X, A)
def test_matrix_derivative_with_inverse():
# Cookbook example 61:
expr = a.T*Inverse(X)*b
assert expr.diff(X) == -Inverse(X).T*a*b.T*Inverse(X).T
# Cookbook example 62:
expr = Determinant(Inverse(X))
# Not implemented yet:
# assert expr.diff(X) == -Determinant(X.inv())*(X.inv()).T
# Cookbook example 63:
expr = Trace(A*Inverse(X)*B)
assert expr.diff(X) == -(X**(-1)*B*A*X**(-1)).T
# Cookbook example 64:
expr = Trace(Inverse(X + A))
assert expr.diff(X) == -(Inverse(X + A)).T**2
def test_matrix_derivative_vectors_and_scalars():
assert x.diff(x) == Identity(k)
assert x[i, 0].diff(x[m, 0]).doit() == KDelta(m, i)
assert x.T.diff(x) == Identity(k)
# Cookbook example 69:
expr = x.T*a
assert expr.diff(x) == a
assert expr[0, 0].diff(x[m, 0]).doit() == a[m, 0]
expr = a.T*x
assert expr.diff(x) == a
# Cookbook example 70:
expr = a.T*X*b
assert expr.diff(X) == a*b.T
# Cookbook example 71:
expr = a.T*X.T*b
assert expr.diff(X) == b*a.T
# Cookbook example 72:
expr = a.T*X*a
assert expr.diff(X) == a*a.T
expr = a.T*X.T*a
assert expr.diff(X) == a*a.T
# Cookbook example 77:
expr = b.T*X.T*X*c
assert expr.diff(X) == X*b*c.T + X*c*b.T
# Cookbook example 78:
expr = (B*x + b).T*C*(D*x + d)
assert expr.diff(x) == B.T*C*(D*x + d) + D.T*C.T*(B*x + b)
# Cookbook example 81:
expr = x.T*B*x
assert expr.diff(x) == B*x + B.T*x
# Cookbook example 82:
expr = b.T*X.T*D*X*c
assert expr.diff(X) == D.T*X*b*c.T + D*X*c*b.T
# Cookbook example 83:
expr = (X*b + c).T*D*(X*b + c)
assert expr.diff(X) == D*(X*b + c)*b.T + D.T*(X*b + c)*b.T
assert str(expr[0, 0].diff(X[m, n]).doit()) == \
'b[n, 0]*Sum((c[_i_1, 0] + Sum(X[_i_1, _i_3]*b[_i_3, 0], (_i_3, 0, k - 1)))*D[_i_1, m], (_i_1, 0, k - 1)) + Sum((c[_i_2, 0] + Sum(X[_i_2, _i_4]*b[_i_4, 0], (_i_4, 0, k - 1)))*D[m, _i_2]*b[n, 0], (_i_2, 0, k - 1))'
def test_matrix_derivatives_of_traces():
expr = Trace(A)*A
I = Identity(k)
assert expr.diff(A) == ArrayAdd(ArrayTensorProduct(I, A), PermuteDims(ArrayTensorProduct(Trace(A)*I, I), Permutation(3)(1, 2)))
assert expr[i, j].diff(A[m, n]).doit() == (
KDelta(i, m)*KDelta(j, n)*Trace(A) +
KDelta(m, n)*A[i, j]
)
## First order:
# Cookbook example 99:
expr = Trace(X)
assert expr.diff(X) == Identity(k)
assert expr.rewrite(Sum).diff(X[m, n]).doit() == KDelta(m, n)
# Cookbook example 100:
expr = Trace(X*A)
assert expr.diff(X) == A.T
assert expr.rewrite(Sum).diff(X[m, n]).doit() == A[n, m]
# Cookbook example 101:
expr = Trace(A*X*B)
assert expr.diff(X) == A.T*B.T
assert expr.rewrite(Sum).diff(X[m, n]).doit().dummy_eq((A.T*B.T)[m, n])
# Cookbook example 102:
expr = Trace(A*X.T*B)
assert expr.diff(X) == B*A
# Cookbook example 103:
expr = Trace(X.T*A)
assert expr.diff(X) == A
# Cookbook example 104:
expr = Trace(A*X.T)
assert expr.diff(X) == A
# Cookbook example 105:
# TODO: TensorProduct is not supported
#expr = Trace(TensorProduct(A, X))
#assert expr.diff(X) == Trace(A)*Identity(k)
## Second order:
# Cookbook example 106:
expr = Trace(X**2)
assert expr.diff(X) == 2*X.T
# Cookbook example 107:
expr = Trace(X**2*B)
assert expr.diff(X) == (X*B + B*X).T
expr = Trace(MatMul(X, X, B))
assert expr.diff(X) == (X*B + B*X).T
# Cookbook example 108:
expr = Trace(X.T*B*X)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 109:
expr = Trace(B*X*X.T)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 110:
expr = Trace(X*X.T*B)
assert expr.diff(X) == B*X + B.T*X
# Cookbook example 111:
expr = Trace(X*B*X.T)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 112:
expr = Trace(B*X.T*X)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 113:
expr = Trace(X.T*X*B)
assert expr.diff(X) == X*B.T + X*B
# Cookbook example 114:
expr = Trace(A*X*B*X)
assert expr.diff(X) == A.T*X.T*B.T + B.T*X.T*A.T
# Cookbook example 115:
expr = Trace(X.T*X)
assert expr.diff(X) == 2*X
expr = Trace(X*X.T)
assert expr.diff(X) == 2*X
# Cookbook example 116:
expr = Trace(B.T*X.T*C*X*B)
assert expr.diff(X) == C.T*X*B*B.T + C*X*B*B.T
# Cookbook example 117:
expr = Trace(X.T*B*X*C)
assert expr.diff(X) == B*X*C + B.T*X*C.T
# Cookbook example 118:
expr = Trace(A*X*B*X.T*C)
assert expr.diff(X) == A.T*C.T*X*B.T + C*A*X*B
# Cookbook example 119:
expr = Trace((A*X*B + C)*(A*X*B + C).T)
assert expr.diff(X) == 2*A.T*(A*X*B + C)*B.T
# Cookbook example 120:
# TODO: no support for TensorProduct.
# expr = Trace(TensorProduct(X, X))
# expr = Trace(X)*Trace(X)
# expr.diff(X) == 2*Trace(X)*Identity(k)
# Higher Order
# Cookbook example 121:
expr = Trace(X**k)
#assert expr.diff(X) == k*(X**(k-1)).T
# Cookbook example 122:
expr = Trace(A*X**k)
#assert expr.diff(X) == # Needs indices
# Cookbook example 123:
expr = Trace(B.T*X.T*C*X*X.T*C*X*B)
assert expr.diff(X) == C*X*X.T*C*X*B*B.T + C.T*X*B*B.T*X.T*C.T*X + C*X*B*B.T*X.T*C*X + C.T*X*X.T*C.T*X*B*B.T
# Other
# Cookbook example 124:
expr = Trace(A*X**(-1)*B)
assert expr.diff(X) == -Inverse(X).T*A.T*B.T*Inverse(X).T
# Cookbook example 125:
expr = Trace(Inverse(X.T*C*X)*A)
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == - X.inv().T*A.T*X.inv()*C.inv().T*X.inv().T - X.inv().T*A*X.inv()*C.inv()*X.inv().T
# Cookbook example 126:
expr = Trace((X.T*C*X).inv()*(X.T*B*X))
assert expr.diff(X) == -2*C*X*(X.T*C*X).inv()*X.T*B*X*(X.T*C*X).inv() + 2*B*X*(X.T*C*X).inv()
# Cookbook example 127:
expr = Trace((A + X.T*C*X).inv()*(X.T*B*X))
# Warning: result in the cookbook is equivalent if B and C are symmetric:
assert expr.diff(X) == B*X*Inverse(A + X.T*C*X) - C*X*Inverse(A + X.T*C*X)*X.T*B*X*Inverse(A + X.T*C*X) - C.T*X*Inverse(A.T + (C*X).T*X)*X.T*B.T*X*Inverse(A.T + (C*X).T*X) + B.T*X*Inverse(A.T + (C*X).T*X)
def test_derivatives_of_complicated_matrix_expr():
expr = a.T*(A*X*(X.T*B + X*A) + B.T*X.T*(a*b.T*(X*D*X.T + X*(X.T*B + A*X)*D*B - X.T*C.T*A)*B + B*(X*D.T + B*A*X*A.T - 3*X*D))*B + 42*X*B*X.T*A.T*(X + X.T))*b
result = (B*(B*A*X*A.T - 3*X*D + X*D.T) + a*b.T*(X*(A*X + X.T*B)*D*B + X*D*X.T - X.T*C.T*A)*B)*B*b*a.T*B.T + B**2*b*a.T*B.T*X.T*a*b.T*X*D + 42*A*X*B.T*X.T*a*b.T + B*D*B**3*b*a.T*B.T*X.T*a*b.T*X + B*b*a.T*A*X + a*b.T*(42*X + 42*X.T)*A*X*B.T + b*a.T*X*B*a*b.T*B.T**2*X*D.T + b*a.T*X*B*a*b.T*B.T**3*D.T*(B.T*X + X.T*A.T) + 42*b*a.T*X*B*X.T*A.T + A.T*(42*X + 42*X.T)*b*a.T*X*B + A.T*B.T**2*X*B*a*b.T*B.T*A + A.T*a*b.T*(A.T*X.T + B.T*X) + A.T*X.T*b*a.T*X*B*a*b.T*B.T**3*D.T + B.T*X*B*a*b.T*B.T*D - 3*B.T*X*B*a*b.T*B.T*D.T - C.T*A*B**2*b*a.T*B.T*X.T*a*b.T + X.T*A.T*a*b.T*A.T
assert expr.diff(X) == result
def test_mixed_deriv_mixed_expressions():
expr = 3*Trace(A)
assert expr.diff(A) == 3*Identity(k)
expr = k
deriv = expr.diff(A)
assert isinstance(deriv, ZeroMatrix)
assert deriv == ZeroMatrix(k, k)
expr = Trace(A)**2
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(A)*A
I = Identity(k)
assert expr.diff(A) == ArrayAdd(ArrayTensorProduct(I, A), PermuteDims(ArrayTensorProduct(Trace(A)*I, I), Permutation(3)(1, 2)))
expr = Trace(Trace(A)*A)
assert expr.diff(A) == (2*Trace(A))*Identity(k)
expr = Trace(Trace(Trace(A)*A)*A)
assert expr.diff(A) == (3*Trace(A)**2)*Identity(k)
def test_derivatives_matrix_norms():
expr = x.T*y
assert expr.diff(x) == y
assert expr[0, 0].diff(x[m, 0]).doit() == y[m, 0]
expr = (x.T*y)**S.Half
assert expr.diff(x) == y/(2*sqrt(x.T*y))
expr = (x.T*x)**S.Half
assert expr.diff(x) == x*(x.T*x)**Rational(-1, 2)
expr = (c.T*a*x.T*b)**S.Half
assert expr.diff(x) == b*a.T*c/sqrt(c.T*a*x.T*b)/2
expr = (c.T*a*x.T*b)**Rational(1, 3)
assert expr.diff(x) == b*a.T*c*(c.T*a*x.T*b)**Rational(-2, 3)/3
expr = (a.T*X*b)**S.Half
assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*b.T
expr = d.T*x*(a.T*X*b)**S.Half*y.T*c
assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*x.T*d*y.T*c*b.T
def test_derivatives_elementwise_applyfunc():
expr = x.applyfunc(tan)
assert expr.diff(x).dummy_eq(
DiagMatrix(x.applyfunc(lambda x: tan(x)**2 + 1)))
assert expr[i, 0].diff(x[m, 0]).doit() == (tan(x[i, 0])**2 + 1)*KDelta(i, m)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (i**2*x).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct((2*i)*x, (i**2*x).applyfunc(cos)))
assert expr[i, 0].diff(i).doit() == 2*i*x[i, 0]*cos(i**2*x[i, 0])
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = (log(i)*A*B).applyfunc(sin)
assert expr.diff(i).dummy_eq(
HadamardProduct(A*B/i, (log(i)*A*B).applyfunc(cos)))
_check_derivative_with_explicit_matrix(expr, i, expr.diff(i))
expr = A*x.applyfunc(exp)
# TODO: restore this result (currently returning the transpose):
# assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(exp))*A.T)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.T*A*x + k*y.applyfunc(sin).T*x
assert expr.diff(x).dummy_eq(A.T*x + A*x + k*y.applyfunc(sin))
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = x.applyfunc(sin).T*y
# TODO: restore (currently returning the traspose):
# assert expr.diff(x).dummy_eq(DiagMatrix(x.applyfunc(cos))*y)
_check_derivative_with_explicit_matrix(expr, x, expr.diff(x))
expr = (a.T * X * b).applyfunc(sin)
assert expr.diff(X).dummy_eq(a*(a.T*X*b).applyfunc(cos)*b.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * X.applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
DiagMatrix(a)*X.applyfunc(cos)*DiagMatrix(b))
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A*X*B).applyfunc(sin) * b
assert expr.diff(X).dummy_eq(
A.T*DiagMatrix(a)*(A*X*B).applyfunc(cos)*DiagMatrix(b)*B.T)
_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T * (A*X*b).applyfunc(sin) * b.T
# TODO: not implemented
#assert expr.diff(X) == ...
#_check_derivative_with_explicit_matrix(expr, X, expr.diff(X))
expr = a.T*A*X.applyfunc(sin)*B*b
assert expr.diff(X).dummy_eq(
HadamardProduct(A.T * a * b.T * B.T, X.applyfunc(cos)))
expr = a.T * (A*X.applyfunc(sin)*B).applyfunc(log) * b
# TODO: wrong
# assert expr.diff(X) == A.T*DiagMatrix(a)*(A*X.applyfunc(sin)*B).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)*B.T
expr = a.T * (X.applyfunc(sin)).applyfunc(log) * b
# TODO: wrong
# assert expr.diff(X) == DiagMatrix(a)*X.applyfunc(sin).applyfunc(Lambda(k, 1/k))*DiagMatrix(b)
def test_derivatives_of_hadamard_expressions():
# Hadamard Product
expr = hadamard_product(a, x, b)
assert expr.diff(x) == DiagMatrix(hadamard_product(b, a))
expr = a.T*hadamard_product(A, X, B)*b
assert expr.diff(X) == HadamardProduct(a*b.T, A, B)
# Hadamard Power
expr = hadamard_power(x, 2)
assert expr.diff(x).doit() == 2*DiagMatrix(x)
expr = hadamard_power(x.T, 2)
assert expr.diff(x).doit() == 2*DiagMatrix(x)
expr = hadamard_power(x, S.Half)
assert expr.diff(x) == S.Half*DiagMatrix(hadamard_power(x, Rational(-1, 2)))
expr = hadamard_power(a.T*X*b, 2)
assert expr.diff(X) == 2*a*a.T*X*b*b.T
expr = hadamard_power(a.T*X*b, S.Half)
assert expr.diff(X) == a/(2*sqrt(a.T*X*b))*b.T
|
11856ce87150347d97a341f808c267442393c9549858f3f838775e3709b0ef79 | from sympy.concrete.summations import Sum
from sympy.core.exprtools import gcd_terms
from sympy.core.function import (diff, expand)
from sympy.core.relational import Eq
from sympy.core.symbol import (Dummy, Symbol)
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.dense import zeros
from sympy.polys.polytools import factor
from sympy.core import (S, symbols, Add, Mul, SympifyError, Rational,
Function)
from sympy.functions import sin, cos, tan, sqrt, cbrt, exp
from sympy.simplify import simplify
from sympy.matrices import (ImmutableMatrix, Inverse, MatAdd, MatMul,
MatPow, Matrix, MatrixExpr, MatrixSymbol, ShapeError,
SparseMatrix, Transpose, Adjoint, NonSquareMatrixError, MatrixSet)
from sympy.matrices.expressions.determinant import Determinant, det
from sympy.matrices.expressions.matexpr import MatrixElement
from sympy.matrices.expressions.special import ZeroMatrix, Identity
from sympy.testing.pytest import raises, XFAIL
n, m, l, k, p = symbols('n m l k p', 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)
w = MatrixSymbol('w', n, 1)
def test_matrix_symbol_creation():
assert MatrixSymbol('A', 2, 2)
assert MatrixSymbol('A', 0, 0)
raises(ValueError, lambda: MatrixSymbol('A', -1, 2))
raises(ValueError, lambda: MatrixSymbol('A', 2.0, 2))
raises(ValueError, lambda: MatrixSymbol('A', 2j, 2))
raises(ValueError, lambda: MatrixSymbol('A', 2, -1))
raises(ValueError, lambda: MatrixSymbol('A', 2, 2.0))
raises(ValueError, lambda: MatrixSymbol('A', 2, 2j))
n = symbols('n')
assert MatrixSymbol('A', n, n)
n = symbols('n', integer=False)
raises(ValueError, lambda: MatrixSymbol('A', n, n))
n = symbols('n', negative=True)
raises(ValueError, lambda: MatrixSymbol('A', n, n))
def test_shape():
assert A.shape == (n, m)
assert (A*B).shape == (n, l)
raises(ShapeError, lambda: B*A)
def test_matexpr():
assert (x*A).shape == A.shape
assert (x*A).__class__ == MatMul
assert 2*A - A - A == ZeroMatrix(*A.shape)
assert (A*B).shape == (n, l)
def test_subs():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', m, l)
assert A.subs(n, m).shape == (m, m)
assert (A*B).subs(B, C) == A*C
assert (A*B).subs(l, n).is_square
A = SparseMatrix([[1, 2], [3, 4]])
B = Matrix([[1, 2], [3, 4]])
C, D = MatrixSymbol('C', 2, 2), MatrixSymbol('D', 2, 2)
assert (C*D).subs({C: A, D: B}) == MatMul(A, B)
def test_addition():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, m)
assert isinstance(A + B, MatAdd)
assert (A + B).shape == A.shape
assert isinstance(A - A + 2*B, MatMul)
raises(ShapeError, lambda: A + B.T)
raises(TypeError, lambda: A + 1)
raises(TypeError, lambda: 5 + A)
raises(TypeError, lambda: 5 - A)
assert A + ZeroMatrix(n, m) - A == ZeroMatrix(n, m)
with raises(TypeError):
ZeroMatrix(n,m) + S.Zero
def test_multiplication():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', n, n)
assert (2*A*B).shape == (n, l)
assert (A*0*B) == ZeroMatrix(n, l)
raises(ShapeError, lambda: B*A)
assert (2*A).shape == A.shape
assert A * ZeroMatrix(m, m) * B == ZeroMatrix(n, l)
assert C * Identity(n) * C.I == Identity(n)
assert B/2 == S.Half*B
raises(NotImplementedError, lambda: 2/B)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
assert Identity(n) * (A + B) == A + B
assert A**2*A == A**3
assert A**2*(A.I)**3 == A.I
assert A**3*(A.I)**2 == A
def test_MatPow():
A = MatrixSymbol('A', n, n)
AA = MatPow(A, 2)
assert AA.exp == 2
assert AA.base == A
assert (A**n).exp == n
assert A**0 == Identity(n)
assert A**1 == A
assert A**2 == AA
assert A**-1 == Inverse(A)
assert (A**-1)**-1 == A
assert (A**2)**3 == A**6
assert A**S.Half == sqrt(A)
assert A**Rational(1, 3) == cbrt(A)
raises(NonSquareMatrixError, lambda: MatrixSymbol('B', 3, 2)**2)
def test_MatrixSymbol():
n, m, t = symbols('n,m,t')
X = MatrixSymbol('X', n, m)
assert X.shape == (n, m)
raises(TypeError, lambda: MatrixSymbol('X', n, m)(t)) # issue 5855
assert X.doit() == X
def test_dense_conversion():
X = MatrixSymbol('X', 2, 2)
assert ImmutableMatrix(X) == ImmutableMatrix(2, 2, lambda i, j: X[i, j])
assert Matrix(X) == Matrix(2, 2, lambda i, j: X[i, j])
def test_free_symbols():
assert (C*D).free_symbols == {C, D}
def test_zero_matmul():
assert isinstance(S.Zero * MatrixSymbol('X', 2, 2), MatrixExpr)
def test_matadd_simplify():
A = MatrixSymbol('A', 1, 1)
assert simplify(MatAdd(A, ImmutableMatrix([[sin(x)**2 + cos(x)**2]]))) == \
MatAdd(A, Matrix([[1]]))
def test_matmul_simplify():
A = MatrixSymbol('A', 1, 1)
assert simplify(MatMul(A, ImmutableMatrix([[sin(x)**2 + cos(x)**2]]))) == \
MatMul(A, Matrix([[1]]))
def test_invariants():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
X = MatrixSymbol('X', n, n)
objs = [Identity(n), ZeroMatrix(m, n), A, MatMul(A, B), MatAdd(A, A),
Transpose(A), Adjoint(A), Inverse(X), MatPow(X, 2), MatPow(X, -1),
MatPow(X, 0)]
for obj in objs:
assert obj == obj.__class__(*obj.args)
def test_indexing():
A = MatrixSymbol('A', n, m)
A[1, 2]
A[l, k]
A[l+1, k+1]
def test_single_indexing():
A = MatrixSymbol('A', 2, 3)
assert A[1] == A[0, 1]
assert A[int(1)] == A[0, 1]
assert A[3] == A[1, 0]
assert list(A[:2, :2]) == [A[0, 0], A[0, 1], A[1, 0], A[1, 1]]
raises(IndexError, lambda: A[6])
raises(IndexError, lambda: A[n])
B = MatrixSymbol('B', n, m)
raises(IndexError, lambda: B[1])
B = MatrixSymbol('B', n, 3)
assert B[3] == B[1, 0]
def test_MatrixElement_commutative():
assert A[0, 1]*A[1, 0] == A[1, 0]*A[0, 1]
def test_MatrixSymbol_determinant():
A = MatrixSymbol('A', 4, 4)
assert A.as_explicit().det() == A[0, 0]*A[1, 1]*A[2, 2]*A[3, 3] - \
A[0, 0]*A[1, 1]*A[2, 3]*A[3, 2] - A[0, 0]*A[1, 2]*A[2, 1]*A[3, 3] + \
A[0, 0]*A[1, 2]*A[2, 3]*A[3, 1] + A[0, 0]*A[1, 3]*A[2, 1]*A[3, 2] - \
A[0, 0]*A[1, 3]*A[2, 2]*A[3, 1] - A[0, 1]*A[1, 0]*A[2, 2]*A[3, 3] + \
A[0, 1]*A[1, 0]*A[2, 3]*A[3, 2] + A[0, 1]*A[1, 2]*A[2, 0]*A[3, 3] - \
A[0, 1]*A[1, 2]*A[2, 3]*A[3, 0] - A[0, 1]*A[1, 3]*A[2, 0]*A[3, 2] + \
A[0, 1]*A[1, 3]*A[2, 2]*A[3, 0] + A[0, 2]*A[1, 0]*A[2, 1]*A[3, 3] - \
A[0, 2]*A[1, 0]*A[2, 3]*A[3, 1] - A[0, 2]*A[1, 1]*A[2, 0]*A[3, 3] + \
A[0, 2]*A[1, 1]*A[2, 3]*A[3, 0] + A[0, 2]*A[1, 3]*A[2, 0]*A[3, 1] - \
A[0, 2]*A[1, 3]*A[2, 1]*A[3, 0] - A[0, 3]*A[1, 0]*A[2, 1]*A[3, 2] + \
A[0, 3]*A[1, 0]*A[2, 2]*A[3, 1] + A[0, 3]*A[1, 1]*A[2, 0]*A[3, 2] - \
A[0, 3]*A[1, 1]*A[2, 2]*A[3, 0] - A[0, 3]*A[1, 2]*A[2, 0]*A[3, 1] + \
A[0, 3]*A[1, 2]*A[2, 1]*A[3, 0]
B = MatrixSymbol('B', 4, 4)
assert Determinant(A + B).doit() == det(A + B) == (A + B).det()
def test_MatrixElement_diff():
assert (A[3, 0]*A[0, 0]).diff(A[0, 0]) == A[3, 0]
def test_MatrixElement_doit():
u = MatrixSymbol('u', 2, 1)
v = ImmutableMatrix([3, 5])
assert u[0, 0].subs(u, v).doit() == v[0, 0]
def test_identity_powers():
M = Identity(n)
assert MatPow(M, 3).doit() == M**3
assert M**n == M
assert MatPow(M, 0).doit() == M**2
assert M**-2 == M
assert MatPow(M, -2).doit() == M**0
N = Identity(3)
assert MatPow(N, 2).doit() == N**n
assert MatPow(N, 3).doit() == N
assert MatPow(N, -2).doit() == N**4
assert MatPow(N, 2).doit() == N**0
def test_Zero_power():
z1 = ZeroMatrix(n, n)
assert z1**4 == z1
raises(ValueError, lambda:z1**-2)
assert z1**0 == Identity(n)
assert MatPow(z1, 2).doit() == z1**2
raises(ValueError, lambda:MatPow(z1, -2).doit())
z2 = ZeroMatrix(3, 3)
assert MatPow(z2, 4).doit() == z2**4
raises(ValueError, lambda:z2**-3)
assert z2**3 == MatPow(z2, 3).doit()
assert z2**0 == Identity(3)
raises(ValueError, lambda:MatPow(z2, -1).doit())
def test_matrixelement_diff():
dexpr = diff((D*w)[k,0], w[p,0])
assert w[k, p].diff(w[k, p]) == 1
assert w[k, p].diff(w[0, 0]) == KroneckerDelta(0, k, (0, n-1))*KroneckerDelta(0, p, (0, 0))
_i_1 = Dummy("_i_1")
assert dexpr.dummy_eq(Sum(KroneckerDelta(_i_1, p, (0, n-1))*D[k, _i_1], (_i_1, 0, n - 1)))
assert dexpr.doit() == D[k, p]
def test_MatrixElement_with_values():
x, y, z, w = symbols("x y z w")
M = Matrix([[x, y], [z, w]])
i, j = symbols("i, j")
Mij = M[i, j]
assert isinstance(Mij, MatrixElement)
Ms = SparseMatrix([[2, 3], [4, 5]])
msij = Ms[i, j]
assert isinstance(msij, MatrixElement)
for oi, oj in [(0, 0), (0, 1), (1, 0), (1, 1)]:
assert Mij.subs({i: oi, j: oj}) == M[oi, oj]
assert msij.subs({i: oi, j: oj}) == Ms[oi, oj]
A = MatrixSymbol("A", 2, 2)
assert A[0, 0].subs(A, M) == x
assert A[i, j].subs(A, M) == M[i, j]
assert M[i, j].subs(M, A) == A[i, j]
assert isinstance(M[3*i - 2, j], MatrixElement)
assert M[3*i - 2, j].subs({i: 1, j: 0}) == M[1, 0]
assert isinstance(M[i, 0], MatrixElement)
assert M[i, 0].subs(i, 0) == M[0, 0]
assert M[0, i].subs(i, 1) == M[0, 1]
assert M[i, j].diff(x) == Matrix([[1, 0], [0, 0]])[i, j]
raises(ValueError, lambda: M[i, 2])
raises(ValueError, lambda: M[i, -1])
raises(ValueError, lambda: M[2, i])
raises(ValueError, lambda: M[-1, i])
def test_inv():
B = MatrixSymbol('B', 3, 3)
assert B.inv() == B**-1
# https://github.com/sympy/sympy/issues/19162
X = MatrixSymbol('X', 1, 1).as_explicit()
assert X.inv() == Matrix([[1/X[0, 0]]])
X = MatrixSymbol('X', 2, 2).as_explicit()
detX = X[0, 0]*X[1, 1] - X[0, 1]*X[1, 0]
invX = Matrix([[ X[1, 1], -X[0, 1]],
[-X[1, 0], X[0, 0]]]) / detX
assert X.inv() == invX
@XFAIL
def test_factor_expand():
A = MatrixSymbol("A", n, n)
B = MatrixSymbol("B", n, n)
expr1 = (A + B)*(C + D)
expr2 = A*C + B*C + A*D + B*D
assert expr1 != expr2
assert expand(expr1) == expr2
assert factor(expr2) == expr1
expr = B**(-1)*(A**(-1)*B**(-1) - A**(-1)*C*B**(-1))**(-1)*A**(-1)
I = Identity(n)
# Ideally we get the first, but we at least don't want a wrong answer
assert factor(expr) in [I - C, B**-1*(A**-1*(I - C)*B**-1)**-1*A**-1]
def test_issue_2749():
A = MatrixSymbol("A", 5, 2)
assert (A.T * A).I.as_explicit() == Matrix([[(A.T * A).I[0, 0], (A.T * A).I[0, 1]], \
[(A.T * A).I[1, 0], (A.T * A).I[1, 1]]])
def test_issue_2750():
x = MatrixSymbol('x', 1, 1)
assert (x.T*x).as_explicit()**-1 == Matrix([[x[0, 0]**(-2)]])
def test_issue_7842():
A = MatrixSymbol('A', 3, 1)
B = MatrixSymbol('B', 2, 1)
assert Eq(A, B) == False
assert Eq(A[1,0], B[1, 0]).func is Eq
A = ZeroMatrix(2, 3)
B = ZeroMatrix(2, 3)
assert Eq(A, B) == True
def test_issue_21195():
t = symbols('t')
x = Function('x')(t)
dx = x.diff(t)
exp1 = cos(x) + cos(x)*dx
exp2 = sin(x) + tan(x)*(dx.diff(t))
exp3 = sin(x)*sin(t)*(dx.diff(t)).diff(t)
A = Matrix([[exp1], [exp2], [exp3]])
B = Matrix([[exp1.diff(x)], [exp2.diff(x)], [exp3.diff(x)]])
assert A.diff(x) == B
def test_MatMul_postprocessor():
z = zeros(2)
z1 = ZeroMatrix(2, 2)
assert Mul(0, z) == Mul(z, 0) in [z, z1]
M = Matrix([[1, 2], [3, 4]])
Mx = Matrix([[x, 2*x], [3*x, 4*x]])
assert Mul(x, M) == Mul(M, x) == Mx
A = MatrixSymbol("A", 2, 2)
assert Mul(A, M) == MatMul(A, M)
assert Mul(M, A) == MatMul(M, A)
# Scalars should be absorbed into constant matrices
a = Mul(x, M, A)
b = Mul(M, x, A)
c = Mul(M, A, x)
assert a == b == c == MatMul(Mx, A)
a = Mul(x, A, M)
b = Mul(A, x, M)
c = Mul(A, M, x)
assert a == b == c == MatMul(A, Mx)
assert Mul(M, M) == M**2
assert Mul(A, M, M) == MatMul(A, M**2)
assert Mul(M, M, A) == MatMul(M**2, A)
assert Mul(M, A, M) == MatMul(M, A, M)
assert Mul(A, x, M, M, x) == MatMul(A, Mx**2)
@XFAIL
def test_MatAdd_postprocessor_xfail():
# This is difficult to get working because of the way that Add processes
# its args.
z = zeros(2)
assert Add(z, S.NaN) == Add(S.NaN, z)
def test_MatAdd_postprocessor():
# Some of these are nonsensical, but we do not raise errors for Add
# because that breaks algorithms that want to replace matrices with dummy
# symbols.
z = zeros(2)
assert Add(0, z) == Add(z, 0) == z
a = Add(S.Infinity, z)
assert a == Add(z, S.Infinity)
assert isinstance(a, Add)
assert a.args == (S.Infinity, z)
a = Add(S.ComplexInfinity, z)
assert a == Add(z, S.ComplexInfinity)
assert isinstance(a, Add)
assert a.args == (S.ComplexInfinity, z)
a = Add(z, S.NaN)
# assert a == Add(S.NaN, z) # See the XFAIL above
assert isinstance(a, Add)
assert a.args == (S.NaN, z)
M = Matrix([[1, 2], [3, 4]])
a = Add(x, M)
assert a == Add(M, x)
assert isinstance(a, Add)
assert a.args == (x, M)
A = MatrixSymbol("A", 2, 2)
assert Add(A, M) == Add(M, A) == A + M
# Scalars should be absorbed into constant matrices (producing an error)
a = Add(x, M, A)
assert a == Add(M, x, A) == Add(M, A, x) == Add(x, A, M) == Add(A, x, M) == Add(A, M, x)
assert isinstance(a, Add)
assert a.args == (x, A + M)
assert Add(M, M) == 2*M
assert Add(M, A, M) == Add(M, M, A) == Add(A, M, M) == A + 2*M
a = Add(A, x, M, M, x)
assert isinstance(a, Add)
assert a.args == (2*x, A + 2*M)
def test_simplify_matrix_expressions():
# Various simplification functions
assert type(gcd_terms(C*D + D*C)) == MatAdd
a = gcd_terms(2*C*D + 4*D*C)
assert type(a) == MatAdd
assert a.args == (2*C*D, 4*D*C)
def test_exp():
A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
expr1 = exp(A)*exp(B)
expr2 = exp(B)*exp(A)
assert expr1 != expr2
assert expr1 - expr2 != 0
assert not isinstance(expr1, exp)
assert not isinstance(expr2, exp)
def test_invalid_args():
raises(SympifyError, lambda: MatrixSymbol(1, 2, 'A'))
def test_matrixsymbol_from_symbol():
# The label should be preserved during doit and subs
A_label = Symbol('A', complex=True)
A = MatrixSymbol(A_label, 2, 2)
A_1 = A.doit()
A_2 = A.subs(2, 3)
assert A_1.args == A.args
assert A_2.args[0] == A.args[0]
def test_as_explicit():
Z = MatrixSymbol('Z', 2, 3)
assert Z.as_explicit() == ImmutableMatrix([
[Z[0, 0], Z[0, 1], Z[0, 2]],
[Z[1, 0], Z[1, 1], Z[1, 2]],
])
raises(ValueError, lambda: A.as_explicit())
def test_MatrixSet():
M = MatrixSet(2, 2, set=S.Reals)
assert M.shape == (2, 2)
assert M.set == S.Reals
X = Matrix([[1, 2], [3, 4]])
assert X in M
X = ZeroMatrix(2, 2)
assert X in M
raises(TypeError, lambda: A in M)
raises(TypeError, lambda: 1 in M)
M = MatrixSet(n, m, set=S.Reals)
assert A in M
raises(TypeError, lambda: C in M)
raises(TypeError, lambda: X in M)
M = MatrixSet(2, 2, set={1, 2, 3})
X = Matrix([[1, 2], [3, 4]])
Y = Matrix([[1, 2]])
assert (X in M) == S.false
assert (Y in M) == S.false
raises(ValueError, lambda: MatrixSet(2, -2, S.Reals))
raises(ValueError, lambda: MatrixSet(2.4, -1, S.Reals))
raises(TypeError, lambda: MatrixSet(2, 2, (1, 2, 3)))
def test_matrixsymbol_solving():
A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
Z = ZeroMatrix(2, 2)
assert -(-A + B) - A + B == Z
assert (-(-A + B) - A + B).simplify() == Z
assert (-(-A + B) - A + B).expand() == Z
assert (-(-A + B) - A + B - Z).simplify() == Z
assert (-(-A + B) - A + B - Z).expand() == Z
|
e943c0fd48422f924f6da40239ab4c9f18469d0f9a20799bde55ea6f0ad2a081 | from sympy.matrices.expressions.factorizations import lu, LofCholesky, qr, svd
from sympy.assumptions.ask import (Q, ask)
from sympy.core.symbol import Symbol
from sympy.matrices.expressions.matexpr import MatrixSymbol
n = Symbol('n')
X = MatrixSymbol('X', n, n)
def test_LU():
L, U = lu(X)
assert L.shape == U.shape == X.shape
assert ask(Q.lower_triangular(L))
assert ask(Q.upper_triangular(U))
def test_Cholesky():
LofCholesky(X)
def test_QR():
Q_, R = qr(X)
assert Q_.shape == R.shape == X.shape
assert ask(Q.orthogonal(Q_))
assert ask(Q.upper_triangular(R))
def test_svd():
U, S, V = svd(X)
assert U.shape == S.shape == V.shape == X.shape
assert ask(Q.orthogonal(U))
assert ask(Q.orthogonal(V))
assert ask(Q.diagonal(S))
|
c306106bb2cd9738410397d4e7480fc84f1ca9a25fd2cdceef43d4c7878d0840 | from sympy.concrete.summations import Sum
from sympy.core.symbol import symbols, Symbol, Dummy
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.dense import eye
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions.hadamard import HadamardPower
from sympy.matrices.expressions.matexpr import (MatrixSymbol,
MatrixExpr, MatrixElement)
from sympy.matrices.expressions.matpow import MatPow
from sympy.matrices.expressions.special import (ZeroMatrix, Identity,
OneMatrix)
from sympy.matrices.expressions.trace import Trace, trace
from sympy.matrices.immutable import ImmutableMatrix
from sympy.tensor.array.expressions.array_expressions import ArrayTensorProduct
from sympy.testing.pytest import XFAIL, raises
k, l, m, n = symbols('k l m n', integer=True)
i, j = symbols('i j', integer=True)
W = MatrixSymbol('W', k, l)
X = MatrixSymbol('X', l, m)
Y = MatrixSymbol('Y', l, m)
Z = MatrixSymbol('Z', m, n)
X1 = MatrixSymbol('X1', m, m)
X2 = MatrixSymbol('X2', m, m)
X3 = MatrixSymbol('X3', m, m)
X4 = MatrixSymbol('X4', m, m)
A = MatrixSymbol('A', 2, 2)
B = MatrixSymbol('B', 2, 2)
x = MatrixSymbol('x', 1, 2)
y = MatrixSymbol('x', 2, 1)
def test_symbolic_indexing():
x12 = X[1, 2]
assert all(s in str(x12) for s in ['1', '2', X.name])
# We don't care about the exact form of this. We do want to make sure
# that all of these features are present
def test_add_index():
assert (X + Y)[i, j] == X[i, j] + Y[i, j]
def test_mul_index():
assert (A*y)[0, 0] == A[0, 0]*y[0, 0] + A[0, 1]*y[1, 0]
assert (A*B).as_mutable() == (A.as_mutable() * B.as_mutable())
X = MatrixSymbol('X', n, m)
Y = MatrixSymbol('Y', m, k)
result = (X*Y)[4,2]
expected = Sum(X[4, i]*Y[i, 2], (i, 0, m - 1))
assert result.args[0].dummy_eq(expected.args[0], i)
assert result.args[1][1:] == expected.args[1][1:]
def test_pow_index():
Q = MatPow(A, 2)
assert Q[0, 0] == A[0, 0]**2 + A[0, 1]*A[1, 0]
n = symbols("n")
Q2 = A**n
assert Q2[0, 0] == 2*(
-sqrt((A[0, 0] + A[1, 1])**2 - 4*A[0, 0]*A[1, 1] +
4*A[0, 1]*A[1, 0])/2 + A[0, 0]/2 + A[1, 1]/2
)**n * \
A[0, 1]*A[1, 0]/(
(sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] +
A[1, 1]**2) + A[0, 0] - A[1, 1])*
sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)
) - 2*(
sqrt((A[0, 0] + A[1, 1])**2 - 4*A[0, 0]*A[1, 1] +
4*A[0, 1]*A[1, 0])/2 + A[0, 0]/2 + A[1, 1]/2
)**n * A[0, 1]*A[1, 0]/(
(-sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] +
A[1, 1]**2) + A[0, 0] - A[1, 1])*
sqrt(A[0, 0]**2 - 2*A[0, 0]*A[1, 1] + 4*A[0, 1]*A[1, 0] + A[1, 1]**2)
)
def test_transpose_index():
assert X.T[i, j] == X[j, i]
def test_Identity_index():
I = Identity(3)
assert I[0, 0] == I[1, 1] == I[2, 2] == 1
assert I[1, 0] == I[0, 1] == I[2, 1] == 0
assert I[i, 0].delta_range == (0, 2)
raises(IndexError, lambda: I[3, 3])
def test_block_index():
I = Identity(3)
Z = ZeroMatrix(3, 3)
B = BlockMatrix([[I, I], [I, I]])
e3 = ImmutableMatrix(eye(3))
BB = BlockMatrix([[e3, e3], [e3, e3]])
assert B[0, 0] == B[3, 0] == B[0, 3] == B[3, 3] == 1
assert B[4, 3] == B[5, 1] == 0
BB = BlockMatrix([[e3, e3], [e3, e3]])
assert B.as_explicit() == BB.as_explicit()
BI = BlockMatrix([[I, Z], [Z, I]])
assert BI.as_explicit().equals(eye(6))
def test_block_index_symbolic():
# Note that these matrices may be zero-sized and indices may be negative, which causes
# all naive simplifications given in the comments to be invalid
A1 = MatrixSymbol('A1', n, k)
A2 = MatrixSymbol('A2', n, l)
A3 = MatrixSymbol('A3', m, k)
A4 = MatrixSymbol('A4', m, l)
A = BlockMatrix([[A1, A2], [A3, A4]])
assert A[0, 0] == MatrixElement(A, 0, 0) # Cannot be A1[0, 0]
assert A[n - 1, k - 1] == A1[n - 1, k - 1]
assert A[n, k] == A4[0, 0]
assert A[n + m - 1, 0] == MatrixElement(A, n + m - 1, 0) # Cannot be A3[m - 1, 0]
assert A[0, k + l - 1] == MatrixElement(A, 0, k + l - 1) # Cannot be A2[0, l - 1]
assert A[n + m - 1, k + l - 1] == MatrixElement(A, n + m - 1, k + l - 1) # Cannot be A4[m - 1, l - 1]
assert A[i, j] == MatrixElement(A, i, j)
assert A[n + i, k + j] == MatrixElement(A, n + i, k + j) # Cannot be A4[i, j]
assert A[n - i - 1, k - j - 1] == MatrixElement(A, n - i - 1, k - j - 1) # Cannot be A1[n - i - 1, k - j - 1]
def test_block_index_symbolic_nonzero():
# All invalid simplifications from test_block_index_symbolic() that become valid if all
# matrices have nonzero size and all indices are nonnegative
k, l, m, n = symbols('k l m n', integer=True, positive=True)
i, j = symbols('i j', integer=True, nonnegative=True)
A1 = MatrixSymbol('A1', n, k)
A2 = MatrixSymbol('A2', n, l)
A3 = MatrixSymbol('A3', m, k)
A4 = MatrixSymbol('A4', m, l)
A = BlockMatrix([[A1, A2], [A3, A4]])
assert A[0, 0] == A1[0, 0]
assert A[n + m - 1, 0] == A3[m - 1, 0]
assert A[0, k + l - 1] == A2[0, l - 1]
assert A[n + m - 1, k + l - 1] == A4[m - 1, l - 1]
assert A[i, j] == MatrixElement(A, i, j)
assert A[n + i, k + j] == A4[i, j]
assert A[n - i - 1, k - j - 1] == A1[n - i - 1, k - j - 1]
assert A[2 * n, 2 * k] == A4[n, k]
def test_block_index_large():
n, m, k = symbols('n m k', integer=True, positive=True)
i = symbols('i', integer=True, nonnegative=True)
A1 = MatrixSymbol('A1', n, n)
A2 = MatrixSymbol('A2', n, m)
A3 = MatrixSymbol('A3', n, k)
A4 = MatrixSymbol('A4', m, n)
A5 = MatrixSymbol('A5', m, m)
A6 = MatrixSymbol('A6', m, k)
A7 = MatrixSymbol('A7', k, n)
A8 = MatrixSymbol('A8', k, m)
A9 = MatrixSymbol('A9', k, k)
A = BlockMatrix([[A1, A2, A3], [A4, A5, A6], [A7, A8, A9]])
assert A[n + i, n + i] == MatrixElement(A, n + i, n + i)
@XFAIL
def test_block_index_symbolic_fail():
# To make this work, symbolic matrix dimensions would need to be somehow assumed nonnegative
# even if the symbols aren't specified as such. Then 2 * n < n would correctly evaluate to
# False in BlockMatrix._entry()
A1 = MatrixSymbol('A1', n, 1)
A2 = MatrixSymbol('A2', m, 1)
A = BlockMatrix([[A1], [A2]])
assert A[2 * n, 0] == A2[n, 0]
def test_slicing():
A.as_explicit()[0, :] # does not raise an error
def test_errors():
raises(IndexError, lambda: Identity(2)[1, 2, 3, 4, 5])
raises(IndexError, lambda: Identity(2)[[1, 2, 3, 4, 5]])
def test_matrix_expression_to_indices():
i, j = symbols("i, j")
i1, i2, i3 = symbols("i_1:4")
def replace_dummies(expr):
repl = {i: Symbol(i.name) for i in expr.atoms(Dummy)}
return expr.xreplace(repl)
expr = W*X*Z
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = Z.T*X.T*W.T
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[j, i2]*X[i2, i1]*Z[i1, i], (i1, 0, m-1), (i2, 0, l-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j), i) == expr
expr = W*X*Z + W*Y*Z
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = 2*W*X*Z + 3*W*Y*Z
assert replace_dummies(expr._entry(i, j)) == \
2*Sum(W[i, i1]*X[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1)) +\
3*Sum(W[i, i1]*Y[i1, i2]*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = W*(X + Y)*Z
assert replace_dummies(expr._entry(i, j)) == \
Sum(W[i, i1]*(X[i1, i2] + Y[i1, i2])*Z[i2, j], (i1, 0, l-1), (i2, 0, m-1))
assert MatrixExpr.from_index_summation(expr._entry(i, j)) == expr
expr = A*B**2*A
#assert replace_dummies(expr._entry(i, j)) == \
# Sum(A[i, i1]*B[i1, i2]*B[i2, i3]*A[i3, j], (i1, 0, 1), (i2, 0, 1), (i3, 0, 1))
# Check that different dummies are used in sub-multiplications:
expr = (X1*X2 + X2*X1)*X3
assert replace_dummies(expr._entry(i, j)) == \
Sum((Sum(X1[i, i2] * X2[i2, i1], (i2, 0, m - 1)) + Sum(X1[i3, i1] * X2[i, i3], (i3, 0, m - 1))) * X3[
i1, j], (i1, 0, m - 1))
def test_matrix_expression_from_index_summation():
from sympy.abc import a,b,c,d
A = MatrixSymbol("A", k, k)
B = MatrixSymbol("B", k, k)
C = MatrixSymbol("C", k, k)
w1 = MatrixSymbol("w1", k, 1)
i0, i1, i2, i3, i4 = symbols("i0:5", cls=Dummy)
expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
expr = Sum(W.T[b,a]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m-1))
assert MatrixExpr.from_index_summation(expr, a) == W*X*Z
expr = Sum(A[b, a]*B[b, c]*C[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixSymbol.from_index_summation(expr, a) == A.T*B*C
expr = Sum(A[b, a]*B[c, b]*C[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
expr = Sum(C[c, d]*A[b, a]*B[c, b], (b, 0, k-1), (c, 0, k-1))
assert MatrixSymbol.from_index_summation(expr, a) == A.T*B.T*C
expr = Sum(A[a, b] + B[a, b], (a, 0, k-1), (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == OneMatrix(1, k)*A*OneMatrix(k, 1) + OneMatrix(1, k)*B*OneMatrix(k, 1)
expr = Sum(A[a, b]**2, (a, 0, k - 1), (b, 0, k - 1))
assert MatrixExpr.from_index_summation(expr, a) == Trace(A * A.T)
expr = Sum(A[a, b]**3, (a, 0, k - 1), (b, 0, k - 1))
assert MatrixExpr.from_index_summation(expr, a) == Trace(HadamardPower(A.T, 2) * A)
expr = Sum((A[a, b] + B[a, b])*C[b, c], (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == (A+B)*C
expr = Sum((A[a, b] + B[b, a])*C[b, c], (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == (A+B.T)*C
expr = Sum(A[a, b]*A[b, c]*A[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A**3
expr = Sum(A[a, b]*A[b, c]*B[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A**2*B
# Parse the trace of a matrix:
expr = Sum(A[a, a], (a, 0, k-1))
assert MatrixExpr.from_index_summation(expr, None) == trace(A)
expr = Sum(A[a, a]*B[b, c]*C[c, d], (a, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, b) == trace(A)*B*C
# Check wrong sum ranges (should raise an exception):
## Case 1: 0 to m instead of 0 to m-1
expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 0, m))
raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))
## Case 2: 1 to m-1 instead of 0 to m-1
expr = Sum(W[a,b]*X[b,c]*Z[c,d], (b, 0, l-1), (c, 1, m-1))
raises(ValueError, lambda: MatrixExpr.from_index_summation(expr, a))
# Parse nested sums:
expr = Sum(A[a, b]*Sum(B[b, c]*C[c, d], (c, 0, k-1)), (b, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A*B*C
# Test Kronecker delta:
expr = Sum(A[a, b]*KroneckerDelta(b, c)*B[c, d], (b, 0, k-1), (c, 0, k-1))
assert MatrixExpr.from_index_summation(expr, a) == A*B
expr = Sum(KroneckerDelta(i1, m)*KroneckerDelta(i2, n)*A[i, i1]*A[j, i2], (i1, 0, k-1), (i2, 0, k-1))
assert MatrixExpr.from_index_summation(expr, m) == ArrayTensorProduct(A.T, A)
# Test numbered indices:
expr = Sum(A[i1, i2]*w1[i2, 0], (i2, 0, k-1))
assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*w1, i1, 0)
expr = Sum(A[i1, i2]*B[i2, 0], (i2, 0, k-1))
assert MatrixExpr.from_index_summation(expr, i1) == MatrixElement(A*B, i1, 0)
|
d1e0faf5f6a81d705cb90ee0d90df57eab3e83247f5df7be82ffbb0d784987f7 | from sympy.assumptions.ask import (Q, ask)
from sympy.core.numbers import (I, Rational)
from sympy.core.singleton import S
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.simplify.simplify import simplify
from sympy.core.symbol import symbols
from sympy.matrices.expressions.fourier import DFT, IDFT
from sympy.matrices import det, Matrix, Identity
from sympy.testing.pytest import raises
def test_dft_creation():
assert DFT(2)
assert DFT(0)
raises(ValueError, lambda: DFT(-1))
raises(ValueError, lambda: DFT(2.0))
raises(ValueError, lambda: DFT(2 + 1j))
n = symbols('n')
assert DFT(n)
n = symbols('n', integer=False)
raises(ValueError, lambda: DFT(n))
n = symbols('n', negative=True)
raises(ValueError, lambda: DFT(n))
def test_dft():
n, i, j = symbols('n i j')
assert DFT(4).shape == (4, 4)
assert ask(Q.unitary(DFT(4)))
assert Abs(simplify(det(Matrix(DFT(4))))) == 1
assert DFT(n)*IDFT(n) == Identity(n)
assert DFT(n)[i, j] == exp(-2*S.Pi*I/n)**(i*j) / sqrt(n)
def test_dft2():
assert DFT(1).as_explicit() == Matrix([[1]])
assert DFT(2).as_explicit() == 1/sqrt(2)*Matrix([[1,1],[1,-1]])
assert DFT(4).as_explicit() == Matrix([[S.Half, S.Half, S.Half, S.Half],
[S.Half, -I/2, Rational(-1,2), I/2],
[S.Half, Rational(-1,2), S.Half, Rational(-1,2)],
[S.Half, I/2, Rational(-1,2), -I/2]])
|
dcbe95b3f62bdfa6bc76022a01e21c3447f4c5c5727b21a100d36ed1d948a702 | from sympy.matrices.expressions import MatrixSymbol
from sympy.matrices.expressions.diagonal import DiagonalMatrix, DiagonalOf, DiagMatrix, diagonalize_vector
from sympy.assumptions.ask import (Q, ask)
from sympy.core.symbol import Symbol
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matmul import MatMul
from sympy.matrices.expressions.special import Identity
from sympy.testing.pytest import raises
n = Symbol('n')
m = Symbol('m')
def test_DiagonalMatrix():
x = MatrixSymbol('x', n, m)
D = DiagonalMatrix(x)
assert D.diagonal_length is None
assert D.shape == (n, m)
x = MatrixSymbol('x', n, n)
D = DiagonalMatrix(x)
assert D.diagonal_length == n
assert D.shape == (n, n)
assert D[1, 2] == 0
assert D[1, 1] == x[1, 1]
i = Symbol('i')
j = Symbol('j')
x = MatrixSymbol('x', 3, 3)
ij = DiagonalMatrix(x)[i, j]
assert ij != 0
assert ij.subs({i:0, j:0}) == x[0, 0]
assert ij.subs({i:0, j:1}) == 0
assert ij.subs({i:1, j:1}) == x[1, 1]
assert ask(Q.diagonal(D)) # affirm that D is diagonal
x = MatrixSymbol('x', n, 3)
D = DiagonalMatrix(x)
assert D.diagonal_length == 3
assert D.shape == (n, 3)
assert D[2, m] == KroneckerDelta(2, m)*x[2, m]
assert D[3, m] == 0
raises(IndexError, lambda: D[m, 3])
x = MatrixSymbol('x', 3, n)
D = DiagonalMatrix(x)
assert D.diagonal_length == 3
assert D.shape == (3, n)
assert D[m, 2] == KroneckerDelta(m, 2)*x[m, 2]
assert D[m, 3] == 0
raises(IndexError, lambda: D[3, m])
x = MatrixSymbol('x', n, m)
D = DiagonalMatrix(x)
assert D.diagonal_length is None
assert D.shape == (n, m)
assert D[m, 4] != 0
x = MatrixSymbol('x', 3, 4)
assert [DiagonalMatrix(x)[i] for i in range(12)] == [
x[0, 0], 0, 0, 0, 0, x[1, 1], 0, 0, 0, 0, x[2, 2], 0]
# shape is retained, issue 12427
assert (
DiagonalMatrix(MatrixSymbol('x', 3, 4))*
DiagonalMatrix(MatrixSymbol('x', 4, 2))).shape == (3, 2)
def test_DiagonalOf():
x = MatrixSymbol('x', n, n)
d = DiagonalOf(x)
assert d.shape == (n, 1)
assert d.diagonal_length == n
assert d[2, 0] == d[2] == x[2, 2]
x = MatrixSymbol('x', n, m)
d = DiagonalOf(x)
assert d.shape == (None, 1)
assert d.diagonal_length is None
assert d[2, 0] == d[2] == x[2, 2]
d = DiagonalOf(MatrixSymbol('x', 4, 3))
assert d.shape == (3, 1)
d = DiagonalOf(MatrixSymbol('x', n, 3))
assert d.shape == (3, 1)
d = DiagonalOf(MatrixSymbol('x', 3, n))
assert d.shape == (3, 1)
x = MatrixSymbol('x', n, m)
assert [DiagonalOf(x)[i] for i in range(4)] ==[
x[0, 0], x[1, 1], x[2, 2], x[3, 3]]
def test_DiagMatrix():
x = MatrixSymbol('x', n, 1)
d = DiagMatrix(x)
assert d.shape == (n, n)
assert d[0, 1] == 0
assert d[0, 0] == x[0, 0]
a = MatrixSymbol('a', 1, 1)
d = diagonalize_vector(a)
assert isinstance(d, MatrixSymbol)
assert a == d
assert diagonalize_vector(Identity(3)) == Identity(3)
assert DiagMatrix(Identity(3)).doit() == Identity(3)
assert isinstance(DiagMatrix(Identity(3)), DiagMatrix)
# A diagonal matrix is equal to its transpose:
assert DiagMatrix(x).T == DiagMatrix(x)
assert diagonalize_vector(x.T) == DiagMatrix(x)
dx = DiagMatrix(x)
assert dx[0, 0] == x[0, 0]
assert dx[1, 1] == x[1, 0]
assert dx[0, 1] == 0
assert dx[0, m] == x[0, 0]*KroneckerDelta(0, m)
z = MatrixSymbol('z', 1, n)
dz = DiagMatrix(z)
assert dz[0, 0] == z[0, 0]
assert dz[1, 1] == z[0, 1]
assert dz[0, 1] == 0
assert dz[0, m] == z[0, m]*KroneckerDelta(0, m)
v = MatrixSymbol('v', 3, 1)
dv = DiagMatrix(v)
assert dv.as_explicit() == Matrix([
[v[0, 0], 0, 0],
[0, v[1, 0], 0],
[0, 0, v[2, 0]],
])
v = MatrixSymbol('v', 1, 3)
dv = DiagMatrix(v)
assert dv.as_explicit() == Matrix([
[v[0, 0], 0, 0],
[0, v[0, 1], 0],
[0, 0, v[0, 2]],
])
dv = DiagMatrix(3*v)
assert dv.args == (3*v,)
assert dv.doit() == 3*DiagMatrix(v)
assert isinstance(dv.doit(), MatMul)
a = MatrixSymbol("a", 3, 1).as_explicit()
expr = DiagMatrix(a)
result = Matrix([
[a[0, 0], 0, 0],
[0, a[1, 0], 0],
[0, 0, a[2, 0]],
])
assert expr.doit() == result
expr = DiagMatrix(a.T)
assert expr.doit() == result
|
9217bf65f6c023f246959a6c7131ead87b72d58c354ce6fb1f84df5c9375d16c | from sympy.core.symbol import symbols, Dummy
from sympy.matrices.expressions.applyfunc import ElementwiseApplyFunction
from sympy.core.function import Lambda
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.trigonometric import sin
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.matmul import MatMul
from sympy.simplify.simplify import simplify
from sympy.testing.pytest import raises
from sympy.matrices.common import ShapeError
X = MatrixSymbol("X", 3, 3)
Y = MatrixSymbol("Y", 3, 3)
k = symbols("k")
Xk = MatrixSymbol("X", k, k)
Xd = X.as_explicit()
x, y, z, t = symbols("x y z t")
def test_applyfunc_matrix():
x = Dummy('x')
double = Lambda(x, x**2)
expr = ElementwiseApplyFunction(double, Xd)
assert isinstance(expr, ElementwiseApplyFunction)
assert expr.doit() == Xd.applyfunc(lambda x: x**2)
assert expr.shape == (3, 3)
assert expr.func(*expr.args) == expr
assert simplify(expr) == expr
assert expr[0, 0] == double(Xd[0, 0])
expr = ElementwiseApplyFunction(double, X)
assert isinstance(expr, ElementwiseApplyFunction)
assert isinstance(expr.doit(), ElementwiseApplyFunction)
assert expr == X.applyfunc(double)
assert expr.func(*expr.args) == expr
expr = ElementwiseApplyFunction(exp, X*Y)
assert expr.expr == X*Y
assert expr.function.dummy_eq(Lambda(x, exp(x)))
assert expr.dummy_eq((X*Y).applyfunc(exp))
assert expr.func(*expr.args) == expr
assert isinstance(X*expr, MatMul)
assert (X*expr).shape == (3, 3)
Z = MatrixSymbol("Z", 2, 3)
assert (Z*expr).shape == (2, 3)
expr = ElementwiseApplyFunction(exp, Z.T)*ElementwiseApplyFunction(exp, Z)
assert expr.shape == (3, 3)
expr = ElementwiseApplyFunction(exp, Z)*ElementwiseApplyFunction(exp, Z.T)
assert expr.shape == (2, 2)
raises(ShapeError, lambda: ElementwiseApplyFunction(exp, Z)*ElementwiseApplyFunction(exp, Z))
M = Matrix([[x, y], [z, t]])
expr = ElementwiseApplyFunction(sin, M)
assert isinstance(expr, ElementwiseApplyFunction)
assert expr.function.dummy_eq(Lambda(x, sin(x)))
assert expr.expr == M
assert expr.doit() == M.applyfunc(sin)
assert expr.doit() == Matrix([[sin(x), sin(y)], [sin(z), sin(t)]])
assert expr.func(*expr.args) == expr
expr = ElementwiseApplyFunction(double, Xk)
assert expr.doit() == expr
assert expr.subs(k, 2).shape == (2, 2)
assert (expr*expr).shape == (k, k)
M = MatrixSymbol("M", k, t)
expr2 = M.T*expr*M
assert isinstance(expr2, MatMul)
assert expr2.args[1] == expr
assert expr2.shape == (t, t)
expr3 = expr*M
assert expr3.shape == (k, t)
raises(ShapeError, lambda: M*expr)
expr1 = ElementwiseApplyFunction(lambda x: x+1, Xk)
expr2 = ElementwiseApplyFunction(lambda x: x, Xk)
assert expr1 != expr2
def test_applyfunc_entry():
af = X.applyfunc(sin)
assert af[0, 0] == sin(X[0, 0])
af = Xd.applyfunc(sin)
assert af[0, 0] == sin(X[0, 0])
def test_applyfunc_as_explicit():
af = X.applyfunc(sin)
assert af.as_explicit() == Matrix([
[sin(X[0, 0]), sin(X[0, 1]), sin(X[0, 2])],
[sin(X[1, 0]), sin(X[1, 1]), sin(X[1, 2])],
[sin(X[2, 0]), sin(X[2, 1]), sin(X[2, 2])],
])
def test_applyfunc_transpose():
af = Xk.applyfunc(sin)
assert af.T.dummy_eq(Xk.T.applyfunc(sin))
def test_applyfunc_shape_11_matrices():
M = MatrixSymbol("M", 1, 1)
double = Lambda(x, x*2)
expr = M.applyfunc(sin)
assert isinstance(expr, ElementwiseApplyFunction)
expr = M.applyfunc(double)
assert isinstance(expr, MatMul)
assert expr == 2*M
|
c580246daea493e9afaa9d19d4ebdfb9a12a43637bcc2d85d1b08fed0b7c7733 | from sympy.core import symbols, S
from sympy.matrices.expressions import MatrixSymbol, Inverse, MatPow, ZeroMatrix, OneMatrix
from sympy.matrices.common import NonSquareMatrixError, NonInvertibleMatrixError
from sympy.matrices import eye, Identity
from sympy.testing.pytest import raises
from sympy.assumptions.ask import Q
from sympy.assumptions.refine import refine
n, m, l = symbols('n m l', integer=True)
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_inverse():
assert Inverse(C).args == (C, S.NegativeOne)
assert Inverse(C).shape == (n, n)
assert Inverse(A*E).shape == (n, n)
assert Inverse(E*A).shape == (m, m)
assert Inverse(C).inverse() == C
assert Inverse(Inverse(C)).doit() == C
assert isinstance(Inverse(Inverse(C)), Inverse)
assert Inverse(*Inverse(E*A).args) == Inverse(E*A)
assert C.inverse().inverse() == C
assert C.inverse()*C == Identity(C.rows)
assert Identity(n).inverse() == Identity(n)
assert (3*Identity(n)).inverse() == Identity(n)/3
# Simplifies Muls if possible (i.e. submatrices are square)
assert (C*D).inverse() == D.I*C.I
# But still works when not possible
assert isinstance((A*E).inverse(), Inverse)
assert Inverse(C*D).doit(inv_expand=False) == Inverse(C*D)
assert Inverse(eye(3)).doit() == eye(3)
assert Inverse(eye(3)).doit(deep=False) == eye(3)
assert OneMatrix(1, 1).I == Identity(1)
assert isinstance(OneMatrix(n, n).I, Inverse)
def test_inverse_non_invertible():
raises(NonSquareMatrixError, lambda: Inverse(A))
raises(NonSquareMatrixError, lambda: Inverse(A*B))
raises(NonSquareMatrixError, lambda: ZeroMatrix(n, m).I)
raises(NonInvertibleMatrixError, lambda: ZeroMatrix(n, n).I)
raises(NonSquareMatrixError, lambda: OneMatrix(n, m).I)
raises(NonInvertibleMatrixError, lambda: OneMatrix(2, 2).I)
def test_refine():
assert refine(C.I, Q.orthogonal(C)) == C.T
def test_inverse_matpow_canonicalization():
A = MatrixSymbol('A', 3, 3)
assert Inverse(MatPow(A, 3)).doit() == MatPow(Inverse(A), 3).doit()
|
7b91fa6afde2a0e9fc8e48f20c1ad3dcd1260a4c876cde991073a5c81f4e60b9 | from sympy.core.mod import Mod
from sympy.core.numbers import I
from sympy.core.symbol import symbols
from sympy.functions.elementary.integers import floor
from sympy.matrices.dense import (Matrix, eye)
from sympy.matrices import MatrixSymbol, Identity
from sympy.matrices.expressions import det, trace
from sympy.matrices.expressions.kronecker import (KroneckerProduct,
kronecker_product,
combine_kronecker)
mat1 = Matrix([[1, 2 * I], [1 + I, 3]])
mat2 = Matrix([[2 * I, 3], [4 * I, 2]])
i, j, k, n, m, o, p, x = symbols('i,j,k,n,m,o,p,x')
Z = MatrixSymbol('Z', n, n)
W = MatrixSymbol('W', m, m)
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, k)
def test_KroneckerProduct():
assert isinstance(KroneckerProduct(A, B), KroneckerProduct)
assert KroneckerProduct(A, B).subs(A, C) == KroneckerProduct(C, B)
assert KroneckerProduct(A, C).shape == (n*m, m*k)
assert (KroneckerProduct(A, C) + KroneckerProduct(-A, C)).is_ZeroMatrix
assert (KroneckerProduct(W, Z) * KroneckerProduct(W.I, Z.I)).is_Identity
def test_KroneckerProduct_identity():
assert KroneckerProduct(Identity(m), Identity(n)) == Identity(m*n)
assert KroneckerProduct(eye(2), eye(3)) == eye(6)
def test_KroneckerProduct_explicit():
X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 2)
kp = KroneckerProduct(X, Y)
assert kp.shape == (4, 4)
assert kp.as_explicit() == Matrix(
[
[X[0, 0]*Y[0, 0], X[0, 0]*Y[0, 1], X[0, 1]*Y[0, 0], X[0, 1]*Y[0, 1]],
[X[0, 0]*Y[1, 0], X[0, 0]*Y[1, 1], X[0, 1]*Y[1, 0], X[0, 1]*Y[1, 1]],
[X[1, 0]*Y[0, 0], X[1, 0]*Y[0, 1], X[1, 1]*Y[0, 0], X[1, 1]*Y[0, 1]],
[X[1, 0]*Y[1, 0], X[1, 0]*Y[1, 1], X[1, 1]*Y[1, 0], X[1, 1]*Y[1, 1]]
]
)
def test_tensor_product_adjoint():
assert KroneckerProduct(I*A, B).adjoint() == \
-I*KroneckerProduct(A.adjoint(), B.adjoint())
assert KroneckerProduct(mat1, mat2).adjoint() == \
kronecker_product(mat1.adjoint(), mat2.adjoint())
def test_tensor_product_conjugate():
assert KroneckerProduct(I*A, B).conjugate() == \
-I*KroneckerProduct(A.conjugate(), B.conjugate())
assert KroneckerProduct(mat1, mat2).conjugate() == \
kronecker_product(mat1.conjugate(), mat2.conjugate())
def test_tensor_product_transpose():
assert KroneckerProduct(I*A, B).transpose() == \
I*KroneckerProduct(A.transpose(), B.transpose())
assert KroneckerProduct(mat1, mat2).transpose() == \
kronecker_product(mat1.transpose(), mat2.transpose())
def test_KroneckerProduct_is_associative():
assert kronecker_product(A, kronecker_product(
B, C)) == kronecker_product(kronecker_product(A, B), C)
assert kronecker_product(A, kronecker_product(
B, C)) == KroneckerProduct(A, B, C)
def test_KroneckerProduct_is_bilinear():
assert kronecker_product(x*A, B) == x*kronecker_product(A, B)
assert kronecker_product(A, x*B) == x*kronecker_product(A, B)
def test_KroneckerProduct_determinant():
kp = kronecker_product(W, Z)
assert det(kp) == det(W)**n * det(Z)**m
def test_KroneckerProduct_trace():
kp = kronecker_product(W, Z)
assert trace(kp) == trace(W)*trace(Z)
def test_KroneckerProduct_isnt_commutative():
assert KroneckerProduct(A, B) != KroneckerProduct(B, A)
assert KroneckerProduct(A, B).is_commutative is False
def test_KroneckerProduct_extracts_commutative_part():
assert kronecker_product(x * A, 2 * B) == x * \
2 * KroneckerProduct(A, B)
def test_KroneckerProduct_inverse():
kp = kronecker_product(W, Z)
assert kp.inverse() == kronecker_product(W.inverse(), Z.inverse())
def test_KroneckerProduct_combine_add():
kp1 = kronecker_product(A, B)
kp2 = kronecker_product(C, W)
assert combine_kronecker(kp1*kp2) == kronecker_product(A*C, B*W)
def test_KroneckerProduct_combine_mul():
X = MatrixSymbol('X', m, n)
Y = MatrixSymbol('Y', m, n)
kp1 = kronecker_product(A, X)
kp2 = kronecker_product(B, Y)
assert combine_kronecker(kp1+kp2) == kronecker_product(A+B, X+Y)
def test_KroneckerProduct_combine_pow():
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', n, n)
assert combine_kronecker(KroneckerProduct(
X, Y)**x) == KroneckerProduct(X**x, Y**x)
assert combine_kronecker(x * KroneckerProduct(X, Y)
** 2) == x * KroneckerProduct(X**2, Y**2)
assert combine_kronecker(
x * (KroneckerProduct(X, Y)**2) * KroneckerProduct(A, B)) == x * KroneckerProduct(X**2 * A, Y**2 * B)
# cannot simplify because of non-square arguments to kronecker product:
assert combine_kronecker(KroneckerProduct(A, B.T) ** m) == KroneckerProduct(A, B.T) ** m
def test_KroneckerProduct_expand():
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', n, n)
assert KroneckerProduct(X + Y, Y + Z).expand(kroneckerproduct=True) == \
KroneckerProduct(X, Y) + KroneckerProduct(X, Z) + \
KroneckerProduct(Y, Y) + KroneckerProduct(Y, Z)
def test_KroneckerProduct_entry():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', o, p)
assert KroneckerProduct(A, B)._entry(i, j) == A[Mod(floor(i/o), n), Mod(floor(j/p), m)]*B[Mod(i, o), Mod(j, p)]
|
209a0b486d2863aa08f107af8f5e7d6fe2272550a7cbd753d43a1eff45a11936 | from sympy.core import Lambda, S, symbols
from sympy.concrete import Sum
from sympy.functions import adjoint, conjugate, transpose
from sympy.matrices import eye, Matrix, ShapeError, ImmutableMatrix
from sympy.matrices.expressions import (
Adjoint, Identity, FunctionMatrix, MatrixExpr, MatrixSymbol, Trace,
ZeroMatrix, trace, MatPow, MatAdd, MatMul
)
from sympy.matrices.expressions.special import OneMatrix
from sympy.testing.pytest import raises
n = symbols('n', integer=True)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
C = MatrixSymbol('C', 3, 4)
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
A / Trace(A) # Make sure this is possible
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(OneMatrix(1, 1)) == 1
assert trace(OneMatrix(2, 2)) == 2
assert trace(OneMatrix(n, n)) == n
assert trace(2*A*B) == 2*Trace(A*B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
raises(TypeError, lambda: Trace(S.One))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
assert Trace(A).is_commutative is True
def test_Trace_A_plus_B():
assert trace(A + B) == Trace(A) + Trace(B)
assert Trace(A + B).arg == MatAdd(A, B)
assert Trace(A + B).doit() == Trace(A) + Trace(B)
def test_Trace_MatAdd_doit():
# See issue #9028
X = ImmutableMatrix([[1, 2, 3]]*3)
Y = MatrixSymbol('Y', 3, 3)
q = MatAdd(X, 2*X, Y, -3*Y)
assert Trace(q).arg == q
assert Trace(q).doit() == 18 - 2*Trace(Y)
def test_Trace_MatPow_doit():
X = Matrix([[1, 2], [3, 4]])
assert Trace(X).doit() == 5
q = MatPow(X, 2)
assert Trace(q).arg == q
assert Trace(q).doit() == 29
def test_Trace_MutableMatrix_plus():
# See issue #9043
X = Matrix([[1, 2], [3, 4]])
assert Trace(X) + Trace(X) == 2*Trace(X)
def test_Trace_doit_deep_False():
X = Matrix([[1, 2], [3, 4]])
q = MatPow(X, 2)
assert Trace(q).doit(deep=False).arg == q
q = MatAdd(X, 2*X)
assert Trace(q).doit(deep=False).arg == q
q = MatMul(X, 2*X)
assert Trace(q).doit(deep=False).arg == q
def test_trace_constant_factor():
# Issue 9052: gave 2*Trace(MatMul(A)) instead of 2*Trace(A)
assert trace(2*A) == 2*Trace(A)
X = ImmutableMatrix([[1, 2], [3, 4]])
assert trace(MatMul(2, X)) == 10
def test_rewrite():
assert isinstance(trace(A).rewrite(Sum), Sum)
def test_trace_normalize():
assert Trace(B*A) != Trace(A*B)
assert Trace(B*A)._normalize() == Trace(A*B)
assert Trace(B*A.T)._normalize() == Trace(A*B.T)
def test_trace_as_explicit():
raises(ValueError, lambda: Trace(A).as_explicit())
X = MatrixSymbol("X", 3, 3)
assert Trace(X).as_explicit() == X[0, 0] + X[1, 1] + X[2, 2]
assert Trace(eye(3)).as_explicit() == 3
|
4fb8b96604375cc266934cff9639e14882098c1af32cfd7f9a07b42d9209f996 | 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
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)
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]]])
|
2da9baafe7123a8923852949c2ce792aafa02f9fe91df795fd7baf9a7e4f9115 | from sympy.core.expr import unchanged
from sympy.core.symbol import Symbol, symbols
from sympy.matrices.immutable import ImmutableDenseMatrix
from sympy.matrices.expressions.companion import CompanionMatrix
from sympy.polys.polytools import Poly
from sympy.testing.pytest import raises
def test_creation():
x = Symbol('x')
y = Symbol('y')
raises(ValueError, lambda: CompanionMatrix(1))
raises(ValueError, lambda: CompanionMatrix(Poly([1], x)))
raises(ValueError, lambda: CompanionMatrix(Poly([2, 1], x)))
raises(ValueError, lambda: CompanionMatrix(Poly(x*y, [x, y])))
assert unchanged(CompanionMatrix, Poly([1, 2, 3], x))
def test_shape():
c0, c1, c2 = symbols('c0:3')
x = Symbol('x')
assert CompanionMatrix(Poly([1, c0], x)).shape == (1, 1)
assert CompanionMatrix(Poly([1, c1, c0], x)).shape == (2, 2)
assert CompanionMatrix(Poly([1, c2, c1, c0], x)).shape == (3, 3)
def test_entry():
c0, c1, c2 = symbols('c0:3')
x = Symbol('x')
A = CompanionMatrix(Poly([1, c2, c1, c0], x))
assert A[0, 0] == 0
assert A[1, 0] == 1
assert A[1, 1] == 0
assert A[2, 1] == 1
assert A[0, 2] == -c0
assert A[1, 2] == -c1
assert A[2, 2] == -c2
def test_as_explicit():
c0, c1, c2 = symbols('c0:3')
x = Symbol('x')
assert CompanionMatrix(Poly([1, c0], x)).as_explicit() == \
ImmutableDenseMatrix([-c0])
assert CompanionMatrix(Poly([1, c1, c0], x)).as_explicit() == \
ImmutableDenseMatrix([[0, -c0], [1, -c1]])
assert CompanionMatrix(Poly([1, c2, c1, c0], x)).as_explicit() == \
ImmutableDenseMatrix([[0, 0, -c0], [1, 0, -c1], [0, 1, -c2]])
|
41e1f2ac1e5fbbcdff86ebfe864f12d1910e2011ffdd5453dc484a5691c2dfb9 | from sympy.core.function import Lambda
from sympy.core.numbers import oo
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.functions.elementary.miscellaneous import (Max, Min)
from sympy.sets.sets import Set
from sympy.core import Basic, Expr, Integer
from sympy.core.numbers import Infinity, NegativeInfinity, Zero
from sympy.multipledispatch import dispatch
from sympy.sets import Interval, FiniteSet, Union, ImageSet
_x, _y = symbols("x y")
@dispatch(Basic, Basic) # type: ignore # noqa:F811
def _set_pow(x, y): # noqa:F811
return None
@dispatch(Set, Set) # type: ignore # noqa:F811
def _set_pow(x, y): # noqa:F811
return ImageSet(Lambda((_x, _y), (_x ** _y)), x, y)
@dispatch(Expr, Expr) # type: ignore # noqa:F811
def _set_pow(x, y): # noqa:F811
return x**y
@dispatch(Interval, Zero) # type: ignore # noqa:F811
def _set_pow(x, z): # noqa:F811
return FiniteSet(S.One)
@dispatch(Interval, Integer) # type: ignore # noqa:F811
def _set_pow(x, exponent): # noqa:F811
"""
Powers in interval arithmetic
https://en.wikipedia.org/wiki/Interval_arithmetic
"""
s1 = x.start**exponent
s2 = x.end**exponent
if ((s2 > s1) if exponent > 0 else (x.end > -x.start)) == True:
left_open = x.left_open
right_open = x.right_open
# TODO: handle unevaluated condition.
sleft = s2
else:
# TODO: `s2 > s1` could be unevaluated.
left_open = x.right_open
right_open = x.left_open
sleft = s1
if x.start.is_positive:
return Interval(
Min(s1, s2),
Max(s1, s2), left_open, right_open)
elif x.end.is_negative:
return Interval(
Min(s1, s2),
Max(s1, s2), left_open, right_open)
# Case where x.start < 0 and x.end > 0:
if exponent.is_odd:
if exponent.is_negative:
if x.start.is_zero:
return Interval(s2, oo, x.right_open)
if x.end.is_zero:
return Interval(-oo, s1, True, x.left_open)
return Union(Interval(-oo, s1, True, x.left_open), Interval(s2, oo, x.right_open))
else:
return Interval(s1, s2, x.left_open, x.right_open)
elif exponent.is_even:
if exponent.is_negative:
if x.start.is_zero:
return Interval(s2, oo, x.right_open)
if x.end.is_zero:
return Interval(s1, oo, x.left_open)
return Interval(0, oo)
else:
return Interval(S.Zero, sleft, S.Zero not in x, left_open)
@dispatch(Interval, Infinity) # type: ignore # noqa:F811
def _set_pow(b, e): # noqa:F811
# TODO: add logic for open intervals?
if b.start.is_nonnegative:
if b.end < 1:
return FiniteSet(S.Zero)
if b.start > 1:
return FiniteSet(S.Infinity)
return Interval(0, oo)
elif b.end.is_negative:
if b.start > -1:
return FiniteSet(S.Zero)
if b.end < -1:
return FiniteSet(-oo, oo)
return Interval(-oo, oo)
else:
if b.start > -1:
if b.end < 1:
return FiniteSet(S.Zero)
return Interval(0, oo)
return Interval(-oo, oo)
@dispatch(Interval, NegativeInfinity) # type: ignore # noqa:F811
def _set_pow(b, e): # noqa:F811
from sympy.sets.setexpr import set_div
return _set_pow(set_div(S.One, b), oo)
|
e006f14ab984903e467df5b569f2bfbf1beeafc986484fb27fb28f0c1b8e0a1b | 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.sets.fancysets import ComplexRegion
from sympy.sets.sets import (FiniteSet, Intersection, Interval, Set, Union)
from sympy.multipledispatch import dispatch
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
@dispatch(ConditionSet, ConditionSet) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return None
@dispatch(ConditionSet, Set) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return ConditionSet(a.sym, a.condition, Intersection(a.base_set, b))
@dispatch(Naturals, Integers) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return a
@dispatch(Naturals, Naturals) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return a if a is S.Naturals else b
@dispatch(Interval, Naturals) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return intersection_sets(b, a)
@dispatch(ComplexRegion, Set) # type: ignore # noqa:F811
def intersection_sets(self, other): # noqa:F811
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)
@dispatch(Integers, Reals) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return a
@dispatch(Range, Interval) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
# 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
from sympy.functions.elementary.integers import floor, ceiling
# 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))
@dispatch(Range, Naturals) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return intersection_sets(a, Interval(b.inf, S.Infinity))
@dispatch(Range, Range) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
# 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
from sympy.functions.elementary.complexes import sign
# 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)
@dispatch(Range, Integers) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return a
@dispatch(ImageSet, Set) # type: ignore # noqa:F811
def intersection_sets(self, other): # noqa:F811
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
@dispatch(ProductSet, ProductSet) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
if len(b.args) != len(a.args):
return S.EmptySet
return ProductSet(*(i.intersect(j) for i, j in zip(a.sets, b.sets)))
@dispatch(Interval, Interval) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
# 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:
start = a.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 = a.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)
@dispatch(EmptySet, Set) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return S.EmptySet
@dispatch(UniversalSet, Set) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return b
@dispatch(FiniteSet, FiniteSet) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return FiniteSet(*(a._elements & b._elements))
@dispatch(FiniteSet, Set) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
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.
@dispatch(Set, Set) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return None
@dispatch(Integers, Rationals) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return a
@dispatch(Naturals, Rationals) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return a
@dispatch(Rationals, Reals) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return a
def _intlike_interval(a, b):
try:
from sympy.functions.elementary.integers import floor, ceiling
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
@dispatch(Integers, Interval) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return _intlike_interval(a, b)
@dispatch(Naturals, Interval) # type: ignore # noqa:F811
def intersection_sets(a, b): # noqa:F811
return _intlike_interval(a, b)
|
b8aec56af1eff5941b9320953e478843a1130b8e4d5ed3d6886b9b48b2c86fab | from sympy.core.numbers import oo
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.core import Basic, Expr
from sympy.core.numbers import Infinity, NegativeInfinity
from sympy.multipledispatch import dispatch
from sympy.sets import Interval, FiniteSet
# XXX: The functions in this module are clearly not tested and are broken in a
# number of ways.
_x, _y = symbols("x y")
@dispatch(Basic, Basic) # type: ignore # noqa:F811
def _set_add(x, y): # noqa:F811
return None
@dispatch(Expr, Expr) # type: ignore # noqa:F811
def _set_add(x, y): # noqa:F811
return x+y
@dispatch(Interval, Interval) # type: ignore # noqa:F811
def _set_add(x, y): # noqa:F811
"""
Additions in interval arithmetic
https://en.wikipedia.org/wiki/Interval_arithmetic
"""
return Interval(x.start + y.start, x.end + y.end,
x.left_open or y.left_open, x.right_open or y.right_open)
@dispatch(Interval, Infinity) # type: ignore # noqa:F811
def _set_add(x, y): # noqa:F811
if x.start is S.NegativeInfinity:
return Interval(-oo, oo)
return FiniteSet({S.Infinity})
@dispatch(Interval, NegativeInfinity) # type: ignore # noqa:F811
def _set_add(x, y): # noqa:F811
if x.end is S.Infinity:
return Interval(-oo, oo)
return FiniteSet({S.NegativeInfinity})
@dispatch(Basic, Basic) # type: ignore
def _set_sub(x, y): # noqa:F811
return None
@dispatch(Expr, Expr) # type: ignore # noqa:F811
def _set_sub(x, y): # noqa:F811
return x-y
@dispatch(Interval, Interval) # type: ignore # noqa:F811
def _set_sub(x, y): # noqa:F811
"""
Subtractions in interval arithmetic
https://en.wikipedia.org/wiki/Interval_arithmetic
"""
return Interval(x.start - y.end, x.end - y.start,
x.left_open or y.right_open, x.right_open or y.left_open)
@dispatch(Interval, Infinity) # type: ignore # noqa:F811
def _set_sub(x, y): # noqa:F811
if x.start is S.NegativeInfinity:
return Interval(-oo, oo)
return FiniteSet(-oo)
@dispatch(Interval, NegativeInfinity) # type: ignore # noqa:F811
def _set_sub(x, y): # noqa:F811
if x.start is S.NegativeInfinity:
return Interval(-oo, oo)
return FiniteSet(-oo)
|
102a4ec59f2e43fded7f443678d6c41be9d8abab34cd21cd743302e7af0d048d | from sympy.core.singleton import S
from sympy.core.sympify import sympify
from sympy.sets.sets import (EmptySet, FiniteSet, Intersection,
Interval, ProductSet, Set, Union, UniversalSet)
from sympy.sets.fancysets import (ComplexRegion, Naturals, Naturals0,
Integers, Rationals, Reals)
from sympy.multipledispatch import dispatch
@dispatch(Naturals0, Naturals) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return a
@dispatch(Rationals, Naturals) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return a
@dispatch(Rationals, Naturals0) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return a
@dispatch(Reals, Naturals) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return a
@dispatch(Reals, Naturals0) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return a
@dispatch(Reals, Rationals) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return a
@dispatch(Integers, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
intersect = Intersection(a, b)
if intersect == a:
return b
elif intersect == b:
return a
@dispatch(ComplexRegion, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
if b.is_subset(S.Reals):
# treat a subset of reals as a complex region
b = ComplexRegion.from_real(b)
if b.is_ComplexRegion:
# a in rectangular form
if (not a.polar) and (not b.polar):
return ComplexRegion(Union(a.sets, b.sets))
# a in polar form
elif a.polar and b.polar:
return ComplexRegion(Union(a.sets, b.sets), polar=True)
return None
@dispatch(EmptySet, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return b
@dispatch(UniversalSet, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return a
@dispatch(ProductSet, ProductSet) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
if b.is_subset(a):
return a
if len(b.sets) != len(a.sets):
return None
if len(a.sets) == 2:
a1, a2 = a.sets
b1, b2 = b.sets
if a1 == b1:
return a1 * Union(a2, b2)
if a2 == b2:
return Union(a1, b1) * a2
return None
@dispatch(ProductSet, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
if b.is_subset(a):
return a
return None
@dispatch(Interval, Interval) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
if a._is_comparable(b):
from sympy.functions.elementary.miscellaneous import Min, Max
# Non-overlapping intervals
end = Min(a.end, b.end)
start = Max(a.start, b.start)
if (end < start or
(end == start and (end not in a and end not in b))):
return None
else:
start = Min(a.start, b.start)
end = Max(a.end, b.end)
left_open = ((a.start != start or a.left_open) and
(b.start != start or b.left_open))
right_open = ((a.end != end or a.right_open) and
(b.end != end or b.right_open))
return Interval(start, end, left_open, right_open)
@dispatch(Interval, UniversalSet) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return S.UniversalSet
@dispatch(Interval, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
# If I have open end points and these endpoints are contained in b
# But only in case, when endpoints are finite. Because
# interval does not contain oo or -oo.
open_left_in_b_and_finite = (a.left_open and
sympify(b.contains(a.start)) is S.true and
a.start.is_finite)
open_right_in_b_and_finite = (a.right_open and
sympify(b.contains(a.end)) is S.true and
a.end.is_finite)
if open_left_in_b_and_finite or open_right_in_b_and_finite:
# Fill in my end points and return
open_left = a.left_open and a.start not in b
open_right = a.right_open and a.end not in b
new_a = Interval(a.start, a.end, open_left, open_right)
return {new_a, b}
return None
@dispatch(FiniteSet, FiniteSet) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return FiniteSet(*(a._elements | b._elements))
@dispatch(FiniteSet, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
# If `b` set contains one of my elements, remove it from `a`
if any(b.contains(x) == True for x in a):
return {
FiniteSet(*[x for x in a if b.contains(x) != True]), b}
return None
@dispatch(Set, Set) # type: ignore # noqa:F811
def union_sets(a, b): # noqa:F811
return None
|
f04f39f3dd9b325d16cc638f1085f8ad5852107f5fddc833363e2b556ffa442a | from sympy.core.singleton import S
from sympy.sets.sets import Set
from sympy.calculus.singularities import singularities
from sympy.core import Expr, Add
from sympy.core.function import Lambda, FunctionClass, diff, expand_mul
from sympy.core.numbers import Float, oo
from sympy.core.symbol import Dummy, symbols, Wild
from sympy.functions.elementary.exponential import exp, log
from sympy.functions.elementary.miscellaneous import Min, Max
from sympy.logic.boolalg import true
from sympy.multipledispatch import dispatch
from sympy.sets import (imageset, Interval, FiniteSet, Union, ImageSet,
Intersection, Range)
from sympy.sets.sets import EmptySet
from sympy.sets.fancysets import Integers, Naturals, Reals
from sympy.functions.elementary.exponential import match_real_imag
_x, _y = symbols("x y")
FunctionUnion = (FunctionClass, Lambda)
@dispatch(FunctionClass, Set) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return None
@dispatch(FunctionUnion, FiniteSet) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return FiniteSet(*map(f, x))
@dispatch(Lambda, Interval) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
from sympy.solvers.solveset import solveset
from sympy.series import limit
from sympy.sets import Complement
# TODO: handle functions with infinitely many solutions (eg, sin, tan)
# TODO: handle multivariate functions
expr = f.expr
if len(expr.free_symbols) > 1 or len(f.variables) != 1:
return
var = f.variables[0]
if not var.is_real:
if expr.subs(var, Dummy(real=True)).is_real is False:
return
if expr.is_Piecewise:
result = S.EmptySet
domain_set = x
for (p_expr, p_cond) in expr.args:
if p_cond is true:
intrvl = domain_set
else:
intrvl = p_cond.as_set()
intrvl = Intersection(domain_set, intrvl)
if p_expr.is_Number:
image = FiniteSet(p_expr)
else:
image = imageset(Lambda(var, p_expr), intrvl)
result = Union(result, image)
# remove the part which has been `imaged`
domain_set = Complement(domain_set, intrvl)
if domain_set is S.EmptySet:
break
return result
if not x.start.is_comparable or not x.end.is_comparable:
return
try:
from sympy.polys.polyutils import _nsort
sing = list(singularities(expr, var, x))
if len(sing) > 1:
sing = _nsort(sing)
except NotImplementedError:
return
if x.left_open:
_start = limit(expr, var, x.start, dir="+")
elif x.start not in sing:
_start = f(x.start)
if x.right_open:
_end = limit(expr, var, x.end, dir="-")
elif x.end not in sing:
_end = f(x.end)
if len(sing) == 0:
soln_expr = solveset(diff(expr, var), var)
if not (isinstance(soln_expr, FiniteSet)
or soln_expr is S.EmptySet):
return
solns = list(soln_expr)
extr = [_start, _end] + [f(i) for i in solns
if i.is_real and i in x]
start, end = Min(*extr), Max(*extr)
left_open, right_open = False, False
if _start <= _end:
# the minimum or maximum value can occur simultaneously
# on both the edge of the interval and in some interior
# point
if start == _start and start not in solns:
left_open = x.left_open
if end == _end and end not in solns:
right_open = x.right_open
else:
if start == _end and start not in solns:
left_open = x.right_open
if end == _start and end not in solns:
right_open = x.left_open
return Interval(start, end, left_open, right_open)
else:
return imageset(f, Interval(x.start, sing[0],
x.left_open, True)) + \
Union(*[imageset(f, Interval(sing[i], sing[i + 1], True, True))
for i in range(0, len(sing) - 1)]) + \
imageset(f, Interval(sing[-1], x.end, True, x.right_open))
@dispatch(FunctionClass, Interval) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
if f == exp:
return Interval(exp(x.start), exp(x.end), x.left_open, x.right_open)
elif f == log:
return Interval(log(x.start), log(x.end), x.left_open, x.right_open)
return ImageSet(Lambda(_x, f(_x)), x)
@dispatch(FunctionUnion, Union) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return Union(*(imageset(f, arg) for arg in x.args))
@dispatch(FunctionUnion, Intersection) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
from sympy.sets.sets import is_function_invertible_in_set
# If the function is invertible, intersect the maps of the sets.
if is_function_invertible_in_set(f, x):
return Intersection(*(imageset(f, arg) for arg in x.args))
else:
return ImageSet(Lambda(_x, f(_x)), x)
@dispatch(FunctionUnion, EmptySet) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return x
@dispatch(FunctionUnion, Set) # type: ignore # noqa:F811
def _set_function(f, x): # noqa:F811
return ImageSet(Lambda(_x, f(_x)), x)
@dispatch(FunctionUnion, Range) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
if not self:
return S.EmptySet
if not isinstance(f.expr, Expr):
return
if self.size == 1:
return FiniteSet(f(self[0]))
if f is S.IdentityFunction:
return self
x = f.variables[0]
expr = f.expr
# handle f that is linear in f's variable
if x not in expr.free_symbols or x in expr.diff(x).free_symbols:
return
if self.start.is_finite:
F = f(self.step*x + self.start) # for i in range(len(self))
else:
F = f(-self.step*x + self[-1])
F = expand_mul(F)
if F != expr:
return imageset(x, F, Range(self.size))
@dispatch(FunctionUnion, Integers) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
expr = f.expr
if not isinstance(expr, Expr):
return
n = f.variables[0]
if expr == abs(n):
return S.Naturals0
# f(x) + c and f(-x) + c cover the same integers
# so choose the form that has the fewest negatives
c = f(0)
fx = f(n) - c
f_x = f(-n) - c
neg_count = lambda e: sum(_.could_extract_minus_sign()
for _ in Add.make_args(e))
if neg_count(f_x) < neg_count(fx):
expr = f_x + c
a = Wild('a', exclude=[n])
b = Wild('b', exclude=[n])
match = expr.match(a*n + b)
if match and match[a] and (
not match[a].atoms(Float) and
not match[b].atoms(Float)):
# canonical shift
a, b = match[a], match[b]
if a in [1, -1]:
# drop integer addends in b
nonint = []
for bi in Add.make_args(b):
if not bi.is_integer:
nonint.append(bi)
b = Add(*nonint)
if b.is_number and a.is_real:
# avoid Mod for complex numbers, #11391
br, bi = match_real_imag(b)
if br and br.is_comparable and a.is_comparable:
br %= a
b = br + S.ImaginaryUnit*bi
elif b.is_number and a.is_imaginary:
br, bi = match_real_imag(b)
ai = a/S.ImaginaryUnit
if bi and bi.is_comparable and ai.is_comparable:
bi %= ai
b = br + S.ImaginaryUnit*bi
expr = a*n + b
if expr != f.expr:
return ImageSet(Lambda(n, expr), S.Integers)
@dispatch(FunctionUnion, Naturals) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
expr = f.expr
if not isinstance(expr, Expr):
return
x = f.variables[0]
if not expr.free_symbols - {x}:
if expr == abs(x):
if self is S.Naturals:
return self
return S.Naturals0
step = expr.coeff(x)
c = expr.subs(x, 0)
if c.is_Integer and step.is_Integer and expr == step*x + c:
if self is S.Naturals:
c += step
if step > 0:
if step == 1:
if c == 0:
return S.Naturals0
elif c == 1:
return S.Naturals
return Range(c, oo, step)
return Range(c, -oo, step)
@dispatch(FunctionUnion, Reals) # type: ignore # noqa:F811
def _set_function(f, self): # noqa:F811
expr = f.expr
if not isinstance(expr, Expr):
return
return _set_function(f, Interval(-oo, oo))
|
40947ad68c0194122edbc3db9bb1fe1342c832080318d00219f2978258e6b64b | from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.logic import fuzzy_and, fuzzy_bool, fuzzy_not, fuzzy_or
from sympy.core.relational import Eq
from sympy.sets.sets import FiniteSet, Interval, Set, Union, ProductSet
from sympy.sets.fancysets import Complexes, Reals, Range, Rationals
from sympy.multipledispatch import dispatch
_inf_sets = [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals, S.Complexes]
@dispatch(Set, Set) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
return None
@dispatch(Interval, Interval) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
# This is correct but can be made more comprehensive...
if fuzzy_bool(a.start < b.start):
return False
if fuzzy_bool(a.end > b.end):
return False
if (b.left_open and not a.left_open and fuzzy_bool(Eq(a.start, b.start))):
return False
if (b.right_open and not a.right_open and fuzzy_bool(Eq(a.end, b.end))):
return False
@dispatch(Interval, FiniteSet) # type: ignore # noqa:F811
def is_subset_sets(a_interval, b_fs): # noqa:F811
# An Interval can only be a subset of a finite set if it is finite
# which can only happen if it has zero measure.
if fuzzy_not(a_interval.measure.is_zero):
return False
@dispatch(Interval, Union) # type: ignore # noqa:F811
def is_subset_sets(a_interval, b_u): # noqa:F811
if all(isinstance(s, (Interval, FiniteSet)) for s in b_u.args):
intervals = [s for s in b_u.args if isinstance(s, Interval)]
if all(fuzzy_bool(a_interval.start < s.start) for s in intervals):
return False
if all(fuzzy_bool(a_interval.end > s.end) for s in intervals):
return False
if a_interval.measure.is_nonzero:
no_overlap = lambda s1, s2: fuzzy_or([
fuzzy_bool(s1.end <= s2.start),
fuzzy_bool(s1.start >= s2.end),
])
if all(no_overlap(s, a_interval) for s in intervals):
return False
@dispatch(Range, Range) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
if a.step == b.step == 1:
return fuzzy_and([fuzzy_bool(a.start >= b.start),
fuzzy_bool(a.stop <= b.stop)])
@dispatch(Range, Interval) # type: ignore # noqa:F811
def is_subset_sets(a_range, b_interval): # noqa:F811
if a_range.step.is_positive:
if b_interval.left_open and a_range.inf.is_finite:
cond_left = a_range.inf > b_interval.left
else:
cond_left = a_range.inf >= b_interval.left
if b_interval.right_open and a_range.sup.is_finite:
cond_right = a_range.sup < b_interval.right
else:
cond_right = a_range.sup <= b_interval.right
return fuzzy_and([cond_left, cond_right])
@dispatch(Range, FiniteSet) # type: ignore # noqa:F811
def is_subset_sets(a_range, b_finiteset): # noqa:F811
try:
a_size = a_range.size
except ValueError:
# symbolic Range of unknown size
return None
if a_size > len(b_finiteset):
return False
elif any(arg.has(Symbol) for arg in a_range.args):
return fuzzy_and(b_finiteset.contains(x) for x in a_range)
else:
# Checking A \ B == EmptySet is more efficient than repeated naive
# membership checks on an arbitrary FiniteSet.
a_set = set(a_range)
b_remaining = len(b_finiteset)
# Symbolic expressions and numbers of unknown type (integer or not) are
# all counted as "candidates", i.e. *potentially* matching some a in
# a_range.
cnt_candidate = 0
for b in b_finiteset:
if b.is_Integer:
a_set.discard(b)
elif fuzzy_not(b.is_integer):
pass
else:
cnt_candidate += 1
b_remaining -= 1
if len(a_set) > b_remaining + cnt_candidate:
return False
if len(a_set) == 0:
return True
return None
@dispatch(Interval, Range) # type: ignore # noqa:F811
def is_subset_sets(a_interval, b_range): # noqa:F811
if a_interval.measure.is_extended_nonzero:
return False
@dispatch(Interval, Rationals) # type: ignore # noqa:F811
def is_subset_sets(a_interval, b_rationals): # noqa:F811
if a_interval.measure.is_extended_nonzero:
return False
@dispatch(Range, Complexes) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
return True
@dispatch(Complexes, Interval) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
return False
@dispatch(Complexes, Range) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
return False
@dispatch(Complexes, Rationals) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
return False
@dispatch(Rationals, Reals) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
return True
@dispatch(Rationals, Range) # type: ignore # noqa:F811
def is_subset_sets(a, b): # noqa:F811
return False
@dispatch(ProductSet, FiniteSet) # type: ignore # noqa:F811
def is_subset_sets(a_ps, b_fs): # noqa:F811
return fuzzy_and(b_fs.contains(x) for x in a_ps)
|
b48be4eff23c2c63de310bc12ca6525ce9d6cbadafeaf6cdd21d44aff432403f | from sympy.core import Basic, Expr
from sympy.core.numbers import oo
from sympy.core.symbol import symbols
from sympy.multipledispatch import dispatch
from sympy.sets.sets import Interval, Set
_x, _y = symbols("x y")
@dispatch(Basic, Basic) # type: ignore # noqa:F811
def _set_mul(x, y): # noqa:F811
return None
@dispatch(Set, Set) # type: ignore # noqa:F811
def _set_mul(x, y): # noqa:F811
return None
@dispatch(Expr, Expr) # type: ignore # noqa:F811
def _set_mul(x, y): # noqa:F811
return x*y
@dispatch(Interval, Interval) # type: ignore # noqa:F811
def _set_mul(x, y): # noqa:F811
"""
Multiplications in interval arithmetic
https://en.wikipedia.org/wiki/Interval_arithmetic
"""
# TODO: some intervals containing 0 and oo will fail as 0*oo returns nan.
comvals = (
(x.start * y.start, bool(x.left_open or y.left_open)),
(x.start * y.end, bool(x.left_open or y.right_open)),
(x.end * y.start, bool(x.right_open or y.left_open)),
(x.end * y.end, bool(x.right_open or y.right_open)),
)
# TODO: handle symbolic intervals
minval, minopen = min(comvals)
maxval, maxopen = max(comvals)
return Interval(
minval,
maxval,
minopen,
maxopen
)
@dispatch(Basic, Basic) # type: ignore # noqa:F811
def _set_div(x, y): # noqa:F811
return None
@dispatch(Expr, Expr) # type: ignore # noqa:F811
def _set_div(x, y): # noqa:F811
return x/y
@dispatch(Set, Set) # type: ignore # noqa:F811 # noqa:F811
def _set_div(x, y): # noqa:F811
return None
@dispatch(Interval, Interval) # type: ignore # noqa:F811
def _set_div(x, y): # noqa:F811
"""
Divisions in interval arithmetic
https://en.wikipedia.org/wiki/Interval_arithmetic
"""
from sympy.sets.setexpr import set_mul
if (y.start*y.end).is_negative:
return Interval(-oo, oo)
if y.start == 0:
s2 = oo
else:
s2 = 1/y.start
if y.end == 0:
s1 = -oo
else:
s1 = 1/y.end
return set_mul(x, Interval(s1, s2, y.right_open, y.left_open))
|
8531cce7f2e309de37026ea6f874d054ca525e4bdc13d894429297e1de288196 | from sympy.core.expr import unchanged
from sympy.sets import (ConditionSet, Intersection, FiniteSet,
EmptySet, Union, Contains, ImageSet)
from sympy.core.function import (Function, Lambda)
from sympy.core.mod import Mod
from sympy.core.numbers import (oo, pi)
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.trigonometric import (asin, sin)
from sympy.logic.boolalg import And
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.sets.sets import Interval
from sympy.testing.pytest import raises, warns_deprecated_sympy
w = Symbol('w')
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
f = Function('f')
def test_CondSet():
sin_sols_principal = ConditionSet(x, Eq(sin(x), 0),
Interval(0, 2*pi, False, True))
assert pi in sin_sols_principal
assert pi/2 not in sin_sols_principal
assert 3*pi not in sin_sols_principal
assert oo not in sin_sols_principal
assert 5 in ConditionSet(x, x**2 > 4, S.Reals)
assert 1 not in ConditionSet(x, x**2 > 4, S.Reals)
# in this case, 0 is not part of the base set so
# it can't be in any subset selected by the condition
assert 0 not in ConditionSet(x, y > 5, Interval(1, 7))
# since 'in' requires a true/false, the following raises
# an error because the given value provides no information
# for the condition to evaluate (since the condition does
# not depend on the dummy symbol): the result is `y > 5`.
# In this case, ConditionSet is just acting like
# Piecewise((Interval(1, 7), y > 5), (S.EmptySet, True)).
raises(TypeError, lambda: 6 in ConditionSet(x, y > 5,
Interval(1, 7)))
X = MatrixSymbol('X', 2, 2)
matrix_set = ConditionSet(X, Eq(X*Matrix([[1, 1], [1, 1]]), X))
Y = Matrix([[0, 0], [0, 0]])
assert matrix_set.contains(Y).doit() is S.true
Z = Matrix([[1, 2], [3, 4]])
assert matrix_set.contains(Z).doit() is S.false
assert isinstance(ConditionSet(x, x < 1, {x, y}).base_set,
FiniteSet)
raises(TypeError, lambda: ConditionSet(x, x + 1, {x, y}))
raises(TypeError, lambda: ConditionSet(x, x, 1))
I = S.Integers
U = S.UniversalSet
C = ConditionSet
assert C(x, False, I) is S.EmptySet
assert C(x, True, I) is I
assert C(x, x < 1, C(x, x < 2, I)
) == C(x, (x < 1) & (x < 2), I)
assert C(y, y < 1, C(x, y < 2, I)
) == C(x, (x < 1) & (y < 2), I), C(y, y < 1, C(x, y < 2, I))
assert C(y, y < 1, C(x, x < 2, I)
) == C(y, (y < 1) & (y < 2), I)
assert C(y, y < 1, C(x, y < x, I)
) == C(x, (x < 1) & (y < x), I)
assert unchanged(C, y, x < 1, C(x, y < x, I))
assert ConditionSet(x, x < 1).base_set is U
# arg checking is not done at instantiation but this
# will raise an error when containment is tested
assert ConditionSet((x,), x < 1).base_set is U
c = ConditionSet((x, y), x < y, I**2)
assert (1, 2) in c
assert (1, pi) not in c
raises(TypeError, lambda: C(x, x > 1, C((x, y), x > 1, I**2)))
# signature mismatch since only 3 args are accepted
raises(TypeError, lambda: C((x, y), x + y < 2, U, U))
def test_CondSet_intersect():
input_conditionset = ConditionSet(x, x**2 > 4, Interval(1, 4, False,
False))
other_domain = Interval(0, 3, False, False)
output_conditionset = ConditionSet(x, x**2 > 4, Interval(
1, 3, False, False))
assert Intersection(input_conditionset, other_domain
) == output_conditionset
def test_issue_9849():
assert ConditionSet(x, Eq(x, x), S.Naturals
) is S.Naturals
assert ConditionSet(x, Eq(Abs(sin(x)), -1), S.Naturals
) == S.EmptySet
def test_simplified_FiniteSet_in_CondSet():
assert ConditionSet(x, And(x < 1, x > -3), FiniteSet(0, 1, 2)
) == FiniteSet(0)
assert ConditionSet(x, x < 0, FiniteSet(0, 1, 2)) == EmptySet
assert ConditionSet(x, And(x < -3), EmptySet) == EmptySet
y = Symbol('y')
assert (ConditionSet(x, And(x > 0), FiniteSet(-1, 0, 1, y)) ==
Union(FiniteSet(1), ConditionSet(x, And(x > 0), FiniteSet(y))))
assert (ConditionSet(x, Eq(Mod(x, 3), 1), FiniteSet(1, 4, 2, y)) ==
Union(FiniteSet(1, 4), ConditionSet(x, Eq(Mod(x, 3), 1),
FiniteSet(y))))
def test_free_symbols():
assert ConditionSet(x, Eq(y, 0), FiniteSet(z)
).free_symbols == {y, z}
assert ConditionSet(x, Eq(x, 0), FiniteSet(z)
).free_symbols == {z}
assert ConditionSet(x, Eq(x, 0), FiniteSet(x, z)
).free_symbols == {x, z}
assert ConditionSet(x, Eq(x, 0), ImageSet(Lambda(y, y**2),
S.Integers)).free_symbols == set()
def test_bound_symbols():
assert ConditionSet(x, Eq(y, 0), FiniteSet(z)
).bound_symbols == [x]
assert ConditionSet(x, Eq(x, 0), FiniteSet(x, y)
).bound_symbols == [x]
assert ConditionSet(x, x < 10, ImageSet(Lambda(y, y**2), S.Integers)
).bound_symbols == [x]
assert ConditionSet(x, x < 10, ConditionSet(y, y > 1, S.Integers)
).bound_symbols == [x]
def test_as_dummy():
_0, _1 = symbols('_0 _1')
assert ConditionSet(x, x < 1, Interval(y, oo)
).as_dummy() == ConditionSet(_0, _0 < 1, Interval(y, oo))
assert ConditionSet(x, x < 1, Interval(x, oo)
).as_dummy() == ConditionSet(_0, _0 < 1, Interval(x, oo))
assert ConditionSet(x, x < 1, ImageSet(Lambda(y, y**2), S.Integers)
).as_dummy() == ConditionSet(
_0, _0 < 1, ImageSet(Lambda(_0, _0**2), S.Integers))
e = ConditionSet((x, y), x <= y, S.Reals**2)
assert e.bound_symbols == [x, y]
assert e.as_dummy() == ConditionSet((_0, _1), _0 <= _1, S.Reals**2)
assert e.as_dummy() == ConditionSet((y, x), y <= x, S.Reals**2
).as_dummy()
def test_subs_CondSet():
s = FiniteSet(z, y)
c = ConditionSet(x, x < 2, s)
assert c.subs(x, y) == c
assert c.subs(z, y) == ConditionSet(x, x < 2, FiniteSet(y))
assert c.xreplace({x: y}) == ConditionSet(y, y < 2, s)
assert ConditionSet(x, x < y, s
).subs(y, w) == ConditionSet(x, x < w, s.subs(y, w))
# if the user uses assumptions that cause the condition
# to evaluate, that can't be helped from SymPy's end
n = Symbol('n', negative=True)
assert ConditionSet(n, 0 < n, S.Integers) is S.EmptySet
p = Symbol('p', positive=True)
assert ConditionSet(n, n < y, S.Integers
).subs(n, x) == ConditionSet(n, n < y, S.Integers)
raises(ValueError, lambda: ConditionSet(
x + 1, x < 1, S.Integers))
assert ConditionSet(
p, n < x, Interval(-5, 5)).subs(x, p) == Interval(-5, 5), ConditionSet(
p, n < x, Interval(-5, 5)).subs(x, p)
assert ConditionSet(
n, n < x, Interval(-oo, 0)).subs(x, p
) == Interval(-oo, 0)
assert ConditionSet(f(x), f(x) < 1, {w, z}
).subs(f(x), y) == ConditionSet(f(x), f(x) < 1, {w, z})
# issue 17341
k = Symbol('k')
img1 = ImageSet(Lambda(k, 2*k*pi + asin(y)), S.Integers)
img2 = ImageSet(Lambda(k, 2*k*pi + asin(S.One/3)), S.Integers)
assert ConditionSet(x, Contains(
y, Interval(-1,1)), img1).subs(y, S.One/3).dummy_eq(img2)
assert (0, 1) in ConditionSet((x, y), x + y < 3, S.Integers**2)
raises(TypeError, lambda: ConditionSet(n, n < -10, Interval(0, 10)))
def test_subs_CondSet_tebr():
with warns_deprecated_sympy():
assert ConditionSet((x, y), {x + 1, x + y}, S.Reals**2) == \
ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2)
def test_dummy_eq():
C = ConditionSet
I = S.Integers
c = C(x, x < 1, I)
assert c.dummy_eq(C(y, y < 1, I))
assert c.dummy_eq(1) == False
assert c.dummy_eq(C(x, x < 1, S.Reals)) == False
c1 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2)
c2 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Reals**2)
c3 = ConditionSet((x, y), Eq(x + 1, 0) & Eq(x + y, 0), S.Complexes**2)
assert c1.dummy_eq(c2)
assert c1.dummy_eq(c3) is False
assert c.dummy_eq(c1) is False
assert c1.dummy_eq(c) is False
# issue 19496
m = Symbol('m')
n = Symbol('n')
a = Symbol('a')
d1 = ImageSet(Lambda(m, m*pi), S.Integers)
d2 = ImageSet(Lambda(n, n*pi), S.Integers)
c1 = ConditionSet(x, Ne(a, 0), d1)
c2 = ConditionSet(x, Ne(a, 0), d2)
assert c1.dummy_eq(c2)
def test_contains():
assert 6 in ConditionSet(x, x > 5, Interval(1, 7))
assert (8 in ConditionSet(x, y > 5, Interval(1, 7))) is False
# `in` should give True or False; in this case there is not
# enough information for that result
raises(TypeError,
lambda: 6 in ConditionSet(x, y > 5, Interval(1, 7)))
# here, there is enough information but the comparison is
# not defined
raises(TypeError, lambda: 0 in ConditionSet(x, 1/x >= 0, S.Reals))
assert ConditionSet(x, y > 5, Interval(1, 7)
).contains(6) == (y > 5)
assert ConditionSet(x, y > 5, Interval(1, 7)
).contains(8) is S.false
assert ConditionSet(x, y > 5, Interval(1, 7)
).contains(w) == And(Contains(w, Interval(1, 7)), y > 5)
# This returns an unevaluated Contains object
# because 1/0 should not be defined for 1 and 0 in the context of
# reals.
assert ConditionSet(x, 1/x >= 0, S.Reals).contains(0) == \
Contains(0, ConditionSet(x, 1/x >= 0, S.Reals), evaluate=False)
c = ConditionSet((x, y), x + y > 1, S.Integers**2)
assert not c.contains(1)
assert c.contains((2, 1))
assert not c.contains((0, 1))
c = ConditionSet((w, (x, y)), w + x + y > 1, S.Integers*S.Integers**2)
assert not c.contains(1)
assert not c.contains((1, 2))
assert not c.contains(((1, 2), 3))
assert not c.contains(((1, 2), (3, 4)))
assert c.contains((1, (3, 4)))
def test_as_relational():
assert ConditionSet((x, y), x > 1, S.Integers**2).as_relational((x, y)
) == (x > 1) & Contains((x, y), S.Integers**2)
assert ConditionSet(x, x > 1, S.Integers).as_relational(x
) == Contains(x, S.Integers) & (x > 1)
def test_flatten():
"""Tests whether there is basic denesting functionality"""
inner = ConditionSet(x, sin(x) + x > 0)
outer = ConditionSet(x, Contains(x, inner), S.Reals)
assert outer == ConditionSet(x, sin(x) + x > 0, S.Reals)
inner = ConditionSet(y, sin(y) + y > 0)
outer = ConditionSet(x, Contains(y, inner), S.Reals)
assert outer != ConditionSet(x, sin(x) + x > 0, S.Reals)
inner = ConditionSet(x, sin(x) + x > 0).intersect(Interval(-1, 1))
outer = ConditionSet(x, Contains(x, inner), S.Reals)
assert outer == ConditionSet(x, sin(x) + x > 0, Interval(-1, 1))
def test_duplicate():
from sympy.core.function import BadSignatureError
# test coverage for line 95 in conditionset.py, check for duplicates in symbols
dup = symbols('a,a')
raises(BadSignatureError, lambda: ConditionSet(dup, x < 0))
|
5be252d9f591cf1848240879d078b99672669dd9327fefebb6bed4982f66124f | 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))))
|
15baec27f8be779bfbbdc3c7a0601cf5361d8722d8286e14409324605c486048 | from sympy.core.expr import unchanged
from sympy.core.numbers import oo
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.sets.contains import Contains
from sympy.sets.sets import (FiniteSet, Interval)
from sympy.testing.pytest import raises
def test_contains_basic():
raises(TypeError, lambda: Contains(S.Integers, 1))
assert Contains(2, S.Integers) is S.true
assert Contains(-2, S.Naturals) is S.false
i = Symbol('i', integer=True)
assert Contains(i, S.Naturals) == Contains(i, S.Naturals, evaluate=False)
def test_issue_6194():
x = Symbol('x')
assert unchanged(Contains, x, Interval(0, 1))
assert Interval(0, 1).contains(x) == (S.Zero <= x) & (x <= 1)
assert Contains(x, FiniteSet(0)) != S.false
assert Contains(x, Interval(1, 1)) != S.false
assert Contains(x, S.Integers) != S.false
def test_issue_10326():
assert Contains(oo, Interval(-oo, oo)) == False
assert Contains(-oo, Interval(-oo, oo)) == False
def test_binary_symbols():
x = Symbol('x')
y = Symbol('y')
z = Symbol('z')
assert Contains(x, FiniteSet(y, Eq(z, True))
).binary_symbols == {y, z}
def test_as_set():
x = Symbol('x')
y = Symbol('y')
# Contains is a BooleanFunction whose value depends on an arg's
# containment in a Set -- rewriting as a Set is not yet implemented
raises(NotImplementedError, lambda:
Contains(x, FiniteSet(y)).as_set())
def test_type_error():
# Pass in a parameter not of type "set"
raises(TypeError, lambda: Contains(2, None))
|
2f80d873a07240c0f0246173242dd885f4ab564e5ab96ee97f8418a49e7163bc | from sympy.sets.ordinals import Ordinal, OmegaPower, ord0, omega
from sympy.testing.pytest import raises
def test_string_ordinals():
assert str(omega) == 'w'
assert str(Ordinal(OmegaPower(5, 3), OmegaPower(3, 2))) == 'w**5*3 + w**3*2'
assert str(Ordinal(OmegaPower(5, 3), OmegaPower(0, 5))) == 'w**5*3 + 5'
assert str(Ordinal(OmegaPower(1, 3), OmegaPower(0, 5))) == 'w*3 + 5'
assert str(Ordinal(OmegaPower(omega + 1, 1), OmegaPower(3, 2))) == 'w**(w + 1) + w**3*2'
def test_addition_with_integers():
assert 3 + Ordinal(OmegaPower(5, 3)) == Ordinal(OmegaPower(5, 3))
assert Ordinal(OmegaPower(5, 3))+3 == Ordinal(OmegaPower(5, 3), OmegaPower(0, 3))
assert Ordinal(OmegaPower(5, 3), OmegaPower(0, 2))+3 == \
Ordinal(OmegaPower(5, 3), OmegaPower(0, 5))
def test_addition_with_ordinals():
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) + Ordinal(OmegaPower(3, 3)) == \
Ordinal(OmegaPower(5, 3), OmegaPower(3, 5))
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) + Ordinal(OmegaPower(4, 2)) == \
Ordinal(OmegaPower(5, 3), OmegaPower(4, 2))
assert Ordinal(OmegaPower(omega, 2), OmegaPower(3, 2)) + Ordinal(OmegaPower(4, 2)) == \
Ordinal(OmegaPower(omega, 2), OmegaPower(4, 2))
def test_comparison():
assert Ordinal(OmegaPower(5, 3)) > Ordinal(OmegaPower(4, 3), OmegaPower(2, 1))
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) < Ordinal(OmegaPower(5, 4))
assert Ordinal(OmegaPower(5, 4)) < Ordinal(OmegaPower(5, 5), OmegaPower(4, 1))
assert Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) == \
Ordinal(OmegaPower(5, 3), OmegaPower(3, 2))
assert not Ordinal(OmegaPower(5, 3), OmegaPower(3, 2)) == Ordinal(OmegaPower(5, 3))
assert Ordinal(OmegaPower(omega, 3)) > Ordinal(OmegaPower(5, 3))
def test_multiplication_with_integers():
w = omega
assert 3*w == w
assert w*9 == Ordinal(OmegaPower(1, 9))
def test_multiplication():
w = omega
assert w*(w + 1) == w*w + w
assert (w + 1)*(w + 1) == w*w + w + 1
assert w*1 == w
assert 1*w == w
assert w*ord0 == ord0
assert ord0*w == ord0
assert w**w == w * w**w
assert (w**w)*w*w == w**(w + 2)
def test_exponentiation():
w = omega
assert w**2 == w*w
assert w**3 == w*w*w
assert w**(w + 1) == Ordinal(OmegaPower(omega + 1, 1))
assert (w**w)*(w**w) == w**(w*2)
def test_comapre_not_instance():
w = OmegaPower(omega + 1, 1)
assert(not (w == None))
assert(not (w < 5))
raises(TypeError, lambda: w < 6.66)
def test_is_successort():
w = Ordinal(OmegaPower(5, 1))
assert not w.is_successor_ordinal
|
64e772e486ad3423f336bb3deb7a9cbba9f622a9c5501b019d13aba946b826b7 |
from sympy.core.expr import unchanged
from sympy.sets.fancysets import (ImageSet, Range, normalize_theta_set,
ComplexRegion)
from sympy.sets.sets import (FiniteSet, Interval, Union, imageset,
Intersection, ProductSet, Contains)
from sympy.sets.conditionset import ConditionSet
from sympy.simplify.simplify import simplify
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.function import Lambda
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.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
# 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*Rational(6, 10)) in c3
assert (S.Half + I*Rational(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_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])
|
d51d9010591b4380600b5ccd1735aef14c730ded07cb8699caedde23c038929e | from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.function import Lambda
from sympy.core.numbers import (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.core.sympify import sympify
from sympy.functions.elementary.miscellaneous import (Max, Min, sqrt)
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import (cos, sin)
from sympy.logic.boolalg import (false, true)
from sympy.matrices.dense import Matrix
from sympy.polys.rootoftools import rootof
from sympy.sets.contains import Contains
from sympy.sets.fancysets import (ImageSet, Range)
from sympy.sets.sets import (Complement, DisjointUnion, FiniteSet, Intersection, Interval, ProductSet, Set, SymmetricDifference, Union, imageset)
from mpmath import mpi
from sympy.core.expr import unchanged
from sympy.core.relational import Eq, Ne, Le, Lt, LessThan
from sympy.logic import And, Or, Xor
from sympy.testing.pytest import raises, XFAIL, warns_deprecated_sympy
from sympy.abc import x, y, z, m, n
EmptySet = S.EmptySet
def test_imageset():
ints = S.Integers
assert imageset(x, x - 1, S.Naturals) is S.Naturals0
assert imageset(x, x + 1, S.Naturals0) is S.Naturals
assert imageset(x, abs(x), S.Naturals0) is S.Naturals0
assert imageset(x, abs(x), S.Naturals) is S.Naturals
assert imageset(x, abs(x), S.Integers) is S.Naturals0
# issue 16878a
r = symbols('r', real=True)
assert imageset(x, (x, x), S.Reals)._contains((1, r)) == None
assert imageset(x, (x, x), S.Reals)._contains((1, 2)) == False
assert (r, r) in imageset(x, (x, x), S.Reals)
assert 1 + I in imageset(x, x + I, S.Reals)
assert {1} not in imageset(x, (x,), S.Reals)
assert (1, 1) not in imageset(x, (x,), S.Reals)
raises(TypeError, lambda: imageset(x, ints))
raises(ValueError, lambda: imageset(x, y, z, ints))
raises(ValueError, lambda: imageset(Lambda(x, cos(x)), y))
assert (1, 2) in imageset(Lambda((x, y), (x, y)), ints, ints)
raises(ValueError, lambda: imageset(Lambda(x, x), ints, ints))
assert imageset(cos, ints) == ImageSet(Lambda(x, cos(x)), ints)
def f(x):
return cos(x)
assert imageset(f, ints) == imageset(x, cos(x), ints)
f = lambda x: cos(x)
assert imageset(f, ints) == ImageSet(Lambda(x, cos(x)), ints)
assert imageset(x, 1, ints) == FiniteSet(1)
assert imageset(x, y, ints) == {y}
assert imageset((x, y), (1, z), ints, S.Reals) == {(1, z)}
clash = Symbol('x', integer=true)
assert (str(imageset(lambda x: x + clash, Interval(-2, 1)).lamda.expr)
in ('x0 + x', 'x + x0'))
x1, x2 = symbols("x1, x2")
assert imageset(lambda x, y:
Add(x, y), Interval(1, 2), Interval(2, 3)).dummy_eq(
ImageSet(Lambda((x1, x2), x1 + x2),
Interval(1, 2), Interval(2, 3)))
def test_is_empty():
for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals,
S.UniversalSet]:
assert s.is_empty is False
assert S.EmptySet.is_empty is True
def test_is_finiteset():
for s in [S.Naturals, S.Naturals0, S.Integers, S.Rationals, S.Reals,
S.UniversalSet]:
assert s.is_finite_set is False
assert S.EmptySet.is_finite_set is True
assert FiniteSet(1, 2).is_finite_set is True
assert Interval(1, 2).is_finite_set is False
assert Interval(x, y).is_finite_set is None
assert ProductSet(FiniteSet(1), FiniteSet(2)).is_finite_set is True
assert ProductSet(FiniteSet(1), Interval(1, 2)).is_finite_set is False
assert ProductSet(FiniteSet(1), Interval(x, y)).is_finite_set is None
assert Union(Interval(0, 1), Interval(2, 3)).is_finite_set is False
assert Union(FiniteSet(1), Interval(2, 3)).is_finite_set is False
assert Union(FiniteSet(1), FiniteSet(2)).is_finite_set is True
assert Union(FiniteSet(1), Interval(x, y)).is_finite_set is None
assert Intersection(Interval(x, y), FiniteSet(1)).is_finite_set is True
assert Intersection(Interval(x, y), Interval(1, 2)).is_finite_set is None
assert Intersection(FiniteSet(x), FiniteSet(y)).is_finite_set is True
assert Complement(FiniteSet(1), Interval(x, y)).is_finite_set is True
assert Complement(Interval(x, y), FiniteSet(1)).is_finite_set is None
assert Complement(Interval(1, 2), FiniteSet(x)).is_finite_set is False
assert DisjointUnion(Interval(-5, 3), FiniteSet(x, y)).is_finite_set is False
assert DisjointUnion(S.EmptySet, FiniteSet(x, y), S.EmptySet).is_finite_set is True
def test_deprecated_is_EmptySet():
with warns_deprecated_sympy():
S.EmptySet.is_EmptySet
def test_interval_arguments():
assert Interval(0, oo) == Interval(0, oo, False, True)
assert Interval(0, oo).right_open is true
assert Interval(-oo, 0) == Interval(-oo, 0, True, False)
assert Interval(-oo, 0).left_open is true
assert Interval(oo, -oo) == S.EmptySet
assert Interval(oo, oo) == S.EmptySet
assert Interval(-oo, -oo) == S.EmptySet
assert Interval(oo, x) == S.EmptySet
assert Interval(oo, oo) == S.EmptySet
assert Interval(x, -oo) == S.EmptySet
assert Interval(x, x) == {x}
assert isinstance(Interval(1, 1), FiniteSet)
e = Sum(x, (x, 1, 3))
assert isinstance(Interval(e, e), FiniteSet)
assert Interval(1, 0) == S.EmptySet
assert Interval(1, 1).measure == 0
assert Interval(1, 1, False, True) == S.EmptySet
assert Interval(1, 1, True, False) == S.EmptySet
assert Interval(1, 1, True, True) == S.EmptySet
assert isinstance(Interval(0, Symbol('a')), Interval)
assert Interval(Symbol('a', real=True, positive=True), 0) == S.EmptySet
raises(ValueError, lambda: Interval(0, S.ImaginaryUnit))
raises(ValueError, lambda: Interval(0, Symbol('z', extended_real=False)))
raises(ValueError, lambda: Interval(x, x + S.ImaginaryUnit))
raises(NotImplementedError, lambda: Interval(0, 1, And(x, y)))
raises(NotImplementedError, lambda: Interval(0, 1, False, And(x, y)))
raises(NotImplementedError, lambda: Interval(0, 1, z, And(x, y)))
def test_interval_symbolic_end_points():
a = Symbol('a', real=True)
assert Union(Interval(0, a), Interval(0, 3)).sup == Max(a, 3)
assert Union(Interval(a, 0), Interval(-3, 0)).inf == Min(-3, a)
assert Interval(0, a).contains(1) == LessThan(1, a)
def test_interval_is_empty():
x, y = symbols('x, y')
r = Symbol('r', real=True)
p = Symbol('p', positive=True)
n = Symbol('n', negative=True)
nn = Symbol('nn', nonnegative=True)
assert Interval(1, 2).is_empty == False
assert Interval(3, 3).is_empty == False # FiniteSet
assert Interval(r, r).is_empty == False # FiniteSet
assert Interval(r, r + nn).is_empty == False
assert Interval(x, x).is_empty == False
assert Interval(1, oo).is_empty == False
assert Interval(-oo, oo).is_empty == False
assert Interval(-oo, 1).is_empty == False
assert Interval(x, y).is_empty == None
assert Interval(r, oo).is_empty == False # real implies finite
assert Interval(n, 0).is_empty == False
assert Interval(n, 0, left_open=True).is_empty == False
assert Interval(p, 0).is_empty == True # EmptySet
assert Interval(nn, 0).is_empty == None
assert Interval(n, p).is_empty == False
assert Interval(0, p, left_open=True).is_empty == False
assert Interval(0, p, right_open=True).is_empty == False
assert Interval(0, nn, left_open=True).is_empty == None
assert Interval(0, nn, right_open=True).is_empty == None
def test_union():
assert Union(Interval(1, 2), Interval(2, 3)) == Interval(1, 3)
assert Union(Interval(1, 2), Interval(2, 3, True)) == Interval(1, 3)
assert Union(Interval(1, 3), Interval(2, 4)) == Interval(1, 4)
assert Union(Interval(1, 2), Interval(1, 3)) == Interval(1, 3)
assert Union(Interval(1, 3), Interval(1, 2)) == Interval(1, 3)
assert Union(Interval(1, 3, False, True), Interval(1, 2)) == \
Interval(1, 3, False, True)
assert Union(Interval(1, 3), Interval(1, 2, False, True)) == Interval(1, 3)
assert Union(Interval(1, 2, True), Interval(1, 3)) == Interval(1, 3)
assert Union(Interval(1, 2, True), Interval(1, 3, True)) == \
Interval(1, 3, True)
assert Union(Interval(1, 2, True), Interval(1, 3, True, True)) == \
Interval(1, 3, True, True)
assert Union(Interval(1, 2, True, True), Interval(1, 3, True)) == \
Interval(1, 3, True)
assert Union(Interval(1, 3), Interval(2, 3)) == Interval(1, 3)
assert Union(Interval(1, 3, False, True), Interval(2, 3)) == \
Interval(1, 3)
assert Union(Interval(1, 2, False, True), Interval(2, 3, True)) != \
Interval(1, 3)
assert Union(Interval(1, 2), S.EmptySet) == Interval(1, 2)
assert Union(S.EmptySet) == S.EmptySet
assert Union(Interval(0, 1), *[FiniteSet(1.0/n) for n in range(1, 10)]) == \
Interval(0, 1)
# issue #18241:
x = Symbol('x')
assert Union(Interval(0, 1), FiniteSet(1, x)) == Union(
Interval(0, 1), FiniteSet(x))
assert unchanged(Union, Interval(0, 1), FiniteSet(2, x))
assert Interval(1, 2).union(Interval(2, 3)) == \
Interval(1, 2) + Interval(2, 3)
assert Interval(1, 2).union(Interval(2, 3)) == Interval(1, 3)
assert Union(Set()) == Set()
assert FiniteSet(1) + FiniteSet(2) + FiniteSet(3) == FiniteSet(1, 2, 3)
assert FiniteSet('ham') + FiniteSet('eggs') == FiniteSet('ham', 'eggs')
assert FiniteSet(1, 2, 3) + S.EmptySet == FiniteSet(1, 2, 3)
assert FiniteSet(1, 2, 3) & FiniteSet(2, 3, 4) == FiniteSet(2, 3)
assert FiniteSet(1, 2, 3) | FiniteSet(2, 3, 4) == FiniteSet(1, 2, 3, 4)
assert FiniteSet(1, 2, 3) & S.EmptySet == S.EmptySet
assert FiniteSet(1, 2, 3) | S.EmptySet == FiniteSet(1, 2, 3)
x = Symbol("x")
y = Symbol("y")
z = Symbol("z")
assert S.EmptySet | FiniteSet(x, FiniteSet(y, z)) == \
FiniteSet(x, FiniteSet(y, z))
# Test that Intervals and FiniteSets play nicely
assert Interval(1, 3) + FiniteSet(2) == Interval(1, 3)
assert Interval(1, 3, True, True) + FiniteSet(3) == \
Interval(1, 3, True, False)
X = Interval(1, 3) + FiniteSet(5)
Y = Interval(1, 2) + FiniteSet(3)
XandY = X.intersect(Y)
assert 2 in X and 3 in X and 3 in XandY
assert XandY.is_subset(X) and XandY.is_subset(Y)
raises(TypeError, lambda: Union(1, 2, 3))
assert X.is_iterable is False
# issue 7843
assert Union(S.EmptySet, FiniteSet(-sqrt(-I), sqrt(-I))) == \
FiniteSet(-sqrt(-I), sqrt(-I))
assert Union(S.Reals, S.Integers) == S.Reals
def test_union_iter():
# Use Range because it is ordered
u = Union(Range(3), Range(5), Range(4), evaluate=False)
# Round robin
assert list(u) == [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4]
def test_union_is_empty():
assert (Interval(x, y) + FiniteSet(1)).is_empty == False
assert (Interval(x, y) + Interval(-x, y)).is_empty == None
def test_difference():
assert Interval(1, 3) - Interval(1, 2) == Interval(2, 3, True)
assert Interval(1, 3) - Interval(2, 3) == Interval(1, 2, False, True)
assert Interval(1, 3, True) - Interval(2, 3) == Interval(1, 2, True, True)
assert Interval(1, 3, True) - Interval(2, 3, True) == \
Interval(1, 2, True, False)
assert Interval(0, 2) - FiniteSet(1) == \
Union(Interval(0, 1, False, True), Interval(1, 2, True, False))
# issue #18119
assert S.Reals - FiniteSet(I) == S.Reals
assert S.Reals - FiniteSet(-I, I) == S.Reals
assert Interval(0, 10) - FiniteSet(-I, I) == Interval(0, 10)
assert Interval(0, 10) - FiniteSet(1, I) == Union(
Interval.Ropen(0, 1), Interval.Lopen(1, 10))
assert S.Reals - FiniteSet(1, 2 + I, x, y**2) == Complement(
Union(Interval.open(-oo, 1), Interval.open(1, oo)), FiniteSet(x, y**2),
evaluate=False)
assert FiniteSet(1, 2, 3) - FiniteSet(2) == FiniteSet(1, 3)
assert FiniteSet('ham', 'eggs') - FiniteSet('eggs') == FiniteSet('ham')
assert FiniteSet(1, 2, 3, 4) - Interval(2, 10, True, False) == \
FiniteSet(1, 2)
assert FiniteSet(1, 2, 3, 4) - S.EmptySet == FiniteSet(1, 2, 3, 4)
assert Union(Interval(0, 2), FiniteSet(2, 3, 4)) - Interval(1, 3) == \
Union(Interval(0, 1, False, True), FiniteSet(4))
assert -1 in S.Reals - S.Naturals
def test_Complement():
A = FiniteSet(1, 3, 4)
B = FiniteSet(3, 4)
C = Interval(1, 3)
D = Interval(1, 2)
assert Complement(A, B, evaluate=False).is_iterable is True
assert Complement(A, C, evaluate=False).is_iterable is True
assert Complement(C, D, evaluate=False).is_iterable is None
assert FiniteSet(*Complement(A, B, evaluate=False)) == FiniteSet(1)
assert FiniteSet(*Complement(A, C, evaluate=False)) == FiniteSet(4)
raises(TypeError, lambda: FiniteSet(*Complement(C, A, evaluate=False)))
assert Complement(Interval(1, 3), Interval(1, 2)) == Interval(2, 3, True)
assert Complement(FiniteSet(1, 3, 4), FiniteSet(3, 4)) == FiniteSet(1)
assert Complement(Union(Interval(0, 2), FiniteSet(2, 3, 4)),
Interval(1, 3)) == \
Union(Interval(0, 1, False, True), FiniteSet(4))
assert not 3 in Complement(Interval(0, 5), Interval(1, 4), evaluate=False)
assert -1 in Complement(S.Reals, S.Naturals, evaluate=False)
assert not 1 in Complement(S.Reals, S.Naturals, evaluate=False)
assert Complement(S.Integers, S.UniversalSet) == EmptySet
assert S.UniversalSet.complement(S.Integers) == EmptySet
assert (not 0 in S.Reals.intersect(S.Integers - FiniteSet(0)))
assert S.EmptySet - S.Integers == S.EmptySet
assert (S.Integers - FiniteSet(0)) - FiniteSet(1) == S.Integers - FiniteSet(0, 1)
assert S.Reals - Union(S.Naturals, FiniteSet(pi)) == \
Intersection(S.Reals - S.Naturals, S.Reals - FiniteSet(pi))
# issue 12712
assert Complement(FiniteSet(x, y, 2), Interval(-10, 10)) == \
Complement(FiniteSet(x, y), Interval(-10, 10))
A = FiniteSet(*symbols('a:c'))
B = FiniteSet(*symbols('d:f'))
assert unchanged(Complement, ProductSet(A, A), B)
A2 = ProductSet(A, A)
B3 = ProductSet(B, B, B)
assert A2 - B3 == A2
assert B3 - A2 == B3
def test_set_operations_nonsets():
'''Tests that e.g. FiniteSet(1) * 2 raises TypeError'''
ops = [
lambda a, b: a + b,
lambda a, b: a - b,
lambda a, b: a * b,
lambda a, b: a / b,
lambda a, b: a // b,
lambda a, b: a | b,
lambda a, b: a & b,
lambda a, b: a ^ b,
# FiniteSet(1) ** 2 gives a ProductSet
#lambda a, b: a ** b,
]
Sx = FiniteSet(x)
Sy = FiniteSet(y)
sets = [
{1},
FiniteSet(1),
Interval(1, 2),
Union(Sx, Interval(1, 2)),
Intersection(Sx, Sy),
Complement(Sx, Sy),
ProductSet(Sx, Sy),
S.EmptySet,
]
nums = [0, 1, 2, S(0), S(1), S(2)]
for si in sets:
for ni in nums:
for op in ops:
raises(TypeError, lambda : op(si, ni))
raises(TypeError, lambda : op(ni, si))
raises(TypeError, lambda: si ** object())
raises(TypeError, lambda: si ** {1})
def test_complement():
assert Complement({1, 2}, {1}) == {2}
assert Interval(0, 1).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, True, True))
assert Interval(0, 1, True, False).complement(S.Reals) == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, True, True))
assert Interval(0, 1, False, True).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, oo, False, True))
assert Interval(0, 1, True, True).complement(S.Reals) == \
Union(Interval(-oo, 0, True, False), Interval(1, oo, False, True))
assert S.UniversalSet.complement(S.EmptySet) == S.EmptySet
assert S.UniversalSet.complement(S.Reals) == S.EmptySet
assert S.UniversalSet.complement(S.UniversalSet) == S.EmptySet
assert S.EmptySet.complement(S.Reals) == S.Reals
assert Union(Interval(0, 1), Interval(2, 3)).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(1, 2, True, True),
Interval(3, oo, True, True))
assert FiniteSet(0).complement(S.Reals) == \
Union(Interval(-oo, 0, True, True), Interval(0, oo, True, True))
assert (FiniteSet(5) + Interval(S.NegativeInfinity,
0)).complement(S.Reals) == \
Interval(0, 5, True, True) + Interval(5, S.Infinity, True, True)
assert FiniteSet(1, 2, 3).complement(S.Reals) == \
Interval(S.NegativeInfinity, 1, True, True) + \
Interval(1, 2, True, True) + Interval(2, 3, True, True) +\
Interval(3, S.Infinity, True, True)
assert FiniteSet(x).complement(S.Reals) == Complement(S.Reals, FiniteSet(x))
assert FiniteSet(0, x).complement(S.Reals) == Complement(Interval(-oo, 0, True, True) +
Interval(0, oo, True, True)
, FiniteSet(x), evaluate=False)
square = Interval(0, 1) * Interval(0, 1)
notsquare = square.complement(S.Reals*S.Reals)
assert all(pt in square for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
assert not any(
pt in notsquare for pt in [(0, 0), (.5, .5), (1, 0), (1, 1)])
assert not any(pt in square for pt in [(-1, 0), (1.5, .5), (10, 10)])
assert all(pt in notsquare for pt in [(-1, 0), (1.5, .5), (10, 10)])
def test_intersect1():
assert all(S.Integers.intersection(i) is i for i in
(S.Naturals, S.Naturals0))
assert all(i.intersection(S.Integers) is i for i in
(S.Naturals, S.Naturals0))
s = S.Naturals0
assert S.Naturals.intersection(s) is S.Naturals
assert s.intersection(S.Naturals) is S.Naturals
x = Symbol('x')
assert Interval(0, 2).intersect(Interval(1, 2)) == Interval(1, 2)
assert Interval(0, 2).intersect(Interval(1, 2, True)) == \
Interval(1, 2, True)
assert Interval(0, 2, True).intersect(Interval(1, 2)) == \
Interval(1, 2, False, False)
assert Interval(0, 2, True, True).intersect(Interval(1, 2)) == \
Interval(1, 2, False, True)
assert Interval(0, 2).intersect(Union(Interval(0, 1), Interval(2, 3))) == \
Union(Interval(0, 1), Interval(2, 2))
assert FiniteSet(1, 2).intersect(FiniteSet(1, 2, 3)) == FiniteSet(1, 2)
assert FiniteSet(1, 2, x).intersect(FiniteSet(x)) == FiniteSet(x)
assert FiniteSet('ham', 'eggs').intersect(FiniteSet('ham')) == \
FiniteSet('ham')
assert FiniteSet(1, 2, 3, 4, 5).intersect(S.EmptySet) == S.EmptySet
assert Interval(0, 5).intersect(FiniteSet(1, 3)) == FiniteSet(1, 3)
assert Interval(0, 1, True, True).intersect(FiniteSet(1)) == S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2)) == \
Union(Interval(1, 1), Interval(2, 2))
assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(0, 2)) == \
Union(Interval(0, 1), Interval(2, 2))
assert Union(Interval(0, 1), Interval(2, 3)).intersect(Interval(1, 2, True, True)) == \
S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersect(S.EmptySet) == \
S.EmptySet
assert Union(Interval(0, 5), FiniteSet('ham')).intersect(FiniteSet(2, 3, 4, 5, 6)) == \
Intersection(FiniteSet(2, 3, 4, 5, 6), Union(FiniteSet('ham'), Interval(0, 5)))
assert Intersection(FiniteSet(1, 2, 3), Interval(2, x), Interval(3, y)) == \
Intersection(FiniteSet(3), Interval(2, x), Interval(3, y), evaluate=False)
assert Intersection(FiniteSet(1, 2), Interval(0, 3), Interval(x, y)) == \
Intersection({1, 2}, Interval(x, y), evaluate=False)
assert Intersection(FiniteSet(1, 2, 4), Interval(0, 3), Interval(x, y)) == \
Intersection({1, 2}, Interval(x, y), evaluate=False)
# XXX: Is the real=True necessary here?
# https://github.com/sympy/sympy/issues/17532
m, n = symbols('m, n', real=True)
assert Intersection(FiniteSet(m), FiniteSet(m, n), Interval(m, m+1)) == \
FiniteSet(m)
# issue 8217
assert Intersection(FiniteSet(x), FiniteSet(y)) == \
Intersection(FiniteSet(x), FiniteSet(y), evaluate=False)
assert FiniteSet(x).intersect(S.Reals) == \
Intersection(S.Reals, FiniteSet(x), evaluate=False)
# tests for the intersection alias
assert Interval(0, 5).intersection(FiniteSet(1, 3)) == FiniteSet(1, 3)
assert Interval(0, 1, True, True).intersection(FiniteSet(1)) == S.EmptySet
assert Union(Interval(0, 1), Interval(2, 3)).intersection(Interval(1, 2)) == \
Union(Interval(1, 1), Interval(2, 2))
def test_intersection():
# iterable
i = Intersection(FiniteSet(1, 2, 3), Interval(2, 5), evaluate=False)
assert i.is_iterable
assert set(i) == {S(2), S(3)}
# challenging intervals
x = Symbol('x', real=True)
i = Intersection(Interval(0, 3), Interval(x, 6))
assert (5 in i) is False
raises(TypeError, lambda: 2 in i)
# Singleton special cases
assert Intersection(Interval(0, 1), S.EmptySet) == S.EmptySet
assert Intersection(Interval(-oo, oo), Interval(-oo, x)) == Interval(-oo, x)
# Products
line = Interval(0, 5)
i = Intersection(line**2, line**3, evaluate=False)
assert (2, 2) not in i
assert (2, 2, 2) not in i
raises(TypeError, lambda: list(i))
a = Intersection(Intersection(S.Integers, S.Naturals, evaluate=False), S.Reals, evaluate=False)
assert a._argset == frozenset([Intersection(S.Naturals, S.Integers, evaluate=False), S.Reals])
assert Intersection(S.Complexes, FiniteSet(S.ComplexInfinity)) == S.EmptySet
# issue 12178
assert Intersection() == S.UniversalSet
# issue 16987
assert Intersection({1}, {1}, {x}) == Intersection({1}, {x})
def test_issue_9623():
n = Symbol('n')
a = S.Reals
b = Interval(0, oo)
c = FiniteSet(n)
assert Intersection(a, b, c) == Intersection(b, c)
assert Intersection(Interval(1, 2), Interval(3, 4), FiniteSet(n)) == EmptySet
def test_is_disjoint():
assert Interval(0, 2).is_disjoint(Interval(1, 2)) == False
assert Interval(0, 2).is_disjoint(Interval(3, 4)) == True
def test_ProductSet__len__():
A = FiniteSet(1, 2)
B = FiniteSet(1, 2, 3)
assert ProductSet(A).__len__() == 2
assert ProductSet(A).__len__() is not S(2)
assert ProductSet(A, B).__len__() == 6
assert ProductSet(A, B).__len__() is not S(6)
def test_ProductSet():
# ProductSet is always a set of Tuples
assert ProductSet(S.Reals) == S.Reals ** 1
assert ProductSet(S.Reals, S.Reals) == S.Reals ** 2
assert ProductSet(S.Reals, S.Reals, S.Reals) == S.Reals ** 3
assert ProductSet(S.Reals) != S.Reals
assert ProductSet(S.Reals, S.Reals) == S.Reals * S.Reals
assert ProductSet(S.Reals, S.Reals, S.Reals) != S.Reals * S.Reals * S.Reals
assert ProductSet(S.Reals, S.Reals, S.Reals) == (S.Reals * S.Reals * S.Reals).flatten()
assert 1 not in ProductSet(S.Reals)
assert (1,) in ProductSet(S.Reals)
assert 1 not in ProductSet(S.Reals, S.Reals)
assert (1, 2) in ProductSet(S.Reals, S.Reals)
assert (1, I) not in ProductSet(S.Reals, S.Reals)
assert (1, 2, 3) in ProductSet(S.Reals, S.Reals, S.Reals)
assert (1, 2, 3) in S.Reals ** 3
assert (1, 2, 3) not in S.Reals * S.Reals * S.Reals
assert ((1, 2), 3) in S.Reals * S.Reals * S.Reals
assert (1, (2, 3)) not in S.Reals * S.Reals * S.Reals
assert (1, (2, 3)) in S.Reals * (S.Reals * S.Reals)
assert ProductSet() == FiniteSet(())
assert ProductSet(S.Reals, S.EmptySet) == S.EmptySet
# See GH-17458
for ni in range(5):
Rn = ProductSet(*(S.Reals,) * ni)
assert (1,) * ni in Rn
assert 1 not in Rn
assert (S.Reals * S.Reals) * S.Reals != S.Reals * (S.Reals * S.Reals)
S1 = S.Reals
S2 = S.Integers
x1 = pi
x2 = 3
assert x1 in S1
assert x2 in S2
assert (x1, x2) in S1 * S2
S3 = S1 * S2
x3 = (x1, x2)
assert x3 in S3
assert (x3, x3) in S3 * S3
assert x3 + x3 not in S3 * S3
raises(ValueError, lambda: S.Reals**-1)
with warns_deprecated_sympy():
ProductSet(FiniteSet(s) for s in range(2))
raises(TypeError, lambda: ProductSet(None))
S1 = FiniteSet(1, 2)
S2 = FiniteSet(3, 4)
S3 = ProductSet(S1, S2)
assert (S3.as_relational(x, y)
== And(S1.as_relational(x), S2.as_relational(y))
== And(Or(Eq(x, 1), Eq(x, 2)), Or(Eq(y, 3), Eq(y, 4))))
raises(ValueError, lambda: S3.as_relational(x))
raises(ValueError, lambda: S3.as_relational(x, 1))
raises(ValueError, lambda: ProductSet(Interval(0, 1)).as_relational(x, y))
Z2 = ProductSet(S.Integers, S.Integers)
assert Z2.contains((1, 2)) is S.true
assert Z2.contains((1,)) is S.false
assert Z2.contains(x) == Contains(x, Z2, evaluate=False)
assert Z2.contains(x).subs(x, 1) is S.false
assert Z2.contains((x, 1)).subs(x, 2) is S.true
assert Z2.contains((x, y)) == Contains((x, y), Z2, evaluate=False)
assert unchanged(Contains, (x, y), Z2)
assert Contains((1, 2), Z2) is S.true
def test_ProductSet_of_single_arg_is_not_arg():
assert unchanged(ProductSet, Interval(0, 1))
assert unchanged(ProductSet, ProductSet(Interval(0, 1)))
def test_ProductSet_is_empty():
assert ProductSet(S.Integers, S.Reals).is_empty == False
assert ProductSet(Interval(x, 1), S.Reals).is_empty == None
def test_interval_subs():
a = Symbol('a', real=True)
assert Interval(0, a).subs(a, 2) == Interval(0, 2)
assert Interval(a, 0).subs(a, 2) == S.EmptySet
def test_interval_to_mpi():
assert Interval(0, 1).to_mpi() == mpi(0, 1)
assert Interval(0, 1, True, False).to_mpi() == mpi(0, 1)
assert type(Interval(0, 1).to_mpi()) == type(mpi(0, 1))
def test_set_evalf():
assert Interval(S(11)/64, S.Half).evalf() == Interval(
Float('0.171875'), Float('0.5'))
assert Interval(x, S.Half, right_open=True).evalf() == Interval(
x, Float('0.5'), right_open=True)
assert Interval(-oo, S.Half).evalf() == Interval(-oo, Float('0.5'))
assert FiniteSet(2, x).evalf() == FiniteSet(Float('2.0'), x)
def test_measure():
a = Symbol('a', real=True)
assert Interval(1, 3).measure == 2
assert Interval(0, a).measure == a
assert Interval(1, a).measure == a - 1
assert Union(Interval(1, 2), Interval(3, 4)).measure == 2
assert Union(Interval(1, 2), Interval(3, 4), FiniteSet(5, 6, 7)).measure \
== 2
assert FiniteSet(1, 2, oo, a, -oo, -5).measure == 0
assert S.EmptySet.measure == 0
square = Interval(0, 10) * Interval(0, 10)
offsetsquare = Interval(5, 15) * Interval(5, 15)
band = Interval(-oo, oo) * Interval(2, 4)
assert square.measure == offsetsquare.measure == 100
assert (square + offsetsquare).measure == 175 # there is some overlap
assert (square - offsetsquare).measure == 75
assert (square * FiniteSet(1, 2, 3)).measure == 0
assert (square.intersect(band)).measure == 20
assert (square + band).measure is oo
assert (band * FiniteSet(1, 2, 3)).measure is nan
def test_is_subset():
assert Interval(0, 1).is_subset(Interval(0, 2)) is True
assert Interval(0, 3).is_subset(Interval(0, 2)) is False
assert Interval(0, 1).is_subset(FiniteSet(0, 1)) is False
assert FiniteSet(1, 2).is_subset(FiniteSet(1, 2, 3, 4))
assert FiniteSet(4, 5).is_subset(FiniteSet(1, 2, 3, 4)) is False
assert FiniteSet(1).is_subset(Interval(0, 2))
assert FiniteSet(1, 2).is_subset(Interval(0, 2, True, True)) is False
assert (Interval(1, 2) + FiniteSet(3)).is_subset(
Interval(0, 2, False, True) + FiniteSet(2, 3))
assert Interval(3, 4).is_subset(Union(Interval(0, 1), Interval(2, 5))) is True
assert Interval(3, 6).is_subset(Union(Interval(0, 1), Interval(2, 5))) is False
assert FiniteSet(1, 2, 3, 4).is_subset(Interval(0, 5)) is True
assert S.EmptySet.is_subset(FiniteSet(1, 2, 3)) is True
assert Interval(0, 1).is_subset(S.EmptySet) is False
assert S.EmptySet.is_subset(S.EmptySet) is True
raises(ValueError, lambda: S.EmptySet.is_subset(1))
# tests for the issubset alias
assert FiniteSet(1, 2, 3, 4).issubset(Interval(0, 5)) is True
assert S.EmptySet.issubset(FiniteSet(1, 2, 3)) is True
assert S.Naturals.is_subset(S.Integers)
assert S.Naturals0.is_subset(S.Integers)
assert FiniteSet(x).is_subset(FiniteSet(y)) is None
assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x)) is True
assert FiniteSet(x).is_subset(FiniteSet(y).subs(y, x+1)) is False
assert Interval(0, 1).is_subset(Interval(0, 1, left_open=True)) is False
assert Interval(-2, 3).is_subset(Union(Interval(-oo, -2), Interval(3, oo))) is False
n = Symbol('n', integer=True)
assert Range(-3, 4, 1).is_subset(FiniteSet(-10, 10)) is False
assert Range(S(10)**100).is_subset(FiniteSet(0, 1, 2)) is False
assert Range(6, 0, -2).is_subset(FiniteSet(2, 4, 6)) is True
assert Range(1, oo).is_subset(FiniteSet(1, 2)) is False
assert Range(-oo, 1).is_subset(FiniteSet(1)) is False
assert Range(3).is_subset(FiniteSet(0, 1, n)) is None
assert Range(n, n + 2).is_subset(FiniteSet(n, n + 1)) is True
assert Range(5).is_subset(Interval(0, 4, right_open=True)) is False
#issue 19513
assert imageset(Lambda(n, 1/n), S.Integers).is_subset(S.Reals) is None
def test_is_proper_subset():
assert Interval(0, 1).is_proper_subset(Interval(0, 2)) is True
assert Interval(0, 3).is_proper_subset(Interval(0, 2)) is False
assert S.EmptySet.is_proper_subset(FiniteSet(1, 2, 3)) is True
raises(ValueError, lambda: Interval(0, 1).is_proper_subset(0))
def test_is_superset():
assert Interval(0, 1).is_superset(Interval(0, 2)) == False
assert Interval(0, 3).is_superset(Interval(0, 2))
assert FiniteSet(1, 2).is_superset(FiniteSet(1, 2, 3, 4)) == False
assert FiniteSet(4, 5).is_superset(FiniteSet(1, 2, 3, 4)) == False
assert FiniteSet(1).is_superset(Interval(0, 2)) == False
assert FiniteSet(1, 2).is_superset(Interval(0, 2, True, True)) == False
assert (Interval(1, 2) + FiniteSet(3)).is_superset(
Interval(0, 2, False, True) + FiniteSet(2, 3)) == False
assert Interval(3, 4).is_superset(Union(Interval(0, 1), Interval(2, 5))) == False
assert FiniteSet(1, 2, 3, 4).is_superset(Interval(0, 5)) == False
assert S.EmptySet.is_superset(FiniteSet(1, 2, 3)) == False
assert Interval(0, 1).is_superset(S.EmptySet) == True
assert S.EmptySet.is_superset(S.EmptySet) == True
raises(ValueError, lambda: S.EmptySet.is_superset(1))
# tests for the issuperset alias
assert Interval(0, 1).issuperset(S.EmptySet) == True
assert S.EmptySet.issuperset(S.EmptySet) == True
def test_is_proper_superset():
assert Interval(0, 1).is_proper_superset(Interval(0, 2)) is False
assert Interval(0, 3).is_proper_superset(Interval(0, 2)) is True
assert FiniteSet(1, 2, 3).is_proper_superset(S.EmptySet) is True
raises(ValueError, lambda: Interval(0, 1).is_proper_superset(0))
def test_contains():
assert Interval(0, 2).contains(1) is S.true
assert Interval(0, 2).contains(3) is S.false
assert Interval(0, 2, True, False).contains(0) is S.false
assert Interval(0, 2, True, False).contains(2) is S.true
assert Interval(0, 2, False, True).contains(0) is S.true
assert Interval(0, 2, False, True).contains(2) is S.false
assert Interval(0, 2, True, True).contains(0) is S.false
assert Interval(0, 2, True, True).contains(2) is S.false
assert (Interval(0, 2) in Interval(0, 2)) is False
assert FiniteSet(1, 2, 3).contains(2) is S.true
assert FiniteSet(1, 2, Symbol('x')).contains(Symbol('x')) is S.true
assert FiniteSet(y)._contains(x) is None
raises(TypeError, lambda: x in FiniteSet(y))
assert FiniteSet({x, y})._contains({x}) is None
assert FiniteSet({x, y}).subs(y, x)._contains({x}) is True
assert FiniteSet({x, y}).subs(y, x+1)._contains({x}) is False
# issue 8197
from sympy.abc import a, b
assert isinstance(FiniteSet(b).contains(-a), Contains)
assert isinstance(FiniteSet(b).contains(a), Contains)
assert isinstance(FiniteSet(a).contains(1), Contains)
raises(TypeError, lambda: 1 in FiniteSet(a))
# issue 8209
rad1 = Pow(Pow(2, Rational(1, 3)) - 1, Rational(1, 3))
rad2 = Pow(Rational(1, 9), Rational(1, 3)) - Pow(Rational(2, 9), Rational(1, 3)) + Pow(Rational(4, 9), Rational(1, 3))
s1 = FiniteSet(rad1)
s2 = FiniteSet(rad2)
assert s1 - s2 == S.EmptySet
items = [1, 2, S.Infinity, S('ham'), -1.1]
fset = FiniteSet(*items)
assert all(item in fset for item in items)
assert all(fset.contains(item) is S.true for item in items)
assert Union(Interval(0, 1), Interval(2, 5)).contains(3) is S.true
assert Union(Interval(0, 1), Interval(2, 5)).contains(6) is S.false
assert Union(Interval(0, 1), FiniteSet(2, 5)).contains(3) is S.false
assert S.EmptySet.contains(1) is S.false
assert FiniteSet(rootof(x**3 + x - 1, 0)).contains(S.Infinity) is S.false
assert rootof(x**5 + x**3 + 1, 0) in S.Reals
assert not rootof(x**5 + x**3 + 1, 1) in S.Reals
# non-bool results
assert Union(Interval(1, 2), Interval(3, 4)).contains(x) == \
Or(And(S.One <= x, x <= 2), And(S(3) <= x, x <= 4))
assert Intersection(Interval(1, x), Interval(2, 3)).contains(y) == \
And(y <= 3, y <= x, S.One <= y, S(2) <= y)
assert (S.Complexes).contains(S.ComplexInfinity) == S.false
def test_interval_symbolic():
x = Symbol('x')
e = Interval(0, 1)
assert e.contains(x) == And(S.Zero <= x, x <= 1)
raises(TypeError, lambda: x in e)
e = Interval(0, 1, True, True)
assert e.contains(x) == And(S.Zero < x, x < 1)
c = Symbol('c', real=False)
assert Interval(x, x + 1).contains(c) == False
e = Symbol('e', extended_real=True)
assert Interval(-oo, oo).contains(e) == And(
S.NegativeInfinity < e, e < S.Infinity)
def test_union_contains():
x = Symbol('x')
i1 = Interval(0, 1)
i2 = Interval(2, 3)
i3 = Union(i1, i2)
assert i3.as_relational(x) == Or(And(S.Zero <= x, x <= 1), And(S(2) <= x, x <= 3))
raises(TypeError, lambda: x in i3)
e = i3.contains(x)
assert e == i3.as_relational(x)
assert e.subs(x, -0.5) is false
assert e.subs(x, 0.5) is true
assert e.subs(x, 1.5) is false
assert e.subs(x, 2.5) is true
assert e.subs(x, 3.5) is false
U = Interval(0, 2, True, True) + Interval(10, oo) + FiniteSet(-1, 2, 5, 6)
assert all(el not in U for el in [0, 4, -oo])
assert all(el in U for el in [2, 5, 10])
def test_is_number():
assert Interval(0, 1).is_number is False
assert Set().is_number is False
def test_Interval_is_left_unbounded():
assert Interval(3, 4).is_left_unbounded is False
assert Interval(-oo, 3).is_left_unbounded is True
assert Interval(Float("-inf"), 3).is_left_unbounded is True
def test_Interval_is_right_unbounded():
assert Interval(3, 4).is_right_unbounded is False
assert Interval(3, oo).is_right_unbounded is True
assert Interval(3, Float("+inf")).is_right_unbounded is True
def test_Interval_as_relational():
x = Symbol('x')
assert Interval(-1, 2, False, False).as_relational(x) == \
And(Le(-1, x), Le(x, 2))
assert Interval(-1, 2, True, False).as_relational(x) == \
And(Lt(-1, x), Le(x, 2))
assert Interval(-1, 2, False, True).as_relational(x) == \
And(Le(-1, x), Lt(x, 2))
assert Interval(-1, 2, True, True).as_relational(x) == \
And(Lt(-1, x), Lt(x, 2))
assert Interval(-oo, 2, right_open=False).as_relational(x) == And(Lt(-oo, x), Le(x, 2))
assert Interval(-oo, 2, right_open=True).as_relational(x) == And(Lt(-oo, x), Lt(x, 2))
assert Interval(-2, oo, left_open=False).as_relational(x) == And(Le(-2, x), Lt(x, oo))
assert Interval(-2, oo, left_open=True).as_relational(x) == And(Lt(-2, x), Lt(x, oo))
assert Interval(-oo, oo).as_relational(x) == And(Lt(-oo, x), Lt(x, oo))
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert Interval(x, y).as_relational(x) == (x <= y)
assert Interval(y, x).as_relational(x) == (y <= x)
def test_Finite_as_relational():
x = Symbol('x')
y = Symbol('y')
assert FiniteSet(1, 2).as_relational(x) == Or(Eq(x, 1), Eq(x, 2))
assert FiniteSet(y, -5).as_relational(x) == Or(Eq(x, y), Eq(x, -5))
def test_Union_as_relational():
x = Symbol('x')
assert (Interval(0, 1) + FiniteSet(2)).as_relational(x) == \
Or(And(Le(0, x), Le(x, 1)), Eq(x, 2))
assert (Interval(0, 1, True, True) + FiniteSet(1)).as_relational(x) == \
And(Lt(0, x), Le(x, 1))
assert Or(x < 0, x > 0).as_set().as_relational(x) == \
And((x > -oo), (x < oo), Ne(x, 0))
assert (Interval.Ropen(1, 3) + Interval.Lopen(3, 5)
).as_relational(x) == And((x > 1), (x < 5), Ne(x, 3))
def test_Intersection_as_relational():
x = Symbol('x')
assert (Intersection(Interval(0, 1), FiniteSet(2),
evaluate=False).as_relational(x)
== And(And(Le(0, x), Le(x, 1)), Eq(x, 2)))
def test_Complement_as_relational():
x = Symbol('x')
expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False)
assert expr.as_relational(x) == \
And(Le(0, x), Le(x, 1), Ne(x, 2))
@XFAIL
def test_Complement_as_relational_fail():
x = Symbol('x')
expr = Complement(Interval(0, 1), FiniteSet(2), evaluate=False)
# XXX This example fails because 0 <= x changes to x >= 0
# during the evaluation.
assert expr.as_relational(x) == \
(0 <= x) & (x <= 1) & Ne(x, 2)
def test_SymmetricDifference_as_relational():
x = Symbol('x')
expr = SymmetricDifference(Interval(0, 1), FiniteSet(2), evaluate=False)
assert expr.as_relational(x) == Xor(Eq(x, 2), Le(0, x) & Le(x, 1))
def test_EmptySet():
assert S.EmptySet.as_relational(Symbol('x')) is S.false
assert S.EmptySet.intersect(S.UniversalSet) == S.EmptySet
assert S.EmptySet.boundary == S.EmptySet
def test_finite_basic():
x = Symbol('x')
A = FiniteSet(1, 2, 3)
B = FiniteSet(3, 4, 5)
AorB = Union(A, B)
AandB = A.intersect(B)
assert A.is_subset(AorB) and B.is_subset(AorB)
assert AandB.is_subset(A)
assert AandB == FiniteSet(3)
assert A.inf == 1 and A.sup == 3
assert AorB.inf == 1 and AorB.sup == 5
assert FiniteSet(x, 1, 5).sup == Max(x, 5)
assert FiniteSet(x, 1, 5).inf == Min(x, 1)
# issue 7335
assert FiniteSet(S.EmptySet) != S.EmptySet
assert FiniteSet(FiniteSet(1, 2, 3)) != FiniteSet(1, 2, 3)
assert FiniteSet((1, 2, 3)) != FiniteSet(1, 2, 3)
# Ensure a variety of types can exist in a FiniteSet
assert FiniteSet((1, 2), Float, A, -5, x, 'eggs', x**2, Interval)
assert (A > B) is False
assert (A >= B) is False
assert (A < B) is False
assert (A <= B) is False
assert AorB > A and AorB > B
assert AorB >= A and AorB >= B
assert A >= A and A <= A
assert A >= AandB and B >= AandB
assert A > AandB and B > AandB
assert FiniteSet(1.0) == FiniteSet(1)
def test_product_basic():
H, T = 'H', 'T'
unit_line = Interval(0, 1)
d6 = FiniteSet(1, 2, 3, 4, 5, 6)
d4 = FiniteSet(1, 2, 3, 4)
coin = FiniteSet(H, T)
square = unit_line * unit_line
assert (0, 0) in square
assert 0 not in square
assert (H, T) in coin ** 2
assert (.5, .5, .5) in (square * unit_line).flatten()
assert ((.5, .5), .5) in square * unit_line
assert (H, 3, 3) in (coin * d6 * d6).flatten()
assert ((H, 3), 3) in coin * d6 * d6
HH, TT = sympify(H), sympify(T)
assert set(coin**2) == {(HH, HH), (HH, TT), (TT, HH), (TT, TT)}
assert (d4*d4).is_subset(d6*d6)
assert square.complement(Interval(-oo, oo)*Interval(-oo, oo)) == Union(
(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True))*Interval(-oo, oo),
Interval(-oo, oo)*(Interval(-oo, 0, True, True) +
Interval(1, oo, True, True)))
assert (Interval(-5, 5)**3).is_subset(Interval(-10, 10)**3)
assert not (Interval(-10, 10)**3).is_subset(Interval(-5, 5)**3)
assert not (Interval(-5, 5)**2).is_subset(Interval(-10, 10)**3)
assert (Interval(.2, .5)*FiniteSet(.5)).is_subset(square) # segment in square
assert len(coin*coin*coin) == 8
assert len(S.EmptySet*S.EmptySet) == 0
assert len(S.EmptySet*coin) == 0
raises(TypeError, lambda: len(coin*Interval(0, 2)))
def test_real():
x = Symbol('x', real=True, finite=True)
I = Interval(0, 5)
J = Interval(10, 20)
A = FiniteSet(1, 2, 30, x, S.Pi)
B = FiniteSet(-4, 0)
C = FiniteSet(100)
D = FiniteSet('Ham', 'Eggs')
assert all(s.is_subset(S.Reals) for s in [I, J, A, B, C])
assert not D.is_subset(S.Reals)
assert all((a + b).is_subset(S.Reals) for a in [I, J, A, B, C] for b in [I, J, A, B, C])
assert not any((a + D).is_subset(S.Reals) for a in [I, J, A, B, C, D])
assert not (I + A + D).is_subset(S.Reals)
def test_supinf():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert (Interval(0, 1) + FiniteSet(2)).sup == 2
assert (Interval(0, 1) + FiniteSet(2)).inf == 0
assert (Interval(0, 1) + FiniteSet(x)).sup == Max(1, x)
assert (Interval(0, 1) + FiniteSet(x)).inf == Min(0, x)
assert FiniteSet(5, 1, x).sup == Max(5, x)
assert FiniteSet(5, 1, x).inf == Min(1, x)
assert FiniteSet(5, 1, x, y).sup == Max(5, x, y)
assert FiniteSet(5, 1, x, y).inf == Min(1, x, y)
assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).sup == \
S.Infinity
assert FiniteSet(5, 1, x, y, S.Infinity, S.NegativeInfinity).inf == \
S.NegativeInfinity
assert FiniteSet('Ham', 'Eggs').sup == Max('Ham', 'Eggs')
def test_universalset():
U = S.UniversalSet
x = Symbol('x')
assert U.as_relational(x) is S.true
assert U.union(Interval(2, 4)) == U
assert U.intersect(Interval(2, 4)) == Interval(2, 4)
assert U.measure is S.Infinity
assert U.boundary == S.EmptySet
assert U.contains(0) is S.true
def test_Union_of_ProductSets_shares():
line = Interval(0, 2)
points = FiniteSet(0, 1, 2)
assert Union(line * line, line * points) == line * line
def test_Interval_free_symbols():
# issue 6211
assert Interval(0, 1).free_symbols == set()
x = Symbol('x', real=True)
assert Interval(0, x).free_symbols == {x}
def test_image_interval():
x = Symbol('x', real=True)
a = Symbol('a', real=True)
assert imageset(x, 2*x, Interval(-2, 1)) == Interval(-4, 2)
assert imageset(x, 2*x, Interval(-2, 1, True, False)) == \
Interval(-4, 2, True, False)
assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
Interval(0, 4, False, True)
assert imageset(x, x**2, Interval(-2, 1)) == Interval(0, 4)
assert imageset(x, x**2, Interval(-2, 1, True, False)) == \
Interval(0, 4, False, True)
assert imageset(x, x**2, Interval(-2, 1, True, True)) == \
Interval(0, 4, False, True)
assert imageset(x, (x - 2)**2, Interval(1, 3)) == Interval(0, 1)
assert imageset(x, 3*x**4 - 26*x**3 + 78*x**2 - 90*x, Interval(0, 4)) == \
Interval(-35, 0) # Multiple Maxima
assert imageset(x, x + 1/x, Interval(-oo, oo)) == Interval(-oo, -2) \
+ Interval(2, oo) # Single Infinite discontinuity
assert imageset(x, 1/x + 1/(x-1)**2, Interval(0, 2, True, False)) == \
Interval(Rational(3, 2), oo, False) # Multiple Infinite discontinuities
# Test for Python lambda
assert imageset(lambda x: 2*x, Interval(-2, 1)) == Interval(-4, 2)
assert imageset(Lambda(x, a*x), Interval(0, 1)) == \
ImageSet(Lambda(x, a*x), Interval(0, 1))
assert imageset(Lambda(x, sin(cos(x))), Interval(0, 1)) == \
ImageSet(Lambda(x, sin(cos(x))), Interval(0, 1))
def test_image_piecewise():
f = Piecewise((x, x <= -1), (1/x**2, x <= 5), (x**3, True))
f1 = Piecewise((0, x <= 1), (1, x <= 2), (2, True))
assert imageset(x, f, Interval(-5, 5)) == Union(Interval(-5, -1), Interval(Rational(1, 25), oo))
assert imageset(x, f1, Interval(1, 2)) == FiniteSet(0, 1)
@XFAIL # See: https://github.com/sympy/sympy/pull/2723#discussion_r8659826
def test_image_Intersection():
x = Symbol('x', real=True)
y = Symbol('y', real=True)
assert imageset(x, x**2, Interval(-2, 0).intersect(Interval(x, y))) == \
Interval(0, 4).intersect(Interval(Min(x**2, y**2), Max(x**2, y**2)))
def test_image_FiniteSet():
x = Symbol('x', real=True)
assert imageset(x, 2*x, FiniteSet(1, 2, 3)) == FiniteSet(2, 4, 6)
def test_image_Union():
x = Symbol('x', real=True)
assert imageset(x, x**2, Interval(-2, 0) + FiniteSet(1, 2, 3)) == \
(Interval(0, 4) + FiniteSet(9))
def test_image_EmptySet():
x = Symbol('x', real=True)
assert imageset(x, 2*x, S.EmptySet) == S.EmptySet
def test_issue_5724_7680():
assert I not in S.Reals # issue 7680
assert Interval(-oo, oo).contains(I) is S.false
def test_boundary():
assert FiniteSet(1).boundary == FiniteSet(1)
assert all(Interval(0, 1, left_open, right_open).boundary == FiniteSet(0, 1)
for left_open in (true, false) for right_open in (true, false))
def test_boundary_Union():
assert (Interval(0, 1) + Interval(2, 3)).boundary == FiniteSet(0, 1, 2, 3)
assert ((Interval(0, 1, False, True)
+ Interval(1, 2, True, False)).boundary == FiniteSet(0, 1, 2))
assert (Interval(0, 1) + FiniteSet(2)).boundary == FiniteSet(0, 1, 2)
assert Union(Interval(0, 10), Interval(5, 15), evaluate=False).boundary \
== FiniteSet(0, 15)
assert Union(Interval(0, 10), Interval(0, 1), evaluate=False).boundary \
== FiniteSet(0, 10)
assert Union(Interval(0, 10, True, True),
Interval(10, 15, True, True), evaluate=False).boundary \
== FiniteSet(0, 10, 15)
@XFAIL
def test_union_boundary_of_joining_sets():
""" Testing the boundary of unions is a hard problem """
assert Union(Interval(0, 10), Interval(10, 15), evaluate=False).boundary \
== FiniteSet(0, 15)
def test_boundary_ProductSet():
open_square = Interval(0, 1, True, True) ** 2
assert open_square.boundary == (FiniteSet(0, 1) * Interval(0, 1)
+ Interval(0, 1) * FiniteSet(0, 1))
second_square = Interval(1, 2, True, True) * Interval(0, 1, True, True)
assert (open_square + second_square).boundary == (
FiniteSet(0, 1) * Interval(0, 1)
+ FiniteSet(1, 2) * Interval(0, 1)
+ Interval(0, 1) * FiniteSet(0, 1)
+ Interval(1, 2) * FiniteSet(0, 1))
def test_boundary_ProductSet_line():
line_in_r2 = Interval(0, 1) * FiniteSet(0)
assert line_in_r2.boundary == line_in_r2
def test_is_open():
assert Interval(0, 1, False, False).is_open is False
assert Interval(0, 1, True, False).is_open is False
assert Interval(0, 1, True, True).is_open is True
assert FiniteSet(1, 2, 3).is_open is False
def test_is_closed():
assert Interval(0, 1, False, False).is_closed is True
assert Interval(0, 1, True, False).is_closed is False
assert FiniteSet(1, 2, 3).is_closed is True
def test_closure():
assert Interval(0, 1, False, True).closure == Interval(0, 1, False, False)
def test_interior():
assert Interval(0, 1, False, True).interior == Interval(0, 1, True, True)
def test_issue_7841():
raises(TypeError, lambda: x in S.Reals)
def test_Eq():
assert Eq(Interval(0, 1), Interval(0, 1))
assert Eq(Interval(0, 1), Interval(0, 2)) == False
s1 = FiniteSet(0, 1)
s2 = FiniteSet(1, 2)
assert Eq(s1, s1)
assert Eq(s1, s2) == False
assert Eq(s1*s2, s1*s2)
assert Eq(s1*s2, s2*s1) == False
assert unchanged(Eq, FiniteSet({x, y}), FiniteSet({x}))
assert Eq(FiniteSet({x, y}).subs(y, x), FiniteSet({x})) is S.true
assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x) is S.true
assert Eq(FiniteSet({x, y}).subs(y, x+1), FiniteSet({x})) is S.false
assert Eq(FiniteSet({x, y}), FiniteSet({x})).subs(y, x+1) is S.false
assert Eq(ProductSet({1}, {2}), Interval(1, 2)) is S.false
assert Eq(ProductSet({1}), ProductSet({1}, {2})) is S.false
assert Eq(FiniteSet(()), FiniteSet(1)) is S.false
assert Eq(ProductSet(), FiniteSet(1)) is S.false
i1 = Interval(0, 1)
i2 = Interval(x, y)
assert unchanged(Eq, ProductSet(i1, i1), ProductSet(i2, i2))
def test_SymmetricDifference():
A = FiniteSet(0, 1, 2, 3, 4, 5)
B = FiniteSet(2, 4, 6, 8, 10)
C = Interval(8, 10)
assert SymmetricDifference(A, B, evaluate=False).is_iterable is True
assert SymmetricDifference(A, C, evaluate=False).is_iterable is None
assert FiniteSet(*SymmetricDifference(A, B, evaluate=False)) == \
FiniteSet(0, 1, 3, 5, 6, 8, 10)
raises(TypeError,
lambda: FiniteSet(*SymmetricDifference(A, C, evaluate=False)))
assert SymmetricDifference(FiniteSet(0, 1, 2, 3, 4, 5), \
FiniteSet(2, 4, 6, 8, 10)) == FiniteSet(0, 1, 3, 5, 6, 8, 10)
assert SymmetricDifference(FiniteSet(2, 3, 4), FiniteSet(2, 3, 4 ,5)) \
== FiniteSet(5)
assert FiniteSet(1, 2, 3, 4, 5) ^ FiniteSet(1, 2, 5, 6) == \
FiniteSet(3, 4, 6)
assert Set(S(1), S(2), S(3)) ^ Set(S(2), S(3), S(4)) == Union(Set(S(1), S(2), S(3)) - Set(S(2), S(3), S(4)), \
Set(S(2), S(3), S(4)) - Set(S(1), S(2), S(3)))
assert Interval(0, 4) ^ Interval(2, 5) == Union(Interval(0, 4) - \
Interval(2, 5), Interval(2, 5) - Interval(0, 4))
def test_issue_9536():
from sympy.functions.elementary.exponential import log
a = Symbol('a', real=True)
assert FiniteSet(log(a)).intersect(S.Reals) == Intersection(S.Reals, FiniteSet(log(a)))
def test_issue_9637():
n = Symbol('n')
a = FiniteSet(n)
b = FiniteSet(2, n)
assert Complement(S.Reals, a) == Complement(S.Reals, a, evaluate=False)
assert Complement(Interval(1, 3), a) == Complement(Interval(1, 3), a, evaluate=False)
assert Complement(Interval(1, 3), b) == \
Complement(Union(Interval(1, 2, False, True), Interval(2, 3, True, False)), a)
assert Complement(a, S.Reals) == Complement(a, S.Reals, evaluate=False)
assert Complement(a, Interval(1, 3)) == Complement(a, Interval(1, 3), evaluate=False)
def test_issue_9808():
# See https://github.com/sympy/sympy/issues/16342
assert Complement(FiniteSet(y), FiniteSet(1)) == Complement(FiniteSet(y), FiniteSet(1), evaluate=False)
assert Complement(FiniteSet(1, 2, x), FiniteSet(x, y, 2, 3)) == \
Complement(FiniteSet(1), FiniteSet(y), evaluate=False)
def test_issue_9956():
assert Union(Interval(-oo, oo), FiniteSet(1)) == Interval(-oo, oo)
assert Interval(-oo, oo).contains(1) is S.true
def test_issue_Symbol_inter():
i = Interval(0, oo)
r = S.Reals
mat = Matrix([0, 0, 0])
assert Intersection(r, i, FiniteSet(m), FiniteSet(m, n)) == \
Intersection(i, FiniteSet(m))
assert Intersection(FiniteSet(1, m, n), FiniteSet(m, n, 2), i) == \
Intersection(i, FiniteSet(m, n))
assert Intersection(FiniteSet(m, n, x), FiniteSet(m, z), r) == \
Intersection(Intersection({m, z}, {m, n, x}), r)
assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, x), r) == \
Intersection(FiniteSet(3, m, n), FiniteSet(m, n, x), r, evaluate=False)
assert Intersection(FiniteSet(m, n, 3), FiniteSet(m, n, 2, 3), r) == \
Intersection(FiniteSet(3, m, n), r)
assert Intersection(r, FiniteSet(mat, 2, n), FiniteSet(0, mat, n)) == \
Intersection(r, FiniteSet(n))
assert Intersection(FiniteSet(sin(x), cos(x)), FiniteSet(sin(x), cos(x), 1), r) == \
Intersection(r, FiniteSet(sin(x), cos(x)))
assert Intersection(FiniteSet(x**2, 1, sin(x)), FiniteSet(x**2, 2, sin(x)), r) == \
Intersection(r, FiniteSet(x**2, sin(x)))
def test_issue_11827():
assert S.Naturals0**4
def test_issue_10113():
f = x**2/(x**2 - 4)
assert imageset(x, f, S.Reals) == Union(Interval(-oo, 0), Interval(1, oo, True, True))
assert imageset(x, f, Interval(-2, 2)) == Interval(-oo, 0)
assert imageset(x, f, Interval(-2, 3)) == Union(Interval(-oo, 0), Interval(Rational(9, 5), oo))
def test_issue_10248():
raises(
TypeError, lambda: list(Intersection(S.Reals, FiniteSet(x)))
)
A = Symbol('A', real=True)
assert list(Intersection(S.Reals, FiniteSet(A))) == [A]
def test_issue_9447():
a = Interval(0, 1) + Interval(2, 3)
assert Complement(S.UniversalSet, a) == Complement(
S.UniversalSet, Union(Interval(0, 1), Interval(2, 3)), evaluate=False)
assert Complement(S.Naturals, a) == Complement(
S.Naturals, Union(Interval(0, 1), Interval(2, 3)), evaluate=False)
def test_issue_10337():
assert (FiniteSet(2) == 3) is False
assert (FiniteSet(2) != 3) is True
raises(TypeError, lambda: FiniteSet(2) < 3)
raises(TypeError, lambda: FiniteSet(2) <= 3)
raises(TypeError, lambda: FiniteSet(2) > 3)
raises(TypeError, lambda: FiniteSet(2) >= 3)
def test_issue_10326():
bad = [
EmptySet,
FiniteSet(1),
Interval(1, 2),
S.ComplexInfinity,
S.ImaginaryUnit,
S.Infinity,
S.NaN,
S.NegativeInfinity,
]
interval = Interval(0, 5)
for i in bad:
assert i not in interval
x = Symbol('x', real=True)
nr = Symbol('nr', extended_real=False)
assert x + 1 in Interval(x, x + 4)
assert nr not in Interval(x, x + 4)
assert Interval(1, 2) in FiniteSet(Interval(0, 5), Interval(1, 2))
assert Interval(-oo, oo).contains(oo) is S.false
assert Interval(-oo, oo).contains(-oo) is S.false
def test_issue_2799():
U = S.UniversalSet
a = Symbol('a', real=True)
inf_interval = Interval(a, oo)
R = S.Reals
assert U + inf_interval == inf_interval + U
assert U + R == R + U
assert R + inf_interval == inf_interval + R
def test_issue_9706():
assert Interval(-oo, 0).closure == Interval(-oo, 0, True, False)
assert Interval(0, oo).closure == Interval(0, oo, False, True)
assert Interval(-oo, oo).closure == Interval(-oo, oo)
def test_issue_8257():
reals_plus_infinity = Union(Interval(-oo, oo), FiniteSet(oo))
reals_plus_negativeinfinity = Union(Interval(-oo, oo), FiniteSet(-oo))
assert Interval(-oo, oo) + FiniteSet(oo) == reals_plus_infinity
assert FiniteSet(oo) + Interval(-oo, oo) == reals_plus_infinity
assert Interval(-oo, oo) + FiniteSet(-oo) == reals_plus_negativeinfinity
assert FiniteSet(-oo) + Interval(-oo, oo) == reals_plus_negativeinfinity
def test_issue_10931():
assert S.Integers - S.Integers == EmptySet
assert S.Integers - S.Reals == EmptySet
def test_issue_11174():
soln = Intersection(Interval(-oo, oo), FiniteSet(-x), evaluate=False)
assert Intersection(FiniteSet(-x), S.Reals) == soln
soln = Intersection(S.Reals, FiniteSet(x), evaluate=False)
assert Intersection(FiniteSet(x), S.Reals) == soln
def test_issue_18505():
assert ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers).contains(0) == \
Contains(0, ImageSet(Lambda(n, sqrt(pi*n/2 - 1 + pi/2)), S.Integers))
def test_finite_set_intersection():
# The following should not produce recursion errors
# Note: some of these are not completely correct. See
# https://github.com/sympy/sympy/issues/16342.
assert Intersection(FiniteSet(-oo, x), FiniteSet(x)) == FiniteSet(x)
assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(0, x)]) == FiniteSet(x)
assert Intersection._handle_finite_sets([FiniteSet(-oo, x), FiniteSet(x)]) == FiniteSet(x)
assert Intersection._handle_finite_sets([FiniteSet(2, 3, x, y), FiniteSet(1, 2, x)]) == \
Intersection._handle_finite_sets([FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)]) == \
Intersection(FiniteSet(1, 2, x), FiniteSet(2, 3, x, y)) == \
Intersection(FiniteSet(1, 2, x), FiniteSet(2, x, y))
assert FiniteSet(1+x-y) & FiniteSet(1) == \
FiniteSet(1) & FiniteSet(1+x-y) == \
Intersection(FiniteSet(1+x-y), FiniteSet(1), evaluate=False)
assert FiniteSet(1) & FiniteSet(x) == FiniteSet(x) & FiniteSet(1) == \
Intersection(FiniteSet(1), FiniteSet(x), evaluate=False)
assert FiniteSet({x}) & FiniteSet({x, y}) == \
Intersection(FiniteSet({x}), FiniteSet({x, y}), evaluate=False)
def test_union_intersection_constructor():
# The actual exception does not matter here, so long as these fail
sets = [FiniteSet(1), FiniteSet(2)]
raises(Exception, lambda: Union(sets))
raises(Exception, lambda: Intersection(sets))
raises(Exception, lambda: Union(tuple(sets)))
raises(Exception, lambda: Intersection(tuple(sets)))
raises(Exception, lambda: Union(i for i in sets))
raises(Exception, lambda: Intersection(i for i in sets))
# Python sets are treated the same as FiniteSet
# The union of a single set (of sets) is the set (of sets) itself
assert Union(set(sets)) == FiniteSet(*sets)
assert Intersection(set(sets)) == FiniteSet(*sets)
assert Union({1}, {2}) == FiniteSet(1, 2)
assert Intersection({1, 2}, {2, 3}) == FiniteSet(2)
def test_Union_contains():
assert zoo not in Union(
Interval.open(-oo, 0), Interval.open(0, oo))
@XFAIL
def test_issue_16878b():
# in intersection_sets for (ImageSet, Set) there is no code
# that handles the base_set of S.Reals like there is
# for Integers
assert imageset(x, (x, x), S.Reals).is_subset(S.Reals**2) is True
def test_DisjointUnion():
assert DisjointUnion(FiniteSet(1, 2, 3), FiniteSet(1, 2, 3), FiniteSet(1, 2, 3)).rewrite(Union) == (FiniteSet(1, 2, 3) * FiniteSet(0, 1, 2))
assert DisjointUnion(Interval(1, 3), Interval(2, 4)).rewrite(Union) == Union(Interval(1, 3) * FiniteSet(0), Interval(2, 4) * FiniteSet(1))
assert DisjointUnion(Interval(0, 5), Interval(0, 5)).rewrite(Union) == Union(Interval(0, 5) * FiniteSet(0), Interval(0, 5) * FiniteSet(1))
assert DisjointUnion(Interval(-1, 2), S.EmptySet, S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(0)
assert DisjointUnion(Interval(-1, 2)).rewrite(Union) == Interval(-1, 2) * FiniteSet(0)
assert DisjointUnion(S.EmptySet, Interval(-1, 2), S.EmptySet).rewrite(Union) == Interval(-1, 2) * FiniteSet(1)
assert DisjointUnion(Interval(-oo, oo)).rewrite(Union) == Interval(-oo, oo) * FiniteSet(0)
assert DisjointUnion(S.EmptySet).rewrite(Union) == S.EmptySet
assert DisjointUnion().rewrite(Union) == S.EmptySet
raises(TypeError, lambda: DisjointUnion(Symbol('n')))
x = Symbol("x")
y = Symbol("y")
z = Symbol("z")
assert DisjointUnion(FiniteSet(x), FiniteSet(y, z)).rewrite(Union) == (FiniteSet(x) * FiniteSet(0)) + (FiniteSet(y, z) * FiniteSet(1))
def test_DisjointUnion_is_empty():
assert DisjointUnion(S.EmptySet).is_empty is True
assert DisjointUnion(S.EmptySet, S.EmptySet).is_empty is True
assert DisjointUnion(S.EmptySet, FiniteSet(1, 2, 3)).is_empty is False
def test_DisjointUnion_is_iterable():
assert DisjointUnion(S.Integers, S.Naturals, S.Rationals).is_iterable is True
assert DisjointUnion(S.EmptySet, S.Reals).is_iterable is False
assert DisjointUnion(FiniteSet(1, 2, 3), S.EmptySet, FiniteSet(x, y)).is_iterable is True
assert DisjointUnion(S.EmptySet, S.EmptySet).is_iterable is False
def test_DisjointUnion_contains():
assert (0, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (1, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (1, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (1, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (2, 0) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (2, 1) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (2, 2) in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 1, 2) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (0, 0.5) not in DisjointUnion(FiniteSet(0.5))
assert (0, 5) not in DisjointUnion(FiniteSet(0, 1, 2), FiniteSet(0, 1, 2), FiniteSet(0, 1, 2))
assert (x, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (y, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (z, 0) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (y, 2) in DisjointUnion(FiniteSet(x, y, z), S.EmptySet, FiniteSet(y))
assert (0.5, 0) in DisjointUnion(Interval(0, 1), Interval(0, 2))
assert (0.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2))
assert (1.5, 0) not in DisjointUnion(Interval(0, 1), Interval(0, 2))
assert (1.5, 1) in DisjointUnion(Interval(0, 1), Interval(0, 2))
def test_DisjointUnion_iter():
D = DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))
it = iter(D)
L1 = [(x, 1), (y, 1), (z, 1)]
L2 = [(3, 0), (5, 0), (7, 0), (9, 0)]
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
nxt = next(it)
assert nxt in L1
L1.remove(nxt)
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
nxt = next(it)
assert nxt in L1
L1.remove(nxt)
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
nxt = next(it)
assert nxt in L1
L1.remove(nxt)
nxt = next(it)
assert nxt in L2
L2.remove(nxt)
raises(StopIteration, lambda: next(it))
raises(ValueError, lambda: iter(DisjointUnion(Interval(0, 1), S.EmptySet)))
def test_DisjointUnion_len():
assert len(DisjointUnion(FiniteSet(3, 5, 7, 9), FiniteSet(x, y, z))) == 7
assert len(DisjointUnion(S.EmptySet, S.EmptySet, FiniteSet(x, y, z), S.EmptySet)) == 3
raises(ValueError, lambda: len(DisjointUnion(Interval(0, 1), S.EmptySet)))
def test_issue_20089():
B = FiniteSet(FiniteSet(1, 2), FiniteSet(1))
assert not 1 in B
assert not 1.0 in B
assert not Eq(1, FiniteSet(1, 2))
assert FiniteSet(1) in B
A = FiniteSet(1, 2)
assert A in B
assert B.issubset(B)
assert not A.issubset(B)
assert 1 in A
C = FiniteSet(FiniteSet(1, 2), FiniteSet(1), 1, 2)
assert A.issubset(C)
assert B.issubset(C)
def test_issue_19378():
a = FiniteSet(1, 2)
b = ProductSet(a, a)
c = FiniteSet((1, 1), (1, 2), (2, 1), (2, 2))
assert b.is_subset(c) is True
d = FiniteSet(1)
assert b.is_subset(d) is False
assert Eq(c, b).simplify() is S.true
assert Eq(a, c).simplify() is S.false
assert Eq({1}, {x}).simplify() == Eq({1}, {x})
def test_intersection_symbolic():
n = Symbol('n')
# These should not throw an error
assert isinstance(Intersection(Range(n), Range(100)), Intersection)
assert isinstance(Intersection(Range(n), Interval(1, 100)), Intersection)
assert isinstance(Intersection(Range(100), Interval(1, n)), Intersection)
@XFAIL
def test_intersection_symbolic_failing():
n = Symbol('n', integer=True, positive=True)
assert Intersection(Range(10, n), Range(4, 500, 5)) == Intersection(
Range(14, n), Range(14, 500, 5))
assert Intersection(Interval(10, n), Range(4, 500, 5)) == Intersection(
Interval(14, n), Range(14, 500, 5))
def test_issue_20379():
#https://github.com/sympy/sympy/issues/20379
x = pi - 3.14159265358979
assert FiniteSet(x).evalf(2) == FiniteSet(Float('3.23108914886517e-15', 2))
def test_finiteset_simplify():
S = FiniteSet(1, cos(1)**2 + sin(1)**2)
assert S.simplify() == {1}
|
3ba895161d82449d592d35af518f29ac512180c626b96ee2aa34a39730e35c7d | from sympy.core.numbers import (I, pi)
from sympy.core.relational import Eq
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.complexes import re
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.trigonometric import (cos, sin, tan)
from sympy.logic.boolalg import (And, Or)
from sympy.plotting.plot_implicit import plot_implicit
from sympy.plotting.plot import unset_show
from tempfile import NamedTemporaryFile, mkdtemp
from sympy.testing.pytest import skip, warns
from sympy.external import import_module
from sympy.testing.tmpfiles import TmpFileManager
import os
#Set plots not to show
unset_show()
def tmp_file(dir=None, name=''):
return NamedTemporaryFile(
suffix='.png', dir=dir, delete=False).name
def plot_and_save(expr, *args, name='', dir=None, **kwargs):
p = plot_implicit(expr, *args, **kwargs)
p.save(tmp_file(dir=dir, name=name))
# Close the plot to avoid a warning from matplotlib
p._backend.close()
def plot_implicit_tests(name):
temp_dir = mkdtemp()
TmpFileManager.tmp_folder(temp_dir)
x = Symbol('x')
y = Symbol('y')
#implicit plot tests
plot_and_save(Eq(y, cos(x)), (x, -5, 5), (y, -2, 2), name=name, dir=temp_dir)
plot_and_save(Eq(y**2, x**3 - x), (x, -5, 5),
(y, -4, 4), name=name, dir=temp_dir)
plot_and_save(y > 1 / x, (x, -5, 5),
(y, -2, 2), name=name, dir=temp_dir)
plot_and_save(y < 1 / tan(x), (x, -5, 5),
(y, -2, 2), name=name, dir=temp_dir)
plot_and_save(y >= 2 * sin(x) * cos(x), (x, -5, 5),
(y, -2, 2), name=name, dir=temp_dir)
plot_and_save(y <= x**2, (x, -3, 3),
(y, -1, 5), name=name, dir=temp_dir)
#Test all input args for plot_implicit
plot_and_save(Eq(y**2, x**3 - x), dir=temp_dir)
plot_and_save(Eq(y**2, x**3 - x), adaptive=False, dir=temp_dir)
plot_and_save(Eq(y**2, x**3 - x), adaptive=False, points=500, dir=temp_dir)
plot_and_save(y > x, (x, -5, 5), dir=temp_dir)
plot_and_save(And(y > exp(x), y > x + 2), dir=temp_dir)
plot_and_save(Or(y > x, y > -x), dir=temp_dir)
plot_and_save(x**2 - 1, (x, -5, 5), dir=temp_dir)
plot_and_save(x**2 - 1, dir=temp_dir)
plot_and_save(y > x, depth=-5, dir=temp_dir)
plot_and_save(y > x, depth=5, dir=temp_dir)
plot_and_save(y > cos(x), adaptive=False, dir=temp_dir)
plot_and_save(y < cos(x), adaptive=False, dir=temp_dir)
plot_and_save(And(y > cos(x), Or(y > x, Eq(y, x))), dir=temp_dir)
plot_and_save(y - cos(pi / x), dir=temp_dir)
#Test plots which cannot be rendered using the adaptive algorithm
with warns(UserWarning, match="Adaptive meshing could not be applied"):
plot_and_save(Eq(y, re(cos(x) + I*sin(x))), name=name, dir=temp_dir)
plot_and_save(x**2 - 1, title='An implicit plot', dir=temp_dir)
def test_line_color():
x, y = symbols('x, y')
p = plot_implicit(x**2 + y**2 - 1, line_color="green", show=False)
assert p._series[0].line_color == "green"
p = plot_implicit(x**2 + y**2 - 1, line_color='r', show=False)
assert p._series[0].line_color == "r"
def test_matplotlib():
matplotlib = import_module('matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
if matplotlib:
try:
plot_implicit_tests('test')
test_line_color()
finally:
TmpFileManager.cleanup()
else:
skip("Matplotlib not the default backend")
def test_region_and():
matplotlib = import_module('matplotlib', min_module_version='1.1.0', catch=(RuntimeError,))
if not matplotlib:
skip("Matplotlib not the default backend")
from matplotlib.testing.compare import compare_images
test_directory = os.path.dirname(os.path.abspath(__file__))
try:
temp_dir = mkdtemp()
TmpFileManager.tmp_folder(temp_dir)
x, y = symbols('x y')
r1 = (x - 1)**2 + y**2 < 2
r2 = (x + 1)**2 + y**2 < 2
test_filename = tmp_file(dir=temp_dir, name="test_region_and")
cmp_filename = os.path.join(test_directory, "test_region_and.png")
p = plot_implicit(r1 & r2, x, y)
p.save(test_filename)
compare_images(cmp_filename, test_filename, 0.005)
test_filename = tmp_file(dir=temp_dir, name="test_region_or")
cmp_filename = os.path.join(test_directory, "test_region_or.png")
p = plot_implicit(r1 | r2, x, y)
p.save(test_filename)
compare_images(cmp_filename, test_filename, 0.005)
test_filename = tmp_file(dir=temp_dir, name="test_region_not")
cmp_filename = os.path.join(test_directory, "test_region_not.png")
p = plot_implicit(~r1, x, y)
p.save(test_filename)
compare_images(cmp_filename, test_filename, 0.005)
test_filename = tmp_file(dir=temp_dir, name="test_region_xor")
cmp_filename = os.path.join(test_directory, "test_region_xor.png")
p = plot_implicit(r1 ^ r2, x, y)
p.save(test_filename)
compare_images(cmp_filename, test_filename, 0.005)
finally:
TmpFileManager.cleanup()
|
90600de17de28868e5dd4bf841b0c6583d80878a315454172f7e344b0e5a7546 | 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
from sympy.utilities import lambdify as lambdify_
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"):
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'
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))
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
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)
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")
|
439eb28b0d86c1971bec0a6da882099a12d6d85a2972f8372eab54e067d9d44b | 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
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'
]
assert lines == list(textplot_str(log(x), 0, 1))
def test_sinc():
x = Symbol('x')
lines = [
' 1 | . . ',
' | . . ',
' | ',
' | . . ',
' | ',
' | . . ',
' | ',
' | ',
' | . . ',
' | ',
' 0.4 |-------------------------------------------------------',
' | . . ',
' | ',
' | . . ',
' | ',
' | ..... ..... ',
' | .. \\ . . / .. ',
' | / \\ / \\ ',
' |/ \\ . . / \\',
' | \\ / \\ / ',
' -0.2 |_______________________________________________________',
' -10 0 10'
]
assert lines == list(textplot_str(sin(x)/x, -10, 10))
def test_imaginary():
x = Symbol('x')
lines = [
' 1 | ..',
' | .. ',
' | ... ',
' | .. ',
' | .. ',
' | .. ',
' | .. ',
' | .. ',
' | .. ',
' | / ',
' 0.5 |----------------------------------/--------------------',
' | .. ',
' | / ',
' | . ',
' | ',
' | . ',
' | . ',
' | ',
' | ',
' | ',
' 0 |_______________________________________________________',
' -1 0 1'
]
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
|
0e3abe7b0566b3f2684ab883a68943861b6db412507493bc043e04d27c8e59c4 | from .plot_interval import PlotInterval
from .plot_object import PlotObject
from .util import parse_option_string
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.geometry.entity import GeometryEntity
from sympy.utilities.iterables import is_sequence
class PlotMode(PlotObject):
"""
Grandparent class for plotting
modes. Serves as interface for
registration, lookup, and init
of modes.
To create a new plot mode,
inherit from PlotModeBase
or one of its children, such
as PlotSurface or PlotCurve.
"""
## Class-level attributes
## used to register and lookup
## plot modes. See PlotModeBase
## for descriptions and usage.
i_vars, d_vars = '', ''
intervals = []
aliases = []
is_default = False
## Draw is the only method here which
## is meant to be overridden in child
## classes, and PlotModeBase provides
## a base implementation.
def draw(self):
raise NotImplementedError()
## Everything else in this file has to
## do with registration and retrieval
## of plot modes. This is where I've
## hidden much of the ugliness of automatic
## plot mode divination...
## Plot mode registry data structures
_mode_alias_list = []
_mode_map = {
1: {1: {}, 2: {}},
2: {1: {}, 2: {}},
3: {1: {}, 2: {}},
} # [d][i][alias_str]: class
_mode_default_map = {
1: {},
2: {},
3: {},
} # [d][i]: class
_i_var_max, _d_var_max = 2, 3
def __new__(cls, *args, **kwargs):
"""
This is the function which interprets
arguments given to Plot.__init__ and
Plot.__setattr__. Returns an initialized
instance of the appropriate child class.
"""
newargs, newkwargs = PlotMode._extract_options(args, kwargs)
mode_arg = newkwargs.get('mode', '')
# Interpret the arguments
d_vars, intervals = PlotMode._interpret_args(newargs)
i_vars = PlotMode._find_i_vars(d_vars, intervals)
i, d = max([len(i_vars), len(intervals)]), len(d_vars)
# Find the appropriate mode
subcls = PlotMode._get_mode(mode_arg, i, d)
# Create the object
o = object.__new__(subcls)
# Do some setup for the mode instance
o.d_vars = d_vars
o._fill_i_vars(i_vars)
o._fill_intervals(intervals)
o.options = newkwargs
return o
@staticmethod
def _get_mode(mode_arg, i_var_count, d_var_count):
"""
Tries to return an appropriate mode class.
Intended to be called only by __new__.
mode_arg
Can be a string or a class. If it is a
PlotMode subclass, it is simply returned.
If it is a string, it can an alias for
a mode or an empty string. In the latter
case, we try to find a default mode for
the i_var_count and d_var_count.
i_var_count
The number of independent variables
needed to evaluate the d_vars.
d_var_count
The number of dependent variables;
usually the number of functions to
be evaluated in plotting.
For example, a Cartesian function y = f(x) has
one i_var (x) and one d_var (y). A parametric
form x,y,z = f(u,v), f(u,v), f(u,v) has two
two i_vars (u,v) and three d_vars (x,y,z).
"""
# if the mode_arg is simply a PlotMode class,
# check that the mode supports the numbers
# of independent and dependent vars, then
# return it
try:
m = None
if issubclass(mode_arg, PlotMode):
m = mode_arg
except TypeError:
pass
if m:
if not m._was_initialized:
raise ValueError(("To use unregistered plot mode %s "
"you must first call %s._init_mode().")
% (m.__name__, m.__name__))
if d_var_count != m.d_var_count:
raise ValueError(("%s can only plot functions "
"with %i dependent variables.")
% (m.__name__,
m.d_var_count))
if i_var_count > m.i_var_count:
raise ValueError(("%s cannot plot functions "
"with more than %i independent "
"variables.")
% (m.__name__,
m.i_var_count))
return m
# If it is a string, there are two possibilities.
if isinstance(mode_arg, str):
i, d = i_var_count, d_var_count
if i > PlotMode._i_var_max:
raise ValueError(var_count_error(True, True))
if d > PlotMode._d_var_max:
raise ValueError(var_count_error(False, True))
# If the string is '', try to find a suitable
# default mode
if not mode_arg:
return PlotMode._get_default_mode(i, d)
# Otherwise, interpret the string as a mode
# alias (e.g. 'cartesian', 'parametric', etc)
else:
return PlotMode._get_aliased_mode(mode_arg, i, d)
else:
raise ValueError("PlotMode argument must be "
"a class or a string")
@staticmethod
def _get_default_mode(i, d, i_vars=-1):
if i_vars == -1:
i_vars = i
try:
return PlotMode._mode_default_map[d][i]
except KeyError:
# Keep looking for modes in higher i var counts
# which support the given d var count until we
# reach the max i_var count.
if i < PlotMode._i_var_max:
return PlotMode._get_default_mode(i + 1, d, i_vars)
else:
raise ValueError(("Couldn't find a default mode "
"for %i independent and %i "
"dependent variables.") % (i_vars, d))
@staticmethod
def _get_aliased_mode(alias, i, d, i_vars=-1):
if i_vars == -1:
i_vars = i
if alias not in PlotMode._mode_alias_list:
raise ValueError(("Couldn't find a mode called"
" %s. Known modes: %s.")
% (alias, ", ".join(PlotMode._mode_alias_list)))
try:
return PlotMode._mode_map[d][i][alias]
except TypeError:
# Keep looking for modes in higher i var counts
# which support the given d var count and alias
# until we reach the max i_var count.
if i < PlotMode._i_var_max:
return PlotMode._get_aliased_mode(alias, i + 1, d, i_vars)
else:
raise ValueError(("Couldn't find a %s mode "
"for %i independent and %i "
"dependent variables.")
% (alias, i_vars, d))
@classmethod
def _register(cls):
"""
Called once for each user-usable plot mode.
For Cartesian2D, it is invoked after the
class definition: Cartesian2D._register()
"""
name = cls.__name__
cls._init_mode()
try:
i, d = cls.i_var_count, cls.d_var_count
# Add the mode to _mode_map under all
# given aliases
for a in cls.aliases:
if a not in PlotMode._mode_alias_list:
# Also track valid aliases, so
# we can quickly know when given
# an invalid one in _get_mode.
PlotMode._mode_alias_list.append(a)
PlotMode._mode_map[d][i][a] = cls
if cls.is_default:
# If this mode was marked as the
# default for this d,i combination,
# also set that.
PlotMode._mode_default_map[d][i] = cls
except Exception as e:
raise RuntimeError(("Failed to register "
"plot mode %s. Reason: %s")
% (name, (str(e))))
@classmethod
def _init_mode(cls):
"""
Initializes the plot mode based on
the 'mode-specific parameters' above.
Only intended to be called by
PlotMode._register(). To use a mode without
registering it, you can directly call
ModeSubclass._init_mode().
"""
def symbols_list(symbol_str):
return [Symbol(s) for s in symbol_str]
# Convert the vars strs into
# lists of symbols.
cls.i_vars = symbols_list(cls.i_vars)
cls.d_vars = symbols_list(cls.d_vars)
# Var count is used often, calculate
# it once here
cls.i_var_count = len(cls.i_vars)
cls.d_var_count = len(cls.d_vars)
if cls.i_var_count > PlotMode._i_var_max:
raise ValueError(var_count_error(True, False))
if cls.d_var_count > PlotMode._d_var_max:
raise ValueError(var_count_error(False, False))
# Try to use first alias as primary_alias
if len(cls.aliases) > 0:
cls.primary_alias = cls.aliases[0]
else:
cls.primary_alias = cls.__name__
di = cls.intervals
if len(di) != cls.i_var_count:
raise ValueError("Plot mode must provide a "
"default interval for each i_var.")
for i in range(cls.i_var_count):
# default intervals must be given [min,max,steps]
# (no var, but they must be in the same order as i_vars)
if len(di[i]) != 3:
raise ValueError("length should be equal to 3")
# Initialize an incomplete interval,
# to later be filled with a var when
# the mode is instantiated.
di[i] = PlotInterval(None, *di[i])
# To prevent people from using modes
# without these required fields set up.
cls._was_initialized = True
_was_initialized = False
## Initializer Helper Methods
@staticmethod
def _find_i_vars(functions, intervals):
i_vars = []
# First, collect i_vars in the
# order they are given in any
# intervals.
for i in intervals:
if i.v is None:
continue
elif i.v in i_vars:
raise ValueError(("Multiple intervals given "
"for %s.") % (str(i.v)))
i_vars.append(i.v)
# Then, find any remaining
# i_vars in given functions
# (aka d_vars)
for f in functions:
for a in f.free_symbols:
if a not in i_vars:
i_vars.append(a)
return i_vars
def _fill_i_vars(self, i_vars):
# copy default i_vars
self.i_vars = [Symbol(str(i)) for i in self.i_vars]
# replace with given i_vars
for i in range(len(i_vars)):
self.i_vars[i] = i_vars[i]
def _fill_intervals(self, intervals):
# copy default intervals
self.intervals = [PlotInterval(i) for i in self.intervals]
# track i_vars used so far
v_used = []
# fill copy of default
# intervals with given info
for i in range(len(intervals)):
self.intervals[i].fill_from(intervals[i])
if self.intervals[i].v is not None:
v_used.append(self.intervals[i].v)
# Find any orphan intervals and
# assign them i_vars
for i in range(len(self.intervals)):
if self.intervals[i].v is None:
u = [v for v in self.i_vars if v not in v_used]
if len(u) == 0:
raise ValueError("length should not be equal to 0")
self.intervals[i].v = u[0]
v_used.append(u[0])
@staticmethod
def _interpret_args(args):
interval_wrong_order = "PlotInterval %s was given before any function(s)."
interpret_error = "Could not interpret %s as a function or interval."
functions, intervals = [], []
if isinstance(args[0], GeometryEntity):
for coords in list(args[0].arbitrary_point()):
functions.append(coords)
intervals.append(PlotInterval.try_parse(args[0].plot_interval()))
else:
for a in args:
i = PlotInterval.try_parse(a)
if i is not None:
if len(functions) == 0:
raise ValueError(interval_wrong_order % (str(i)))
else:
intervals.append(i)
else:
if is_sequence(a, include=str):
raise ValueError(interpret_error % (str(a)))
try:
f = sympify(a)
functions.append(f)
except TypeError:
raise ValueError(interpret_error % str(a))
return functions, intervals
@staticmethod
def _extract_options(args, kwargs):
newkwargs, newargs = {}, []
for a in args:
if isinstance(a, str):
newkwargs = dict(newkwargs, **parse_option_string(a))
else:
newargs.append(a)
newkwargs = dict(newkwargs, **kwargs)
return newargs, newkwargs
def var_count_error(is_independent, is_plotting):
"""
Used to format an error message which differs
slightly in 4 places.
"""
if is_plotting:
v = "Plotting"
else:
v = "Registering plot modes"
if is_independent:
n, s = PlotMode._i_var_max, "independent"
else:
n, s = PlotMode._d_var_max, "dependent"
return ("%s with more than %i %s variables "
"is not supported.") % (v, n, s)
|
eca64bb337b87feae24326ea545fd4484a217b14534bb9fa4650903fb85a2a2b | from threading import RLock
# it is sufficient to import "pyglet" here once
try:
import pyglet.gl as pgl
except ImportError:
raise ImportError("pyglet is required for plotting.\n "
"visit http://www.pyglet.org/")
from sympy.core.numbers import Integer
from sympy.external.gmpy import SYMPY_INTS
from sympy.geometry.entity import GeometryEntity
from sympy.plotting.pygletplot.plot_axes import PlotAxes
from sympy.plotting.pygletplot.plot_mode import PlotMode
from sympy.plotting.pygletplot.plot_object import PlotObject
from sympy.plotting.pygletplot.plot_window import PlotWindow
from sympy.plotting.pygletplot.util import parse_option_string
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.iterables import is_sequence
from time import sleep
from os import getcwd, listdir
import ctypes
@doctest_depends_on(modules=('pyglet',))
class PygletPlot:
"""
Plot Examples
=============
See examples/advanced/pyglet_plotting.py for many more examples.
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy.abc import x, y, z
>>> Plot(x*y**3-y*x**3)
[0]: -x**3*y + x*y**3, 'mode=cartesian'
>>> p = Plot()
>>> p[1] = x*y
>>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
>>> p = Plot()
>>> p[1] = x**2+y**2
>>> p[2] = -x**2-y**2
Variable Intervals
==================
The basic format is [var, min, max, steps], but the
syntax is flexible and arguments left out are taken
from the defaults for the current coordinate mode:
>>> Plot(x**2) # implies [x,-5,5,100]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [], []) # [x,-1,1,40], [y,-1,1,40]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2-y**2, [100], [100]) # [x,-1,1,100], [y,-1,1,100]
[0]: x**2 - y**2, 'mode=cartesian'
>>> Plot(x**2, [x,-13,13,100])
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [-13,13]) # [x,-13,13,100]
[0]: x**2, 'mode=cartesian'
>>> Plot(x**2, [x,-13,13]) # [x,-13,13,10]
[0]: x**2, 'mode=cartesian'
>>> Plot(1*x, [], [x], mode='cylindrical')
... # [unbound_theta,0,2*Pi,40], [x,-1,1,20]
[0]: x, 'mode=cartesian'
Coordinate Modes
================
Plot supports several curvilinear coordinate modes, and
they independent for each plotted function. You can specify
a coordinate mode explicitly with the 'mode' named argument,
but it can be automatically determined for Cartesian or
parametric plots, and therefore must only be specified for
polar, cylindrical, and spherical modes.
Specifically, Plot(function arguments) and Plot[n] =
(function arguments) will interpret your arguments as a
Cartesian plot if you provide one function and a parametric
plot if you provide two or three functions. Similarly, the
arguments will be interpreted as a curve if one variable is
used, and a surface if two are used.
Supported mode names by number of variables:
1: parametric, cartesian, polar
2: parametric, cartesian, cylindrical = polar, spherical
>>> Plot(1, mode='spherical')
Calculator-like Interface
=========================
>>> p = Plot(visible=False)
>>> f = x**2
>>> p[1] = f
>>> p[2] = f.diff(x)
>>> p[3] = f.diff(x).diff(x)
>>> p
[1]: x**2, 'mode=cartesian'
[2]: 2*x, 'mode=cartesian'
[3]: 2, 'mode=cartesian'
>>> p.show()
>>> p.clear()
>>> p
<blank plot>
>>> p[1] = x**2+y**2
>>> p[1].style = 'solid'
>>> p[2] = -x**2-y**2
>>> p[2].style = 'wireframe'
>>> p[1].color = z, (0.4,0.4,0.9), (0.9,0.4,0.4)
>>> p[1].style = 'both'
>>> p[2].style = 'both'
>>> p.close()
Plot Window Keyboard Controls
=============================
Screen Rotation:
X,Y axis Arrow Keys, A,S,D,W, Numpad 4,6,8,2
Z axis Q,E, Numpad 7,9
Model Rotation:
Z axis Z,C, Numpad 1,3
Zoom: R,F, PgUp,PgDn, Numpad +,-
Reset Camera: X, Numpad 5
Camera Presets:
XY F1
XZ F2
YZ F3
Perspective F4
Sensitivity Modifier: SHIFT
Axes Toggle:
Visible F5
Colors F6
Close Window: ESCAPE
=============================
"""
@doctest_depends_on(modules=('pyglet',))
def __init__(self, *fargs, **win_args):
"""
Positional Arguments
====================
Any given positional arguments are used to
initialize a plot function at index 1. In
other words...
>>> from sympy.plotting.pygletplot import PygletPlot as Plot
>>> from sympy.abc import x
>>> p = Plot(x**2, visible=False)
...is equivalent to...
>>> p = Plot(visible=False)
>>> p[1] = x**2
Note that in earlier versions of the plotting
module, you were able to specify multiple
functions in the initializer. This functionality
has been dropped in favor of better automatic
plot plot_mode detection.
Named Arguments
===============
axes
An option string of the form
"key1=value1; key2 = value2" which
can use the following options:
style = ordinate
none OR frame OR box OR ordinate
stride = 0.25
val OR (val_x, val_y, val_z)
overlay = True (draw on top of plot)
True OR False
colored = False (False uses Black,
True uses colors
R,G,B = X,Y,Z)
True OR False
label_axes = False (display axis names
at endpoints)
True OR False
visible = True (show immediately
True OR False
The following named arguments are passed as
arguments to window initialization:
antialiasing = True
True OR False
ortho = False
True OR False
invert_mouse_zoom = False
True OR False
"""
# Register the plot modes
from . import plot_modes # noqa
self._win_args = win_args
self._window = None
self._render_lock = RLock()
self._functions = {}
self._pobjects = []
self._screenshot = ScreenShot(self)
axe_options = parse_option_string(win_args.pop('axes', ''))
self.axes = PlotAxes(**axe_options)
self._pobjects.append(self.axes)
self[0] = fargs
if win_args.get('visible', True):
self.show()
## Window Interfaces
def show(self):
"""
Creates and displays a plot window, or activates it
(gives it focus) if it has already been created.
"""
if self._window and not self._window.has_exit:
self._window.activate()
else:
self._win_args['visible'] = True
self.axes.reset_resources()
#if hasattr(self, '_doctest_depends_on'):
# self._win_args['runfromdoctester'] = True
self._window = PlotWindow(self, **self._win_args)
def close(self):
"""
Closes the plot window.
"""
if self._window:
self._window.close()
def saveimage(self, outfile=None, format='', size=(600, 500)):
"""
Saves a screen capture of the plot window to an
image file.
If outfile is given, it can either be a path
or a file object. Otherwise a png image will
be saved to the current working directory.
If the format is omitted, it is determined from
the filename extension.
"""
self._screenshot.save(outfile, format, size)
## Function List Interfaces
def clear(self):
"""
Clears the function list of this plot.
"""
self._render_lock.acquire()
self._functions = {}
self.adjust_all_bounds()
self._render_lock.release()
def __getitem__(self, i):
"""
Returns the function at position i in the
function list.
"""
return self._functions[i]
def __setitem__(self, i, args):
"""
Parses and adds a PlotMode to the function
list.
"""
if not (isinstance(i, (SYMPY_INTS, Integer)) and i >= 0):
raise ValueError("Function index must "
"be an integer >= 0.")
if isinstance(args, PlotObject):
f = args
else:
if (not is_sequence(args)) or isinstance(args, GeometryEntity):
args = [args]
if len(args) == 0:
return # no arguments given
kwargs = dict(bounds_callback=self.adjust_all_bounds)
f = PlotMode(*args, **kwargs)
if f:
self._render_lock.acquire()
self._functions[i] = f
self._render_lock.release()
else:
raise ValueError("Failed to parse '%s'."
% ', '.join(str(a) for a in args))
def __delitem__(self, i):
"""
Removes the function in the function list at
position i.
"""
self._render_lock.acquire()
del self._functions[i]
self.adjust_all_bounds()
self._render_lock.release()
def firstavailableindex(self):
"""
Returns the first unused index in the function list.
"""
i = 0
self._render_lock.acquire()
while i in self._functions:
i += 1
self._render_lock.release()
return i
def append(self, *args):
"""
Parses and adds a PlotMode to the function
list at the first available index.
"""
self.__setitem__(self.firstavailableindex(), args)
def __len__(self):
"""
Returns the number of functions in the function list.
"""
return len(self._functions)
def __iter__(self):
"""
Allows iteration of the function list.
"""
return self._functions.itervalues()
def __repr__(self):
return str(self)
def __str__(self):
"""
Returns a string containing a new-line separated
list of the functions in the function list.
"""
s = ""
if len(self._functions) == 0:
s += "<blank plot>"
else:
self._render_lock.acquire()
s += "\n".join(["%s[%i]: %s" % ("", i, str(self._functions[i]))
for i in self._functions])
self._render_lock.release()
return s
def adjust_all_bounds(self):
self._render_lock.acquire()
self.axes.reset_bounding_box()
for f in self._functions:
self.axes.adjust_bounds(self._functions[f].bounds)
self._render_lock.release()
def wait_for_calculations(self):
sleep(0)
self._render_lock.acquire()
for f in self._functions:
a = self._functions[f]._get_calculating_verts
b = self._functions[f]._get_calculating_cverts
while a() or b():
sleep(0)
self._render_lock.release()
class ScreenShot:
def __init__(self, plot):
self._plot = plot
self.screenshot_requested = False
self.outfile = None
self.format = ''
self.invisibleMode = False
self.flag = 0
def __bool__(self):
return self.screenshot_requested
def _execute_saving(self):
if self.flag < 3:
self.flag += 1
return
size_x, size_y = self._plot._window.get_size()
size = size_x*size_y*4*ctypes.sizeof(ctypes.c_ubyte)
image = ctypes.create_string_buffer(size)
pgl.glReadPixels(0, 0, size_x, size_y, pgl.GL_RGBA, pgl.GL_UNSIGNED_BYTE, image)
from PIL import Image
im = Image.frombuffer('RGBA', (size_x, size_y),
image.raw, 'raw', 'RGBA', 0, 1)
im.transpose(Image.FLIP_TOP_BOTTOM).save(self.outfile, self.format)
self.flag = 0
self.screenshot_requested = False
if self.invisibleMode:
self._plot._window.close()
def save(self, outfile=None, format='', size=(600, 500)):
self.outfile = outfile
self.format = format
self.size = size
self.screenshot_requested = True
if not self._plot._window or self._plot._window.has_exit:
self._plot._win_args['visible'] = False
self._plot._win_args['width'] = size[0]
self._plot._win_args['height'] = size[1]
self._plot.axes.reset_resources()
self._plot._window = PlotWindow(self._plot, **self._plot._win_args)
self.invisibleMode = True
if self.outfile is None:
self.outfile = self._create_unique_path()
print(self.outfile)
def _create_unique_path(self):
cwd = getcwd()
l = listdir(cwd)
path = ''
i = 0
while True:
if not 'plot_%s.png' % i in l:
path = cwd + '/plot_%s.png' % i
break
i += 1
return path
|
11ce4626ba2eaa4f48f42c462e080bf105febb55ec58fa7b75c4d1110656c15d | import pyglet.gl as pgl
from sympy.core import S
from sympy.plotting.pygletplot.color_scheme import ColorScheme
from sympy.plotting.pygletplot.plot_mode import PlotMode
from sympy.utilities.iterables import is_sequence
from time import sleep
from threading import Thread, Event, RLock
import warnings
class PlotModeBase(PlotMode):
"""
Intended parent class for plotting
modes. Provides base functionality
in conjunction with its parent,
PlotMode.
"""
##
## Class-Level Attributes
##
"""
The following attributes are meant
to be set at the class level, and serve
as parameters to the plot mode registry
(in PlotMode). See plot_modes.py for
concrete examples.
"""
"""
i_vars
'x' for Cartesian2D
'xy' for Cartesian3D
etc.
d_vars
'y' for Cartesian2D
'r' for Polar
etc.
"""
i_vars, d_vars = '', ''
"""
intervals
Default intervals for each i_var, and in the
same order. Specified [min, max, steps].
No variable can be given (it is bound later).
"""
intervals = []
"""
aliases
A list of strings which can be used to
access this mode.
'cartesian' for Cartesian2D and Cartesian3D
'polar' for Polar
'cylindrical', 'polar' for Cylindrical
Note that _init_mode chooses the first alias
in the list as the mode's primary_alias, which
will be displayed to the end user in certain
contexts.
"""
aliases = []
"""
is_default
Whether to set this mode as the default
for arguments passed to PlotMode() containing
the same number of d_vars as this mode and
at most the same number of i_vars.
"""
is_default = False
"""
All of the above attributes are defined in PlotMode.
The following ones are specific to PlotModeBase.
"""
"""
A list of the render styles. Do not modify.
"""
styles = {'wireframe': 1, 'solid': 2, 'both': 3}
"""
style_override
Always use this style if not blank.
"""
style_override = ''
"""
default_wireframe_color
default_solid_color
Can be used when color is None or being calculated.
Used by PlotCurve and PlotSurface, but not anywhere
in PlotModeBase.
"""
default_wireframe_color = (0.85, 0.85, 0.85)
default_solid_color = (0.6, 0.6, 0.9)
default_rot_preset = 'xy'
##
## Instance-Level Attributes
##
## 'Abstract' member functions
def _get_evaluator(self):
if self.use_lambda_eval:
try:
e = self._get_lambda_evaluator()
return e
except Exception:
warnings.warn("\nWarning: creating lambda evaluator failed. "
"Falling back on SymPy subs evaluator.")
return self._get_sympy_evaluator()
def _get_sympy_evaluator(self):
raise NotImplementedError()
def _get_lambda_evaluator(self):
raise NotImplementedError()
def _on_calculate_verts(self):
raise NotImplementedError()
def _on_calculate_cverts(self):
raise NotImplementedError()
## Base member functions
def __init__(self, *args, bounds_callback=None, **kwargs):
self.verts = []
self.cverts = []
self.bounds = [[S.Infinity, S.NegativeInfinity, 0],
[S.Infinity, S.NegativeInfinity, 0],
[S.Infinity, S.NegativeInfinity, 0]]
self.cbounds = [[S.Infinity, S.NegativeInfinity, 0],
[S.Infinity, S.NegativeInfinity, 0],
[S.Infinity, S.NegativeInfinity, 0]]
self._draw_lock = RLock()
self._calculating_verts = Event()
self._calculating_cverts = Event()
self._calculating_verts_pos = 0.0
self._calculating_verts_len = 0.0
self._calculating_cverts_pos = 0.0
self._calculating_cverts_len = 0.0
self._max_render_stack_size = 3
self._draw_wireframe = [-1]
self._draw_solid = [-1]
self._style = None
self._color = None
self.predraw = []
self.postdraw = []
self.use_lambda_eval = self.options.pop('use_sympy_eval', None) is None
self.style = self.options.pop('style', '')
self.color = self.options.pop('color', 'rainbow')
self.bounds_callback = bounds_callback
self._on_calculate()
def synchronized(f):
def w(self, *args, **kwargs):
self._draw_lock.acquire()
try:
r = f(self, *args, **kwargs)
return r
finally:
self._draw_lock.release()
return w
@synchronized
def push_wireframe(self, function):
"""
Push a function which performs gl commands
used to build a display list. (The list is
built outside of the function)
"""
assert callable(function)
self._draw_wireframe.append(function)
if len(self._draw_wireframe) > self._max_render_stack_size:
del self._draw_wireframe[1] # leave marker element
@synchronized
def push_solid(self, function):
"""
Push a function which performs gl commands
used to build a display list. (The list is
built outside of the function)
"""
assert callable(function)
self._draw_solid.append(function)
if len(self._draw_solid) > self._max_render_stack_size:
del self._draw_solid[1] # leave marker element
def _create_display_list(self, function):
dl = pgl.glGenLists(1)
pgl.glNewList(dl, pgl.GL_COMPILE)
function()
pgl.glEndList()
return dl
def _render_stack_top(self, render_stack):
top = render_stack[-1]
if top == -1:
return -1 # nothing to display
elif callable(top):
dl = self._create_display_list(top)
render_stack[-1] = (dl, top)
return dl # display newly added list
elif len(top) == 2:
if pgl.GL_TRUE == pgl.glIsList(top[0]):
return top[0] # display stored list
dl = self._create_display_list(top[1])
render_stack[-1] = (dl, top[1])
return dl # display regenerated list
def _draw_solid_display_list(self, dl):
pgl.glPushAttrib(pgl.GL_ENABLE_BIT | pgl.GL_POLYGON_BIT)
pgl.glPolygonMode(pgl.GL_FRONT_AND_BACK, pgl.GL_FILL)
pgl.glCallList(dl)
pgl.glPopAttrib()
def _draw_wireframe_display_list(self, dl):
pgl.glPushAttrib(pgl.GL_ENABLE_BIT | pgl.GL_POLYGON_BIT)
pgl.glPolygonMode(pgl.GL_FRONT_AND_BACK, pgl.GL_LINE)
pgl.glEnable(pgl.GL_POLYGON_OFFSET_LINE)
pgl.glPolygonOffset(-0.005, -50.0)
pgl.glCallList(dl)
pgl.glPopAttrib()
@synchronized
def draw(self):
for f in self.predraw:
if callable(f):
f()
if self.style_override:
style = self.styles[self.style_override]
else:
style = self.styles[self._style]
# Draw solid component if style includes solid
if style & 2:
dl = self._render_stack_top(self._draw_solid)
if dl > 0 and pgl.GL_TRUE == pgl.glIsList(dl):
self._draw_solid_display_list(dl)
# Draw wireframe component if style includes wireframe
if style & 1:
dl = self._render_stack_top(self._draw_wireframe)
if dl > 0 and pgl.GL_TRUE == pgl.glIsList(dl):
self._draw_wireframe_display_list(dl)
for f in self.postdraw:
if callable(f):
f()
def _on_change_color(self, color):
Thread(target=self._calculate_cverts).start()
def _on_calculate(self):
Thread(target=self._calculate_all).start()
def _calculate_all(self):
self._calculate_verts()
self._calculate_cverts()
def _calculate_verts(self):
if self._calculating_verts.isSet():
return
self._calculating_verts.set()
try:
self._on_calculate_verts()
finally:
self._calculating_verts.clear()
if callable(self.bounds_callback):
self.bounds_callback()
def _calculate_cverts(self):
if self._calculating_verts.isSet():
return
while self._calculating_cverts.isSet():
sleep(0) # wait for previous calculation
self._calculating_cverts.set()
try:
self._on_calculate_cverts()
finally:
self._calculating_cverts.clear()
def _get_calculating_verts(self):
return self._calculating_verts.isSet()
def _get_calculating_verts_pos(self):
return self._calculating_verts_pos
def _get_calculating_verts_len(self):
return self._calculating_verts_len
def _get_calculating_cverts(self):
return self._calculating_cverts.isSet()
def _get_calculating_cverts_pos(self):
return self._calculating_cverts_pos
def _get_calculating_cverts_len(self):
return self._calculating_cverts_len
## Property handlers
def _get_style(self):
return self._style
@synchronized
def _set_style(self, v):
if v is None:
return
if v == '':
step_max = 0
for i in self.intervals:
if i.v_steps is None:
continue
step_max = max([step_max, int(i.v_steps)])
v = ['both', 'solid'][step_max > 40]
if v not in self.styles:
raise ValueError("v should be there in self.styles")
if v == self._style:
return
self._style = v
def _get_color(self):
return self._color
@synchronized
def _set_color(self, v):
try:
if v is not None:
if is_sequence(v):
v = ColorScheme(*v)
else:
v = ColorScheme(v)
if repr(v) == repr(self._color):
return
self._on_change_color(v)
self._color = v
except Exception as e:
raise RuntimeError("Color change failed. "
"Reason: %s" % (str(e)))
style = property(_get_style, _set_style)
color = property(_get_color, _set_color)
calculating_verts = property(_get_calculating_verts)
calculating_verts_pos = property(_get_calculating_verts_pos)
calculating_verts_len = property(_get_calculating_verts_len)
calculating_cverts = property(_get_calculating_cverts)
calculating_cverts_pos = property(_get_calculating_cverts_pos)
calculating_cverts_len = property(_get_calculating_cverts_len)
## String representations
def __str__(self):
f = ", ".join(str(d) for d in self.d_vars)
o = "'mode=%s'" % (self.primary_alias)
return ", ".join([f, o])
def __repr__(self):
f = ", ".join(str(d) for d in self.d_vars)
i = ", ".join(str(i) for i in self.intervals)
d = [('mode', self.primary_alias),
('color', str(self.color)),
('style', str(self.style))]
o = "'%s'" % ("; ".join("%s=%s" % (k, v)
for k, v in d if v != 'None'))
return ", ".join([f, i, o])
|
225799aec1d7b5d1d7380133d514237992ae89889fe0e2b4bd8b84efb9aa3aa7 | from sympy.core.basic import Basic
from sympy.core.symbol import (Symbol, symbols)
from sympy.utilities.lambdify import lambdify
from .util import interpolate, rinterpolate, create_bounds, update_bounds
from sympy.utilities.iterables import sift
class ColorGradient:
colors = [0.4, 0.4, 0.4], [0.9, 0.9, 0.9]
intervals = 0.0, 1.0
def __init__(self, *args):
if len(args) == 2:
self.colors = list(args)
self.intervals = [0.0, 1.0]
elif len(args) > 0:
if len(args) % 2 != 0:
raise ValueError("len(args) should be even")
self.colors = [args[i] for i in range(1, len(args), 2)]
self.intervals = [args[i] for i in range(0, len(args), 2)]
assert len(self.colors) == len(self.intervals)
def copy(self):
c = ColorGradient()
c.colors = [e[::] for e in self.colors]
c.intervals = self.intervals[::]
return c
def _find_interval(self, v):
m = len(self.intervals)
i = 0
while i < m - 1 and self.intervals[i] <= v:
i += 1
return i
def _interpolate_axis(self, axis, v):
i = self._find_interval(v)
v = rinterpolate(self.intervals[i - 1], self.intervals[i], v)
return interpolate(self.colors[i - 1][axis], self.colors[i][axis], v)
def __call__(self, r, g, b):
c = self._interpolate_axis
return c(0, r), c(1, g), c(2, b)
default_color_schemes = {} # defined at the bottom of this file
class ColorScheme:
def __init__(self, *args, **kwargs):
self.args = args
self.f, self.gradient = None, ColorGradient()
if len(args) == 1 and not isinstance(args[0], Basic) and callable(args[0]):
self.f = args[0]
elif len(args) == 1 and isinstance(args[0], str):
if args[0] in default_color_schemes:
cs = default_color_schemes[args[0]]
self.f, self.gradient = cs.f, cs.gradient.copy()
else:
self.f = lambdify('x,y,z,u,v', args[0])
else:
self.f, self.gradient = self._interpret_args(args)
self._test_color_function()
if not isinstance(self.gradient, ColorGradient):
raise ValueError("Color gradient not properly initialized. "
"(Not a ColorGradient instance.)")
def _interpret_args(self, args):
f, gradient = None, self.gradient
atoms, lists = self._sort_args(args)
s = self._pop_symbol_list(lists)
s = self._fill_in_vars(s)
# prepare the error message for lambdification failure
f_str = ', '.join(str(fa) for fa in atoms)
s_str = (str(sa) for sa in s)
s_str = ', '.join(sa for sa in s_str if sa.find('unbound') < 0)
f_error = ValueError("Could not interpret arguments "
"%s as functions of %s." % (f_str, s_str))
# try to lambdify args
if len(atoms) == 1:
fv = atoms[0]
try:
f = lambdify(s, [fv, fv, fv])
except TypeError:
raise f_error
elif len(atoms) == 3:
fr, fg, fb = atoms
try:
f = lambdify(s, [fr, fg, fb])
except TypeError:
raise f_error
else:
raise ValueError("A ColorScheme must provide 1 or 3 "
"functions in x, y, z, u, and/or v.")
# try to intrepret any given color information
if len(lists) == 0:
gargs = []
elif len(lists) == 1:
gargs = lists[0]
elif len(lists) == 2:
try:
(r1, g1, b1), (r2, g2, b2) = lists
except TypeError:
raise ValueError("If two color arguments are given, "
"they must be given in the format "
"(r1, g1, b1), (r2, g2, b2).")
gargs = lists
elif len(lists) == 3:
try:
(r1, r2), (g1, g2), (b1, b2) = lists
except Exception:
raise ValueError("If three color arguments are given, "
"they must be given in the format "
"(r1, r2), (g1, g2), (b1, b2). To create "
"a multi-step gradient, use the syntax "
"[0, colorStart, step1, color1, ..., 1, "
"colorEnd].")
gargs = [[r1, g1, b1], [r2, g2, b2]]
else:
raise ValueError("Don't know what to do with collection "
"arguments %s." % (', '.join(str(l) for l in lists)))
if gargs:
try:
gradient = ColorGradient(*gargs)
except Exception as ex:
raise ValueError(("Could not initialize a gradient "
"with arguments %s. Inner "
"exception: %s") % (gargs, str(ex)))
return f, gradient
def _pop_symbol_list(self, lists):
symbol_lists = []
for l in lists:
mark = True
for s in l:
if s is not None and not isinstance(s, Symbol):
mark = False
break
if mark:
lists.remove(l)
symbol_lists.append(l)
if len(symbol_lists) == 1:
return symbol_lists[0]
elif len(symbol_lists) == 0:
return []
else:
raise ValueError("Only one list of Symbols "
"can be given for a color scheme.")
def _fill_in_vars(self, args):
defaults = symbols('x,y,z,u,v')
v_error = ValueError("Could not find what to plot.")
if len(args) == 0:
return defaults
if not isinstance(args, (tuple, list)):
raise v_error
if len(args) == 0:
return defaults
for s in args:
if s is not None and not isinstance(s, Symbol):
raise v_error
# when vars are given explicitly, any vars
# not given are marked 'unbound' as to not
# be accidentally used in an expression
vars = [Symbol('unbound%i' % (i)) for i in range(1, 6)]
# interpret as t
if len(args) == 1:
vars[3] = args[0]
# interpret as u,v
elif len(args) == 2:
if args[0] is not None:
vars[3] = args[0]
if args[1] is not None:
vars[4] = args[1]
# interpret as x,y,z
elif len(args) >= 3:
# allow some of x,y,z to be
# left unbound if not given
if args[0] is not None:
vars[0] = args[0]
if args[1] is not None:
vars[1] = args[1]
if args[2] is not None:
vars[2] = args[2]
# interpret the rest as t
if len(args) >= 4:
vars[3] = args[3]
# ...or u,v
if len(args) >= 5:
vars[4] = args[4]
return vars
def _sort_args(self, args):
lists, atoms = sift(args,
lambda a: isinstance(a, (tuple, list)), binary=True)
return atoms, lists
def _test_color_function(self):
if not callable(self.f):
raise ValueError("Color function is not callable.")
try:
result = self.f(0, 0, 0, 0, 0)
if len(result) != 3:
raise ValueError("length should be equal to 3")
except TypeError:
raise ValueError("Color function needs to accept x,y,z,u,v, "
"as arguments even if it doesn't use all of them.")
except AssertionError:
raise ValueError("Color function needs to return 3-tuple r,g,b.")
except Exception:
pass # color function probably not valid at 0,0,0,0,0
def __call__(self, x, y, z, u, v):
try:
return self.f(x, y, z, u, v)
except Exception:
return None
def apply_to_curve(self, verts, u_set, set_len=None, inc_pos=None):
"""
Apply this color scheme to a
set of vertices over a single
independent variable u.
"""
bounds = create_bounds()
cverts = list()
if callable(set_len):
set_len(len(u_set)*2)
# calculate f() = r,g,b for each vert
# and find the min and max for r,g,b
for _u in range(len(u_set)):
if verts[_u] is None:
cverts.append(None)
else:
x, y, z = verts[_u]
u, v = u_set[_u], None
c = self(x, y, z, u, v)
if c is not None:
c = list(c)
update_bounds(bounds, c)
cverts.append(c)
if callable(inc_pos):
inc_pos()
# scale and apply gradient
for _u in range(len(u_set)):
if cverts[_u] is not None:
for _c in range(3):
# scale from [f_min, f_max] to [0,1]
cverts[_u][_c] = rinterpolate(bounds[_c][0], bounds[_c][1],
cverts[_u][_c])
# apply gradient
cverts[_u] = self.gradient(*cverts[_u])
if callable(inc_pos):
inc_pos()
return cverts
def apply_to_surface(self, verts, u_set, v_set, set_len=None, inc_pos=None):
"""
Apply this color scheme to a
set of vertices over two
independent variables u and v.
"""
bounds = create_bounds()
cverts = list()
if callable(set_len):
set_len(len(u_set)*len(v_set)*2)
# calculate f() = r,g,b for each vert
# and find the min and max for r,g,b
for _u in range(len(u_set)):
column = list()
for _v in range(len(v_set)):
if verts[_u][_v] is None:
column.append(None)
else:
x, y, z = verts[_u][_v]
u, v = u_set[_u], v_set[_v]
c = self(x, y, z, u, v)
if c is not None:
c = list(c)
update_bounds(bounds, c)
column.append(c)
if callable(inc_pos):
inc_pos()
cverts.append(column)
# scale and apply gradient
for _u in range(len(u_set)):
for _v in range(len(v_set)):
if cverts[_u][_v] is not None:
# scale from [f_min, f_max] to [0,1]
for _c in range(3):
cverts[_u][_v][_c] = rinterpolate(bounds[_c][0],
bounds[_c][1], cverts[_u][_v][_c])
# apply gradient
cverts[_u][_v] = self.gradient(*cverts[_u][_v])
if callable(inc_pos):
inc_pos()
return cverts
def str_base(self):
return ", ".join(str(a) for a in self.args)
def __repr__(self):
return "%s" % (self.str_base())
x, y, z, t, u, v = symbols('x,y,z,t,u,v')
default_color_schemes['rainbow'] = ColorScheme(z, y, x)
default_color_schemes['zfade'] = ColorScheme(z, (0.4, 0.4, 0.97),
(0.97, 0.4, 0.4), (None, None, z))
default_color_schemes['zfade3'] = ColorScheme(z, (None, None, z),
[0.00, (0.2, 0.2, 1.0),
0.35, (0.2, 0.8, 0.4),
0.50, (0.3, 0.9, 0.3),
0.65, (0.4, 0.8, 0.2),
1.00, (1.0, 0.2, 0.2)])
default_color_schemes['zfade4'] = ColorScheme(z, (None, None, z),
[0.0, (0.3, 0.3, 1.0),
0.30, (0.3, 1.0, 0.3),
0.55, (0.95, 1.0, 0.2),
0.65, (1.0, 0.95, 0.2),
0.85, (1.0, 0.7, 0.2),
1.0, (1.0, 0.3, 0.2)])
|
768aabdb0c67ac40f83090e5931434ec616f69a817dfa1d46fbee5cbfbc8b8a4 | from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.core.numbers import Integer
class PlotInterval:
"""
"""
_v, _v_min, _v_max, _v_steps = None, None, None, None
def require_all_args(f):
def check(self, *args, **kwargs):
for g in [self._v, self._v_min, self._v_max, self._v_steps]:
if g is None:
raise ValueError("PlotInterval is incomplete.")
return f(self, *args, **kwargs)
return check
def __init__(self, *args):
if len(args) == 1:
if isinstance(args[0], PlotInterval):
self.fill_from(args[0])
return
elif isinstance(args[0], str):
try:
args = eval(args[0])
except TypeError:
s_eval_error = "Could not interpret string %s."
raise ValueError(s_eval_error % (args[0]))
elif isinstance(args[0], (tuple, list)):
args = args[0]
else:
raise ValueError("Not an interval.")
if not isinstance(args, (tuple, list)) or len(args) > 4:
f_error = "PlotInterval must be a tuple or list of length 4 or less."
raise ValueError(f_error)
args = list(args)
if len(args) > 0 and (args[0] is None or isinstance(args[0], Symbol)):
self.v = args.pop(0)
if len(args) in [2, 3]:
self.v_min = args.pop(0)
self.v_max = args.pop(0)
if len(args) == 1:
self.v_steps = args.pop(0)
elif len(args) == 1:
self.v_steps = args.pop(0)
def get_v(self):
return self._v
def set_v(self, v):
if v is None:
self._v = None
return
if not isinstance(v, Symbol):
raise ValueError("v must be a SymPy Symbol.")
self._v = v
def get_v_min(self):
return self._v_min
def set_v_min(self, v_min):
if v_min is None:
self._v_min = None
return
try:
self._v_min = sympify(v_min)
float(self._v_min.evalf())
except TypeError:
raise ValueError("v_min could not be interpreted as a number.")
def get_v_max(self):
return self._v_max
def set_v_max(self, v_max):
if v_max is None:
self._v_max = None
return
try:
self._v_max = sympify(v_max)
float(self._v_max.evalf())
except TypeError:
raise ValueError("v_max could not be interpreted as a number.")
def get_v_steps(self):
return self._v_steps
def set_v_steps(self, v_steps):
if v_steps is None:
self._v_steps = None
return
if isinstance(v_steps, int):
v_steps = Integer(v_steps)
elif not isinstance(v_steps, Integer):
raise ValueError("v_steps must be an int or SymPy Integer.")
if v_steps <= S.Zero:
raise ValueError("v_steps must be positive.")
self._v_steps = v_steps
@require_all_args
def get_v_len(self):
return self.v_steps + 1
v = property(get_v, set_v)
v_min = property(get_v_min, set_v_min)
v_max = property(get_v_max, set_v_max)
v_steps = property(get_v_steps, set_v_steps)
v_len = property(get_v_len)
def fill_from(self, b):
if b.v is not None:
self.v = b.v
if b.v_min is not None:
self.v_min = b.v_min
if b.v_max is not None:
self.v_max = b.v_max
if b.v_steps is not None:
self.v_steps = b.v_steps
@staticmethod
def try_parse(*args):
"""
Returns a PlotInterval if args can be interpreted
as such, otherwise None.
"""
if len(args) == 1 and isinstance(args[0], PlotInterval):
return args[0]
try:
return PlotInterval(*args)
except ValueError:
return None
def _str_base(self):
return ",".join([str(self.v), str(self.v_min),
str(self.v_max), str(self.v_steps)])
def __repr__(self):
"""
A string representing the interval in class constructor form.
"""
return "PlotInterval(%s)" % (self._str_base())
def __str__(self):
"""
A string representing the interval in list form.
"""
return "[%s]" % (self._str_base())
@require_all_args
def assert_complete(self):
pass
@require_all_args
def vrange(self):
"""
Yields v_steps+1 SymPy numbers ranging from
v_min to v_max.
"""
d = (self.v_max - self.v_min) / self.v_steps
for i in range(self.v_steps + 1):
a = self.v_min + (d * Integer(i))
yield a
@require_all_args
def vrange2(self):
"""
Yields v_steps pairs of SymPy numbers ranging from
(v_min, v_min + step) to (v_max - step, v_max).
"""
d = (self.v_max - self.v_min) / self.v_steps
a = self.v_min + (d * S.Zero)
for i in range(self.v_steps):
b = self.v_min + (d * Integer(i + 1))
yield a, b
a = b
def frange(self):
for i in self.vrange():
yield float(i.evalf())
|
80acc94a1cfa3976ba10729aafc4ae1d6ac42d5ce7c60fecd9de480509247981 | from sympy.utilities.lambdify import lambdify
from sympy.core.numbers import pi
from sympy.functions import sin, cos
from sympy.plotting.pygletplot.plot_curve import PlotCurve
from sympy.plotting.pygletplot.plot_surface import PlotSurface
from math import sin as p_sin
from math import cos as p_cos
def float_vec3(f):
def inner(*args):
v = f(*args)
return float(v[0]), float(v[1]), float(v[2])
return inner
class Cartesian2D(PlotCurve):
i_vars, d_vars = 'x', 'y'
intervals = [[-5, 5, 100]]
aliases = ['cartesian']
is_default = True
def _get_sympy_evaluator(self):
fy = self.d_vars[0]
x = self.t_interval.v
@float_vec3
def e(_x):
return (_x, fy.subs(x, _x), 0.0)
return e
def _get_lambda_evaluator(self):
fy = self.d_vars[0]
x = self.t_interval.v
return lambdify([x], [x, fy, 0.0])
class Cartesian3D(PlotSurface):
i_vars, d_vars = 'xy', 'z'
intervals = [[-1, 1, 40], [-1, 1, 40]]
aliases = ['cartesian', 'monge']
is_default = True
def _get_sympy_evaluator(self):
fz = self.d_vars[0]
x = self.u_interval.v
y = self.v_interval.v
@float_vec3
def e(_x, _y):
return (_x, _y, fz.subs(x, _x).subs(y, _y))
return e
def _get_lambda_evaluator(self):
fz = self.d_vars[0]
x = self.u_interval.v
y = self.v_interval.v
return lambdify([x, y], [x, y, fz])
class ParametricCurve2D(PlotCurve):
i_vars, d_vars = 't', 'xy'
intervals = [[0, 2*pi, 100]]
aliases = ['parametric']
is_default = True
def _get_sympy_evaluator(self):
fx, fy = self.d_vars
t = self.t_interval.v
@float_vec3
def e(_t):
return (fx.subs(t, _t), fy.subs(t, _t), 0.0)
return e
def _get_lambda_evaluator(self):
fx, fy = self.d_vars
t = self.t_interval.v
return lambdify([t], [fx, fy, 0.0])
class ParametricCurve3D(PlotCurve):
i_vars, d_vars = 't', 'xyz'
intervals = [[0, 2*pi, 100]]
aliases = ['parametric']
is_default = True
def _get_sympy_evaluator(self):
fx, fy, fz = self.d_vars
t = self.t_interval.v
@float_vec3
def e(_t):
return (fx.subs(t, _t), fy.subs(t, _t), fz.subs(t, _t))
return e
def _get_lambda_evaluator(self):
fx, fy, fz = self.d_vars
t = self.t_interval.v
return lambdify([t], [fx, fy, fz])
class ParametricSurface(PlotSurface):
i_vars, d_vars = 'uv', 'xyz'
intervals = [[-1, 1, 40], [-1, 1, 40]]
aliases = ['parametric']
is_default = True
def _get_sympy_evaluator(self):
fx, fy, fz = self.d_vars
u = self.u_interval.v
v = self.v_interval.v
@float_vec3
def e(_u, _v):
return (fx.subs(u, _u).subs(v, _v),
fy.subs(u, _u).subs(v, _v),
fz.subs(u, _u).subs(v, _v))
return e
def _get_lambda_evaluator(self):
fx, fy, fz = self.d_vars
u = self.u_interval.v
v = self.v_interval.v
return lambdify([u, v], [fx, fy, fz])
class Polar(PlotCurve):
i_vars, d_vars = 't', 'r'
intervals = [[0, 2*pi, 100]]
aliases = ['polar']
is_default = False
def _get_sympy_evaluator(self):
fr = self.d_vars[0]
t = self.t_interval.v
def e(_t):
_r = float(fr.subs(t, _t))
return (_r*p_cos(_t), _r*p_sin(_t), 0.0)
return e
def _get_lambda_evaluator(self):
fr = self.d_vars[0]
t = self.t_interval.v
fx, fy = fr*cos(t), fr*sin(t)
return lambdify([t], [fx, fy, 0.0])
class Cylindrical(PlotSurface):
i_vars, d_vars = 'th', 'r'
intervals = [[0, 2*pi, 40], [-1, 1, 20]]
aliases = ['cylindrical', 'polar']
is_default = False
def _get_sympy_evaluator(self):
fr = self.d_vars[0]
t = self.u_interval.v
h = self.v_interval.v
def e(_t, _h):
_r = float(fr.subs(t, _t).subs(h, _h))
return (_r*p_cos(_t), _r*p_sin(_t), _h)
return e
def _get_lambda_evaluator(self):
fr = self.d_vars[0]
t = self.u_interval.v
h = self.v_interval.v
fx, fy = fr*cos(t), fr*sin(t)
return lambdify([t, h], [fx, fy, h])
class Spherical(PlotSurface):
i_vars, d_vars = 'tp', 'r'
intervals = [[0, 2*pi, 40], [0, pi, 20]]
aliases = ['spherical']
is_default = False
def _get_sympy_evaluator(self):
fr = self.d_vars[0]
t = self.u_interval.v
p = self.v_interval.v
def e(_t, _p):
_r = float(fr.subs(t, _t).subs(p, _p))
return (_r*p_cos(_t)*p_sin(_p),
_r*p_sin(_t)*p_sin(_p),
_r*p_cos(_p))
return e
def _get_lambda_evaluator(self):
fr = self.d_vars[0]
t = self.u_interval.v
p = self.v_interval.v
fx = fr * cos(t) * sin(p)
fy = fr * sin(t) * sin(p)
fz = fr * cos(p)
return lambdify([t, p], [fx, fy, fz])
Cartesian2D._register()
Cartesian3D._register()
ParametricCurve2D._register()
ParametricCurve3D._register()
ParametricSurface._register()
Polar._register()
Cylindrical._register()
Spherical._register()
|
d758d221b274ffeaabf99f352551579b9e30e828c7b035967e0050400ba91041 | try:
from ctypes import c_float, c_int, c_double
except ImportError:
pass
import pyglet.gl as pgl
from sympy.core import S
def get_model_matrix(array_type=c_float, glGetMethod=pgl.glGetFloatv):
"""
Returns the current modelview matrix.
"""
m = (array_type*16)()
glGetMethod(pgl.GL_MODELVIEW_MATRIX, m)
return m
def get_projection_matrix(array_type=c_float, glGetMethod=pgl.glGetFloatv):
"""
Returns the current modelview matrix.
"""
m = (array_type*16)()
glGetMethod(pgl.GL_PROJECTION_MATRIX, m)
return m
def get_viewport():
"""
Returns the current viewport.
"""
m = (c_int*4)()
pgl.glGetIntegerv(pgl.GL_VIEWPORT, m)
return m
def get_direction_vectors():
m = get_model_matrix()
return ((m[0], m[4], m[8]),
(m[1], m[5], m[9]),
(m[2], m[6], m[10]))
def get_view_direction_vectors():
m = get_model_matrix()
return ((m[0], m[1], m[2]),
(m[4], m[5], m[6]),
(m[8], m[9], m[10]))
def get_basis_vectors():
return ((1, 0, 0), (0, 1, 0), (0, 0, 1))
def screen_to_model(x, y, z):
m = get_model_matrix(c_double, pgl.glGetDoublev)
p = get_projection_matrix(c_double, pgl.glGetDoublev)
w = get_viewport()
mx, my, mz = c_double(), c_double(), c_double()
pgl.gluUnProject(x, y, z, m, p, w, mx, my, mz)
return float(mx.value), float(my.value), float(mz.value)
def model_to_screen(x, y, z):
m = get_model_matrix(c_double, pgl.glGetDoublev)
p = get_projection_matrix(c_double, pgl.glGetDoublev)
w = get_viewport()
mx, my, mz = c_double(), c_double(), c_double()
pgl.gluProject(x, y, z, m, p, w, mx, my, mz)
return float(mx.value), float(my.value), float(mz.value)
def vec_subs(a, b):
return tuple(a[i] - b[i] for i in range(len(a)))
def billboard_matrix():
"""
Removes rotational components of
current matrix so that primitives
are always drawn facing the viewer.
|1|0|0|x|
|0|1|0|x|
|0|0|1|x| (x means left unchanged)
|x|x|x|x|
"""
m = get_model_matrix()
# XXX: for i in range(11): m[i] = i ?
m[0] = 1
m[1] = 0
m[2] = 0
m[4] = 0
m[5] = 1
m[6] = 0
m[8] = 0
m[9] = 0
m[10] = 1
pgl.glLoadMatrixf(m)
def create_bounds():
return [[S.Infinity, S.NegativeInfinity, 0],
[S.Infinity, S.NegativeInfinity, 0],
[S.Infinity, S.NegativeInfinity, 0]]
def update_bounds(b, v):
if v is None:
return
for axis in range(3):
b[axis][0] = min([b[axis][0], v[axis]])
b[axis][1] = max([b[axis][1], v[axis]])
def interpolate(a_min, a_max, a_ratio):
return a_min + a_ratio * (a_max - a_min)
def rinterpolate(a_min, a_max, a_value):
a_range = a_max - a_min
if a_max == a_min:
a_range = 1.0
return (a_value - a_min) / float(a_range)
def interpolate_color(color1, color2, ratio):
return tuple(interpolate(color1[i], color2[i], ratio) for i in range(3))
def scale_value(v, v_min, v_len):
return (v - v_min) / v_len
def scale_value_list(flist):
v_min, v_max = min(flist), max(flist)
v_len = v_max - v_min
return list(scale_value(f, v_min, v_len) for f in flist)
def strided_range(r_min, r_max, stride, max_steps=50):
o_min, o_max = r_min, r_max
if abs(r_min - r_max) < 0.001:
return []
try:
range(int(r_min - r_max))
except (TypeError, OverflowError):
return []
if r_min > r_max:
raise ValueError("r_min cannot be greater than r_max")
r_min_s = (r_min % stride)
r_max_s = stride - (r_max % stride)
if abs(r_max_s - stride) < 0.001:
r_max_s = 0.0
r_min -= r_min_s
r_max += r_max_s
r_steps = int((r_max - r_min)/stride)
if max_steps and r_steps > max_steps:
return strided_range(o_min, o_max, stride*2)
return [r_min] + list(r_min + e*stride for e in range(1, r_steps + 1)) + [r_max]
def parse_option_string(s):
if not isinstance(s, str):
return None
options = {}
for token in s.split(';'):
pieces = token.split('=')
if len(pieces) == 1:
option, value = pieces[0], ""
elif len(pieces) == 2:
option, value = pieces
else:
raise ValueError("Plot option string '%s' is malformed." % (s))
options[option.strip()] = value.strip()
return options
def dot_product(v1, v2):
return sum(v1[i]*v2[i] for i in range(3))
def vec_sub(v1, v2):
return tuple(v1[i] - v2[i] for i in range(3))
def vec_mag(v):
return sum(v[i]**2 for i in range(3))**(0.5)
|
137a3927d8a9be20bb2b145f108109657b46ce6869b85d2b3e27b49e4a0871df | from time import perf_counter
import pyglet.gl as pgl
from sympy.plotting.pygletplot.managed_window import ManagedWindow
from sympy.plotting.pygletplot.plot_camera import PlotCamera
from sympy.plotting.pygletplot.plot_controller import PlotController
class PlotWindow(ManagedWindow):
def __init__(self, plot, antialiasing=True, ortho=False,
invert_mouse_zoom=False, linewidth=1.5, caption="SymPy Plot",
**kwargs):
"""
Named Arguments
===============
antialiasing = True
True OR False
ortho = False
True OR False
invert_mouse_zoom = False
True OR False
"""
self.plot = plot
self.camera = None
self._calculating = False
self.antialiasing = antialiasing
self.ortho = ortho
self.invert_mouse_zoom = invert_mouse_zoom
self.linewidth = linewidth
self.title = caption
self.last_caption_update = 0
self.caption_update_interval = 0.2
self.drawing_first_object = True
super().__init__(**kwargs)
def setup(self):
self.camera = PlotCamera(self, ortho=self.ortho)
self.controller = PlotController(self,
invert_mouse_zoom=self.invert_mouse_zoom)
self.push_handlers(self.controller)
pgl.glClearColor(1.0, 1.0, 1.0, 0.0)
pgl.glClearDepth(1.0)
pgl.glDepthFunc(pgl.GL_LESS)
pgl.glEnable(pgl.GL_DEPTH_TEST)
pgl.glEnable(pgl.GL_LINE_SMOOTH)
pgl.glShadeModel(pgl.GL_SMOOTH)
pgl.glLineWidth(self.linewidth)
pgl.glEnable(pgl.GL_BLEND)
pgl.glBlendFunc(pgl.GL_SRC_ALPHA, pgl.GL_ONE_MINUS_SRC_ALPHA)
if self.antialiasing:
pgl.glHint(pgl.GL_LINE_SMOOTH_HINT, pgl.GL_NICEST)
pgl.glHint(pgl.GL_POLYGON_SMOOTH_HINT, pgl.GL_NICEST)
self.camera.setup_projection()
def on_resize(self, w, h):
super().on_resize(w, h)
if self.camera is not None:
self.camera.setup_projection()
def update(self, dt):
self.controller.update(dt)
def draw(self):
self.plot._render_lock.acquire()
self.camera.apply_transformation()
calc_verts_pos, calc_verts_len = 0, 0
calc_cverts_pos, calc_cverts_len = 0, 0
should_update_caption = (perf_counter() - self.last_caption_update >
self.caption_update_interval)
if len(self.plot._functions.values()) == 0:
self.drawing_first_object = True
try:
dict.iteritems
except AttributeError:
# Python 3
iterfunctions = iter(self.plot._functions.values())
else:
# Python 2
iterfunctions = self.plot._functions.itervalues()
for r in iterfunctions:
if self.drawing_first_object:
self.camera.set_rot_preset(r.default_rot_preset)
self.drawing_first_object = False
pgl.glPushMatrix()
r._draw()
pgl.glPopMatrix()
# might as well do this while we are
# iterating and have the lock rather
# than locking and iterating twice
# per frame:
if should_update_caption:
try:
if r.calculating_verts:
calc_verts_pos += r.calculating_verts_pos
calc_verts_len += r.calculating_verts_len
if r.calculating_cverts:
calc_cverts_pos += r.calculating_cverts_pos
calc_cverts_len += r.calculating_cverts_len
except ValueError:
pass
for r in self.plot._pobjects:
pgl.glPushMatrix()
r._draw()
pgl.glPopMatrix()
if should_update_caption:
self.update_caption(calc_verts_pos, calc_verts_len,
calc_cverts_pos, calc_cverts_len)
self.last_caption_update = perf_counter()
if self.plot._screenshot:
self.plot._screenshot._execute_saving()
self.plot._render_lock.release()
def update_caption(self, calc_verts_pos, calc_verts_len,
calc_cverts_pos, calc_cverts_len):
caption = self.title
if calc_verts_len or calc_cverts_len:
caption += " (calculating"
if calc_verts_len > 0:
p = (calc_verts_pos / calc_verts_len) * 100
caption += " vertices %i%%" % (p)
if calc_cverts_len > 0:
p = (calc_cverts_pos / calc_cverts_len) * 100
caption += " colors %i%%" % (p)
caption += ")"
if self.caption != caption:
self.set_caption(caption)
|
43d60df16d077753de7e51cbbcebd267e87178b5382aade75052d80c2d31d04f | import pyglet.gl as pgl
from pyglet import font
from sympy.core import S
from sympy.plotting.pygletplot.plot_object import PlotObject
from sympy.plotting.pygletplot.util import billboard_matrix, dot_product, \
get_direction_vectors, strided_range, vec_mag, vec_sub
from sympy.utilities.iterables import is_sequence
class PlotAxes(PlotObject):
def __init__(self, *args,
style='', none=None, frame=None, box=None, ordinate=None,
stride=0.25,
visible='', overlay='', colored='', label_axes='', label_ticks='',
tick_length=0.1,
font_face='Arial', font_size=28,
**kwargs):
# initialize style parameter
style = style.lower()
# allow alias kwargs to override style kwarg
if none is not None:
style = 'none'
if frame is not None:
style = 'frame'
if box is not None:
style = 'box'
if ordinate is not None:
style = 'ordinate'
if style in ['', 'ordinate']:
self._render_object = PlotAxesOrdinate(self)
elif style in ['frame', 'box']:
self._render_object = PlotAxesFrame(self)
elif style in ['none']:
self._render_object = None
else:
raise ValueError(("Unrecognized axes style %s.") % (style))
# initialize stride parameter
try:
stride = eval(stride)
except TypeError:
pass
if is_sequence(stride):
if len(stride) != 3:
raise ValueError("length should be equal to 3")
self._stride = stride
else:
self._stride = [stride, stride, stride]
self._tick_length = float(tick_length)
# setup bounding box and ticks
self._origin = [0, 0, 0]
self.reset_bounding_box()
def flexible_boolean(input, default):
if input in [True, False]:
return input
if input in ('f', 'F', 'false', 'False'):
return False
if input in ('t', 'T', 'true', 'True'):
return True
return default
# initialize remaining parameters
self.visible = flexible_boolean(kwargs, True)
self._overlay = flexible_boolean(overlay, True)
self._colored = flexible_boolean(colored, False)
self._label_axes = flexible_boolean(label_axes, False)
self._label_ticks = flexible_boolean(label_ticks, True)
# setup label font
self.font_face = font_face
self.font_size = font_size
# this is also used to reinit the
# font on window close/reopen
self.reset_resources()
def reset_resources(self):
self.label_font = None
def reset_bounding_box(self):
self._bounding_box = [[None, None], [None, None], [None, None]]
self._axis_ticks = [[], [], []]
def draw(self):
if self._render_object:
pgl.glPushAttrib(pgl.GL_ENABLE_BIT | pgl.GL_POLYGON_BIT | pgl.GL_DEPTH_BUFFER_BIT)
if self._overlay:
pgl.glDisable(pgl.GL_DEPTH_TEST)
self._render_object.draw()
pgl.glPopAttrib()
def adjust_bounds(self, child_bounds):
b = self._bounding_box
c = child_bounds
for i in range(3):
if abs(c[i][0]) is S.Infinity or abs(c[i][1]) is S.Infinity:
continue
b[i][0] = c[i][0] if b[i][0] is None else min([b[i][0], c[i][0]])
b[i][1] = c[i][1] if b[i][1] is None else max([b[i][1], c[i][1]])
self._bounding_box = b
self._recalculate_axis_ticks(i)
def _recalculate_axis_ticks(self, axis):
b = self._bounding_box
if b[axis][0] is None or b[axis][1] is None:
self._axis_ticks[axis] = []
else:
self._axis_ticks[axis] = strided_range(b[axis][0], b[axis][1],
self._stride[axis])
def toggle_visible(self):
self.visible = not self.visible
def toggle_colors(self):
self._colored = not self._colored
class PlotAxesBase(PlotObject):
def __init__(self, parent_axes):
self._p = parent_axes
def draw(self):
color = [([0.2, 0.1, 0.3], [0.2, 0.1, 0.3], [0.2, 0.1, 0.3]),
([0.9, 0.3, 0.5], [0.5, 1.0, 0.5], [0.3, 0.3, 0.9])][self._p._colored]
self.draw_background(color)
self.draw_axis(2, color[2])
self.draw_axis(1, color[1])
self.draw_axis(0, color[0])
def draw_background(self, color):
pass # optional
def draw_axis(self, axis, color):
raise NotImplementedError()
def draw_text(self, text, position, color, scale=1.0):
if len(color) == 3:
color = (color[0], color[1], color[2], 1.0)
if self._p.label_font is None:
self._p.label_font = font.load(self._p.font_face,
self._p.font_size,
bold=True, italic=False)
label = font.Text(self._p.label_font, text,
color=color,
valign=font.Text.BASELINE,
halign=font.Text.CENTER)
pgl.glPushMatrix()
pgl.glTranslatef(*position)
billboard_matrix()
scale_factor = 0.005 * scale
pgl.glScalef(scale_factor, scale_factor, scale_factor)
pgl.glColor4f(0, 0, 0, 0)
label.draw()
pgl.glPopMatrix()
def draw_line(self, v, color):
o = self._p._origin
pgl.glBegin(pgl.GL_LINES)
pgl.glColor3f(*color)
pgl.glVertex3f(v[0][0] + o[0], v[0][1] + o[1], v[0][2] + o[2])
pgl.glVertex3f(v[1][0] + o[0], v[1][1] + o[1], v[1][2] + o[2])
pgl.glEnd()
class PlotAxesOrdinate(PlotAxesBase):
def __init__(self, parent_axes):
super().__init__(parent_axes)
def draw_axis(self, axis, color):
ticks = self._p._axis_ticks[axis]
radius = self._p._tick_length / 2.0
if len(ticks) < 2:
return
# calculate the vector for this axis
axis_lines = [[0, 0, 0], [0, 0, 0]]
axis_lines[0][axis], axis_lines[1][axis] = ticks[0], ticks[-1]
axis_vector = vec_sub(axis_lines[1], axis_lines[0])
# calculate angle to the z direction vector
pos_z = get_direction_vectors()[2]
d = abs(dot_product(axis_vector, pos_z))
d = d / vec_mag(axis_vector)
# don't draw labels if we're looking down the axis
labels_visible = abs(d - 1.0) > 0.02
# draw the ticks and labels
for tick in ticks:
self.draw_tick_line(axis, color, radius, tick, labels_visible)
# draw the axis line and labels
self.draw_axis_line(axis, color, ticks[0], ticks[-1], labels_visible)
def draw_axis_line(self, axis, color, a_min, a_max, labels_visible):
axis_line = [[0, 0, 0], [0, 0, 0]]
axis_line[0][axis], axis_line[1][axis] = a_min, a_max
self.draw_line(axis_line, color)
if labels_visible:
self.draw_axis_line_labels(axis, color, axis_line)
def draw_axis_line_labels(self, axis, color, axis_line):
if not self._p._label_axes:
return
axis_labels = [axis_line[0][::], axis_line[1][::]]
axis_labels[0][axis] -= 0.3
axis_labels[1][axis] += 0.3
a_str = ['X', 'Y', 'Z'][axis]
self.draw_text("-" + a_str, axis_labels[0], color)
self.draw_text("+" + a_str, axis_labels[1], color)
def draw_tick_line(self, axis, color, radius, tick, labels_visible):
tick_axis = {0: 1, 1: 0, 2: 1}[axis]
tick_line = [[0, 0, 0], [0, 0, 0]]
tick_line[0][axis] = tick_line[1][axis] = tick
tick_line[0][tick_axis], tick_line[1][tick_axis] = -radius, radius
self.draw_line(tick_line, color)
if labels_visible:
self.draw_tick_line_label(axis, color, radius, tick)
def draw_tick_line_label(self, axis, color, radius, tick):
if not self._p._label_axes:
return
tick_label_vector = [0, 0, 0]
tick_label_vector[axis] = tick
tick_label_vector[{0: 1, 1: 0, 2: 1}[axis]] = [-1, 1, 1][
axis] * radius * 3.5
self.draw_text(str(tick), tick_label_vector, color, scale=0.5)
class PlotAxesFrame(PlotAxesBase):
def __init__(self, parent_axes):
super().__init__(parent_axes)
def draw_background(self, color):
pass
def draw_axis(self, axis, color):
raise NotImplementedError()
|
3a26e42392348bafcda457751c4978f1dfd1e0f456a1f528392e0761006454e7 | """
Interval Arithmetic for plotting.
This module does not implement interval arithmetic accurately and
hence cannot be used for purposes other than plotting. If you want
to use interval arithmetic, use mpmath's interval arithmetic.
The module implements interval arithmetic using numpy and
python floating points. The rounding up and down is not handled
and hence this is not an accurate implementation of interval
arithmetic.
The module uses numpy for speed which cannot be achieved with mpmath.
"""
# Q: Why use numpy? Why not simply use mpmath's interval arithmetic?
# A: mpmath's interval arithmetic simulates a floating point unit
# and hence is slow, while numpy evaluations are orders of magnitude
# faster.
# Q: Why create a separate class for intervals? Why not use SymPy's
# Interval Sets?
# A: The functionalities that will be required for plotting is quite
# different from what Interval Sets implement.
# Q: Why is rounding up and down according to IEEE754 not handled?
# A: It is not possible to do it in both numpy and python. An external
# library has to used, which defeats the whole purpose i.e., speed. Also
# rounding is handled for very few functions in those libraries.
# Q Will my plots be affected?
# A It will not affect most of the plots. The interval arithmetic
# module based suffers the same problems as that of floating point
# arithmetic.
from sympy.core.logic import fuzzy_and
from sympy.simplify.simplify import nsimplify
from .interval_membership import intervalMembership
class interval:
""" Represents an interval containing floating points as start and
end of the interval
The is_valid variable tracks whether the interval obtained as the
result of the function is in the domain and is continuous.
- True: Represents the interval result of a function is continuous and
in the domain of the function.
- False: The interval argument of the function was not in the domain of
the function, hence the is_valid of the result interval is False
- None: The function was not continuous over the interval or
the function's argument interval is partly in the domain of the
function
A comparison between an interval and a real number, or a
comparison between two intervals may return ``intervalMembership``
of two 3-valued logic values.
"""
def __init__(self, *args, is_valid=True, **kwargs):
self.is_valid = is_valid
if len(args) == 1:
if isinstance(args[0], interval):
self.start, self.end = args[0].start, args[0].end
else:
self.start = float(args[0])
self.end = float(args[0])
elif len(args) == 2:
if args[0] < args[1]:
self.start = float(args[0])
self.end = float(args[1])
else:
self.start = float(args[1])
self.end = float(args[0])
else:
raise ValueError("interval takes a maximum of two float values "
"as arguments")
@property
def mid(self):
return (self.start + self.end) / 2.0
@property
def width(self):
return self.end - self.start
def __repr__(self):
return "interval(%f, %f)" % (self.start, self.end)
def __str__(self):
return "[%f, %f]" % (self.start, self.end)
def __lt__(self, other):
if isinstance(other, (int, float)):
if self.end < other:
return intervalMembership(True, self.is_valid)
elif self.start > other:
return intervalMembership(False, self.is_valid)
else:
return intervalMembership(None, self.is_valid)
elif isinstance(other, interval):
valid = fuzzy_and([self.is_valid, other.is_valid])
if self.end < other. start:
return intervalMembership(True, valid)
if self.start > other.end:
return intervalMembership(False, valid)
return intervalMembership(None, valid)
else:
return NotImplemented
def __gt__(self, other):
if isinstance(other, (int, float)):
if self.start > other:
return intervalMembership(True, self.is_valid)
elif self.end < other:
return intervalMembership(False, self.is_valid)
else:
return intervalMembership(None, self.is_valid)
elif isinstance(other, interval):
return other.__lt__(self)
else:
return NotImplemented
def __eq__(self, other):
if isinstance(other, (int, float)):
if self.start == other and self.end == other:
return intervalMembership(True, self.is_valid)
if other in self:
return intervalMembership(None, self.is_valid)
else:
return intervalMembership(False, self.is_valid)
if isinstance(other, interval):
valid = fuzzy_and([self.is_valid, other.is_valid])
if self.start == other.start and self.end == other.end:
return intervalMembership(True, valid)
elif self.__lt__(other)[0] is not None:
return intervalMembership(False, valid)
else:
return intervalMembership(None, valid)
else:
return NotImplemented
def __ne__(self, other):
if isinstance(other, (int, float)):
if self.start == other and self.end == other:
return intervalMembership(False, self.is_valid)
if other in self:
return intervalMembership(None, self.is_valid)
else:
return intervalMembership(True, self.is_valid)
if isinstance(other, interval):
valid = fuzzy_and([self.is_valid, other.is_valid])
if self.start == other.start and self.end == other.end:
return intervalMembership(False, valid)
if not self.__lt__(other)[0] is None:
return intervalMembership(True, valid)
return intervalMembership(None, valid)
else:
return NotImplemented
def __le__(self, other):
if isinstance(other, (int, float)):
if self.end <= other:
return intervalMembership(True, self.is_valid)
if self.start > other:
return intervalMembership(False, self.is_valid)
else:
return intervalMembership(None, self.is_valid)
if isinstance(other, interval):
valid = fuzzy_and([self.is_valid, other.is_valid])
if self.end <= other.start:
return intervalMembership(True, valid)
if self.start > other.end:
return intervalMembership(False, valid)
return intervalMembership(None, valid)
else:
return NotImplemented
def __ge__(self, other):
if isinstance(other, (int, float)):
if self.start >= other:
return intervalMembership(True, self.is_valid)
elif self.end < other:
return intervalMembership(False, self.is_valid)
else:
return intervalMembership(None, self.is_valid)
elif isinstance(other, interval):
return other.__le__(self)
def __add__(self, other):
if isinstance(other, (int, float)):
if self.is_valid:
return interval(self.start + other, self.end + other)
else:
start = self.start + other
end = self.end + other
return interval(start, end, is_valid=self.is_valid)
elif isinstance(other, interval):
start = self.start + other.start
end = self.end + other.end
valid = fuzzy_and([self.is_valid, other.is_valid])
return interval(start, end, is_valid=valid)
else:
return NotImplemented
__radd__ = __add__
def __sub__(self, other):
if isinstance(other, (int, float)):
start = self.start - other
end = self.end - other
return interval(start, end, is_valid=self.is_valid)
elif isinstance(other, interval):
start = self.start - other.end
end = self.end - other.start
valid = fuzzy_and([self.is_valid, other.is_valid])
return interval(start, end, is_valid=valid)
else:
return NotImplemented
def __rsub__(self, other):
if isinstance(other, (int, float)):
start = other - self.end
end = other - self.start
return interval(start, end, is_valid=self.is_valid)
elif isinstance(other, interval):
return other.__sub__(self)
else:
return NotImplemented
def __neg__(self):
if self.is_valid:
return interval(-self.end, -self.start)
else:
return interval(-self.end, -self.start, is_valid=self.is_valid)
def __mul__(self, other):
if isinstance(other, interval):
if self.is_valid is False or other.is_valid is False:
return interval(-float('inf'), float('inf'), is_valid=False)
elif self.is_valid is None or other.is_valid is None:
return interval(-float('inf'), float('inf'), is_valid=None)
else:
inters = []
inters.append(self.start * other.start)
inters.append(self.end * other.start)
inters.append(self.start * other.end)
inters.append(self.end * other.end)
start = min(inters)
end = max(inters)
return interval(start, end)
elif isinstance(other, (int, float)):
return interval(self.start*other, self.end*other, is_valid=self.is_valid)
else:
return NotImplemented
__rmul__ = __mul__
def __contains__(self, other):
if isinstance(other, (int, float)):
return self.start <= other and self.end >= other
else:
return self.start <= other.start and other.end <= self.end
def __rtruediv__(self, other):
if isinstance(other, (int, float)):
other = interval(other)
return other.__truediv__(self)
elif isinstance(other, interval):
return other.__truediv__(self)
else:
return NotImplemented
def __truediv__(self, other):
# Both None and False are handled
if not self.is_valid:
# Don't divide as the value is not valid
return interval(-float('inf'), float('inf'), is_valid=self.is_valid)
if isinstance(other, (int, float)):
if other == 0:
# Divide by zero encountered. valid nowhere
return interval(-float('inf'), float('inf'), is_valid=False)
else:
return interval(self.start / other, self.end / other)
elif isinstance(other, interval):
if other.is_valid is False or self.is_valid is False:
return interval(-float('inf'), float('inf'), is_valid=False)
elif other.is_valid is None or self.is_valid is None:
return interval(-float('inf'), float('inf'), is_valid=None)
else:
# denominator contains both signs, i.e. being divided by zero
# return the whole real line with is_valid = None
if 0 in other:
return interval(-float('inf'), float('inf'), is_valid=None)
# denominator negative
this = self
if other.end < 0:
this = -this
other = -other
# denominator positive
inters = []
inters.append(this.start / other.start)
inters.append(this.end / other.start)
inters.append(this.start / other.end)
inters.append(this.end / other.end)
start = max(inters)
end = min(inters)
return interval(start, end)
else:
return NotImplemented
def __pow__(self, other):
# Implements only power to an integer.
from .lib_interval import exp, log
if not self.is_valid:
return self
if isinstance(other, interval):
return exp(other * log(self))
elif isinstance(other, (float, int)):
if other < 0:
return 1 / self.__pow__(abs(other))
else:
if int(other) == other:
return _pow_int(self, other)
else:
return _pow_float(self, other)
else:
return NotImplemented
def __rpow__(self, other):
if isinstance(other, (float, int)):
if not self.is_valid:
#Don't do anything
return self
elif other < 0:
if self.width > 0:
return interval(-float('inf'), float('inf'), is_valid=False)
else:
power_rational = nsimplify(self.start)
num, denom = power_rational.as_numer_denom()
if denom % 2 == 0:
return interval(-float('inf'), float('inf'),
is_valid=False)
else:
start = -abs(other)**self.start
end = start
return interval(start, end)
else:
return interval(other**self.start, other**self.end)
elif isinstance(other, interval):
return other.__pow__(self)
else:
return NotImplemented
def __hash__(self):
return hash((self.is_valid, self.start, self.end))
def _pow_float(inter, power):
"""Evaluates an interval raised to a floating point."""
power_rational = nsimplify(power)
num, denom = power_rational.as_numer_denom()
if num % 2 == 0:
start = abs(inter.start)**power
end = abs(inter.end)**power
if start < 0:
ret = interval(0, max(start, end))
else:
ret = interval(start, end)
return ret
elif denom % 2 == 0:
if inter.end < 0:
return interval(-float('inf'), float('inf'), is_valid=False)
elif inter.start < 0:
return interval(0, inter.end**power, is_valid=None)
else:
return interval(inter.start**power, inter.end**power)
else:
if inter.start < 0:
start = -abs(inter.start)**power
else:
start = inter.start**power
if inter.end < 0:
end = -abs(inter.end)**power
else:
end = inter.end**power
return interval(start, end, is_valid=inter.is_valid)
def _pow_int(inter, power):
"""Evaluates an interval raised to an integer power"""
power = int(power)
if power & 1:
return interval(inter.start**power, inter.end**power)
else:
if inter.start < 0 and inter.end > 0:
start = 0
end = max(inter.start**power, inter.end**power)
return interval(start, end)
else:
return interval(inter.start**power, inter.end**power)
|
362b2347ecae05125f42f8dc6f6d2c913477ca27765373212875bab32f11135d | from sympy.external.importtools import import_module
disabled = False
# if pyglet.gl fails to import, e.g. opengl is missing, we disable the tests
pyglet_gl = import_module("pyglet.gl", catch=(OSError,))
pyglet_window = import_module("pyglet.window", catch=(OSError,))
if not pyglet_gl or not pyglet_window:
disabled = True
from sympy.core.symbol import symbols
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.trigonometric import (cos, sin)
x, y, z = symbols('x, y, z')
def test_plot_2d():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(x, [x, -5, 5, 4], visible=False)
p.wait_for_calculations()
def test_plot_2d_discontinuous():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(1/x, [x, -1, 1, 2], visible=False)
p.wait_for_calculations()
def test_plot_3d():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(x*y, [x, -5, 5, 5], [y, -5, 5, 5], visible=False)
p.wait_for_calculations()
def test_plot_3d_discontinuous():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(1/x, [x, -3, 3, 6], [y, -1, 1, 1], visible=False)
p.wait_for_calculations()
def test_plot_2d_polar():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(1/x, [x, -1, 1, 4], 'mode=polar', visible=False)
p.wait_for_calculations()
def test_plot_3d_cylinder():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(
1/y, [x, 0, 6.282, 4], [y, -1, 1, 4], 'mode=polar;style=solid',
visible=False)
p.wait_for_calculations()
def test_plot_3d_spherical():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(
1, [x, 0, 6.282, 4], [y, 0, 3.141,
4], 'mode=spherical;style=wireframe',
visible=False)
p.wait_for_calculations()
def test_plot_2d_parametric():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(sin(x), cos(x), [x, 0, 6.282, 4], visible=False)
p.wait_for_calculations()
def test_plot_3d_parametric():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(sin(x), cos(x), x/5.0, [x, 0, 6.282, 4], visible=False)
p.wait_for_calculations()
def _test_plot_log():
from sympy.plotting.pygletplot import PygletPlot
p = PygletPlot(log(x), [x, 0, 6.282, 4], 'mode=polar', visible=False)
p.wait_for_calculations()
def test_plot_integral():
# Make sure it doesn't treat x as an independent variable
from sympy.plotting.pygletplot import PygletPlot
from sympy.integrals.integrals import Integral
p = PygletPlot(Integral(z*x, (x, 1, z), (z, 1, y)), visible=False)
p.wait_for_calculations()
|
59b070b94af577832ebd36e5bf85a4ec5927d70ce0e303f501227ac72efd9e01 | #!/usr/bin/env python3
import os
from os.path import dirname, join, basename, normpath
from os import chdir
import shutil
from helpers import run
ROOTDIR = dirname(dirname(__file__))
DOCSDIR = join(ROOTDIR, 'doc')
def main(version, outputdir):
os.makedirs(outputdir, exist_ok=True)
build_html(DOCSDIR, outputdir, version)
build_latex(DOCSDIR, outputdir, version)
def build_html(docsdir, outputdir, version):
run('make', 'clean', cwd=docsdir)
run('make', 'html', cwd=docsdir)
builddir = join(docsdir, '_build')
docsname = 'sympy-docs-html-%s' % (version,)
zipname = docsname + '.zip'
cwd = os.getcwd()
try:
chdir(builddir)
shutil.move('html', docsname)
run('zip', '-9lr', zipname, docsname)
finally:
chdir(cwd)
shutil.move(join(builddir, zipname), join(outputdir, zipname))
def build_latex(docsdir, outputdir, version):
run('make', 'clean', cwd=docsdir)
run('make', 'latexpdf', cwd=docsdir)
srcfilename = 'sympy-%s.pdf' % (version,)
dstfilename = 'sympy-docs-pdf-%s.pdf' % (version,)
src = join('doc', '_build', 'latex', srcfilename)
dst = join(outputdir, dstfilename)
shutil.copyfile(src, dst)
if __name__ == "__main__":
import sys
main(*sys.argv[1:])
|
4dd6f1f61276ea1f60f6dc8beb1a04c8c25559512f003152952b586fd6ad23c4 | """
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, 7):
raise ImportError("Python version 3.7 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,
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, as_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__ = [
# 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', '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',
'as_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',
))
|
5fbb51697e7257cd5d787d6d7ed090013f5e88136eea71242dc280a36d235efe | #
# 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', 'sympylive', 'sphinx_reredirects',
'sphinx.ext.graphviz', 'matplotlib.sphinxext.plot_directive',
'myst_parser'
]
redirects = {
"install.rst": "guides/getting_started/install.html",
"documentation-style-guide.rst": "guides/contributing/documentation-style-guide.html",
"gotchas.rst": "explanation/gotchas.html",
"special_topics/classification.rst": "explanation/classification.html",
"special_topics/finite_diff_derivatives.rst": "explanation/finite_diff_derivatives.html",
"special_topics/intro.rst": "explanation/index.html",
"special_topics/index.rst": "explanation/index.html",
"modules/index.rst": "reference/public/index.html",
"modules/physics/index.rst": "reference/physics/index.html",
}
# 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 = 2
# 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.
pygments_style = 'sphinx'
# 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_logo = '_static/sympylogo.png'
html_favicon = '../_build/logo/sympy-notailtext-favicon.ico'
# See http://www.sphinx-doc.org/en/master/theming.html#builtin-themes
# 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'
# 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'
# 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'
# 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
|
6e36ec0088cb874bcb07e728e1a6ee82b67db04a1d12718cac1c37550fdb912f | """
Continuous Random Variables - Prebuilt variables
Contains
========
Arcsin
Benini
Beta
BetaNoncentral
BetaPrime
BoundedPareto
Cauchy
Chi
ChiNoncentral
ChiSquared
Dagum
Erlang
ExGaussian
Exponential
ExponentialPower
FDistribution
FisherZ
Frechet
Gamma
GammaInverse
Gumbel
Gompertz
Kumaraswamy
Laplace
Levy
LogCauchy
Logistic
LogLogistic
LogitNormal
LogNormal
Lomax
Maxwell
Moyal
Nakagami
Normal
Pareto
PowerFunction
QuadraticU
RaisedCosine
Rayleigh
Reciprocal
ShiftedGompertz
StudentT
Trapezoidal
Triangular
Uniform
UniformSum
VonMises
Wald
Weibull
WignerSemicircle
"""
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.trigonometric import (atan, cos, sin, tan)
from sympy.functions.special.bessel import (besseli, besselj, besselk)
from sympy.functions.special.beta_functions import beta as beta_fn
from sympy.concrete.summations import Sum
from sympy.core.basic import Basic
from sympy.core.function import Lambda
from sympy.core.numbers import (I, Rational, pi)
from sympy.core.relational import (Eq, Ne)
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.complexes import (Abs, sign)
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.hyperbolic import sinh
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import asin
from sympy.functions.special.error_functions import (erf, erfc, erfi, erfinv, expint)
from sympy.functions.special.gamma_functions import (gamma, lowergamma, uppergamma)
from sympy.functions.special.hyper import hyper
from sympy.integrals.integrals import integrate
from sympy.logic.boolalg import And
from sympy.sets.sets import Interval
from sympy.matrices import MatrixBase
from sympy.stats.crv import SingleContinuousPSpace, SingleContinuousDistribution
from sympy.stats.rv import _value_check, is_random
oo = S.Infinity
__all__ = ['ContinuousRV',
'Arcsin',
'Benini',
'Beta',
'BetaNoncentral',
'BetaPrime',
'BoundedPareto',
'Cauchy',
'Chi',
'ChiNoncentral',
'ChiSquared',
'Dagum',
'Erlang',
'ExGaussian',
'Exponential',
'ExponentialPower',
'FDistribution',
'FisherZ',
'Frechet',
'Gamma',
'GammaInverse',
'Gompertz',
'Gumbel',
'Kumaraswamy',
'Laplace',
'Levy',
'LogCauchy',
'Logistic',
'LogLogistic',
'LogitNormal',
'LogNormal',
'Lomax',
'Maxwell',
'Moyal',
'Nakagami',
'Normal',
'GaussianInverse',
'Pareto',
'PowerFunction',
'QuadraticU',
'RaisedCosine',
'Rayleigh',
'Reciprocal',
'StudentT',
'ShiftedGompertz',
'Trapezoidal',
'Triangular',
'Uniform',
'UniformSum',
'VonMises',
'Wald',
'Weibull',
'WignerSemicircle',
]
@is_random.register(MatrixBase)
def _(x):
return any(is_random(i) for i in x)
def rv(symbol, cls, args, **kwargs):
args = list(map(sympify, args))
dist = cls(*args)
if kwargs.pop('check', True):
dist.check(*args)
pspace = SingleContinuousPSpace(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 ContinuousDistributionHandmade(SingleContinuousDistribution):
_argnames = ('pdf',)
def __new__(cls, pdf, set=Interval(-oo, oo)):
return Basic.__new__(cls, pdf, set)
@property
def set(self):
return self.args[1]
@staticmethod
def check(pdf, set):
x = Dummy('x')
val = integrate(pdf(x), (x, set))
_value_check(Eq(val, 1) != S.false, "The pdf on the given set is incorrect.")
def ContinuousRV(symbol, density, set=Interval(-oo, oo), **kwargs):
"""
Create a Continuous Random Variable given the following:
Parameters
==========
symbol : Symbol
Represents name of the random variable.
density : Expression containing symbol
Represents probability density function.
set : set/Interval
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.
Returns
=======
RandomSymbol
Many common continuous random variable types are already implemented.
This function should be necessary only very rarely.
Examples
========
>>> from sympy import Symbol, sqrt, exp, pi
>>> from sympy.stats import ContinuousRV, P, E
>>> x = Symbol("x")
>>> pdf = sqrt(2)*exp(-x**2/2)/(2*sqrt(pi)) # Normal distribution
>>> X = ContinuousRV(x, pdf)
>>> E(X)
0
>>> P(X>0)
1/2
"""
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, ContinuousDistributionHandmade, (pdf, set), **kwargs)
########################################
# Continuous Probability Distributions #
########################################
#-------------------------------------------------------------------------------
# Arcsin distribution ----------------------------------------------------------
class ArcsinDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
@property
def set(self):
return Interval(self.a, self.b)
def pdf(self, x):
a, b = self.a, self.b
return 1/(pi*sqrt((x - a)*(b - x)))
def _cdf(self, x):
a, b = self.a, self.b
return Piecewise(
(S.Zero, x < a),
(2*asin(sqrt((x - a)/(b - a)))/pi, x <= b),
(S.One, True))
def Arcsin(name, a=0, b=1):
r"""
Create a Continuous Random Variable with an arcsin distribution.
The density of the arcsin distribution is given by
.. math::
f(x) := \frac{1}{\pi\sqrt{(x-a)(b-x)}}
with :math:`x \in (a,b)`. It must hold that :math:`-\infty < a < b < \infty`.
Parameters
==========
a : Real number, the left interval boundary
b : Real number, the right interval boundary
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Arcsin, density, cdf
>>> from sympy import Symbol
>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")
>>> X = Arcsin("x", a, b)
>>> density(X)(z)
1/(pi*sqrt((-a + z)*(b - z)))
>>> cdf(X)(z)
Piecewise((0, a > z),
(2*asin(sqrt((-a + z)/(-a + b)))/pi, b >= z),
(1, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Arcsine_distribution
"""
return rv(name, ArcsinDistribution, (a, b))
#-------------------------------------------------------------------------------
# Benini distribution ----------------------------------------------------------
class BeniniDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta', 'sigma')
@staticmethod
def check(alpha, beta, sigma):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
_value_check(sigma > 0, "Scale parameter Sigma must be positive.")
@property
def set(self):
return Interval(self.sigma, oo)
def pdf(self, x):
alpha, beta, sigma = self.alpha, self.beta, self.sigma
return (exp(-alpha*log(x/sigma) - beta*log(x/sigma)**2)
*(alpha/x + 2*beta*log(x/sigma)/x))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function of the '
'Benini distribution does not exist.')
def Benini(name, alpha, beta, sigma):
r"""
Create a Continuous Random Variable with a Benini distribution.
The density of the Benini distribution is given by
.. math::
f(x) := e^{-\alpha\log{\frac{x}{\sigma}}
-\beta\log^2\left[{\frac{x}{\sigma}}\right]}
\left(\frac{\alpha}{x}+\frac{2\beta\log{\frac{x}{\sigma}}}{x}\right)
This is a heavy-tailed distribution and is also known as the log-Rayleigh
distribution.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
sigma : Real number, `\sigma > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Benini, density, cdf
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> X = Benini("x", alpha, beta, sigma)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/ / z \\ / z \ 2/ z \
| 2*beta*log|-----|| - alpha*log|-----| - beta*log |-----|
|alpha \sigma/| \sigma/ \sigma/
|----- + -----------------|*e
\ z z /
>>> cdf(X)(z)
Piecewise((1 - exp(-alpha*log(z/sigma) - beta*log(z/sigma)**2), sigma <= z),
(0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Benini_distribution
.. [2] http://reference.wolfram.com/legacy/v8/ref/BeniniDistribution.html
"""
return rv(name, BeniniDistribution, (alpha, beta, sigma))
#-------------------------------------------------------------------------------
# Beta distribution ------------------------------------------------------------
class BetaDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
set = Interval(0, 1)
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
def pdf(self, x):
alpha, beta = self.alpha, self.beta
return x**(alpha - 1) * (1 - x)**(beta - 1) / beta_fn(alpha, beta)
def _characteristic_function(self, t):
return hyper((self.alpha,), (self.alpha + self.beta,), I*t)
def _moment_generating_function(self, t):
return hyper((self.alpha,), (self.alpha + self.beta,), t)
def Beta(name, alpha, beta):
r"""
Create a Continuous Random Variable with a Beta distribution.
The density of the Beta distribution is given by
.. math::
f(x) := \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}
with :math:`x \in [0,1]`.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Beta, density, E, variance
>>> from sympy import Symbol, simplify, pprint, factor
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")
>>> X = Beta("x", alpha, beta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
alpha - 1 beta - 1
z *(1 - z)
--------------------------
B(alpha, beta)
>>> simplify(E(X))
alpha/(alpha + beta)
>>> factor(simplify(variance(X)))
alpha*beta/((alpha + beta)**2*(alpha + beta + 1))
References
==========
.. [1] https://en.wikipedia.org/wiki/Beta_distribution
.. [2] http://mathworld.wolfram.com/BetaDistribution.html
"""
return rv(name, BetaDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
# Noncentral Beta distribution ------------------------------------------------------------
class BetaNoncentralDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta', 'lamda')
set = Interval(0, 1)
@staticmethod
def check(alpha, beta, lamda):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
_value_check(lamda >= 0, "Noncentrality parameter Lambda must be positive")
def pdf(self, x):
alpha, beta, lamda = self.alpha, self.beta, self.lamda
k = Dummy("k")
return Sum(exp(-lamda / 2) * (lamda / 2)**k * x**(alpha + k - 1) *(
1 - x)**(beta - 1) / (factorial(k) * beta_fn(alpha + k, beta)), (k, 0, oo))
def BetaNoncentral(name, alpha, beta, lamda):
r"""
Create a Continuous Random Variable with a Type I Noncentral Beta distribution.
The density of the Noncentral Beta distribution is given by
.. math::
f(x) := \sum_{k=0}^\infty e^{-\lambda/2}\frac{(\lambda/2)^k}{k!}
\frac{x^{\alpha+k-1}(1-x)^{\beta-1}}{\mathrm{B}(\alpha+k,\beta)}
with :math:`x \in [0,1]`.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
lamda: Real number, `\lambda >= 0`, noncentrality parameter
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import BetaNoncentral, density, cdf
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> lamda = Symbol("lamda", nonnegative=True)
>>> z = Symbol("z")
>>> X = BetaNoncentral("x", alpha, beta, lamda)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
oo
_____
\ `
\ -lamda
\ k -------
\ k + alpha - 1 /lamda\ beta - 1 2
) z *|-----| *(1 - z) *e
/ \ 2 /
/ ------------------------------------------------
/ B(k + alpha, beta)*k!
/____,
k = 0
Compute cdf with specific 'x', 'alpha', 'beta' and 'lamda' values as follows :
>>> cdf(BetaNoncentral("x", 1, 1, 1), evaluate=False)(2).doit()
2*exp(1/2)
The argument evaluate=False prevents an attempt at evaluation
of the sum for general x, before the argument 2 is passed.
References
==========
.. [1] https://en.wikipedia.org/wiki/Noncentral_beta_distribution
.. [2] https://reference.wolfram.com/language/ref/NoncentralBetaDistribution.html
"""
return rv(name, BetaNoncentralDistribution, (alpha, beta, lamda))
#-------------------------------------------------------------------------------
# Beta prime distribution ------------------------------------------------------
class BetaPrimeDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Shape parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
set = Interval(0, oo)
def pdf(self, x):
alpha, beta = self.alpha, self.beta
return x**(alpha - 1)*(1 + x)**(-alpha - beta)/beta_fn(alpha, beta)
def BetaPrime(name, alpha, beta):
r"""
Create a continuous random variable with a Beta prime distribution.
The density of the Beta prime distribution is given by
.. math::
f(x) := \frac{x^{\alpha-1} (1+x)^{-\alpha -\beta}}{B(\alpha,\beta)}
with :math:`x > 0`.
Parameters
==========
alpha : Real number, `\alpha > 0`, a shape
beta : Real number, `\beta > 0`, a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import BetaPrime, density
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")
>>> X = BetaPrime("x", alpha, beta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
alpha - 1 -alpha - beta
z *(z + 1)
-------------------------------
B(alpha, beta)
References
==========
.. [1] https://en.wikipedia.org/wiki/Beta_prime_distribution
.. [2] http://mathworld.wolfram.com/BetaPrimeDistribution.html
"""
return rv(name, BetaPrimeDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
# Bounded Pareto Distribution --------------------------------------------------
class BoundedParetoDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'left', 'right')
@property
def set(self):
return Interval(self.left, self.right)
@staticmethod
def check(alpha, left, right):
_value_check (alpha.is_positive, "Shape must be positive.")
_value_check (left.is_positive, "Left value should be positive.")
_value_check (right > left, "Right should be greater than left.")
def pdf(self, x):
alpha, left, right = self.alpha, self.left, self.right
num = alpha * (left**alpha) * x**(- alpha -1)
den = 1 - (left/right)**alpha
return num/den
def BoundedPareto(name, alpha, left, right):
r"""
Create a continuous random variable with a Bounded Pareto distribution.
The density of the Bounded Pareto distribution is given by
.. math::
f(x) := \frac{\alpha L^{\alpha}x^{-\alpha-1}}{1-(\frac{L}{H})^{\alpha}}
Parameters
==========
alpha : Real Number, `alpha > 0`
Shape parameter
left : Real Number, `left > 0`
Location parameter
right : Real Number, `right > left`
Location parameter
Examples
========
>>> from sympy.stats import BoundedPareto, density, cdf, E
>>> from sympy import symbols
>>> L, H = symbols('L, H', positive=True)
>>> X = BoundedPareto('X', 2, L, H)
>>> x = symbols('x')
>>> density(X)(x)
2*L**2/(x**3*(1 - L**2/H**2))
>>> cdf(X)(x)
Piecewise((-H**2*L**2/(x**2*(H**2 - L**2)) + H**2/(H**2 - L**2), L <= x), (0, True))
>>> E(X).simplify()
2*H*L/(H + L)
Returns
=======
RandomSymbol
References
==========
.. [1] https://en.wikipedia.org/wiki/Pareto_distribution#Bounded_Pareto_distribution
"""
return rv (name, BoundedParetoDistribution, (alpha, left, right))
# ------------------------------------------------------------------------------
# Cauchy distribution ----------------------------------------------------------
class CauchyDistribution(SingleContinuousDistribution):
_argnames = ('x0', 'gamma')
@staticmethod
def check(x0, gamma):
_value_check(gamma > 0, "Scale parameter Gamma must be positive.")
_value_check(x0.is_real, "Location parameter must be real.")
def pdf(self, x):
return 1/(pi*self.gamma*(1 + ((x - self.x0)/self.gamma)**2))
def _cdf(self, x):
x0, gamma = self.x0, self.gamma
return (1/pi)*atan((x - x0)/gamma) + S.Half
def _characteristic_function(self, t):
return exp(self.x0 * I * t - self.gamma * Abs(t))
def _moment_generating_function(self, t):
raise NotImplementedError("The moment generating function for the "
"Cauchy distribution does not exist.")
def _quantile(self, p):
return self.x0 + self.gamma*tan(pi*(p - S.Half))
def Cauchy(name, x0, gamma):
r"""
Create a continuous random variable with a Cauchy distribution.
The density of the Cauchy distribution is given by
.. math::
f(x) := \frac{1}{\pi \gamma [1 + {(\frac{x-x_0}{\gamma})}^2]}
Parameters
==========
x0 : Real number, the location
gamma : Real number, `\gamma > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Cauchy, density
>>> from sympy import Symbol
>>> x0 = Symbol("x0")
>>> gamma = Symbol("gamma", positive=True)
>>> z = Symbol("z")
>>> X = Cauchy("x", x0, gamma)
>>> density(X)(z)
1/(pi*gamma*(1 + (-x0 + z)**2/gamma**2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Cauchy_distribution
.. [2] http://mathworld.wolfram.com/CauchyDistribution.html
"""
return rv(name, CauchyDistribution, (x0, gamma))
#-------------------------------------------------------------------------------
# Chi distribution -------------------------------------------------------------
class ChiDistribution(SingleContinuousDistribution):
_argnames = ('k',)
@staticmethod
def check(k):
_value_check(k > 0, "Number of degrees of freedom (k) must be positive.")
_value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.")
set = Interval(0, oo)
def pdf(self, x):
return 2**(1 - self.k/2)*x**(self.k - 1)*exp(-x**2/2)/gamma(self.k/2)
def _characteristic_function(self, t):
k = self.k
part_1 = hyper((k/2,), (S.Half,), -t**2/2)
part_2 = I*t*sqrt(2)*gamma((k+1)/2)/gamma(k/2)
part_3 = hyper(((k+1)/2,), (Rational(3, 2),), -t**2/2)
return part_1 + part_2*part_3
def _moment_generating_function(self, t):
k = self.k
part_1 = hyper((k / 2,), (S.Half,), t ** 2 / 2)
part_2 = t * sqrt(2) * gamma((k + 1) / 2) / gamma(k / 2)
part_3 = hyper(((k + 1) / 2,), (S(3) / 2,), t ** 2 / 2)
return part_1 + part_2 * part_3
def Chi(name, k):
r"""
Create a continuous random variable with a Chi distribution.
The density of the Chi distribution is given by
.. math::
f(x) := \frac{2^{1-k/2}x^{k-1}e^{-x^2/2}}{\Gamma(k/2)}
with :math:`x \geq 0`.
Parameters
==========
k : Positive integer, The number of degrees of freedom
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Chi, density, E
>>> from sympy import Symbol, simplify
>>> k = Symbol("k", integer=True)
>>> z = Symbol("z")
>>> X = Chi("x", k)
>>> density(X)(z)
2**(1 - k/2)*z**(k - 1)*exp(-z**2/2)/gamma(k/2)
>>> simplify(E(X))
sqrt(2)*gamma(k/2 + 1/2)/gamma(k/2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Chi_distribution
.. [2] http://mathworld.wolfram.com/ChiDistribution.html
"""
return rv(name, ChiDistribution, (k,))
#-------------------------------------------------------------------------------
# Non-central Chi distribution -------------------------------------------------
class ChiNoncentralDistribution(SingleContinuousDistribution):
_argnames = ('k', 'l')
@staticmethod
def check(k, l):
_value_check(k > 0, "Number of degrees of freedom (k) must be positive.")
_value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.")
_value_check(l > 0, "Shift parameter Lambda must be positive.")
set = Interval(0, oo)
def pdf(self, x):
k, l = self.k, self.l
return exp(-(x**2+l**2)/2)*x**k*l / (l*x)**(k/2) * besseli(k/2-1, l*x)
def ChiNoncentral(name, k, l):
r"""
Create a continuous random variable with a non-central Chi distribution.
Explanation
===========
The density of the non-central Chi distribution is given by
.. math::
f(x) := \frac{e^{-(x^2+\lambda^2)/2} x^k\lambda}
{(\lambda x)^{k/2}} I_{k/2-1}(\lambda x)
with `x \geq 0`. Here, `I_\nu (x)` is the
:ref:`modified Bessel function of the first kind <besseli>`.
Parameters
==========
k : A positive Integer, $k > 0$
The number of degrees of freedom.
lambda : Real number, `\lambda > 0`
Shift parameter.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ChiNoncentral, density
>>> from sympy import Symbol
>>> k = Symbol("k", integer=True)
>>> l = Symbol("l")
>>> z = Symbol("z")
>>> X = ChiNoncentral("x", k, l)
>>> density(X)(z)
l*z**k*exp(-l**2/2 - z**2/2)*besseli(k/2 - 1, l*z)/(l*z)**(k/2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Noncentral_chi_distribution
"""
return rv(name, ChiNoncentralDistribution, (k, l))
#-------------------------------------------------------------------------------
# Chi squared distribution -----------------------------------------------------
class ChiSquaredDistribution(SingleContinuousDistribution):
_argnames = ('k',)
@staticmethod
def check(k):
_value_check(k > 0, "Number of degrees of freedom (k) must be positive.")
_value_check(k.is_integer, "Number of degrees of freedom (k) must be an integer.")
set = Interval(0, oo)
def pdf(self, x):
k = self.k
return 1/(2**(k/2)*gamma(k/2))*x**(k/2 - 1)*exp(-x/2)
def _cdf(self, x):
k = self.k
return Piecewise(
(S.One/gamma(k/2)*lowergamma(k/2, x/2), x >= 0),
(0, True)
)
def _characteristic_function(self, t):
return (1 - 2*I*t)**(-self.k/2)
def _moment_generating_function(self, t):
return (1 - 2*t)**(-self.k/2)
def ChiSquared(name, k):
r"""
Create a continuous random variable with a Chi-squared distribution.
Explanation
===========
The density of the Chi-squared distribution is given by
.. math::
f(x) := \frac{1}{2^{\frac{k}{2}}\Gamma\left(\frac{k}{2}\right)}
x^{\frac{k}{2}-1} e^{-\frac{x}{2}}
with :math:`x \geq 0`.
Parameters
==========
k : Positive integer
The number of degrees of freedom.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ChiSquared, density, E, variance, moment
>>> from sympy import Symbol
>>> k = Symbol("k", integer=True, positive=True)
>>> z = Symbol("z")
>>> X = ChiSquared("x", k)
>>> density(X)(z)
z**(k/2 - 1)*exp(-z/2)/(2**(k/2)*gamma(k/2))
>>> E(X)
k
>>> variance(X)
2*k
>>> moment(X, 3)
k**3 + 6*k**2 + 8*k
References
==========
.. [1] https://en.wikipedia.org/wiki/Chi_squared_distribution
.. [2] http://mathworld.wolfram.com/Chi-SquaredDistribution.html
"""
return rv(name, ChiSquaredDistribution, (k, ))
#-------------------------------------------------------------------------------
# Dagum distribution -----------------------------------------------------------
class DagumDistribution(SingleContinuousDistribution):
_argnames = ('p', 'a', 'b')
set = Interval(0, oo)
@staticmethod
def check(p, a, b):
_value_check(p > 0, "Shape parameter p must be positive.")
_value_check(a > 0, "Shape parameter a must be positive.")
_value_check(b > 0, "Scale parameter b must be positive.")
def pdf(self, x):
p, a, b = self.p, self.a, self.b
return a*p/x*((x/b)**(a*p)/(((x/b)**a + 1)**(p + 1)))
def _cdf(self, x):
p, a, b = self.p, self.a, self.b
return Piecewise(((S.One + (S(x)/b)**-a)**-p, x>=0),
(S.Zero, True))
def Dagum(name, p, a, b):
r"""
Create a continuous random variable with a Dagum distribution.
Explanation
===========
The density of the Dagum distribution is given by
.. math::
f(x) := \frac{a p}{x} \left( \frac{\left(\tfrac{x}{b}\right)^{a p}}
{\left(\left(\tfrac{x}{b}\right)^a + 1 \right)^{p+1}} \right)
with :math:`x > 0`.
Parameters
==========
p : Real number
``p > 0``, a shape.
a : Real number
``a > 0``, a shape.
b : Real number
``b > 0``, a scale.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Dagum, density, cdf
>>> from sympy import Symbol
>>> p = Symbol("p", positive=True)
>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Dagum("x", p, a, b)
>>> density(X)(z)
a*p*(z/b)**(a*p)*((z/b)**a + 1)**(-p - 1)/z
>>> cdf(X)(z)
Piecewise(((1 + (z/b)**(-a))**(-p), z >= 0), (0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Dagum_distribution
"""
return rv(name, DagumDistribution, (p, a, b))
#-------------------------------------------------------------------------------
# Erlang distribution ----------------------------------------------------------
def Erlang(name, k, l):
r"""
Create a continuous random variable with an Erlang distribution.
Explanation
===========
The density of the Erlang distribution is given by
.. math::
f(x) := \frac{\lambda^k x^{k-1} e^{-\lambda x}}{(k-1)!}
with :math:`x \in [0,\infty]`.
Parameters
==========
k : Positive integer
l : Real number, `\lambda > 0`, the rate
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Erlang, density, cdf, E, variance
>>> from sympy import Symbol, simplify, pprint
>>> k = Symbol("k", integer=True, positive=True)
>>> l = Symbol("l", positive=True)
>>> z = Symbol("z")
>>> X = Erlang("x", k, l)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
k k - 1 -l*z
l *z *e
---------------
Gamma(k)
>>> C = cdf(X)(z)
>>> pprint(C, use_unicode=False)
/lowergamma(k, l*z)
|------------------ for z > 0
< Gamma(k)
|
\ 0 otherwise
>>> E(X)
k/l
>>> simplify(variance(X))
k/l**2
References
==========
.. [1] https://en.wikipedia.org/wiki/Erlang_distribution
.. [2] http://mathworld.wolfram.com/ErlangDistribution.html
"""
return rv(name, GammaDistribution, (k, S.One/l))
# -------------------------------------------------------------------------------
# ExGaussian distribution -----------------------------------------------------
class ExGaussianDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'std', 'rate')
set = Interval(-oo, oo)
@staticmethod
def check(mean, std, rate):
_value_check(
std > 0, "Standard deviation of ExGaussian must be positive.")
_value_check(rate > 0, "Rate of ExGaussian must be positive.")
def pdf(self, x):
mean, std, rate = self.mean, self.std, self.rate
term1 = rate/2
term2 = exp(rate * (2 * mean + rate * std**2 - 2*x)/2)
term3 = erfc((mean + rate*std**2 - x)/(sqrt(2)*std))
return term1*term2*term3
def _cdf(self, x):
from sympy.stats import cdf
mean, std, rate = self.mean, self.std, self.rate
u = rate*(x - mean)
v = rate*std
GaussianCDF1 = cdf(Normal('x', 0, v))(u)
GaussianCDF2 = cdf(Normal('x', v**2, v))(u)
return GaussianCDF1 - exp(-u + (v**2/2) + log(GaussianCDF2))
def _characteristic_function(self, t):
mean, std, rate = self.mean, self.std, self.rate
term1 = (1 - I*t/rate)**(-1)
term2 = exp(I*mean*t - std**2*t**2/2)
return term1 * term2
def _moment_generating_function(self, t):
mean, std, rate = self.mean, self.std, self.rate
term1 = (1 - t/rate)**(-1)
term2 = exp(mean*t + std**2*t**2/2)
return term1*term2
def ExGaussian(name, mean, std, rate):
r"""
Create a continuous random variable with an Exponentially modified
Gaussian (EMG) distribution.
Explanation
===========
The density of the exponentially modified Gaussian distribution is given by
.. math::
f(x) := \frac{\lambda}{2}e^{\frac{\lambda}{2}(2\mu+\lambda\sigma^2-2x)}
\text{erfc}(\frac{\mu + \lambda\sigma^2 - x}{\sqrt{2}\sigma})
with $x > 0$. Note that the expected value is `1/\lambda`.
Parameters
==========
mu : A Real number, the mean of Gaussian component
std: A positive Real number,
:math: `\sigma^2 > 0` the variance of Gaussian component
lambda: A positive Real number,
:math: `\lambda > 0` the rate of Exponential component
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ExGaussian, density, cdf, E
>>> from sympy.stats import variance, skewness
>>> from sympy import Symbol, pprint, simplify
>>> mean = Symbol("mu")
>>> std = Symbol("sigma", positive=True)
>>> rate = Symbol("lamda", positive=True)
>>> z = Symbol("z")
>>> X = ExGaussian("x", mean, std, rate)
>>> pprint(density(X)(z), use_unicode=False)
/ 2 \
lamda*\lamda*sigma + 2*mu - 2*z/
--------------------------------- / ___ / 2 \\
2 |\/ 2 *\lamda*sigma + mu - z/|
lamda*e *erfc|-----------------------------|
\ 2*sigma /
----------------------------------------------------------------------------
2
>>> cdf(X)(z)
-(erf(sqrt(2)*(-lamda**2*sigma**2 + lamda*(-mu + z))/(2*lamda*sigma))/2 + 1/2)*exp(lamda**2*sigma**2/2 - lamda*(-mu + z)) + erf(sqrt(2)*(-mu + z)/(2*sigma))/2 + 1/2
>>> E(X)
(lamda*mu + 1)/lamda
>>> simplify(variance(X))
sigma**2 + lamda**(-2)
>>> simplify(skewness(X))
2/(lamda**2*sigma**2 + 1)**(3/2)
References
==========
.. [1] https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution
"""
return rv(name, ExGaussianDistribution, (mean, std, rate))
#-------------------------------------------------------------------------------
# Exponential distribution -----------------------------------------------------
class ExponentialDistribution(SingleContinuousDistribution):
_argnames = ('rate',)
set = Interval(0, oo)
@staticmethod
def check(rate):
_value_check(rate > 0, "Rate must be positive.")
def pdf(self, x):
return self.rate * exp(-self.rate*x)
def _cdf(self, x):
return Piecewise(
(S.One - exp(-self.rate*x), x >= 0),
(0, True),
)
def _characteristic_function(self, t):
rate = self.rate
return rate / (rate - I*t)
def _moment_generating_function(self, t):
rate = self.rate
return rate / (rate - t)
def _quantile(self, p):
return -log(1-p)/self.rate
def Exponential(name, rate):
r"""
Create a continuous random variable with an Exponential distribution.
Explanation
===========
The density of the exponential distribution is given by
.. math::
f(x) := \lambda \exp(-\lambda x)
with $x > 0$. Note that the expected value is `1/\lambda`.
Parameters
==========
rate : A positive Real number, `\lambda > 0`, the rate (or inverse scale/inverse mean)
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Exponential, density, cdf, E
>>> from sympy.stats import variance, std, skewness, quantile
>>> from sympy import Symbol
>>> l = Symbol("lambda", positive=True)
>>> z = Symbol("z")
>>> p = Symbol("p")
>>> X = Exponential("x", l)
>>> density(X)(z)
lambda*exp(-lambda*z)
>>> cdf(X)(z)
Piecewise((1 - exp(-lambda*z), z >= 0), (0, True))
>>> quantile(X)(p)
-log(1 - p)/lambda
>>> E(X)
1/lambda
>>> variance(X)
lambda**(-2)
>>> skewness(X)
2
>>> X = Exponential('x', 10)
>>> density(X)(z)
10*exp(-10*z)
>>> E(X)
1/10
>>> std(X)
1/10
References
==========
.. [1] https://en.wikipedia.org/wiki/Exponential_distribution
.. [2] http://mathworld.wolfram.com/ExponentialDistribution.html
"""
return rv(name, ExponentialDistribution, (rate, ))
# -------------------------------------------------------------------------------
# Exponential Power distribution -----------------------------------------------------
class ExponentialPowerDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'alpha', 'beta')
set = Interval(-oo, oo)
@staticmethod
def check(mu, alpha, beta):
_value_check(alpha > 0, "Scale parameter alpha must be positive.")
_value_check(beta > 0, "Shape parameter beta must be positive.")
def pdf(self, x):
mu, alpha, beta = self.mu, self.alpha, self.beta
num = beta*exp(-(Abs(x - mu)/alpha)**beta)
den = 2*alpha*gamma(1/beta)
return num/den
def _cdf(self, x):
mu, alpha, beta = self.mu, self.alpha, self.beta
num = lowergamma(1/beta, (Abs(x - mu) / alpha)**beta)
den = 2*gamma(1/beta)
return sign(x - mu)*num/den + S.Half
def ExponentialPower(name, mu, alpha, beta):
r"""
Create a Continuous Random Variable with Exponential Power distribution.
This distribution is known also as Generalized Normal
distribution version 1.
Explanation
===========
The density of the Exponential Power distribution is given by
.. math::
f(x) := \frac{\beta}{2\alpha\Gamma(\frac{1}{\beta})}
e^{{-(\frac{|x - \mu|}{\alpha})^{\beta}}}
with :math:`x \in [ - \infty, \infty ]`.
Parameters
==========
mu : Real number
A location.
alpha : Real number,``alpha > 0``
A scale.
beta : Real number, ``beta > 0``
A shape.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ExponentialPower, density, cdf
>>> from sympy import Symbol, pprint
>>> z = Symbol("z")
>>> mu = Symbol("mu")
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> X = ExponentialPower("x", mu, alpha, beta)
>>> pprint(density(X)(z), use_unicode=False)
beta
/|mu - z|\
-|--------|
\ alpha /
beta*e
---------------------
/ 1 \
2*alpha*Gamma|----|
\beta/
>>> cdf(X)(z)
1/2 + lowergamma(1/beta, (Abs(mu - z)/alpha)**beta)*sign(-mu + z)/(2*gamma(1/beta))
References
==========
.. [1] https://reference.wolfram.com/language/ref/ExponentialPowerDistribution.html
.. [2] https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
"""
return rv(name, ExponentialPowerDistribution, (mu, alpha, beta))
#-------------------------------------------------------------------------------
# F distribution ---------------------------------------------------------------
class FDistributionDistribution(SingleContinuousDistribution):
_argnames = ('d1', 'd2')
set = Interval(0, oo)
@staticmethod
def check(d1, d2):
_value_check((d1 > 0, d1.is_integer),
"Degrees of freedom d1 must be positive integer.")
_value_check((d2 > 0, d2.is_integer),
"Degrees of freedom d2 must be positive integer.")
def pdf(self, x):
d1, d2 = self.d1, self.d2
return (sqrt((d1*x)**d1*d2**d2 / (d1*x+d2)**(d1+d2))
/ (x * beta_fn(d1/2, d2/2)))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function for the '
'F-distribution does not exist.')
def FDistribution(name, d1, d2):
r"""
Create a continuous random variable with a F distribution.
Explanation
===========
The density of the F distribution is given by
.. math::
f(x) := \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}
{(d_1 x + d_2)^{d_1 + d_2}}}}
{x \mathrm{B} \left(\frac{d_1}{2}, \frac{d_2}{2}\right)}
with :math:`x > 0`.
Parameters
==========
d1 : `d_1 > 0`, where d_1 is the degrees of freedom (n_1 - 1)
d2 : `d_2 > 0`, where d_2 is the degrees of freedom (n_2 - 1)
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import FDistribution, density
>>> from sympy import Symbol, pprint
>>> d1 = Symbol("d1", positive=True)
>>> d2 = Symbol("d2", positive=True)
>>> z = Symbol("z")
>>> X = FDistribution("x", d1, d2)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
d2
-- ______________________________
2 / d1 -d1 - d2
d2 *\/ (d1*z) *(d1*z + d2)
--------------------------------------
/d1 d2\
z*B|--, --|
\2 2 /
References
==========
.. [1] https://en.wikipedia.org/wiki/F-distribution
.. [2] http://mathworld.wolfram.com/F-Distribution.html
"""
return rv(name, FDistributionDistribution, (d1, d2))
#-------------------------------------------------------------------------------
# Fisher Z distribution --------------------------------------------------------
class FisherZDistribution(SingleContinuousDistribution):
_argnames = ('d1', 'd2')
set = Interval(-oo, oo)
@staticmethod
def check(d1, d2):
_value_check(d1 > 0, "Degree of freedom d1 must be positive.")
_value_check(d2 > 0, "Degree of freedom d2 must be positive.")
def pdf(self, x):
d1, d2 = self.d1, self.d2
return (2*d1**(d1/2)*d2**(d2/2) / beta_fn(d1/2, d2/2) *
exp(d1*x) / (d1*exp(2*x)+d2)**((d1+d2)/2))
def FisherZ(name, d1, d2):
r"""
Create a Continuous Random Variable with an Fisher's Z distribution.
Explanation
===========
The density of the Fisher's Z distribution is given by
.. math::
f(x) := \frac{2d_1^{d_1/2} d_2^{d_2/2}} {\mathrm{B}(d_1/2, d_2/2)}
\frac{e^{d_1z}}{\left(d_1e^{2z}+d_2\right)^{\left(d_1+d_2\right)/2}}
.. TODO - What is the difference between these degrees of freedom?
Parameters
==========
d1 : ``d_1 > 0``
Degree of freedom.
d2 : ``d_2 > 0``
Degree of freedom.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import FisherZ, density
>>> from sympy import Symbol, pprint
>>> d1 = Symbol("d1", positive=True)
>>> d2 = Symbol("d2", positive=True)
>>> z = Symbol("z")
>>> X = FisherZ("x", d1, d2)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
d1 d2
d1 d2 - -- - --
-- -- 2 2
2 2 / 2*z \ d1*z
2*d1 *d2 *\d1*e + d2/ *e
-----------------------------------------
/d1 d2\
B|--, --|
\2 2 /
References
==========
.. [1] https://en.wikipedia.org/wiki/Fisher%27s_z-distribution
.. [2] http://mathworld.wolfram.com/Fishersz-Distribution.html
"""
return rv(name, FisherZDistribution, (d1, d2))
#-------------------------------------------------------------------------------
# Frechet distribution ---------------------------------------------------------
class FrechetDistribution(SingleContinuousDistribution):
_argnames = ('a', 's', 'm')
set = Interval(0, oo)
@staticmethod
def check(a, s, m):
_value_check(a > 0, "Shape parameter alpha must be positive.")
_value_check(s > 0, "Scale parameter s must be positive.")
def __new__(cls, a, s=1, m=0):
a, s, m = list(map(sympify, (a, s, m)))
return Basic.__new__(cls, a, s, m)
def pdf(self, x):
a, s, m = self.a, self.s, self.m
return a/s * ((x-m)/s)**(-1-a) * exp(-((x-m)/s)**(-a))
def _cdf(self, x):
a, s, m = self.a, self.s, self.m
return Piecewise((exp(-((x-m)/s)**(-a)), x >= m),
(S.Zero, True))
def Frechet(name, a, s=1, m=0):
r"""
Create a continuous random variable with a Frechet distribution.
Explanation
===========
The density of the Frechet distribution is given by
.. math::
f(x) := \frac{\alpha}{s} \left(\frac{x-m}{s}\right)^{-1-\alpha}
e^{-(\frac{x-m}{s})^{-\alpha}}
with :math:`x \geq m`.
Parameters
==========
a : Real number, :math:`a \in \left(0, \infty\right)` the shape
s : Real number, :math:`s \in \left(0, \infty\right)` the scale
m : Real number, :math:`m \in \left(-\infty, \infty\right)` the minimum
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Frechet, density, cdf
>>> from sympy import Symbol
>>> a = Symbol("a", positive=True)
>>> s = Symbol("s", positive=True)
>>> m = Symbol("m", real=True)
>>> z = Symbol("z")
>>> X = Frechet("x", a, s, m)
>>> density(X)(z)
a*((-m + z)/s)**(-a - 1)*exp(-1/((-m + z)/s)**a)/s
>>> cdf(X)(z)
Piecewise((exp(-1/((-m + z)/s)**a), m <= z), (0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution
"""
return rv(name, FrechetDistribution, (a, s, m))
#-------------------------------------------------------------------------------
# Gamma distribution -----------------------------------------------------------
class GammaDistribution(SingleContinuousDistribution):
_argnames = ('k', 'theta')
set = Interval(0, oo)
@staticmethod
def check(k, theta):
_value_check(k > 0, "k must be positive")
_value_check(theta > 0, "Theta must be positive")
def pdf(self, x):
k, theta = self.k, self.theta
return x**(k - 1) * exp(-x/theta) / (gamma(k)*theta**k)
def _cdf(self, x):
k, theta = self.k, self.theta
return Piecewise(
(lowergamma(k, S(x)/theta)/gamma(k), x > 0),
(S.Zero, True))
def _characteristic_function(self, t):
return (1 - self.theta*I*t)**(-self.k)
def _moment_generating_function(self, t):
return (1- self.theta*t)**(-self.k)
def Gamma(name, k, theta):
r"""
Create a continuous random variable with a Gamma distribution.
Explanation
===========
The density of the Gamma distribution is given by
.. math::
f(x) := \frac{1}{\Gamma(k) \theta^k} x^{k - 1} e^{-\frac{x}{\theta}}
with :math:`x \in [0,1]`.
Parameters
==========
k : Real number, ``k > 0``, a shape
theta : Real number, `\theta > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Gamma, density, cdf, E, variance
>>> from sympy import Symbol, pprint, simplify
>>> k = Symbol("k", positive=True)
>>> theta = Symbol("theta", positive=True)
>>> z = Symbol("z")
>>> X = Gamma("x", k, theta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
-z
-----
-k k - 1 theta
theta *z *e
---------------------
Gamma(k)
>>> C = cdf(X, meijerg=True)(z)
>>> pprint(C, use_unicode=False)
/ / z \
|k*lowergamma|k, -----|
| \ theta/
<---------------------- for z >= 0
| Gamma(k + 1)
|
\ 0 otherwise
>>> E(X)
k*theta
>>> V = simplify(variance(X))
>>> pprint(V, use_unicode=False)
2
k*theta
References
==========
.. [1] https://en.wikipedia.org/wiki/Gamma_distribution
.. [2] http://mathworld.wolfram.com/GammaDistribution.html
"""
return rv(name, GammaDistribution, (k, theta))
#-------------------------------------------------------------------------------
# Inverse Gamma distribution ---------------------------------------------------
class GammaInverseDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
set = Interval(0, oo)
@staticmethod
def check(a, b):
_value_check(a > 0, "alpha must be positive")
_value_check(b > 0, "beta must be positive")
def pdf(self, x):
a, b = self.a, self.b
return b**a/gamma(a) * x**(-a-1) * exp(-b/x)
def _cdf(self, x):
a, b = self.a, self.b
return Piecewise((uppergamma(a,b/x)/gamma(a), x > 0),
(S.Zero, True))
def _characteristic_function(self, t):
a, b = self.a, self.b
return 2 * (-I*b*t)**(a/2) * besselk(a, sqrt(-4*I*b*t)) / gamma(a)
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function for the '
'gamma inverse distribution does not exist.')
def GammaInverse(name, a, b):
r"""
Create a continuous random variable with an inverse Gamma distribution.
Explanation
===========
The density of the inverse Gamma distribution is given by
.. math::
f(x) := \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1}
\exp\left(\frac{-\beta}{x}\right)
with :math:`x > 0`.
Parameters
==========
a : Real number, `a > 0` a shape
b : Real number, `b > 0` a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import GammaInverse, density, cdf
>>> from sympy import Symbol, pprint
>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = GammaInverse("x", a, b)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
-b
---
a -a - 1 z
b *z *e
---------------
Gamma(a)
>>> cdf(X)(z)
Piecewise((uppergamma(a, b/z)/gamma(a), z > 0), (0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse-gamma_distribution
"""
return rv(name, GammaInverseDistribution, (a, b))
#-------------------------------------------------------------------------------
# Gumbel distribution (Maximum and Minimum) --------------------------------------------------------
class GumbelDistribution(SingleContinuousDistribution):
_argnames = ('beta', 'mu', 'minimum')
set = Interval(-oo, oo)
@staticmethod
def check(beta, mu, minimum):
_value_check(beta > 0, "Scale parameter beta must be positive.")
def pdf(self, x):
beta, mu = self.beta, self.mu
z = (x - mu)/beta
f_max = (1/beta)*exp(-z - exp(-z))
f_min = (1/beta)*exp(z - exp(z))
return Piecewise((f_min, self.minimum), (f_max, not self.minimum))
def _cdf(self, x):
beta, mu = self.beta, self.mu
z = (x - mu)/beta
F_max = exp(-exp(-z))
F_min = 1 - exp(-exp(z))
return Piecewise((F_min, self.minimum), (F_max, not self.minimum))
def _characteristic_function(self, t):
cf_max = gamma(1 - I*self.beta*t) * exp(I*self.mu*t)
cf_min = gamma(1 + I*self.beta*t) * exp(I*self.mu*t)
return Piecewise((cf_min, self.minimum), (cf_max, not self.minimum))
def _moment_generating_function(self, t):
mgf_max = gamma(1 - self.beta*t) * exp(self.mu*t)
mgf_min = gamma(1 + self.beta*t) * exp(self.mu*t)
return Piecewise((mgf_min, self.minimum), (mgf_max, not self.minimum))
def Gumbel(name, beta, mu, minimum=False):
r"""
Create a Continuous Random Variable with Gumbel distribution.
Explanation
===========
The density of the Gumbel distribution is given by
For Maximum
.. math::
f(x) := \dfrac{1}{\beta} \exp \left( -\dfrac{x-\mu}{\beta}
- \exp \left( -\dfrac{x - \mu}{\beta} \right) \right)
with :math:`x \in [ - \infty, \infty ]`.
For Minimum
.. math::
f(x) := \frac{e^{- e^{\frac{- \mu + x}{\beta}} + \frac{- \mu + x}{\beta}}}{\beta}
with :math:`x \in [ - \infty, \infty ]`.
Parameters
==========
mu : Real number, 'mu' is a location
beta : Real number, 'beta > 0' is a scale
minimum : Boolean, by default, False, set to True for enabling minimum distribution
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Gumbel, density, cdf
>>> from sympy import Symbol
>>> x = Symbol("x")
>>> mu = Symbol("mu")
>>> beta = Symbol("beta", positive=True)
>>> X = Gumbel("x", beta, mu)
>>> density(X)(x)
exp(-exp(-(-mu + x)/beta) - (-mu + x)/beta)/beta
>>> cdf(X)(x)
exp(-exp(-(-mu + x)/beta))
References
==========
.. [1] http://mathworld.wolfram.com/GumbelDistribution.html
.. [2] https://en.wikipedia.org/wiki/Gumbel_distribution
.. [3] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_max.html
.. [4] http://www.mathwave.com/help/easyfit/html/analyses/distributions/gumbel_min.html
"""
return rv(name, GumbelDistribution, (beta, mu, minimum))
#-------------------------------------------------------------------------------
# Gompertz distribution --------------------------------------------------------
class GompertzDistribution(SingleContinuousDistribution):
_argnames = ('b', 'eta')
set = Interval(0, oo)
@staticmethod
def check(b, eta):
_value_check(b > 0, "b must be positive")
_value_check(eta > 0, "eta must be positive")
def pdf(self, x):
eta, b = self.eta, self.b
return b*eta*exp(b*x)*exp(eta)*exp(-eta*exp(b*x))
def _cdf(self, x):
eta, b = self.eta, self.b
return 1 - exp(eta)*exp(-eta*exp(b*x))
def _moment_generating_function(self, t):
eta, b = self.eta, self.b
return eta * exp(eta) * expint(t/b, eta)
def Gompertz(name, b, eta):
r"""
Create a Continuous Random Variable with Gompertz distribution.
Explanation
===========
The density of the Gompertz distribution is given by
.. math::
f(x) := b \eta e^{b x} e^{\eta} \exp \left(-\eta e^{bx} \right)
with :math: 'x \in [0, \inf)'.
Parameters
==========
b: Real number, 'b > 0' a scale
eta: Real number, 'eta > 0' a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Gompertz, density
>>> from sympy import Symbol
>>> b = Symbol("b", positive=True)
>>> eta = Symbol("eta", positive=True)
>>> z = Symbol("z")
>>> X = Gompertz("x", b, eta)
>>> density(X)(z)
b*eta*exp(eta)*exp(b*z)*exp(-eta*exp(b*z))
References
==========
.. [1] https://en.wikipedia.org/wiki/Gompertz_distribution
"""
return rv(name, GompertzDistribution, (b, eta))
#-------------------------------------------------------------------------------
# Kumaraswamy distribution -----------------------------------------------------
class KumaraswamyDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
set = Interval(0, oo)
@staticmethod
def check(a, b):
_value_check(a > 0, "a must be positive")
_value_check(b > 0, "b must be positive")
def pdf(self, x):
a, b = self.a, self.b
return a * b * x**(a-1) * (1-x**a)**(b-1)
def _cdf(self, x):
a, b = self.a, self.b
return Piecewise(
(S.Zero, x < S.Zero),
(1 - (1 - x**a)**b, x <= S.One),
(S.One, True))
def Kumaraswamy(name, a, b):
r"""
Create a Continuous Random Variable with a Kumaraswamy distribution.
Explanation
===========
The density of the Kumaraswamy distribution is given by
.. math::
f(x) := a b x^{a-1} (1-x^a)^{b-1}
with :math:`x \in [0,1]`.
Parameters
==========
a : Real number, ``a > 0`` a shape
b : Real number, ``b > 0`` a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Kumaraswamy, density, cdf
>>> from sympy import Symbol, pprint
>>> a = Symbol("a", positive=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Kumaraswamy("x", a, b)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
b - 1
a - 1 / a\
a*b*z *\1 - z /
>>> cdf(X)(z)
Piecewise((0, z < 0), (1 - (1 - z**a)**b, z <= 1), (1, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Kumaraswamy_distribution
"""
return rv(name, KumaraswamyDistribution, (a, b))
#-------------------------------------------------------------------------------
# Laplace distribution ---------------------------------------------------------
class LaplaceDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'b')
set = Interval(-oo, oo)
@staticmethod
def check(mu, b):
_value_check(b > 0, "Scale parameter b must be positive.")
_value_check(mu.is_real, "Location parameter mu should be real")
def pdf(self, x):
mu, b = self.mu, self.b
return 1/(2*b)*exp(-Abs(x - mu)/b)
def _cdf(self, x):
mu, b = self.mu, self.b
return Piecewise(
(S.Half*exp((x - mu)/b), x < mu),
(S.One - S.Half*exp(-(x - mu)/b), x >= mu)
)
def _characteristic_function(self, t):
return exp(self.mu*I*t) / (1 + self.b**2*t**2)
def _moment_generating_function(self, t):
return exp(self.mu*t) / (1 - self.b**2*t**2)
def Laplace(name, mu, b):
r"""
Create a continuous random variable with a Laplace distribution.
Explanation
===========
The density of the Laplace distribution is given by
.. math::
f(x) := \frac{1}{2 b} \exp \left(-\frac{|x-\mu|}b \right)
Parameters
==========
mu : Real number or a list/matrix, the location (mean) or the
location vector
b : Real number or a positive definite matrix, representing a scale
or the covariance matrix.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Laplace, density, cdf
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu")
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Laplace("x", mu, b)
>>> density(X)(z)
exp(-Abs(mu - z)/b)/(2*b)
>>> cdf(X)(z)
Piecewise((exp((-mu + z)/b)/2, mu > z), (1 - exp((mu - z)/b)/2, True))
>>> L = Laplace('L', [1, 2], [[1, 0], [0, 1]])
>>> pprint(density(L)(1, 2), use_unicode=False)
5 / ____\
e *besselk\0, \/ 35 /
---------------------
pi
References
==========
.. [1] https://en.wikipedia.org/wiki/Laplace_distribution
.. [2] http://mathworld.wolfram.com/LaplaceDistribution.html
"""
if isinstance(mu, (list, MatrixBase)) and\
isinstance(b, (list, MatrixBase)):
from sympy.stats.joint_rv_types import MultivariateLaplace
return MultivariateLaplace(name, mu, b)
return rv(name, LaplaceDistribution, (mu, b))
#-------------------------------------------------------------------------------
# Levy distribution ---------------------------------------------------------
class LevyDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'c')
@property
def set(self):
return Interval(self.mu, oo)
@staticmethod
def check(mu, c):
_value_check(c > 0, "c (scale parameter) must be positive")
_value_check(mu.is_real, "mu (location paramater) must be real")
def pdf(self, x):
mu, c = self.mu, self.c
return sqrt(c/(2*pi))*exp(-c/(2*(x - mu)))/((x - mu)**(S.One + S.Half))
def _cdf(self, x):
mu, c = self.mu, self.c
return erfc(sqrt(c/(2*(x - mu))))
def _characteristic_function(self, t):
mu, c = self.mu, self.c
return exp(I * mu * t - sqrt(-2 * I * c * t))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function of Levy distribution does not exist.')
def Levy(name, mu, c):
r"""
Create a continuous random variable with a Levy distribution.
The density of the Levy distribution is given by
.. math::
f(x) := \sqrt(\frac{c}{2 \pi}) \frac{\exp -\frac{c}{2 (x - \mu)}}{(x - \mu)^{3/2}}
Parameters
==========
mu : Real number
The location parameter.
c : Real number, ``c > 0``
A scale parameter.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Levy, density, cdf
>>> from sympy import Symbol
>>> mu = Symbol("mu", real=True)
>>> c = Symbol("c", positive=True)
>>> z = Symbol("z")
>>> X = Levy("x", mu, c)
>>> density(X)(z)
sqrt(2)*sqrt(c)*exp(-c/(-2*mu + 2*z))/(2*sqrt(pi)*(-mu + z)**(3/2))
>>> cdf(X)(z)
erfc(sqrt(c)*sqrt(1/(-2*mu + 2*z)))
References
==========
.. [1] https://en.wikipedia.org/wiki/L%C3%A9vy_distribution
.. [2] http://mathworld.wolfram.com/LevyDistribution.html
"""
return rv(name, LevyDistribution, (mu, c))
#-------------------------------------------------------------------------------
# Log-Cauchy distribution --------------------------------------------------------
class LogCauchyDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'sigma')
set = Interval.open(0, oo)
@staticmethod
def check(mu, sigma):
_value_check((sigma > 0) != False, "Scale parameter Gamma must be positive.")
_value_check(mu.is_real != False, "Location parameter must be real.")
def pdf(self, x):
mu, sigma = self.mu, self.sigma
return 1/(x*pi)*(sigma/((log(x) - mu)**2 + sigma**2))
def _cdf(self, x):
mu, sigma = self.mu, self.sigma
return (1/pi)*atan((log(x) - mu)/sigma) + S.Half
def _characteristic_function(self, t):
raise NotImplementedError("The characteristic function for the "
"Log-Cauchy distribution does not exist.")
def _moment_generating_function(self, t):
raise NotImplementedError("The moment generating function for the "
"Log-Cauchy distribution does not exist.")
def LogCauchy(name, mu, sigma):
r"""
Create a continuous random variable with a Log-Cauchy distribution.
The density of the Log-Cauchy distribution is given by
.. math::
f(x) := \frac{1}{\pi x} \frac{\sigma}{(log(x)-\mu^2) + \sigma^2}
Parameters
==========
mu : Real number, the location
sigma : Real number, `\sigma > 0`, a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogCauchy, density, cdf
>>> from sympy import Symbol, S
>>> mu = 2
>>> sigma = S.One / 5
>>> z = Symbol("z")
>>> X = LogCauchy("x", mu, sigma)
>>> density(X)(z)
1/(5*pi*z*((log(z) - 2)**2 + 1/25))
>>> cdf(X)(z)
atan(5*log(z) - 10)/pi + 1/2
References
==========
.. [1] https://en.wikipedia.org/wiki/Log-Cauchy_distribution
"""
return rv(name, LogCauchyDistribution, (mu, sigma))
#-------------------------------------------------------------------------------
# Logistic distribution --------------------------------------------------------
class LogisticDistribution(SingleContinuousDistribution):
_argnames = ('mu', 's')
set = Interval(-oo, oo)
@staticmethod
def check(mu, s):
_value_check(s > 0, "Scale parameter s must be positive.")
def pdf(self, x):
mu, s = self.mu, self.s
return exp(-(x - mu)/s)/(s*(1 + exp(-(x - mu)/s))**2)
def _cdf(self, x):
mu, s = self.mu, self.s
return S.One/(1 + exp(-(x - mu)/s))
def _characteristic_function(self, t):
return Piecewise((exp(I*t*self.mu) * pi*self.s*t / sinh(pi*self.s*t), Ne(t, 0)), (S.One, True))
def _moment_generating_function(self, t):
return exp(self.mu*t) * beta_fn(1 - self.s*t, 1 + self.s*t)
def _quantile(self, p):
return self.mu - self.s*log(-S.One + S.One/p)
def Logistic(name, mu, s):
r"""
Create a continuous random variable with a logistic distribution.
Explanation
===========
The density of the logistic distribution is given by
.. math::
f(x) := \frac{e^{-(x-\mu)/s}} {s\left(1+e^{-(x-\mu)/s}\right)^2}
Parameters
==========
mu : Real number, the location (mean)
s : Real number, `s > 0` a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Logistic, density, cdf
>>> from sympy import Symbol
>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")
>>> X = Logistic("x", mu, s)
>>> density(X)(z)
exp((mu - z)/s)/(s*(exp((mu - z)/s) + 1)**2)
>>> cdf(X)(z)
1/(exp((mu - z)/s) + 1)
References
==========
.. [1] https://en.wikipedia.org/wiki/Logistic_distribution
.. [2] http://mathworld.wolfram.com/LogisticDistribution.html
"""
return rv(name, LogisticDistribution, (mu, s))
#-------------------------------------------------------------------------------
# Log-logistic distribution --------------------------------------------------------
class LogLogisticDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
set = Interval(0, oo)
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Scale parameter Alpha must be positive.")
_value_check(beta > 0, "Shape parameter Beta must be positive.")
def pdf(self, x):
a, b = self.alpha, self.beta
return ((b/a)*(x/a)**(b - 1))/(1 + (x/a)**b)**2
def _cdf(self, x):
a, b = self.alpha, self.beta
return 1/(1 + (x/a)**(-b))
def _quantile(self, p):
a, b = self.alpha, self.beta
return a*((p/(1 - p))**(1/b))
def expectation(self, expr, var, **kwargs):
a, b = self.args
return Piecewise((S.NaN, b <= 1), (pi*a/(b*sin(pi/b)), True))
def LogLogistic(name, alpha, beta):
r"""
Create a continuous random variable with a log-logistic distribution.
The distribution is unimodal when ``beta > 1``.
Explanation
===========
The density of the log-logistic distribution is given by
.. math::
f(x) := \frac{(\frac{\beta}{\alpha})(\frac{x}{\alpha})^{\beta - 1}}
{(1 + (\frac{x}{\alpha})^{\beta})^2}
Parameters
==========
alpha : Real number, `\alpha > 0`, scale parameter and median of distribution
beta : Real number, `\beta > 0` a shape parameter
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogLogistic, density, cdf, quantile
>>> from sympy import Symbol, pprint
>>> alpha = Symbol("alpha", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> p = Symbol("p")
>>> z = Symbol("z", positive=True)
>>> X = LogLogistic("x", alpha, beta)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
beta - 1
/ z \
beta*|-----|
\alpha/
------------------------
2
/ beta \
|/ z \ |
alpha*||-----| + 1|
\\alpha/ /
>>> cdf(X)(z)
1/(1 + (z/alpha)**(-beta))
>>> quantile(X)(p)
alpha*(p/(1 - p))**(1/beta)
References
==========
.. [1] https://en.wikipedia.org/wiki/Log-logistic_distribution
"""
return rv(name, LogLogisticDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
#Logit-Normal distribution------------------------------------------------------
class LogitNormalDistribution(SingleContinuousDistribution):
_argnames = ('mu', 's')
set = Interval.open(0, 1)
@staticmethod
def check(mu, s):
_value_check((s ** 2).is_real is not False and s ** 2 > 0, "Squared scale parameter s must be positive.")
_value_check(mu.is_real is not False, "Location parameter must be real")
def _logit(self, x):
return log(x / (1 - x))
def pdf(self, x):
mu, s = self.mu, self.s
return exp(-(self._logit(x) - mu)**2/(2*s**2))*(S.One/sqrt(2*pi*(s**2)))*(1/(x*(1 - x)))
def _cdf(self, x):
mu, s = self.mu, self.s
return (S.One/2)*(1 + erf((self._logit(x) - mu)/(sqrt(2*s**2))))
def LogitNormal(name, mu, s):
r"""
Create a continuous random variable with a Logit-Normal distribution.
The density of the logistic distribution is given by
.. math::
f(x) := \frac{1}{s \sqrt{2 \pi}} \frac{1}{x(1 - x)} e^{- \frac{(logit(x) - \mu)^2}{s^2}}
where logit(x) = \log(\frac{x}{1 - x})
Parameters
==========
mu : Real number, the location (mean)
s : Real number, `s > 0` a scale
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogitNormal, density, cdf
>>> from sympy import Symbol,pprint
>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")
>>> X = LogitNormal("x",mu,s)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
/ / z \\
-|-mu + log|-----||
\ \1 - z//
---------------------
2
___ 2*s
\/ 2 *e
----------------------------
____
2*\/ pi *s*z*(1 - z)
>>> density(X)(z)
sqrt(2)*exp(-(-mu + log(z/(1 - z)))**2/(2*s**2))/(2*sqrt(pi)*s*z*(1 - z))
>>> cdf(X)(z)
erf(sqrt(2)*(-mu + log(z/(1 - z)))/(2*s))/2 + 1/2
References
==========
.. [1] https://en.wikipedia.org/wiki/Logit-normal_distribution
"""
return rv(name, LogitNormalDistribution, (mu, s))
#-------------------------------------------------------------------------------
# Log Normal distribution ------------------------------------------------------
class LogNormalDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'std')
set = Interval(0, oo)
@staticmethod
def check(mean, std):
_value_check(std > 0, "Parameter std must be positive.")
def pdf(self, x):
mean, std = self.mean, self.std
return exp(-(log(x) - mean)**2 / (2*std**2)) / (x*sqrt(2*pi)*std)
def _cdf(self, x):
mean, std = self.mean, self.std
return Piecewise(
(S.Half + S.Half*erf((log(x) - mean)/sqrt(2)/std), x > 0),
(S.Zero, True)
)
def _moment_generating_function(self, t):
raise NotImplementedError('Moment generating function of the log-normal distribution is not defined.')
def LogNormal(name, mean, std):
r"""
Create a continuous random variable with a log-normal distribution.
Explanation
===========
The density of the log-normal distribution is given by
.. math::
f(x) := \frac{1}{x\sqrt{2\pi\sigma^2}}
e^{-\frac{\left(\ln x-\mu\right)^2}{2\sigma^2}}
with :math:`x \geq 0`.
Parameters
==========
mu : Real number
The log-scale.
sigma : Real number
A shape. ($\sigma^2 > 0$)
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import LogNormal, density
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu", real=True)
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> X = LogNormal("x", mu, sigma)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-(-mu + log(z))
-----------------
2
___ 2*sigma
\/ 2 *e
------------------------
____
2*\/ pi *sigma*z
>>> X = LogNormal('x', 0, 1) # Mean 0, standard deviation 1
>>> density(X)(z)
sqrt(2)*exp(-log(z)**2/2)/(2*sqrt(pi)*z)
References
==========
.. [1] https://en.wikipedia.org/wiki/Lognormal
.. [2] http://mathworld.wolfram.com/LogNormalDistribution.html
"""
return rv(name, LogNormalDistribution, (mean, std))
#-------------------------------------------------------------------------------
# Lomax Distribution -----------------------------------------------------------
class LomaxDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'lamda',)
set = Interval(0, oo)
@staticmethod
def check(alpha, lamda):
_value_check(alpha.is_real, "Shape parameter should be real.")
_value_check(lamda.is_real, "Scale parameter should be real.")
_value_check(alpha.is_positive, "Shape parameter should be positive.")
_value_check(lamda.is_positive, "Scale parameter should be positive.")
def pdf(self, x):
lamba, alpha = self.lamda, self.alpha
return (alpha/lamba) * (S.One + x/lamba)**(-alpha-1)
def Lomax(name, alpha, lamda):
r"""
Create a continuous random variable with a Lomax distribution.
Explanation
===========
The density of the Lomax distribution is given by
.. math::
f(x) := \frac{\alpha}{\lambda}\left[1+\frac{x}{\lambda}\right]^{-(\alpha+1)}
Parameters
==========
alpha : Real Number, `alpha > 0`
Shape parameter
lamda : Real Number, `lamda > 0`
Scale parameter
Examples
========
>>> from sympy.stats import Lomax, density, cdf, E
>>> from sympy import symbols
>>> a, l = symbols('a, l', positive=True)
>>> X = Lomax('X', a, l)
>>> x = symbols('x')
>>> density(X)(x)
a*(1 + x/l)**(-a - 1)/l
>>> cdf(X)(x)
Piecewise((1 - 1/(1 + x/l)**a, x >= 0), (0, True))
>>> a = 2
>>> X = Lomax('X', a, l)
>>> E(X)
l
Returns
=======
RandomSymbol
References
==========
.. [1] https://en.wikipedia.org/wiki/Lomax_distribution
"""
return rv(name, LomaxDistribution, (alpha, lamda))
#-------------------------------------------------------------------------------
# Maxwell distribution ---------------------------------------------------------
class MaxwellDistribution(SingleContinuousDistribution):
_argnames = ('a',)
set = Interval(0, oo)
@staticmethod
def check(a):
_value_check(a > 0, "Parameter a must be positive.")
def pdf(self, x):
a = self.a
return sqrt(2/pi)*x**2*exp(-x**2/(2*a**2))/a**3
def _cdf(self, x):
a = self.a
return erf(sqrt(2)*x/(2*a)) - sqrt(2)*x*exp(-x**2/(2*a**2))/(sqrt(pi)*a)
def Maxwell(name, a):
r"""
Create a continuous random variable with a Maxwell distribution.
Explanation
===========
The density of the Maxwell distribution is given by
.. math::
f(x) := \sqrt{\frac{2}{\pi}} \frac{x^2 e^{-x^2/(2a^2)}}{a^3}
with :math:`x \geq 0`.
.. TODO - what does the parameter mean?
Parameters
==========
a : Real number, `a > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Maxwell, density, E, variance
>>> from sympy import Symbol, simplify
>>> a = Symbol("a", positive=True)
>>> z = Symbol("z")
>>> X = Maxwell("x", a)
>>> density(X)(z)
sqrt(2)*z**2*exp(-z**2/(2*a**2))/(sqrt(pi)*a**3)
>>> E(X)
2*sqrt(2)*a/sqrt(pi)
>>> simplify(variance(X))
a**2*(-8 + 3*pi)/pi
References
==========
.. [1] https://en.wikipedia.org/wiki/Maxwell_distribution
.. [2] http://mathworld.wolfram.com/MaxwellDistribution.html
"""
return rv(name, MaxwellDistribution, (a, ))
#-------------------------------------------------------------------------------
# Moyal Distribution -----------------------------------------------------------
class MoyalDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'sigma')
@staticmethod
def check(mu, sigma):
_value_check(mu.is_real, "Location parameter must be real.")
_value_check(sigma.is_real and sigma > 0, "Scale parameter must be real\
and positive.")
def pdf(self, x):
mu, sigma = self.mu, self.sigma
num = exp(-(exp(-(x - mu)/sigma) + (x - mu)/(sigma))/2)
den = (sqrt(2*pi) * sigma)
return num/den
def _characteristic_function(self, t):
mu, sigma = self.mu, self.sigma
term1 = exp(I*t*mu)
term2 = (2**(-I*sigma*t) * gamma(Rational(1, 2) - I*t*sigma))
return (term1 * term2)/sqrt(pi)
def _moment_generating_function(self, t):
mu, sigma = self.mu, self.sigma
term1 = exp(t*mu)
term2 = (2**(-1*sigma*t) * gamma(Rational(1, 2) - t*sigma))
return (term1 * term2)/sqrt(pi)
def Moyal(name, mu, sigma):
r"""
Create a continuous random variable with a Moyal distribution.
Explanation
===========
The density of the Moyal distribution is given by
.. math::
f(x) := \frac{\exp-\frac{1}{2}\exp-\frac{x-\mu}{\sigma}-\frac{x-\mu}{2\sigma}}{\sqrt{2\pi}\sigma}
with :math:`x \in \mathbb{R}`.
Parameters
==========
mu : Real number
Location parameter
sigma : Real positive number
Scale parameter
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Moyal, density, cdf
>>> from sympy import Symbol, simplify
>>> mu = Symbol("mu", real=True)
>>> sigma = Symbol("sigma", positive=True, real=True)
>>> z = Symbol("z")
>>> X = Moyal("x", mu, sigma)
>>> density(X)(z)
sqrt(2)*exp(-exp((mu - z)/sigma)/2 - (-mu + z)/(2*sigma))/(2*sqrt(pi)*sigma)
>>> simplify(cdf(X)(z))
1 - erf(sqrt(2)*exp((mu - z)/(2*sigma))/2)
References
==========
.. [1] https://reference.wolfram.com/language/ref/MoyalDistribution.html
.. [2] http://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf
"""
return rv(name, MoyalDistribution, (mu, sigma))
#-------------------------------------------------------------------------------
# Nakagami distribution --------------------------------------------------------
class NakagamiDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'omega')
set = Interval(0, oo)
@staticmethod
def check(mu, omega):
_value_check(mu >= S.Half, "Shape parameter mu must be greater than equal to 1/2.")
_value_check(omega > 0, "Spread parameter omega must be positive.")
def pdf(self, x):
mu, omega = self.mu, self.omega
return 2*mu**mu/(gamma(mu)*omega**mu)*x**(2*mu - 1)*exp(-mu/omega*x**2)
def _cdf(self, x):
mu, omega = self.mu, self.omega
return Piecewise(
(lowergamma(mu, (mu/omega)*x**2)/gamma(mu), x > 0),
(S.Zero, True))
def Nakagami(name, mu, omega):
r"""
Create a continuous random variable with a Nakagami distribution.
Explanation
===========
The density of the Nakagami distribution is given by
.. math::
f(x) := \frac{2\mu^\mu}{\Gamma(\mu)\omega^\mu} x^{2\mu-1}
\exp\left(-\frac{\mu}{\omega}x^2 \right)
with :math:`x > 0`.
Parameters
==========
mu : Real number, `\mu \geq \frac{1}{2}` a shape
omega : Real number, `\omega > 0`, the spread
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Nakagami, density, E, variance, cdf
>>> from sympy import Symbol, simplify, pprint
>>> mu = Symbol("mu", positive=True)
>>> omega = Symbol("omega", positive=True)
>>> z = Symbol("z")
>>> X = Nakagami("x", mu, omega)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-mu*z
-------
mu -mu 2*mu - 1 omega
2*mu *omega *z *e
----------------------------------
Gamma(mu)
>>> simplify(E(X))
sqrt(mu)*sqrt(omega)*gamma(mu + 1/2)/gamma(mu + 1)
>>> V = simplify(variance(X))
>>> pprint(V, use_unicode=False)
2
omega*Gamma (mu + 1/2)
omega - -----------------------
Gamma(mu)*Gamma(mu + 1)
>>> cdf(X)(z)
Piecewise((lowergamma(mu, mu*z**2/omega)/gamma(mu), z > 0),
(0, True))
References
==========
.. [1] https://en.wikipedia.org/wiki/Nakagami_distribution
"""
return rv(name, NakagamiDistribution, (mu, omega))
#-------------------------------------------------------------------------------
# Normal distribution ----------------------------------------------------------
class NormalDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'std')
@staticmethod
def check(mean, std):
_value_check(std > 0, "Standard deviation must be positive")
def pdf(self, x):
return exp(-(x - self.mean)**2 / (2*self.std**2)) / (sqrt(2*pi)*self.std)
def _cdf(self, x):
mean, std = self.mean, self.std
return erf(sqrt(2)*(-mean + x)/(2*std))/2 + S.Half
def _characteristic_function(self, t):
mean, std = self.mean, self.std
return exp(I*mean*t - std**2*t**2/2)
def _moment_generating_function(self, t):
mean, std = self.mean, self.std
return exp(mean*t + std**2*t**2/2)
def _quantile(self, p):
mean, std = self.mean, self.std
return mean + std*sqrt(2)*erfinv(2*p - 1)
def Normal(name, mean, std):
r"""
Create a continuous random variable with a Normal distribution.
Explanation
===========
The density of the Normal distribution is given by
.. math::
f(x) := \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{(x-\mu)^2}{2\sigma^2} }
Parameters
==========
mu : Real number or a list representing the mean or the mean vector
sigma : Real number or a positive definite square matrix,
:math:`\sigma^2 > 0` the variance
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Normal, density, E, std, cdf, skewness, quantile, marginal_distribution
>>> from sympy import Symbol, simplify, pprint
>>> mu = Symbol("mu")
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> y = Symbol("y")
>>> p = Symbol("p")
>>> X = Normal("x", mu, sigma)
>>> density(X)(z)
sqrt(2)*exp(-(-mu + z)**2/(2*sigma**2))/(2*sqrt(pi)*sigma)
>>> C = simplify(cdf(X))(z) # it needs a little more help...
>>> pprint(C, use_unicode=False)
/ ___ \
|\/ 2 *(-mu + z)|
erf|---------------|
\ 2*sigma / 1
-------------------- + -
2 2
>>> quantile(X)(p)
mu + sqrt(2)*sigma*erfinv(2*p - 1)
>>> simplify(skewness(X))
0
>>> X = Normal("x", 0, 1) # Mean 0, standard deviation 1
>>> density(X)(z)
sqrt(2)*exp(-z**2/2)/(2*sqrt(pi))
>>> E(2*X + 1)
1
>>> simplify(std(2*X + 1))
2
>>> m = Normal('X', [1, 2], [[2, 1], [1, 2]])
>>> pprint(density(m)(y, z), use_unicode=False)
2 2
y y*z z
- -- + --- - -- + z - 1
___ 3 3 3
\/ 3 *e
------------------------------
6*pi
>>> marginal_distribution(m, m[0])(1)
1/(2*sqrt(pi))
References
==========
.. [1] https://en.wikipedia.org/wiki/Normal_distribution
.. [2] http://mathworld.wolfram.com/NormalDistributionFunction.html
"""
if isinstance(mean, list) or getattr(mean, 'is_Matrix', False) and\
isinstance(std, list) or getattr(std, 'is_Matrix', False):
from sympy.stats.joint_rv_types import MultivariateNormal
return MultivariateNormal(name, mean, std)
return rv(name, NormalDistribution, (mean, std))
#-------------------------------------------------------------------------------
# Inverse Gaussian distribution ----------------------------------------------------------
class GaussianInverseDistribution(SingleContinuousDistribution):
_argnames = ('mean', 'shape')
@property
def set(self):
return Interval(0, oo)
@staticmethod
def check(mean, shape):
_value_check(shape > 0, "Shape parameter must be positive")
_value_check(mean > 0, "Mean must be positive")
def pdf(self, x):
mu, s = self.mean, self.shape
return exp(-s*(x - mu)**2 / (2*x*mu**2)) * sqrt(s/(2*pi*x**3))
def _cdf(self, x):
from sympy.stats import cdf
mu, s = self.mean, self.shape
stdNormalcdf = cdf(Normal('x', 0, 1))
first_term = stdNormalcdf(sqrt(s/x) * ((x/mu) - S.One))
second_term = exp(2*s/mu) * stdNormalcdf(-sqrt(s/x)*(x/mu + S.One))
return first_term + second_term
def _characteristic_function(self, t):
mu, s = self.mean, self.shape
return exp((s/mu)*(1 - sqrt(1 - (2*mu**2*I*t)/s)))
def _moment_generating_function(self, t):
mu, s = self.mean, self.shape
return exp((s/mu)*(1 - sqrt(1 - (2*mu**2*t)/s)))
def GaussianInverse(name, mean, shape):
r"""
Create a continuous random variable with an Inverse Gaussian distribution.
Inverse Gaussian distribution is also known as Wald distribution.
Explanation
===========
The density of the Inverse Gaussian distribution is given by
.. math::
f(x) := \sqrt{\frac{\lambda}{2\pi x^3}} e^{-\frac{\lambda(x-\mu)^2}{2x\mu^2}}
Parameters
==========
mu :
Positive number representing the mean.
lambda :
Positive number representing the shape parameter.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import GaussianInverse, density, E, std, skewness
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu", positive=True)
>>> lamda = Symbol("lambda", positive=True)
>>> z = Symbol("z", positive=True)
>>> X = GaussianInverse("x", mu, lamda)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
2
-lambda*(-mu + z)
-------------------
2
___ ________ 2*mu *z
\/ 2 *\/ lambda *e
-------------------------------------
____ 3/2
2*\/ pi *z
>>> E(X)
mu
>>> std(X).expand()
mu**(3/2)/sqrt(lambda)
>>> skewness(X).expand()
3*sqrt(mu)/sqrt(lambda)
References
==========
.. [1] https://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
.. [2] http://mathworld.wolfram.com/InverseGaussianDistribution.html
"""
return rv(name, GaussianInverseDistribution, (mean, shape))
Wald = GaussianInverse
#-------------------------------------------------------------------------------
# Pareto distribution ----------------------------------------------------------
class ParetoDistribution(SingleContinuousDistribution):
_argnames = ('xm', 'alpha')
@property
def set(self):
return Interval(self.xm, oo)
@staticmethod
def check(xm, alpha):
_value_check(xm > 0, "Xm must be positive")
_value_check(alpha > 0, "Alpha must be positive")
def pdf(self, x):
xm, alpha = self.xm, self.alpha
return alpha * xm**alpha / x**(alpha + 1)
def _cdf(self, x):
xm, alpha = self.xm, self.alpha
return Piecewise(
(S.One - xm**alpha/x**alpha, x>=xm),
(0, True),
)
def _moment_generating_function(self, t):
xm, alpha = self.xm, self.alpha
return alpha * (-xm*t)**alpha * uppergamma(-alpha, -xm*t)
def _characteristic_function(self, t):
xm, alpha = self.xm, self.alpha
return alpha * (-I * xm * t) ** alpha * uppergamma(-alpha, -I * xm * t)
def Pareto(name, xm, alpha):
r"""
Create a continuous random variable with the Pareto distribution.
Explanation
===========
The density of the Pareto distribution is given by
.. math::
f(x) := \frac{\alpha\,x_m^\alpha}{x^{\alpha+1}}
with :math:`x \in [x_m,\infty]`.
Parameters
==========
xm : Real number, `x_m > 0`, a scale
alpha : Real number, `\alpha > 0`, a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Pareto, density
>>> from sympy import Symbol
>>> xm = Symbol("xm", positive=True)
>>> beta = Symbol("beta", positive=True)
>>> z = Symbol("z")
>>> X = Pareto("x", xm, beta)
>>> density(X)(z)
beta*xm**beta*z**(-beta - 1)
References
==========
.. [1] https://en.wikipedia.org/wiki/Pareto_distribution
.. [2] http://mathworld.wolfram.com/ParetoDistribution.html
"""
return rv(name, ParetoDistribution, (xm, alpha))
#-------------------------------------------------------------------------------
# PowerFunction distribution ---------------------------------------------------
class PowerFunctionDistribution(SingleContinuousDistribution):
_argnames=('alpha','a','b')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(alpha, a, b):
_value_check(a.is_real, "Continuous Boundary parameter should be real.")
_value_check(b.is_real, "Continuous Boundary parameter should be real.")
_value_check(a < b, " 'a' the left Boundary must be smaller than 'b' the right Boundary." )
_value_check(alpha.is_positive, "Continuous Shape parameter should be positive.")
def pdf(self, x):
alpha, a, b = self.alpha, self.a, self.b
num = alpha*(x - a)**(alpha - 1)
den = (b - a)**alpha
return num/den
def PowerFunction(name, alpha, a, b):
r"""
Creates a continuous random variable with a Power Function Distribution.
Explanation
===========
The density of PowerFunction distribution is given by
.. math::
f(x) := \frac{{\alpha}(x - a)^{\alpha - 1}}{(b - a)^{\alpha}}
with :math:`x \in [a,b]`.
Parameters
==========
alpha: Positive number, `0 < alpha` the shape paramater
a : Real number, :math:`-\infty < a` the left boundary
b : Real number, :math:`a < b < \infty` the right boundary
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import PowerFunction, density, cdf, E, variance
>>> from sympy import Symbol
>>> alpha = Symbol("alpha", positive=True)
>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")
>>> X = PowerFunction("X", 2, a, b)
>>> density(X)(z)
(-2*a + 2*z)/(-a + b)**2
>>> cdf(X)(z)
Piecewise((a**2/(a**2 - 2*a*b + b**2) - 2*a*z/(a**2 - 2*a*b + b**2) +
z**2/(a**2 - 2*a*b + b**2), a <= z), (0, True))
>>> alpha = 2
>>> a = 0
>>> b = 1
>>> Y = PowerFunction("Y", alpha, a, b)
>>> E(Y)
2/3
>>> variance(Y)
1/18
References
==========
.. [1] http://www.mathwave.com/help/easyfit/html/analyses/distributions/power_func.html
"""
return rv(name, PowerFunctionDistribution, (alpha, a, b))
#-------------------------------------------------------------------------------
# QuadraticU distribution ------------------------------------------------------
class QuadraticUDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(a, b):
_value_check(b > a, "Parameter b must be in range (%s, oo)."%(a))
def pdf(self, x):
a, b = self.a, self.b
alpha = 12 / (b-a)**3
beta = (a+b) / 2
return Piecewise(
(alpha * (x-beta)**2, And(a<=x, x<=b)),
(S.Zero, True))
def _moment_generating_function(self, t):
a, b = self.a, self.b
return -3 * (exp(a*t) * (4 + (a**2 + 2*a*(-2 + b) + b**2) * t) \
- exp(b*t) * (4 + (-4*b + (a + b)**2) * t)) / ((a-b)**3 * t**2)
def _characteristic_function(self, t):
a, b = self.a, self.b
return -3*I*(exp(I*a*t*exp(I*b*t)) * (4*I - (-4*b + (a+b)**2)*t)) \
/ ((a-b)**3 * t**2)
def QuadraticU(name, a, b):
r"""
Create a Continuous Random Variable with a U-quadratic distribution.
Explanation
===========
The density of the U-quadratic distribution is given by
.. math::
f(x) := \alpha (x-\beta)^2
with :math:`x \in [a,b]`.
Parameters
==========
a : Real number
b : Real number, :math:`a < b`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import QuadraticU, density
>>> from sympy import Symbol, pprint
>>> a = Symbol("a", real=True)
>>> b = Symbol("b", real=True)
>>> z = Symbol("z")
>>> X = QuadraticU("x", a, b)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/ 2
| / a b \
|12*|- - - - + z|
| \ 2 2 /
<----------------- for And(b >= z, a <= z)
| 3
| (-a + b)
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/U-quadratic_distribution
"""
return rv(name, QuadraticUDistribution, (a, b))
#-------------------------------------------------------------------------------
# RaisedCosine distribution ----------------------------------------------------
class RaisedCosineDistribution(SingleContinuousDistribution):
_argnames = ('mu', 's')
@property
def set(self):
return Interval(self.mu - self.s, self.mu + self.s)
@staticmethod
def check(mu, s):
_value_check(s > 0, "s must be positive")
def pdf(self, x):
mu, s = self.mu, self.s
return Piecewise(
((1+cos(pi*(x-mu)/s)) / (2*s), And(mu-s<=x, x<=mu+s)),
(S.Zero, True))
def _characteristic_function(self, t):
mu, s = self.mu, self.s
return Piecewise((exp(-I*pi*mu/s)/2, Eq(t, -pi/s)),
(exp(I*pi*mu/s)/2, Eq(t, pi/s)),
(pi**2*sin(s*t)*exp(I*mu*t) / (s*t*(pi**2 - s**2*t**2)), True))
def _moment_generating_function(self, t):
mu, s = self.mu, self.s
return pi**2 * sinh(s*t) * exp(mu*t) / (s*t*(pi**2 + s**2*t**2))
def RaisedCosine(name, mu, s):
r"""
Create a Continuous Random Variable with a raised cosine distribution.
Explanation
===========
The density of the raised cosine distribution is given by
.. math::
f(x) := \frac{1}{2s}\left(1+\cos\left(\frac{x-\mu}{s}\pi\right)\right)
with :math:`x \in [\mu-s,\mu+s]`.
Parameters
==========
mu : Real number
s : Real number, `s > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import RaisedCosine, density
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu", real=True)
>>> s = Symbol("s", positive=True)
>>> z = Symbol("z")
>>> X = RaisedCosine("x", mu, s)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
/ /pi*(-mu + z)\
|cos|------------| + 1
| \ s /
<--------------------- for And(z >= mu - s, z <= mu + s)
| 2*s
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/Raised_cosine_distribution
"""
return rv(name, RaisedCosineDistribution, (mu, s))
#-------------------------------------------------------------------------------
# Rayleigh distribution --------------------------------------------------------
class RayleighDistribution(SingleContinuousDistribution):
_argnames = ('sigma',)
set = Interval(0, oo)
@staticmethod
def check(sigma):
_value_check(sigma > 0, "Scale parameter sigma must be positive.")
def pdf(self, x):
sigma = self.sigma
return x/sigma**2*exp(-x**2/(2*sigma**2))
def _cdf(self, x):
sigma = self.sigma
return 1 - exp(-(x**2/(2*sigma**2)))
def _characteristic_function(self, t):
sigma = self.sigma
return 1 - sigma*t*exp(-sigma**2*t**2/2) * sqrt(pi/2) * (erfi(sigma*t/sqrt(2)) - I)
def _moment_generating_function(self, t):
sigma = self.sigma
return 1 + sigma*t*exp(sigma**2*t**2/2) * sqrt(pi/2) * (erf(sigma*t/sqrt(2)) + 1)
def Rayleigh(name, sigma):
r"""
Create a continuous random variable with a Rayleigh distribution.
Explanation
===========
The density of the Rayleigh distribution is given by
.. math ::
f(x) := \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}
with :math:`x > 0`.
Parameters
==========
sigma : Real number, `\sigma > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Rayleigh, density, E, variance
>>> from sympy import Symbol
>>> sigma = Symbol("sigma", positive=True)
>>> z = Symbol("z")
>>> X = Rayleigh("x", sigma)
>>> density(X)(z)
z*exp(-z**2/(2*sigma**2))/sigma**2
>>> E(X)
sqrt(2)*sqrt(pi)*sigma/2
>>> variance(X)
-pi*sigma**2/2 + 2*sigma**2
References
==========
.. [1] https://en.wikipedia.org/wiki/Rayleigh_distribution
.. [2] http://mathworld.wolfram.com/RayleighDistribution.html
"""
return rv(name, RayleighDistribution, (sigma, ))
#-------------------------------------------------------------------------------
# Reciprocal distribution --------------------------------------------------------
class ReciprocalDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(a, b):
_value_check(a > 0, "Parameter > 0. a = %s"%a)
_value_check((a < b),
"Parameter b must be in range (%s, +oo]. b = %s"%(a, b))
def pdf(self, x):
a, b = self.a, self.b
return 1/(x*(log(b) - log(a)))
def Reciprocal(name, a, b):
r"""Creates a continuous random variable with a reciprocal distribution.
Parameters
==========
a : Real number, :math:`0 < a`
b : Real number, :math:`a < b`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Reciprocal, density, cdf
>>> from sympy import symbols
>>> a, b, x = symbols('a, b, x', positive=True)
>>> R = Reciprocal('R', a, b)
>>> density(R)(x)
1/(x*(-log(a) + log(b)))
>>> cdf(R)(x)
Piecewise((log(a)/(log(a) - log(b)) - log(x)/(log(a) - log(b)), a <= x), (0, True))
Reference
=========
.. [1] https://en.wikipedia.org/wiki/Reciprocal_distribution
"""
return rv(name, ReciprocalDistribution, (a, b))
#-------------------------------------------------------------------------------
# Shifted Gompertz distribution ------------------------------------------------
class ShiftedGompertzDistribution(SingleContinuousDistribution):
_argnames = ('b', 'eta')
set = Interval(0, oo)
@staticmethod
def check(b, eta):
_value_check(b > 0, "b must be positive")
_value_check(eta > 0, "eta must be positive")
def pdf(self, x):
b, eta = self.b, self.eta
return b*exp(-b*x)*exp(-eta*exp(-b*x))*(1+eta*(1-exp(-b*x)))
def ShiftedGompertz(name, b, eta):
r"""
Create a continuous random variable with a Shifted Gompertz distribution.
Explanation
===========
The density of the Shifted Gompertz distribution is given by
.. math::
f(x) := b e^{-b x} e^{-\eta \exp(-b x)} \left[1 + \eta(1 - e^(-bx)) \right]
with :math: 'x \in [0, \inf)'.
Parameters
==========
b: Real number, 'b > 0' a scale
eta: Real number, 'eta > 0' a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import ShiftedGompertz, density
>>> from sympy import Symbol
>>> b = Symbol("b", positive=True)
>>> eta = Symbol("eta", positive=True)
>>> x = Symbol("x")
>>> X = ShiftedGompertz("x", b, eta)
>>> density(X)(x)
b*(eta*(1 - exp(-b*x)) + 1)*exp(-b*x)*exp(-eta*exp(-b*x))
References
==========
.. [1] https://en.wikipedia.org/wiki/Shifted_Gompertz_distribution
"""
return rv(name, ShiftedGompertzDistribution, (b, eta))
#-------------------------------------------------------------------------------
# StudentT distribution --------------------------------------------------------
class StudentTDistribution(SingleContinuousDistribution):
_argnames = ('nu',)
set = Interval(-oo, oo)
@staticmethod
def check(nu):
_value_check(nu > 0, "Degrees of freedom nu must be positive.")
def pdf(self, x):
nu = self.nu
return 1/(sqrt(nu)*beta_fn(S.Half, nu/2))*(1 + x**2/nu)**(-(nu + 1)/2)
def _cdf(self, x):
nu = self.nu
return S.Half + x*gamma((nu+1)/2)*hyper((S.Half, (nu+1)/2),
(Rational(3, 2),), -x**2/nu)/(sqrt(pi*nu)*gamma(nu/2))
def _moment_generating_function(self, t):
raise NotImplementedError('The moment generating function for the Student-T distribution is undefined.')
def StudentT(name, nu):
r"""
Create a continuous random variable with a student's t distribution.
Explanation
===========
The density of the student's t distribution is given by
.. math::
f(x) := \frac{\Gamma \left(\frac{\nu+1}{2} \right)}
{\sqrt{\nu\pi}\Gamma \left(\frac{\nu}{2} \right)}
\left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}
Parameters
==========
nu : Real number, `\nu > 0`, the degrees of freedom
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import StudentT, density, cdf
>>> from sympy import Symbol, pprint
>>> nu = Symbol("nu", positive=True)
>>> z = Symbol("z")
>>> X = StudentT("x", nu)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
nu 1
- -- - -
2 2
/ 2\
| z |
|1 + --|
\ nu/
-----------------
____ / nu\
\/ nu *B|1/2, --|
\ 2 /
>>> cdf(X)(z)
1/2 + z*gamma(nu/2 + 1/2)*hyper((1/2, nu/2 + 1/2), (3/2,),
-z**2/nu)/(sqrt(pi)*sqrt(nu)*gamma(nu/2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Student_t-distribution
.. [2] http://mathworld.wolfram.com/Studentst-Distribution.html
"""
return rv(name, StudentTDistribution, (nu, ))
#-------------------------------------------------------------------------------
# Trapezoidal distribution ------------------------------------------------------
class TrapezoidalDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b', 'c', 'd')
@property
def set(self):
return Interval(self.a, self.d)
@staticmethod
def check(a, b, c, d):
_value_check(a < d, "Lower bound parameter a < %s. a = %s"%(d, a))
_value_check((a <= b, b < c),
"Level start parameter b must be in range [%s, %s). b = %s"%(a, c, b))
_value_check((b < c, c <= d),
"Level end parameter c must be in range (%s, %s]. c = %s"%(b, d, c))
_value_check(d >= c, "Upper bound parameter d > %s. d = %s"%(c, d))
def pdf(self, x):
a, b, c, d = self.a, self.b, self.c, self.d
return Piecewise(
(2*(x-a) / ((b-a)*(d+c-a-b)), And(a <= x, x < b)),
(2 / (d+c-a-b), And(b <= x, x < c)),
(2*(d-x) / ((d-c)*(d+c-a-b)), And(c <= x, x <= d)),
(S.Zero, True))
def Trapezoidal(name, a, b, c, d):
r"""
Create a continuous random variable with a trapezoidal distribution.
Explanation
===========
The density of the trapezoidal distribution is given by
.. math::
f(x) := \begin{cases}
0 & \mathrm{for\ } x < a, \\
\frac{2(x-a)}{(b-a)(d+c-a-b)} & \mathrm{for\ } a \le x < b, \\
\frac{2}{d+c-a-b} & \mathrm{for\ } b \le x < c, \\
\frac{2(d-x)}{(d-c)(d+c-a-b)} & \mathrm{for\ } c \le x < d, \\
0 & \mathrm{for\ } d < x.
\end{cases}
Parameters
==========
a : Real number, :math:`a < d`
b : Real number, :math:`a <= b < c`
c : Real number, :math:`b < c <= d`
d : Real number
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Trapezoidal, density
>>> from sympy import Symbol, pprint
>>> a = Symbol("a")
>>> b = Symbol("b")
>>> c = Symbol("c")
>>> d = Symbol("d")
>>> z = Symbol("z")
>>> X = Trapezoidal("x", a,b,c,d)
>>> pprint(density(X)(z), use_unicode=False)
/ -2*a + 2*z
|------------------------- for And(a <= z, b > z)
|(-a + b)*(-a - b + c + d)
|
| 2
| -------------- for And(b <= z, c > z)
< -a - b + c + d
|
| 2*d - 2*z
|------------------------- for And(d >= z, c <= z)
|(-c + d)*(-a - b + c + d)
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/Trapezoidal_distribution
"""
return rv(name, TrapezoidalDistribution, (a, b, c, d))
#-------------------------------------------------------------------------------
# Triangular distribution ------------------------------------------------------
class TriangularDistribution(SingleContinuousDistribution):
_argnames = ('a', 'b', 'c')
@property
def set(self):
return Interval(self.a, self.b)
@staticmethod
def check(a, b, c):
_value_check(b > a, "Parameter b > %s. b = %s"%(a, b))
_value_check((a <= c, c <= b),
"Parameter c must be in range [%s, %s]. c = %s"%(a, b, c))
def pdf(self, x):
a, b, c = self.a, self.b, self.c
return Piecewise(
(2*(x - a)/((b - a)*(c - a)), And(a <= x, x < c)),
(2/(b - a), Eq(x, c)),
(2*(b - x)/((b - a)*(b - c)), And(c < x, x <= b)),
(S.Zero, True))
def _characteristic_function(self, t):
a, b, c = self.a, self.b, self.c
return -2 *((b-c) * exp(I*a*t) - (b-a) * exp(I*c*t) + (c-a) * exp(I*b*t)) / ((b-a)*(c-a)*(b-c)*t**2)
def _moment_generating_function(self, t):
a, b, c = self.a, self.b, self.c
return 2 * ((b - c) * exp(a * t) - (b - a) * exp(c * t) + (c - a) * exp(b * t)) / (
(b - a) * (c - a) * (b - c) * t ** 2)
def Triangular(name, a, b, c):
r"""
Create a continuous random variable with a triangular distribution.
Explanation
===========
The density of the triangular distribution is given by
.. math::
f(x) := \begin{cases}
0 & \mathrm{for\ } x < a, \\
\frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x < c, \\
\frac{2}{b-a} & \mathrm{for\ } x = c, \\
\frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\
0 & \mathrm{for\ } b < x.
\end{cases}
Parameters
==========
a : Real number, :math:`a \in \left(-\infty, \infty\right)`
b : Real number, :math:`a < b`
c : Real number, :math:`a \leq c \leq b`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Triangular, density
>>> from sympy import Symbol, pprint
>>> a = Symbol("a")
>>> b = Symbol("b")
>>> c = Symbol("c")
>>> z = Symbol("z")
>>> X = Triangular("x", a,b,c)
>>> pprint(density(X)(z), use_unicode=False)
/ -2*a + 2*z
|----------------- for And(a <= z, c > z)
|(-a + b)*(-a + c)
|
| 2
| ------ for c = z
< -a + b
|
| 2*b - 2*z
|---------------- for And(b >= z, c < z)
|(-a + b)*(b - c)
|
\ 0 otherwise
References
==========
.. [1] https://en.wikipedia.org/wiki/Triangular_distribution
.. [2] http://mathworld.wolfram.com/TriangularDistribution.html
"""
return rv(name, TriangularDistribution, (a, b, c))
#-------------------------------------------------------------------------------
# Uniform distribution ---------------------------------------------------------
class UniformDistribution(SingleContinuousDistribution):
_argnames = ('left', 'right')
@property
def set(self):
return Interval(self.left, self.right)
@staticmethod
def check(left, right):
_value_check(left < right, "Lower limit should be less than Upper limit.")
def pdf(self, x):
left, right = self.left, self.right
return Piecewise(
(S.One/(right - left), And(left <= x, x <= right)),
(S.Zero, True)
)
def _cdf(self, x):
left, right = self.left, self.right
return Piecewise(
(S.Zero, x < left),
((x - left)/(right - left), x <= right),
(S.One, True)
)
def _characteristic_function(self, t):
left, right = self.left, self.right
return Piecewise(((exp(I*t*right) - exp(I*t*left)) / (I*t*(right - left)), Ne(t, 0)),
(S.One, True))
def _moment_generating_function(self, t):
left, right = self.left, self.right
return Piecewise(((exp(t*right) - exp(t*left)) / (t * (right - left)), Ne(t, 0)),
(S.One, True))
def expectation(self, expr, var, **kwargs):
from sympy.functions.elementary.miscellaneous import (Max, Min)
kwargs['evaluate'] = True
result = SingleContinuousDistribution.expectation(self, expr, var, **kwargs)
result = result.subs({Max(self.left, self.right): self.right,
Min(self.left, self.right): self.left})
return result
def Uniform(name, left, right):
r"""
Create a continuous random variable with a uniform distribution.
Explanation
===========
The density of the uniform distribution is given by
.. math::
f(x) := \begin{cases}
\frac{1}{b - a} & \text{for } x \in [a,b] \\
0 & \text{otherwise}
\end{cases}
with :math:`x \in [a,b]`.
Parameters
==========
a : Real number, :math:`-\infty < a` the left boundary
b : Real number, :math:`a < b < \infty` the right boundary
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Uniform, density, cdf, E, variance
>>> from sympy import Symbol, simplify
>>> a = Symbol("a", negative=True)
>>> b = Symbol("b", positive=True)
>>> z = Symbol("z")
>>> X = Uniform("x", a, b)
>>> density(X)(z)
Piecewise((1/(-a + b), (b >= z) & (a <= z)), (0, True))
>>> cdf(X)(z)
Piecewise((0, a > z), ((-a + z)/(-a + b), b >= z), (1, True))
>>> E(X)
a/2 + b/2
>>> simplify(variance(X))
a**2/12 - a*b/6 + b**2/12
References
==========
.. [1] https://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29
.. [2] http://mathworld.wolfram.com/UniformDistribution.html
"""
return rv(name, UniformDistribution, (left, right))
#-------------------------------------------------------------------------------
# UniformSum distribution ------------------------------------------------------
class UniformSumDistribution(SingleContinuousDistribution):
_argnames = ('n',)
@property
def set(self):
return Interval(0, self.n)
@staticmethod
def check(n):
_value_check((n > 0, n.is_integer),
"Parameter n must be positive integer.")
def pdf(self, x):
n = self.n
k = Dummy("k")
return 1/factorial(
n - 1)*Sum((-1)**k*binomial(n, k)*(x - k)**(n - 1), (k, 0, floor(x)))
def _cdf(self, x):
n = self.n
k = Dummy("k")
return Piecewise((S.Zero, x < 0),
(1/factorial(n)*Sum((-1)**k*binomial(n, k)*(x - k)**(n),
(k, 0, floor(x))), x <= n),
(S.One, True))
def _characteristic_function(self, t):
return ((exp(I*t) - 1) / (I*t))**self.n
def _moment_generating_function(self, t):
return ((exp(t) - 1) / t)**self.n
def UniformSum(name, n):
r"""
Create a continuous random variable with an Irwin-Hall distribution.
Explanation
===========
The probability distribution function depends on a single parameter
$n$ which is an integer.
The density of the Irwin-Hall distribution is given by
.. math ::
f(x) := \frac{1}{(n-1)!}\sum_{k=0}^{\left\lfloor x\right\rfloor}(-1)^k
\binom{n}{k}(x-k)^{n-1}
Parameters
==========
n : A positive Integer, `n > 0`
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import UniformSum, density, cdf
>>> from sympy import Symbol, pprint
>>> n = Symbol("n", integer=True)
>>> z = Symbol("z")
>>> X = UniformSum("x", n)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
floor(z)
___
\ `
\ k n - 1 /n\
) (-1) *(-k + z) *| |
/ \k/
/__,
k = 0
--------------------------------
(n - 1)!
>>> cdf(X)(z)
Piecewise((0, z < 0), (Sum((-1)**_k*(-_k + z)**n*binomial(n, _k),
(_k, 0, floor(z)))/factorial(n), n >= z), (1, True))
Compute cdf with specific 'x' and 'n' values as follows :
>>> cdf(UniformSum("x", 5), evaluate=False)(2).doit()
9/40
The argument evaluate=False prevents an attempt at evaluation
of the sum for general n, before the argument 2 is passed.
References
==========
.. [1] https://en.wikipedia.org/wiki/Uniform_sum_distribution
.. [2] http://mathworld.wolfram.com/UniformSumDistribution.html
"""
return rv(name, UniformSumDistribution, (n, ))
#-------------------------------------------------------------------------------
# VonMises distribution --------------------------------------------------------
class VonMisesDistribution(SingleContinuousDistribution):
_argnames = ('mu', 'k')
set = Interval(0, 2*pi)
@staticmethod
def check(mu, k):
_value_check(k > 0, "k must be positive")
def pdf(self, x):
mu, k = self.mu, self.k
return exp(k*cos(x-mu)) / (2*pi*besseli(0, k))
def VonMises(name, mu, k):
r"""
Create a Continuous Random Variable with a von Mises distribution.
Explanation
===========
The density of the von Mises distribution is given by
.. math::
f(x) := \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}
with :math:`x \in [0,2\pi]`.
Parameters
==========
mu : Real number
Measure of location.
k : Real number
Measure of concentration.
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import VonMises, density
>>> from sympy import Symbol, pprint
>>> mu = Symbol("mu")
>>> k = Symbol("k", positive=True)
>>> z = Symbol("z")
>>> X = VonMises("x", mu, k)
>>> D = density(X)(z)
>>> pprint(D, use_unicode=False)
k*cos(mu - z)
e
------------------
2*pi*besseli(0, k)
References
==========
.. [1] https://en.wikipedia.org/wiki/Von_Mises_distribution
.. [2] http://mathworld.wolfram.com/vonMisesDistribution.html
"""
return rv(name, VonMisesDistribution, (mu, k))
#-------------------------------------------------------------------------------
# Weibull distribution ---------------------------------------------------------
class WeibullDistribution(SingleContinuousDistribution):
_argnames = ('alpha', 'beta')
set = Interval(0, oo)
@staticmethod
def check(alpha, beta):
_value_check(alpha > 0, "Alpha must be positive")
_value_check(beta > 0, "Beta must be positive")
def pdf(self, x):
alpha, beta = self.alpha, self.beta
return beta * (x/alpha)**(beta - 1) * exp(-(x/alpha)**beta) / alpha
def Weibull(name, alpha, beta):
r"""
Create a continuous random variable with a Weibull distribution.
Explanation
===========
The density of the Weibull distribution is given by
.. math::
f(x) := \begin{cases}
\frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}
e^{-(x/\lambda)^{k}} & x\geq0\\
0 & x<0
\end{cases}
Parameters
==========
lambda : Real number, :math:`\lambda > 0` a scale
k : Real number, ``k > 0`` a shape
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import Weibull, density, E, variance
>>> from sympy import Symbol, simplify
>>> l = Symbol("lambda", positive=True)
>>> k = Symbol("k", positive=True)
>>> z = Symbol("z")
>>> X = Weibull("x", l, k)
>>> density(X)(z)
k*(z/lambda)**(k - 1)*exp(-(z/lambda)**k)/lambda
>>> simplify(E(X))
lambda*gamma(1 + 1/k)
>>> simplify(variance(X))
lambda**2*(-gamma(1 + 1/k)**2 + gamma(1 + 2/k))
References
==========
.. [1] https://en.wikipedia.org/wiki/Weibull_distribution
.. [2] http://mathworld.wolfram.com/WeibullDistribution.html
"""
return rv(name, WeibullDistribution, (alpha, beta))
#-------------------------------------------------------------------------------
# Wigner semicircle distribution -----------------------------------------------
class WignerSemicircleDistribution(SingleContinuousDistribution):
_argnames = ('R',)
@property
def set(self):
return Interval(-self.R, self.R)
@staticmethod
def check(R):
_value_check(R > 0, "Radius R must be positive.")
def pdf(self, x):
R = self.R
return 2/(pi*R**2)*sqrt(R**2 - x**2)
def _characteristic_function(self, t):
return Piecewise((2 * besselj(1, self.R*t) / (self.R*t), Ne(t, 0)),
(S.One, True))
def _moment_generating_function(self, t):
return Piecewise((2 * besseli(1, self.R*t) / (self.R*t), Ne(t, 0)),
(S.One, True))
def WignerSemicircle(name, R):
r"""
Create a continuous random variable with a Wigner semicircle distribution.
Explanation
===========
The density of the Wigner semicircle distribution is given by
.. math::
f(x) := \frac2{\pi R^2}\,\sqrt{R^2-x^2}
with :math:`x \in [-R,R]`.
Parameters
==========
R : Real number, `R > 0`, the radius
Returns
=======
A `RandomSymbol`.
Examples
========
>>> from sympy.stats import WignerSemicircle, density, E
>>> from sympy import Symbol
>>> R = Symbol("R", positive=True)
>>> z = Symbol("z")
>>> X = WignerSemicircle("x", R)
>>> density(X)(z)
2*sqrt(R**2 - z**2)/(pi*R**2)
>>> E(X)
0
References
==========
.. [1] https://en.wikipedia.org/wiki/Wigner_semicircle_distribution
.. [2] http://mathworld.wolfram.com/WignersSemicircleLaw.html
"""
return rv(name, WignerSemicircleDistribution, (R,))
|
755378d5fe88927c5795e9b0f58fdd5daced2454e517bf06f39b15624362f12a | from __future__ import print_function, division
import random
import itertools
from typing import (Sequence as tSequence, Union as tUnion, List as tList,
Tuple as tTuple, Set as tSet)
from sympy.concrete.summations import Sum
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, Lambda)
from sympy.core.mul import Mul
from sympy.core.numbers import (Integer, Rational, igcd, oo, pi)
from sympy.core.relational import (Eq, Ge, Gt, Le, Lt, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol)
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.integers import ceiling
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.special.gamma_functions import gamma
from sympy.logic.boolalg import (And, Not, Or)
from sympy.matrices.common import NonSquareMatrixError
from sympy.matrices.dense import (Matrix, eye, ones, zeros)
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.matrices.expressions.special import Identity
from sympy.matrices.immutable import ImmutableMatrix
from sympy.sets.conditionset import ConditionSet
from sympy.sets.contains import Contains
from sympy.sets.fancysets import Range
from sympy.sets.sets import (FiniteSet, Intersection, Interval, Set, Union)
from sympy.solvers.solveset import linsolve
from sympy.tensor.indexed import (Indexed, IndexedBase)
from sympy.core.relational import Relational
from sympy.logic.boolalg import Boolean
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import strongly_connected_components
from sympy.stats.joint_rv import JointDistribution
from sympy.stats.joint_rv_types import JointDistributionHandmade
from sympy.stats.rv import (RandomIndexedSymbol, random_symbols, RandomSymbol,
_symbol_converter, _value_check, pspace, given,
dependent, is_random, sample_iter, Distribution,
Density)
from sympy.stats.stochastic_process import StochasticPSpace
from sympy.stats.symbolic_probability import Probability, Expectation
from sympy.stats.frv_types import Bernoulli, BernoulliDistribution, FiniteRV
from sympy.stats.drv_types import Poisson, PoissonDistribution
from sympy.stats.crv_types import Normal, NormalDistribution, Gamma, GammaDistribution
from sympy.core.sympify import _sympify, sympify
EmptySet = S.EmptySet
__all__ = [
'StochasticProcess',
'DiscreteTimeStochasticProcess',
'DiscreteMarkovChain',
'TransitionMatrixOf',
'StochasticStateSpaceOf',
'GeneratorMatrixOf',
'ContinuousMarkovChain',
'BernoulliProcess',
'PoissonProcess',
'WienerProcess',
'GammaProcess'
]
@is_random.register(Indexed)
def _(x):
return is_random(x.base)
@is_random.register(RandomIndexedSymbol) # type: ignore
def _(x):
return True
def _set_converter(itr):
"""
Helper function for converting list/tuple/set to Set.
If parameter is not an instance of list/tuple/set then
no operation is performed.
Returns
=======
Set
The argument converted to Set.
Raises
======
TypeError
If the argument is not an instance of list/tuple/set.
"""
if isinstance(itr, (list, tuple, set)):
itr = FiniteSet(*itr)
if not isinstance(itr, Set):
raise TypeError("%s is not an instance of list/tuple/set."%(itr))
return itr
def _state_converter(itr: tSequence) -> tUnion[Tuple, Range]:
"""
Helper function for converting list/tuple/set/Range/Tuple/FiniteSet
to tuple/Range.
"""
itr_ret: tUnion[Tuple, Range]
if isinstance(itr, (Tuple, set, FiniteSet)):
itr_ret = Tuple(*(sympify(i) if isinstance(i, str) else i for i in itr))
elif isinstance(itr, (list, tuple)):
# check if states are unique
if len(set(itr)) != len(itr):
raise ValueError('The state space must have unique elements.')
itr_ret = Tuple(*(sympify(i) if isinstance(i, str) else i for i in itr))
elif isinstance(itr, Range):
# the only ordered set in SymPy I know of
# try to convert to tuple
try:
itr_ret = Tuple(*(sympify(i) if isinstance(i, str) else i for i in itr))
except (TypeError, ValueError):
itr_ret = itr
else:
raise TypeError("%s is not an instance of list/tuple/set/Range/Tuple/FiniteSet." % (itr))
return itr_ret
def _sym_sympify(arg):
"""
Converts an arbitrary expression to a type that can be used inside SymPy.
As generally strings are unwise to use in the expressions,
it returns the Symbol of argument if the string type argument is passed.
Parameters
=========
arg: The parameter to be converted to be used in SymPy.
Returns
=======
The converted parameter.
"""
if isinstance(arg, str):
return Symbol(arg)
else:
return _sympify(arg)
def _matrix_checks(matrix):
if not isinstance(matrix, (Matrix, MatrixSymbol, ImmutableMatrix)):
raise TypeError("Transition probabilities either should "
"be a Matrix or a MatrixSymbol.")
if matrix.shape[0] != matrix.shape[1]:
raise NonSquareMatrixError("%s is not a square matrix"%(matrix))
if isinstance(matrix, Matrix):
matrix = ImmutableMatrix(matrix.tolist())
return matrix
class StochasticProcess(Basic):
"""
Base class for all the stochastic processes whether
discrete or continuous.
Parameters
==========
sym: Symbol or str
state_space: Set
The state space of the stochastic process, by default S.Reals.
For discrete sets it is zero indexed.
See Also
========
DiscreteTimeStochasticProcess
"""
index_set = S.Reals
def __new__(cls, sym, state_space=S.Reals, **kwargs):
sym = _symbol_converter(sym)
state_space = _set_converter(state_space)
return Basic.__new__(cls, sym, state_space)
@property
def symbol(self):
return self.args[0]
@property
def state_space(self) -> tUnion[FiniteSet, Range]:
if not isinstance(self.args[1], (FiniteSet, Range)):
assert isinstance(self.args[1], Tuple)
return FiniteSet(*self.args[1])
return self.args[1]
def _deprecation_warn_distribution(self):
SymPyDeprecationWarning(
feature="Calling distribution with RandomIndexedSymbol",
useinstead="distribution with just timestamp as argument",
issue=20078,
deprecated_since_version="1.7.1"
).warn()
def distribution(self, key=None):
if key is None:
self._deprecation_warn_distribution()
return Distribution()
def density(self, x):
return Density()
def __call__(self, time):
"""
Overridden in ContinuousTimeStochasticProcess.
"""
raise NotImplementedError("Use [] for indexing discrete time stochastic process.")
def __getitem__(self, time):
"""
Overridden in DiscreteTimeStochasticProcess.
"""
raise NotImplementedError("Use () for indexing continuous time stochastic process.")
def probability(self, condition):
raise NotImplementedError()
def joint_distribution(self, *args):
"""
Computes the joint distribution of the random indexed variables.
Parameters
==========
args: iterable
The finite list of random indexed variables/the key of a stochastic
process whose joint distribution has to be computed.
Returns
=======
JointDistribution
The joint distribution of the list of random indexed variables.
An unevaluated object is returned if it is not possible to
compute the joint distribution.
Raises
======
ValueError: When the arguments passed are not of type RandomIndexSymbol
or Number.
"""
args = list(args)
for i, arg in enumerate(args):
if S(arg).is_Number:
if self.index_set.is_subset(S.Integers):
args[i] = self.__getitem__(arg)
else:
args[i] = self.__call__(arg)
elif not isinstance(arg, RandomIndexedSymbol):
raise ValueError("Expected a RandomIndexedSymbol or "
"key not %s"%(type(arg)))
if args[0].pspace.distribution == Distribution():
return JointDistribution(*args)
density = Lambda(tuple(args),
expr=Mul.fromiter(arg.pspace.process.density(arg) for arg in args))
return JointDistributionHandmade(density)
def expectation(self, condition, given_condition):
raise NotImplementedError("Abstract method for expectation queries.")
def sample(self):
raise NotImplementedError("Abstract method for sampling queries.")
class DiscreteTimeStochasticProcess(StochasticProcess):
"""
Base class for all discrete stochastic processes.
"""
def __getitem__(self, time):
"""
For indexing discrete time stochastic processes.
Returns
=======
RandomIndexedSymbol
"""
time = sympify(time)
if not time.is_symbol and time not in self.index_set:
raise IndexError("%s is not in the index set of %s"%(time, self.symbol))
idx_obj = Indexed(self.symbol, time)
pspace_obj = StochasticPSpace(self.symbol, self, self.distribution(time))
return RandomIndexedSymbol(idx_obj, pspace_obj)
class ContinuousTimeStochasticProcess(StochasticProcess):
"""
Base class for all continuous time stochastic process.
"""
def __call__(self, time):
"""
For indexing continuous time stochastic processes.
Returns
=======
RandomIndexedSymbol
"""
time = sympify(time)
if not time.is_symbol and time not in self.index_set:
raise IndexError("%s is not in the index set of %s"%(time, self.symbol))
func_obj = Function(self.symbol)(time)
pspace_obj = StochasticPSpace(self.symbol, self, self.distribution(time))
return RandomIndexedSymbol(func_obj, pspace_obj)
class TransitionMatrixOf(Boolean):
"""
Assumes that the matrix is the transition matrix
of the process.
"""
def __new__(cls, process, matrix):
if not isinstance(process, DiscreteMarkovChain):
raise ValueError("Currently only DiscreteMarkovChain "
"support TransitionMatrixOf.")
matrix = _matrix_checks(matrix)
return Basic.__new__(cls, process, matrix)
process = property(lambda self: self.args[0])
matrix = property(lambda self: self.args[1])
class GeneratorMatrixOf(TransitionMatrixOf):
"""
Assumes that the matrix is the generator matrix
of the process.
"""
def __new__(cls, process, matrix):
if not isinstance(process, ContinuousMarkovChain):
raise ValueError("Currently only ContinuousMarkovChain "
"support GeneratorMatrixOf.")
matrix = _matrix_checks(matrix)
return Basic.__new__(cls, process, matrix)
class StochasticStateSpaceOf(Boolean):
def __new__(cls, process, state_space):
if not isinstance(process, (DiscreteMarkovChain, ContinuousMarkovChain)):
raise ValueError("Currently only DiscreteMarkovChain and ContinuousMarkovChain "
"support StochasticStateSpaceOf.")
state_space = _state_converter(state_space)
if isinstance(state_space, Range):
ss_size = ceiling((state_space.stop - state_space.start) / state_space.step)
else:
ss_size = len(state_space)
state_index = Range(ss_size)
return Basic.__new__(cls, process, state_index)
process = property(lambda self: self.args[0])
state_index = property(lambda self: self.args[1])
class MarkovProcess(StochasticProcess):
"""
Contains methods that handle queries
common to Markov processes.
"""
@property
def number_of_states(self) -> tUnion[Integer, Symbol]:
"""
The number of states in the Markov Chain.
"""
return _sympify(self.args[2].shape[0]) # type: ignore
@property
def _state_index(self):
"""
Returns state index as Range.
"""
return self.args[1]
@classmethod
def _sanity_checks(cls, state_space, trans_probs):
# Try to never have None as state_space or trans_probs.
# This helps a lot if we get it done at the start.
if (state_space is None) and (trans_probs is None):
_n = Dummy('n', integer=True, nonnegative=True)
state_space = _state_converter(Range(_n))
trans_probs = _matrix_checks(MatrixSymbol('_T', _n, _n))
elif state_space is None:
trans_probs = _matrix_checks(trans_probs)
state_space = _state_converter(Range(trans_probs.shape[0]))
elif trans_probs is None:
state_space = _state_converter(state_space)
if isinstance(state_space, Range):
_n = ceiling((state_space.stop - state_space.start) / state_space.step)
else:
_n = len(state_space)
trans_probs = MatrixSymbol('_T', _n, _n)
else:
state_space = _state_converter(state_space)
trans_probs = _matrix_checks(trans_probs)
# Range object doesn't want to give a symbolic size
# so we do it ourselves.
if isinstance(state_space, Range):
ss_size = ceiling((state_space.stop - state_space.start) / state_space.step)
else:
ss_size = len(state_space)
if ss_size != trans_probs.shape[0]:
raise ValueError('The size of the state space and the number of '
'rows of the transition matrix must be the same.')
return state_space, trans_probs
def _extract_information(self, given_condition):
"""
Helper function to extract information, like,
transition matrix/generator matrix, state space, etc.
"""
if isinstance(self, DiscreteMarkovChain):
trans_probs = self.transition_probabilities
state_index = self._state_index
elif isinstance(self, ContinuousMarkovChain):
trans_probs = self.generator_matrix
state_index = self._state_index
if isinstance(given_condition, And):
gcs = given_condition.args
given_condition = S.true
for gc in gcs:
if isinstance(gc, TransitionMatrixOf):
trans_probs = gc.matrix
if isinstance(gc, StochasticStateSpaceOf):
state_index = gc.state_index
if isinstance(gc, Relational):
given_condition = given_condition & gc
if isinstance(given_condition, TransitionMatrixOf):
trans_probs = given_condition.matrix
given_condition = S.true
if isinstance(given_condition, StochasticStateSpaceOf):
state_index = given_condition.state_index
given_condition = S.true
return trans_probs, state_index, given_condition
def _check_trans_probs(self, trans_probs, row_sum=1):
"""
Helper function for checking the validity of transition
probabilities.
"""
if not isinstance(trans_probs, MatrixSymbol):
rows = trans_probs.tolist()
for row in rows:
if (sum(row) - row_sum) != 0:
raise ValueError("Values in a row must sum to %s. "
"If you are using Float or floats then please use Rational."%(row_sum))
def _work_out_state_index(self, state_index, given_condition, trans_probs):
"""
Helper function to extract state space if there
is a random symbol in the given condition.
"""
# if given condition is None, then there is no need to work out
# state_space from random variables
if given_condition != None:
rand_var = list(given_condition.atoms(RandomSymbol) -
given_condition.atoms(RandomIndexedSymbol))
if len(rand_var) == 1:
state_index = rand_var[0].pspace.set
# `not None` is `True`. So the old test fails for symbolic sizes.
# Need to build the statement differently.
sym_cond = not self.number_of_states.is_Integer
cond1 = not sym_cond and len(state_index) != trans_probs.shape[0]
if cond1:
raise ValueError("state space is not compatible with the transition probabilities.")
if not isinstance(trans_probs.shape[0], Symbol):
state_index = FiniteSet(*[i for i in range(trans_probs.shape[0])])
return state_index
@cacheit
def _preprocess(self, given_condition, evaluate):
"""
Helper function for pre-processing the information.
"""
is_insufficient = False
if not evaluate: # avoid pre-processing if the result is not to be evaluated
return (True, None, None, None)
# extracting transition matrix and state space
trans_probs, state_index, given_condition = self._extract_information(given_condition)
# given_condition does not have sufficient information
# for computations
if trans_probs is None or \
given_condition is None:
is_insufficient = True
else:
# checking transition probabilities
if isinstance(self, DiscreteMarkovChain):
self._check_trans_probs(trans_probs, row_sum=1)
elif isinstance(self, ContinuousMarkovChain):
self._check_trans_probs(trans_probs, row_sum=0)
# working out state space
state_index = self._work_out_state_index(state_index, given_condition, trans_probs)
return is_insufficient, trans_probs, state_index, given_condition
def replace_with_index(self, condition):
if isinstance(condition, Relational):
lhs, rhs = condition.lhs, condition.rhs
if not isinstance(lhs, RandomIndexedSymbol):
lhs, rhs = rhs, lhs
condition = type(condition)(self.index_of.get(lhs, lhs),
self.index_of.get(rhs, rhs))
return condition
def probability(self, condition, given_condition=None, evaluate=True, **kwargs):
"""
Handles probability queries for Markov process.
Parameters
==========
condition: Relational
given_condition: Relational/And
Returns
=======
Probability
If the information is not sufficient.
Expr
In all other cases.
Note
====
Any information passed at the time of query overrides
any information passed at the time of object creation like
transition probabilities, state space.
Pass the transition matrix using TransitionMatrixOf,
generator matrix using GeneratorMatrixOf and state space
using StochasticStateSpaceOf in given_condition using & or And.
"""
check, mat, state_index, new_given_condition = \
self._preprocess(given_condition, evaluate)
rv = list(condition.atoms(RandomIndexedSymbol))
symbolic = False
for sym in rv:
if sym.key.is_symbol:
symbolic = True
break
if check:
return Probability(condition, new_given_condition)
if isinstance(self, ContinuousMarkovChain):
trans_probs = self.transition_probabilities(mat)
elif isinstance(self, DiscreteMarkovChain):
trans_probs = mat
condition = self.replace_with_index(condition)
given_condition = self.replace_with_index(given_condition)
new_given_condition = self.replace_with_index(new_given_condition)
if isinstance(condition, Relational):
if isinstance(new_given_condition, And):
gcs = new_given_condition.args
else:
gcs = (new_given_condition, )
min_key_rv = list(new_given_condition.atoms(RandomIndexedSymbol))
if len(min_key_rv):
min_key_rv = min_key_rv[0]
for r in rv:
if min_key_rv.key.is_symbol or r.key.is_symbol:
continue
if min_key_rv.key > r.key:
return Probability(condition)
else:
min_key_rv = None
return Probability(condition)
if symbolic:
return self._symbolic_probability(condition, new_given_condition, rv, min_key_rv)
if len(rv) > 1:
rv[0] = condition.lhs
rv[1] = condition.rhs
if rv[0].key < rv[1].key:
rv[0], rv[1] = rv[1], rv[0]
if isinstance(condition, Gt):
condition = Lt(condition.lhs, condition.rhs)
elif isinstance(condition, Lt):
condition = Gt(condition.lhs, condition.rhs)
elif isinstance(condition, Ge):
condition = Le(condition.lhs, condition.rhs)
elif isinstance(condition, Le):
condition = Ge(condition.lhs, condition.rhs)
s = Rational(0, 1)
n = len(self.state_space)
if isinstance(condition, (Eq, Ne)):
for i in range(0, n):
s += self.probability(Eq(rv[0], i), Eq(rv[1], i)) * self.probability(Eq(rv[1], i), new_given_condition)
return s if isinstance(condition, Eq) else 1 - s
else:
upper = 0
greater = False
if isinstance(condition, (Ge, Lt)):
upper = 1
if isinstance(condition, (Ge, Gt)):
greater = True
for i in range(0, n):
if i <= n//2:
for j in range(0, i + upper):
s += self.probability(Eq(rv[0], i), Eq(rv[1], j)) * self.probability(Eq(rv[1], j), new_given_condition)
else:
s += self.probability(Eq(rv[0], i), new_given_condition)
for j in range(i + upper, n):
s -= self.probability(Eq(rv[0], i), Eq(rv[1], j)) * self.probability(Eq(rv[1], j), new_given_condition)
return s if greater else 1 - s
rv = rv[0]
states = condition.as_set()
prob, gstate = dict(), None
for gc in gcs:
if gc.has(min_key_rv):
if gc.has(Probability):
p, gp = (gc.rhs, gc.lhs) if isinstance(gc.lhs, Probability) \
else (gc.lhs, gc.rhs)
gr = gp.args[0]
gset = Intersection(gr.as_set(), state_index)
gstate = list(gset)[0]
prob[gset] = p
else:
_, gstate = (gc.lhs.key, gc.rhs) if isinstance(gc.lhs, RandomIndexedSymbol) \
else (gc.rhs.key, gc.lhs)
if not all(k in self.index_set for k in (rv.key, min_key_rv.key)):
raise IndexError("The timestamps of the process are not in it's index set.")
states = Intersection(states, state_index) if not isinstance(self.number_of_states, Symbol) else states
for state in Union(states, FiniteSet(gstate)):
if not state.is_Integer or Ge(state, mat.shape[0]) is True:
raise IndexError("No information is available for (%s, %s) in "
"transition probabilities of shape, (%s, %s). "
"State space is zero indexed."
%(gstate, state, mat.shape[0], mat.shape[1]))
if prob:
gstates = Union(*prob.keys())
if len(gstates) == 1:
gstate = list(gstates)[0]
gprob = list(prob.values())[0]
prob[gstates] = gprob
elif len(gstates) == len(state_index) - 1:
gstate = list(state_index - gstates)[0]
gprob = S.One - sum(prob.values())
prob[state_index - gstates] = gprob
else:
raise ValueError("Conflicting information.")
else:
gprob = S.One
if min_key_rv == rv:
return sum([prob[FiniteSet(state)] for state in states])
if isinstance(self, ContinuousMarkovChain):
return gprob * sum([trans_probs(rv.key - min_key_rv.key).__getitem__((gstate, state))
for state in states])
if isinstance(self, DiscreteMarkovChain):
return gprob * sum([(trans_probs**(rv.key - min_key_rv.key)).__getitem__((gstate, state))
for state in states])
if isinstance(condition, Not):
expr = condition.args[0]
return S.One - self.probability(expr, given_condition, evaluate, **kwargs)
if isinstance(condition, And):
compute_later, state2cond, conds = [], dict(), condition.args
for expr in conds:
if isinstance(expr, Relational):
ris = list(expr.atoms(RandomIndexedSymbol))[0]
if state2cond.get(ris, None) is None:
state2cond[ris] = S.true
state2cond[ris] &= expr
else:
compute_later.append(expr)
ris = []
for ri in state2cond:
ris.append(ri)
cset = Intersection(state2cond[ri].as_set(), state_index)
if len(cset) == 0:
return S.Zero
state2cond[ri] = cset.as_relational(ri)
sorted_ris = sorted(ris, key=lambda ri: ri.key)
prod = self.probability(state2cond[sorted_ris[0]], given_condition, evaluate, **kwargs)
for i in range(1, len(sorted_ris)):
ri, prev_ri = sorted_ris[i], sorted_ris[i-1]
if not isinstance(state2cond[ri], Eq):
raise ValueError("The process is in multiple states at %s, unable to determine the probability."%(ri))
mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat)
prod *= self.probability(state2cond[ri], state2cond[prev_ri]
& mat_of
& StochasticStateSpaceOf(self, state_index),
evaluate, **kwargs)
for expr in compute_later:
prod *= self.probability(expr, given_condition, evaluate, **kwargs)
return prod
if isinstance(condition, Or):
return sum([self.probability(expr, given_condition, evaluate, **kwargs)
for expr in condition.args])
raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been "
"implemented yet."%(condition, given_condition))
def _symbolic_probability(self, condition, new_given_condition, rv, min_key_rv):
#Function to calculate probability for queries with symbols
if isinstance(condition, Relational):
curr_state = new_given_condition.rhs if isinstance(new_given_condition.lhs, RandomIndexedSymbol) \
else new_given_condition.lhs
next_state = condition.rhs if isinstance(condition.lhs, RandomIndexedSymbol) \
else condition.lhs
if isinstance(condition, (Eq, Ne)):
if isinstance(self, DiscreteMarkovChain):
P = self.transition_probabilities**(rv[0].key - min_key_rv.key)
else:
P = exp(self.generator_matrix*(rv[0].key - min_key_rv.key))
prob = P[curr_state, next_state] if isinstance(condition, Eq) else 1 - P[curr_state, next_state]
return Piecewise((prob, rv[0].key > min_key_rv.key), (Probability(condition), True))
else:
upper = 1
greater = False
if isinstance(condition, (Ge, Lt)):
upper = 0
if isinstance(condition, (Ge, Gt)):
greater = True
k = Dummy('k')
condition = Eq(condition.lhs, k) if isinstance(condition.lhs, RandomIndexedSymbol)\
else Eq(condition.rhs, k)
total = Sum(self.probability(condition, new_given_condition), (k, next_state + upper, self.state_space._sup))
return Piecewise((total, rv[0].key > min_key_rv.key), (Probability(condition), True)) if greater\
else Piecewise((1 - total, rv[0].key > min_key_rv.key), (Probability(condition), True))
else:
return Probability(condition, new_given_condition)
def expectation(self, expr, condition=None, evaluate=True, **kwargs):
"""
Handles expectation queries for markov process.
Parameters
==========
expr: RandomIndexedSymbol, Relational, Logic
Condition for which expectation has to be computed. Must
contain a RandomIndexedSymbol of the process.
condition: Relational, Logic
The given conditions under which computations should be done.
Returns
=======
Expectation
Unevaluated object if computations cannot be done due to
insufficient information.
Expr
In all other cases when the computations are successful.
Note
====
Any information passed at the time of query overrides
any information passed at the time of object creation like
transition probabilities, state space.
Pass the transition matrix using TransitionMatrixOf,
generator matrix using GeneratorMatrixOf and state space
using StochasticStateSpaceOf in given_condition using & or And.
"""
check, mat, state_index, condition = \
self._preprocess(condition, evaluate)
if check:
return Expectation(expr, condition)
rvs = random_symbols(expr)
if isinstance(expr, Expr) and isinstance(condition, Eq) \
and len(rvs) == 1:
# handle queries similar to E(f(X[i]), Eq(X[i-m], <some-state>))
condition=self.replace_with_index(condition)
state_index=self.replace_with_index(state_index)
rv = list(rvs)[0]
lhsg, rhsg = condition.lhs, condition.rhs
if not isinstance(lhsg, RandomIndexedSymbol):
lhsg, rhsg = (rhsg, lhsg)
if rhsg not in state_index:
raise ValueError("%s state is not in the state space."%(rhsg))
if rv.key < lhsg.key:
raise ValueError("Incorrect given condition is given, expectation "
"time %s < time %s"%(rv.key, rv.key))
mat_of = TransitionMatrixOf(self, mat) if isinstance(self, DiscreteMarkovChain) else GeneratorMatrixOf(self, mat)
cond = condition & mat_of & \
StochasticStateSpaceOf(self, state_index)
func = lambda s: self.probability(Eq(rv, s), cond) * expr.subs(rv, self._state_index[s])
return sum([func(s) for s in state_index])
raise NotImplementedError("Mechanism for handling (%s, %s) queries hasn't been "
"implemented yet."%(expr, condition))
class DiscreteMarkovChain(DiscreteTimeStochasticProcess, MarkovProcess):
"""
Represents a finite discrete time-homogeneous Markov chain.
This type of Markov Chain can be uniquely characterised by
its (ordered) state space and its one-step transition probability
matrix.
Parameters
==========
sym:
The name given to the Markov Chain
state_space:
Optional, by default, Range(n)
trans_probs:
Optional, by default, MatrixSymbol('_T', n, n)
Examples
========
>>> from sympy.stats import DiscreteMarkovChain, TransitionMatrixOf, P, E
>>> from sympy import Matrix, MatrixSymbol, Eq, symbols
>>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
>>> YS = DiscreteMarkovChain("Y")
>>> Y.state_space
{0, 1, 2}
>>> Y.transition_probabilities
Matrix([
[0.5, 0.2, 0.3],
[0.2, 0.5, 0.3],
[0.2, 0.3, 0.5]])
>>> TS = MatrixSymbol('T', 3, 3)
>>> P(Eq(YS[3], 2), Eq(YS[1], 1) & TransitionMatrixOf(YS, TS))
T[0, 2]*T[1, 0] + T[1, 1]*T[1, 2] + T[1, 2]*T[2, 2]
>>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2)
0.36
Probabilities will be calculated based on indexes rather
than state names. For example, with the Sunny-Cloudy-Rainy
model with string state names:
>>> from sympy.core.symbol import Str
>>> Y = DiscreteMarkovChain("Y", [Str('Sunny'), Str('Cloudy'), Str('Rainy')], T)
>>> P(Eq(Y[3], 2), Eq(Y[1], 1)).round(2)
0.36
This gives the same answer as the ``[0, 1, 2]`` state space.
Currently, there is no support for state names within probability
and expectation statements. Here is a work-around using ``Str``:
>>> P(Eq(Str('Rainy'), Y[3]), Eq(Y[1], Str('Cloudy'))).round(2)
0.36
Symbol state names can also be used:
>>> sunny, cloudy, rainy = symbols('Sunny, Cloudy, Rainy')
>>> Y = DiscreteMarkovChain("Y", [sunny, cloudy, rainy], T)
>>> P(Eq(Y[3], rainy), Eq(Y[1], cloudy)).round(2)
0.36
Expectations will be calculated as follows:
>>> E(Y[3], Eq(Y[1], cloudy))
0.38*Cloudy + 0.36*Rainy + 0.26*Sunny
Probability of expressions with multiple RandomIndexedSymbols
can also be calculated provided there is only 1 RandomIndexedSymbol
in the given condition. It is always better to use Rational instead
of floating point numbers for the probabilities in the
transition matrix to avoid errors.
>>> from sympy import Gt, Le, Rational
>>> T = Matrix([[Rational(5, 10), Rational(3, 10), Rational(2, 10)], [Rational(2, 10), Rational(7, 10), Rational(1, 10)], [Rational(3, 10), Rational(3, 10), Rational(4, 10)]])
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
>>> P(Eq(Y[3], Y[1]), Eq(Y[0], 0)).round(3)
0.409
>>> P(Gt(Y[3], Y[1]), Eq(Y[0], 0)).round(2)
0.36
>>> P(Le(Y[15], Y[10]), Eq(Y[8], 2)).round(7)
0.6963328
Symbolic probability queries are also supported
>>> from sympy import symbols, Matrix, Rational, Eq, Gt
>>> from sympy.stats import P, DiscreteMarkovChain
>>> a, b, c, d = symbols('a b c d')
>>> T = Matrix([[Rational(1, 10), Rational(4, 10), Rational(5, 10)], [Rational(3, 10), Rational(4, 10), Rational(3, 10)], [Rational(7, 10), Rational(2, 10), Rational(1, 10)]])
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
>>> query = P(Eq(Y[a], b), Eq(Y[c], d))
>>> query.subs({a:10, b:2, c:5, d:1}).round(4)
0.3096
>>> P(Eq(Y[10], 2), Eq(Y[5], 1)).evalf().round(4)
0.3096
>>> query_gt = P(Gt(Y[a], b), Eq(Y[c], d))
>>> query_gt.subs({a:21, b:0, c:5, d:0}).evalf().round(5)
0.64705
>>> P(Gt(Y[21], 0), Eq(Y[5], 0)).round(5)
0.64705
There is limited support for arbitrarily sized states:
>>> n = symbols('n', nonnegative=True, integer=True)
>>> T = MatrixSymbol('T', n, n)
>>> Y = DiscreteMarkovChain("Y", trans_probs=T)
>>> Y.state_space
Range(0, n, 1)
>>> query = P(Eq(Y[a], b), Eq(Y[c], d))
>>> query.subs({a:10, b:2, c:5, d:1})
(T**5)[1, 2]
References
==========
.. [1] https://en.wikipedia.org/wiki/Markov_chain#Discrete-time_Markov_chain
.. [2] https://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter11.pdf
"""
index_set = S.Naturals0
def __new__(cls, sym, state_space=None, trans_probs=None):
sym = _symbol_converter(sym)
state_space, trans_probs = MarkovProcess._sanity_checks(state_space, trans_probs)
obj = Basic.__new__(cls, sym, state_space, trans_probs) # type: ignore
indices = dict()
if isinstance(obj.number_of_states, Integer):
for index, state in enumerate(obj._state_index):
indices[state] = index
obj.index_of = indices
return obj
@property
def transition_probabilities(self):
"""
Transition probabilities of discrete Markov chain,
either an instance of Matrix or MatrixSymbol.
"""
return self.args[2]
def communication_classes(self) -> tList[tTuple[tList[Basic], Boolean, Integer]]:
"""
Returns the list of communication classes that partition
the states of the markov chain.
A communication class is defined to be a set of states
such that every state in that set is reachable from
every other state in that set. Due to its properties
this forms a class in the mathematical sense.
Communication classes are also known as recurrence
classes.
Returns
=======
classes
The ``classes`` are a list of tuples. Each
tuple represents a single communication class
with its properties. The first element in the
tuple is the list of states in the class, the
second element is whether the class is recurrent
and the third element is the period of the
communication class.
Examples
========
>>> from sympy.stats import DiscreteMarkovChain
>>> from sympy import Matrix
>>> T = Matrix([[0, 1, 0],
... [1, 0, 0],
... [1, 0, 0]])
>>> X = DiscreteMarkovChain('X', [1, 2, 3], T)
>>> classes = X.communication_classes()
>>> for states, is_recurrent, period in classes:
... states, is_recurrent, period
([1, 2], True, 2)
([3], False, 1)
From this we can see that states ``1`` and ``2``
communicate, are recurrent and have a period
of 2. We can also see state ``3`` is transient
with a period of 1.
Notes
=====
The algorithm used is of order ``O(n**2)`` where
``n`` is the number of states in the markov chain.
It uses Tarjan's algorithm to find the classes
themselves and then it uses a breadth-first search
algorithm to find each class's periodicity.
Most of the algorithm's components approach ``O(n)``
as the matrix becomes more and more sparse.
References
==========
.. [1] http://www.columbia.edu/~ww2040/4701Sum07/4701-06-Notes-MCII.pdf
.. [2] http://cecas.clemson.edu/~shierd/Shier/markov.pdf
.. [3] https://ujcontent.uj.ac.za/vital/access/services/Download/uj:7506/CONTENT1
.. [4] https://www.mathworks.com/help/econ/dtmc.classify.html
"""
n = self.number_of_states
T = self.transition_probabilities
if isinstance(T, MatrixSymbol):
raise NotImplementedError("Cannot perform the operation with a symbolic matrix.")
# begin Tarjan's algorithm
V = Range(n)
# don't use state names. Rather use state
# indexes since we use them for matrix
# indexing here and later onward
E = [(i, j) for i in V for j in V if T[i, j] != 0]
classes = strongly_connected_components((V, E))
# end Tarjan's algorithm
recurrence = []
periods = []
for class_ in classes:
# begin recurrent check (similar to self._check_trans_probs())
submatrix = T[class_, class_] # get the submatrix with those states
is_recurrent = S.true
rows = submatrix.tolist()
for row in rows:
if (sum(row) - 1) != 0:
is_recurrent = S.false
break
recurrence.append(is_recurrent)
# end recurrent check
# begin breadth-first search
non_tree_edge_values: tSet[int] = set()
visited = {class_[0]}
newly_visited = {class_[0]}
level = {class_[0]: 0}
current_level = 0
done = False # imitate a do-while loop
while not done: # runs at most len(class_) times
done = len(visited) == len(class_)
current_level += 1
# this loop and the while loop above run a combined len(class_) number of times.
# so this triple nested loop runs through each of the n states once.
for i in newly_visited:
# the loop below runs len(class_) number of times
# complexity is around about O(n * avg(len(class_)))
newly_visited = {j for j in class_ if T[i, j] != 0}
new_tree_edges = newly_visited.difference(visited)
for j in new_tree_edges:
level[j] = current_level
new_non_tree_edges = newly_visited.intersection(visited)
new_non_tree_edge_values = {level[i]-level[j]+1 for j in new_non_tree_edges}
non_tree_edge_values = non_tree_edge_values.union(new_non_tree_edge_values)
visited = visited.union(new_tree_edges)
# igcd needs at least 2 arguments
positive_ntev = {val_e for val_e in non_tree_edge_values if val_e > 0}
if len(positive_ntev) == 0:
periods.append(len(class_))
elif len(positive_ntev) == 1:
periods.append(positive_ntev.pop())
else:
periods.append(igcd(*positive_ntev))
# end breadth-first search
# convert back to the user's state names
classes = [[_sympify(self._state_index[i]) for i in class_] for class_ in classes]
return list(zip(classes, recurrence, map(Integer,periods)))
def fundamental_matrix(self):
"""
Each entry fundamental matrix can be interpreted as
the expected number of times the chains is in state j
if it started in state i.
References
==========
.. [1] https://lips.cs.princeton.edu/the-fundamental-matrix-of-a-finite-markov-chain/
"""
_, _, _, Q = self.decompose()
if Q.shape[0] > 0: # if non-ergodic
I = eye(Q.shape[0])
if (I - Q).det() == 0:
raise ValueError("The fundamental matrix doesn't exist.")
return (I - Q).inv().as_immutable()
else: # if ergodic
P = self.transition_probabilities
I = eye(P.shape[0])
w = self.fixed_row_vector()
W = Matrix([list(w) for i in range(0, P.shape[0])])
if (I - P + W).det() == 0:
raise ValueError("The fundamental matrix doesn't exist.")
return (I - P + W).inv().as_immutable()
def absorbing_probabilities(self):
"""
Computes the absorbing probabilities, i.e.,
the ij-th entry of the matrix denotes the
probability of Markov chain being absorbed
in state j starting from state i.
"""
_, _, R, _ = self.decompose()
N = self.fundamental_matrix()
if R is None or N is None:
return None
return N*R
def absorbing_probabilites(self):
SymPyDeprecationWarning(
feature="absorbing_probabilites",
useinstead="absorbing_probabilities",
issue=20042,
deprecated_since_version="1.7"
).warn()
return self.absorbing_probabilities()
def is_regular(self):
tuples = self.communication_classes()
if len(tuples) == 0:
return S.false # not defined for a 0x0 matrix
classes, _, periods = list(zip(*tuples))
return And(len(classes) == 1, periods[0] == 1)
def is_ergodic(self):
tuples = self.communication_classes()
if len(tuples) == 0:
return S.false # not defined for a 0x0 matrix
classes, _, _ = list(zip(*tuples))
return S(len(classes) == 1)
def is_absorbing_state(self, state):
trans_probs = self.transition_probabilities
if isinstance(trans_probs, ImmutableMatrix) and \
state < trans_probs.shape[0]:
return S(trans_probs[state, state]) is S.One
def is_absorbing_chain(self):
states, A, B, C = self.decompose()
r = A.shape[0]
return And(r > 0, A == Identity(r).as_explicit())
def stationary_distribution(self, condition_set=False) -> tUnion[ImmutableMatrix, ConditionSet, Lambda]:
r"""
The stationary distribution is any row vector, p, that solves p = pP,
is row stochastic and each element in p must be nonnegative.
That means in matrix form: :math:`(P-I)^T p^T = 0` and
:math:`(1, \dots, 1) p = 1`
where ``P`` is the one-step transition matrix.
All time-homogeneous Markov Chains with a finite state space
have at least one stationary distribution. In addition, if
a finite time-homogeneous Markov Chain is irreducible, the
stationary distribution is unique.
Parameters
==========
condition_set : bool
If the chain has a symbolic size or transition matrix,
it will return a ``Lambda`` if ``False`` and return a
``ConditionSet`` if ``True``.
Examples
========
>>> from sympy.stats import DiscreteMarkovChain
>>> from sympy import Matrix, S
An irreducible Markov Chain
>>> T = Matrix([[S(1)/2, S(1)/2, 0],
... [S(4)/5, S(1)/5, 0],
... [1, 0, 0]])
>>> X = DiscreteMarkovChain('X', trans_probs=T)
>>> X.stationary_distribution()
Matrix([[8/13, 5/13, 0]])
A reducible Markov Chain
>>> T = Matrix([[S(1)/2, S(1)/2, 0],
... [S(4)/5, S(1)/5, 0],
... [0, 0, 1]])
>>> X = DiscreteMarkovChain('X', trans_probs=T)
>>> X.stationary_distribution()
Matrix([[8/13 - 8*tau0/13, 5/13 - 5*tau0/13, tau0]])
>>> Y = DiscreteMarkovChain('Y')
>>> Y.stationary_distribution()
Lambda((wm, _T), Eq(wm*_T, wm))
>>> Y.stationary_distribution(condition_set=True)
ConditionSet(wm, Eq(wm*_T, wm))
References
==========
.. [1] https://www.probabilitycourse.com/chapter11/11_2_6_stationary_and_limiting_distributions.php
.. [2] https://galton.uchicago.edu/~yibi/teaching/stat317/2014/Lectures/Lecture4_6up.pdf
See Also
========
sympy.stats.DiscreteMarkovChain.limiting_distribution
"""
trans_probs = self.transition_probabilities
n = self.number_of_states
if n == 0:
return ImmutableMatrix(Matrix([[]]))
# symbolic matrix version
if isinstance(trans_probs, MatrixSymbol):
wm = MatrixSymbol('wm', 1, n)
if condition_set:
return ConditionSet(wm, Eq(wm * trans_probs, wm))
else:
return Lambda((wm, trans_probs), Eq(wm * trans_probs, wm))
# numeric matrix version
a = Matrix(trans_probs - Identity(n)).T
a[0, 0:n] = ones(1, n) # type: ignore
b = zeros(n, 1)
b[0, 0] = 1
soln = list(linsolve((a, b)))[0]
return ImmutableMatrix([[sol for sol in soln]])
def fixed_row_vector(self):
"""
A wrapper for ``stationary_distribution()``.
"""
return self.stationary_distribution()
@property
def limiting_distribution(self):
"""
The fixed row vector is the limiting
distribution of a discrete Markov chain.
"""
return self.fixed_row_vector()
def decompose(self) -> tTuple[tList[Basic], ImmutableMatrix, ImmutableMatrix, ImmutableMatrix]:
"""
Decomposes the transition matrix into submatrices with
special properties.
The transition matrix can be decomposed into 4 submatrices:
- A - the submatrix from recurrent states to recurrent states.
- B - the submatrix from transient to recurrent states.
- C - the submatrix from transient to transient states.
- O - the submatrix of zeros for recurrent to transient states.
Returns
=======
states, A, B, C
``states`` - a list of state names with the first being
the recurrent states and the last being
the transient states in the order
of the row names of A and then the row names of C.
``A`` - the submatrix from recurrent states to recurrent states.
``B`` - the submatrix from transient to recurrent states.
``C`` - the submatrix from transient to transient states.
Examples
========
>>> from sympy.stats import DiscreteMarkovChain
>>> from sympy import Matrix, S
One can decompose this chain for example:
>>> T = Matrix([[S(1)/2, S(1)/2, 0, 0, 0],
... [S(2)/5, S(1)/5, S(2)/5, 0, 0],
... [0, 0, 1, 0, 0],
... [0, 0, S(1)/2, S(1)/2, 0],
... [S(1)/2, 0, 0, 0, S(1)/2]])
>>> X = DiscreteMarkovChain('X', trans_probs=T)
>>> states, A, B, C = X.decompose()
>>> states
[2, 0, 1, 3, 4]
>>> A # recurrent to recurrent
Matrix([[1]])
>>> B # transient to recurrent
Matrix([
[ 0],
[2/5],
[1/2],
[ 0]])
>>> C # transient to transient
Matrix([
[1/2, 1/2, 0, 0],
[2/5, 1/5, 0, 0],
[ 0, 0, 1/2, 0],
[1/2, 0, 0, 1/2]])
This means that state 2 is the only absorbing state
(since A is a 1x1 matrix). B is a 4x1 matrix since
the 4 remaining transient states all merge into reccurent
state 2. And C is the 4x4 matrix that shows how the
transient states 0, 1, 3, 4 all interact.
See Also
========
sympy.stats.DiscreteMarkovChain.communication_classes
sympy.stats.DiscreteMarkovChain.canonical_form
References
==========
.. [1] https://en.wikipedia.org/wiki/Absorbing_Markov_chain
.. [2] http://people.brandeis.edu/~igusa/Math56aS08/Math56a_S08_notes015.pdf
"""
trans_probs = self.transition_probabilities
classes = self.communication_classes()
r_states = []
t_states = []
for states, recurrent, period in classes:
if recurrent:
r_states += states
else:
t_states += states
states = r_states + t_states
indexes = [self.index_of[state] for state in states] # type: ignore
A = Matrix(len(r_states), len(r_states),
lambda i, j: trans_probs[indexes[i], indexes[j]])
B = Matrix(len(t_states), len(r_states),
lambda i, j: trans_probs[indexes[len(r_states) + i], indexes[j]])
C = Matrix(len(t_states), len(t_states),
lambda i, j: trans_probs[indexes[len(r_states) + i], indexes[len(r_states) + j]])
return states, A.as_immutable(), B.as_immutable(), C.as_immutable()
def canonical_form(self) -> tTuple[tList[Basic], ImmutableMatrix]:
"""
Reorders the one-step transition matrix
so that recurrent states appear first and transient
states appear last. Other representations include inserting
transient states first and recurrent states last.
Returns
=======
states, P_new
``states`` is the list that describes the order of the
new states in the matrix
so that the ith element in ``states`` is the state of the
ith row of A.
``P_new`` is the new transition matrix in canonical form.
Examples
========
>>> from sympy.stats import DiscreteMarkovChain
>>> from sympy import Matrix, S
You can convert your chain into canonical form:
>>> T = Matrix([[S(1)/2, S(1)/2, 0, 0, 0],
... [S(2)/5, S(1)/5, S(2)/5, 0, 0],
... [0, 0, 1, 0, 0],
... [0, 0, S(1)/2, S(1)/2, 0],
... [S(1)/2, 0, 0, 0, S(1)/2]])
>>> X = DiscreteMarkovChain('X', list(range(1, 6)), trans_probs=T)
>>> states, new_matrix = X.canonical_form()
>>> states
[3, 1, 2, 4, 5]
>>> new_matrix
Matrix([
[ 1, 0, 0, 0, 0],
[ 0, 1/2, 1/2, 0, 0],
[2/5, 2/5, 1/5, 0, 0],
[1/2, 0, 0, 1/2, 0],
[ 0, 1/2, 0, 0, 1/2]])
The new states are [3, 1, 2, 4, 5] and you can
create a new chain with this and its canonical
form will remain the same (since it is already
in canonical form).
>>> X = DiscreteMarkovChain('X', states, new_matrix)
>>> states, new_matrix = X.canonical_form()
>>> states
[3, 1, 2, 4, 5]
>>> new_matrix
Matrix([
[ 1, 0, 0, 0, 0],
[ 0, 1/2, 1/2, 0, 0],
[2/5, 2/5, 1/5, 0, 0],
[1/2, 0, 0, 1/2, 0],
[ 0, 1/2, 0, 0, 1/2]])
This is not limited to absorbing chains:
>>> T = Matrix([[0, 5, 5, 0, 0],
... [0, 0, 0, 10, 0],
... [5, 0, 5, 0, 0],
... [0, 10, 0, 0, 0],
... [0, 3, 0, 3, 4]])/10
>>> X = DiscreteMarkovChain('X', trans_probs=T)
>>> states, new_matrix = X.canonical_form()
>>> states
[1, 3, 0, 2, 4]
>>> new_matrix
Matrix([
[ 0, 1, 0, 0, 0],
[ 1, 0, 0, 0, 0],
[ 1/2, 0, 0, 1/2, 0],
[ 0, 0, 1/2, 1/2, 0],
[3/10, 3/10, 0, 0, 2/5]])
See Also
========
sympy.stats.DiscreteMarkovChain.communication_classes
sympy.stats.DiscreteMarkovChain.decompose
References
==========
.. [1] https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470316887.app1
.. [2] http://www.columbia.edu/~ww2040/6711F12/lect1023big.pdf
"""
states, A, B, C = self.decompose()
O = zeros(A.shape[0], C.shape[1])
return states, BlockMatrix([[A, O], [B, C]]).as_explicit()
def sample(self):
"""
Returns
=======
sample: iterator object
iterator object containing the sample
"""
if not isinstance(self.transition_probabilities, (Matrix, ImmutableMatrix)):
raise ValueError("Transition Matrix must be provided for sampling")
Tlist = self.transition_probabilities.tolist()
samps = [random.choice(list(self.state_space))]
yield samps[0]
time = 1
densities = {}
for state in self.state_space:
states = list(self.state_space)
densities[state] = {states[i]: Tlist[state][i]
for i in range(len(states))}
while time < S.Infinity:
samps.append((next(sample_iter(FiniteRV("_", densities[samps[time - 1]])))))
yield samps[time]
time += 1
class ContinuousMarkovChain(ContinuousTimeStochasticProcess, MarkovProcess):
"""
Represents continuous time Markov chain.
Parameters
==========
sym: Symbol/str
state_space: Set
Optional, by default, S.Reals
gen_mat: Matrix/ImmutableMatrix/MatrixSymbol
Optional, by default, None
Examples
========
>>> from sympy.stats import ContinuousMarkovChain, P
>>> from sympy import Matrix, S, Eq, Gt
>>> G = Matrix([[-S(1), S(1)], [S(1), -S(1)]])
>>> C = ContinuousMarkovChain('C', state_space=[0, 1], gen_mat=G)
>>> C.limiting_distribution()
Matrix([[1/2, 1/2]])
>>> C.state_space
{0, 1}
>>> C.generator_matrix
Matrix([
[-1, 1],
[ 1, -1]])
Probability queries are supported
>>> P(Eq(C(1.96), 0), Eq(C(0.78), 1)).round(5)
0.45279
>>> P(Gt(C(1.7), 0), Eq(C(0.82), 1)).round(5)
0.58602
Probability of expressions with multiple RandomIndexedSymbols
can also be calculated provided there is only 1 RandomIndexedSymbol
in the given condition. It is always better to use Rational instead
of floating point numbers for the probabilities in the
generator matrix to avoid errors.
>>> from sympy import Gt, Le, Rational
>>> G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]])
>>> C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G)
>>> P(Eq(C(3.92), C(1.75)), Eq(C(0.46), 0)).round(5)
0.37933
>>> P(Gt(C(3.92), C(1.75)), Eq(C(0.46), 0)).round(5)
0.34211
>>> P(Le(C(1.57), C(3.14)), Eq(C(1.22), 1)).round(4)
0.7143
Symbolic probability queries are also supported
>>> from sympy import S, symbols, Matrix, Rational, Eq, Gt
>>> from sympy.stats import P, ContinuousMarkovChain
>>> a,b,c,d = symbols('a b c d')
>>> G = Matrix([[-S(1), Rational(1, 10), Rational(9, 10)], [Rational(2, 5), -S(1), Rational(3, 5)], [Rational(1, 2), Rational(1, 2), -S(1)]])
>>> C = ContinuousMarkovChain('C', state_space=[0, 1, 2], gen_mat=G)
>>> query = P(Eq(C(a), b), Eq(C(c), d))
>>> query.subs({a:3.65, b:2, c:1.78, d:1}).evalf().round(10)
0.4002723175
>>> P(Eq(C(3.65), 2), Eq(C(1.78), 1)).round(10)
0.4002723175
>>> query_gt = P(Gt(C(a), b), Eq(C(c), d))
>>> query_gt.subs({a:43.2, b:0, c:3.29, d:2}).evalf().round(10)
0.6832579186
>>> P(Gt(C(43.2), 0), Eq(C(3.29), 2)).round(10)
0.6832579186
References
==========
.. [1] https://en.wikipedia.org/wiki/Markov_chain#Continuous-time_Markov_chain
.. [2] http://u.math.biu.ac.il/~amirgi/CTMCnotes.pdf
"""
index_set = S.Reals
def __new__(cls, sym, state_space=None, gen_mat=None):
sym = _symbol_converter(sym)
state_space, gen_mat = MarkovProcess._sanity_checks(state_space, gen_mat)
obj = Basic.__new__(cls, sym, state_space, gen_mat)
indices = dict()
if isinstance(obj.number_of_states, Integer):
for index, state in enumerate(obj.state_space):
indices[state] = index
obj.index_of = indices
return obj
@property
def generator_matrix(self):
return self.args[2]
@cacheit
def transition_probabilities(self, gen_mat=None):
t = Dummy('t')
if isinstance(gen_mat, (Matrix, ImmutableMatrix)) and \
gen_mat.is_diagonalizable():
# for faster computation use diagonalized generator matrix
Q, D = gen_mat.diagonalize()
return Lambda(t, Q*exp(t*D)*Q.inv())
if gen_mat != None:
return Lambda(t, exp(t*gen_mat))
def limiting_distribution(self):
gen_mat = self.generator_matrix
if gen_mat is None:
return None
if isinstance(gen_mat, MatrixSymbol):
wm = MatrixSymbol('wm', 1, gen_mat.shape[0])
return Lambda((wm, gen_mat), Eq(wm*gen_mat, wm))
w = IndexedBase('w')
wi = [w[i] for i in range(gen_mat.shape[0])]
wm = Matrix([wi])
eqs = (wm*gen_mat).tolist()[0]
eqs.append(sum(wi) - 1)
soln = list(linsolve(eqs, wi))[0]
return ImmutableMatrix([[sol for sol in soln]])
class BernoulliProcess(DiscreteTimeStochasticProcess):
"""
The Bernoulli process consists of repeated
independent Bernoulli process trials with the same parameter `p`.
It's assumed that the probability `p` applies to every
trial and that the outcomes of each trial
are independent of all the rest. Therefore Bernoulli Processs
is Discrete State and Discrete Time Stochastic Process.
Parameters
==========
sym: Symbol/str
success: Integer/str
The event which is considered to be success, by default is 1.
failure: Integer/str
The event which is considered to be failure, by default is 0.
p: Real Number between 0 and 1
Represents the probability of getting success.
Examples
========
>>> from sympy.stats import BernoulliProcess, P, E
>>> from sympy import Eq, Gt
>>> B = BernoulliProcess("B", p=0.7, success=1, failure=0)
>>> B.state_space
{0, 1}
>>> (B.p).round(2)
0.70
>>> B.success
1
>>> B.failure
0
>>> X = B[1] + B[2] + B[3]
>>> P(Eq(X, 0)).round(2)
0.03
>>> P(Eq(X, 2)).round(2)
0.44
>>> P(Eq(X, 4)).round(2)
0
>>> P(Gt(X, 1)).round(2)
0.78
>>> P(Eq(B[1], 0) & Eq(B[2], 1) & Eq(B[3], 0) & Eq(B[4], 1)).round(2)
0.04
>>> B.joint_distribution(B[1], B[2])
JointDistributionHandmade(Lambda((B[1], B[2]), Piecewise((0.7, Eq(B[1], 1)),
(0.3, Eq(B[1], 0)), (0, True))*Piecewise((0.7, Eq(B[2], 1)), (0.3, Eq(B[2], 0)),
(0, True))))
>>> E(2*B[1] + B[2]).round(2)
2.10
>>> P(B[1] < 1).round(2)
0.30
References
==========
.. [1] https://en.wikipedia.org/wiki/Bernoulli_process
.. [2] https://mathcs.clarku.edu/~djoyce/ma217/bernoulli.pdf
"""
index_set = S.Naturals0
def __new__(cls, sym, p, success=1, failure=0):
_value_check(p >= 0 and p <= 1, 'Value of p must be between 0 and 1.')
sym = _symbol_converter(sym)
p = _sympify(p)
success = _sym_sympify(success)
failure = _sym_sympify(failure)
return Basic.__new__(cls, sym, p, success, failure)
@property
def symbol(self):
return self.args[0]
@property
def p(self):
return self.args[1]
@property
def success(self):
return self.args[2]
@property
def failure(self):
return self.args[3]
@property
def state_space(self):
return _set_converter([self.success, self.failure])
def distribution(self, key=None):
if key is None:
self._deprecation_warn_distribution()
return BernoulliDistribution(self.p)
return BernoulliDistribution(self.p, self.success, self.failure)
def simple_rv(self, rv):
return Bernoulli(rv.name, p=self.p,
succ=self.success, fail=self.failure)
def expectation(self, expr, condition=None, evaluate=True, **kwargs):
"""
Computes expectation.
Parameters
==========
expr: RandomIndexedSymbol, Relational, Logic
Condition for which expectation has to be computed. Must
contain a RandomIndexedSymbol of the process.
condition: Relational, Logic
The given conditions under which computations should be done.
Returns
=======
Expectation of the RandomIndexedSymbol.
"""
return _SubstituteRV._expectation(expr, condition, evaluate, **kwargs)
def probability(self, condition, given_condition=None, evaluate=True, **kwargs):
"""
Computes probability.
Parameters
==========
condition: Relational
Condition for which probability has to be computed. Must
contain a RandomIndexedSymbol of the process.
given_condition: Relational/And
The given conditions under which computations should be done.
Returns
=======
Probability of the condition.
"""
return _SubstituteRV._probability(condition, given_condition, evaluate, **kwargs)
def density(self, x):
return Piecewise((self.p, Eq(x, self.success)),
(1 - self.p, Eq(x, self.failure)),
(S.Zero, True))
class _SubstituteRV:
"""
Internal class to handle the queries of expectation and probability
by substitution.
"""
@staticmethod
def _rvindexed_subs(expr, condition=None):
"""
Substitutes the RandomIndexedSymbol with the RandomSymbol with
same name, distribution and probability as RandomIndexedSymbol.
Parameters
==========
expr: RandomIndexedSymbol, Relational, Logic
Condition for which expectation has to be computed. Must
contain a RandomIndexedSymbol of the process.
condition: Relational, Logic
The given conditions under which computations should be done.
"""
rvs_expr = random_symbols(expr)
if len(rvs_expr) != 0:
swapdict_expr = {}
for rv in rvs_expr:
if isinstance(rv, RandomIndexedSymbol):
newrv = rv.pspace.process.simple_rv(rv) # substitute with equivalent simple rv
swapdict_expr[rv] = newrv
expr = expr.subs(swapdict_expr)
rvs_cond = random_symbols(condition)
if len(rvs_cond)!=0:
swapdict_cond = {}
for rv in rvs_cond:
if isinstance(rv, RandomIndexedSymbol):
newrv = rv.pspace.process.simple_rv(rv)
swapdict_cond[rv] = newrv
condition = condition.subs(swapdict_cond)
return expr, condition
@classmethod
def _expectation(self, expr, condition=None, evaluate=True, **kwargs):
"""
Internal method for computing expectation of indexed RV.
Parameters
==========
expr: RandomIndexedSymbol, Relational, Logic
Condition for which expectation has to be computed. Must
contain a RandomIndexedSymbol of the process.
condition: Relational, Logic
The given conditions under which computations should be done.
Returns
=======
Expectation of the RandomIndexedSymbol.
"""
new_expr, new_condition = self._rvindexed_subs(expr, condition)
if not is_random(new_expr):
return new_expr
new_pspace = pspace(new_expr)
if new_condition is not None:
new_expr = given(new_expr, new_condition)
if new_expr.is_Add: # As E is Linear
return Add(*[new_pspace.compute_expectation(
expr=arg, evaluate=evaluate, **kwargs)
for arg in new_expr.args])
return new_pspace.compute_expectation(
new_expr, evaluate=evaluate, **kwargs)
@classmethod
def _probability(self, condition, given_condition=None, evaluate=True, **kwargs):
"""
Internal method for computing probability of indexed RV
Parameters
==========
condition: Relational
Condition for which probability has to be computed. Must
contain a RandomIndexedSymbol of the process.
given_condition: Relational/And
The given conditions under which computations should be done.
Returns
=======
Probability of the condition.
"""
new_condition, new_givencondition = self._rvindexed_subs(condition, given_condition)
if isinstance(new_givencondition, RandomSymbol):
condrv = random_symbols(new_condition)
if len(condrv) == 1 and condrv[0] == new_givencondition:
return BernoulliDistribution(self._probability(new_condition), 0, 1)
if any(dependent(rv, new_givencondition) for rv in condrv):
return Probability(new_condition, new_givencondition)
else:
return self._probability(new_condition)
if new_givencondition is not None and \
not isinstance(new_givencondition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (new_givencondition))
if new_givencondition == False or new_condition == False:
return S.Zero
if new_condition == True:
return S.One
if not isinstance(new_condition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (new_condition))
if new_givencondition is not None: # If there is a condition
# Recompute on new conditional expr
return self._probability(given(new_condition, new_givencondition, **kwargs), **kwargs)
result = pspace(new_condition).probability(new_condition, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def get_timerv_swaps(expr, condition):
"""
Finds the appropriate interval for each time stamp in expr by parsing
the given condition and returns intervals for each timestamp and
dictionary that maps variable time-stamped Random Indexed Symbol to its
corresponding Random Indexed variable with fixed time stamp.
Parameters
==========
expr: SymPy Expression
Expression containing Random Indexed Symbols with variable time stamps
condition: Relational/Boolean Expression
Expression containing time bounds of variable time stamps in expr
Examples
========
>>> from sympy.stats.stochastic_process_types import get_timerv_swaps, PoissonProcess
>>> from sympy import symbols, Contains, Interval
>>> x, t, d = symbols('x t d', positive=True)
>>> X = PoissonProcess("X", 3)
>>> get_timerv_swaps(x*X(t), Contains(t, Interval.Lopen(0, 1)))
([Interval.Lopen(0, 1)], {X(t): X(1)})
>>> get_timerv_swaps((X(t)**2 + X(d)**2), Contains(t, Interval.Lopen(0, 1))
... & Contains(d, Interval.Ropen(1, 4))) # doctest: +SKIP
([Interval.Ropen(1, 4), Interval.Lopen(0, 1)], {X(d): X(3), X(t): X(1)})
Returns
=======
intervals: list
List of Intervals/FiniteSet on which each time stamp is defined
rv_swap: dict
Dictionary mapping variable time Random Indexed Symbol to constant time
Random Indexed Variable
"""
if not isinstance(condition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (condition))
expr_syms = list(expr.atoms(RandomIndexedSymbol))
if isinstance(condition, (And, Or)):
given_cond_args = condition.args
else: # single condition
given_cond_args = (condition, )
rv_swap = {}
intervals = []
for expr_sym in expr_syms:
for arg in given_cond_args:
if arg.has(expr_sym.key) and isinstance(expr_sym.key, Symbol):
intv = _set_converter(arg.args[1])
diff_key = intv._sup - intv._inf
if diff_key == oo:
raise ValueError("%s should have finite bounds" % str(expr_sym.name))
elif diff_key == S.Zero: # has singleton set
diff_key = intv._sup
rv_swap[expr_sym] = expr_sym.subs({expr_sym.key: diff_key})
intervals.append(intv)
return intervals, rv_swap
class CountingProcess(ContinuousTimeStochasticProcess):
"""
This class handles the common methods of the Counting Processes
such as Poisson, Wiener and Gamma Processes
"""
index_set = _set_converter(Interval(0, oo))
@property
def symbol(self):
return self.args[0]
def expectation(self, expr, condition=None, evaluate=True, **kwargs):
"""
Computes expectation
Parameters
==========
expr: RandomIndexedSymbol, Relational, Logic
Condition for which expectation has to be computed. Must
contain a RandomIndexedSymbol of the process.
condition: Relational, Boolean
The given conditions under which computations should be done, i.e,
the intervals on which each variable time stamp in expr is defined
Returns
=======
Expectation of the given expr
"""
if condition is not None:
intervals, rv_swap = get_timerv_swaps(expr, condition)
# they are independent when they have non-overlapping intervals
if len(intervals) == 1 or all(Intersection(*intv_comb) == EmptySet
for intv_comb in itertools.combinations(intervals, 2)):
if expr.is_Add:
return Add.fromiter(self.expectation(arg, condition)
for arg in expr.args)
expr = expr.subs(rv_swap)
else:
return Expectation(expr, condition)
return _SubstituteRV._expectation(expr, evaluate=evaluate, **kwargs)
def _solve_argwith_tworvs(self, arg):
if arg.args[0].key >= arg.args[1].key or isinstance(arg, Eq):
diff_key = abs(arg.args[0].key - arg.args[1].key)
rv = arg.args[0]
arg = arg.__class__(rv.pspace.process(diff_key), 0)
else:
diff_key = arg.args[1].key - arg.args[0].key
rv = arg.args[1]
arg = arg.__class__(rv.pspace.process(diff_key), 0)
return arg
def _solve_numerical(self, condition, given_condition=None):
if isinstance(condition, And):
args_list = list(condition.args)
else:
args_list = [condition]
if given_condition is not None:
if isinstance(given_condition, And):
args_list.extend(list(given_condition.args))
else:
args_list.extend([given_condition])
# sort the args based on timestamp to get the independent increments in
# each segment using all the condition args as well as given_condition args
args_list = sorted(args_list, key=lambda x: x.args[0].key)
result = []
cond_args = list(condition.args) if isinstance(condition, And) else [condition]
if args_list[0] in cond_args and not (is_random(args_list[0].args[0])
and is_random(args_list[0].args[1])):
result.append(_SubstituteRV._probability(args_list[0]))
if is_random(args_list[0].args[0]) and is_random(args_list[0].args[1]):
arg = self._solve_argwith_tworvs(args_list[0])
result.append(_SubstituteRV._probability(arg))
for i in range(len(args_list) - 1):
curr, nex = args_list[i], args_list[i + 1]
diff_key = nex.args[0].key - curr.args[0].key
working_set = curr.args[0].pspace.process.state_space
if curr.args[1] > nex.args[1]: #impossible condition so return 0
result.append(0)
break
if isinstance(curr, Eq):
working_set = Intersection(working_set, Interval.Lopen(curr.args[1], oo))
else:
working_set = Intersection(working_set, curr.as_set())
if isinstance(nex, Eq):
working_set = Intersection(working_set, Interval(-oo, nex.args[1]))
else:
working_set = Intersection(working_set, nex.as_set())
if working_set == EmptySet:
rv = Eq(curr.args[0].pspace.process(diff_key), 0)
result.append(_SubstituteRV._probability(rv))
else:
if working_set.is_finite_set:
if isinstance(curr, Eq) and isinstance(nex, Eq):
rv = Eq(curr.args[0].pspace.process(diff_key), len(working_set))
result.append(_SubstituteRV._probability(rv))
elif isinstance(curr, Eq) ^ isinstance(nex, Eq):
result.append(Add.fromiter(_SubstituteRV._probability(Eq(
curr.args[0].pspace.process(diff_key), x))
for x in range(len(working_set))))
else:
n = len(working_set)
result.append(Add.fromiter((n - x)*_SubstituteRV._probability(Eq(
curr.args[0].pspace.process(diff_key), x)) for x in range(n)))
else:
result.append(_SubstituteRV._probability(
curr.args[0].pspace.process(diff_key) <= working_set._sup - working_set._inf))
return Mul.fromiter(result)
def probability(self, condition, given_condition=None, evaluate=True, **kwargs):
"""
Computes probability.
Parameters
==========
condition: Relational
Condition for which probability has to be computed. Must
contain a RandomIndexedSymbol of the process.
given_condition: Relational, Boolean
The given conditions under which computations should be done, i.e,
the intervals on which each variable time stamp in expr is defined
Returns
=======
Probability of the condition
"""
check_numeric = True
if isinstance(condition, (And, Or)):
cond_args = condition.args
else:
cond_args = (condition, )
# check that condition args are numeric or not
if not all(arg.args[0].key.is_number for arg in cond_args):
check_numeric = False
if given_condition is not None:
check_given_numeric = True
if isinstance(given_condition, (And, Or)):
given_cond_args = given_condition.args
else:
given_cond_args = (given_condition, )
# check that given condition args are numeric or not
if given_condition.has(Contains):
check_given_numeric = False
# Handle numerical queries
if check_numeric and check_given_numeric:
res = []
if isinstance(condition, Or):
res.append(Add.fromiter(self._solve_numerical(arg, given_condition)
for arg in condition.args))
if isinstance(given_condition, Or):
res.append(Add.fromiter(self._solve_numerical(condition, arg)
for arg in given_condition.args))
if res:
return Add.fromiter(res)
return self._solve_numerical(condition, given_condition)
# No numeric queries, go by Contains?... then check that all the
# given condition are in form of `Contains`
if not all(arg.has(Contains) for arg in given_cond_args):
raise ValueError("If given condition is passed with `Contains`, then "
"please pass the evaluated condition with its corresponding information "
"in terms of intervals of each time stamp to be passed in given condition.")
intervals, rv_swap = get_timerv_swaps(condition, given_condition)
# they are independent when they have non-overlapping intervals
if len(intervals) == 1 or all(Intersection(*intv_comb) == EmptySet
for intv_comb in itertools.combinations(intervals, 2)):
if isinstance(condition, And):
return Mul.fromiter(self.probability(arg, given_condition)
for arg in condition.args)
elif isinstance(condition, Or):
return Add.fromiter(self.probability(arg, given_condition)
for arg in condition.args)
condition = condition.subs(rv_swap)
else:
return Probability(condition, given_condition)
if check_numeric:
return self._solve_numerical(condition)
return _SubstituteRV._probability(condition, evaluate=evaluate, **kwargs)
class PoissonProcess(CountingProcess):
"""
The Poisson process is a counting process. It is usually used in scenarios
where we are counting the occurrences of certain events that appear
to happen at a certain rate, but completely at random.
Parameters
==========
sym: Symbol/str
lamda: Positive number
Rate of the process, ``lamda > 0``
Examples
========
>>> from sympy.stats import PoissonProcess, P, E
>>> from sympy import symbols, Eq, Ne, Contains, Interval
>>> X = PoissonProcess("X", lamda=3)
>>> X.state_space
Naturals0
>>> X.lamda
3
>>> t1, t2 = symbols('t1 t2', positive=True)
>>> P(X(t1) < 4)
(9*t1**3/2 + 9*t1**2/2 + 3*t1 + 1)*exp(-3*t1)
>>> P(Eq(X(t1), 2) | Ne(X(t1), 4), Contains(t1, Interval.Ropen(2, 4)))
1 - 36*exp(-6)
>>> P(Eq(X(t1), 2) & Eq(X(t2), 3), Contains(t1, Interval.Lopen(0, 2))
... & Contains(t2, Interval.Lopen(2, 4)))
648*exp(-12)
>>> E(X(t1))
3*t1
>>> E(X(t1)**2 + 2*X(t2), Contains(t1, Interval.Lopen(0, 1))
... & Contains(t2, Interval.Lopen(1, 2)))
18
>>> P(X(3) < 1, Eq(X(1), 0))
exp(-6)
>>> P(Eq(X(4), 3), Eq(X(2), 3))
exp(-6)
>>> P(X(2) <= 3, X(1) > 1)
5*exp(-3)
Merging two Poisson Processes
>>> Y = PoissonProcess("Y", lamda=4)
>>> Z = X + Y
>>> Z.lamda
7
Splitting a Poisson Process into two independent Poisson Processes
>>> N, M = Z.split(l1=2, l2=5)
>>> N.lamda, M.lamda
(2, 5)
References
==========
.. [1] https://www.probabilitycourse.com/chapter11/11_0_0_intro.php
.. [2] https://en.wikipedia.org/wiki/Poisson_point_process
"""
def __new__(cls, sym, lamda):
_value_check(lamda > 0, 'lamda should be a positive number.')
sym = _symbol_converter(sym)
lamda = _sympify(lamda)
return Basic.__new__(cls, sym, lamda)
@property
def lamda(self):
return self.args[1]
@property
def state_space(self):
return S.Naturals0
def distribution(self, key):
if isinstance(key, RandomIndexedSymbol):
self._deprecation_warn_distribution()
return PoissonDistribution(self.lamda*key.key)
return PoissonDistribution(self.lamda*key)
def density(self, x):
return (self.lamda*x.key)**x / factorial(x) * exp(-(self.lamda*x.key))
def simple_rv(self, rv):
return Poisson(rv.name, lamda=self.lamda*rv.key)
def __add__(self, other):
if not isinstance(other, PoissonProcess):
raise ValueError("Only instances of Poisson Process can be merged")
return PoissonProcess(Dummy(self.symbol.name + other.symbol.name),
self.lamda + other.lamda)
def split(self, l1, l2):
if _sympify(l1 + l2) != self.lamda:
raise ValueError("Sum of l1 and l2 should be %s" % str(self.lamda))
return PoissonProcess(Dummy("l1"), l1), PoissonProcess(Dummy("l2"), l2)
class WienerProcess(CountingProcess):
"""
The Wiener process is a real valued continuous-time stochastic process.
In physics it is used to study Brownian motion and therefore also known as
Brownian Motion.
Parameters
==========
sym: Symbol/str
Examples
========
>>> from sympy.stats import WienerProcess, P, E
>>> from sympy import symbols, Contains, Interval
>>> X = WienerProcess("X")
>>> X.state_space
Reals
>>> t1, t2 = symbols('t1 t2', positive=True)
>>> P(X(t1) < 7).simplify()
erf(7*sqrt(2)/(2*sqrt(t1)))/2 + 1/2
>>> P((X(t1) > 2) | (X(t1) < 4), Contains(t1, Interval.Ropen(2, 4))).simplify()
-erf(1)/2 + erf(2)/2 + 1
>>> E(X(t1))
0
>>> E(X(t1) + 2*X(t2), Contains(t1, Interval.Lopen(0, 1))
... & Contains(t2, Interval.Lopen(1, 2)))
0
References
==========
.. [1] https://www.probabilitycourse.com/chapter11/11_4_0_brownian_motion_wiener_process.php
.. [2] https://en.wikipedia.org/wiki/Wiener_process
"""
def __new__(cls, sym):
sym = _symbol_converter(sym)
return Basic.__new__(cls, sym)
@property
def state_space(self):
return S.Reals
def distribution(self, key):
if isinstance(key, RandomIndexedSymbol):
self._deprecation_warn_distribution()
return NormalDistribution(0, sqrt(key.key))
return NormalDistribution(0, sqrt(key))
def density(self, x):
return exp(-x**2/(2*x.key)) / (sqrt(2*pi)*sqrt(x.key))
def simple_rv(self, rv):
return Normal(rv.name, 0, sqrt(rv.key))
class GammaProcess(CountingProcess):
"""
A Gamma process is a random process with independent gamma distributed
increments. It is a pure-jump increasing Levy process.
Parameters
==========
sym: Symbol/str
lamda: Positive number
Jump size of the process, ``lamda > 0``
gamma: Positive number
Rate of jump arrivals, ``gamma > 0``
Examples
========
>>> from sympy.stats import GammaProcess, E, P, variance
>>> from sympy import symbols, Contains, Interval, Not
>>> t, d, x, l, g = symbols('t d x l g', positive=True)
>>> X = GammaProcess("X", l, g)
>>> E(X(t))
g*t/l
>>> variance(X(t)).simplify()
g*t/l**2
>>> X = GammaProcess('X', 1, 2)
>>> P(X(t) < 1).simplify()
lowergamma(2*t, 1)/gamma(2*t)
>>> P(Not((X(t) < 5) & (X(d) > 3)), Contains(t, Interval.Ropen(2, 4)) &
... Contains(d, Interval.Lopen(7, 8))).simplify()
-4*exp(-3) + 472*exp(-8)/3 + 1
>>> E(X(2) + x*E(X(5)))
10*x + 4
References
==========
.. [1] https://en.wikipedia.org/wiki/Gamma_process
"""
def __new__(cls, sym, lamda, gamma):
_value_check(lamda > 0, 'lamda should be a positive number')
_value_check(gamma > 0, 'gamma should be a positive number')
sym = _symbol_converter(sym)
gamma = _sympify(gamma)
lamda = _sympify(lamda)
return Basic.__new__(cls, sym, lamda, gamma)
@property
def lamda(self):
return self.args[1]
@property
def gamma(self):
return self.args[2]
@property
def state_space(self):
return _set_converter(Interval(0, oo))
def distribution(self, key):
if isinstance(key, RandomIndexedSymbol):
self._deprecation_warn_distribution()
return GammaDistribution(self.gamma*key.key, 1/self.lamda)
return GammaDistribution(self.gamma*key, 1/self.lamda)
def density(self, x):
k = self.gamma*x.key
theta = 1/self.lamda
return x**(k - 1) * exp(-x/theta) / (gamma(k)*theta**k)
def simple_rv(self, rv):
return Gamma(rv.name, self.gamma*rv.key, 1/self.lamda)
|
73e738911dd1c2d1e17bf2646cd4547d6ccd21cfe8b5c6116403e0492409355d | from sympy.core.basic import Basic
from sympy.core.mul import prod
from sympy.core.numbers import pi
from sympy.core.singleton import S
from sympy.functions.elementary.exponential import exp
from sympy.functions.special.gamma_functions import multigamma
from sympy.core.sympify import sympify, _sympify
from sympy.matrices import (ImmutableMatrix, Inverse, Trace, Determinant,
MatrixSymbol, MatrixBase, Transpose, MatrixSet,
matrix2numpy)
from sympy.stats.rv import (_value_check, RandomMatrixSymbol, NamedArgsMixin, PSpace,
_symbol_converter, MatrixDomain, Distribution)
from sympy.external import import_module
################################################################################
#------------------------Matrix Probability Space------------------------------#
################################################################################
class MatrixPSpace(PSpace):
"""
Represents probability space for
Matrix Distributions.
"""
def __new__(cls, sym, distribution, dim_n, dim_m):
sym = _symbol_converter(sym)
dim_n, dim_m = _sympify(dim_n), _sympify(dim_m)
if not (dim_n.is_integer and dim_m.is_integer):
raise ValueError("Dimensions should be integers")
return Basic.__new__(cls, sym, distribution, dim_n, dim_m)
distribution = property(lambda self: self.args[1])
symbol = property(lambda self: self.args[0])
@property
def domain(self):
return MatrixDomain(self.symbol, self.distribution.set)
@property
def value(self):
return RandomMatrixSymbol(self.symbol, self.args[2], self.args[3], self)
@property
def values(self):
return {self.value}
def compute_density(self, expr, *args):
rms = expr.atoms(RandomMatrixSymbol)
if len(rms) > 1 or (not isinstance(expr, RandomMatrixSymbol)):
raise NotImplementedError("Currently, no algorithm has been "
"implemented to handle general expressions containing "
"multiple matrix distributions.")
return self.distribution.pdf(expr)
def sample(self, size=(), library='scipy', seed=None):
"""
Internal sample method
Returns dictionary mapping RandomMatrixSymbol to realization value.
"""
return {self.value: self.distribution.sample(size, library=library, seed=seed)}
def rv(symbol, cls, args):
args = list(map(sympify, args))
dist = cls(*args)
dist.check(*args)
dim = dist.dimension
pspace = MatrixPSpace(symbol, dist, dim[0], dim[1])
return pspace.value
class SampleMatrixScipy:
"""Returns the sample from scipy of the given distribution"""
def __new__(cls, dist, size, seed=None):
return cls._sample_scipy(dist, size, seed)
@classmethod
def _sample_scipy(cls, dist, size, seed):
"""Sample from SciPy."""
from scipy import stats as scipy_stats
import numpy
scipy_rv_map = {
'WishartDistribution': lambda dist, size, rand_state: scipy_stats.wishart.rvs(
df=int(dist.n), scale=matrix2numpy(dist.scale_matrix, float), size=size),
'MatrixNormalDistribution': lambda dist, size, rand_state: scipy_stats.matrix_normal.rvs(
mean=matrix2numpy(dist.location_matrix, float),
rowcov=matrix2numpy(dist.scale_matrix_1, float),
colcov=matrix2numpy(dist.scale_matrix_2, float), size=size, random_state=rand_state)
}
sample_shape = {
'WishartDistribution': lambda dist: dist.scale_matrix.shape,
'MatrixNormalDistribution' : lambda dist: dist.location_matrix.shape
}
dist_list = scipy_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
samp = scipy_rv_map[dist.__class__.__name__](dist, prod(size), rand_state)
return samp.reshape(size + sample_shape[dist.__class__.__name__](dist))
class SampleMatrixNumpy:
"""Returns the sample from numpy of the given distribution"""
### TODO: Add tests after adding matrix distributions in numpy_rv_map
def __new__(cls, dist, size, seed=None):
return cls._sample_numpy(dist, size, seed)
@classmethod
def _sample_numpy(cls, dist, size, seed):
"""Sample from NumPy."""
numpy_rv_map = {
}
sample_shape = {
}
dist_list = numpy_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
import numpy
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
samp = numpy_rv_map[dist.__class__.__name__](dist, prod(size), rand_state)
return samp.reshape(size + sample_shape[dist.__class__.__name__](dist))
class SampleMatrixPymc:
"""Returns the sample from pymc3 of the given distribution"""
def __new__(cls, dist, size, seed=None):
return cls._sample_pymc3(dist, size, seed)
@classmethod
def _sample_pymc3(cls, dist, size, seed):
"""Sample from PyMC3."""
import pymc3
pymc3_rv_map = {
'MatrixNormalDistribution': lambda dist: pymc3.MatrixNormal('X',
mu=matrix2numpy(dist.location_matrix, float),
rowcov=matrix2numpy(dist.scale_matrix_1, float),
colcov=matrix2numpy(dist.scale_matrix_2, float),
shape=dist.location_matrix.shape),
'WishartDistribution': lambda dist: pymc3.WishartBartlett('X',
nu=int(dist.n), S=matrix2numpy(dist.scale_matrix, float))
}
sample_shape = {
'WishartDistribution': lambda dist: dist.scale_matrix.shape,
'MatrixNormalDistribution' : lambda dist: dist.location_matrix.shape
}
dist_list = pymc3_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
import logging
logging.getLogger("pymc3").setLevel(logging.ERROR)
with pymc3.Model():
pymc3_rv_map[dist.__class__.__name__](dist)
samps = pymc3.sample(draws=prod(size), chains=1, progressbar=False, random_seed=seed, return_inferencedata=False, compute_convergence_checks=False)['X']
return samps.reshape(size + sample_shape[dist.__class__.__name__](dist))
_get_sample_class_matrixrv = {
'scipy': SampleMatrixScipy,
'pymc3': SampleMatrixPymc,
'numpy': SampleMatrixNumpy
}
################################################################################
#-------------------------Matrix Distribution----------------------------------#
################################################################################
class MatrixDistribution(Distribution, NamedArgsMixin):
"""
Abstract class for Matrix Distribution.
"""
def __new__(cls, *args):
args = [ImmutableMatrix(arg) if isinstance(arg, list)
else _sympify(arg) for arg in args]
return Basic.__new__(cls, *args)
@staticmethod
def check(*args):
pass
def __call__(self, expr):
if isinstance(expr, list):
expr = ImmutableMatrix(expr)
return self.pdf(expr)
def sample(self, size=(), library='scipy', seed=None):
"""
Internal sample method
Returns dictionary mapping RandomSymbol to realization value.
"""
libraries = ['scipy', 'numpy', 'pymc3']
if library not in libraries:
raise NotImplementedError("Sampling from %s is not supported yet."
% str(library))
if not import_module(library):
raise ValueError("Failed to import %s" % library)
samps = _get_sample_class_matrixrv[library](self, size, seed)
if samps is not None:
return samps
raise NotImplementedError(
"Sampling for %s is not currently implemented from %s"
% (self.__class__.__name__, library)
)
################################################################################
#------------------------Matrix Distribution Types-----------------------------#
################################################################################
#-------------------------------------------------------------------------------
# Matrix Gamma distribution ----------------------------------------------------
class MatrixGammaDistribution(MatrixDistribution):
_argnames = ('alpha', 'beta', 'scale_matrix')
@staticmethod
def check(alpha, beta, scale_matrix):
if not isinstance(scale_matrix, MatrixSymbol):
_value_check(scale_matrix.is_positive_definite, "The shape "
"matrix must be positive definite.")
_value_check(scale_matrix.is_square, "Should "
"be square matrix")
_value_check(alpha.is_positive, "Shape parameter should be positive.")
_value_check(beta.is_positive, "Scale parameter should be positive.")
@property
def set(self):
k = self.scale_matrix.shape[0]
return MatrixSet(k, k, S.Reals)
@property
def dimension(self):
return self.scale_matrix.shape
def pdf(self, x):
alpha, beta, scale_matrix = self.alpha, self.beta, self.scale_matrix
p = scale_matrix.shape[0]
if isinstance(x, list):
x = ImmutableMatrix(x)
if not isinstance(x, (MatrixBase, MatrixSymbol)):
raise ValueError("%s should be an isinstance of Matrix "
"or MatrixSymbol" % str(x))
sigma_inv_x = - Inverse(scale_matrix)*x / beta
term1 = exp(Trace(sigma_inv_x))/((beta**(p*alpha)) * multigamma(alpha, p))
term2 = (Determinant(scale_matrix))**(-alpha)
term3 = (Determinant(x))**(alpha - S(p + 1)/2)
return term1 * term2 * term3
def MatrixGamma(symbol, alpha, beta, scale_matrix):
"""
Creates a random variable with Matrix Gamma Distribution.
The density of the said distribution can be found at [1].
Parameters
==========
alpha: Positive Real number
Shape Parameter
beta: Positive Real number
Scale Parameter
scale_matrix: Positive definite real square matrix
Scale Matrix
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import density, MatrixGamma
>>> from sympy import MatrixSymbol, symbols
>>> a, b = symbols('a b', positive=True)
>>> M = MatrixGamma('M', a, b, [[2, 1], [1, 2]])
>>> X = MatrixSymbol('X', 2, 2)
>>> density(M)(X).doit()
exp(Trace(Matrix([
[-2/3, 1/3],
[ 1/3, -2/3]])*X)/b)*Determinant(X)**(a - 3/2)/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2))
>>> density(M)([[1, 0], [0, 1]]).doit()
exp(-4/(3*b))/(3**a*sqrt(pi)*b**(2*a)*gamma(a)*gamma(a - 1/2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Matrix_gamma_distribution
"""
if isinstance(scale_matrix, list):
scale_matrix = ImmutableMatrix(scale_matrix)
return rv(symbol, MatrixGammaDistribution, (alpha, beta, scale_matrix))
#-------------------------------------------------------------------------------
# Wishart Distribution ---------------------------------------------------------
class WishartDistribution(MatrixDistribution):
_argnames = ('n', 'scale_matrix')
@staticmethod
def check(n, scale_matrix):
if not isinstance(scale_matrix, MatrixSymbol):
_value_check(scale_matrix.is_positive_definite, "The shape "
"matrix must be positive definite.")
_value_check(scale_matrix.is_square, "Should "
"be square matrix")
_value_check(n.is_positive, "Shape parameter should be positive.")
@property
def set(self):
k = self.scale_matrix.shape[0]
return MatrixSet(k, k, S.Reals)
@property
def dimension(self):
return self.scale_matrix.shape
def pdf(self, x):
n, scale_matrix = self.n, self.scale_matrix
p = scale_matrix.shape[0]
if isinstance(x, list):
x = ImmutableMatrix(x)
if not isinstance(x, (MatrixBase, MatrixSymbol)):
raise ValueError("%s should be an isinstance of Matrix "
"or MatrixSymbol" % str(x))
sigma_inv_x = - Inverse(scale_matrix)*x / S(2)
term1 = exp(Trace(sigma_inv_x))/((2**(p*n/S(2))) * multigamma(n/S(2), p))
term2 = (Determinant(scale_matrix))**(-n/S(2))
term3 = (Determinant(x))**(S(n - p - 1)/2)
return term1 * term2 * term3
def Wishart(symbol, n, scale_matrix):
"""
Creates a random variable with Wishart Distribution.
The density of the said distribution can be found at [1].
Parameters
==========
n: Positive Real number
Represents degrees of freedom
scale_matrix: Positive definite real square matrix
Scale Matrix
Returns
=======
RandomSymbol
Examples
========
>>> from sympy.stats import density, Wishart
>>> from sympy import MatrixSymbol, symbols
>>> n = symbols('n', positive=True)
>>> W = Wishart('W', n, [[2, 1], [1, 2]])
>>> X = MatrixSymbol('X', 2, 2)
>>> density(W)(X).doit()
exp(Trace(Matrix([
[-1/3, 1/6],
[ 1/6, -1/3]])*X))*Determinant(X)**(n/2 - 3/2)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2))
>>> density(W)([[1, 0], [0, 1]]).doit()
exp(-2/3)/(2**n*3**(n/2)*sqrt(pi)*gamma(n/2)*gamma(n/2 - 1/2))
References
==========
.. [1] https://en.wikipedia.org/wiki/Wishart_distribution
"""
if isinstance(scale_matrix, list):
scale_matrix = ImmutableMatrix(scale_matrix)
return rv(symbol, WishartDistribution, (n, scale_matrix))
#-------------------------------------------------------------------------------
# Matrix Normal distribution ---------------------------------------------------
class MatrixNormalDistribution(MatrixDistribution):
_argnames = ('location_matrix', 'scale_matrix_1', 'scale_matrix_2')
@staticmethod
def check(location_matrix, scale_matrix_1, scale_matrix_2):
if not isinstance(scale_matrix_1, MatrixSymbol):
_value_check(scale_matrix_1.is_positive_definite, "The shape "
"matrix must be positive definite.")
if not isinstance(scale_matrix_2, MatrixSymbol):
_value_check(scale_matrix_2.is_positive_definite, "The shape "
"matrix must be positive definite.")
_value_check(scale_matrix_1.is_square, "Scale matrix 1 should be "
"be square matrix")
_value_check(scale_matrix_2.is_square, "Scale matrix 2 should be "
"be square matrix")
n = location_matrix.shape[0]
p = location_matrix.shape[1]
_value_check(scale_matrix_1.shape[0] == n, "Scale matrix 1 should be"
" of shape %s x %s"% (str(n), str(n)))
_value_check(scale_matrix_2.shape[0] == p, "Scale matrix 2 should be"
" of shape %s x %s"% (str(p), str(p)))
@property
def set(self):
n, p = self.location_matrix.shape
return MatrixSet(n, p, S.Reals)
@property
def dimension(self):
return self.location_matrix.shape
def pdf(self, x):
M, U, V = self.location_matrix, self.scale_matrix_1, self.scale_matrix_2
n, p = M.shape
if isinstance(x, list):
x = ImmutableMatrix(x)
if not isinstance(x, (MatrixBase, MatrixSymbol)):
raise ValueError("%s should be an isinstance of Matrix "
"or MatrixSymbol" % str(x))
term1 = Inverse(V)*Transpose(x - M)*Inverse(U)*(x - M)
num = exp(-Trace(term1)/S(2))
den = (2*pi)**(S(n*p)/2) * Determinant(U)**S(p)/2 * Determinant(V)**S(n)/2
return num/den
def MatrixNormal(symbol, location_matrix, scale_matrix_1, scale_matrix_2):
"""
Creates a random variable with Matrix Normal Distribution.
The density of the said distribution can be found at [1].
Parameters
==========
location_matrix: Real ``n x p`` matrix
Represents degrees of freedom
scale_matrix_1: Positive definite matrix
Scale Matrix of shape ``n x n``
scale_matrix_2: Positive definite matrix
Scale Matrix of shape ``p x p``
Returns
=======
RandomSymbol
Examples
========
>>> from sympy import MatrixSymbol
>>> from sympy.stats import density, MatrixNormal
>>> M = MatrixNormal('M', [[1, 2]], [1], [[1, 0], [0, 1]])
>>> X = MatrixSymbol('X', 1, 2)
>>> density(M)(X).doit()
2*exp(-Trace((Matrix([
[-1],
[-2]]) + X.T)*(Matrix([[-1, -2]]) + X))/2)/pi
>>> density(M)([[3, 4]]).doit()
2*exp(-4)/pi
References
==========
.. [1] https://en.wikipedia.org/wiki/Matrix_normal_distribution
"""
if isinstance(location_matrix, list):
location_matrix = ImmutableMatrix(location_matrix)
if isinstance(scale_matrix_1, list):
scale_matrix_1 = ImmutableMatrix(scale_matrix_1)
if isinstance(scale_matrix_2, list):
scale_matrix_2 = ImmutableMatrix(scale_matrix_2)
args = (location_matrix, scale_matrix_1, scale_matrix_2)
return rv(symbol, MatrixNormalDistribution, args)
#-------------------------------------------------------------------------------
# Matrix Student's T distribution ---------------------------------------------------
class MatrixStudentTDistribution(MatrixDistribution):
_argnames = ('nu', 'location_matrix', 'scale_matrix_1', 'scale_matrix_2')
@staticmethod
def check(nu, location_matrix, scale_matrix_1, scale_matrix_2):
if not isinstance(scale_matrix_1, MatrixSymbol):
_value_check(scale_matrix_1.is_positive_definite != False, "The shape "
"matrix must be positive definite.")
if not isinstance(scale_matrix_2, MatrixSymbol):
_value_check(scale_matrix_2.is_positive_definite != False, "The shape "
"matrix must be positive definite.")
_value_check(scale_matrix_1.is_square != False, "Scale matrix 1 should be "
"be square matrix")
_value_check(scale_matrix_2.is_square != False, "Scale matrix 2 should be "
"be square matrix")
n = location_matrix.shape[0]
p = location_matrix.shape[1]
_value_check(scale_matrix_1.shape[0] == p, "Scale matrix 1 should be"
" of shape %s x %s" % (str(p), str(p)))
_value_check(scale_matrix_2.shape[0] == n, "Scale matrix 2 should be"
" of shape %s x %s" % (str(n), str(n)))
_value_check(nu.is_positive != False, "Degrees of freedom must be positive")
@property
def set(self):
n, p = self.location_matrix.shape
return MatrixSet(n, p, S.Reals)
@property
def dimension(self):
return self.location_matrix.shape
def pdf(self, x):
from sympy.matrices.dense import eye
if isinstance(x, list):
x = ImmutableMatrix(x)
if not isinstance(x, (MatrixBase, MatrixSymbol)):
raise ValueError("%s should be an isinstance of Matrix "
"or MatrixSymbol" % str(x))
nu, M, Omega, Sigma = self.nu, self.location_matrix, self.scale_matrix_1, self.scale_matrix_2
n, p = M.shape
K = multigamma((nu + n + p - 1)/2, p) * Determinant(Omega)**(-n/2) * Determinant(Sigma)**(-p/2) \
/ ((pi)**(n*p/2) * multigamma((nu + p - 1)/2, p))
return K * (Determinant(eye(n) + Inverse(Sigma)*(x - M)*Inverse(Omega)*Transpose(x - M))) \
**(-(nu + n + p -1)/2)
def MatrixStudentT(symbol, nu, location_matrix, scale_matrix_1, scale_matrix_2):
"""
Creates a random variable with Matrix Gamma Distribution.
The density of the said distribution can be found at [1].
Parameters
==========
nu: Positive Real number
degrees of freedom
location_matrix: Positive definite real square matrix
Location Matrix of shape ``n x p``
scale_matrix_1: Positive definite real square matrix
Scale Matrix of shape ``p x p``
scale_matrix_2: Positive definite real square matrix
Scale Matrix of shape ``n x n``
Returns
=======
RandomSymbol
Examples
========
>>> from sympy import MatrixSymbol,symbols
>>> from sympy.stats import density, MatrixStudentT
>>> v = symbols('v',positive=True)
>>> M = MatrixStudentT('M', v, [[1, 2]], [[1, 0], [0, 1]], [1])
>>> X = MatrixSymbol('X', 1, 2)
>>> density(M)(X)
gamma(v/2 + 1)*Determinant((Matrix([[-1, -2]]) + X)*(Matrix([
[-1],
[-2]]) + X.T) + Matrix([[1]]))**(-v/2 - 1)/(pi**1.0*gamma(v/2)*Determinant(Matrix([[1]]))**1.0*Determinant(Matrix([
[1, 0],
[0, 1]]))**0.5)
References
==========
.. [1] https://en.wikipedia.org/wiki/Matrix_t-distribution
"""
if isinstance(location_matrix, list):
location_matrix = ImmutableMatrix(location_matrix)
if isinstance(scale_matrix_1, list):
scale_matrix_1 = ImmutableMatrix(scale_matrix_1)
if isinstance(scale_matrix_2, list):
scale_matrix_2 = ImmutableMatrix(scale_matrix_2)
args = (nu, location_matrix, scale_matrix_1, scale_matrix_2)
return rv(symbol, MatrixStudentTDistribution, args)
|
366ccb0b924c0d62364ef1fcb58ea0a3c958c879aa719f81eec4883bbcfba009 | from sympy.core.basic import Basic
from sympy.stats.rv import PSpace, _symbol_converter, RandomMatrixSymbol
class RandomMatrixPSpace(PSpace):
"""
Represents probability space for
random matrices. It contains the mechanics
for handling the API calls for random matrices.
"""
def __new__(cls, sym, model=None):
sym = _symbol_converter(sym)
if model:
return Basic.__new__(cls, sym, model)
else:
return Basic.__new__(cls, sym)
@property
def model(self):
try:
return self.args[1]
except IndexError:
return None
def compute_density(self, expr, *args):
rms = expr.atoms(RandomMatrixSymbol)
if len(rms) > 2 or (not isinstance(expr, RandomMatrixSymbol)):
raise NotImplementedError("Currently, no algorithm has been "
"implemented to handle general expressions containing "
"multiple random matrices.")
return self.model.density(expr)
|
186f0449ab86abfd03a2e95d74ca813aff0fbb444ebada80b8682cf32c0a4124 | from sympy.sets import FiniteSet
from sympy.core.numbers import Rational
from sympy.core.relational import Eq
from sympy.core.symbol import Dummy
from sympy.functions.combinatorial.factorials import FallingFactorial
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.piecewise import piecewise_fold
from sympy.integrals.integrals import Integral
from sympy.solvers.solveset import solveset
from .rv import (probability, expectation, density, where, given, pspace, cdf, PSpace,
characteristic_function, sample, sample_iter, random_symbols, independent, dependent,
sampling_density, moment_generating_function, quantile, is_random,
sample_stochastic_process)
__all__ = ['P', 'E', 'H', 'density', 'where', 'given', 'sample', 'cdf',
'characteristic_function', 'pspace', 'sample_iter', 'variance', 'std',
'skewness', 'kurtosis', 'covariance', 'dependent', 'entropy', 'median',
'independent', 'random_symbols', 'correlation', 'factorial_moment',
'moment', 'cmoment', 'sampling_density', 'moment_generating_function',
'smoment', 'quantile', 'sample_stochastic_process']
def moment(X, n, c=0, condition=None, *, evaluate=True, **kwargs):
"""
Return the nth moment of a random expression about c.
.. math::
moment(X, c, n) = E((X-c)^{n})
Default value of c is 0.
Examples
========
>>> from sympy.stats import Die, moment, E
>>> X = Die('X', 6)
>>> moment(X, 1, 6)
-5/2
>>> moment(X, 2)
91/6
>>> moment(X, 1) == E(X)
True
"""
from sympy.stats.symbolic_probability import Moment
if evaluate:
return Moment(X, n, c, condition).doit()
return Moment(X, n, c, condition).rewrite(Integral)
def variance(X, condition=None, **kwargs):
"""
Variance of a random expression.
.. math::
variance(X) = E((X-E(X))^{2})
Examples
========
>>> from sympy.stats import Die, Bernoulli, variance
>>> from sympy import simplify, Symbol
>>> X = Die('X', 6)
>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)
>>> variance(2*X)
35/3
>>> simplify(variance(B))
p*(1 - p)
"""
if is_random(X) and pspace(X) == PSpace():
from sympy.stats.symbolic_probability import Variance
return Variance(X, condition)
return cmoment(X, 2, condition, **kwargs)
def standard_deviation(X, condition=None, **kwargs):
r"""
Standard Deviation of a random expression
.. math::
std(X) = \sqrt(E((X-E(X))^{2}))
Examples
========
>>> from sympy.stats import Bernoulli, std
>>> from sympy import Symbol, simplify
>>> p = Symbol('p')
>>> B = Bernoulli('B', p, 1, 0)
>>> simplify(std(B))
sqrt(p*(1 - p))
"""
return sqrt(variance(X, condition, **kwargs))
std = standard_deviation
def entropy(expr, condition=None, **kwargs):
"""
Calculuates entropy of a probability distribution.
Parameters
==========
expression : the random expression whose entropy is to be calculated
condition : optional, to specify conditions on random expression
b: base of the logarithm, optional
By default, it is taken as Euler's number
Returns
=======
result : Entropy of the expression, a constant
Examples
========
>>> from sympy.stats import Normal, Die, entropy
>>> X = Normal('X', 0, 1)
>>> entropy(X)
log(2)/2 + 1/2 + log(pi)/2
>>> D = Die('D', 4)
>>> entropy(D)
log(4)
References
==========
.. [1] https://en.wikipedia.org/wiki/Entropy_(information_theory)
.. [2] https://www.crmarsh.com/static/pdf/Charles_Marsh_Continuous_Entropy.pdf
.. [3] http://www.math.uconn.edu/~kconrad/blurbs/analysis/entropypost.pdf
"""
pdf = density(expr, condition, **kwargs)
base = kwargs.get('b', exp(1))
if isinstance(pdf, dict):
return sum([-prob*log(prob, base) for prob in pdf.values()])
return expectation(-log(pdf(expr), base))
def covariance(X, Y, condition=None, **kwargs):
"""
Covariance of two random expressions.
Explanation
===========
The expectation that the two variables will rise and fall together
.. math::
covariance(X,Y) = E((X-E(X)) (Y-E(Y)))
Examples
========
>>> from sympy.stats import Exponential, covariance
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)
>>> covariance(X, X)
lambda**(-2)
>>> covariance(X, Y)
0
>>> covariance(X, Y + rate*X)
1/lambda
"""
if (is_random(X) and pspace(X) == PSpace()) or (is_random(Y) and pspace(Y) == PSpace()):
from sympy.stats.symbolic_probability import Covariance
return Covariance(X, Y, condition)
return expectation(
(X - expectation(X, condition, **kwargs)) *
(Y - expectation(Y, condition, **kwargs)),
condition, **kwargs)
def correlation(X, Y, condition=None, **kwargs):
r"""
Correlation of two random expressions, also known as correlation
coefficient or Pearson's correlation.
Explanation
===========
The normalized expectation that the two variables will rise
and fall together
.. math::
correlation(X,Y) = E((X-E(X))(Y-E(Y)) / (\sigma_x \sigma_y))
Examples
========
>>> from sympy.stats import Exponential, correlation
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True)
>>> X = Exponential('X', rate)
>>> Y = Exponential('Y', rate)
>>> correlation(X, X)
1
>>> correlation(X, Y)
0
>>> correlation(X, Y + rate*X)
1/sqrt(1 + lambda**(-2))
"""
return covariance(X, Y, condition, **kwargs)/(std(X, condition, **kwargs)
* std(Y, condition, **kwargs))
def cmoment(X, n, condition=None, *, evaluate=True, **kwargs):
"""
Return the nth central moment of a random expression about its mean.
.. math::
cmoment(X, n) = E((X - E(X))^{n})
Examples
========
>>> from sympy.stats import Die, cmoment, variance
>>> X = Die('X', 6)
>>> cmoment(X, 3)
0
>>> cmoment(X, 2)
35/12
>>> cmoment(X, 2) == variance(X)
True
"""
from sympy.stats.symbolic_probability import CentralMoment
if evaluate:
return CentralMoment(X, n, condition).doit()
return CentralMoment(X, n, condition).rewrite(Integral)
def smoment(X, n, condition=None, **kwargs):
r"""
Return the nth Standardized moment of a random expression.
.. math::
smoment(X, n) = E(((X - \mu)/\sigma_X)^{n})
Examples
========
>>> from sympy.stats import skewness, Exponential, smoment
>>> from sympy import Symbol
>>> rate = Symbol('lambda', positive=True, real=True)
>>> Y = Exponential('Y', rate)
>>> smoment(Y, 4)
9
>>> smoment(Y, 4) == smoment(3*Y, 4)
True
>>> smoment(Y, 3) == skewness(Y)
True
"""
sigma = std(X, condition, **kwargs)
return (1/sigma)**n*cmoment(X, n, condition, **kwargs)
def skewness(X, condition=None, **kwargs):
r"""
Measure of the asymmetry of the probability distribution.
Explanation
===========
Positive skew indicates that most of the values lie to the right of
the mean.
.. math::
skewness(X) = E(((X - E(X))/\sigma_X)^{3})
Parameters
==========
condition : Expr containing RandomSymbols
A conditional expression. skewness(X, X>0) is skewness of X given X > 0
Examples
========
>>> from sympy.stats import skewness, Exponential, Normal
>>> from sympy import Symbol
>>> X = Normal('X', 0, 1)
>>> skewness(X)
0
>>> skewness(X, X > 0) # find skewness given X > 0
(-sqrt(2)/sqrt(pi) + 4*sqrt(2)/pi**(3/2))/(1 - 2/pi)**(3/2)
>>> rate = Symbol('lambda', positive=True, real=True)
>>> Y = Exponential('Y', rate)
>>> skewness(Y)
2
"""
return smoment(X, 3, condition=condition, **kwargs)
def kurtosis(X, condition=None, **kwargs):
r"""
Characterizes the tails/outliers of a probability distribution.
Explanation
===========
Kurtosis of any univariate normal distribution is 3. Kurtosis less than
3 means that the distribution produces fewer and less extreme outliers
than the normal distribution.
.. math::
kurtosis(X) = E(((X - E(X))/\sigma_X)^{4})
Parameters
==========
condition : Expr containing RandomSymbols
A conditional expression. kurtosis(X, X>0) is kurtosis of X given X > 0
Examples
========
>>> from sympy.stats import kurtosis, Exponential, Normal
>>> from sympy import Symbol
>>> X = Normal('X', 0, 1)
>>> kurtosis(X)
3
>>> kurtosis(X, X > 0) # find kurtosis given X > 0
(-4/pi - 12/pi**2 + 3)/(1 - 2/pi)**2
>>> rate = Symbol('lamda', positive=True, real=True)
>>> Y = Exponential('Y', rate)
>>> kurtosis(Y)
9
References
==========
.. [1] https://en.wikipedia.org/wiki/Kurtosis
.. [2] http://mathworld.wolfram.com/Kurtosis.html
"""
return smoment(X, 4, condition=condition, **kwargs)
def factorial_moment(X, n, condition=None, **kwargs):
"""
The factorial moment is a mathematical quantity defined as the expectation
or average of the falling factorial of a random variable.
.. math::
factorial-moment(X, n) = E(X(X - 1)(X - 2)...(X - n + 1))
Parameters
==========
n: A natural number, n-th factorial moment.
condition : Expr containing RandomSymbols
A conditional expression.
Examples
========
>>> from sympy.stats import factorial_moment, Poisson, Binomial
>>> from sympy import Symbol, S
>>> lamda = Symbol('lamda')
>>> X = Poisson('X', lamda)
>>> factorial_moment(X, 2)
lamda**2
>>> Y = Binomial('Y', 2, S.Half)
>>> factorial_moment(Y, 2)
1/2
>>> factorial_moment(Y, 2, Y > 1) # find factorial moment for Y > 1
2
References
==========
.. [1] https://en.wikipedia.org/wiki/Factorial_moment
.. [2] http://mathworld.wolfram.com/FactorialMoment.html
"""
return expectation(FallingFactorial(X, n), condition=condition, **kwargs)
def median(X, evaluate=True, **kwargs):
r"""
Calculuates the median of the probability distribution.
Explanation
===========
Mathematically, median of Probability distribution is defined as all those
values of `m` for which the following condition is satisfied
.. math::
P(X\leq m) \geq \frac{1}{2} \text{ and} \text{ } P(X\geq m)\geq \frac{1}{2}
Parameters
==========
X: The random expression whose median is to be calculated.
Returns
=======
The FiniteSet or an Interval which contains the median of the
random expression.
Examples
========
>>> from sympy.stats import Normal, Die, median
>>> N = Normal('N', 3, 1)
>>> median(N)
{3}
>>> D = Die('D')
>>> median(D)
{3, 4}
References
==========
.. [1] https://en.wikipedia.org/wiki/Median#Probability_distributions
"""
if not is_random(X):
return X
from sympy.stats.crv import ContinuousPSpace
from sympy.stats.drv import DiscretePSpace
from sympy.stats.frv import FinitePSpace
if isinstance(pspace(X), FinitePSpace):
cdf = pspace(X).compute_cdf(X)
result = []
for key, value in cdf.items():
if value>= Rational(1, 2) and (1 - value) + \
pspace(X).probability(Eq(X, key)) >= Rational(1, 2):
result.append(key)
return FiniteSet(*result)
if isinstance(pspace(X), (ContinuousPSpace, DiscretePSpace)):
cdf = pspace(X).compute_cdf(X)
x = Dummy('x')
result = solveset(piecewise_fold(cdf(x) - Rational(1, 2)), x, pspace(X).set)
return result
raise NotImplementedError("The median of %s is not implemeted."%str(pspace(X)))
def coskewness(X, Y, Z, condition=None, **kwargs):
r"""
Calculates the co-skewness of three random variables.
Explanation
===========
Mathematically Coskewness is defined as
.. math::
coskewness(X,Y,Z)=\frac{E[(X-E[X]) * (Y-E[Y]) * (Z-E[Z])]} {\sigma_{X}\sigma_{Y}\sigma_{Z}}
Parameters
==========
X : RandomSymbol
Random Variable used to calculate coskewness
Y : RandomSymbol
Random Variable used to calculate coskewness
Z : RandomSymbol
Random Variable used to calculate coskewness
condition : Expr containing RandomSymbols
A conditional expression
Examples
========
>>> from sympy.stats import coskewness, Exponential, skewness
>>> from sympy import symbols
>>> p = symbols('p', positive=True)
>>> X = Exponential('X', p)
>>> Y = Exponential('Y', 2*p)
>>> coskewness(X, Y, Y)
0
>>> coskewness(X, Y + X, Y + 2*X)
16*sqrt(85)/85
>>> coskewness(X + 2*Y, Y + X, Y + 2*X, X > 3)
9*sqrt(170)/85
>>> coskewness(Y, Y, Y) == skewness(Y)
True
>>> coskewness(X, Y + p*X, Y + 2*p*X)
4/(sqrt(1 + 1/(4*p**2))*sqrt(4 + 1/(4*p**2)))
Returns
=======
coskewness : The coskewness of the three random variables
References
==========
.. [1] https://en.wikipedia.org/wiki/Coskewness
"""
num = expectation((X - expectation(X, condition, **kwargs)) \
* (Y - expectation(Y, condition, **kwargs)) \
* (Z - expectation(Z, condition, **kwargs)), condition, **kwargs)
den = std(X, condition, **kwargs) * std(Y, condition, **kwargs) \
* std(Z, condition, **kwargs)
return num/den
P = probability
E = expectation
H = entropy
|
b3aca7487015743fbda3221466553cf56cd642e95057c0e1879d9290df3882a7 | """
Main Random Variables Module
Defines abstract random variable type.
Contains interfaces for probability space object (PSpace) as well as standard
operators, P, E, sample, density, where, quantile
See Also
========
sympy.stats.crv
sympy.stats.frv
sympy.stats.rv_interface
"""
from functools import singledispatch
from typing import Tuple as tTuple
from sympy.core.add import Add
from sympy.core.basic import Basic
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import (Function, Lambda)
from sympy.core.logic import fuzzy_and
from sympy.core.mul import (Mul, prod)
from sympy.core.relational import (Eq, Ne)
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol)
from sympy.core.sympify import sympify
from sympy.functions.special.delta_functions import DiracDelta
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.logic.boolalg import (And, Or)
from sympy.matrices.expressions.matexpr import MatrixSymbol
from sympy.tensor.indexed import Indexed
from sympy.utilities.lambdify import lambdify
from sympy.core.relational import Relational
from sympy.core.sympify import _sympify
from sympy.sets.sets import FiniteSet, ProductSet, Intersection
from sympy.solvers.solveset import solveset
from sympy.external import import_module
from sympy.utilities.misc import filldedent
from sympy.utilities.decorator import doctest_depends_on
from sympy.utilities.exceptions import SymPyDeprecationWarning
from sympy.utilities.iterables import iterable
import warnings
x = Symbol('x')
@singledispatch
def is_random(x):
return False
@is_random.register(Basic)
def _(x):
atoms = x.free_symbols
return any(is_random(i) for i in atoms)
class RandomDomain(Basic):
"""
Represents a set of variables and the values which they can take.
See Also
========
sympy.stats.crv.ContinuousDomain
sympy.stats.frv.FiniteDomain
"""
is_ProductDomain = False
is_Finite = False
is_Continuous = False
is_Discrete = False
def __new__(cls, symbols, *args):
symbols = FiniteSet(*symbols)
return Basic.__new__(cls, symbols, *args)
@property
def symbols(self):
return self.args[0]
@property
def set(self):
return self.args[1]
def __contains__(self, other):
raise NotImplementedError()
def compute_expectation(self, expr):
raise NotImplementedError()
class SingleDomain(RandomDomain):
"""
A single variable and its domain.
See Also
========
sympy.stats.crv.SingleContinuousDomain
sympy.stats.frv.SingleFiniteDomain
"""
def __new__(cls, symbol, set):
assert symbol.is_Symbol
return Basic.__new__(cls, symbol, set)
@property
def symbol(self):
return self.args[0]
@property
def symbols(self):
return FiniteSet(self.symbol)
def __contains__(self, other):
if len(other) != 1:
return False
sym, val = tuple(other)[0]
return self.symbol == sym and val in self.set
class MatrixDomain(RandomDomain):
"""
A Random Matrix variable and its domain.
"""
def __new__(cls, symbol, set):
symbol, set = _symbol_converter(symbol), _sympify(set)
return Basic.__new__(cls, symbol, set)
@property
def symbol(self):
return self.args[0]
@property
def symbols(self):
return FiniteSet(self.symbol)
class ConditionalDomain(RandomDomain):
"""
A RandomDomain with an attached condition.
See Also
========
sympy.stats.crv.ConditionalContinuousDomain
sympy.stats.frv.ConditionalFiniteDomain
"""
def __new__(cls, fulldomain, condition):
condition = condition.xreplace({rs: rs.symbol
for rs in random_symbols(condition)})
return Basic.__new__(cls, fulldomain, condition)
@property
def symbols(self):
return self.fulldomain.symbols
@property
def fulldomain(self):
return self.args[0]
@property
def condition(self):
return self.args[1]
@property
def set(self):
raise NotImplementedError("Set of Conditional Domain not Implemented")
def as_boolean(self):
return And(self.fulldomain.as_boolean(), self.condition)
class PSpace(Basic):
"""
A Probability Space.
Explanation
===========
Probability Spaces encode processes that equal different values
probabilistically. These underly Random Symbols which occur in SymPy
expressions and contain the mechanics to evaluate statistical statements.
See Also
========
sympy.stats.crv.ContinuousPSpace
sympy.stats.frv.FinitePSpace
"""
is_Finite = None # type: bool
is_Continuous = None # type: bool
is_Discrete = None # type: bool
is_real = None # type: bool
@property
def domain(self):
return self.args[0]
@property
def density(self):
return self.args[1]
@property
def values(self):
return frozenset(RandomSymbol(sym, self) for sym in self.symbols)
@property
def symbols(self):
return self.domain.symbols
def where(self, condition):
raise NotImplementedError()
def compute_density(self, expr):
raise NotImplementedError()
def sample(self, size=(), library='scipy', seed=None):
raise NotImplementedError()
def probability(self, condition):
raise NotImplementedError()
def compute_expectation(self, expr):
raise NotImplementedError()
class SinglePSpace(PSpace):
"""
Represents the probabilities of a set of random events that can be
attributed to a single variable/symbol.
"""
def __new__(cls, s, distribution):
s = _symbol_converter(s)
return Basic.__new__(cls, s, distribution)
@property
def value(self):
return RandomSymbol(self.symbol, self)
@property
def symbol(self):
return self.args[0]
@property
def distribution(self):
return self.args[1]
@property
def pdf(self):
return self.distribution.pdf(self.symbol)
class RandomSymbol(Expr):
"""
Random Symbols represent ProbabilitySpaces in SymPy Expressions.
In principle they can take on any value that their symbol can take on
within the associated PSpace with probability determined by the PSpace
Density.
Explanation
===========
Random Symbols contain pspace and symbol properties.
The pspace property points to the represented Probability Space
The symbol is a standard SymPy Symbol that is used in that probability space
for example in defining a density.
You can form normal SymPy expressions using RandomSymbols and operate on
those expressions with the Functions
E - Expectation of a random expression
P - Probability of a condition
density - Probability Density of an expression
given - A new random expression (with new random symbols) given a condition
An object of the RandomSymbol type should almost never be created by the
user. They tend to be created instead by the PSpace class's value method.
Traditionally a user doesn't even do this but instead calls one of the
convenience functions Normal, Exponential, Coin, Die, FiniteRV, etc....
"""
def __new__(cls, symbol, pspace=None):
from sympy.stats.joint_rv import JointRandomSymbol
if pspace is None:
# Allow single arg, representing pspace == PSpace()
pspace = PSpace()
symbol = _symbol_converter(symbol)
if not isinstance(pspace, PSpace):
raise TypeError("pspace variable should be of type PSpace")
if cls == JointRandomSymbol and isinstance(pspace, SinglePSpace):
cls = RandomSymbol
return Basic.__new__(cls, symbol, pspace)
is_finite = True
is_symbol = True
is_Atom = True
_diff_wrt = True
pspace = property(lambda self: self.args[1])
symbol = property(lambda self: self.args[0])
name = property(lambda self: self.symbol.name)
def _eval_is_positive(self):
return self.symbol.is_positive
def _eval_is_integer(self):
return self.symbol.is_integer
def _eval_is_real(self):
return self.symbol.is_real or self.pspace.is_real
@property
def is_commutative(self):
return self.symbol.is_commutative
@property
def free_symbols(self):
return {self}
class RandomIndexedSymbol(RandomSymbol):
def __new__(cls, idx_obj, pspace=None):
if pspace is None:
# Allow single arg, representing pspace == PSpace()
pspace = PSpace()
if not isinstance(idx_obj, (Indexed, Function)):
raise TypeError("An Function or Indexed object is expected not %s"%(idx_obj))
return Basic.__new__(cls, idx_obj, pspace)
symbol = property(lambda self: self.args[0])
name = property(lambda self: str(self.args[0]))
@property
def key(self):
if isinstance(self.symbol, Indexed):
return self.symbol.args[1]
elif isinstance(self.symbol, Function):
return self.symbol.args[0]
@property
def free_symbols(self):
if self.key.free_symbols:
free_syms = self.key.free_symbols
free_syms.add(self)
return free_syms
return {self}
@property
def pspace(self):
return self.args[1]
class RandomMatrixSymbol(RandomSymbol, MatrixSymbol): # type: ignore
def __new__(cls, symbol, n, m, pspace=None):
n, m = _sympify(n), _sympify(m)
symbol = _symbol_converter(symbol)
if pspace is None:
# Allow single arg, representing pspace == PSpace()
pspace = PSpace()
return Basic.__new__(cls, symbol, n, m, pspace)
symbol = property(lambda self: self.args[0])
pspace = property(lambda self: self.args[3])
class ProductPSpace(PSpace):
"""
Abstract class for representing probability spaces with multiple random
variables.
See Also
========
sympy.stats.rv.IndependentProductPSpace
sympy.stats.joint_rv.JointPSpace
"""
pass
class IndependentProductPSpace(ProductPSpace):
"""
A probability space resulting from the merger of two independent probability
spaces.
Often created using the function, pspace.
"""
def __new__(cls, *spaces):
rs_space_dict = {}
for space in spaces:
for value in space.values:
rs_space_dict[value] = space
symbols = FiniteSet(*[val.symbol for val in rs_space_dict.keys()])
# Overlapping symbols
from sympy.stats.joint_rv import MarginalDistribution
from sympy.stats.compound_rv import CompoundDistribution
if len(symbols) < sum(len(space.symbols) for space in spaces if not
isinstance(space.distribution, (
CompoundDistribution, MarginalDistribution))):
raise ValueError("Overlapping Random Variables")
if all(space.is_Finite for space in spaces):
from sympy.stats.frv import ProductFinitePSpace
cls = ProductFinitePSpace
obj = Basic.__new__(cls, *FiniteSet(*spaces))
return obj
@property
def pdf(self):
p = Mul(*[space.pdf for space in self.spaces])
return p.subs({rv: rv.symbol for rv in self.values})
@property
def rs_space_dict(self):
d = {}
for space in self.spaces:
for value in space.values:
d[value] = space
return d
@property
def symbols(self):
return FiniteSet(*[val.symbol for val in self.rs_space_dict.keys()])
@property
def spaces(self):
return FiniteSet(*self.args)
@property
def values(self):
return sumsets(space.values for space in self.spaces)
def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
rvs = rvs or self.values
rvs = frozenset(rvs)
for space in self.spaces:
expr = space.compute_expectation(expr, rvs & space.values, evaluate=False, **kwargs)
if evaluate and hasattr(expr, 'doit'):
return expr.doit(**kwargs)
return expr
@property
def domain(self):
return ProductDomain(*[space.domain for space in self.spaces])
@property
def density(self):
raise NotImplementedError("Density not available for ProductSpaces")
def sample(self, size=(), library='scipy', seed=None):
return {k: v for space in self.spaces
for k, v in space.sample(size=size, library=library, seed=seed).items()}
def probability(self, condition, **kwargs):
cond_inv = False
if isinstance(condition, Ne):
condition = Eq(condition.args[0], condition.args[1])
cond_inv = True
elif isinstance(condition, And): # they are independent
return Mul(*[self.probability(arg) for arg in condition.args])
elif isinstance(condition, Or): # they are independent
return Add(*[self.probability(arg) for arg in condition.args])
expr = condition.lhs - condition.rhs
rvs = random_symbols(expr)
dens = self.compute_density(expr)
if any(pspace(rv).is_Continuous for rv in rvs):
from sympy.stats.crv import SingleContinuousPSpace
from sympy.stats.crv_types import ContinuousDistributionHandmade
if expr in self.values:
# Marginalize all other random symbols out of the density
randomsymbols = tuple(set(self.values) - frozenset([expr]))
symbols = tuple(rs.symbol for rs in randomsymbols)
pdf = self.domain.integrate(self.pdf, symbols, **kwargs)
return Lambda(expr.symbol, pdf)
dens = ContinuousDistributionHandmade(dens)
z = Dummy('z', real=True)
space = SingleContinuousPSpace(z, dens)
result = space.probability(condition.__class__(space.value, 0))
else:
from sympy.stats.drv import SingleDiscretePSpace
from sympy.stats.drv_types import DiscreteDistributionHandmade
dens = DiscreteDistributionHandmade(dens)
z = Dummy('z', integer=True)
space = SingleDiscretePSpace(z, dens)
result = space.probability(condition.__class__(space.value, 0))
return result if not cond_inv else S.One - result
def compute_density(self, expr, **kwargs):
rvs = random_symbols(expr)
if any(pspace(rv).is_Continuous for rv in rvs):
z = Dummy('z', real=True)
expr = self.compute_expectation(DiracDelta(expr - z),
**kwargs)
else:
z = Dummy('z', integer=True)
expr = self.compute_expectation(KroneckerDelta(expr, z),
**kwargs)
return Lambda(z, expr)
def compute_cdf(self, expr, **kwargs):
raise ValueError("CDF not well defined on multivariate expressions")
def conditional_space(self, condition, normalize=True, **kwargs):
rvs = random_symbols(condition)
condition = condition.xreplace({rv: rv.symbol for rv in self.values})
pspaces = [pspace(rv) for rv in rvs]
if any(ps.is_Continuous for ps in pspaces):
from sympy.stats.crv import (ConditionalContinuousDomain,
ContinuousPSpace)
space = ContinuousPSpace
domain = ConditionalContinuousDomain(self.domain, condition)
elif any(ps.is_Discrete for ps in pspaces):
from sympy.stats.drv import (ConditionalDiscreteDomain,
DiscretePSpace)
space = DiscretePSpace
domain = ConditionalDiscreteDomain(self.domain, condition)
elif all(ps.is_Finite for ps in pspaces):
from sympy.stats.frv import FinitePSpace
return FinitePSpace.conditional_space(self, condition)
if normalize:
replacement = {rv: Dummy(str(rv)) for rv in self.symbols}
norm = domain.compute_expectation(self.pdf, **kwargs)
pdf = self.pdf / norm.xreplace(replacement)
# XXX: Converting symbols from set to tuple. The order matters to
# Lambda though so we shouldn't be starting with a set here...
density = Lambda(tuple(domain.symbols), pdf)
return space(domain, density)
class ProductDomain(RandomDomain):
"""
A domain resulting from the merger of two independent domains.
See Also
========
sympy.stats.crv.ProductContinuousDomain
sympy.stats.frv.ProductFiniteDomain
"""
is_ProductDomain = True
def __new__(cls, *domains):
# Flatten any product of products
domains2 = []
for domain in domains:
if not domain.is_ProductDomain:
domains2.append(domain)
else:
domains2.extend(domain.domains)
domains2 = FiniteSet(*domains2)
if all(domain.is_Finite for domain in domains2):
from sympy.stats.frv import ProductFiniteDomain
cls = ProductFiniteDomain
if all(domain.is_Continuous for domain in domains2):
from sympy.stats.crv import ProductContinuousDomain
cls = ProductContinuousDomain
if all(domain.is_Discrete for domain in domains2):
from sympy.stats.drv import ProductDiscreteDomain
cls = ProductDiscreteDomain
return Basic.__new__(cls, *domains2)
@property
def sym_domain_dict(self):
return {symbol: domain for domain in self.domains
for symbol in domain.symbols}
@property
def symbols(self):
return FiniteSet(*[sym for domain in self.domains
for sym in domain.symbols])
@property
def domains(self):
return self.args
@property
def set(self):
return ProductSet(*(domain.set for domain in self.domains))
def __contains__(self, other):
# Split event into each subdomain
for domain in self.domains:
# Collect the parts of this event which associate to this domain
elem = frozenset([item for item in other
if sympify(domain.symbols.contains(item[0]))
is S.true])
# Test this sub-event
if elem not in domain:
return False
# All subevents passed
return True
def as_boolean(self):
return And(*[domain.as_boolean() for domain in self.domains])
def random_symbols(expr):
"""
Returns all RandomSymbols within a SymPy Expression.
"""
atoms = getattr(expr, 'atoms', None)
if atoms is not None:
comp = lambda rv: rv.symbol.name
l = list(atoms(RandomSymbol))
return sorted(l, key=comp)
else:
return []
def pspace(expr):
"""
Returns the underlying Probability Space of a random expression.
For internal use.
Examples
========
>>> from sympy.stats import pspace, Normal
>>> X = Normal('X', 0, 1)
>>> pspace(2*X + 1) == X.pspace
True
"""
expr = sympify(expr)
if isinstance(expr, RandomSymbol) and expr.pspace is not None:
return expr.pspace
if expr.has(RandomMatrixSymbol):
rm = list(expr.atoms(RandomMatrixSymbol))[0]
return rm.pspace
rvs = random_symbols(expr)
if not rvs:
raise ValueError("Expression containing Random Variable expected, not %s" % (expr))
# If only one space present
if all(rv.pspace == rvs[0].pspace for rv in rvs):
return rvs[0].pspace
from sympy.stats.compound_rv import CompoundPSpace
from sympy.stats.stochastic_process import StochasticPSpace
for rv in rvs:
if isinstance(rv.pspace, (CompoundPSpace, StochasticPSpace)):
return rv.pspace
# Otherwise make a product space
return IndependentProductPSpace(*[rv.pspace for rv in rvs])
def sumsets(sets):
"""
Union of sets
"""
return frozenset().union(*sets)
def rs_swap(a, b):
"""
Build a dictionary to swap RandomSymbols based on their underlying symbol.
i.e.
if ``X = ('x', pspace1)``
and ``Y = ('x', pspace2)``
then ``X`` and ``Y`` match and the key, value pair
``{X:Y}`` will appear in the result
Inputs: collections a and b of random variables which share common symbols
Output: dict mapping RVs in a to RVs in b
"""
d = {}
for rsa in a:
d[rsa] = [rsb for rsb in b if rsa.symbol == rsb.symbol][0]
return d
def given(expr, condition=None, **kwargs):
r""" Conditional Random Expression.
Explanation
===========
From a random expression and a condition on that expression creates a new
probability space from the condition and returns the same expression on that
conditional probability space.
Examples
========
>>> from sympy.stats import given, density, Die
>>> X = Die('X', 6)
>>> Y = given(X, X > 3)
>>> density(Y).dict
{4: 1/3, 5: 1/3, 6: 1/3}
Following convention, if the condition is a random symbol then that symbol
is considered fixed.
>>> from sympy.stats import Normal
>>> from sympy import pprint
>>> from sympy.abc import z
>>> X = Normal('X', 0, 1)
>>> Y = Normal('Y', 0, 1)
>>> pprint(density(X + Y, Y)(z), use_unicode=False)
2
-(-Y + z)
-----------
___ 2
\/ 2 *e
------------------
____
2*\/ pi
"""
if not is_random(condition) or pspace_independent(expr, condition):
return expr
if isinstance(condition, RandomSymbol):
condition = Eq(condition, condition.symbol)
condsymbols = random_symbols(condition)
if (isinstance(condition, Eq) and len(condsymbols) == 1 and
not isinstance(pspace(expr).domain, ConditionalDomain)):
rv = tuple(condsymbols)[0]
results = solveset(condition, rv)
if isinstance(results, Intersection) and S.Reals in results.args:
results = list(results.args[1])
sums = 0
for res in results:
temp = expr.subs(rv, res)
if temp == True:
return True
if temp != False:
# XXX: This seems nonsensical but preserves existing behaviour
# after the change that Relational is no longer a subclass of
# Expr. Here expr is sometimes Relational and sometimes Expr
# but we are trying to add them with +=. This needs to be
# fixed somehow.
if sums == 0 and isinstance(expr, Relational):
sums = expr.subs(rv, res)
else:
sums += expr.subs(rv, res)
if sums == 0:
return False
return sums
# Get full probability space of both the expression and the condition
fullspace = pspace(Tuple(expr, condition))
# Build new space given the condition
space = fullspace.conditional_space(condition, **kwargs)
# Dictionary to swap out RandomSymbols in expr with new RandomSymbols
# That point to the new conditional space
swapdict = rs_swap(fullspace.values, space.values)
# Swap random variables in the expression
expr = expr.xreplace(swapdict)
return expr
def expectation(expr, condition=None, numsamples=None, evaluate=True, **kwargs):
"""
Returns the expected value of a random expression.
Parameters
==========
expr : Expr containing RandomSymbols
The expression of which you want to compute the expectation value
given : Expr containing RandomSymbols
A conditional expression. E(X, X>0) is expectation of X given X > 0
numsamples : int
Enables sampling and approximates the expectation with this many samples
evalf : Bool (defaults to True)
If sampling return a number rather than a complex expression
evaluate : Bool (defaults to True)
In case of continuous systems return unevaluated integral
Examples
========
>>> from sympy.stats import E, Die
>>> X = Die('X', 6)
>>> E(X)
7/2
>>> E(2*X + 1)
8
>>> E(X, X > 3) # Expectation of X given that it is above 3
5
"""
if not is_random(expr): # expr isn't random?
return expr
kwargs['numsamples'] = numsamples
from sympy.stats.symbolic_probability import Expectation
if evaluate:
return Expectation(expr, condition).doit(**kwargs)
return Expectation(expr, condition)
def probability(condition, given_condition=None, numsamples=None,
evaluate=True, **kwargs):
"""
Probability that a condition is true, optionally given a second condition.
Parameters
==========
condition : Combination of Relationals containing RandomSymbols
The condition of which you want to compute the probability
given_condition : Combination of Relationals containing RandomSymbols
A conditional expression. P(X > 1, X > 0) is expectation of X > 1
given X > 0
numsamples : int
Enables sampling and approximates the probability with this many samples
evaluate : Bool (defaults to True)
In case of continuous systems return unevaluated integral
Examples
========
>>> from sympy.stats import P, Die
>>> from sympy import Eq
>>> X, Y = Die('X', 6), Die('Y', 6)
>>> P(X > 3)
1/2
>>> P(Eq(X, 5), X > 2) # Probability that X == 5 given that X > 2
1/4
>>> P(X > Y)
5/12
"""
kwargs['numsamples'] = numsamples
from sympy.stats.symbolic_probability import Probability
if evaluate:
return Probability(condition, given_condition).doit(**kwargs)
### TODO: Remove the user warnings in the future releases
message = ("Since version 1.7, using `evaluate=False` returns `Probability` "
"object. If you want unevaluated Integral/Sum use "
"`P(condition, given_condition, evaluate=False).rewrite(Integral)`")
warnings.warn(filldedent(message))
return Probability(condition, given_condition)
class Density(Basic):
expr = property(lambda self: self.args[0])
def __new__(cls, expr, condition = None):
expr = _sympify(expr)
if condition is None:
obj = Basic.__new__(cls, expr)
else:
condition = _sympify(condition)
obj = Basic.__new__(cls, expr, condition)
return obj
@property
def condition(self):
if len(self.args) > 1:
return self.args[1]
else:
return None
def doit(self, evaluate=True, **kwargs):
from sympy.stats.random_matrix import RandomMatrixPSpace
from sympy.stats.joint_rv import JointPSpace
from sympy.stats.matrix_distributions import MatrixPSpace
from sympy.stats.compound_rv import CompoundPSpace
from sympy.stats.frv import SingleFiniteDistribution
expr, condition = self.expr, self.condition
if isinstance(expr, SingleFiniteDistribution):
return expr.dict
if condition is not None:
# Recompute on new conditional expr
expr = given(expr, condition, **kwargs)
if not random_symbols(expr):
return Lambda(x, DiracDelta(x - expr))
if isinstance(expr, RandomSymbol):
if isinstance(expr.pspace, (SinglePSpace, JointPSpace, MatrixPSpace)) and \
hasattr(expr.pspace, 'distribution'):
return expr.pspace.distribution
elif isinstance(expr.pspace, RandomMatrixPSpace):
return expr.pspace.model
if isinstance(pspace(expr), CompoundPSpace):
kwargs['compound_evaluate'] = evaluate
result = pspace(expr).compute_density(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def density(expr, condition=None, evaluate=True, numsamples=None, **kwargs):
"""
Probability density of a random expression, optionally given a second
condition.
Explanation
===========
This density will take on different forms for different types of
probability spaces. Discrete variables produce Dicts. Continuous
variables produce Lambdas.
Parameters
==========
expr : Expr containing RandomSymbols
The expression of which you want to compute the density value
condition : Relational containing RandomSymbols
A conditional expression. density(X > 1, X > 0) is density of X > 1
given X > 0
numsamples : int
Enables sampling and approximates the density with this many samples
Examples
========
>>> from sympy.stats import density, Die, Normal
>>> from sympy import Symbol
>>> x = Symbol('x')
>>> D = Die('D', 6)
>>> X = Normal(x, 0, 1)
>>> density(D).dict
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}
>>> density(2*D).dict
{2: 1/6, 4: 1/6, 6: 1/6, 8: 1/6, 10: 1/6, 12: 1/6}
>>> density(X)(x)
sqrt(2)*exp(-x**2/2)/(2*sqrt(pi))
"""
if numsamples:
return sampling_density(expr, condition, numsamples=numsamples,
**kwargs)
return Density(expr, condition).doit(evaluate=evaluate, **kwargs)
def cdf(expr, condition=None, evaluate=True, **kwargs):
"""
Cumulative Distribution Function of a random expression.
optionally given a second condition.
Explanation
===========
This density will take on different forms for different types of
probability spaces.
Discrete variables produce Dicts.
Continuous variables produce Lambdas.
Examples
========
>>> from sympy.stats import density, Die, Normal, cdf
>>> D = Die('D', 6)
>>> X = Normal('X', 0, 1)
>>> density(D).dict
{1: 1/6, 2: 1/6, 3: 1/6, 4: 1/6, 5: 1/6, 6: 1/6}
>>> cdf(D)
{1: 1/6, 2: 1/3, 3: 1/2, 4: 2/3, 5: 5/6, 6: 1}
>>> cdf(3*D, D > 2)
{9: 1/4, 12: 1/2, 15: 3/4, 18: 1}
>>> cdf(X)
Lambda(_z, erf(sqrt(2)*_z/2)/2 + 1/2)
"""
if condition is not None: # If there is a condition
# Recompute on new conditional expr
return cdf(given(expr, condition, **kwargs), **kwargs)
# Otherwise pass work off to the ProbabilitySpace
result = pspace(expr).compute_cdf(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def characteristic_function(expr, condition=None, evaluate=True, **kwargs):
"""
Characteristic function of a random expression, optionally given a second condition.
Returns a Lambda.
Examples
========
>>> from sympy.stats import Normal, DiscreteUniform, Poisson, characteristic_function
>>> X = Normal('X', 0, 1)
>>> characteristic_function(X)
Lambda(_t, exp(-_t**2/2))
>>> Y = DiscreteUniform('Y', [1, 2, 7])
>>> characteristic_function(Y)
Lambda(_t, exp(7*_t*I)/3 + exp(2*_t*I)/3 + exp(_t*I)/3)
>>> Z = Poisson('Z', 2)
>>> characteristic_function(Z)
Lambda(_t, exp(2*exp(_t*I) - 2))
"""
if condition is not None:
return characteristic_function(given(expr, condition, **kwargs), **kwargs)
result = pspace(expr).compute_characteristic_function(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def moment_generating_function(expr, condition=None, evaluate=True, **kwargs):
if condition is not None:
return moment_generating_function(given(expr, condition, **kwargs), **kwargs)
result = pspace(expr).compute_moment_generating_function(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def where(condition, given_condition=None, **kwargs):
"""
Returns the domain where a condition is True.
Examples
========
>>> from sympy.stats import where, Die, Normal
>>> from sympy import And
>>> D1, D2 = Die('a', 6), Die('b', 6)
>>> a, b = D1.symbol, D2.symbol
>>> X = Normal('x', 0, 1)
>>> where(X**2<1)
Domain: (-1 < x) & (x < 1)
>>> where(X**2<1).set
Interval.open(-1, 1)
>>> where(And(D1<=D2, D2<3))
Domain: (Eq(a, 1) & Eq(b, 1)) | (Eq(a, 1) & Eq(b, 2)) | (Eq(a, 2) & Eq(b, 2))
"""
if given_condition is not None: # If there is a condition
# Recompute on new conditional expr
return where(given(condition, given_condition, **kwargs), **kwargs)
# Otherwise pass work off to the ProbabilitySpace
return pspace(condition).where(condition, **kwargs)
@doctest_depends_on(modules=('scipy',))
def sample(expr, condition=None, size=(), library='scipy',
numsamples=1, seed=None, **kwargs):
"""
A realization of the random expression.
Parameters
==========
expr : Expression of random variables
Expression from which sample is extracted
condition : Expr containing RandomSymbols
A conditional expression
size : int, tuple
Represents size of each sample in numsamples
library : str
- 'scipy' : Sample using scipy
- 'numpy' : Sample using numpy
- 'pymc3' : Sample using PyMC3
Choose any of the available options to sample from as string,
by default is 'scipy'
numsamples : int
Number of samples, each with size as ``size``. The ``numsamples`` parameter is
deprecated and is only provided for compatibility with v1.8. Use a list comprehension
or an additional dimension in ``size`` instead.
seed :
An object to be used as seed by the given external library for sampling `expr`.
Following is the list of possible types of object for the supported libraries,
- 'scipy': int, numpy.random.RandomState, numpy.random.Generator
- 'numpy': int, numpy.random.RandomState, numpy.random.Generator
- 'pymc3': int
Optional, by default None, in which case seed settings
related to the given library will be used.
No modifications to environment's global seed settings
are done by this argument.
Returns
=======
sample: float/list/numpy.ndarray
one sample or a collection of samples of the random expression.
- sample(X) returns float/numpy.float64/numpy.int64 object.
- sample(X, size=int/tuple) returns numpy.ndarray object.
Examples
========
>>> from sympy.stats import Die, sample, Normal, Geometric
>>> X, Y, Z = Die('X', 6), Die('Y', 6), Die('Z', 6) # Finite Random Variable
>>> die_roll = sample(X + Y + Z)
>>> die_roll # doctest: +SKIP
3
>>> N = Normal('N', 3, 4) # Continuous Random Variable
>>> samp = sample(N)
>>> samp in N.pspace.domain.set
True
>>> samp = sample(N, N>0)
>>> samp > 0
True
>>> samp_list = sample(N, size=4)
>>> [sam in N.pspace.domain.set for sam in samp_list]
[True, True, True, True]
>>> sample(N, size = (2,3)) # doctest: +SKIP
array([[5.42519758, 6.40207856, 4.94991743],
[1.85819627, 6.83403519, 1.9412172 ]])
>>> G = Geometric('G', 0.5) # Discrete Random Variable
>>> samp_list = sample(G, size=3)
>>> samp_list # doctest: +SKIP
[1, 3, 2]
>>> [sam in G.pspace.domain.set for sam in samp_list]
[True, True, True]
>>> MN = Normal("MN", [3, 4], [[2, 1], [1, 2]]) # Joint Random Variable
>>> samp_list = sample(MN, size=4)
>>> samp_list # doctest: +SKIP
[array([2.85768055, 3.38954165]),
array([4.11163337, 4.3176591 ]),
array([0.79115232, 1.63232916]),
array([4.01747268, 3.96716083])]
>>> [tuple(sam) in MN.pspace.domain.set for sam in samp_list]
[True, True, True, True]
.. versionchanged:: 1.7.0
sample used to return an iterator containing the samples instead of value.
.. versionchanged:: 1.9.0
sample returns values or array of values instead of an iterator and numsamples is deprecated.
"""
iterator = sample_iter(expr, condition, size=size, library=library,
numsamples=numsamples, seed=seed)
if numsamples != 1:
SymPyDeprecationWarning(
feature="numsamples parameter",
issue=21723,
deprecated_since_version="1.9",
useinstead="a list comprehension or an additional dimension in ``size``").warn()
return [next(iterator) for i in range(numsamples)]
return next(iterator)
def quantile(expr, evaluate=True, **kwargs):
r"""
Return the :math:`p^{th}` order quantile of a probability distribution.
Explanation
===========
Quantile is defined as the value at which the probability of the random
variable is less than or equal to the given probability.
..math::
Q(p) = inf{x \in (-\infty, \infty) such that p <= F(x)}
Examples
========
>>> from sympy.stats import quantile, Die, Exponential
>>> from sympy import Symbol, pprint
>>> p = Symbol("p")
>>> l = Symbol("lambda", positive=True)
>>> X = Exponential("x", l)
>>> quantile(X)(p)
-log(1 - p)/lambda
>>> D = Die("d", 6)
>>> pprint(quantile(D)(p), use_unicode=False)
/nan for Or(p > 1, p < 0)
|
| 1 for p <= 1/6
|
| 2 for p <= 1/3
|
< 3 for p <= 1/2
|
| 4 for p <= 2/3
|
| 5 for p <= 5/6
|
\ 6 for p <= 1
"""
result = pspace(expr).compute_quantile(expr, **kwargs)
if evaluate and hasattr(result, 'doit'):
return result.doit()
else:
return result
def sample_iter(expr, condition=None, size=(), library='scipy',
numsamples=S.Infinity, seed=None, **kwargs):
"""
Returns an iterator of realizations from the expression given a condition.
Parameters
==========
expr: Expr
Random expression to be realized
condition: Expr, optional
A conditional expression
size : int, tuple
Represents size of each sample in numsamples
numsamples: integer, optional
Length of the iterator (defaults to infinity)
seed :
An object to be used as seed by the given external library for sampling `expr`.
Following is the list of possible types of object for the supported libraries,
- 'scipy': int, numpy.random.RandomState, numpy.random.Generator
- 'numpy': int, numpy.random.RandomState, numpy.random.Generator
- 'pymc3': int
Optional, by default None, in which case seed settings
related to the given library will be used.
No modifications to environment's global seed settings
are done by this argument.
Examples
========
>>> from sympy.stats import Normal, sample_iter
>>> X = Normal('X', 0, 1)
>>> expr = X*X + 3
>>> iterator = sample_iter(expr, numsamples=3) # doctest: +SKIP
>>> list(iterator) # doctest: +SKIP
[12, 4, 7]
Returns
=======
sample_iter: iterator object
iterator object containing the sample/samples of given expr
See Also
========
sample
sampling_P
sampling_E
"""
from sympy.stats.joint_rv import JointRandomSymbol
if not import_module(library):
raise ValueError("Failed to import %s" % library)
if condition is not None:
ps = pspace(Tuple(expr, condition))
else:
ps = pspace(expr)
rvs = list(ps.values)
if isinstance(expr, JointRandomSymbol):
expr = expr.subs({expr: RandomSymbol(expr.symbol, expr.pspace)})
else:
sub = {}
for arg in expr.args:
if isinstance(arg, JointRandomSymbol):
sub[arg] = RandomSymbol(arg.symbol, arg.pspace)
expr = expr.subs(sub)
def fn_subs(*args):
return expr.subs({rv: arg for rv, arg in zip(rvs, args)})
def given_fn_subs(*args):
if condition is not None:
return condition.subs({rv: arg for rv, arg in zip(rvs, args)})
return False
if library == 'pymc3':
# Currently unable to lambdify in pymc3
# TODO : Remove 'pymc3' when lambdify accepts 'pymc3' as module
fn = lambdify(rvs, expr, **kwargs)
else:
fn = lambdify(rvs, expr, modules=library, **kwargs)
if condition is not None:
given_fn = lambdify(rvs, condition, **kwargs)
def return_generator_infinite():
count = 0
_size = (1,)+((size,) if isinstance(size, int) else size)
while count < numsamples:
d = ps.sample(size=_size, library=library, seed=seed) # a dictionary that maps RVs to values
args = [d[rv][0] for rv in rvs]
if condition is not None: # Check that these values satisfy the condition
# TODO: Replace the try-except block with only given_fn(*args)
# once lambdify works with unevaluated SymPy objects.
try:
gd = given_fn(*args)
except (NameError, TypeError):
gd = given_fn_subs(*args)
if gd != True and gd != False:
raise ValueError(
"Conditions must not contain free symbols")
if not gd: # If the values don't satisfy then try again
continue
yield fn(*args)
count += 1
def return_generator_finite():
faulty = True
while faulty:
d = ps.sample(size=(numsamples,) + ((size,) if isinstance(size, int) else size),
library=library, seed=seed) # a dictionary that maps RVs to values
faulty = False
count = 0
while count < numsamples and not faulty:
args = [d[rv][count] for rv in rvs]
if condition is not None: # Check that these values satisfy the condition
# TODO: Replace the try-except block with only given_fn(*args)
# once lambdify works with unevaluated SymPy objects.
try:
gd = given_fn(*args)
except (NameError, TypeError):
gd = given_fn_subs(*args)
if gd != True and gd != False:
raise ValueError(
"Conditions must not contain free symbols")
if not gd: # If the values don't satisfy then try again
faulty = True
count += 1
count = 0
while count < numsamples:
args = [d[rv][count] for rv in rvs]
# TODO: Replace the try-except block with only fn(*args)
# once lambdify works with unevaluated SymPy objects.
try:
yield fn(*args)
except (NameError, TypeError):
yield fn_subs(*args)
count += 1
if numsamples is S.Infinity:
return return_generator_infinite()
return return_generator_finite()
def sample_iter_lambdify(expr, condition=None, size=(),
numsamples=S.Infinity, seed=None, **kwargs):
return sample_iter(expr, condition=condition, size=size,
numsamples=numsamples, seed=seed, **kwargs)
def sample_iter_subs(expr, condition=None, size=(),
numsamples=S.Infinity, seed=None, **kwargs):
return sample_iter(expr, condition=condition, size=size,
numsamples=numsamples, seed=seed, **kwargs)
def sampling_P(condition, given_condition=None, library='scipy', numsamples=1,
evalf=True, seed=None, **kwargs):
"""
Sampling version of P.
See Also
========
P
sampling_E
sampling_density
"""
count_true = 0
count_false = 0
samples = sample_iter(condition, given_condition, library=library,
numsamples=numsamples, seed=seed, **kwargs)
for sample in samples:
if sample:
count_true += 1
else:
count_false += 1
result = S(count_true) / numsamples
if evalf:
return result.evalf()
else:
return result
def sampling_E(expr, given_condition=None, library='scipy', numsamples=1,
evalf=True, seed=None, **kwargs):
"""
Sampling version of E.
See Also
========
P
sampling_P
sampling_density
"""
samples = list(sample_iter(expr, given_condition, library=library,
numsamples=numsamples, seed=seed, **kwargs))
result = Add(*[samp for samp in samples]) / numsamples
if evalf:
return result.evalf()
else:
return result
def sampling_density(expr, given_condition=None, library='scipy',
numsamples=1, seed=None, **kwargs):
"""
Sampling version of density.
See Also
========
density
sampling_P
sampling_E
"""
results = {}
for result in sample_iter(expr, given_condition, library=library,
numsamples=numsamples, seed=seed, **kwargs):
results[result] = results.get(result, 0) + 1
return results
def dependent(a, b):
"""
Dependence of two random expressions.
Two expressions are independent if knowledge of one does not change
computations on the other.
Examples
========
>>> from sympy.stats import Normal, dependent, given
>>> from sympy import Tuple, Eq
>>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
>>> dependent(X, Y)
False
>>> dependent(2*X + Y, -Y)
True
>>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3))
>>> dependent(X, Y)
True
See Also
========
independent
"""
if pspace_independent(a, b):
return False
z = Symbol('z', real=True)
# Dependent if density is unchanged when one is given information about
# the other
return (density(a, Eq(b, z)) != density(a) or
density(b, Eq(a, z)) != density(b))
def independent(a, b):
"""
Independence of two random expressions.
Two expressions are independent if knowledge of one does not change
computations on the other.
Examples
========
>>> from sympy.stats import Normal, independent, given
>>> from sympy import Tuple, Eq
>>> X, Y = Normal('X', 0, 1), Normal('Y', 0, 1)
>>> independent(X, Y)
True
>>> independent(2*X + Y, -Y)
False
>>> X, Y = given(Tuple(X, Y), Eq(X + Y, 3))
>>> independent(X, Y)
False
See Also
========
dependent
"""
return not dependent(a, b)
def pspace_independent(a, b):
"""
Tests for independence between a and b by checking if their PSpaces have
overlapping symbols. This is a sufficient but not necessary condition for
independence and is intended to be used internally.
Notes
=====
pspace_independent(a, b) implies independent(a, b)
independent(a, b) does not imply pspace_independent(a, b)
"""
a_symbols = set(pspace(b).symbols)
b_symbols = set(pspace(a).symbols)
if len(set(random_symbols(a)).intersection(random_symbols(b))) != 0:
return False
if len(a_symbols.intersection(b_symbols)) == 0:
return True
return None
def rv_subs(expr, symbols=None):
"""
Given a random expression replace all random variables with their symbols.
If symbols keyword is given restrict the swap to only the symbols listed.
"""
if symbols is None:
symbols = random_symbols(expr)
if not symbols:
return expr
swapdict = {rv: rv.symbol for rv in symbols}
return expr.subs(swapdict)
class NamedArgsMixin:
_argnames = () # type: tTuple[str, ...]
def __getattr__(self, attr):
try:
return self.args[self._argnames.index(attr)]
except ValueError:
raise AttributeError("'%s' object has no attribute '%s'" % (
type(self).__name__, attr))
class Distribution(Basic):
def sample(self, size=(), library='scipy', seed=None):
""" A random realization from the distribution """
module = import_module(library)
if library in {'scipy', 'numpy', 'pymc3'} and module is None:
raise ValueError("Failed to import %s" % library)
if library == 'scipy':
# scipy does not require map as it can handle using custom distributions.
# However, we will still use a map where we can.
# TODO: do this for drv.py and frv.py if necessary.
# TODO: add more distributions here if there are more
# See links below referring to sections beginning with "A common parametrization..."
# I will remove all these comments if everything is ok.
from sympy.stats.sampling.sample_scipy import do_sample_scipy
import numpy
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
samps = do_sample_scipy(self, size, rand_state)
elif library == 'numpy':
from sympy.stats.sampling.sample_numpy import do_sample_numpy
import numpy
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
_size = None if size == () else size
samps = do_sample_numpy(self, _size, rand_state)
elif library == 'pymc3':
from sympy.stats.sampling.sample_pymc3 import do_sample_pymc3
import logging
logging.getLogger("pymc3").setLevel(logging.ERROR)
import pymc3
with pymc3.Model():
if do_sample_pymc3(self):
samps = pymc3.sample(draws=prod(size), chains=1, compute_convergence_checks=False,
progressbar=False, random_seed=seed, return_inferencedata=False)[:]['X']
samps = samps.reshape(size)
else:
samps = None
else:
raise NotImplementedError("Sampling from %s is not supported yet."
% str(library))
if samps is not None:
return samps
raise NotImplementedError(
"Sampling for %s is not currently implemented from %s"
% (self, library))
def _value_check(condition, message):
"""
Raise a ValueError with message if condition is False, else
return True if all conditions were True, else False.
Examples
========
>>> from sympy.stats.rv import _value_check
>>> from sympy.abc import a, b, c
>>> from sympy import And, Dummy
>>> _value_check(2 < 3, '')
True
Here, the condition is not False, but it doesn't evaluate to True
so False is returned (but no error is raised). So checking if the
return value is True or False will tell you if all conditions were
evaluated.
>>> _value_check(a < b, '')
False
In this case the condition is False so an error is raised:
>>> r = Dummy(real=True)
>>> _value_check(r < r - 1, 'condition is not true')
Traceback (most recent call last):
...
ValueError: condition is not true
If no condition of many conditions must be False, they can be
checked by passing them as an iterable:
>>> _value_check((a < 0, b < 0, c < 0), '')
False
The iterable can be a generator, too:
>>> _value_check((i < 0 for i in (a, b, c)), '')
False
The following are equivalent to the above but do not pass
an iterable:
>>> all(_value_check(i < 0, '') for i in (a, b, c))
False
>>> _value_check(And(a < 0, b < 0, c < 0), '')
False
"""
if not iterable(condition):
condition = [condition]
truth = fuzzy_and(condition)
if truth == False:
raise ValueError(message)
return truth == True
def _symbol_converter(sym):
"""
Casts the parameter to Symbol if it is 'str'
otherwise no operation is performed on it.
Parameters
==========
sym
The parameter to be converted.
Returns
=======
Symbol
the parameter converted to Symbol.
Raises
======
TypeError
If the parameter is not an instance of both str and
Symbol.
Examples
========
>>> from sympy import Symbol
>>> from sympy.stats.rv import _symbol_converter
>>> s = _symbol_converter('s')
>>> isinstance(s, Symbol)
True
>>> _symbol_converter(1)
Traceback (most recent call last):
...
TypeError: 1 is neither a Symbol nor a string
>>> r = Symbol('r')
>>> isinstance(r, Symbol)
True
"""
if isinstance(sym, str):
sym = Symbol(sym)
if not isinstance(sym, Symbol):
raise TypeError("%s is neither a Symbol nor a string"%(sym))
return sym
def sample_stochastic_process(process):
"""
This function is used to sample from stochastic process.
Parameters
==========
process: StochasticProcess
Process used to extract the samples. It must be an instance of
StochasticProcess
Examples
========
>>> from sympy.stats import sample_stochastic_process, DiscreteMarkovChain
>>> from sympy import Matrix
>>> T = Matrix([[0.5, 0.2, 0.3],[0.2, 0.5, 0.3],[0.2, 0.3, 0.5]])
>>> Y = DiscreteMarkovChain("Y", [0, 1, 2], T)
>>> next(sample_stochastic_process(Y)) in Y.state_space # doctest: +SKIP
True
>>> next(sample_stochastic_process(Y)) # doctest: +SKIP
0
>>> next(sample_stochastic_process(Y)) # doctest: +SKIP
2
Returns
=======
sample: iterator object
iterator object containing the sample of given process
"""
from sympy.stats.stochastic_process_types import StochasticProcess
if not isinstance(process, StochasticProcess):
raise ValueError("Process must be an instance of Stochastic Process")
return process.sample()
|
b2ff97f45fe5b52895416a025771667c762c1b104993bf6e09d5d4db6dedebe4 | """
Joint Random Variables Module
See Also
========
sympy.stats.rv
sympy.stats.frv
sympy.stats.crv
sympy.stats.drv
"""
from sympy.core.basic import Basic
from sympy.core.function import Lambda
from sympy.core.mul import prod
from sympy.core.singleton import S
from sympy.core.symbol import (Dummy, Symbol)
from sympy.core.sympify import sympify
from sympy.sets.sets import ProductSet
from sympy.tensor.indexed import Indexed
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum, summation
from sympy.core.containers import Tuple
from sympy.integrals.integrals import Integral, integrate
from sympy.matrices import ImmutableMatrix, matrix2numpy, list2numpy
from sympy.stats.crv import SingleContinuousDistribution, SingleContinuousPSpace
from sympy.stats.drv import SingleDiscreteDistribution, SingleDiscretePSpace
from sympy.stats.rv import (ProductPSpace, NamedArgsMixin, Distribution,
ProductDomain, RandomSymbol, random_symbols,
SingleDomain, _symbol_converter)
from sympy.utilities.iterables import iterable
from sympy.utilities.misc import filldedent
from sympy.external import import_module
# __all__ = ['marginal_distribution']
class JointPSpace(ProductPSpace):
"""
Represents a joint probability space. Represented using symbols for
each component and a distribution.
"""
def __new__(cls, sym, dist):
if isinstance(dist, SingleContinuousDistribution):
return SingleContinuousPSpace(sym, dist)
if isinstance(dist, SingleDiscreteDistribution):
return SingleDiscretePSpace(sym, dist)
sym = _symbol_converter(sym)
return Basic.__new__(cls, sym, dist)
@property
def set(self):
return self.domain.set
@property
def symbol(self):
return self.args[0]
@property
def distribution(self):
return self.args[1]
@property
def value(self):
return JointRandomSymbol(self.symbol, self)
@property
def component_count(self):
_set = self.distribution.set
if isinstance(_set, ProductSet):
return S(len(_set.args))
elif isinstance(_set, Product):
return _set.limits[0][-1]
return S.One
@property
def pdf(self):
sym = [Indexed(self.symbol, i) for i in range(self.component_count)]
return self.distribution(*sym)
@property
def domain(self):
rvs = random_symbols(self.distribution)
if not rvs:
return SingleDomain(self.symbol, self.distribution.set)
return ProductDomain(*[rv.pspace.domain for rv in rvs])
def component_domain(self, index):
return self.set.args[index]
def marginal_distribution(self, *indices):
count = self.component_count
if count.atoms(Symbol):
raise ValueError("Marginal distributions cannot be computed "
"for symbolic dimensions. It is a work under progress.")
orig = [Indexed(self.symbol, i) for i in range(count)]
all_syms = [Symbol(str(i)) for i in orig]
replace_dict = dict(zip(all_syms, orig))
sym = tuple(Symbol(str(Indexed(self.symbol, i))) for i in indices)
limits = list([i,] for i in all_syms if i not in sym)
index = 0
for i in range(count):
if i not in indices:
limits[index].append(self.distribution.set.args[i])
limits[index] = tuple(limits[index])
index += 1
if self.distribution.is_Continuous:
f = Lambda(sym, integrate(self.distribution(*all_syms), *limits))
elif self.distribution.is_Discrete:
f = Lambda(sym, summation(self.distribution(*all_syms), *limits))
return f.xreplace(replace_dict)
def compute_expectation(self, expr, rvs=None, evaluate=False, **kwargs):
syms = tuple(self.value[i] for i in range(self.component_count))
rvs = rvs or syms
if not any(i in rvs for i in syms):
return expr
expr = expr*self.pdf
for rv in rvs:
if isinstance(rv, Indexed):
expr = expr.xreplace({rv: Indexed(str(rv.base), rv.args[1])})
elif isinstance(rv, RandomSymbol):
expr = expr.xreplace({rv: rv.symbol})
if self.value in random_symbols(expr):
raise NotImplementedError(filldedent('''
Expectations of expression with unindexed joint random symbols
cannot be calculated yet.'''))
limits = tuple((Indexed(str(rv.base),rv.args[1]),
self.distribution.set.args[rv.args[1]]) for rv in syms)
return Integral(expr, *limits)
def where(self, condition):
raise NotImplementedError()
def compute_density(self, expr):
raise NotImplementedError()
def sample(self, size=(), library='scipy', seed=None):
"""
Internal sample method
Returns dictionary mapping RandomSymbol to realization value.
"""
return {RandomSymbol(self.symbol, self): self.distribution.sample(size,
library=library, seed=seed)}
def probability(self, condition):
raise NotImplementedError()
class SampleJointScipy:
"""Returns the sample from scipy of the given distribution"""
def __new__(cls, dist, size, seed=None):
return cls._sample_scipy(dist, size, seed)
@classmethod
def _sample_scipy(cls, dist, size, seed):
"""Sample from SciPy."""
import numpy
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
from scipy import stats as scipy_stats
scipy_rv_map = {
'MultivariateNormalDistribution': lambda dist, size: scipy_stats.multivariate_normal.rvs(
mean=matrix2numpy(dist.mu).flatten(),
cov=matrix2numpy(dist.sigma), size=size, random_state=rand_state),
'MultivariateBetaDistribution': lambda dist, size: scipy_stats.dirichlet.rvs(
alpha=list2numpy(dist.alpha, float).flatten(), size=size, random_state=rand_state),
'MultinomialDistribution': lambda dist, size: scipy_stats.multinomial.rvs(
n=int(dist.n), p=list2numpy(dist.p, float).flatten(), size=size, random_state=rand_state)
}
sample_shape = {
'MultivariateNormalDistribution': lambda dist: matrix2numpy(dist.mu).flatten().shape,
'MultivariateBetaDistribution': lambda dist: list2numpy(dist.alpha).flatten().shape,
'MultinomialDistribution': lambda dist: list2numpy(dist.p).flatten().shape
}
dist_list = scipy_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
samples = scipy_rv_map[dist.__class__.__name__](dist, size)
return samples.reshape(size + sample_shape[dist.__class__.__name__](dist))
class SampleJointNumpy:
"""Returns the sample from numpy of the given distribution"""
def __new__(cls, dist, size, seed=None):
return cls._sample_numpy(dist, size, seed)
@classmethod
def _sample_numpy(cls, dist, size, seed):
"""Sample from NumPy."""
import numpy
if seed is None or isinstance(seed, int):
rand_state = numpy.random.default_rng(seed=seed)
else:
rand_state = seed
numpy_rv_map = {
'MultivariateNormalDistribution': lambda dist, size: rand_state.multivariate_normal(
mean=matrix2numpy(dist.mu, float).flatten(),
cov=matrix2numpy(dist.sigma, float), size=size),
'MultivariateBetaDistribution': lambda dist, size: rand_state.dirichlet(
alpha=list2numpy(dist.alpha, float).flatten(), size=size),
'MultinomialDistribution': lambda dist, size: rand_state.multinomial(
n=int(dist.n), pvals=list2numpy(dist.p, float).flatten(), size=size)
}
sample_shape = {
'MultivariateNormalDistribution': lambda dist: matrix2numpy(dist.mu).flatten().shape,
'MultivariateBetaDistribution': lambda dist: list2numpy(dist.alpha).flatten().shape,
'MultinomialDistribution': lambda dist: list2numpy(dist.p).flatten().shape
}
dist_list = numpy_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
samples = numpy_rv_map[dist.__class__.__name__](dist, prod(size))
return samples.reshape(size + sample_shape[dist.__class__.__name__](dist))
class SampleJointPymc:
"""Returns the sample from pymc3 of the given distribution"""
def __new__(cls, dist, size, seed=None):
return cls._sample_pymc3(dist, size, seed)
@classmethod
def _sample_pymc3(cls, dist, size, seed):
"""Sample from PyMC3."""
import pymc3
pymc3_rv_map = {
'MultivariateNormalDistribution': lambda dist:
pymc3.MvNormal('X', mu=matrix2numpy(dist.mu, float).flatten(),
cov=matrix2numpy(dist.sigma, float), shape=(1, dist.mu.shape[0])),
'MultivariateBetaDistribution': lambda dist:
pymc3.Dirichlet('X', a=list2numpy(dist.alpha, float).flatten()),
'MultinomialDistribution': lambda dist:
pymc3.Multinomial('X', n=int(dist.n),
p=list2numpy(dist.p, float).flatten(), shape=(1, len(dist.p)))
}
sample_shape = {
'MultivariateNormalDistribution': lambda dist: matrix2numpy(dist.mu).flatten().shape,
'MultivariateBetaDistribution': lambda dist: list2numpy(dist.alpha).flatten().shape,
'MultinomialDistribution': lambda dist: list2numpy(dist.p).flatten().shape
}
dist_list = pymc3_rv_map.keys()
if dist.__class__.__name__ not in dist_list:
return None
import logging
logging.getLogger("pymc3").setLevel(logging.ERROR)
with pymc3.Model():
pymc3_rv_map[dist.__class__.__name__](dist)
samples = pymc3.sample(draws=prod(size), chains=1, progressbar=False, random_seed=seed, return_inferencedata=False, compute_convergence_checks=False)[:]['X']
return samples.reshape(size + sample_shape[dist.__class__.__name__](dist))
_get_sample_class_jrv = {
'scipy': SampleJointScipy,
'pymc3': SampleJointPymc,
'numpy': SampleJointNumpy
}
class JointDistribution(Distribution, NamedArgsMixin):
"""
Represented by the random variables part of the joint distribution.
Contains methods for PDF, CDF, sampling, marginal densities, etc.
"""
_argnames = ('pdf', )
def __new__(cls, *args):
args = list(map(sympify, args))
for i in range(len(args)):
if isinstance(args[i], list):
args[i] = ImmutableMatrix(args[i])
return Basic.__new__(cls, *args)
@property
def domain(self):
return ProductDomain(self.symbols)
@property
def pdf(self):
return self.density.args[1]
def cdf(self, other):
if not isinstance(other, dict):
raise ValueError("%s should be of type dict, got %s"%(other, type(other)))
rvs = other.keys()
_set = self.domain.set.sets
expr = self.pdf(tuple(i.args[0] for i in self.symbols))
for i in range(len(other)):
if rvs[i].is_Continuous:
density = Integral(expr, (rvs[i], _set[i].inf,
other[rvs[i]]))
elif rvs[i].is_Discrete:
density = Sum(expr, (rvs[i], _set[i].inf,
other[rvs[i]]))
return density
def sample(self, size=(), library='scipy', seed=None):
""" A random realization from the distribution """
libraries = ['scipy', 'numpy', 'pymc3']
if library not in libraries:
raise NotImplementedError("Sampling from %s is not supported yet."
% str(library))
if not import_module(library):
raise ValueError("Failed to import %s" % library)
samps = _get_sample_class_jrv[library](self, size, seed=seed)
if samps is not None:
return samps
raise NotImplementedError(
"Sampling for %s is not currently implemented from %s"
% (self.__class__.__name__, library)
)
def __call__(self, *args):
return self.pdf(*args)
class JointRandomSymbol(RandomSymbol):
"""
Representation of random symbols with joint probability distributions
to allow indexing."
"""
def __getitem__(self, key):
if isinstance(self.pspace, JointPSpace):
if (self.pspace.component_count <= key) == True:
raise ValueError("Index keys for %s can only up to %s." %
(self.name, self.pspace.component_count - 1))
return Indexed(self, key)
class MarginalDistribution(Distribution):
"""
Represents the marginal distribution of a joint probability space.
Initialised using a probability distribution and random variables(or
their indexed components) which should be a part of the resultant
distribution.
"""
def __new__(cls, dist, *rvs):
if len(rvs) == 1 and iterable(rvs[0]):
rvs = tuple(rvs[0])
if not all(isinstance(rv, (Indexed, RandomSymbol)) for rv in rvs):
raise ValueError(filldedent('''Marginal distribution can be
intitialised only in terms of random variables or indexed random
variables'''))
rvs = Tuple.fromiter(rv for rv in rvs)
if not isinstance(dist, JointDistribution) and len(random_symbols(dist)) == 0:
return dist
return Basic.__new__(cls, dist, rvs)
def check(self):
pass
@property
def set(self):
rvs = [i for i in self.args[1] if isinstance(i, RandomSymbol)]
return ProductSet(*[rv.pspace.set for rv in rvs])
@property
def symbols(self):
rvs = self.args[1]
return {rv.pspace.symbol for rv in rvs}
def pdf(self, *x):
expr, rvs = self.args[0], self.args[1]
marginalise_out = [i for i in random_symbols(expr) if i not in rvs]
if isinstance(expr, JointDistribution):
count = len(expr.domain.args)
x = Dummy('x', real=True)
syms = tuple(Indexed(x, i) for i in count)
expr = expr.pdf(syms)
else:
syms = tuple(rv.pspace.symbol if isinstance(rv, RandomSymbol) else rv.args[0] for rv in rvs)
return Lambda(syms, self.compute_pdf(expr, marginalise_out))(*x)
def compute_pdf(self, expr, rvs):
for rv in rvs:
lpdf = 1
if isinstance(rv, RandomSymbol):
lpdf = rv.pspace.pdf
expr = self.marginalise_out(expr*lpdf, rv)
return expr
def marginalise_out(self, expr, rv):
from sympy.concrete.summations import Sum
if isinstance(rv, RandomSymbol):
dom = rv.pspace.set
elif isinstance(rv, Indexed):
dom = rv.base.component_domain(
rv.pspace.component_domain(rv.args[1]))
expr = expr.xreplace({rv: rv.pspace.symbol})
if rv.pspace.is_Continuous:
#TODO: Modify to support integration
#for all kinds of sets.
expr = Integral(expr, (rv.pspace.symbol, dom))
elif rv.pspace.is_Discrete:
#incorporate this into `Sum`/`summation`
if dom in (S.Integers, S.Naturals, S.Naturals0):
dom = (dom.inf, dom.sup)
expr = Sum(expr, (rv.pspace.symbol, dom))
return expr
def __call__(self, *args):
return self.pdf(*args)
|
cbb7808f73e55b847118c4c9e8d96fd8ff1c9df23d2628f58c776afb3a65bad3 | import itertools
from sympy.concrete.summations import Sum
from sympy.core.add import Add
from sympy.core.expr import Expr
from sympy.core.function import expand as _expand
from sympy.core.mul import Mul
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import Not
from sympy.core.parameters import global_parameters
from sympy.core.sorting import default_sort_key
from sympy.core.sympify import _sympify
from sympy.core.relational import Relational
from sympy.logic.boolalg import Boolean
from sympy.stats import variance, covariance
from sympy.stats.rv import (RandomSymbol, pspace, dependent,
given, sampling_E, RandomIndexedSymbol, is_random,
PSpace, sampling_P, random_symbols)
__all__ = ['Probability', 'Expectation', 'Variance', 'Covariance']
@is_random.register(Expr)
def _(x):
atoms = x.free_symbols
if len(atoms) == 1 and next(iter(atoms)) == x:
return False
return any(is_random(i) for i in atoms)
@is_random.register(RandomSymbol) # type: ignore
def _(x):
return True
class Probability(Expr):
"""
Symbolic expression for the probability.
Examples
========
>>> from sympy.stats import Probability, Normal
>>> from sympy import Integral
>>> X = Normal("X", 0, 1)
>>> prob = Probability(X > 1)
>>> prob
Probability(X > 1)
Integral representation:
>>> prob.rewrite(Integral)
Integral(sqrt(2)*exp(-_z**2/2)/(2*sqrt(pi)), (_z, 1, oo))
Evaluation of the integral:
>>> prob.evaluate_integral()
sqrt(2)*(-sqrt(2)*sqrt(pi)*erf(sqrt(2)/2) + sqrt(2)*sqrt(pi))/(4*sqrt(pi))
"""
def __new__(cls, prob, condition=None, **kwargs):
prob = _sympify(prob)
if condition is None:
obj = Expr.__new__(cls, prob)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, prob, condition)
obj._condition = condition
return obj
def doit(self, **hints):
condition = self.args[0]
given_condition = self._condition
numsamples = hints.get('numsamples', False)
for_rewrite = not hints.get('for_rewrite', False)
if isinstance(condition, Not):
return S.One - self.func(condition.args[0], given_condition,
evaluate=for_rewrite).doit(**hints)
if condition.has(RandomIndexedSymbol):
return pspace(condition).probability(condition, given_condition,
evaluate=for_rewrite)
if isinstance(given_condition, RandomSymbol):
condrv = random_symbols(condition)
if len(condrv) == 1 and condrv[0] == given_condition:
from sympy.stats.frv_types import BernoulliDistribution
return BernoulliDistribution(self.func(condition).doit(**hints), 0, 1)
if any(dependent(rv, given_condition) for rv in condrv):
return Probability(condition, given_condition)
else:
return Probability(condition).doit()
if given_condition is not None and \
not isinstance(given_condition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (given_condition))
if given_condition == False or condition is S.false:
return S.Zero
if not isinstance(condition, (Relational, Boolean)):
raise ValueError("%s is not a relational or combination of relationals"
% (condition))
if condition is S.true:
return S.One
if numsamples:
return sampling_P(condition, given_condition, numsamples=numsamples)
if given_condition is not None: # If there is a condition
# Recompute on new conditional expr
return Probability(given(condition, given_condition)).doit()
# Otherwise pass work off to the ProbabilitySpace
if pspace(condition) == PSpace():
return Probability(condition, given_condition)
result = pspace(condition).probability(condition)
if hasattr(result, 'doit') and for_rewrite:
return result.doit()
else:
return result
def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs):
return self.func(arg, condition=condition).doit(for_rewrite=True)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral
def evaluate_integral(self):
return self.rewrite(Integral).doit()
class Expectation(Expr):
"""
Symbolic expression for the expectation.
Examples
========
>>> from sympy.stats import Expectation, Normal, Probability, Poisson
>>> from sympy import symbols, Integral, Sum
>>> mu = symbols("mu")
>>> sigma = symbols("sigma", positive=True)
>>> X = Normal("X", mu, sigma)
>>> Expectation(X)
Expectation(X)
>>> Expectation(X).evaluate_integral().simplify()
mu
To get the integral expression of the expectation:
>>> Expectation(X).rewrite(Integral)
Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))
The same integral expression, in more abstract terms:
>>> Expectation(X).rewrite(Probability)
Integral(x*Probability(Eq(X, x)), (x, -oo, oo))
To get the Summation expression of the expectation for discrete random variables:
>>> lamda = symbols('lamda', positive=True)
>>> Z = Poisson('Z', lamda)
>>> Expectation(Z).rewrite(Sum)
Sum(Z*lamda**Z*exp(-lamda)/factorial(Z), (Z, 0, oo))
This class is aware of some properties of the expectation:
>>> from sympy.abc import a
>>> Expectation(a*X)
Expectation(a*X)
>>> Y = Normal("Y", 1, 2)
>>> Expectation(X + Y)
Expectation(X + Y)
To expand the ``Expectation`` into its expression, use ``expand()``:
>>> Expectation(X + Y).expand()
Expectation(X) + Expectation(Y)
>>> Expectation(a*X + Y).expand()
a*Expectation(X) + Expectation(Y)
>>> Expectation(a*X + Y)
Expectation(a*X + Y)
>>> Expectation((X + Y)*(X - Y)).expand()
Expectation(X**2) - Expectation(Y**2)
To evaluate the ``Expectation``, use ``doit()``:
>>> Expectation(X + Y).doit()
mu + 1
>>> Expectation(X + Expectation(Y + Expectation(2*X))).doit()
3*mu + 1
To prevent evaluating nested ``Expectation``, use ``doit(deep=False)``
>>> Expectation(X + Expectation(Y)).doit(deep=False)
mu + Expectation(Expectation(Y))
>>> Expectation(X + Expectation(Y + Expectation(2*X))).doit(deep=False)
mu + Expectation(Expectation(Y + Expectation(2*X)))
"""
def __new__(cls, expr, condition=None, **kwargs):
expr = _sympify(expr)
if expr.is_Matrix:
from sympy.stats.symbolic_multivariate_probability import ExpectationMatrix
return ExpectationMatrix(expr, condition)
if condition is None:
if not is_random(expr):
return expr
obj = Expr.__new__(cls, expr)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, expr, condition)
obj._condition = condition
return obj
def expand(self, **hints):
expr = self.args[0]
condition = self._condition
if not is_random(expr):
return expr
if isinstance(expr, Add):
return Add.fromiter(Expectation(a, condition=condition).expand()
for a in expr.args)
expand_expr = _expand(expr)
if isinstance(expand_expr, Add):
return Add.fromiter(Expectation(a, condition=condition).expand()
for a in expand_expr.args)
elif isinstance(expr, Mul):
rv = []
nonrv = []
for a in expr.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
return Mul.fromiter(nonrv)*Expectation(Mul.fromiter(rv), condition=condition)
return self
def doit(self, **hints):
deep = hints.get('deep', True)
condition = self._condition
expr = self.args[0]
numsamples = hints.get('numsamples', False)
for_rewrite = not hints.get('for_rewrite', False)
if deep:
expr = expr.doit(**hints)
if not is_random(expr) or isinstance(expr, Expectation): # expr isn't random?
return expr
if numsamples: # Computing by monte carlo sampling?
evalf = hints.get('evalf', True)
return sampling_E(expr, condition, numsamples=numsamples, evalf=evalf)
if expr.has(RandomIndexedSymbol):
return pspace(expr).compute_expectation(expr, condition)
# Create new expr and recompute E
if condition is not None: # If there is a condition
return self.func(given(expr, condition)).doit(**hints)
# A few known statements for efficiency
if expr.is_Add: # We know that E is Linear
return Add(*[self.func(arg, condition).doit(**hints)
if not isinstance(arg, Expectation) else self.func(arg, condition)
for arg in expr.args])
if expr.is_Mul:
if expr.atoms(Expectation):
return expr
if pspace(expr) == PSpace():
return self.func(expr)
# Otherwise case is simple, pass work off to the ProbabilitySpace
result = pspace(expr).compute_expectation(expr, evaluate=for_rewrite)
if hasattr(result, 'doit') and for_rewrite:
return result.doit(**hints)
else:
return result
def _eval_rewrite_as_Probability(self, arg, condition=None, **kwargs):
rvs = arg.atoms(RandomSymbol)
if len(rvs) > 1:
raise NotImplementedError()
if len(rvs) == 0:
return arg
rv = rvs.pop()
if rv.pspace is None:
raise ValueError("Probability space not known")
symbol = rv.symbol
if symbol.name[0].isupper():
symbol = Symbol(symbol.name.lower())
else :
symbol = Symbol(symbol.name + "_1")
if rv.pspace.is_Continuous:
return Integral(arg.replace(rv, symbol)*Probability(Eq(rv, symbol), condition), (symbol, rv.pspace.domain.set.inf, rv.pspace.domain.set.sup))
else:
if rv.pspace.is_Finite:
raise NotImplementedError
else:
return Sum(arg.replace(rv, symbol)*Probability(Eq(rv, symbol), condition), (symbol, rv.pspace.domain.set.inf, rv.pspace.set.sup))
def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs):
return self.func(arg, condition=condition).doit(deep=False, for_rewrite=True)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral # For discrete this will be Sum
def evaluate_integral(self):
return self.rewrite(Integral).doit()
evaluate_sum = evaluate_integral
class Variance(Expr):
"""
Symbolic expression for the variance.
Examples
========
>>> from sympy import symbols, Integral
>>> from sympy.stats import Normal, Expectation, Variance, Probability
>>> mu = symbols("mu", positive=True)
>>> sigma = symbols("sigma", positive=True)
>>> X = Normal("X", mu, sigma)
>>> Variance(X)
Variance(X)
>>> Variance(X).evaluate_integral()
sigma**2
Integral representation of the underlying calculations:
>>> Variance(X).rewrite(Integral)
Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**2*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))
Integral representation, without expanding the PDF:
>>> Variance(X).rewrite(Probability)
-Integral(x*Probability(Eq(X, x)), (x, -oo, oo))**2 + Integral(x**2*Probability(Eq(X, x)), (x, -oo, oo))
Rewrite the variance in terms of the expectation
>>> Variance(X).rewrite(Expectation)
-Expectation(X)**2 + Expectation(X**2)
Some transformations based on the properties of the variance may happen:
>>> from sympy.abc import a
>>> Y = Normal("Y", 0, 1)
>>> Variance(a*X)
Variance(a*X)
To expand the variance in its expression, use ``expand()``:
>>> Variance(a*X).expand()
a**2*Variance(X)
>>> Variance(X + Y)
Variance(X + Y)
>>> Variance(X + Y).expand()
2*Covariance(X, Y) + Variance(X) + Variance(Y)
"""
def __new__(cls, arg, condition=None, **kwargs):
arg = _sympify(arg)
if arg.is_Matrix:
from sympy.stats.symbolic_multivariate_probability import VarianceMatrix
return VarianceMatrix(arg, condition)
if condition is None:
obj = Expr.__new__(cls, arg)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, arg, condition)
obj._condition = condition
return obj
def expand(self, **hints):
arg = self.args[0]
condition = self._condition
if not is_random(arg):
return S.Zero
if isinstance(arg, RandomSymbol):
return self
elif isinstance(arg, Add):
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
variances = Add(*map(lambda xv: Variance(xv, condition).expand(), rv))
map_to_covar = lambda x: 2*Covariance(*x, condition=condition).expand()
covariances = Add(*map(map_to_covar, itertools.combinations(rv, 2)))
return variances + covariances
elif isinstance(arg, Mul):
nonrv = []
rv = []
for a in arg.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a**2)
if len(rv) == 0:
return S.Zero
return Mul.fromiter(nonrv)*Variance(Mul.fromiter(rv), condition)
# this expression contains a RandomSymbol somehow:
return self
def _eval_rewrite_as_Expectation(self, arg, condition=None, **kwargs):
e1 = Expectation(arg**2, condition)
e2 = Expectation(arg, condition)**2
return e1 - e2
def _eval_rewrite_as_Probability(self, arg, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Probability)
def _eval_rewrite_as_Integral(self, arg, condition=None, **kwargs):
return variance(self.args[0], self._condition, evaluate=False)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral
def evaluate_integral(self):
return self.rewrite(Integral).doit()
class Covariance(Expr):
"""
Symbolic expression for the covariance.
Examples
========
>>> from sympy.stats import Covariance
>>> from sympy.stats import Normal
>>> X = Normal("X", 3, 2)
>>> Y = Normal("Y", 0, 1)
>>> Z = Normal("Z", 0, 1)
>>> W = Normal("W", 0, 1)
>>> cexpr = Covariance(X, Y)
>>> cexpr
Covariance(X, Y)
Evaluate the covariance, `X` and `Y` are independent,
therefore zero is the result:
>>> cexpr.evaluate_integral()
0
Rewrite the covariance expression in terms of expectations:
>>> from sympy.stats import Expectation
>>> cexpr.rewrite(Expectation)
Expectation(X*Y) - Expectation(X)*Expectation(Y)
In order to expand the argument, use ``expand()``:
>>> from sympy.abc import a, b, c, d
>>> Covariance(a*X + b*Y, c*Z + d*W)
Covariance(a*X + b*Y, c*Z + d*W)
>>> Covariance(a*X + b*Y, c*Z + d*W).expand()
a*c*Covariance(X, Z) + a*d*Covariance(W, X) + b*c*Covariance(Y, Z) + b*d*Covariance(W, Y)
This class is aware of some properties of the covariance:
>>> Covariance(X, X).expand()
Variance(X)
>>> Covariance(a*X, b*Y).expand()
a*b*Covariance(X, Y)
"""
def __new__(cls, arg1, arg2, condition=None, **kwargs):
arg1 = _sympify(arg1)
arg2 = _sympify(arg2)
if arg1.is_Matrix or arg2.is_Matrix:
from sympy.stats.symbolic_multivariate_probability import CrossCovarianceMatrix
return CrossCovarianceMatrix(arg1, arg2, condition)
if kwargs.pop('evaluate', global_parameters.evaluate):
arg1, arg2 = sorted([arg1, arg2], key=default_sort_key)
if condition is None:
obj = Expr.__new__(cls, arg1, arg2)
else:
condition = _sympify(condition)
obj = Expr.__new__(cls, arg1, arg2, condition)
obj._condition = condition
return obj
def expand(self, **hints):
arg1 = self.args[0]
arg2 = self.args[1]
condition = self._condition
if arg1 == arg2:
return Variance(arg1, condition).expand()
if not is_random(arg1):
return S.Zero
if not is_random(arg2):
return S.Zero
arg1, arg2 = sorted([arg1, arg2], key=default_sort_key)
if isinstance(arg1, RandomSymbol) and isinstance(arg2, RandomSymbol):
return Covariance(arg1, arg2, condition)
coeff_rv_list1 = self._expand_single_argument(arg1.expand())
coeff_rv_list2 = self._expand_single_argument(arg2.expand())
addends = [a*b*Covariance(*sorted([r1, r2], key=default_sort_key), condition=condition)
for (a, r1) in coeff_rv_list1 for (b, r2) in coeff_rv_list2]
return Add.fromiter(addends)
@classmethod
def _expand_single_argument(cls, expr):
# return (coefficient, random_symbol) pairs:
if isinstance(expr, RandomSymbol):
return [(S.One, expr)]
elif isinstance(expr, Add):
outval = []
for a in expr.args:
if isinstance(a, Mul):
outval.append(cls._get_mul_nonrv_rv_tuple(a))
elif is_random(a):
outval.append((S.One, a))
return outval
elif isinstance(expr, Mul):
return [cls._get_mul_nonrv_rv_tuple(expr)]
elif is_random(expr):
return [(S.One, expr)]
@classmethod
def _get_mul_nonrv_rv_tuple(cls, m):
rv = []
nonrv = []
for a in m.args:
if is_random(a):
rv.append(a)
else:
nonrv.append(a)
return (Mul.fromiter(nonrv), Mul.fromiter(rv))
def _eval_rewrite_as_Expectation(self, arg1, arg2, condition=None, **kwargs):
e1 = Expectation(arg1*arg2, condition)
e2 = Expectation(arg1, condition)*Expectation(arg2, condition)
return e1 - e2
def _eval_rewrite_as_Probability(self, arg1, arg2, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Probability)
def _eval_rewrite_as_Integral(self, arg1, arg2, condition=None, **kwargs):
return covariance(self.args[0], self.args[1], self._condition, evaluate=False)
_eval_rewrite_as_Sum = _eval_rewrite_as_Integral
def evaluate_integral(self):
return self.rewrite(Integral).doit()
class Moment(Expr):
"""
Symbolic class for Moment
Examples
========
>>> from sympy import Symbol, Integral
>>> from sympy.stats import Normal, Expectation, Probability, Moment
>>> mu = Symbol('mu', real=True)
>>> sigma = Symbol('sigma', positive=True)
>>> X = Normal('X', mu, sigma)
>>> M = Moment(X, 3, 1)
To evaluate the result of Moment use `doit`:
>>> M.doit()
mu**3 - 3*mu**2 + 3*mu*sigma**2 + 3*mu - 3*sigma**2 - 1
Rewrite the Moment expression in terms of Expectation:
>>> M.rewrite(Expectation)
Expectation((X - 1)**3)
Rewrite the Moment expression in terms of Probability:
>>> M.rewrite(Probability)
Integral((x - 1)**3*Probability(Eq(X, x)), (x, -oo, oo))
Rewrite the Moment expression in terms of Integral:
>>> M.rewrite(Integral)
Integral(sqrt(2)*(X - 1)**3*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))
"""
def __new__(cls, X, n, c=0, condition=None, **kwargs):
X = _sympify(X)
n = _sympify(n)
c = _sympify(c)
if condition is not None:
condition = _sympify(condition)
return super().__new__(cls, X, n, c, condition)
else:
return super().__new__(cls, X, n, c)
def doit(self, **hints):
return self.rewrite(Expectation).doit(**hints)
def _eval_rewrite_as_Expectation(self, X, n, c=0, condition=None, **kwargs):
return Expectation((X - c)**n, condition)
def _eval_rewrite_as_Probability(self, X, n, c=0, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Probability)
def _eval_rewrite_as_Integral(self, X, n, c=0, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Integral)
class CentralMoment(Expr):
"""
Symbolic class Central Moment
Examples
========
>>> from sympy import Symbol, Integral
>>> from sympy.stats import Normal, Expectation, Probability, CentralMoment
>>> mu = Symbol('mu', real=True)
>>> sigma = Symbol('sigma', positive=True)
>>> X = Normal('X', mu, sigma)
>>> CM = CentralMoment(X, 4)
To evaluate the result of CentralMoment use `doit`:
>>> CM.doit().simplify()
3*sigma**4
Rewrite the CentralMoment expression in terms of Expectation:
>>> CM.rewrite(Expectation)
Expectation((X - Expectation(X))**4)
Rewrite the CentralMoment expression in terms of Probability:
>>> CM.rewrite(Probability)
Integral((x - Integral(x*Probability(True), (x, -oo, oo)))**4*Probability(Eq(X, x)), (x, -oo, oo))
Rewrite the CentralMoment expression in terms of Integral:
>>> CM.rewrite(Integral)
Integral(sqrt(2)*(X - Integral(sqrt(2)*X*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo)))**4*exp(-(X - mu)**2/(2*sigma**2))/(2*sqrt(pi)*sigma), (X, -oo, oo))
"""
def __new__(cls, X, n, condition=None, **kwargs):
X = _sympify(X)
n = _sympify(n)
if condition is not None:
condition = _sympify(condition)
return super().__new__(cls, X, n, condition)
else:
return super().__new__(cls, X, n)
def doit(self, **hints):
return self.rewrite(Expectation).doit(**hints)
def _eval_rewrite_as_Expectation(self, X, n, condition=None, **kwargs):
mu = Expectation(X, condition, **kwargs)
return Moment(X, n, mu, condition, **kwargs).rewrite(Expectation)
def _eval_rewrite_as_Probability(self, X, n, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Probability)
def _eval_rewrite_as_Integral(self, X, n, condition=None, **kwargs):
return self.rewrite(Expectation).rewrite(Integral)
|
50c0f77987ff3e886dd2ac63747dbf6b11b0c0e37a719a354a9ef4d028b53353 | #!/usr/bin/env python
from sympy.core.random import random
from sympy.core.numbers import (I, Integer, pi)
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import sin
from sympy.polys.polytools import factor
from sympy.simplify.simplify import simplify
from sympy.abc import x, y, z
from timeit import default_timer as clock
def bench_R1():
"real(f(f(f(f(f(f(f(f(f(f(i/2)))))))))))"
def f(z):
return sqrt(Integer(1)/3)*z**2 + I/3
f(f(f(f(f(f(f(f(f(f(I/2)))))))))).as_real_imag()[0]
def bench_R2():
"Hermite polynomial hermite(15, y)"
def hermite(n, y):
if n == 1:
return 2*y
if n == 0:
return 1
return (2*y*hermite(n - 1, y) - 2*(n - 1)*hermite(n - 2, y)).expand()
hermite(15, y)
def bench_R3():
"a = [bool(f==f) for _ in range(10)]"
f = x + y + z
[bool(f == f) for _ in range(10)]
def bench_R4():
# we don't have Tuples
pass
def bench_R5():
"blowup(L, 8); L=uniq(L)"
def blowup(L, n):
for i in range(n):
L.append( (L[i] + L[i + 1]) * L[i + 2] )
def uniq(x):
v = set(x)
return v
L = [x, y, z]
blowup(L, 8)
L = uniq(L)
def bench_R6():
"sum(simplify((x+sin(i))/x+(x-sin(i))/x) for i in range(100))"
sum(simplify((x + sin(i))/x + (x - sin(i))/x) for i in range(100))
def bench_R7():
"[f.subs(x, random()) for _ in range(10**4)]"
f = x**24 + 34*x**12 + 45*x**3 + 9*x**18 + 34*x**10 + 32*x**21
[f.subs(x, random()) for _ in range(10**4)]
def bench_R8():
"right(x^2,0,5,10^4)"
def right(f, a, b, n):
a = sympify(a)
b = sympify(b)
n = sympify(n)
x = f.atoms(Symbol).pop()
Deltax = (b - a)/n
c = a
est = 0
for i in range(n):
c += Deltax
est += f.subs(x, c)
return est*Deltax
right(x**2, 0, 5, 10**4)
def _bench_R9():
"factor(x^20 - pi^5*y^20)"
factor(x**20 - pi**5*y**20)
def bench_R10():
"v = [-pi,-pi+1/10..,pi]"
def srange(min, max, step):
v = [min]
while (max - v[-1]).evalf() > 0:
v.append(v[-1] + step)
return v[:-1]
srange(-pi, pi, sympify(1)/10)
def bench_R11():
"a = [random() + random()*I for w in [0..1000]]"
[random() + random()*I for w in range(1000)]
def bench_S1():
"e=(x+y+z+1)**7;f=e*(e+1);f.expand()"
e = (x + y + z + 1)**7
f = e*(e + 1)
f.expand()
if __name__ == '__main__':
benchmarks = [
bench_R1,
bench_R2,
bench_R3,
bench_R5,
bench_R6,
bench_R7,
bench_R8,
#_bench_R9,
bench_R10,
bench_R11,
#bench_S1,
]
report = []
for b in benchmarks:
t = clock()
b()
t = clock() - t
print("%s%65s: %f" % (b.__name__, b.__doc__, t))
|
75235d661a1cf1ae21e74c11b5b3b12f0a668148ae305a5eeb2b532d445ca656 | # conceal the implicit import from the code quality tester
from sympy.core.numbers import (oo, pi)
from sympy.core.symbol import (Symbol, symbols)
from sympy.functions.elementary.exponential import exp
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.special.bessel import besseli
from sympy.functions.special.gamma_functions import gamma
from sympy.integrals.integrals import integrate
from sympy.integrals.transforms import (mellin_transform,
inverse_fourier_transform, inverse_mellin_transform,
laplace_transform, inverse_laplace_transform, fourier_transform)
LT = laplace_transform
FT = fourier_transform
MT = mellin_transform
IFT = inverse_fourier_transform
ILT = inverse_laplace_transform
IMT = inverse_mellin_transform
from sympy.abc import x, y
nu, beta, rho = symbols('nu beta rho')
apos, bpos, cpos, dpos, posk, p = symbols('a b c d k p', positive=True)
k = Symbol('k', real=True)
negk = Symbol('k', negative=True)
mu1, mu2 = symbols('mu1 mu2', real=True, nonzero=True, finite=True)
sigma1, sigma2 = symbols('sigma1 sigma2', real=True, nonzero=True,
finite=True, positive=True)
rate = Symbol('lambda', positive=True)
def normal(x, mu, sigma):
return 1/sqrt(2*pi*sigma**2)*exp(-(x - mu)**2/2/sigma**2)
def exponential(x, rate):
return rate*exp(-rate*x)
alpha, beta = symbols('alpha beta', positive=True)
betadist = x**(alpha - 1)*(1 + x)**(-alpha - beta)*gamma(alpha + beta) \
/gamma(alpha)/gamma(beta)
kint = Symbol('k', integer=True, positive=True)
chi = 2**(1 - kint/2)*x**(kint - 1)*exp(-x**2/2)/gamma(kint/2)
chisquared = 2**(-k/2)/gamma(k/2)*x**(k/2 - 1)*exp(-x/2)
dagum = apos*p/x*(x/bpos)**(apos*p)/(1 + x**apos/bpos**apos)**(p + 1)
d1, d2 = symbols('d1 d2', positive=True)
f = sqrt(((d1*x)**d1 * d2**d2)/(d1*x + d2)**(d1 + d2))/x \
/gamma(d1/2)/gamma(d2/2)*gamma((d1 + d2)/2)
nupos, sigmapos = symbols('nu sigma', positive=True)
rice = x/sigmapos**2*exp(-(x**2 + nupos**2)/2/sigmapos**2)*besseli(0, x*
nupos/sigmapos**2)
mu = Symbol('mu', real=True)
laplace = exp(-abs(x - mu)/bpos)/2/bpos
u = Symbol('u', polar=True)
tpos = Symbol('t', positive=True)
def E(expr):
integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(x, 0, oo), (y, -oo, oo), meijerg=True)
integrate(expr*exponential(x, rate)*normal(y, mu1, sigma1),
(y, -oo, oo), (x, 0, oo), meijerg=True)
bench = [
'MT(x**nu*Heaviside(x - 1), x, s)',
'MT(x**nu*Heaviside(1 - x), x, s)',
'MT((1-x)**(beta - 1)*Heaviside(1-x), x, s)',
'MT((x-1)**(beta - 1)*Heaviside(x-1), x, s)',
'MT((1+x)**(-rho), x, s)',
'MT(abs(1-x)**(-rho), x, s)',
'MT((1-x)**(beta-1)*Heaviside(1-x) + a*(x-1)**(beta-1)*Heaviside(x-1), x, s)',
'MT((x**a-b**a)/(x-b), x, s)',
'MT((x**a-bpos**a)/(x-bpos), x, s)',
'MT(exp(-x), x, s)',
'MT(exp(-1/x), x, s)',
'MT(log(x)**4*Heaviside(1-x), x, s)',
'MT(log(x)**3*Heaviside(x-1), x, s)',
'MT(log(x + 1), x, s)',
'MT(log(1/x + 1), x, s)',
'MT(log(abs(1 - x)), x, s)',
'MT(log(abs(1 - 1/x)), x, s)',
'MT(log(x)/(x+1), x, s)',
'MT(log(x)**2/(x+1), x, s)',
'MT(log(x)/(x+1)**2, x, s)',
'MT(erf(sqrt(x)), x, s)',
'MT(besselj(a, 2*sqrt(x)), x, s)',
'MT(sin(sqrt(x))*besselj(a, sqrt(x)), x, s)',
'MT(cos(sqrt(x))*besselj(a, sqrt(x)), x, s)',
'MT(besselj(a, sqrt(x))**2, x, s)',
'MT(besselj(a, sqrt(x))*besselj(-a, sqrt(x)), x, s)',
'MT(besselj(a - 1, sqrt(x))*besselj(a, sqrt(x)), x, s)',
'MT(besselj(a, sqrt(x))*besselj(b, sqrt(x)), x, s)',
'MT(besselj(a, sqrt(x))**2 + besselj(-a, sqrt(x))**2, x, s)',
'MT(bessely(a, 2*sqrt(x)), x, s)',
'MT(sin(sqrt(x))*bessely(a, sqrt(x)), x, s)',
'MT(cos(sqrt(x))*bessely(a, sqrt(x)), x, s)',
'MT(besselj(a, sqrt(x))*bessely(a, sqrt(x)), x, s)',
'MT(besselj(a, sqrt(x))*bessely(b, sqrt(x)), x, s)',
'MT(bessely(a, sqrt(x))**2, x, s)',
'MT(besselk(a, 2*sqrt(x)), x, s)',
'MT(besselj(a, 2*sqrt(2*sqrt(x)))*besselk(a, 2*sqrt(2*sqrt(x))), x, s)',
'MT(besseli(a, sqrt(x))*besselk(a, sqrt(x)), x, s)',
'MT(besseli(b, sqrt(x))*besselk(a, sqrt(x)), x, s)',
'MT(exp(-x/2)*besselk(a, x/2), x, s)',
# later: ILT, IMT
'LT((t-apos)**bpos*exp(-cpos*(t-apos))*Heaviside(t-apos), t, s)',
'LT(t**apos, t, s)',
'LT(Heaviside(t), t, s)',
'LT(Heaviside(t - apos), t, s)',
'LT(1 - exp(-apos*t), t, s)',
'LT((exp(2*t)-1)*exp(-bpos - t)*Heaviside(t)/2, t, s, noconds=True)',
'LT(exp(t), t, s)',
'LT(exp(2*t), t, s)',
'LT(exp(apos*t), t, s)',
'LT(log(t/apos), t, s)',
'LT(erf(t), t, s)',
'LT(sin(apos*t), t, s)',
'LT(cos(apos*t), t, s)',
'LT(exp(-apos*t)*sin(bpos*t), t, s)',
'LT(exp(-apos*t)*cos(bpos*t), t, s)',
'LT(besselj(0, t), t, s, noconds=True)',
'LT(besselj(1, t), t, s, noconds=True)',
'FT(Heaviside(1 - abs(2*apos*x)), x, k)',
'FT(Heaviside(1-abs(apos*x))*(1-abs(apos*x)), x, k)',
'FT(exp(-apos*x)*Heaviside(x), x, k)',
'IFT(1/(apos + 2*pi*I*x), x, posk, noconds=False)',
'IFT(1/(apos + 2*pi*I*x), x, -posk, noconds=False)',
'IFT(1/(apos + 2*pi*I*x), x, negk)',
'FT(x*exp(-apos*x)*Heaviside(x), x, k)',
'FT(exp(-apos*x)*sin(bpos*x)*Heaviside(x), x, k)',
'FT(exp(-apos*x**2), x, k)',
'IFT(sqrt(pi/apos)*exp(-(pi*k)**2/apos), k, x)',
'FT(exp(-apos*abs(x)), x, k)',
'integrate(normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)',
'integrate(x*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)',
'integrate(x**2*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)',
'integrate(x**3*normal(x, mu1, sigma1), (x, -oo, oo), meijerg=True)',
'integrate(normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate(x*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate(y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate(x*y*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate((x+y+1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate((x+y-1)*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate(x**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate(y**2*normal(x, mu1, sigma1)*normal(y, mu2, sigma2),'
' (x, -oo, oo), (y, -oo, oo), meijerg=True)',
'integrate(exponential(x, rate), (x, 0, oo), meijerg=True)',
'integrate(x*exponential(x, rate), (x, 0, oo), meijerg=True)',
'integrate(x**2*exponential(x, rate), (x, 0, oo), meijerg=True)',
'E(1)',
'E(x*y)',
'E(x*y**2)',
'E((x+y+1)**2)',
'E(x+y+1)',
'E((x+y-1)**2)',
'integrate(betadist, (x, 0, oo), meijerg=True)',
'integrate(x*betadist, (x, 0, oo), meijerg=True)',
'integrate(x**2*betadist, (x, 0, oo), meijerg=True)',
'integrate(chi, (x, 0, oo), meijerg=True)',
'integrate(x*chi, (x, 0, oo), meijerg=True)',
'integrate(x**2*chi, (x, 0, oo), meijerg=True)',
'integrate(chisquared, (x, 0, oo), meijerg=True)',
'integrate(x*chisquared, (x, 0, oo), meijerg=True)',
'integrate(x**2*chisquared, (x, 0, oo), meijerg=True)',
'integrate(((x-k)/sqrt(2*k))**3*chisquared, (x, 0, oo), meijerg=True)',
'integrate(dagum, (x, 0, oo), meijerg=True)',
'integrate(x*dagum, (x, 0, oo), meijerg=True)',
'integrate(x**2*dagum, (x, 0, oo), meijerg=True)',
'integrate(f, (x, 0, oo), meijerg=True)',
'integrate(x*f, (x, 0, oo), meijerg=True)',
'integrate(x**2*f, (x, 0, oo), meijerg=True)',
'integrate(rice, (x, 0, oo), meijerg=True)',
'integrate(laplace, (x, -oo, oo), meijerg=True)',
'integrate(x*laplace, (x, -oo, oo), meijerg=True)',
'integrate(x**2*laplace, (x, -oo, oo), meijerg=True)',
'integrate(log(x) * x**(k-1) * exp(-x) / gamma(k), (x, 0, oo))',
'integrate(sin(z*x)*(x**2-1)**(-(y+S(1)/2)), (x, 1, oo), meijerg=True)',
'integrate(besselj(0,x)*besselj(1,x)*exp(-x**2), (x, 0, oo), meijerg=True)',
'integrate(besselj(0,x)*besselj(1,x)*besselk(0,x), (x, 0, oo), meijerg=True)',
'integrate(besselj(0,x)*besselj(1,x)*exp(-x**2), (x, 0, oo), meijerg=True)',
'integrate(besselj(a,x)*besselj(b,x)/x, (x,0,oo), meijerg=True)',
'hyperexpand(meijerg((-s - a/2 + 1, -s + a/2 + 1), (-a/2 - S(1)/2, -s + a/2 + S(3)/2), (a/2, -a/2), (-a/2 - S(1)/2, -s + a/2 + S(3)/2), 1))',
"gammasimp(S('2**(2*s)*(-pi*gamma(-a + 1)*gamma(a + 1)*gamma(-a - s + 1)*gamma(-a + s - 1/2)*gamma(a - s + 3/2)*gamma(a + s + 1)/(a*(a + s)) - gamma(-a - 1/2)*gamma(-a + 1)*gamma(a + 1)*gamma(a + 3/2)*gamma(-s + 3/2)*gamma(s - 1/2)*gamma(-a + s + 1)*gamma(a - s + 1)/(a*(-a + s)))*gamma(-2*s + 1)*gamma(s + 1)/(pi*s*gamma(-a - 1/2)*gamma(a + 3/2)*gamma(-s + 1)*gamma(-s + 3/2)*gamma(s - 1/2)*gamma(-a - s + 1)*gamma(-a + s - 1/2)*gamma(a - s + 1)*gamma(a - s + 3/2))'))",
'mellin_transform(E1(x), x, s)',
'inverse_mellin_transform(gamma(s)/s, s, x, (0, oo))',
'mellin_transform(expint(a, x), x, s)',
'mellin_transform(Si(x), x, s)',
'inverse_mellin_transform(-2**s*sqrt(pi)*gamma((s + 1)/2)/(2*s*gamma(-s/2 + 1)), s, x, (-1, 0))',
'mellin_transform(Ci(sqrt(x)), x, s)',
'inverse_mellin_transform(-4**s*sqrt(pi)*gamma(s)/(2*s*gamma(-s + S(1)/2)),s, u, (0, 1))',
'laplace_transform(Ci(x), x, s)',
'laplace_transform(expint(a, x), x, s)',
'laplace_transform(expint(1, x), x, s)',
'laplace_transform(expint(2, x), x, s)',
'inverse_laplace_transform(-log(1 + s**2)/2/s, s, u)',
'inverse_laplace_transform(log(s + 1)/s, s, x)',
'inverse_laplace_transform((s - log(s + 1))/s**2, s, x)',
'laplace_transform(Chi(x), x, s)',
'laplace_transform(Shi(x), x, s)',
'integrate(exp(-z*x)/x, (x, 1, oo), meijerg=True, conds="none")',
'integrate(exp(-z*x)/x**2, (x, 1, oo), meijerg=True, conds="none")',
'integrate(exp(-z*x)/x**3, (x, 1, oo), meijerg=True,conds="none")',
'integrate(-cos(x)/x, (x, tpos, oo), meijerg=True)',
'integrate(-sin(x)/x, (x, tpos, oo), meijerg=True)',
'integrate(sin(x)/x, (x, 0, z), meijerg=True)',
'integrate(sinh(x)/x, (x, 0, z), meijerg=True)',
'integrate(exp(-x)/x, x, meijerg=True)',
'integrate(exp(-x)/x**2, x, meijerg=True)',
'integrate(cos(u)/u, u, meijerg=True)',
'integrate(cosh(u)/u, u, meijerg=True)',
'integrate(expint(1, x), x, meijerg=True)',
'integrate(expint(2, x), x, meijerg=True)',
'integrate(Si(x), x, meijerg=True)',
'integrate(Ci(u), u, meijerg=True)',
'integrate(Shi(x), x, meijerg=True)',
'integrate(Chi(u), u, meijerg=True)',
'integrate(Si(x)*exp(-x), (x, 0, oo), meijerg=True)',
'integrate(expint(1, x)*sin(x), (x, 0, oo), meijerg=True)'
]
from time import time
from sympy.core.cache import clear_cache
import sys
timings = []
if __name__ == '__main__':
for n, string in enumerate(bench):
clear_cache()
_t = time()
exec(string)
_t = time() - _t
timings += [(_t, string)]
sys.stdout.write('.')
sys.stdout.flush()
if n % (len(bench) // 10) == 0:
sys.stdout.write('%s' % (10*n // len(bench)))
print()
timings.sort(key=lambda x: -x[0])
for ti, string in timings:
print('%.2fs %s' % (ti, string))
|
d7e0200a080a32c35aae8e9500114d2a0249206af83c3b3c05998ff4ebba8ef5 | from sympy.core.function import Function
from sympy.core.numbers import igcd, igcdex, mod_inverse
from sympy.core.power import isqrt
from sympy.core.singleton import S
from sympy.polys.domains import ZZ
from .primetest import isprime
from .factor_ import factorint, trailing, totient, multiplicity
from sympy.utilities.misc import as_int
from sympy.core.random import _randint, randint
from itertools import cycle, product
def n_order(a, n):
"""Returns the order of ``a`` modulo ``n``.
The order of ``a`` modulo ``n`` is the smallest integer
``k`` such that ``a**k`` leaves a remainder of 1 with ``n``.
Examples
========
>>> from sympy.ntheory import n_order
>>> n_order(3, 7)
6
>>> n_order(4, 7)
3
"""
from collections import defaultdict
a, n = as_int(a), as_int(n)
if igcd(a, n) != 1:
raise ValueError("The two numbers should be relatively prime")
factors = defaultdict(int)
f = factorint(n)
for px, kx in f.items():
if kx > 1:
factors[px] += kx - 1
fpx = factorint(px - 1)
for py, ky in fpx.items():
factors[py] += ky
group_order = 1
for px, kx in factors.items():
group_order *= px**kx
order = 1
if a > n:
a = a % n
for p, e in factors.items():
exponent = group_order
for f in range(e + 1):
if pow(a, exponent, n) != 1:
order *= p ** (e - f + 1)
break
exponent = exponent // p
return order
def _primitive_root_prime_iter(p):
"""
Generates the primitive roots for a prime ``p``
Examples
========
>>> from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter
>>> list(_primitive_root_prime_iter(19))
[2, 3, 10, 13, 14, 15]
References
==========
.. [1] W. Stein "Elementary Number Theory" (2011), page 44
"""
# it is assumed that p is an int
v = [(p - 1) // i for i in factorint(p - 1).keys()]
a = 2
while a < p:
for pw in v:
# a TypeError below may indicate that p was not an int
if pow(a, pw, p) == 1:
break
else:
yield a
a += 1
def primitive_root(p):
"""
Returns the smallest primitive root or None
Parameters
==========
p : positive integer
Examples
========
>>> from sympy.ntheory.residue_ntheory import primitive_root
>>> primitive_root(19)
2
References
==========
.. [1] W. Stein "Elementary Number Theory" (2011), page 44
.. [2] P. Hackman "Elementary Number Theory" (2009), Chapter C
"""
p = as_int(p)
if p < 1:
raise ValueError('p is required to be positive')
if p <= 2:
return 1
f = factorint(p)
if len(f) > 2:
return None
if len(f) == 2:
if 2 not in f or f[2] > 1:
return None
# case p = 2*p1**k, p1 prime
for p1, e1 in f.items():
if p1 != 2:
break
i = 1
while i < p:
i += 2
if i % p1 == 0:
continue
if is_primitive_root(i, p):
return i
else:
if 2 in f:
if p == 4:
return 3
return None
p1, n = list(f.items())[0]
if n > 1:
# see Ref [2], page 81
g = primitive_root(p1)
if is_primitive_root(g, p1**2):
return g
else:
for i in range(2, g + p1 + 1):
if igcd(i, p) == 1 and is_primitive_root(i, p):
return i
return next(_primitive_root_prime_iter(p))
def is_primitive_root(a, p):
"""
Returns True if ``a`` is a primitive root of ``p``
``a`` is said to be the primitive root of ``p`` if gcd(a, p) == 1 and
totient(p) is the smallest positive number s.t.
a**totient(p) cong 1 mod(p)
Examples
========
>>> from sympy.ntheory import is_primitive_root, n_order, totient
>>> is_primitive_root(3, 10)
True
>>> is_primitive_root(9, 10)
False
>>> n_order(3, 10) == totient(10)
True
>>> n_order(9, 10) == totient(10)
False
"""
a, p = as_int(a), as_int(p)
if igcd(a, p) != 1:
raise ValueError("The two numbers should be relatively prime")
if a > p:
a = a % p
return n_order(a, p) == totient(p)
def _sqrt_mod_tonelli_shanks(a, p):
"""
Returns the square root in the case of ``p`` prime with ``p == 1 (mod 8)``
References
==========
.. [1] R. Crandall and C. Pomerance "Prime Numbers", 2nt Ed., page 101
"""
s = trailing(p - 1)
t = p >> s
# find a non-quadratic residue
while 1:
d = randint(2, p - 1)
r = legendre_symbol(d, p)
if r == -1:
break
#assert legendre_symbol(d, p) == -1
A = pow(a, t, p)
D = pow(d, t, p)
m = 0
for i in range(s):
adm = A*pow(D, m, p) % p
adm = pow(adm, 2**(s - 1 - i), p)
if adm % p == p - 1:
m += 2**i
#assert A*pow(D, m, p) % p == 1
x = pow(a, (t + 1)//2, p)*pow(D, m//2, p) % p
return x
def sqrt_mod(a, p, all_roots=False):
"""
Find a root of ``x**2 = a mod p``
Parameters
==========
a : integer
p : positive integer
all_roots : if True the list of roots is returned or None
Notes
=====
If there is no root it is returned None; else the returned root
is less or equal to ``p // 2``; in general is not the smallest one.
It is returned ``p // 2`` only if it is the only root.
Use ``all_roots`` only when it is expected that all the roots fit
in memory; otherwise use ``sqrt_mod_iter``.
Examples
========
>>> from sympy.ntheory import sqrt_mod
>>> sqrt_mod(11, 43)
21
>>> sqrt_mod(17, 32, True)
[7, 9, 23, 25]
"""
if all_roots:
return sorted(list(sqrt_mod_iter(a, p)))
try:
p = abs(as_int(p))
it = sqrt_mod_iter(a, p)
r = next(it)
if r > p // 2:
return p - r
elif r < p // 2:
return r
else:
try:
r = next(it)
if r > p // 2:
return p - r
except StopIteration:
pass
return r
except StopIteration:
return None
def _product(*iters):
"""
Cartesian product generator
Notes
=====
Unlike itertools.product, it works also with iterables which do not fit
in memory. See http://bugs.python.org/issue10109
Author: Fernando Sumudu
with small changes
"""
inf_iters = tuple(cycle(enumerate(it)) for it in iters)
num_iters = len(inf_iters)
cur_val = [None]*num_iters
first_v = True
while True:
i, p = 0, num_iters
while p and not i:
p -= 1
i, cur_val[p] = next(inf_iters[p])
if not p and not i:
if first_v:
first_v = False
else:
break
yield cur_val
def sqrt_mod_iter(a, p, domain=int):
"""
Iterate over solutions to ``x**2 = a mod p``
Parameters
==========
a : integer
p : positive integer
domain : integer domain, ``int``, ``ZZ`` or ``Integer``
Examples
========
>>> from sympy.ntheory.residue_ntheory import sqrt_mod_iter
>>> list(sqrt_mod_iter(11, 43))
[21, 22]
"""
from sympy.polys.galoistools import gf_crt1, gf_crt2
a, p = as_int(a), abs(as_int(p))
if isprime(p):
a = a % p
if a == 0:
res = _sqrt_mod1(a, p, 1)
else:
res = _sqrt_mod_prime_power(a, p, 1)
if res:
if domain is ZZ:
yield from res
else:
for x in res:
yield domain(x)
else:
f = factorint(p)
v = []
pv = []
for px, ex in f.items():
if a % px == 0:
rx = _sqrt_mod1(a, px, ex)
if not rx:
return
else:
rx = _sqrt_mod_prime_power(a, px, ex)
if not rx:
return
v.append(rx)
pv.append(px**ex)
mm, e, s = gf_crt1(pv, ZZ)
if domain is ZZ:
for vx in _product(*v):
r = gf_crt2(vx, pv, mm, e, s, ZZ)
yield r
else:
for vx in _product(*v):
r = gf_crt2(vx, pv, mm, e, s, ZZ)
yield domain(r)
def _sqrt_mod_prime_power(a, p, k):
"""
Find the solutions to ``x**2 = a mod p**k`` when ``a % p != 0``
Parameters
==========
a : integer
p : prime number
k : positive integer
Examples
========
>>> from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
>>> _sqrt_mod_prime_power(11, 43, 1)
[21, 22]
References
==========
.. [1] P. Hackman "Elementary Number Theory" (2009), page 160
.. [2] http://www.numbertheory.org/php/squareroot.html
.. [3] [Gathen99]_
"""
pk = p**k
a = a % pk
if k == 1:
if p == 2:
return [ZZ(a)]
if not (a % p < 2 or pow(a, (p - 1) // 2, p) == 1):
return None
if p % 4 == 3:
res = pow(a, (p + 1) // 4, p)
elif p % 8 == 5:
sign = pow(a, (p - 1) // 4, p)
if sign == 1:
res = pow(a, (p + 3) // 8, p)
else:
b = pow(4*a, (p - 5) // 8, p)
x = (2*a*b) % p
if pow(x, 2, p) == a:
res = x
else:
res = _sqrt_mod_tonelli_shanks(a, p)
# ``_sqrt_mod_tonelli_shanks(a, p)`` is not deterministic;
# sort to get always the same result
return sorted([ZZ(res), ZZ(p - res)])
if k > 1:
# see Ref.[2]
if p == 2:
if a % 8 != 1:
return None
if k <= 3:
s = set()
for i in range(0, pk, 4):
s.add(1 + i)
s.add(-1 + i)
return list(s)
# according to Ref.[2] for k > 2 there are two solutions
# (mod 2**k-1), that is four solutions (mod 2**k), which can be
# obtained from the roots of x**2 = 0 (mod 8)
rv = [ZZ(1), ZZ(3), ZZ(5), ZZ(7)]
# hensel lift them to solutions of x**2 = 0 (mod 2**k)
# if r**2 - a = 0 mod 2**nx but not mod 2**(nx+1)
# then r + 2**(nx - 1) is a root mod 2**(nx+1)
n = 3
res = []
for r in rv:
nx = n
while nx < k:
r1 = (r**2 - a) >> nx
if r1 % 2:
r = r + (1 << (nx - 1))
#assert (r**2 - a)% (1 << (nx + 1)) == 0
nx += 1
if r not in res:
res.append(r)
x = r + (1 << (k - 1))
#assert (x**2 - a) % pk == 0
if x < (1 << nx) and x not in res:
if (x**2 - a) % pk == 0:
res.append(x)
return res
rv = _sqrt_mod_prime_power(a, p, 1)
if not rv:
return None
r = rv[0]
fr = r**2 - a
# hensel lifting with Newton iteration, see Ref.[3] chapter 9
# with f(x) = x**2 - a; one has f'(a) != 0 (mod p) for p != 2
n = 1
px = p
while 1:
n1 = n
n1 *= 2
if n1 > k:
break
n = n1
px = px**2
frinv = igcdex(2*r, px)[0]
r = (r - fr*frinv) % px
fr = r**2 - a
if n < k:
px = p**k
frinv = igcdex(2*r, px)[0]
r = (r - fr*frinv) % px
return [r, px - r]
def _sqrt_mod1(a, p, n):
"""
Find solution to ``x**2 == a mod p**n`` when ``a % p == 0``
see http://www.numbertheory.org/php/squareroot.html
"""
pn = p**n
a = a % pn
if a == 0:
# case gcd(a, p**k) = p**n
m = n // 2
if n % 2 == 1:
pm1 = p**(m + 1)
def _iter0a():
i = 0
while i < pn:
yield i
i += pm1
return _iter0a()
else:
pm = p**m
def _iter0b():
i = 0
while i < pn:
yield i
i += pm
return _iter0b()
# case gcd(a, p**k) = p**r, r < n
f = factorint(a)
r = f[p]
if r % 2 == 1:
return None
m = r // 2
a1 = a >> r
if p == 2:
if n - r == 1:
pnm1 = 1 << (n - m + 1)
pm1 = 1 << (m + 1)
def _iter1():
k = 1 << (m + 2)
i = 1 << m
while i < pnm1:
j = i
while j < pn:
yield j
j += k
i += pm1
return _iter1()
if n - r == 2:
res = _sqrt_mod_prime_power(a1, p, n - r)
if res is None:
return None
pnm = 1 << (n - m)
def _iter2():
s = set()
for r in res:
i = 0
while i < pn:
x = (r << m) + i
if x not in s:
s.add(x)
yield x
i += pnm
return _iter2()
if n - r > 2:
res = _sqrt_mod_prime_power(a1, p, n - r)
if res is None:
return None
pnm1 = 1 << (n - m - 1)
def _iter3():
s = set()
for r in res:
i = 0
while i < pn:
x = ((r << m) + i) % pn
if x not in s:
s.add(x)
yield x
i += pnm1
return _iter3()
else:
m = r // 2
a1 = a // p**r
res1 = _sqrt_mod_prime_power(a1, p, n - r)
if res1 is None:
return None
pm = p**m
pnr = p**(n-r)
pnm = p**(n-m)
def _iter4():
s = set()
pm = p**m
for rx in res1:
i = 0
while i < pnm:
x = ((rx + i) % pn)
if x not in s:
s.add(x)
yield x*pm
i += pnr
return _iter4()
def is_quad_residue(a, p):
"""
Returns True if ``a`` (mod ``p``) is in the set of squares mod ``p``,
i.e a % p in set([i**2 % p for i in range(p)]). If ``p`` is an odd
prime, an iterative method is used to make the determination:
>>> from sympy.ntheory import is_quad_residue
>>> sorted(set([i**2 % 7 for i in range(7)]))
[0, 1, 2, 4]
>>> [j for j in range(7) if is_quad_residue(j, 7)]
[0, 1, 2, 4]
See Also
========
legendre_symbol, jacobi_symbol
"""
a, p = as_int(a), as_int(p)
if p < 1:
raise ValueError('p must be > 0')
if a >= p or a < 0:
a = a % p
if a < 2 or p < 3:
return True
if not isprime(p):
if p % 2 and jacobi_symbol(a, p) == -1:
return False
r = sqrt_mod(a, p)
if r is None:
return False
else:
return True
return pow(a, (p - 1) // 2, p) == 1
def is_nthpow_residue(a, n, m):
"""
Returns True if ``x**n == a (mod m)`` has solutions.
References
==========
.. [1] P. Hackman "Elementary Number Theory" (2009), page 76
"""
a = a % m
a, n, m = as_int(a), as_int(n), as_int(m)
if m <= 0:
raise ValueError('m must be > 0')
if n < 0:
raise ValueError('n must be >= 0')
if n == 0:
if m == 1:
return False
return a == 1
if a == 0:
return True
if n == 1:
return True
if n == 2:
return is_quad_residue(a, m)
return _is_nthpow_residue_bign(a, n, m)
def _is_nthpow_residue_bign(a, n, m):
"""Returns True if ``x**n == a (mod m)`` has solutions for n > 2."""
# assert n > 2
# assert a > 0 and m > 0
if primitive_root(m) is None or igcd(a, m) != 1:
# assert m >= 8
for prime, power in factorint(m).items():
if not _is_nthpow_residue_bign_prime_power(a, n, prime, power):
return False
return True
f = totient(m)
k = f // igcd(f, n)
return pow(a, k, m) == 1
def _is_nthpow_residue_bign_prime_power(a, n, p, k):
"""Returns True/False if a solution for ``x**n == a (mod(p**k))``
does/doesn't exist."""
# assert a > 0
# assert n > 2
# assert p is prime
# assert k > 0
if a % p:
if p != 2:
return _is_nthpow_residue_bign(a, n, pow(p, k))
if n & 1:
return True
c = trailing(n)
return a % pow(2, min(c + 2, k)) == 1
else:
a %= pow(p, k)
if not a:
return True
mu = multiplicity(p, a)
if mu % n:
return False
pm = pow(p, mu)
return _is_nthpow_residue_bign_prime_power(a//pm, n, p, k - mu)
def _nthroot_mod2(s, q, p):
f = factorint(q)
v = []
for b, e in f.items():
v.extend([b]*e)
for qx in v:
s = _nthroot_mod1(s, qx, p, False)
return s
def _nthroot_mod1(s, q, p, all_roots):
"""
Root of ``x**q = s mod p``, ``p`` prime and ``q`` divides ``p - 1``
References
==========
.. [1] A. M. Johnston "A Generalized qth Root Algorithm"
"""
g = primitive_root(p)
if not isprime(q):
r = _nthroot_mod2(s, q, p)
else:
f = p - 1
assert (p - 1) % q == 0
# determine k
k = 0
while f % q == 0:
k += 1
f = f // q
# find z, x, r1
f1 = igcdex(-f, q)[0] % q
z = f*f1
x = (1 + z) // q
r1 = pow(s, x, p)
s1 = pow(s, f, p)
h = pow(g, f*q, p)
t = discrete_log(p, s1, h)
g2 = pow(g, z*t, p)
g3 = igcdex(g2, p)[0]
r = r1*g3 % p
#assert pow(r, q, p) == s
res = [r]
h = pow(g, (p - 1) // q, p)
#assert pow(h, q, p) == 1
hx = r
for i in range(q - 1):
hx = (hx*h) % p
res.append(hx)
if all_roots:
res.sort()
return res
return min(res)
def _help(m, prime_modulo_method, diff_method, expr_val):
"""
Helper function for _nthroot_mod_composite and polynomial_congruence.
Parameters
==========
m : positive integer
prime_modulo_method : function to calculate the root of the congruence
equation for the prime divisors of m
diff_method : function to calculate derivative of expression at any
given point
expr_val : function to calculate value of the expression at any
given point
"""
from sympy.ntheory.modular import crt
f = factorint(m)
dd = {}
for p, e in f.items():
tot_roots = set()
if e == 1:
tot_roots.update(prime_modulo_method(p))
else:
for root in prime_modulo_method(p):
diff = diff_method(root, p)
if diff != 0:
ppow = p
m_inv = mod_inverse(diff, p)
for j in range(1, e):
ppow *= p
root = (root - expr_val(root, ppow) * m_inv) % ppow
tot_roots.add(root)
else:
new_base = p
roots_in_base = {root}
while new_base < pow(p, e):
new_base *= p
new_roots = set()
for k in roots_in_base:
if expr_val(k, new_base)!= 0:
continue
while k not in new_roots:
new_roots.add(k)
k = (k + (new_base // p)) % new_base
roots_in_base = new_roots
tot_roots = tot_roots | roots_in_base
if tot_roots == set():
return []
dd[pow(p, e)] = tot_roots
a = []
m = []
for x, y in dd.items():
m.append(x)
a.append(list(y))
return sorted({crt(m, list(i))[0] for i in product(*a)})
def _nthroot_mod_composite(a, n, m):
"""
Find the solutions to ``x**n = a mod m`` when m is not prime.
"""
return _help(m,
lambda p: nthroot_mod(a, n, p, True),
lambda root, p: (pow(root, n - 1, p) * (n % p)) % p,
lambda root, p: (pow(root, n, p) - a) % p)
def nthroot_mod(a, n, p, all_roots=False):
"""
Find the solutions to ``x**n = a mod p``
Parameters
==========
a : integer
n : positive integer
p : positive integer
all_roots : if False returns the smallest root, else the list of roots
Examples
========
>>> from sympy.ntheory.residue_ntheory import nthroot_mod
>>> nthroot_mod(11, 4, 19)
8
>>> nthroot_mod(11, 4, 19, True)
[8, 11]
>>> nthroot_mod(68, 3, 109)
23
"""
a = a % p
a, n, p = as_int(a), as_int(n), as_int(p)
if n == 2:
return sqrt_mod(a, p, all_roots)
# see Hackman "Elementary Number Theory" (2009), page 76
if not isprime(p):
return _nthroot_mod_composite(a, n, p)
if a % p == 0:
return [0]
if not is_nthpow_residue(a, n, p):
return [] if all_roots else None
if (p - 1) % n == 0:
return _nthroot_mod1(a, n, p, all_roots)
# The roots of ``x**n - a = 0 (mod p)`` are roots of
# ``gcd(x**n - a, x**(p - 1) - 1) = 0 (mod p)``
pa = n
pb = p - 1
b = 1
if pa < pb:
a, pa, b, pb = b, pb, a, pa
while pb:
# x**pa - a = 0; x**pb - b = 0
# x**pa - a = x**(q*pb + r) - a = (x**pb)**q * x**r - a =
# b**q * x**r - a; x**r - c = 0; c = b**-q * a mod p
q, r = divmod(pa, pb)
c = pow(b, q, p)
c = igcdex(c, p)[0]
c = (c * a) % p
pa, pb = pb, r
a, b = b, c
if pa == 1:
if all_roots:
res = [a]
else:
res = a
elif pa == 2:
return sqrt_mod(a, p, all_roots)
else:
res = _nthroot_mod1(a, pa, p, all_roots)
return res
def quadratic_residues(p):
"""
Returns the list of quadratic residues.
Examples
========
>>> from sympy.ntheory.residue_ntheory import quadratic_residues
>>> quadratic_residues(7)
[0, 1, 2, 4]
"""
p = as_int(p)
r = set()
for i in range(p // 2 + 1):
r.add(pow(i, 2, p))
return sorted(list(r))
def legendre_symbol(a, p):
r"""
Returns the Legendre symbol `(a / p)`.
For an integer ``a`` and an odd prime ``p``, the Legendre symbol is
defined as
.. math ::
\genfrac(){}{}{a}{p} = \begin{cases}
0 & \text{if } p \text{ divides } a\\
1 & \text{if } a \text{ is a quadratic residue modulo } p\\
-1 & \text{if } a \text{ is a quadratic nonresidue modulo } p
\end{cases}
Parameters
==========
a : integer
p : odd prime
Examples
========
>>> from sympy.ntheory import legendre_symbol
>>> [legendre_symbol(i, 7) for i in range(7)]
[0, 1, 1, -1, 1, -1, -1]
>>> sorted(set([i**2 % 7 for i in range(7)]))
[0, 1, 2, 4]
See Also
========
is_quad_residue, jacobi_symbol
"""
a, p = as_int(a), as_int(p)
if not isprime(p) or p == 2:
raise ValueError("p should be an odd prime")
a = a % p
if not a:
return 0
if pow(a, (p - 1) // 2, p) == 1:
return 1
return -1
def jacobi_symbol(m, n):
r"""
Returns the Jacobi symbol `(m / n)`.
For any integer ``m`` and any positive odd integer ``n`` the Jacobi symbol
is defined as the product of the Legendre symbols corresponding to the
prime factors of ``n``:
.. math ::
\genfrac(){}{}{m}{n} =
\genfrac(){}{}{m}{p^{1}}^{\alpha_1}
\genfrac(){}{}{m}{p^{2}}^{\alpha_2}
...
\genfrac(){}{}{m}{p^{k}}^{\alpha_k}
\text{ where } n =
p_1^{\alpha_1}
p_2^{\alpha_2}
...
p_k^{\alpha_k}
Like the Legendre symbol, if the Jacobi symbol `\genfrac(){}{}{m}{n} = -1`
then ``m`` is a quadratic nonresidue modulo ``n``.
But, unlike the Legendre symbol, if the Jacobi symbol
`\genfrac(){}{}{m}{n} = 1` then ``m`` may or may not be a quadratic residue
modulo ``n``.
Parameters
==========
m : integer
n : odd positive integer
Examples
========
>>> from sympy.ntheory import jacobi_symbol, legendre_symbol
>>> from sympy import S
>>> jacobi_symbol(45, 77)
-1
>>> jacobi_symbol(60, 121)
1
The relationship between the ``jacobi_symbol`` and ``legendre_symbol`` can
be demonstrated as follows:
>>> L = legendre_symbol
>>> S(45).factors()
{3: 2, 5: 1}
>>> jacobi_symbol(7, 45) == L(7, 3)**2 * L(7, 5)**1
True
See Also
========
is_quad_residue, legendre_symbol
"""
m, n = as_int(m), as_int(n)
if n < 0 or not n % 2:
raise ValueError("n should be an odd positive integer")
if m < 0 or m > n:
m %= n
if not m:
return int(n == 1)
if n == 1 or m == 1:
return 1
if igcd(m, n) != 1:
return 0
j = 1
if m < 0:
m = -m
if n % 4 == 3:
j = -j
while m != 0:
while m % 2 == 0 and m > 0:
m >>= 1
if n % 8 in [3, 5]:
j = -j
m, n = n, m
if m % 4 == n % 4 == 3:
j = -j
m %= n
if n != 1:
j = 0
return j
class mobius(Function):
"""
Mobius function maps natural number to {-1, 0, 1}
It is defined as follows:
1) `1` if `n = 1`.
2) `0` if `n` has a squared prime factor.
3) `(-1)^k` if `n` is a square-free positive integer with `k`
number of prime factors.
It is an important multiplicative function in number theory
and combinatorics. It has applications in mathematical series,
algebraic number theory and also physics (Fermion operator has very
concrete realization with Mobius Function model).
Parameters
==========
n : positive integer
Examples
========
>>> from sympy.ntheory import mobius
>>> mobius(13*7)
1
>>> mobius(1)
1
>>> mobius(13*7*5)
-1
>>> mobius(13**2)
0
References
==========
.. [1] https://en.wikipedia.org/wiki/M%C3%B6bius_function
.. [2] Thomas Koshy "Elementary Number Theory with Applications"
"""
@classmethod
def eval(cls, n):
if n.is_integer:
if n.is_positive is not True:
raise ValueError("n should be a positive integer")
else:
raise TypeError("n should be an integer")
if n.is_prime:
return S.NegativeOne
elif n is S.One:
return S.One
elif n.is_Integer:
a = factorint(n)
if any(i > 1 for i in a.values()):
return S.Zero
return S.NegativeOne**len(a)
def _discrete_log_trial_mul(n, a, b, order=None):
"""
Trial multiplication algorithm for computing the discrete logarithm of
``a`` to the base ``b`` modulo ``n``.
The algorithm finds the discrete logarithm using exhaustive search. This
naive method is used as fallback algorithm of ``discrete_log`` when the
group order is very small.
Examples
========
>>> from sympy.ntheory.residue_ntheory import _discrete_log_trial_mul
>>> _discrete_log_trial_mul(41, 15, 7)
3
See Also
========
discrete_log
References
==========
.. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
Vanstone, S. A. (1997).
"""
a %= n
b %= n
if order is None:
order = n
x = 1
for i in range(order):
if x == a:
return i
x = x * b % n
raise ValueError("Log does not exist")
def _discrete_log_shanks_steps(n, a, b, order=None):
"""
Baby-step giant-step algorithm for computing the discrete logarithm of
``a`` to the base ``b`` modulo ``n``.
The algorithm is a time-memory trade-off of the method of exhaustive
search. It uses `O(sqrt(m))` memory, where `m` is the group order.
Examples
========
>>> from sympy.ntheory.residue_ntheory import _discrete_log_shanks_steps
>>> _discrete_log_shanks_steps(41, 15, 7)
3
See Also
========
discrete_log
References
==========
.. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
Vanstone, S. A. (1997).
"""
a %= n
b %= n
if order is None:
order = n_order(b, n)
m = isqrt(order) + 1
T = dict()
x = 1
for i in range(m):
T[x] = i
x = x * b % n
z = mod_inverse(b, n)
z = pow(z, m, n)
x = a
for i in range(m):
if x in T:
return i * m + T[x]
x = x * z % n
raise ValueError("Log does not exist")
def _discrete_log_pollard_rho(n, a, b, order=None, retries=10, rseed=None):
"""
Pollard's Rho algorithm for computing the discrete logarithm of ``a`` to
the base ``b`` modulo ``n``.
It is a randomized algorithm with the same expected running time as
``_discrete_log_shanks_steps``, but requires a negligible amount of memory.
Examples
========
>>> from sympy.ntheory.residue_ntheory import _discrete_log_pollard_rho
>>> _discrete_log_pollard_rho(227, 3**7, 3)
7
See Also
========
discrete_log
References
==========
.. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
Vanstone, S. A. (1997).
"""
a %= n
b %= n
if order is None:
order = n_order(b, n)
randint = _randint(rseed)
for i in range(retries):
aa = randint(1, order - 1)
ba = randint(1, order - 1)
xa = pow(b, aa, n) * pow(a, ba, n) % n
c = xa % 3
if c == 0:
xb = a * xa % n
ab = aa
bb = (ba + 1) % order
elif c == 1:
xb = xa * xa % n
ab = (aa + aa) % order
bb = (ba + ba) % order
else:
xb = b * xa % n
ab = (aa + 1) % order
bb = ba
for j in range(order):
c = xa % 3
if c == 0:
xa = a * xa % n
ba = (ba + 1) % order
elif c == 1:
xa = xa * xa % n
aa = (aa + aa) % order
ba = (ba + ba) % order
else:
xa = b * xa % n
aa = (aa + 1) % order
c = xb % 3
if c == 0:
xb = a * xb % n
bb = (bb + 1) % order
elif c == 1:
xb = xb * xb % n
ab = (ab + ab) % order
bb = (bb + bb) % order
else:
xb = b * xb % n
ab = (ab + 1) % order
c = xb % 3
if c == 0:
xb = a * xb % n
bb = (bb + 1) % order
elif c == 1:
xb = xb * xb % n
ab = (ab + ab) % order
bb = (bb + bb) % order
else:
xb = b * xb % n
ab = (ab + 1) % order
if xa == xb:
r = (ba - bb) % order
try:
e = mod_inverse(r, order) * (ab - aa) % order
if (pow(b, e, n) - a) % n == 0:
return e
except ValueError:
pass
break
raise ValueError("Pollard's Rho failed to find logarithm")
def _discrete_log_pohlig_hellman(n, a, b, order=None):
"""
Pohlig-Hellman algorithm for computing the discrete logarithm of ``a`` to
the base ``b`` modulo ``n``.
In order to compute the discrete logarithm, the algorithm takes advantage
of the factorization of the group order. It is more efficient when the
group order factors into many small primes.
Examples
========
>>> from sympy.ntheory.residue_ntheory import _discrete_log_pohlig_hellman
>>> _discrete_log_pohlig_hellman(251, 210, 71)
197
See Also
========
discrete_log
References
==========
.. [1] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
Vanstone, S. A. (1997).
"""
from .modular import crt
a %= n
b %= n
if order is None:
order = n_order(b, n)
f = factorint(order)
l = [0] * len(f)
for i, (pi, ri) in enumerate(f.items()):
for j in range(ri):
gj = pow(b, l[i], n)
aj = pow(a * mod_inverse(gj, n), order // pi**(j + 1), n)
bj = pow(b, order // pi, n)
cj = discrete_log(n, aj, bj, pi, True)
l[i] += cj * pi**j
d, _ = crt([pi**ri for pi, ri in f.items()], l)
return d
def discrete_log(n, a, b, order=None, prime_order=None):
"""
Compute the discrete logarithm of ``a`` to the base ``b`` modulo ``n``.
This is a recursive function to reduce the discrete logarithm problem in
cyclic groups of composite order to the problem in cyclic groups of prime
order.
It employs different algorithms depending on the problem (subgroup order
size, prime order or not):
* Trial multiplication
* Baby-step giant-step
* Pollard's Rho
* Pohlig-Hellman
Examples
========
>>> from sympy.ntheory import discrete_log
>>> discrete_log(41, 15, 7)
3
References
==========
.. [1] http://mathworld.wolfram.com/DiscreteLogarithm.html
.. [2] "Handbook of applied cryptography", Menezes, A. J., Van, O. P. C., &
Vanstone, S. A. (1997).
"""
n, a, b = as_int(n), as_int(a), as_int(b)
if order is None:
order = n_order(b, n)
if prime_order is None:
prime_order = isprime(order)
if order < 1000:
return _discrete_log_trial_mul(n, a, b, order)
elif prime_order:
if order < 1000000000000:
return _discrete_log_shanks_steps(n, a, b, order)
return _discrete_log_pollard_rho(n, a, b, order)
return _discrete_log_pohlig_hellman(n, a, b, order)
def quadratic_congruence(a, b, c, p):
"""
Find the solutions to ``a x**2 + b x + c = 0 mod p
a : integer
b : integer
c : integer
p : positive integer
"""
from sympy.polys.galoistools import linear_congruence
a = as_int(a)
b = as_int(b)
c = as_int(c)
p = as_int(p)
a = a % p
b = b % p
c = c % p
if a == 0:
return linear_congruence(b, -c, p)
if p == 2:
roots = []
if c % 2 == 0:
roots.append(0)
if (a + b + c) % 2 == 0:
roots.append(1)
return roots
if isprime(p):
inv_a = mod_inverse(a, p)
b *= inv_a
c *= inv_a
if b % 2 == 1:
b = b + p
d = ((b * b) // 4 - c) % p
y = sqrt_mod(d, p, all_roots=True)
res = set()
for i in y:
res.add((i - b // 2) % p)
return sorted(res)
y = sqrt_mod(b * b - 4 * a * c, 4 * a * p, all_roots=True)
res = set()
for i in y:
root = linear_congruence(2 * a, i - b, 4 * a * p)
for j in root:
res.add(j % p)
return sorted(res)
def _polynomial_congruence_prime(coefficients, p):
"""A helper function used by polynomial_congruence.
It returns the root of a polynomial modulo prime number
by naive search from [0, p).
Parameters
==========
coefficients : list of integers
p : prime number
"""
roots = []
rank = len(coefficients)
for i in range(0, p):
f_val = 0
for coeff in range(0,rank - 1):
f_val = (f_val + pow(i, int(rank - coeff - 1), p) * coefficients[coeff]) % p
f_val = f_val + coefficients[-1]
if f_val % p == 0:
roots.append(i)
return roots
def _diff_poly(root, coefficients, p):
"""A helper function used by polynomial_congruence.
It returns the derivative of the polynomial evaluated at the
root (mod p).
Parameters
==========
coefficients : list of integers
p : prime number
root : integer
"""
diff = 0
rank = len(coefficients)
for coeff in range(0, rank - 1):
if not coefficients[coeff]:
continue
diff = (diff + pow(root, rank - coeff - 2, p)*(rank - coeff - 1)*
coefficients[coeff]) % p
return diff % p
def _val_poly(root, coefficients, p):
"""A helper function used by polynomial_congruence.
It returns value of the polynomial at root (mod p).
Parameters
==========
coefficients : list of integers
p : prime number
root : integer
"""
rank = len(coefficients)
f_val = 0
for coeff in range(0, rank - 1):
f_val = (f_val + pow(root, rank - coeff - 1, p)*
coefficients[coeff]) % p
f_val = f_val + coefficients[-1]
return f_val % p
def _valid_expr(expr):
"""
return coefficients of expr if it is a univariate polynomial
with integer coefficients else raise a ValueError.
"""
if not expr.is_polynomial():
raise ValueError("The expression should be a polynomial")
from sympy.polys import Poly
polynomial = Poly(expr)
if not polynomial.is_univariate:
raise ValueError("The expression should be univariate")
if not polynomial.domain == ZZ:
raise ValueError("The expression should should have integer coefficients")
return polynomial.all_coeffs()
def polynomial_congruence(expr, m):
"""
Find the solutions to a polynomial congruence equation modulo m.
Parameters
==========
coefficients : Coefficients of the Polynomial
m : positive integer
Examples
========
>>> from sympy.ntheory import polynomial_congruence
>>> from sympy.abc import x
>>> expr = x**6 - 2*x**5 -35
>>> polynomial_congruence(expr, 6125)
[3257]
"""
coefficients = _valid_expr(expr)
coefficients = [num % m for num in coefficients]
rank = len(coefficients)
if rank == 3:
return quadratic_congruence(*coefficients, m)
if rank == 2:
return quadratic_congruence(0, *coefficients, m)
if coefficients[0] == 1 and 1 + coefficients[-1] == sum(coefficients):
return nthroot_mod(-coefficients[-1], rank - 1, m, True)
if isprime(m):
return _polynomial_congruence_prime(coefficients, m)
return _help(m,
lambda p: _polynomial_congruence_prime(coefficients, p),
lambda root, p: _diff_poly(root, coefficients, p),
lambda root, p: _val_poly(root, coefficients, p))
|
871208216d9e35e5db21dbb18d62c050f6134617c71e99c90e602802080ffbfa | """
Integer factorization
"""
from collections import defaultdict
import random
import math
from sympy.core import sympify
from sympy.core.containers import Dict
from sympy.core.evalf import bitcount
from sympy.core.expr import Expr
from sympy.core.function import Function
from sympy.core.logic import fuzzy_and
from sympy.core.mul import Mul, prod
from sympy.core.numbers import igcd, ilcm, Rational, Integer
from sympy.core.power import integer_nthroot, Pow, integer_log
from sympy.core.singleton import S
from sympy.external.gmpy import SYMPY_INTS
from .primetest import isprime
from .generate import sieve, primerange, nextprime
from .digits import digits
from sympy.utilities.iterables import flatten
from sympy.utilities.misc import as_int, filldedent
from .ecm import _ecm_one_factor
# Note: This list should be updated whenever new Mersenne primes are found.
# Refer: https://www.mersenne.org/
MERSENNE_PRIME_EXPONENTS = (2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203,
2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049,
216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583,
25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933)
# compute more when needed for i in Mersenne prime exponents
PERFECT = [6] # 2**(i-1)*(2**i-1)
MERSENNES = [3] # 2**i - 1
def _ismersenneprime(n):
global MERSENNES
j = len(MERSENNES)
while n > MERSENNES[-1] and j < len(MERSENNE_PRIME_EXPONENTS):
# conservatively grow the list
MERSENNES.append(2**MERSENNE_PRIME_EXPONENTS[j] - 1)
j += 1
return n in MERSENNES
def _isperfect(n):
global PERFECT
if n % 2 == 0:
j = len(PERFECT)
while n > PERFECT[-1] and j < len(MERSENNE_PRIME_EXPONENTS):
# conservatively grow the list
t = 2**(MERSENNE_PRIME_EXPONENTS[j] - 1)
PERFECT.append(t*(2*t - 1))
j += 1
return n in PERFECT
small_trailing = [0] * 256
for j in range(1,8):
small_trailing[1<<j::1<<(j+1)] = [j] * (1<<(7-j))
def smoothness(n):
"""
Return the B-smooth and B-power smooth values of n.
The smoothness of n is the largest prime factor of n; the power-
smoothness is the largest divisor raised to its multiplicity.
Examples
========
>>> from sympy.ntheory.factor_ import smoothness
>>> smoothness(2**7*3**2)
(3, 128)
>>> smoothness(2**4*13)
(13, 16)
>>> smoothness(2)
(2, 2)
See Also
========
factorint, smoothness_p
"""
if n == 1:
return (1, 1) # not prime, but otherwise this causes headaches
facs = factorint(n)
return max(facs), max(m**facs[m] for m in facs)
def smoothness_p(n, m=-1, power=0, visual=None):
"""
Return a list of [m, (p, (M, sm(p + m), psm(p + m)))...]
where:
1. p**M is the base-p divisor of n
2. sm(p + m) is the smoothness of p + m (m = -1 by default)
3. psm(p + m) is the power smoothness of p + m
The list is sorted according to smoothness (default) or by power smoothness
if power=1.
The smoothness of the numbers to the left (m = -1) or right (m = 1) of a
factor govern the results that are obtained from the p +/- 1 type factoring
methods.
>>> from sympy.ntheory.factor_ import smoothness_p, factorint
>>> smoothness_p(10431, m=1)
(1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))])
>>> smoothness_p(10431)
(-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))])
>>> smoothness_p(10431, power=1)
(-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))])
If visual=True then an annotated string will be returned:
>>> print(smoothness_p(21477639576571, visual=1))
p**i=4410317**1 has p-1 B=1787, B-pow=1787
p**i=4869863**1 has p-1 B=2434931, B-pow=2434931
This string can also be generated directly from a factorization dictionary
and vice versa:
>>> factorint(17*9)
{3: 2, 17: 1}
>>> smoothness_p(_)
'p**i=3**2 has p-1 B=2, B-pow=2\\np**i=17**1 has p-1 B=2, B-pow=16'
>>> smoothness_p(_)
{3: 2, 17: 1}
The table of the output logic is:
====== ====== ======= =======
| Visual
------ ----------------------
Input True False other
====== ====== ======= =======
dict str tuple str
str str tuple dict
tuple str tuple str
n str tuple tuple
mul str tuple tuple
====== ====== ======= =======
See Also
========
factorint, smoothness
"""
# visual must be True, False or other (stored as None)
if visual in (1, 0):
visual = bool(visual)
elif visual not in (True, False):
visual = None
if isinstance(n, str):
if visual:
return n
d = {}
for li in n.splitlines():
k, v = [int(i) for i in
li.split('has')[0].split('=')[1].split('**')]
d[k] = v
if visual is not True and visual is not False:
return d
return smoothness_p(d, visual=False)
elif not isinstance(n, tuple):
facs = factorint(n, visual=False)
if power:
k = -1
else:
k = 1
if isinstance(n, tuple):
rv = n
else:
rv = (m, sorted([(f,
tuple([M] + list(smoothness(f + m))))
for f, M in [i for i in facs.items()]],
key=lambda x: (x[1][k], x[0])))
if visual is False or (visual is not True) and (type(n) in [int, Mul]):
return rv
lines = []
for dat in rv[1]:
dat = flatten(dat)
dat.insert(2, m)
lines.append('p**i=%i**%i has p%+i B=%i, B-pow=%i' % tuple(dat))
return '\n'.join(lines)
def trailing(n):
"""Count the number of trailing zero digits in the binary
representation of n, i.e. determine the largest power of 2
that divides n.
Examples
========
>>> from sympy import trailing
>>> trailing(128)
7
>>> trailing(63)
0
"""
n = abs(int(n))
if not n:
return 0
low_byte = n & 0xff
if low_byte:
return small_trailing[low_byte]
# 2**m is quick for z up through 2**30
z = bitcount(n) - 1
if isinstance(z, SYMPY_INTS):
if n == 1 << z:
return z
if z < 300:
# fixed 8-byte reduction
t = 8
n >>= 8
while not n & 0xff:
n >>= 8
t += 8
return t + small_trailing[n & 0xff]
# binary reduction important when there might be a large
# number of trailing 0s
t = 0
p = 8
while not n & 1:
while not n & ((1 << p) - 1):
n >>= p
t += p
p *= 2
p //= 2
return t
def multiplicity(p, n):
"""
Find the greatest integer m such that p**m divides n.
Examples
========
>>> from sympy import multiplicity, Rational
>>> [multiplicity(5, n) for n in [8, 5, 25, 125, 250]]
[0, 1, 2, 3, 3]
>>> multiplicity(3, Rational(1, 9))
-2
Note: when checking for the multiplicity of a number in a
large factorial it is most efficient to send it as an unevaluated
factorial or to call ``multiplicity_in_factorial`` directly:
>>> from sympy.ntheory import multiplicity_in_factorial
>>> from sympy import factorial
>>> p = factorial(25)
>>> n = 2**100
>>> nfac = factorial(n, evaluate=False)
>>> multiplicity(p, nfac)
52818775009509558395695966887
>>> _ == multiplicity_in_factorial(p, n)
True
"""
from sympy.functions.combinatorial.factorials import factorial
try:
p, n = as_int(p), as_int(n)
except ValueError:
if all(isinstance(i, (SYMPY_INTS, Rational)) for i in (p, n)):
p = Rational(p)
n = Rational(n)
if p.q == 1:
if n.p == 1:
return -multiplicity(p.p, n.q)
return multiplicity(p.p, n.p) - multiplicity(p.p, n.q)
elif p.p == 1:
return multiplicity(p.q, n.q)
else:
like = min(
multiplicity(p.p, n.p),
multiplicity(p.q, n.q))
cross = min(
multiplicity(p.q, n.p),
multiplicity(p.p, n.q))
return like - cross
elif (isinstance(p, (SYMPY_INTS, Integer)) and
isinstance(n, factorial) and
isinstance(n.args[0], Integer) and
n.args[0] >= 0):
return multiplicity_in_factorial(p, n.args[0])
raise ValueError('expecting ints or fractions, got %s and %s' % (p, n))
if n == 0:
raise ValueError('no such integer exists: multiplicity of %s is not-defined' %(n))
if p == 2:
return trailing(n)
if p < 2:
raise ValueError('p must be an integer, 2 or larger, but got %s' % p)
if p == n:
return 1
m = 0
n, rem = divmod(n, p)
while not rem:
m += 1
if m > 5:
# The multiplicity could be very large. Better
# to increment in powers of two
e = 2
while 1:
ppow = p**e
if ppow < n:
nnew, rem = divmod(n, ppow)
if not rem:
m += e
e *= 2
n = nnew
continue
return m + multiplicity(p, n)
n, rem = divmod(n, p)
return m
def multiplicity_in_factorial(p, n):
"""return the largest integer ``m`` such that ``p**m`` divides ``n!``
without calculating the factorial of ``n``.
Examples
========
>>> from sympy.ntheory import multiplicity_in_factorial
>>> from sympy import factorial
>>> multiplicity_in_factorial(2, 3)
1
An instructive use of this is to tell how many trailing zeros
a given factorial has. For example, there are 6 in 25!:
>>> factorial(25)
15511210043330985984000000
>>> multiplicity_in_factorial(10, 25)
6
For large factorials, it is much faster/feasible to use
this function rather than computing the actual factorial:
>>> multiplicity_in_factorial(factorial(25), 2**100)
52818775009509558395695966887
"""
p, n = as_int(p), as_int(n)
if p <= 0:
raise ValueError('expecting positive integer got %s' % p )
if n < 0:
raise ValueError('expecting non-negative integer got %s' % n )
factors = factorint(p)
# keep only the largest of a given multiplicity since those
# of a given multiplicity will be goverened by the behavior
# of the largest factor
test = defaultdict(int)
for k, v in factors.items():
test[v] = max(k, test[v])
keep = set(test.values())
# remove others from factors
for k in list(factors.keys()):
if k not in keep:
factors.pop(k)
mp = S.Infinity
for i in factors:
# multiplicity of i in n! is
mi = (n - (sum(digits(n, i)) - i))//(i - 1)
# multiplicity of p in n! depends on multiplicity
# of prime `i` in p, so we floor divide by factors[i]
# and keep it if smaller than the multiplicity of p
# seen so far
mp = min(mp, mi//factors[i])
return mp
def perfect_power(n, candidates=None, big=True, factor=True):
"""
Return ``(b, e)`` such that ``n`` == ``b**e`` if ``n`` is a unique
perfect power with ``e > 1``, else ``False`` (e.g. 1 is not a
perfect power). A ValueError is raised if ``n`` is not Rational.
By default, the base is recursively decomposed and the exponents
collected so the largest possible ``e`` is sought. If ``big=False``
then the smallest possible ``e`` (thus prime) will be chosen.
If ``factor=True`` then simultaneous factorization of ``n`` is
attempted since finding a factor indicates the only possible root
for ``n``. This is True by default since only a few small factors will
be tested in the course of searching for the perfect power.
The use of ``candidates`` is primarily for internal use; if provided,
False will be returned if ``n`` cannot be written as a power with one
of the candidates as an exponent and factoring (beyond testing for
a factor of 2) will not be attempted.
Examples
========
>>> from sympy import perfect_power, Rational
>>> perfect_power(16)
(2, 4)
>>> perfect_power(16, big=False)
(4, 2)
Negative numbers can only have odd perfect powers:
>>> perfect_power(-4)
False
>>> perfect_power(-8)
(-2, 3)
Rationals are also recognized:
>>> perfect_power(Rational(1, 2)**3)
(1/2, 3)
>>> perfect_power(Rational(-3, 2)**3)
(-3/2, 3)
Notes
=====
To know whether an integer is a perfect power of 2 use
>>> is2pow = lambda n: bool(n and not n & (n - 1))
>>> [(i, is2pow(i)) for i in range(5)]
[(0, False), (1, True), (2, True), (3, False), (4, True)]
It is not necessary to provide ``candidates``. When provided
it will be assumed that they are ints. The first one that is
larger than the computed maximum possible exponent will signal
failure for the routine.
>>> perfect_power(3**8, [9])
False
>>> perfect_power(3**8, [2, 4, 8])
(3, 8)
>>> perfect_power(3**8, [4, 8], big=False)
(9, 4)
See Also
========
sympy.core.power.integer_nthroot
sympy.ntheory.primetest.is_square
"""
if isinstance(n, Rational) and not n.is_Integer:
p, q = n.as_numer_denom()
if p is S.One:
pp = perfect_power(q)
if pp:
pp = (n.func(1, pp[0]), pp[1])
else:
pp = perfect_power(p)
if pp:
num, e = pp
pq = perfect_power(q, [e])
if pq:
den, _ = pq
pp = n.func(num, den), e
return pp
n = as_int(n)
if n < 0:
pp = perfect_power(-n)
if pp:
b, e = pp
if e % 2:
return -b, e
return False
if n <= 3:
# no unique exponent for 0, 1
# 2 and 3 have exponents of 1
return False
logn = math.log(n, 2)
max_possible = int(logn) + 2 # only check values less than this
not_square = n % 10 in [2, 3, 7, 8] # squares cannot end in 2, 3, 7, 8
min_possible = 2 + not_square
if not candidates:
candidates = primerange(min_possible, max_possible)
else:
candidates = sorted([i for i in candidates
if min_possible <= i < max_possible])
if n%2 == 0:
e = trailing(n)
candidates = [i for i in candidates if e%i == 0]
if big:
candidates = reversed(candidates)
for e in candidates:
r, ok = integer_nthroot(n, e)
if ok:
return (r, e)
return False
def _factors():
rv = 2 + n % 2
while True:
yield rv
rv = nextprime(rv)
for fac, e in zip(_factors(), candidates):
# see if there is a factor present
if factor and n % fac == 0:
# find what the potential power is
if fac == 2:
e = trailing(n)
else:
e = multiplicity(fac, n)
# if it's a trivial power we are done
if e == 1:
return False
# maybe the e-th root of n is exact
r, exact = integer_nthroot(n, e)
if not exact:
# Having a factor, we know that e is the maximal
# possible value for a root of n.
# If n = fac**e*m can be written as a perfect
# power then see if m can be written as r**E where
# gcd(e, E) != 1 so n = (fac**(e//E)*r)**E
m = n//fac**e
rE = perfect_power(m, candidates=divisors(e, generator=True))
if not rE:
return False
else:
r, E = rE
r, e = fac**(e//E)*r, E
if not big:
e0 = primefactors(e)
if e0[0] != e:
r, e = r**(e//e0[0]), e0[0]
return r, e
# Weed out downright impossible candidates
if logn/e < 40:
b = 2.0**(logn/e)
if abs(int(b + 0.5) - b) > 0.01:
continue
# now see if the plausible e makes a perfect power
r, exact = integer_nthroot(n, e)
if exact:
if big:
m = perfect_power(r, big=big, factor=factor)
if m:
r, e = m[0], e*m[1]
return int(r), e
return False
def pollard_rho(n, s=2, a=1, retries=5, seed=1234, max_steps=None, F=None):
r"""
Use Pollard's rho method to try to extract a nontrivial factor
of ``n``. The returned factor may be a composite number. If no
factor is found, ``None`` is returned.
The algorithm generates pseudo-random values of x with a generator
function, replacing x with F(x). If F is not supplied then the
function x**2 + ``a`` is used. The first value supplied to F(x) is ``s``.
Upon failure (if ``retries`` is > 0) a new ``a`` and ``s`` will be
supplied; the ``a`` will be ignored if F was supplied.
The sequence of numbers generated by such functions generally have a
a lead-up to some number and then loop around back to that number and
begin to repeat the sequence, e.g. 1, 2, 3, 4, 5, 3, 4, 5 -- this leader
and loop look a bit like the Greek letter rho, and thus the name, 'rho'.
For a given function, very different leader-loop values can be obtained
so it is a good idea to allow for retries:
>>> from sympy.ntheory.generate import cycle_length
>>> n = 16843009
>>> F = lambda x:(2048*pow(x, 2, n) + 32767) % n
>>> for s in range(5):
... print('loop length = %4i; leader length = %3i' % next(cycle_length(F, s)))
...
loop length = 2489; leader length = 42
loop length = 78; leader length = 120
loop length = 1482; leader length = 99
loop length = 1482; leader length = 285
loop length = 1482; leader length = 100
Here is an explicit example where there is a two element leadup to
a sequence of 3 numbers (11, 14, 4) that then repeat:
>>> x=2
>>> for i in range(9):
... x=(x**2+12)%17
... print(x)
...
16
13
11
14
4
11
14
4
11
>>> next(cycle_length(lambda x: (x**2+12)%17, 2))
(3, 2)
>>> list(cycle_length(lambda x: (x**2+12)%17, 2, values=True))
[16, 13, 11, 14, 4]
Instead of checking the differences of all generated values for a gcd
with n, only the kth and 2*kth numbers are checked, e.g. 1st and 2nd,
2nd and 4th, 3rd and 6th until it has been detected that the loop has been
traversed. Loops may be many thousands of steps long before rho finds a
factor or reports failure. If ``max_steps`` is specified, the iteration
is cancelled with a failure after the specified number of steps.
Examples
========
>>> from sympy import pollard_rho
>>> n=16843009
>>> F=lambda x:(2048*pow(x,2,n) + 32767) % n
>>> pollard_rho(n, F=F)
257
Use the default setting with a bad value of ``a`` and no retries:
>>> pollard_rho(n, a=n-2, retries=0)
If retries is > 0 then perhaps the problem will correct itself when
new values are generated for a:
>>> pollard_rho(n, a=n-2, retries=1)
257
References
==========
.. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 229-231
"""
n = int(n)
if n < 5:
raise ValueError('pollard_rho should receive n > 4')
prng = random.Random(seed + retries)
V = s
for i in range(retries + 1):
U = V
if not F:
F = lambda x: (pow(x, 2, n) + a) % n
j = 0
while 1:
if max_steps and (j > max_steps):
break
j += 1
U = F(U)
V = F(F(V)) # V is 2x further along than U
g = igcd(U - V, n)
if g == 1:
continue
if g == n:
break
return int(g)
V = prng.randint(0, n - 1)
a = prng.randint(1, n - 3) # for x**2 + a, a%n should not be 0 or -2
F = None
return None
def pollard_pm1(n, B=10, a=2, retries=0, seed=1234):
"""
Use Pollard's p-1 method to try to extract a nontrivial factor
of ``n``. Either a divisor (perhaps composite) or ``None`` is returned.
The value of ``a`` is the base that is used in the test gcd(a**M - 1, n).
The default is 2. If ``retries`` > 0 then if no factor is found after the
first attempt, a new ``a`` will be generated randomly (using the ``seed``)
and the process repeated.
Note: the value of M is lcm(1..B) = reduce(ilcm, range(2, B + 1)).
A search is made for factors next to even numbers having a power smoothness
less than ``B``. Choosing a larger B increases the likelihood of finding a
larger factor but takes longer. Whether a factor of n is found or not
depends on ``a`` and the power smoothness of the even number just less than
the factor p (hence the name p - 1).
Although some discussion of what constitutes a good ``a`` some
descriptions are hard to interpret. At the modular.math site referenced
below it is stated that if gcd(a**M - 1, n) = N then a**M % q**r is 1
for every prime power divisor of N. But consider the following:
>>> from sympy.ntheory.factor_ import smoothness_p, pollard_pm1
>>> n=257*1009
>>> smoothness_p(n)
(-1, [(257, (1, 2, 256)), (1009, (1, 7, 16))])
So we should (and can) find a root with B=16:
>>> pollard_pm1(n, B=16, a=3)
1009
If we attempt to increase B to 256 we find that it doesn't work:
>>> pollard_pm1(n, B=256)
>>>
But if the value of ``a`` is changed we find that only multiples of
257 work, e.g.:
>>> pollard_pm1(n, B=256, a=257)
1009
Checking different ``a`` values shows that all the ones that didn't
work had a gcd value not equal to ``n`` but equal to one of the
factors:
>>> from sympy import ilcm, igcd, factorint, Pow
>>> M = 1
>>> for i in range(2, 256):
... M = ilcm(M, i)
...
>>> set([igcd(pow(a, M, n) - 1, n) for a in range(2, 256) if
... igcd(pow(a, M, n) - 1, n) != n])
{1009}
But does aM % d for every divisor of n give 1?
>>> aM = pow(255, M, n)
>>> [(d, aM%Pow(*d.args)) for d in factorint(n, visual=True).args]
[(257**1, 1), (1009**1, 1)]
No, only one of them. So perhaps the principle is that a root will
be found for a given value of B provided that:
1) the power smoothness of the p - 1 value next to the root
does not exceed B
2) a**M % p != 1 for any of the divisors of n.
By trying more than one ``a`` it is possible that one of them
will yield a factor.
Examples
========
With the default smoothness bound, this number cannot be cracked:
>>> from sympy.ntheory import pollard_pm1
>>> pollard_pm1(21477639576571)
Increasing the smoothness bound helps:
>>> pollard_pm1(21477639576571, B=2000)
4410317
Looking at the smoothness of the factors of this number we find:
>>> from sympy.ntheory.factor_ import smoothness_p, factorint
>>> print(smoothness_p(21477639576571, visual=1))
p**i=4410317**1 has p-1 B=1787, B-pow=1787
p**i=4869863**1 has p-1 B=2434931, B-pow=2434931
The B and B-pow are the same for the p - 1 factorizations of the divisors
because those factorizations had a very large prime factor:
>>> factorint(4410317 - 1)
{2: 2, 617: 1, 1787: 1}
>>> factorint(4869863-1)
{2: 1, 2434931: 1}
Note that until B reaches the B-pow value of 1787, the number is not cracked;
>>> pollard_pm1(21477639576571, B=1786)
>>> pollard_pm1(21477639576571, B=1787)
4410317
The B value has to do with the factors of the number next to the divisor,
not the divisors themselves. A worst case scenario is that the number next
to the factor p has a large prime divisisor or is a perfect power. If these
conditions apply then the power-smoothness will be about p/2 or p. The more
realistic is that there will be a large prime factor next to p requiring
a B value on the order of p/2. Although primes may have been searched for
up to this level, the p/2 is a factor of p - 1, something that we do not
know. The modular.math reference below states that 15% of numbers in the
range of 10**15 to 15**15 + 10**4 are 10**6 power smooth so a B of 10**6
will fail 85% of the time in that range. From 10**8 to 10**8 + 10**3 the
percentages are nearly reversed...but in that range the simple trial
division is quite fast.
References
==========
.. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers:
A Computational Perspective", Springer, 2nd edition, 236-238
.. [2] http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node81.html
.. [3] https://www.cs.toronto.edu/~yuvalf/Factorization.pdf
"""
n = int(n)
if n < 4 or B < 3:
raise ValueError('pollard_pm1 should receive n > 3 and B > 2')
prng = random.Random(seed + B)
# computing a**lcm(1,2,3,..B) % n for B > 2
# it looks weird, but it's right: primes run [2, B]
# and the answer's not right until the loop is done.
for i in range(retries + 1):
aM = a
for p in sieve.primerange(2, B + 1):
e = int(math.log(B, p))
aM = pow(aM, pow(p, e), n)
g = igcd(aM - 1, n)
if 1 < g < n:
return int(g)
# get a new a:
# since the exponent, lcm(1..B), is even, if we allow 'a' to be 'n-1'
# then (n - 1)**even % n will be 1 which will give a g of 0 and 1 will
# give a zero, too, so we set the range as [2, n-2]. Some references
# say 'a' should be coprime to n, but either will detect factors.
a = prng.randint(2, n - 2)
def _trial(factors, n, candidates, verbose=False):
"""
Helper function for integer factorization. Trial factors ``n`
against all integers given in the sequence ``candidates``
and updates the dict ``factors`` in-place. Returns the reduced
value of ``n`` and a flag indicating whether any factors were found.
"""
if verbose:
factors0 = list(factors.keys())
nfactors = len(factors)
for d in candidates:
if n % d == 0:
m = multiplicity(d, n)
n //= d**m
factors[d] = m
if verbose:
for k in sorted(set(factors).difference(set(factors0))):
print(factor_msg % (k, factors[k]))
return int(n), len(factors) != nfactors
def _check_termination(factors, n, limitp1, use_trial, use_rho, use_pm1,
verbose):
"""
Helper function for integer factorization. Checks if ``n``
is a prime or a perfect power, and in those cases updates
the factorization and raises ``StopIteration``.
"""
if verbose:
print('Check for termination')
# since we've already been factoring there is no need to do
# simultaneous factoring with the power check
p = perfect_power(n, factor=False)
if p is not False:
base, exp = p
if limitp1:
limit = limitp1 - 1
else:
limit = limitp1
facs = factorint(base, limit, use_trial, use_rho, use_pm1,
verbose=False)
for b, e in facs.items():
if verbose:
print(factor_msg % (b, e))
factors[b] = exp*e
raise StopIteration
if isprime(n):
factors[int(n)] = 1
raise StopIteration
if n == 1:
raise StopIteration
trial_int_msg = "Trial division with ints [%i ... %i] and fail_max=%i"
trial_msg = "Trial division with primes [%i ... %i]"
rho_msg = "Pollard's rho with retries %i, max_steps %i and seed %i"
pm1_msg = "Pollard's p-1 with smoothness bound %i and seed %i"
ecm_msg = "Elliptic Curve with B1 bound %i, B2 bound %i, num_curves %i"
factor_msg = '\t%i ** %i'
fermat_msg = 'Close factors satisying Fermat condition found.'
complete_msg = 'Factorization is complete.'
def _factorint_small(factors, n, limit, fail_max):
"""
Return the value of n and either a 0 (indicating that factorization up
to the limit was complete) or else the next near-prime that would have
been tested.
Factoring stops if there are fail_max unsuccessful tests in a row.
If factors of n were found they will be in the factors dictionary as
{factor: multiplicity} and the returned value of n will have had those
factors removed. The factors dictionary is modified in-place.
"""
def done(n, d):
"""return n, d if the sqrt(n) wasn't reached yet, else
n, 0 indicating that factoring is done.
"""
if d*d <= n:
return n, d
return n, 0
d = 2
m = trailing(n)
if m:
factors[d] = m
n >>= m
d = 3
if limit < d:
if n > 1:
factors[n] = 1
return done(n, d)
# reduce
m = 0
while n % d == 0:
n //= d
m += 1
if m == 20:
mm = multiplicity(d, n)
m += mm
n //= d**mm
break
if m:
factors[d] = m
# when d*d exceeds maxx or n we are done; if limit**2 is greater
# than n then maxx is set to zero so the value of n will flag the finish
if limit*limit > n:
maxx = 0
else:
maxx = limit*limit
dd = maxx or n
d = 5
fails = 0
while fails < fail_max:
if d*d > dd:
break
# d = 6*i - 1
# reduce
m = 0
while n % d == 0:
n //= d
m += 1
if m == 20:
mm = multiplicity(d, n)
m += mm
n //= d**mm
break
if m:
factors[d] = m
dd = maxx or n
fails = 0
else:
fails += 1
d += 2
if d*d > dd:
break
# d = 6*i - 1
# reduce
m = 0
while n % d == 0:
n //= d
m += 1
if m == 20:
mm = multiplicity(d, n)
m += mm
n //= d**mm
break
if m:
factors[d] = m
dd = maxx or n
fails = 0
else:
fails += 1
# d = 6*(i + 1) - 1
d += 4
return done(n, d)
def factorint(n, limit=None, use_trial=True, use_rho=True, use_pm1=True,
use_ecm=True, verbose=False, visual=None, multiple=False):
r"""
Given a positive integer ``n``, ``factorint(n)`` returns a dict containing
the prime factors of ``n`` as keys and their respective multiplicities
as values. For example:
>>> from sympy.ntheory import factorint
>>> factorint(2000) # 2000 = (2**4) * (5**3)
{2: 4, 5: 3}
>>> factorint(65537) # This number is prime
{65537: 1}
For input less than 2, factorint behaves as follows:
- ``factorint(1)`` returns the empty factorization, ``{}``
- ``factorint(0)`` returns ``{0:1}``
- ``factorint(-n)`` adds ``-1:1`` to the factors and then factors ``n``
Partial Factorization:
If ``limit`` (> 3) is specified, the search is stopped after performing
trial division up to (and including) the limit (or taking a
corresponding number of rho/p-1 steps). This is useful if one has
a large number and only is interested in finding small factors (if
any). Note that setting a limit does not prevent larger factors
from being found early; it simply means that the largest factor may
be composite. Since checking for perfect power is relatively cheap, it is
done regardless of the limit setting.
This number, for example, has two small factors and a huge
semi-prime factor that cannot be reduced easily:
>>> from sympy.ntheory import isprime
>>> a = 1407633717262338957430697921446883
>>> f = factorint(a, limit=10000)
>>> f == {991: 1, int(202916782076162456022877024859): 1, 7: 1}
True
>>> isprime(max(f))
False
This number has a small factor and a residual perfect power whose
base is greater than the limit:
>>> factorint(3*101**7, limit=5)
{3: 1, 101: 7}
List of Factors:
If ``multiple`` is set to ``True`` then a list containing the
prime factors including multiplicities is returned.
>>> factorint(24, multiple=True)
[2, 2, 2, 3]
Visual Factorization:
If ``visual`` is set to ``True``, then it will return a visual
factorization of the integer. For example:
>>> from sympy import pprint
>>> pprint(factorint(4200, visual=True))
3 1 2 1
2 *3 *5 *7
Note that this is achieved by using the evaluate=False flag in Mul
and Pow. If you do other manipulations with an expression where
evaluate=False, it may evaluate. Therefore, you should use the
visual option only for visualization, and use the normal dictionary
returned by visual=False if you want to perform operations on the
factors.
You can easily switch between the two forms by sending them back to
factorint:
>>> from sympy import Mul
>>> regular = factorint(1764); regular
{2: 2, 3: 2, 7: 2}
>>> pprint(factorint(regular))
2 2 2
2 *3 *7
>>> visual = factorint(1764, visual=True); pprint(visual)
2 2 2
2 *3 *7
>>> print(factorint(visual))
{2: 2, 3: 2, 7: 2}
If you want to send a number to be factored in a partially factored form
you can do so with a dictionary or unevaluated expression:
>>> factorint(factorint({4: 2, 12: 3})) # twice to toggle to dict form
{2: 10, 3: 3}
>>> factorint(Mul(4, 12, evaluate=False))
{2: 4, 3: 1}
The table of the output logic is:
====== ====== ======= =======
Visual
------ ----------------------
Input True False other
====== ====== ======= =======
dict mul dict mul
n mul dict dict
mul mul dict dict
====== ====== ======= =======
Notes
=====
Algorithm:
The function switches between multiple algorithms. Trial division
quickly finds small factors (of the order 1-5 digits), and finds
all large factors if given enough time. The Pollard rho and p-1
algorithms are used to find large factors ahead of time; they
will often find factors of the order of 10 digits within a few
seconds:
>>> factors = factorint(12345678910111213141516)
>>> for base, exp in sorted(factors.items()):
... print('%s %s' % (base, exp))
...
2 2
2507191691 1
1231026625769 1
Any of these methods can optionally be disabled with the following
boolean parameters:
- ``use_trial``: Toggle use of trial division
- ``use_rho``: Toggle use of Pollard's rho method
- ``use_pm1``: Toggle use of Pollard's p-1 method
``factorint`` also periodically checks if the remaining part is
a prime number or a perfect power, and in those cases stops.
For unevaluated factorial, it uses Legendre's formula(theorem).
If ``verbose`` is set to ``True``, detailed progress is printed.
See Also
========
smoothness, smoothness_p, divisors
"""
if isinstance(n, Dict):
n = dict(n)
if multiple:
fac = factorint(n, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose, visual=False, multiple=False)
factorlist = sum(([p] * fac[p] if fac[p] > 0 else [S.One/p]*(-fac[p])
for p in sorted(fac)), [])
return factorlist
factordict = {}
if visual and not isinstance(n, (Mul, dict)):
factordict = factorint(n, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose, visual=False)
elif isinstance(n, Mul):
factordict = {int(k): int(v) for k, v in
n.as_powers_dict().items()}
elif isinstance(n, dict):
factordict = n
if factordict and isinstance(n, (Mul, dict)):
# check it
for key in list(factordict.keys()):
if isprime(key):
continue
e = factordict.pop(key)
d = factorint(key, limit=limit, use_trial=use_trial, use_rho=use_rho,
use_pm1=use_pm1, verbose=verbose, visual=False)
for k, v in d.items():
if k in factordict:
factordict[k] += v*e
else:
factordict[k] = v*e
if visual or (type(n) is dict and
visual is not True and
visual is not False):
if factordict == {}:
return S.One
if -1 in factordict:
factordict.pop(-1)
args = [S.NegativeOne]
else:
args = []
args.extend([Pow(*i, evaluate=False)
for i in sorted(factordict.items())])
return Mul(*args, evaluate=False)
elif isinstance(n, dict) or isinstance(n, Mul):
return factordict
assert use_trial or use_rho or use_pm1 or use_ecm
from sympy.functions.combinatorial.factorials import factorial
if isinstance(n, factorial):
x = as_int(n.args[0])
if x >= 20:
factors = {}
m = 2 # to initialize the if condition below
for p in sieve.primerange(2, x + 1):
if m > 1:
m, q = 0, x // p
while q != 0:
m += q
q //= p
factors[p] = m
if factors and verbose:
for k in sorted(factors):
print(factor_msg % (k, factors[k]))
if verbose:
print(complete_msg)
return factors
else:
# if n < 20!, direct computation is faster
# since it uses a lookup table
n = n.func(x)
n = as_int(n)
if limit:
limit = int(limit)
use_ecm = False
# special cases
if n < 0:
factors = factorint(
-n, limit=limit, use_trial=use_trial, use_rho=use_rho,
use_pm1=use_pm1, verbose=verbose, visual=False)
factors[-1] = 1
return factors
if limit and limit < 2:
if n == 1:
return {}
return {n: 1}
elif n < 10:
# doing this we are assured of getting a limit > 2
# when we have to compute it later
return [{0: 1}, {}, {2: 1}, {3: 1}, {2: 2}, {5: 1},
{2: 1, 3: 1}, {7: 1}, {2: 3}, {3: 2}][n]
factors = {}
# do simplistic factorization
if verbose:
sn = str(n)
if len(sn) > 50:
print('Factoring %s' % sn[:5] + \
'..(%i other digits)..' % (len(sn) - 10) + sn[-5:])
else:
print('Factoring', n)
if use_trial:
# this is the preliminary factorization for small factors
small = 2**15
fail_max = 600
small = min(small, limit or small)
if verbose:
print(trial_int_msg % (2, small, fail_max))
n, next_p = _factorint_small(factors, n, small, fail_max)
else:
next_p = 2
if factors and verbose:
for k in sorted(factors):
print(factor_msg % (k, factors[k]))
if next_p == 0:
if n > 1:
factors[int(n)] = 1
if verbose:
print(complete_msg)
return factors
# continue with more advanced factorization methods
# first check if the simplistic run didn't finish
# because of the limit and check for a perfect
# power before exiting
try:
if limit and next_p > limit:
if verbose:
print('Exceeded limit:', limit)
_check_termination(factors, n, limit, use_trial, use_rho, use_pm1,
verbose)
if n > 1:
factors[int(n)] = 1
return factors
else:
# Before quitting (or continuing on)...
# ...do a Fermat test since it's so easy and we need the
# square root anyway. Finding 2 factors is easy if they are
# "close enough." This is the big root equivalent of dividing by
# 2, 3, 5.
sqrt_n = integer_nthroot(n, 2)[0]
a = sqrt_n + 1
a2 = a**2
b2 = a2 - n
for i in range(3):
b, fermat = integer_nthroot(b2, 2)
if fermat:
break
b2 += 2*a + 1 # equiv to (a + 1)**2 - n
a += 1
if fermat:
if verbose:
print(fermat_msg)
if limit:
limit -= 1
for r in [a - b, a + b]:
facs = factorint(r, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose)
for k, v in facs.items():
factors[k] = factors.get(k, 0) + v
raise StopIteration
# ...see if factorization can be terminated
_check_termination(factors, n, limit, use_trial, use_rho, use_pm1,
verbose)
except StopIteration:
if verbose:
print(complete_msg)
return factors
# these are the limits for trial division which will
# be attempted in parallel with pollard methods
low, high = next_p, 2*next_p
limit = limit or sqrt_n
# add 1 to make sure limit is reached in primerange calls
limit += 1
iteration = 0
while 1:
try:
high_ = high
if limit < high_:
high_ = limit
# Trial division
if use_trial:
if verbose:
print(trial_msg % (low, high_))
ps = sieve.primerange(low, high_)
n, found_trial = _trial(factors, n, ps, verbose)
if found_trial:
_check_termination(factors, n, limit, use_trial, use_rho,
use_pm1, verbose)
else:
found_trial = False
if high > limit:
if verbose:
print('Exceeded limit:', limit)
if n > 1:
factors[int(n)] = 1
raise StopIteration
# Only used advanced methods when no small factors were found
if not found_trial:
if (use_pm1 or use_rho):
high_root = max(int(math.log(high_**0.7)), low, 3)
# Pollard p-1
if use_pm1:
if verbose:
print(pm1_msg % (high_root, high_))
c = pollard_pm1(n, B=high_root, seed=high_)
if c:
# factor it and let _trial do the update
ps = factorint(c, limit=limit - 1,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
use_ecm=use_ecm,
verbose=verbose)
n, _ = _trial(factors, n, ps, verbose=False)
_check_termination(factors, n, limit, use_trial,
use_rho, use_pm1, verbose)
# Pollard rho
if use_rho:
max_steps = high_root
if verbose:
print(rho_msg % (1, max_steps, high_))
c = pollard_rho(n, retries=1, max_steps=max_steps,
seed=high_)
if c:
# factor it and let _trial do the update
ps = factorint(c, limit=limit - 1,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
use_ecm=use_ecm,
verbose=verbose)
n, _ = _trial(factors, n, ps, verbose=False)
_check_termination(factors, n, limit, use_trial,
use_rho, use_pm1, verbose)
except StopIteration:
if verbose:
print(complete_msg)
return factors
#Use subexponential algorithms if use_ecm
#Use pollard algorithms for finding small factors for 3 iterations
#if after small factors the number of digits of n is >= 20 then use ecm
iteration += 1
if use_ecm and iteration >= 3 and len(str(n)) >= 25:
break
low, high = high, high*2
B1 = 10000
B2 = 100*B1
num_curves = 50
while(1):
if verbose:
print(ecm_msg % (B1, B2, num_curves))
while(1):
try:
factor = _ecm_one_factor(n, B1, B2, num_curves)
ps = factorint(factor, limit=limit - 1,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
use_ecm=use_ecm,
verbose=verbose)
n, _ = _trial(factors, n, ps, verbose=False)
_check_termination(factors, n, limit, use_trial,
use_rho, use_pm1, verbose)
except ValueError:
break
except StopIteration:
if verbose:
print(complete_msg)
return factors
B1 *= 5
B2 = 100*B1
num_curves *= 4
def factorrat(rat, limit=None, use_trial=True, use_rho=True, use_pm1=True,
verbose=False, visual=None, multiple=False):
r"""
Given a Rational ``r``, ``factorrat(r)`` returns a dict containing
the prime factors of ``r`` as keys and their respective multiplicities
as values. For example:
>>> from sympy import factorrat, S
>>> factorrat(S(8)/9) # 8/9 = (2**3) * (3**-2)
{2: 3, 3: -2}
>>> factorrat(S(-1)/987) # -1/789 = -1 * (3**-1) * (7**-1) * (47**-1)
{-1: 1, 3: -1, 7: -1, 47: -1}
Please see the docstring for ``factorint`` for detailed explanations
and examples of the following keywords:
- ``limit``: Integer limit up to which trial division is done
- ``use_trial``: Toggle use of trial division
- ``use_rho``: Toggle use of Pollard's rho method
- ``use_pm1``: Toggle use of Pollard's p-1 method
- ``verbose``: Toggle detailed printing of progress
- ``multiple``: Toggle returning a list of factors or dict
- ``visual``: Toggle product form of output
"""
if multiple:
fac = factorrat(rat, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose, visual=False, multiple=False)
factorlist = sum(([p] * fac[p] if fac[p] > 0 else [S.One/p]*(-fac[p])
for p, _ in sorted(fac.items(),
key=lambda elem: elem[0]
if elem[1] > 0
else 1/elem[0])), [])
return factorlist
f = factorint(rat.p, limit=limit, use_trial=use_trial,
use_rho=use_rho, use_pm1=use_pm1,
verbose=verbose).copy()
f = defaultdict(int, f)
for p, e in factorint(rat.q, limit=limit,
use_trial=use_trial,
use_rho=use_rho,
use_pm1=use_pm1,
verbose=verbose).items():
f[p] += -e
if len(f) > 1 and 1 in f:
del f[1]
if not visual:
return dict(f)
else:
if -1 in f:
f.pop(-1)
args = [S.NegativeOne]
else:
args = []
args.extend([Pow(*i, evaluate=False)
for i in sorted(f.items())])
return Mul(*args, evaluate=False)
def primefactors(n, limit=None, verbose=False):
"""Return a sorted list of n's prime factors, ignoring multiplicity
and any composite factor that remains if the limit was set too low
for complete factorization. Unlike factorint(), primefactors() does
not return -1 or 0.
Examples
========
>>> from sympy.ntheory import primefactors, factorint, isprime
>>> primefactors(6)
[2, 3]
>>> primefactors(-5)
[5]
>>> sorted(factorint(123456).items())
[(2, 6), (3, 1), (643, 1)]
>>> primefactors(123456)
[2, 3, 643]
>>> sorted(factorint(10000000001, limit=200).items())
[(101, 1), (99009901, 1)]
>>> isprime(99009901)
False
>>> primefactors(10000000001, limit=300)
[101]
See Also
========
divisors
"""
n = int(n)
factors = sorted(factorint(n, limit=limit, verbose=verbose).keys())
s = [f for f in factors[:-1:] if f not in [-1, 0, 1]]
if factors and isprime(factors[-1]):
s += [factors[-1]]
return s
def _divisors(n, proper=False):
"""Helper function for divisors which generates the divisors."""
factordict = factorint(n)
ps = sorted(factordict.keys())
def rec_gen(n=0):
if n == len(ps):
yield 1
else:
pows = [1]
for j in range(factordict[ps[n]]):
pows.append(pows[-1] * ps[n])
for q in rec_gen(n + 1):
for p in pows:
yield p * q
if proper:
for p in rec_gen():
if p != n:
yield p
else:
yield from rec_gen()
def divisors(n, generator=False, proper=False):
r"""
Return all divisors of n sorted from 1..n by default.
If generator is ``True`` an unordered generator is returned.
The number of divisors of n can be quite large if there are many
prime factors (counting repeated factors). If only the number of
factors is desired use divisor_count(n).
Examples
========
>>> from sympy import divisors, divisor_count
>>> divisors(24)
[1, 2, 3, 4, 6, 8, 12, 24]
>>> divisor_count(24)
8
>>> list(divisors(120, generator=True))
[1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60, 120]
Notes
=====
This is a slightly modified version of Tim Peters referenced at:
https://stackoverflow.com/questions/1010381/python-factorization
See Also
========
primefactors, factorint, divisor_count
"""
n = as_int(abs(n))
if isprime(n):
if proper:
return [1]
return [1, n]
if n == 1:
if proper:
return []
return [1]
if n == 0:
return []
rv = _divisors(n, proper)
if not generator:
return sorted(rv)
return rv
def divisor_count(n, modulus=1, proper=False):
"""
Return the number of divisors of ``n``. If ``modulus`` is not 1 then only
those that are divisible by ``modulus`` are counted. If ``proper`` is True
then the divisor of ``n`` will not be counted.
Examples
========
>>> from sympy import divisor_count
>>> divisor_count(6)
4
>>> divisor_count(6, 2)
2
>>> divisor_count(6, proper=True)
3
See Also
========
factorint, divisors, totient, proper_divisor_count
"""
if not modulus:
return 0
elif modulus != 1:
n, r = divmod(n, modulus)
if r:
return 0
if n == 0:
return 0
n = Mul(*[v + 1 for k, v in factorint(n).items() if k > 1])
if n and proper:
n -= 1
return n
def proper_divisors(n, generator=False):
"""
Return all divisors of n except n, sorted by default.
If generator is ``True`` an unordered generator is returned.
Examples
========
>>> from sympy import proper_divisors, proper_divisor_count
>>> proper_divisors(24)
[1, 2, 3, 4, 6, 8, 12]
>>> proper_divisor_count(24)
7
>>> list(proper_divisors(120, generator=True))
[1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60]
See Also
========
factorint, divisors, proper_divisor_count
"""
return divisors(n, generator=generator, proper=True)
def proper_divisor_count(n, modulus=1):
"""
Return the number of proper divisors of ``n``.
Examples
========
>>> from sympy import proper_divisor_count
>>> proper_divisor_count(6)
3
>>> proper_divisor_count(6, modulus=2)
1
See Also
========
divisors, proper_divisors, divisor_count
"""
return divisor_count(n, modulus=modulus, proper=True)
def _udivisors(n):
"""Helper function for udivisors which generates the unitary divisors."""
factorpows = [p**e for p, e in factorint(n).items()]
for i in range(2**len(factorpows)):
d, j, k = 1, i, 0
while j:
if (j & 1):
d *= factorpows[k]
j >>= 1
k += 1
yield d
def udivisors(n, generator=False):
r"""
Return all unitary divisors of n sorted from 1..n by default.
If generator is ``True`` an unordered generator is returned.
The number of unitary divisors of n can be quite large if there are many
prime factors. If only the number of unitary divisors is desired use
udivisor_count(n).
Examples
========
>>> from sympy.ntheory.factor_ import udivisors, udivisor_count
>>> udivisors(15)
[1, 3, 5, 15]
>>> udivisor_count(15)
4
>>> sorted(udivisors(120, generator=True))
[1, 3, 5, 8, 15, 24, 40, 120]
See Also
========
primefactors, factorint, divisors, divisor_count, udivisor_count
References
==========
.. [1] https://en.wikipedia.org/wiki/Unitary_divisor
.. [2] http://mathworld.wolfram.com/UnitaryDivisor.html
"""
n = as_int(abs(n))
if isprime(n):
return [1, n]
if n == 1:
return [1]
if n == 0:
return []
rv = _udivisors(n)
if not generator:
return sorted(rv)
return rv
def udivisor_count(n):
"""
Return the number of unitary divisors of ``n``.
Parameters
==========
n : integer
Examples
========
>>> from sympy.ntheory.factor_ import udivisor_count
>>> udivisor_count(120)
8
See Also
========
factorint, divisors, udivisors, divisor_count, totient
References
==========
.. [1] http://mathworld.wolfram.com/UnitaryDivisorFunction.html
"""
if n == 0:
return 0
return 2**len([p for p in factorint(n) if p > 1])
def _antidivisors(n):
"""Helper function for antidivisors which generates the antidivisors."""
for d in _divisors(n):
y = 2*d
if n > y and n % y:
yield y
for d in _divisors(2*n-1):
if n > d >= 2 and n % d:
yield d
for d in _divisors(2*n+1):
if n > d >= 2 and n % d:
yield d
def antidivisors(n, generator=False):
r"""
Return all antidivisors of n sorted from 1..n by default.
Antidivisors [1]_ of n are numbers that do not divide n by the largest
possible margin. If generator is True an unordered generator is returned.
Examples
========
>>> from sympy.ntheory.factor_ import antidivisors
>>> antidivisors(24)
[7, 16]
>>> sorted(antidivisors(128, generator=True))
[3, 5, 15, 17, 51, 85]
See Also
========
primefactors, factorint, divisors, divisor_count, antidivisor_count
References
==========
.. [1] definition is described in https://oeis.org/A066272/a066272a.html
"""
n = as_int(abs(n))
if n <= 2:
return []
rv = _antidivisors(n)
if not generator:
return sorted(rv)
return rv
def antidivisor_count(n):
"""
Return the number of antidivisors [1]_ of ``n``.
Parameters
==========
n : integer
Examples
========
>>> from sympy.ntheory.factor_ import antidivisor_count
>>> antidivisor_count(13)
4
>>> antidivisor_count(27)
5
See Also
========
factorint, divisors, antidivisors, divisor_count, totient
References
==========
.. [1] formula from https://oeis.org/A066272
"""
n = as_int(abs(n))
if n <= 2:
return 0
return divisor_count(2*n - 1) + divisor_count(2*n + 1) + \
divisor_count(n) - divisor_count(n, 2) - 5
class totient(Function):
r"""
Calculate the Euler totient function phi(n)
``totient(n)`` or `\phi(n)` is the number of positive integers `\leq` n
that are relatively prime to n.
Parameters
==========
n : integer
Examples
========
>>> from sympy.ntheory import totient
>>> totient(1)
1
>>> totient(25)
20
>>> totient(45) == totient(5)*totient(9)
True
See Also
========
divisor_count
References
==========
.. [1] https://en.wikipedia.org/wiki/Euler%27s_totient_function
.. [2] http://mathworld.wolfram.com/TotientFunction.html
"""
@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n < 1:
raise ValueError("n must be a positive integer")
factors = factorint(n)
return cls._from_factors(factors)
elif not isinstance(n, Expr) or (n.is_integer is False) or (n.is_positive is False):
raise ValueError("n must be a positive integer")
def _eval_is_integer(self):
return fuzzy_and([self.args[0].is_integer, self.args[0].is_positive])
@classmethod
def _from_distinct_primes(self, *args):
"""Subroutine to compute totient from the list of assumed
distinct primes
Examples
========
>>> from sympy.ntheory.factor_ import totient
>>> totient._from_distinct_primes(5, 7)
24
"""
from functools import reduce
return reduce(lambda i, j: i * (j-1), args, 1)
@classmethod
def _from_factors(self, factors):
"""Subroutine to compute totient from already-computed factors
Examples
========
>>> from sympy.ntheory.factor_ import totient
>>> totient._from_factors({5: 2})
20
"""
t = 1
for p, k in factors.items():
t *= (p - 1) * p**(k - 1)
return t
class reduced_totient(Function):
r"""
Calculate the Carmichael reduced totient function lambda(n)
``reduced_totient(n)`` or `\lambda(n)` is the smallest m > 0 such that
`k^m \equiv 1 \mod n` for all k relatively prime to n.
Examples
========
>>> from sympy.ntheory import reduced_totient
>>> reduced_totient(1)
1
>>> reduced_totient(8)
2
>>> reduced_totient(30)
4
See Also
========
totient
References
==========
.. [1] https://en.wikipedia.org/wiki/Carmichael_function
.. [2] http://mathworld.wolfram.com/CarmichaelFunction.html
"""
@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n < 1:
raise ValueError("n must be a positive integer")
factors = factorint(n)
return cls._from_factors(factors)
@classmethod
def _from_factors(self, factors):
"""Subroutine to compute totient from already-computed factors
"""
t = 1
for p, k in factors.items():
if p == 2 and k > 2:
t = ilcm(t, 2**(k - 2))
else:
t = ilcm(t, (p - 1) * p**(k - 1))
return t
@classmethod
def _from_distinct_primes(self, *args):
"""Subroutine to compute totient from the list of assumed
distinct primes
"""
args = [p - 1 for p in args]
return ilcm(*args)
def _eval_is_integer(self):
return fuzzy_and([self.args[0].is_integer, self.args[0].is_positive])
class divisor_sigma(Function):
r"""
Calculate the divisor function `\sigma_k(n)` for positive integer n
``divisor_sigma(n, k)`` is equal to ``sum([x**k for x in divisors(n)])``
If n's prime factorization is:
.. math ::
n = \prod_{i=1}^\omega p_i^{m_i},
then
.. math ::
\sigma_k(n) = \prod_{i=1}^\omega (1+p_i^k+p_i^{2k}+\cdots
+ p_i^{m_ik}).
Parameters
==========
n : integer
k : integer, optional
power of divisors in the sum
for k = 0, 1:
``divisor_sigma(n, 0)`` is equal to ``divisor_count(n)``
``divisor_sigma(n, 1)`` is equal to ``sum(divisors(n))``
Default for k is 1.
Examples
========
>>> from sympy.ntheory import divisor_sigma
>>> divisor_sigma(18, 0)
6
>>> divisor_sigma(39, 1)
56
>>> divisor_sigma(12, 2)
210
>>> divisor_sigma(37)
38
See Also
========
divisor_count, totient, divisors, factorint
References
==========
.. [1] https://en.wikipedia.org/wiki/Divisor_function
"""
@classmethod
def eval(cls, n, k=1):
n = sympify(n)
k = sympify(k)
if n.is_prime:
return 1 + n**k
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
elif k.is_Integer:
k = int(k)
return Integer(prod(
(p**(k*(e + 1)) - 1)//(p**k - 1) if k != 0
else e + 1 for p, e in factorint(n).items()))
else:
return Mul(*[(p**(k*(e + 1)) - 1)/(p**k - 1) if k != 0
else e + 1 for p, e in factorint(n).items()])
if n.is_integer: # symbolic case
args = []
for p, e in (_.as_base_exp() for _ in Mul.make_args(n)):
if p.is_prime and e.is_positive:
args.append((p**(k*(e + 1)) - 1)/(p**k - 1) if
k != 0 else e + 1)
else:
return
return Mul(*args)
def core(n, t=2):
r"""
Calculate core(n, t) = `core_t(n)` of a positive integer n
``core_2(n)`` is equal to the squarefree part of n
If n's prime factorization is:
.. math ::
n = \prod_{i=1}^\omega p_i^{m_i},
then
.. math ::
core_t(n) = \prod_{i=1}^\omega p_i^{m_i \mod t}.
Parameters
==========
n : integer
t : integer
core(n, t) calculates the t-th power free part of n
``core(n, 2)`` is the squarefree part of ``n``
``core(n, 3)`` is the cubefree part of ``n``
Default for t is 2.
Examples
========
>>> from sympy.ntheory.factor_ import core
>>> core(24, 2)
6
>>> core(9424, 3)
1178
>>> core(379238)
379238
>>> core(15**11, 10)
15
See Also
========
factorint, sympy.solvers.diophantine.diophantine.square_factor
References
==========
.. [1] https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core
"""
n = as_int(n)
t = as_int(t)
if n <= 0:
raise ValueError("n must be a positive integer")
elif t <= 1:
raise ValueError("t must be >= 2")
else:
y = 1
for p, e in factorint(n).items():
y *= p**(e % t)
return y
class udivisor_sigma(Function):
r"""
Calculate the unitary divisor function `\sigma_k^*(n)` for positive integer n
``udivisor_sigma(n, k)`` is equal to ``sum([x**k for x in udivisors(n)])``
If n's prime factorization is:
.. math ::
n = \prod_{i=1}^\omega p_i^{m_i},
then
.. math ::
\sigma_k^*(n) = \prod_{i=1}^\omega (1+ p_i^{m_ik}).
Parameters
==========
k : power of divisors in the sum
for k = 0, 1:
``udivisor_sigma(n, 0)`` is equal to ``udivisor_count(n)``
``udivisor_sigma(n, 1)`` is equal to ``sum(udivisors(n))``
Default for k is 1.
Examples
========
>>> from sympy.ntheory.factor_ import udivisor_sigma
>>> udivisor_sigma(18, 0)
4
>>> udivisor_sigma(74, 1)
114
>>> udivisor_sigma(36, 3)
47450
>>> udivisor_sigma(111)
152
See Also
========
divisor_count, totient, divisors, udivisors, udivisor_count, divisor_sigma,
factorint
References
==========
.. [1] http://mathworld.wolfram.com/UnitaryDivisorFunction.html
"""
@classmethod
def eval(cls, n, k=1):
n = sympify(n)
k = sympify(k)
if n.is_prime:
return 1 + n**k
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
else:
return Mul(*[1+p**(k*e) for p, e in factorint(n).items()])
class primenu(Function):
r"""
Calculate the number of distinct prime factors for a positive integer n.
If n's prime factorization is:
.. math ::
n = \prod_{i=1}^k p_i^{m_i},
then ``primenu(n)`` or `\nu(n)` is:
.. math ::
\nu(n) = k.
Examples
========
>>> from sympy.ntheory.factor_ import primenu
>>> primenu(1)
0
>>> primenu(30)
3
See Also
========
factorint
References
==========
.. [1] http://mathworld.wolfram.com/PrimeFactor.html
"""
@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
else:
return len(factorint(n).keys())
class primeomega(Function):
r"""
Calculate the number of prime factors counting multiplicities for a
positive integer n.
If n's prime factorization is:
.. math ::
n = \prod_{i=1}^k p_i^{m_i},
then ``primeomega(n)`` or `\Omega(n)` is:
.. math ::
\Omega(n) = \sum_{i=1}^k m_i.
Examples
========
>>> from sympy.ntheory.factor_ import primeomega
>>> primeomega(1)
0
>>> primeomega(20)
3
See Also
========
factorint
References
==========
.. [1] http://mathworld.wolfram.com/PrimeFactor.html
"""
@classmethod
def eval(cls, n):
n = sympify(n)
if n.is_Integer:
if n <= 0:
raise ValueError("n must be a positive integer")
else:
return sum(factorint(n).values())
def mersenne_prime_exponent(nth):
"""Returns the exponent ``i`` for the nth Mersenne prime (which
has the form `2^i - 1`).
Examples
========
>>> from sympy.ntheory.factor_ import mersenne_prime_exponent
>>> mersenne_prime_exponent(1)
2
>>> mersenne_prime_exponent(20)
4423
"""
n = as_int(nth)
if n < 1:
raise ValueError("nth must be a positive integer; mersenne_prime_exponent(1) == 2")
if n > 51:
raise ValueError("There are only 51 perfect numbers; nth must be less than or equal to 51")
return MERSENNE_PRIME_EXPONENTS[n - 1]
def is_perfect(n):
"""Returns True if ``n`` is a perfect number, else False.
A perfect number is equal to the sum of its positive, proper divisors.
Examples
========
>>> from sympy.ntheory.factor_ import is_perfect, divisors, divisor_sigma
>>> is_perfect(20)
False
>>> is_perfect(6)
True
>>> 6 == divisor_sigma(6) - 6 == sum(divisors(6)[:-1])
True
References
==========
.. [1] http://mathworld.wolfram.com/PerfectNumber.html
.. [2] https://en.wikipedia.org/wiki/Perfect_number
"""
n = as_int(n)
if _isperfect(n):
return True
# all perfect numbers for Mersenne primes with exponents
# less than or equal to 43112609 are known
iknow = MERSENNE_PRIME_EXPONENTS.index(43112609)
if iknow <= len(PERFECT) - 1 and n <= PERFECT[iknow]:
# there may be gaps between this and larger known values
# so only conclude in the range for which all values
# are known
return False
if n%2 == 0:
last2 = n % 100
if last2 != 28 and last2 % 10 != 6:
return False
r, b = integer_nthroot(1 + 8*n, 2)
if not b:
return False
m, x = divmod(1 + r, 4)
if x:
return False
e, b = integer_log(m, 2)
if not b:
return False
else:
if n < 10**2000: # http://www.lirmm.fr/~ochem/opn/
return False
if n % 105 == 0: # not divis by 105
return False
if not any(n%m == r for m, r in [(12, 1), (468, 117), (324, 81)]):
return False
# there are many criteria that the factor structure of n
# must meet; since we will have to factor it to test the
# structure we will have the factors and can then check
# to see whether it is a perfect number or not. So we
# skip the structure checks and go straight to the final
# test below.
rv = divisor_sigma(n) - n
if rv == n:
if n%2 == 0:
raise ValueError(filldedent('''
This even number is perfect and is associated with a
Mersenne Prime, 2^%s - 1. It should be
added to SymPy.''' % (e + 1)))
else:
raise ValueError(filldedent('''In 1888, Sylvester stated: "
...a prolonged meditation on the subject has satisfied
me that the existence of any one such [odd perfect number]
-- its escape, so to say, from the complex web of conditions
which hem it in on all sides -- would be little short of a
miracle." I guess SymPy just found that miracle and it
factors like this: %s''' % factorint(n)))
def is_mersenne_prime(n):
"""Returns True if ``n`` is a Mersenne prime, else False.
A Mersenne prime is a prime number having the form `2^i - 1`.
Examples
========
>>> from sympy.ntheory.factor_ import is_mersenne_prime
>>> is_mersenne_prime(6)
False
>>> is_mersenne_prime(127)
True
References
==========
.. [1] http://mathworld.wolfram.com/MersennePrime.html
"""
n = as_int(n)
if _ismersenneprime(n):
return True
if not isprime(n):
return False
r, b = integer_log(n + 1, 2)
if not b:
return False
raise ValueError(filldedent('''
This Mersenne Prime, 2^%s - 1, should
be added to SymPy's known values.''' % r))
def abundance(n):
"""Returns the difference between the sum of the positive
proper divisors of a number and the number.
Examples
========
>>> from sympy.ntheory import abundance, is_perfect, is_abundant
>>> abundance(6)
0
>>> is_perfect(6)
True
>>> abundance(10)
-2
>>> is_abundant(10)
False
"""
return divisor_sigma(n, 1) - 2 * n
def is_abundant(n):
"""Returns True if ``n`` is an abundant number, else False.
A abundant number is smaller than the sum of its positive proper divisors.
Examples
========
>>> from sympy.ntheory.factor_ import is_abundant
>>> is_abundant(20)
True
>>> is_abundant(15)
False
References
==========
.. [1] http://mathworld.wolfram.com/AbundantNumber.html
"""
n = as_int(n)
if is_perfect(n):
return False
return n % 6 == 0 or bool(abundance(n) > 0)
def is_deficient(n):
"""Returns True if ``n`` is a deficient number, else False.
A deficient number is greater than the sum of its positive proper divisors.
Examples
========
>>> from sympy.ntheory.factor_ import is_deficient
>>> is_deficient(20)
False
>>> is_deficient(15)
True
References
==========
.. [1] http://mathworld.wolfram.com/DeficientNumber.html
"""
n = as_int(n)
if is_perfect(n):
return False
return bool(abundance(n) < 0)
def is_amicable(m, n):
"""Returns True if the numbers `m` and `n` are "amicable", else False.
Amicable numbers are two different numbers so related that the sum
of the proper divisors of each is equal to that of the other.
Examples
========
>>> from sympy.ntheory.factor_ import is_amicable, divisor_sigma
>>> is_amicable(220, 284)
True
>>> divisor_sigma(220) == divisor_sigma(284)
True
References
==========
.. [1] https://en.wikipedia.org/wiki/Amicable_numbers
"""
if m == n:
return False
a, b = map(lambda i: divisor_sigma(i), (m, n))
return a == b == (m + n)
def dra(n, b):
"""
Returns the additive digital root of a natural number ``n`` in base ``b``
which is a single digit value obtained by an iterative process of summing
digits, on each iteration using the result from the previous iteration to
compute a digit sum.
Examples
========
>>> from sympy.ntheory.factor_ import dra
>>> dra(3110, 12)
8
References
==========
.. [1] https://en.wikipedia.org/wiki/Digital_root
"""
num = abs(as_int(n))
b = as_int(b)
if b <= 1:
raise ValueError("Base should be an integer greater than 1")
if num == 0:
return 0
return (1 + (num - 1) % (b - 1))
def drm(n, b):
"""
Returns the multiplicative digital root of a natural number ``n`` in a given
base ``b`` which is a single digit value obtained by an iterative process of
multiplying digits, on each iteration using the result from the previous
iteration to compute the digit multiplication.
Examples
========
>>> from sympy.ntheory.factor_ import drm
>>> drm(9876, 10)
0
>>> drm(49, 10)
8
References
==========
.. [1] http://mathworld.wolfram.com/MultiplicativeDigitalRoot.html
"""
n = abs(as_int(n))
b = as_int(b)
if b <= 1:
raise ValueError("Base should be an integer greater than 1")
while n > b:
mul = 1
while n > 1:
n, r = divmod(n, b)
if r == 0:
return 0
mul *= r
n = mul
return n
|
23afdedfc33aea659daf86ff39acd5a992aec7909bbbf00b79bc7ee19090fe7b | from sympy.core.random import randrange, choice
from math import log
from sympy.ntheory import primefactors
from sympy.core.symbol import Symbol
from sympy.ntheory.factor_ import (factorint, multiplicity)
from sympy.combinatorics import Permutation
from sympy.combinatorics.permutations import (_af_commutes_with, _af_invert,
_af_rmul, _af_rmuln, _af_pow, Cycle)
from sympy.combinatorics.util import (_check_cycles_alt_sym,
_distribute_gens_by_base, _orbits_transversals_from_bsgs,
_handle_precomputed_bsgs, _base_ordering, _strong_gens_from_distr,
_strip, _strip_af)
from sympy.core import Basic
from sympy.functions.combinatorial.factorials import factorial
from sympy.ntheory import sieve
from sympy.utilities.iterables import has_variety, is_sequence, uniq
from sympy.core.random import _randrange
from itertools import islice
from sympy.core.sympify import _sympify
rmul = Permutation.rmul_with_af
_af_new = Permutation._af_new
class PermutationGroup(Basic):
r"""The class defining a Permutation group.
Explanation
===========
``PermutationGroup([p1, p2, ..., pn])`` returns the permutation group
generated by the list of permutations. This group can be supplied
to Polyhedron if one desires to decorate the elements to which the
indices of the permutation refer.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.polyhedron import Polyhedron
>>> from sympy.combinatorics.perm_groups import PermutationGroup
The permutations corresponding to motion of the front, right and
bottom face of a $2 \times 2$ Rubik's cube are defined:
>>> F = Permutation(2, 19, 21, 8)(3, 17, 20, 10)(4, 6, 7, 5)
>>> R = Permutation(1, 5, 21, 14)(3, 7, 23, 12)(8, 10, 11, 9)
>>> D = Permutation(6, 18, 14, 10)(7, 19, 15, 11)(20, 22, 23, 21)
These are passed as permutations to PermutationGroup:
>>> G = PermutationGroup(F, R, D)
>>> G.order()
3674160
The group can be supplied to a Polyhedron in order to track the
objects being moved. An example involving the $2 \times 2$ Rubik's cube is
given there, but here is a simple demonstration:
>>> a = Permutation(2, 1)
>>> b = Permutation(1, 0)
>>> G = PermutationGroup(a, b)
>>> P = Polyhedron(list('ABC'), pgroup=G)
>>> P.corners
(A, B, C)
>>> P.rotate(0) # apply permutation 0
>>> P.corners
(A, C, B)
>>> P.reset()
>>> P.corners
(A, B, C)
Or one can make a permutation as a product of selected permutations
and apply them to an iterable directly:
>>> P10 = G.make_perm([0, 1])
>>> P10('ABC')
['C', 'A', 'B']
See Also
========
sympy.combinatorics.polyhedron.Polyhedron,
sympy.combinatorics.permutations.Permutation
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
.. [2] Seress, A.
"Permutation Group Algorithms"
.. [3] https://en.wikipedia.org/wiki/Schreier_vector
.. [4] https://en.wikipedia.org/wiki/Nielsen_transformation#Product_replacement_algorithm
.. [5] Frank Celler, Charles R.Leedham-Green, Scott H.Murray,
Alice C.Niemeyer, and E.A.O'Brien. "Generating Random
Elements of a Finite Group"
.. [6] https://en.wikipedia.org/wiki/Block_%28permutation_group_theory%29
.. [7] http://www.algorithmist.com/index.php/Union_Find
.. [8] https://en.wikipedia.org/wiki/Multiply_transitive_group#Multiply_transitive_groups
.. [9] https://en.wikipedia.org/wiki/Center_%28group_theory%29
.. [10] https://en.wikipedia.org/wiki/Centralizer_and_normalizer
.. [11] http://groupprops.subwiki.org/wiki/Derived_subgroup
.. [12] https://en.wikipedia.org/wiki/Nilpotent_group
.. [13] http://www.math.colostate.edu/~hulpke/CGT/cgtnotes.pdf
.. [14] https://www.gap-system.org/Manuals/doc/ref/manual.pdf
"""
is_group = True
def __new__(cls, *args, dups=True, **kwargs):
"""The default constructor. Accepts Cycle and Permutation forms.
Removes duplicates unless ``dups`` keyword is ``False``.
"""
if not args:
args = [Permutation()]
else:
args = list(args[0] if is_sequence(args[0]) else args)
if not args:
args = [Permutation()]
if any(isinstance(a, Cycle) for a in args):
args = [Permutation(a) for a in args]
if has_variety(a.size for a in args):
degree = kwargs.pop('degree', None)
if degree is None:
degree = max(a.size for a in args)
for i in range(len(args)):
if args[i].size != degree:
args[i] = Permutation(args[i], size=degree)
if dups:
args = list(uniq([_af_new(list(a)) for a in args]))
if len(args) > 1:
args = [g for g in args if not g.is_identity]
return Basic.__new__(cls, *args, **kwargs)
def __init__(self, *args, **kwargs):
self._generators = list(self.args)
self._order = None
self._center = []
self._is_abelian = None
self._is_transitive = None
self._is_sym = None
self._is_alt = None
self._is_primitive = None
self._is_nilpotent = None
self._is_solvable = None
self._is_trivial = None
self._transitivity_degree = None
self._max_div = None
self._is_perfect = None
self._is_cyclic = None
self._r = len(self._generators)
self._degree = self._generators[0].size
# these attributes are assigned after running schreier_sims
self._base = []
self._strong_gens = []
self._strong_gens_slp = []
self._basic_orbits = []
self._transversals = []
self._transversal_slp = []
# these attributes are assigned after running _random_pr_init
self._random_gens = []
# finite presentation of the group as an instance of `FpGroup`
self._fp_presentation = None
def __getitem__(self, i):
return self._generators[i]
def __contains__(self, i):
"""Return ``True`` if *i* is contained in PermutationGroup.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = Permutation(1, 2, 3)
>>> Permutation(3) in PermutationGroup(p)
True
"""
if not isinstance(i, Permutation):
raise TypeError("A PermutationGroup contains only Permutations as "
"elements, not elements of type %s" % type(i))
return self.contains(i)
def __len__(self):
return len(self._generators)
def equals(self, other):
"""Return ``True`` if PermutationGroup generated by elements in the
group are same i.e they represent the same PermutationGroup.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> p = Permutation(0, 1, 2, 3, 4, 5)
>>> G = PermutationGroup([p, p**2])
>>> H = PermutationGroup([p**2, p])
>>> G.generators == H.generators
False
>>> G.equals(H)
True
"""
if not isinstance(other, PermutationGroup):
return False
set_self_gens = set(self.generators)
set_other_gens = set(other.generators)
# before reaching the general case there are also certain
# optimisation and obvious cases requiring less or no actual
# computation.
if set_self_gens == set_other_gens:
return True
# in the most general case it will check that each generator of
# one group belongs to the other PermutationGroup and vice-versa
for gen1 in set_self_gens:
if not other.contains(gen1):
return False
for gen2 in set_other_gens:
if not self.contains(gen2):
return False
return True
def __mul__(self, other):
"""
Return the direct product of two permutation groups as a permutation
group.
Explanation
===========
This implementation realizes the direct product by shifting the index
set for the generators of the second group: so if we have ``G`` acting
on ``n1`` points and ``H`` acting on ``n2`` points, ``G*H`` acts on
``n1 + n2`` points.
Examples
========
>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> G = CyclicGroup(5)
>>> H = G*G
>>> H
PermutationGroup([
(9)(0 1 2 3 4),
(5 6 7 8 9)])
>>> H.order()
25
"""
if isinstance(other, Permutation):
return Coset(other, self, dir='+')
gens1 = [perm._array_form for perm in self.generators]
gens2 = [perm._array_form for perm in other.generators]
n1 = self._degree
n2 = other._degree
start = list(range(n1))
end = list(range(n1, n1 + n2))
for i in range(len(gens2)):
gens2[i] = [x + n1 for x in gens2[i]]
gens2 = [start + gen for gen in gens2]
gens1 = [gen + end for gen in gens1]
together = gens1 + gens2
gens = [_af_new(x) for x in together]
return PermutationGroup(gens)
def _random_pr_init(self, r, n, _random_prec_n=None):
r"""Initialize random generators for the product replacement algorithm.
Explanation
===========
The implementation uses a modification of the original product
replacement algorithm due to Leedham-Green, as described in [1],
pp. 69-71; also, see [2], pp. 27-29 for a detailed theoretical
analysis of the original product replacement algorithm, and [4].
The product replacement algorithm is used for producing random,
uniformly distributed elements of a group `G` with a set of generators
`S`. For the initialization ``_random_pr_init``, a list ``R`` of
`\max\{r, |S|\}` group generators is created as the attribute
``G._random_gens``, repeating elements of `S` if necessary, and the
identity element of `G` is appended to ``R`` - we shall refer to this
last element as the accumulator. Then the function ``random_pr()``
is called ``n`` times, randomizing the list ``R`` while preserving
the generation of `G` by ``R``. The function ``random_pr()`` itself
takes two random elements ``g, h`` among all elements of ``R`` but
the accumulator and replaces ``g`` with a randomly chosen element
from `\{gh, g(~h), hg, (~h)g\}`. Then the accumulator is multiplied
by whatever ``g`` was replaced by. The new value of the accumulator is
then returned by ``random_pr()``.
The elements returned will eventually (for ``n`` large enough) become
uniformly distributed across `G` ([5]). For practical purposes however,
the values ``n = 50, r = 11`` are suggested in [1].
Notes
=====
THIS FUNCTION HAS SIDE EFFECTS: it changes the attribute
self._random_gens
See Also
========
random_pr
"""
deg = self.degree
random_gens = [x._array_form for x in self.generators]
k = len(random_gens)
if k < r:
for i in range(k, r):
random_gens.append(random_gens[i - k])
acc = list(range(deg))
random_gens.append(acc)
self._random_gens = random_gens
# handle randomized input for testing purposes
if _random_prec_n is None:
for i in range(n):
self.random_pr()
else:
for i in range(n):
self.random_pr(_random_prec=_random_prec_n[i])
def _union_find_merge(self, first, second, ranks, parents, not_rep):
"""Merges two classes in a union-find data structure.
Explanation
===========
Used in the implementation of Atkinson's algorithm as suggested in [1],
pp. 83-87. The class merging process uses union by rank as an
optimization. ([7])
Notes
=====
THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives,
``parents``, the list of class sizes, ``ranks``, and the list of
elements that are not representatives, ``not_rep``, are changed due to
class merging.
See Also
========
minimal_block, _union_find_rep
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
.. [7] http://www.algorithmist.com/index.php/Union_Find
"""
rep_first = self._union_find_rep(first, parents)
rep_second = self._union_find_rep(second, parents)
if rep_first != rep_second:
# union by rank
if ranks[rep_first] >= ranks[rep_second]:
new_1, new_2 = rep_first, rep_second
else:
new_1, new_2 = rep_second, rep_first
total_rank = ranks[new_1] + ranks[new_2]
if total_rank > self.max_div:
return -1
parents[new_2] = new_1
ranks[new_1] = total_rank
not_rep.append(new_2)
return 1
return 0
def _union_find_rep(self, num, parents):
"""Find representative of a class in a union-find data structure.
Explanation
===========
Used in the implementation of Atkinson's algorithm as suggested in [1],
pp. 83-87. After the representative of the class to which ``num``
belongs is found, path compression is performed as an optimization
([7]).
Notes
=====
THIS FUNCTION HAS SIDE EFFECTS: the list of class representatives,
``parents``, is altered due to path compression.
See Also
========
minimal_block, _union_find_merge
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
.. [7] http://www.algorithmist.com/index.php/Union_Find
"""
rep, parent = num, parents[num]
while parent != rep:
rep = parent
parent = parents[rep]
# path compression
temp, parent = num, parents[num]
while parent != rep:
parents[temp] = rep
temp = parent
parent = parents[temp]
return rep
@property
def base(self):
r"""Return a base from the Schreier-Sims algorithm.
Explanation
===========
For a permutation group `G`, a base is a sequence of points
`B = (b_1, b_2, \dots, b_k)` such that no element of `G` apart
from the identity fixes all the points in `B`. The concepts of
a base and strong generating set and their applications are
discussed in depth in [1], pp. 87-89 and [2], pp. 55-57.
An alternative way to think of `B` is that it gives the
indices of the stabilizer cosets that contain more than the
identity permutation.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> G = PermutationGroup([Permutation(0, 1, 3)(2, 4)])
>>> G.base
[0, 2]
See Also
========
strong_gens, basic_transversals, basic_orbits, basic_stabilizers
"""
if self._base == []:
self.schreier_sims()
return self._base
def baseswap(self, base, strong_gens, pos, randomized=False,
transversals=None, basic_orbits=None, strong_gens_distr=None):
r"""Swap two consecutive base points in base and strong generating set.
Explanation
===========
If a base for a group `G` is given by `(b_1, b_2, \dots, b_k)`, this
function returns a base `(b_1, b_2, \dots, b_{i+1}, b_i, \dots, b_k)`,
where `i` is given by ``pos``, and a strong generating set relative
to that base. The original base and strong generating set are not
modified.
The randomized version (default) is of Las Vegas type.
Parameters
==========
base, strong_gens
The base and strong generating set.
pos
The position at which swapping is performed.
randomized
A switch between randomized and deterministic version.
transversals
The transversals for the basic orbits, if known.
basic_orbits
The basic orbits, if known.
strong_gens_distr
The strong generators distributed by basic stabilizers, if known.
Returns
=======
(base, strong_gens)
``base`` is the new base, and ``strong_gens`` is a generating set
relative to it.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> S = SymmetricGroup(4)
>>> S.schreier_sims()
>>> S.base
[0, 1, 2]
>>> base, gens = S.baseswap(S.base, S.strong_gens, 1, randomized=False)
>>> base, gens
([0, 2, 1],
[(0 1 2 3), (3)(0 1), (1 3 2),
(2 3), (1 3)])
check that base, gens is a BSGS
>>> S1 = PermutationGroup(gens)
>>> _verify_bsgs(S1, base, gens)
True
See Also
========
schreier_sims
Notes
=====
The deterministic version of the algorithm is discussed in
[1], pp. 102-103; the randomized version is discussed in [1], p.103, and
[2], p.98. It is of Las Vegas type.
Notice that [1] contains a mistake in the pseudocode and
discussion of BASESWAP: on line 3 of the pseudocode,
`|\beta_{i+1}^{\left\langle T\right\rangle}|` should be replaced by
`|\beta_{i}^{\left\langle T\right\rangle}|`, and the same for the
discussion of the algorithm.
"""
# construct the basic orbits, generators for the stabilizer chain
# and transversal elements from whatever was provided
transversals, basic_orbits, strong_gens_distr = \
_handle_precomputed_bsgs(base, strong_gens, transversals,
basic_orbits, strong_gens_distr)
base_len = len(base)
degree = self.degree
# size of orbit of base[pos] under the stabilizer we seek to insert
# in the stabilizer chain at position pos + 1
size = len(basic_orbits[pos])*len(basic_orbits[pos + 1]) \
//len(_orbit(degree, strong_gens_distr[pos], base[pos + 1]))
# initialize the wanted stabilizer by a subgroup
if pos + 2 > base_len - 1:
T = []
else:
T = strong_gens_distr[pos + 2][:]
# randomized version
if randomized is True:
stab_pos = PermutationGroup(strong_gens_distr[pos])
schreier_vector = stab_pos.schreier_vector(base[pos + 1])
# add random elements of the stabilizer until they generate it
while len(_orbit(degree, T, base[pos])) != size:
new = stab_pos.random_stab(base[pos + 1],
schreier_vector=schreier_vector)
T.append(new)
# deterministic version
else:
Gamma = set(basic_orbits[pos])
Gamma.remove(base[pos])
if base[pos + 1] in Gamma:
Gamma.remove(base[pos + 1])
# add elements of the stabilizer until they generate it by
# ruling out member of the basic orbit of base[pos] along the way
while len(_orbit(degree, T, base[pos])) != size:
gamma = next(iter(Gamma))
x = transversals[pos][gamma]
temp = x._array_form.index(base[pos + 1]) # (~x)(base[pos + 1])
if temp not in basic_orbits[pos + 1]:
Gamma = Gamma - _orbit(degree, T, gamma)
else:
y = transversals[pos + 1][temp]
el = rmul(x, y)
if el(base[pos]) not in _orbit(degree, T, base[pos]):
T.append(el)
Gamma = Gamma - _orbit(degree, T, base[pos])
# build the new base and strong generating set
strong_gens_new_distr = strong_gens_distr[:]
strong_gens_new_distr[pos + 1] = T
base_new = base[:]
base_new[pos], base_new[pos + 1] = base_new[pos + 1], base_new[pos]
strong_gens_new = _strong_gens_from_distr(strong_gens_new_distr)
for gen in T:
if gen not in strong_gens_new:
strong_gens_new.append(gen)
return base_new, strong_gens_new
@property
def basic_orbits(self):
r"""
Return the basic orbits relative to a base and strong generating set.
Explanation
===========
If `(b_1, b_2, \dots, b_k)` is a base for a group `G`, and
`G^{(i)} = G_{b_1, b_2, \dots, b_{i-1}}` is the ``i``-th basic stabilizer
(so that `G^{(1)} = G`), the ``i``-th basic orbit relative to this base
is the orbit of `b_i` under `G^{(i)}`. See [1], pp. 87-89 for more
information.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(4)
>>> S.basic_orbits
[[0, 1, 2, 3], [1, 2, 3], [2, 3]]
See Also
========
base, strong_gens, basic_transversals, basic_stabilizers
"""
if self._basic_orbits == []:
self.schreier_sims()
return self._basic_orbits
@property
def basic_stabilizers(self):
r"""
Return a chain of stabilizers relative to a base and strong generating
set.
Explanation
===========
The ``i``-th basic stabilizer `G^{(i)}` relative to a base
`(b_1, b_2, \dots, b_k)` is `G_{b_1, b_2, \dots, b_{i-1}}`. For more
information, see [1], pp. 87-89.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> A = AlternatingGroup(4)
>>> A.schreier_sims()
>>> A.base
[0, 1]
>>> for g in A.basic_stabilizers:
... print(g)
...
PermutationGroup([
(3)(0 1 2),
(1 2 3)])
PermutationGroup([
(1 2 3)])
See Also
========
base, strong_gens, basic_orbits, basic_transversals
"""
if self._transversals == []:
self.schreier_sims()
strong_gens = self._strong_gens
base = self._base
if not base: # e.g. if self is trivial
return []
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_stabilizers = []
for gens in strong_gens_distr:
basic_stabilizers.append(PermutationGroup(gens))
return basic_stabilizers
@property
def basic_transversals(self):
"""
Return basic transversals relative to a base and strong generating set.
Explanation
===========
The basic transversals are transversals of the basic orbits. They
are provided as a list of dictionaries, each dictionary having
keys - the elements of one of the basic orbits, and values - the
corresponding transversal elements. See [1], pp. 87-89 for more
information.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> A = AlternatingGroup(4)
>>> A.basic_transversals
[{0: (3), 1: (3)(0 1 2), 2: (3)(0 2 1), 3: (0 3 1)}, {1: (3), 2: (1 2 3), 3: (1 3 2)}]
See Also
========
strong_gens, base, basic_orbits, basic_stabilizers
"""
if self._transversals == []:
self.schreier_sims()
return self._transversals
def composition_series(self):
r"""
Return the composition series for a group as a list
of permutation groups.
Explanation
===========
The composition series for a group `G` is defined as a
subnormal series `G = H_0 > H_1 > H_2 \ldots` A composition
series is a subnormal series such that each factor group
`H(i+1) / H(i)` is simple.
A subnormal series is a composition series only if it is of
maximum length.
The algorithm works as follows:
Starting with the derived series the idea is to fill
the gap between `G = der[i]` and `H = der[i+1]` for each
`i` independently. Since, all subgroups of the abelian group
`G/H` are normal so, first step is to take the generators
`g` of `G` and add them to generators of `H` one by one.
The factor groups formed are not simple in general. Each
group is obtained from the previous one by adding one
generator `g`, if the previous group is denoted by `H`
then the next group `K` is generated by `g` and `H`.
The factor group `K/H` is cyclic and it's order is
`K.order()//G.order()`. The series is then extended between
`K` and `H` by groups generated by powers of `g` and `H`.
The series formed is then prepended to the already existing
series.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> S = SymmetricGroup(12)
>>> G = S.sylow_subgroup(2)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1]
>>> G = S.sylow_subgroup(3)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[243, 81, 27, 9, 3, 1]
>>> G = CyclicGroup(12)
>>> C = G.composition_series()
>>> [H.order() for H in C]
[12, 6, 3, 1]
"""
der = self.derived_series()
if not all(g.is_identity for g in der[-1].generators):
raise NotImplementedError('Group should be solvable')
series = []
for i in range(len(der)-1):
H = der[i+1]
up_seg = []
for g in der[i].generators:
K = PermutationGroup([g] + H.generators)
order = K.order() // H.order()
down_seg = []
for p, e in factorint(order).items():
for _ in range(e):
down_seg.append(PermutationGroup([g] + H.generators))
g = g**p
up_seg = down_seg + up_seg
H = K
up_seg[0] = der[i]
series.extend(up_seg)
series.append(der[-1])
return series
def coset_transversal(self, H):
"""Return a transversal of the right cosets of self by its subgroup H
using the second method described in [1], Subsection 4.6.7
"""
if not H.is_subgroup(self):
raise ValueError("The argument must be a subgroup")
if H.order() == 1:
return self._elements
self._schreier_sims(base=H.base) # make G.base an extension of H.base
base = self.base
base_ordering = _base_ordering(base, self.degree)
identity = Permutation(self.degree - 1)
transversals = self.basic_transversals[:]
# transversals is a list of dictionaries. Get rid of the keys
# so that it is a list of lists and sort each list in
# the increasing order of base[l]^x
for l, t in enumerate(transversals):
transversals[l] = sorted(t.values(),
key = lambda x: base_ordering[base[l]^x])
orbits = H.basic_orbits
h_stabs = H.basic_stabilizers
g_stabs = self.basic_stabilizers
indices = [x.order()//y.order() for x, y in zip(g_stabs, h_stabs)]
# T^(l) should be a right transversal of H^(l) in G^(l) for
# 1<=l<=len(base). While H^(l) is the trivial group, T^(l)
# contains all the elements of G^(l) so we might just as well
# start with l = len(h_stabs)-1
if len(g_stabs) > len(h_stabs):
T = g_stabs[len(h_stabs)]._elements
else:
T = [identity]
l = len(h_stabs)-1
t_len = len(T)
while l > -1:
T_next = []
for u in transversals[l]:
if u == identity:
continue
b = base_ordering[base[l]^u]
for t in T:
p = t*u
if all(base_ordering[h^p] >= b for h in orbits[l]):
T_next.append(p)
if t_len + len(T_next) == indices[l]:
break
if t_len + len(T_next) == indices[l]:
break
T += T_next
t_len += len(T_next)
l -= 1
T.remove(identity)
T = [identity] + T
return T
def _coset_representative(self, g, H):
"""Return the representative of Hg from the transversal that
would be computed by ``self.coset_transversal(H)``.
"""
if H.order() == 1:
return g
# The base of self must be an extension of H.base.
if not(self.base[:len(H.base)] == H.base):
self._schreier_sims(base=H.base)
orbits = H.basic_orbits[:]
h_transversals = [list(_.values()) for _ in H.basic_transversals]
transversals = [list(_.values()) for _ in self.basic_transversals]
base = self.base
base_ordering = _base_ordering(base, self.degree)
def step(l, x):
gamma = sorted(orbits[l], key = lambda y: base_ordering[y^x])[0]
i = [base[l]^h for h in h_transversals[l]].index(gamma)
x = h_transversals[l][i]*x
if l < len(orbits)-1:
for u in transversals[l]:
if base[l]^u == base[l]^x:
break
x = step(l+1, x*u**-1)*u
return x
return step(0, g)
def coset_table(self, H):
"""Return the standardised (right) coset table of self in H as
a list of lists.
"""
# Maybe this should be made to return an instance of CosetTable
# from fp_groups.py but the class would need to be changed first
# to be compatible with PermutationGroups
from itertools import chain, product
if not H.is_subgroup(self):
raise ValueError("The argument must be a subgroup")
T = self.coset_transversal(H)
n = len(T)
A = list(chain.from_iterable((gen, gen**-1)
for gen in self.generators))
table = []
for i in range(n):
row = [self._coset_representative(T[i]*x, H) for x in A]
row = [T.index(r) for r in row]
table.append(row)
# standardize (this is the same as the algorithm used in coset_table)
# If CosetTable is made compatible with PermutationGroups, this
# should be replaced by table.standardize()
A = range(len(A))
gamma = 1
for alpha, a in product(range(n), A):
beta = table[alpha][a]
if beta >= gamma:
if beta > gamma:
for x in A:
z = table[gamma][x]
table[gamma][x] = table[beta][x]
table[beta][x] = z
for i in range(n):
if table[i][x] == beta:
table[i][x] = gamma
elif table[i][x] == gamma:
table[i][x] = beta
gamma += 1
if gamma >= n-1:
return table
def center(self):
r"""
Return the center of a permutation group.
Explanation
===========
The center for a group `G` is defined as
`Z(G) = \{z\in G | \forall g\in G, zg = gz \}`,
the set of elements of `G` that commute with all elements of `G`.
It is equal to the centralizer of `G` inside `G`, and is naturally a
subgroup of `G` ([9]).
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(4)
>>> G = D.center()
>>> G.order()
2
See Also
========
centralizer
Notes
=====
This is a naive implementation that is a straightforward application
of ``.centralizer()``
"""
return self.centralizer(self)
def centralizer(self, other):
r"""
Return the centralizer of a group/set/element.
Explanation
===========
The centralizer of a set of permutations ``S`` inside
a group ``G`` is the set of elements of ``G`` that commute with all
elements of ``S``::
`C_G(S) = \{ g \in G | gs = sg \forall s \in S\}` ([10])
Usually, ``S`` is a subset of ``G``, but if ``G`` is a proper subgroup of
the full symmetric group, we allow for ``S`` to have elements outside
``G``.
It is naturally a subgroup of ``G``; the centralizer of a permutation
group is equal to the centralizer of any set of generators for that
group, since any element commuting with the generators commutes with
any product of the generators.
Parameters
==========
other
a permutation group/list of permutations/single permutation
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup)
>>> S = SymmetricGroup(6)
>>> C = CyclicGroup(6)
>>> H = S.centralizer(C)
>>> H.is_subgroup(C)
True
See Also
========
subgroup_search
Notes
=====
The implementation is an application of ``.subgroup_search()`` with
tests using a specific base for the group ``G``.
"""
if hasattr(other, 'generators'):
if other.is_trivial or self.is_trivial:
return self
degree = self.degree
identity = _af_new(list(range(degree)))
orbits = other.orbits()
num_orbits = len(orbits)
orbits.sort(key=lambda x: -len(x))
long_base = []
orbit_reps = [None]*num_orbits
orbit_reps_indices = [None]*num_orbits
orbit_descr = [None]*degree
for i in range(num_orbits):
orbit = list(orbits[i])
orbit_reps[i] = orbit[0]
orbit_reps_indices[i] = len(long_base)
for point in orbit:
orbit_descr[point] = i
long_base = long_base + orbit
base, strong_gens = self.schreier_sims_incremental(base=long_base)
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
i = 0
for i in range(len(base)):
if strong_gens_distr[i] == [identity]:
break
base = base[:i]
base_len = i
for j in range(num_orbits):
if base[base_len - 1] in orbits[j]:
break
rel_orbits = orbits[: j + 1]
num_rel_orbits = len(rel_orbits)
transversals = [None]*num_rel_orbits
for j in range(num_rel_orbits):
rep = orbit_reps[j]
transversals[j] = dict(
other.orbit_transversal(rep, pairs=True))
trivial_test = lambda x: True
tests = [None]*base_len
for l in range(base_len):
if base[l] in orbit_reps:
tests[l] = trivial_test
else:
def test(computed_words, l=l):
g = computed_words[l]
rep_orb_index = orbit_descr[base[l]]
rep = orbit_reps[rep_orb_index]
im = g._array_form[base[l]]
im_rep = g._array_form[rep]
tr_el = transversals[rep_orb_index][base[l]]
# using the definition of transversal,
# base[l]^g = rep^(tr_el*g);
# if g belongs to the centralizer, then
# base[l]^g = (rep^g)^tr_el
return im == tr_el._array_form[im_rep]
tests[l] = test
def prop(g):
return [rmul(g, gen) for gen in other.generators] == \
[rmul(gen, g) for gen in other.generators]
return self.subgroup_search(prop, base=base,
strong_gens=strong_gens, tests=tests)
elif hasattr(other, '__getitem__'):
gens = list(other)
return self.centralizer(PermutationGroup(gens))
elif hasattr(other, 'array_form'):
return self.centralizer(PermutationGroup([other]))
def commutator(self, G, H):
"""
Return the commutator of two subgroups.
Explanation
===========
For a permutation group ``K`` and subgroups ``G``, ``H``, the
commutator of ``G`` and ``H`` is defined as the group generated
by all the commutators `[g, h] = hgh^{-1}g^{-1}` for ``g`` in ``G`` and
``h`` in ``H``. It is naturally a subgroup of ``K`` ([1], p.27).
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> S = SymmetricGroup(5)
>>> A = AlternatingGroup(5)
>>> G = S.commutator(S, A)
>>> G.is_subgroup(A)
True
See Also
========
derived_subgroup
Notes
=====
The commutator of two subgroups `H, G` is equal to the normal closure
of the commutators of all the generators, i.e. `hgh^{-1}g^{-1}` for `h`
a generator of `H` and `g` a generator of `G` ([1], p.28)
"""
ggens = G.generators
hgens = H.generators
commutators = []
for ggen in ggens:
for hgen in hgens:
commutator = rmul(hgen, ggen, ~hgen, ~ggen)
if commutator not in commutators:
commutators.append(commutator)
res = self.normal_closure(commutators)
return res
def coset_factor(self, g, factor_index=False):
"""Return ``G``'s (self's) coset factorization of ``g``
Explanation
===========
If ``g`` is an element of ``G`` then it can be written as the product
of permutations drawn from the Schreier-Sims coset decomposition,
The permutations returned in ``f`` are those for which
the product gives ``g``: ``g = f[n]*...f[1]*f[0]`` where ``n = len(B)``
and ``B = G.base``. f[i] is one of the permutations in
``self._basic_orbits[i]``.
If factor_index==True,
returns a tuple ``[b[0],..,b[n]]``, where ``b[i]``
belongs to ``self._basic_orbits[i]``
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation(0, 1, 3, 7, 6, 4)(2, 5)
>>> b = Permutation(0, 1, 3, 2)(4, 5, 7, 6)
>>> G = PermutationGroup([a, b])
Define g:
>>> g = Permutation(7)(1, 2, 4)(3, 6, 5)
Confirm that it is an element of G:
>>> G.contains(g)
True
Thus, it can be written as a product of factors (up to
3) drawn from u. See below that a factor from u1 and u2
and the Identity permutation have been used:
>>> f = G.coset_factor(g)
>>> f[2]*f[1]*f[0] == g
True
>>> f1 = G.coset_factor(g, True); f1
[0, 4, 4]
>>> tr = G.basic_transversals
>>> f[0] == tr[0][f1[0]]
True
If g is not an element of G then [] is returned:
>>> c = Permutation(5, 6, 7)
>>> G.coset_factor(c)
[]
See Also
========
sympy.combinatorics.util._strip
"""
if isinstance(g, (Cycle, Permutation)):
g = g.list()
if len(g) != self._degree:
# this could either adjust the size or return [] immediately
# but we don't choose between the two and just signal a possible
# error
raise ValueError('g should be the same size as permutations of G')
I = list(range(self._degree))
basic_orbits = self.basic_orbits
transversals = self._transversals
factors = []
base = self.base
h = g
for i in range(len(base)):
beta = h[base[i]]
if beta == base[i]:
factors.append(beta)
continue
if beta not in basic_orbits[i]:
return []
u = transversals[i][beta]._array_form
h = _af_rmul(_af_invert(u), h)
factors.append(beta)
if h != I:
return []
if factor_index:
return factors
tr = self.basic_transversals
factors = [tr[i][factors[i]] for i in range(len(base))]
return factors
def generator_product(self, g, original=False):
r'''
Return a list of strong generators `[s1, \dots, sn]`
s.t `g = sn \times \dots \times s1`. If ``original=True``, make the
list contain only the original group generators
'''
product = []
if g.is_identity:
return []
if g in self.strong_gens:
if not original or g in self.generators:
return [g]
else:
slp = self._strong_gens_slp[g]
for s in slp:
product.extend(self.generator_product(s, original=True))
return product
elif g**-1 in self.strong_gens:
g = g**-1
if not original or g in self.generators:
return [g**-1]
else:
slp = self._strong_gens_slp[g]
for s in slp:
product.extend(self.generator_product(s, original=True))
l = len(product)
product = [product[l-i-1]**-1 for i in range(l)]
return product
f = self.coset_factor(g, True)
for i, j in enumerate(f):
slp = self._transversal_slp[i][j]
for s in slp:
if not original:
product.append(self.strong_gens[s])
else:
s = self.strong_gens[s]
product.extend(self.generator_product(s, original=True))
return product
def coset_rank(self, g):
"""rank using Schreier-Sims representation.
Explanation
===========
The coset rank of ``g`` is the ordering number in which
it appears in the lexicographic listing according to the
coset decomposition
The ordering is the same as in G.generate(method='coset').
If ``g`` does not belong to the group it returns None.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation(0, 1, 3, 7, 6, 4)(2, 5)
>>> b = Permutation(0, 1, 3, 2)(4, 5, 7, 6)
>>> G = PermutationGroup([a, b])
>>> c = Permutation(7)(2, 4)(3, 5)
>>> G.coset_rank(c)
16
>>> G.coset_unrank(16)
(7)(2 4)(3 5)
See Also
========
coset_factor
"""
factors = self.coset_factor(g, True)
if not factors:
return None
rank = 0
b = 1
transversals = self._transversals
base = self._base
basic_orbits = self._basic_orbits
for i in range(len(base)):
k = factors[i]
j = basic_orbits[i].index(k)
rank += b*j
b = b*len(transversals[i])
return rank
def coset_unrank(self, rank, af=False):
"""unrank using Schreier-Sims representation
coset_unrank is the inverse operation of coset_rank
if 0 <= rank < order; otherwise it returns None.
"""
if rank < 0 or rank >= self.order():
return None
base = self.base
transversals = self.basic_transversals
basic_orbits = self.basic_orbits
m = len(base)
v = [0]*m
for i in range(m):
rank, c = divmod(rank, len(transversals[i]))
v[i] = basic_orbits[i][c]
a = [transversals[i][v[i]]._array_form for i in range(m)]
h = _af_rmuln(*a)
if af:
return h
else:
return _af_new(h)
@property
def degree(self):
"""Returns the size of the permutations in the group.
Explanation
===========
The number of permutations comprising the group is given by
``len(group)``; the number of permutations that can be generated
by the group is given by ``group.order()``.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 0, 2])
>>> G = PermutationGroup([a])
>>> G.degree
3
>>> len(G)
1
>>> G.order()
2
>>> list(G.generate())
[(2), (2)(0 1)]
See Also
========
order
"""
return self._degree
@property
def identity(self):
'''
Return the identity element of the permutation group.
'''
return _af_new(list(range(self.degree)))
@property
def elements(self):
"""Returns all the elements of the permutation group as a set
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2))
>>> p.elements
{(1 2 3), (1 3 2), (1 3), (2 3), (3), (3)(1 2)}
"""
return set(self._elements)
@property
def _elements(self):
"""Returns all the elements of the permutation group as a list
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> p = PermutationGroup(Permutation(1, 3), Permutation(1, 2))
>>> p._elements
[(3), (3)(1 2), (1 3), (2 3), (1 2 3), (1 3 2)]
"""
return list(islice(self.generate(), None))
def derived_series(self):
r"""Return the derived series for the group.
Explanation
===========
The derived series for a group `G` is defined as
`G = G_0 > G_1 > G_2 > \ldots` where `G_i = [G_{i-1}, G_{i-1}]`,
i.e. `G_i` is the derived subgroup of `G_{i-1}`, for
`i\in\mathbb{N}`. When we have `G_k = G_{k-1}` for some
`k\in\mathbb{N}`, the series terminates.
Returns
=======
A list of permutation groups containing the members of the derived
series in the order `G = G_0, G_1, G_2, \ldots`.
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup, DihedralGroup)
>>> A = AlternatingGroup(5)
>>> len(A.derived_series())
1
>>> S = SymmetricGroup(4)
>>> len(S.derived_series())
4
>>> S.derived_series()[1].is_subgroup(AlternatingGroup(4))
True
>>> S.derived_series()[2].is_subgroup(DihedralGroup(2))
True
See Also
========
derived_subgroup
"""
res = [self]
current = self
nxt = self.derived_subgroup()
while not current.is_subgroup(nxt):
res.append(nxt)
current = nxt
nxt = nxt.derived_subgroup()
return res
def derived_subgroup(self):
r"""Compute the derived subgroup.
Explanation
===========
The derived subgroup, or commutator subgroup is the subgroup generated
by all commutators `[g, h] = hgh^{-1}g^{-1}` for `g, h\in G` ; it is
equal to the normal closure of the set of commutators of the generators
([1], p.28, [11]).
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 0, 2, 4, 3])
>>> b = Permutation([0, 1, 3, 2, 4])
>>> G = PermutationGroup([a, b])
>>> C = G.derived_subgroup()
>>> list(C.generate(af=True))
[[0, 1, 2, 3, 4], [0, 1, 3, 4, 2], [0, 1, 4, 2, 3]]
See Also
========
derived_series
"""
r = self._r
gens = [p._array_form for p in self.generators]
set_commutators = set()
degree = self._degree
rng = list(range(degree))
for i in range(r):
for j in range(r):
p1 = gens[i]
p2 = gens[j]
c = list(range(degree))
for k in rng:
c[p2[p1[k]]] = p1[p2[k]]
ct = tuple(c)
if ct not in set_commutators:
set_commutators.add(ct)
cms = [_af_new(p) for p in set_commutators]
G2 = self.normal_closure(cms)
return G2
def generate(self, method="coset", af=False):
"""Return iterator to generate the elements of the group.
Explanation
===========
Iteration is done with one of these methods::
method='coset' using the Schreier-Sims coset representation
method='dimino' using the Dimino method
If ``af = True`` it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics import PermutationGroup
>>> from sympy.combinatorics.polyhedron import tetrahedron
The permutation group given in the tetrahedron object is also
true groups:
>>> G = tetrahedron.pgroup
>>> G.is_group
True
Also the group generated by the permutations in the tetrahedron
pgroup -- even the first two -- is a proper group:
>>> H = PermutationGroup(G[0], G[1])
>>> J = PermutationGroup(list(H.generate())); J
PermutationGroup([
(0 1)(2 3),
(1 2 3),
(1 3 2),
(0 3 1),
(0 2 3),
(0 3)(1 2),
(0 1 3),
(3)(0 2 1),
(0 3 2),
(3)(0 1 2),
(0 2)(1 3)])
>>> _.is_group
True
"""
if method == "coset":
return self.generate_schreier_sims(af)
elif method == "dimino":
return self.generate_dimino(af)
else:
raise NotImplementedError('No generation defined for %s' % method)
def generate_dimino(self, af=False):
"""Yield group elements using Dimino's algorithm.
If ``af == True`` it yields the array form of the permutations.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_dimino(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 2, 3, 1],
[0, 1, 3, 2], [0, 3, 2, 1], [0, 3, 1, 2]]
References
==========
.. [1] The Implementation of Various Algorithms for Permutation Groups in
the Computer Algebra System: AXIOM, N.J. Doye, M.Sc. Thesis
"""
idn = list(range(self.degree))
order = 0
element_list = [idn]
set_element_list = {tuple(idn)}
if af:
yield idn
else:
yield _af_new(idn)
gens = [p._array_form for p in self.generators]
for i in range(len(gens)):
# D elements of the subgroup G_i generated by gens[:i]
D = element_list[:]
N = [idn]
while N:
A = N
N = []
for a in A:
for g in gens[:i + 1]:
ag = _af_rmul(a, g)
if tuple(ag) not in set_element_list:
# produce G_i*g
for d in D:
order += 1
ap = _af_rmul(d, ag)
if af:
yield ap
else:
p = _af_new(ap)
yield p
element_list.append(ap)
set_element_list.add(tuple(ap))
N.append(ap)
self._order = len(element_list)
def generate_schreier_sims(self, af=False):
"""Yield group elements using the Schreier-Sims representation
in coset_rank order
If ``af = True`` it yields the array form of the permutations
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([0, 2, 3, 1])
>>> g = PermutationGroup([a, b])
>>> list(g.generate_schreier_sims(af=True))
[[0, 1, 2, 3], [0, 2, 1, 3], [0, 3, 2, 1],
[0, 1, 3, 2], [0, 2, 3, 1], [0, 3, 1, 2]]
"""
n = self._degree
u = self.basic_transversals
basic_orbits = self._basic_orbits
if len(u) == 0:
for x in self.generators:
if af:
yield x._array_form
else:
yield x
return
if len(u) == 1:
for i in basic_orbits[0]:
if af:
yield u[0][i]._array_form
else:
yield u[0][i]
return
u = list(reversed(u))
basic_orbits = basic_orbits[::-1]
# stg stack of group elements
stg = [list(range(n))]
posmax = [len(x) for x in u]
n1 = len(posmax) - 1
pos = [0]*n1
h = 0
while 1:
# backtrack when finished iterating over coset
if pos[h] >= posmax[h]:
if h == 0:
return
pos[h] = 0
h -= 1
stg.pop()
continue
p = _af_rmul(u[h][basic_orbits[h][pos[h]]]._array_form, stg[-1])
pos[h] += 1
stg.append(p)
h += 1
if h == n1:
if af:
for i in basic_orbits[-1]:
p = _af_rmul(u[-1][i]._array_form, stg[-1])
yield p
else:
for i in basic_orbits[-1]:
p = _af_rmul(u[-1][i]._array_form, stg[-1])
p1 = _af_new(p)
yield p1
stg.pop()
h -= 1
@property
def generators(self):
"""Returns the generators of the group.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.generators
[(1 2), (2)(0 1)]
"""
return self._generators
def contains(self, g, strict=True):
"""Test if permutation ``g`` belong to self, ``G``.
Explanation
===========
If ``g`` is an element of ``G`` it can be written as a product
of factors drawn from the cosets of ``G``'s stabilizers. To see
if ``g`` is one of the actual generators defining the group use
``G.has(g)``.
If ``strict`` is not ``True``, ``g`` will be resized, if necessary,
to match the size of permutations in ``self``.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation(1, 2)
>>> b = Permutation(2, 3, 1)
>>> G = PermutationGroup(a, b, degree=5)
>>> G.contains(G[0]) # trivial check
True
>>> elem = Permutation([[2, 3]], size=5)
>>> G.contains(elem)
True
>>> G.contains(Permutation(4)(0, 1, 2, 3))
False
If strict is False, a permutation will be resized, if
necessary:
>>> H = PermutationGroup(Permutation(5))
>>> H.contains(Permutation(3))
False
>>> H.contains(Permutation(3), strict=False)
True
To test if a given permutation is present in the group:
>>> elem in G.generators
False
>>> G.has(elem)
False
See Also
========
coset_factor, sympy.core.basic.Basic.has, __contains__
"""
if not isinstance(g, Permutation):
return False
if g.size != self.degree:
if strict:
return False
g = Permutation(g, size=self.degree)
if g in self.generators:
return True
return bool(self.coset_factor(g.array_form, True))
@property
def is_perfect(self):
"""Return ``True`` if the group is perfect.
A group is perfect if it equals to its derived subgroup.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation(1,2,3)(4,5)
>>> b = Permutation(1,2,3,4,5)
>>> G = PermutationGroup([a, b])
>>> G.is_perfect
False
"""
if self._is_perfect is None:
self._is_perfect = self.equals(self.derived_subgroup())
return self._is_perfect
@property
def is_abelian(self):
"""Test if the group is Abelian.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.is_abelian
False
>>> a = Permutation([0, 2, 1])
>>> G = PermutationGroup([a])
>>> G.is_abelian
True
"""
if self._is_abelian is not None:
return self._is_abelian
self._is_abelian = True
gens = [p._array_form for p in self.generators]
for x in gens:
for y in gens:
if y <= x:
continue
if not _af_commutes_with(x, y):
self._is_abelian = False
return False
return True
def abelian_invariants(self):
"""
Returns the abelian invariants for the given group.
Let ``G`` be a nontrivial finite abelian group. Then G is isomorphic to
the direct product of finitely many nontrivial cyclic groups of
prime-power order.
Explanation
===========
The prime-powers that occur as the orders of the factors are uniquely
determined by G. More precisely, the primes that occur in the orders of the
factors in any such decomposition of ``G`` are exactly the primes that divide
``|G|`` and for any such prime ``p``, if the orders of the factors that are
p-groups in one such decomposition of ``G`` are ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``,
then the orders of the factors that are p-groups in any such decomposition of ``G``
are ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``.
The uniquely determined integers ``p^{t_1} >= p^{t_2} >= ... p^{t_r}``, taken
for all primes that divide ``|G|`` are called the invariants of the nontrivial
group ``G`` as suggested in ([14], p. 542).
Notes
=====
We adopt the convention that the invariants of a trivial group are [].
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.abelian_invariants()
[2]
>>> from sympy.combinatorics.named_groups import CyclicGroup
>>> G = CyclicGroup(7)
>>> G.abelian_invariants()
[7]
"""
if self.is_trivial:
return []
gns = self.generators
inv = []
G = self
H = G.derived_subgroup()
Hgens = H.generators
for p in primefactors(G.order()):
ranks = []
while True:
pows = []
for g in gns:
elm = g**p
if not H.contains(elm):
pows.append(elm)
K = PermutationGroup(Hgens + pows) if pows else H
r = G.order()//K.order()
G = K
gns = pows
if r == 1:
break
ranks.append(multiplicity(p, r))
if ranks:
pows = [1]*ranks[0]
for i in ranks:
for j in range(0, i):
pows[j] = pows[j]*p
inv.extend(pows)
inv.sort()
return inv
def is_elementary(self, p):
"""Return ``True`` if the group is elementary abelian. An elementary
abelian group is a finite abelian group, where every nontrivial
element has order `p`, where `p` is a prime.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> G = PermutationGroup([a])
>>> G.is_elementary(2)
True
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([3, 1, 2, 0])
>>> G = PermutationGroup([a, b])
>>> G.is_elementary(2)
True
>>> G.is_elementary(3)
False
"""
return self.is_abelian and all(g.order() == p for g in self.generators)
def _eval_is_alt_sym_naive(self, only_sym=False, only_alt=False):
"""A naive test using the group order."""
if only_sym and only_alt:
raise ValueError(
"Both {} and {} cannot be set to True"
.format(only_sym, only_alt))
n = self.degree
sym_order = 1
for i in range(2, n+1):
sym_order *= i
order = self.order()
if order == sym_order:
self._is_sym = True
self._is_alt = False
if only_alt:
return False
return True
elif 2*order == sym_order:
self._is_sym = False
self._is_alt = True
if only_sym:
return False
return True
return False
def _eval_is_alt_sym_monte_carlo(self, eps=0.05, perms=None):
"""A test using monte-carlo algorithm.
Parameters
==========
eps : float, optional
The criterion for the incorrect ``False`` return.
perms : list[Permutation], optional
If explicitly given, it tests over the given candidats
for testing.
If ``None``, it randomly computes ``N_eps`` and chooses
``N_eps`` sample of the permutation from the group.
See Also
========
_check_cycles_alt_sym
"""
if perms is None:
n = self.degree
if n < 17:
c_n = 0.34
else:
c_n = 0.57
d_n = (c_n*log(2))/log(n)
N_eps = int(-log(eps)/d_n)
perms = (self.random_pr() for i in range(N_eps))
return self._eval_is_alt_sym_monte_carlo(perms=perms)
for perm in perms:
if _check_cycles_alt_sym(perm):
return True
return False
def is_alt_sym(self, eps=0.05, _random_prec=None):
r"""Monte Carlo test for the symmetric/alternating group for degrees
>= 8.
Explanation
===========
More specifically, it is one-sided Monte Carlo with the
answer True (i.e., G is symmetric/alternating) guaranteed to be
correct, and the answer False being incorrect with probability eps.
For degree < 8, the order of the group is checked so the test
is deterministic.
Notes
=====
The algorithm itself uses some nontrivial results from group theory and
number theory:
1) If a transitive group ``G`` of degree ``n`` contains an element
with a cycle of length ``n/2 < p < n-2`` for ``p`` a prime, ``G`` is the
symmetric or alternating group ([1], pp. 81-82)
2) The proportion of elements in the symmetric/alternating group having
the property described in 1) is approximately `\log(2)/\log(n)`
([1], p.82; [2], pp. 226-227).
The helper function ``_check_cycles_alt_sym`` is used to
go over the cycles in a permutation and look for ones satisfying 1).
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.is_alt_sym()
False
See Also
========
_check_cycles_alt_sym
"""
if _random_prec is not None:
N_eps = _random_prec['N_eps']
perms= (_random_prec[i] for i in range(N_eps))
return self._eval_is_alt_sym_monte_carlo(perms=perms)
if self._is_sym or self._is_alt:
return True
if self._is_sym is False and self._is_alt is False:
return False
n = self.degree
if n < 8:
return self._eval_is_alt_sym_naive()
elif self.is_transitive():
return self._eval_is_alt_sym_monte_carlo(eps=eps)
self._is_sym, self._is_alt = False, False
return False
@property
def is_nilpotent(self):
"""Test if the group is nilpotent.
Explanation
===========
A group `G` is nilpotent if it has a central series of finite length.
Alternatively, `G` is nilpotent if its lower central series terminates
with the trivial group. Every nilpotent group is also solvable
([1], p.29, [12]).
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup)
>>> C = CyclicGroup(6)
>>> C.is_nilpotent
True
>>> S = SymmetricGroup(5)
>>> S.is_nilpotent
False
See Also
========
lower_central_series, is_solvable
"""
if self._is_nilpotent is None:
lcs = self.lower_central_series()
terminator = lcs[len(lcs) - 1]
gens = terminator.generators
degree = self.degree
identity = _af_new(list(range(degree)))
if all(g == identity for g in gens):
self._is_solvable = True
self._is_nilpotent = True
return True
else:
self._is_nilpotent = False
return False
else:
return self._is_nilpotent
def is_normal(self, gr, strict=True):
"""Test if ``G=self`` is a normal subgroup of ``gr``.
Explanation
===========
G is normal in gr if
for each g2 in G, g1 in gr, ``g = g1*g2*g1**-1`` belongs to G
It is sufficient to check this for each g1 in gr.generators and
g2 in G.generators.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G1 = PermutationGroup([a, Permutation([2, 0, 1])])
>>> G1.is_normal(G)
True
"""
if not self.is_subgroup(gr, strict=strict):
return False
d_self = self.degree
d_gr = gr.degree
if self.is_trivial and (d_self == d_gr or not strict):
return True
if self._is_abelian:
return True
new_self = self.copy()
if not strict and d_self != d_gr:
if d_self < d_gr:
new_self = PermGroup(new_self.generators + [Permutation(d_gr - 1)])
else:
gr = PermGroup(gr.generators + [Permutation(d_self - 1)])
gens2 = [p._array_form for p in new_self.generators]
gens1 = [p._array_form for p in gr.generators]
for g1 in gens1:
for g2 in gens2:
p = _af_rmuln(g1, g2, _af_invert(g1))
if not new_self.coset_factor(p, True):
return False
return True
def is_primitive(self, randomized=True):
r"""Test if a group is primitive.
Explanation
===========
A permutation group ``G`` acting on a set ``S`` is called primitive if
``S`` contains no nontrivial block under the action of ``G``
(a block is nontrivial if its cardinality is more than ``1``).
Notes
=====
The algorithm is described in [1], p.83, and uses the function
minimal_block to search for blocks of the form `\{0, k\}` for ``k``
ranging over representatives for the orbits of `G_0`, the stabilizer of
``0``. This algorithm has complexity `O(n^2)` where ``n`` is the degree
of the group, and will perform badly if `G_0` is small.
There are two implementations offered: one finds `G_0`
deterministically using the function ``stabilizer``, and the other
(default) produces random elements of `G_0` using ``random_stab``,
hoping that they generate a subgroup of `G_0` with not too many more
orbits than `G_0` (this is suggested in [1], p.83). Behavior is changed
by the ``randomized`` flag.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.is_primitive()
False
See Also
========
minimal_block, random_stab
"""
if self._is_primitive is not None:
return self._is_primitive
if self.is_transitive() is False:
return False
if randomized:
random_stab_gens = []
v = self.schreier_vector(0)
for _ in range(len(self)):
random_stab_gens.append(self.random_stab(0, v))
stab = PermutationGroup(random_stab_gens)
else:
stab = self.stabilizer(0)
orbits = stab.orbits()
for orb in orbits:
x = orb.pop()
if x != 0 and any(e != 0 for e in self.minimal_block([0, x])):
self._is_primitive = False
return False
self._is_primitive = True
return True
def minimal_blocks(self, randomized=True):
'''
For a transitive group, return the list of all minimal
block systems. If a group is intransitive, return `False`.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> DihedralGroup(6).minimal_blocks()
[[0, 1, 0, 1, 0, 1], [0, 1, 2, 0, 1, 2]]
>>> G = PermutationGroup(Permutation(1,2,5))
>>> G.minimal_blocks()
False
See Also
========
minimal_block, is_transitive, is_primitive
'''
def _number_blocks(blocks):
# number the blocks of a block system
# in order and return the number of
# blocks and the tuple with the
# reordering
n = len(blocks)
appeared = {}
m = 0
b = [None]*n
for i in range(n):
if blocks[i] not in appeared:
appeared[blocks[i]] = m
b[i] = m
m += 1
else:
b[i] = appeared[blocks[i]]
return tuple(b), m
if not self.is_transitive():
return False
blocks = []
num_blocks = []
rep_blocks = []
if randomized:
random_stab_gens = []
v = self.schreier_vector(0)
for i in range(len(self)):
random_stab_gens.append(self.random_stab(0, v))
stab = PermutationGroup(random_stab_gens)
else:
stab = self.stabilizer(0)
orbits = stab.orbits()
for orb in orbits:
x = orb.pop()
if x != 0:
block = self.minimal_block([0, x])
num_block, _ = _number_blocks(block)
# a representative block (containing 0)
rep = {j for j in range(self.degree) if num_block[j] == 0}
# check if the system is minimal with
# respect to the already discovere ones
minimal = True
blocks_remove_mask = [False] * len(blocks)
for i, r in enumerate(rep_blocks):
if len(r) > len(rep) and rep.issubset(r):
# i-th block system is not minimal
blocks_remove_mask[i] = True
elif len(r) < len(rep) and r.issubset(rep):
# the system being checked is not minimal
minimal = False
break
# remove non-minimal representative blocks
blocks = [b for i, b in enumerate(blocks) if not blocks_remove_mask[i]]
num_blocks = [n for i, n in enumerate(num_blocks) if not blocks_remove_mask[i]]
rep_blocks = [r for i, r in enumerate(rep_blocks) if not blocks_remove_mask[i]]
if minimal and num_block not in num_blocks:
blocks.append(block)
num_blocks.append(num_block)
rep_blocks.append(rep)
return blocks
@property
def is_solvable(self):
"""Test if the group is solvable.
``G`` is solvable if its derived series terminates with the trivial
group ([1], p.29).
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(3)
>>> S.is_solvable
True
See Also
========
is_nilpotent, derived_series
"""
if self._is_solvable is None:
if self.order() % 2 != 0:
return True
ds = self.derived_series()
terminator = ds[len(ds) - 1]
gens = terminator.generators
degree = self.degree
identity = _af_new(list(range(degree)))
if all(g == identity for g in gens):
self._is_solvable = True
return True
else:
self._is_solvable = False
return False
else:
return self._is_solvable
def is_subgroup(self, G, strict=True):
"""Return ``True`` if all elements of ``self`` belong to ``G``.
If ``strict`` is ``False`` then if ``self``'s degree is smaller
than ``G``'s, the elements will be resized to have the same degree.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup)
Testing is strict by default: the degree of each group must be the
same:
>>> p = Permutation(0, 1, 2, 3, 4, 5)
>>> G1 = PermutationGroup([Permutation(0, 1, 2), Permutation(0, 1)])
>>> G2 = PermutationGroup([Permutation(0, 2), Permutation(0, 1, 2)])
>>> G3 = PermutationGroup([p, p**2])
>>> assert G1.order() == G2.order() == G3.order() == 6
>>> G1.is_subgroup(G2)
True
>>> G1.is_subgroup(G3)
False
>>> G3.is_subgroup(PermutationGroup(G3[1]))
False
>>> G3.is_subgroup(PermutationGroup(G3[0]))
True
To ignore the size, set ``strict`` to ``False``:
>>> S3 = SymmetricGroup(3)
>>> S5 = SymmetricGroup(5)
>>> S3.is_subgroup(S5, strict=False)
True
>>> C7 = CyclicGroup(7)
>>> G = S5*C7
>>> S5.is_subgroup(G, False)
True
>>> C7.is_subgroup(G, 0)
False
"""
if isinstance(G, SymmetricPermutationGroup):
if self.degree != G.degree:
return False
return True
if not isinstance(G, PermutationGroup):
return False
if self == G or self.generators[0]==Permutation():
return True
if G.order() % self.order() != 0:
return False
if self.degree == G.degree or \
(self.degree < G.degree and not strict):
gens = self.generators
else:
return False
return all(G.contains(g, strict=strict) for g in gens)
@property
def is_polycyclic(self):
"""Return ``True`` if a group is polycyclic. A group is polycyclic if
it has a subnormal series with cyclic factors. For finite groups,
this is the same as if the group is solvable.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> G = PermutationGroup([a, b])
>>> G.is_polycyclic
True
"""
return self.is_solvable
def is_transitive(self, strict=True):
"""Test if the group is transitive.
Explanation
===========
A group is transitive if it has a single orbit.
If ``strict`` is ``False`` the group is transitive if it has
a single orbit of length different from 1.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1, 3])
>>> b = Permutation([2, 0, 1, 3])
>>> G1 = PermutationGroup([a, b])
>>> G1.is_transitive()
False
>>> G1.is_transitive(strict=False)
True
>>> c = Permutation([2, 3, 0, 1])
>>> G2 = PermutationGroup([a, c])
>>> G2.is_transitive()
True
>>> d = Permutation([1, 0, 2, 3])
>>> e = Permutation([0, 1, 3, 2])
>>> G3 = PermutationGroup([d, e])
>>> G3.is_transitive() or G3.is_transitive(strict=False)
False
"""
if self._is_transitive: # strict or not, if True then True
return self._is_transitive
if strict:
if self._is_transitive is not None: # we only store strict=True
return self._is_transitive
ans = len(self.orbit(0)) == self.degree
self._is_transitive = ans
return ans
got_orb = False
for x in self.orbits():
if len(x) > 1:
if got_orb:
return False
got_orb = True
return got_orb
@property
def is_trivial(self):
"""Test if the group is the trivial group.
This is true if the group contains only the identity permutation.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> G = PermutationGroup([Permutation([0, 1, 2])])
>>> G.is_trivial
True
"""
if self._is_trivial is None:
self._is_trivial = len(self) == 1 and self[0].is_Identity
return self._is_trivial
def lower_central_series(self):
r"""Return the lower central series for the group.
The lower central series for a group `G` is the series
`G = G_0 > G_1 > G_2 > \ldots` where
`G_k = [G, G_{k-1}]`, i.e. every term after the first is equal to the
commutator of `G` and the previous term in `G1` ([1], p.29).
Returns
=======
A list of permutation groups in the order `G = G_0, G_1, G_2, \ldots`
Examples
========
>>> from sympy.combinatorics.named_groups import (AlternatingGroup,
... DihedralGroup)
>>> A = AlternatingGroup(4)
>>> len(A.lower_central_series())
2
>>> A.lower_central_series()[1].is_subgroup(DihedralGroup(2))
True
See Also
========
commutator, derived_series
"""
res = [self]
current = self
nxt = self.commutator(self, current)
while not current.is_subgroup(nxt):
res.append(nxt)
current = nxt
nxt = self.commutator(self, current)
return res
@property
def max_div(self):
"""Maximum proper divisor of the degree of a permutation group.
Explanation
===========
Obviously, this is the degree divided by its minimal proper divisor
(larger than ``1``, if one exists). As it is guaranteed to be prime,
the ``sieve`` from ``sympy.ntheory`` is used.
This function is also used as an optimization tool for the functions
``minimal_block`` and ``_union_find_merge``.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> G = PermutationGroup([Permutation([0, 2, 1, 3])])
>>> G.max_div
2
See Also
========
minimal_block, _union_find_merge
"""
if self._max_div is not None:
return self._max_div
n = self.degree
if n == 1:
return 1
for x in sieve:
if n % x == 0:
d = n//x
self._max_div = d
return d
def minimal_block(self, points):
r"""For a transitive group, finds the block system generated by
``points``.
Explanation
===========
If a group ``G`` acts on a set ``S``, a nonempty subset ``B`` of ``S``
is called a block under the action of ``G`` if for all ``g`` in ``G``
we have ``gB = B`` (``g`` fixes ``B``) or ``gB`` and ``B`` have no
common points (``g`` moves ``B`` entirely). ([1], p.23; [6]).
The distinct translates ``gB`` of a block ``B`` for ``g`` in ``G``
partition the set ``S`` and this set of translates is known as a block
system. Moreover, we obviously have that all blocks in the partition
have the same size, hence the block size divides ``|S|`` ([1], p.23).
A ``G``-congruence is an equivalence relation ``~`` on the set ``S``
such that ``a ~ b`` implies ``g(a) ~ g(b)`` for all ``g`` in ``G``.
For a transitive group, the equivalence classes of a ``G``-congruence
and the blocks of a block system are the same thing ([1], p.23).
The algorithm below checks the group for transitivity, and then finds
the ``G``-congruence generated by the pairs ``(p_0, p_1), (p_0, p_2),
..., (p_0,p_{k-1})`` which is the same as finding the maximal block
system (i.e., the one with minimum block size) such that
``p_0, ..., p_{k-1}`` are in the same block ([1], p.83).
It is an implementation of Atkinson's algorithm, as suggested in [1],
and manipulates an equivalence relation on the set ``S`` using a
union-find data structure. The running time is just above
`O(|points||S|)`. ([1], pp. 83-87; [7]).
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(10)
>>> D.minimal_block([0, 5])
[0, 1, 2, 3, 4, 0, 1, 2, 3, 4]
>>> D.minimal_block([0, 1])
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
See Also
========
_union_find_rep, _union_find_merge, is_transitive, is_primitive
"""
if not self.is_transitive():
return False
n = self.degree
gens = self.generators
# initialize the list of equivalence class representatives
parents = list(range(n))
ranks = [1]*n
not_rep = []
k = len(points)
# the block size must divide the degree of the group
if k > self.max_div:
return [0]*n
for i in range(k - 1):
parents[points[i + 1]] = points[0]
not_rep.append(points[i + 1])
ranks[points[0]] = k
i = 0
len_not_rep = k - 1
while i < len_not_rep:
gamma = not_rep[i]
i += 1
for gen in gens:
# find has side effects: performs path compression on the list
# of representatives
delta = self._union_find_rep(gamma, parents)
# union has side effects: performs union by rank on the list
# of representatives
temp = self._union_find_merge(gen(gamma), gen(delta), ranks,
parents, not_rep)
if temp == -1:
return [0]*n
len_not_rep += temp
for i in range(n):
# force path compression to get the final state of the equivalence
# relation
self._union_find_rep(i, parents)
# rewrite result so that block representatives are minimal
new_reps = {}
return [new_reps.setdefault(r, i) for i, r in enumerate(parents)]
def conjugacy_class(self, x):
r"""Return the conjugacy class of an element in the group.
Explanation
===========
The conjugacy class of an element ``g`` in a group ``G`` is the set of
elements ``x`` in ``G`` that are conjugate with ``g``, i.e. for which
``g = xax^{-1}``
for some ``a`` in ``G``.
Note that conjugacy is an equivalence relation, and therefore that
conjugacy classes are partitions of ``G``. For a list of all the
conjugacy classes of the group, use the conjugacy_classes() method.
In a permutation group, each conjugacy class corresponds to a particular
`cycle structure': for example, in ``S_3``, the conjugacy classes are:
* the identity class, ``{()}``
* all transpositions, ``{(1 2), (1 3), (2 3)}``
* all 3-cycles, ``{(1 2 3), (1 3 2)}``
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S3 = SymmetricGroup(3)
>>> S3.conjugacy_class(Permutation(0, 1, 2))
{(0 1 2), (0 2 1)}
Notes
=====
This procedure computes the conjugacy class directly by finding the
orbit of the element under conjugation in G. This algorithm is only
feasible for permutation groups of relatively small order, but is like
the orbit() function itself in that respect.
"""
# Ref: "Computing the conjugacy classes of finite groups"; Butler, G.
# Groups '93 Galway/St Andrews; edited by Campbell, C. M.
new_class = {x}
last_iteration = new_class
while len(last_iteration) > 0:
this_iteration = set()
for y in last_iteration:
for s in self.generators:
conjugated = s * y * (~s)
if conjugated not in new_class:
this_iteration.add(conjugated)
new_class.update(last_iteration)
last_iteration = this_iteration
return new_class
def conjugacy_classes(self):
r"""Return the conjugacy classes of the group.
Explanation
===========
As described in the documentation for the .conjugacy_class() function,
conjugacy is an equivalence relation on a group G which partitions the
set of elements. This method returns a list of all these conjugacy
classes of G.
Examples
========
>>> from sympy.combinatorics import SymmetricGroup
>>> SymmetricGroup(3).conjugacy_classes()
[{(2)}, {(0 1 2), (0 2 1)}, {(0 2), (1 2), (2)(0 1)}]
"""
identity = _af_new(list(range(self.degree)))
known_elements = {identity}
classes = [known_elements.copy()]
for x in self.generate():
if x not in known_elements:
new_class = self.conjugacy_class(x)
classes.append(new_class)
known_elements.update(new_class)
return classes
def normal_closure(self, other, k=10):
r"""Return the normal closure of a subgroup/set of permutations.
Explanation
===========
If ``S`` is a subset of a group ``G``, the normal closure of ``A`` in ``G``
is defined as the intersection of all normal subgroups of ``G`` that
contain ``A`` ([1], p.14). Alternatively, it is the group generated by
the conjugates ``x^{-1}yx`` for ``x`` a generator of ``G`` and ``y`` a
generator of the subgroup ``\left\langle S\right\rangle`` generated by
``S`` (for some chosen generating set for ``\left\langle S\right\rangle``)
([1], p.73).
Parameters
==========
other
a subgroup/list of permutations/single permutation
k
an implementation-specific parameter that determines the number
of conjugates that are adjoined to ``other`` at once
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... CyclicGroup, AlternatingGroup)
>>> S = SymmetricGroup(5)
>>> C = CyclicGroup(5)
>>> G = S.normal_closure(C)
>>> G.order()
60
>>> G.is_subgroup(AlternatingGroup(5))
True
See Also
========
commutator, derived_subgroup, random_pr
Notes
=====
The algorithm is described in [1], pp. 73-74; it makes use of the
generation of random elements for permutation groups by the product
replacement algorithm.
"""
if hasattr(other, 'generators'):
degree = self.degree
identity = _af_new(list(range(degree)))
if all(g == identity for g in other.generators):
return other
Z = PermutationGroup(other.generators[:])
base, strong_gens = Z.schreier_sims_incremental()
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_orbits, basic_transversals = \
_orbits_transversals_from_bsgs(base, strong_gens_distr)
self._random_pr_init(r=10, n=20)
_loop = True
while _loop:
Z._random_pr_init(r=10, n=10)
for _ in range(k):
g = self.random_pr()
h = Z.random_pr()
conj = h^g
res = _strip(conj, base, basic_orbits, basic_transversals)
if res[0] != identity or res[1] != len(base) + 1:
gens = Z.generators
gens.append(conj)
Z = PermutationGroup(gens)
strong_gens.append(conj)
temp_base, temp_strong_gens = \
Z.schreier_sims_incremental(base, strong_gens)
base, strong_gens = temp_base, temp_strong_gens
strong_gens_distr = \
_distribute_gens_by_base(base, strong_gens)
basic_orbits, basic_transversals = \
_orbits_transversals_from_bsgs(base,
strong_gens_distr)
_loop = False
for g in self.generators:
for h in Z.generators:
conj = h^g
res = _strip(conj, base, basic_orbits,
basic_transversals)
if res[0] != identity or res[1] != len(base) + 1:
_loop = True
break
if _loop:
break
return Z
elif hasattr(other, '__getitem__'):
return self.normal_closure(PermutationGroup(other))
elif hasattr(other, 'array_form'):
return self.normal_closure(PermutationGroup([other]))
def orbit(self, alpha, action='tuples'):
r"""Compute the orbit of alpha `\{g(\alpha) | g \in G\}` as a set.
Explanation
===========
The time complexity of the algorithm used here is `O(|Orb|*r)` where
`|Orb|` is the size of the orbit and ``r`` is the number of generators of
the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21.
Here alpha can be a single point, or a list of points.
If alpha is a single point, the ordinary orbit is computed.
if alpha is a list of points, there are three available options:
'union' - computes the union of the orbits of the points in the list
'tuples' - computes the orbit of the list interpreted as an ordered
tuple under the group action ( i.e., g((1,2,3)) = (g(1), g(2), g(3)) )
'sets' - computes the orbit of the list interpreted as a sets
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 2, 0, 4, 5, 6, 3])
>>> G = PermutationGroup([a])
>>> G.orbit(0)
{0, 1, 2}
>>> G.orbit([0, 4], 'union')
{0, 1, 2, 3, 4, 5, 6}
See Also
========
orbit_transversal
"""
return _orbit(self.degree, self.generators, alpha, action)
def orbit_rep(self, alpha, beta, schreier_vector=None):
"""Return a group element which sends ``alpha`` to ``beta``.
Explanation
===========
If ``beta`` is not in the orbit of ``alpha``, the function returns
``False``. This implementation makes use of the schreier vector.
For a proof of correctness, see [1], p.80
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> G = AlternatingGroup(5)
>>> G.orbit_rep(0, 4)
(0 4 1 2 3)
See Also
========
schreier_vector
"""
if schreier_vector is None:
schreier_vector = self.schreier_vector(alpha)
if schreier_vector[beta] is None:
return False
k = schreier_vector[beta]
gens = [x._array_form for x in self.generators]
a = []
while k != -1:
a.append(gens[k])
beta = gens[k].index(beta) # beta = (~gens[k])(beta)
k = schreier_vector[beta]
if a:
return _af_new(_af_rmuln(*a))
else:
return _af_new(list(range(self._degree)))
def orbit_transversal(self, alpha, pairs=False):
r"""Computes a transversal for the orbit of ``alpha`` as a set.
Explanation
===========
For a permutation group `G`, a transversal for the orbit
`Orb = \{g(\alpha) | g \in G\}` is a set
`\{g_\beta | g_\beta(\alpha) = \beta\}` for `\beta \in Orb`.
Note that there may be more than one possible transversal.
If ``pairs`` is set to ``True``, it returns the list of pairs
`(\beta, g_\beta)`. For a proof of correctness, see [1], p.79
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> G.orbit_transversal(0)
[(5), (0 1 2 3 4 5), (0 5)(1 4)(2 3), (0 2 4)(1 3 5), (5)(0 4)(1 3), (0 3)(1 4)(2 5)]
See Also
========
orbit
"""
return _orbit_transversal(self._degree, self.generators, alpha, pairs)
def orbits(self, rep=False):
"""Return the orbits of ``self``, ordered according to lowest element
in each orbit.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation(1, 5)(2, 3)(4, 0, 6)
>>> b = Permutation(1, 5)(3, 4)(2, 6, 0)
>>> G = PermutationGroup([a, b])
>>> G.orbits()
[{0, 2, 3, 4, 6}, {1, 5}]
"""
return _orbits(self._degree, self._generators)
def order(self):
"""Return the order of the group: the number of permutations that
can be generated from elements of the group.
The number of permutations comprising the group is given by
``len(group)``; the length of each permutation in the group is
given by ``group.size``.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([1, 0, 2])
>>> G = PermutationGroup([a])
>>> G.degree
3
>>> len(G)
1
>>> G.order()
2
>>> list(G.generate())
[(2), (2)(0 1)]
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.order()
6
See Also
========
degree
"""
if self._order is not None:
return self._order
if self._is_sym:
n = self._degree
self._order = factorial(n)
return self._order
if self._is_alt:
n = self._degree
self._order = factorial(n)/2
return self._order
basic_transversals = self.basic_transversals
m = 1
for x in basic_transversals:
m *= len(x)
self._order = m
return m
def index(self, H):
"""
Returns the index of a permutation group.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation(1,2,3)
>>> b =Permutation(3)
>>> G = PermutationGroup([a])
>>> H = PermutationGroup([b])
>>> G.index(H)
3
"""
if H.is_subgroup(self):
return self.order()//H.order()
@property
def is_symmetric(self):
"""Return ``True`` if the group is symmetric.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> g = SymmetricGroup(5)
>>> g.is_symmetric
True
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> g = PermutationGroup(
... Permutation(0, 1, 2, 3, 4),
... Permutation(2, 3))
>>> g.is_symmetric
True
Notes
=====
This uses a naive test involving the computation of the full
group order.
If you need more quicker taxonomy for large groups, you can use
:meth:`PermutationGroup.is_alt_sym`.
However, :meth:`PermutationGroup.is_alt_sym` may not be accurate
and is not able to distinguish between an alternating group and
a symmetric group.
See Also
========
is_alt_sym
"""
_is_sym = self._is_sym
if _is_sym is not None:
return _is_sym
n = self.degree
if n >= 8:
if self.is_transitive():
_is_alt_sym = self._eval_is_alt_sym_monte_carlo()
if _is_alt_sym:
if any(g.is_odd for g in self.generators):
self._is_sym, self._is_alt = True, False
return True
self._is_sym, self._is_alt = False, True
return False
return self._eval_is_alt_sym_naive(only_sym=True)
self._is_sym, self._is_alt = False, False
return False
return self._eval_is_alt_sym_naive(only_sym=True)
@property
def is_alternating(self):
"""Return ``True`` if the group is alternating.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> g = AlternatingGroup(5)
>>> g.is_alternating
True
>>> from sympy.combinatorics import Permutation, PermutationGroup
>>> g = PermutationGroup(
... Permutation(0, 1, 2, 3, 4),
... Permutation(2, 3, 4))
>>> g.is_alternating
True
Notes
=====
This uses a naive test involving the computation of the full
group order.
If you need more quicker taxonomy for large groups, you can use
:meth:`PermutationGroup.is_alt_sym`.
However, :meth:`PermutationGroup.is_alt_sym` may not be accurate
and is not able to distinguish between an alternating group and
a symmetric group.
See Also
========
is_alt_sym
"""
_is_alt = self._is_alt
if _is_alt is not None:
return _is_alt
n = self.degree
if n >= 8:
if self.is_transitive():
_is_alt_sym = self._eval_is_alt_sym_monte_carlo()
if _is_alt_sym:
if all(g.is_even for g in self.generators):
self._is_sym, self._is_alt = False, True
return True
self._is_sym, self._is_alt = True, False
return False
return self._eval_is_alt_sym_naive(only_alt=True)
self._is_sym, self._is_alt = False, False
return False
return self._eval_is_alt_sym_naive(only_alt=True)
@classmethod
def _distinct_primes_lemma(cls, primes):
"""Subroutine to test if there is only one cyclic group for the
order."""
primes = sorted(primes)
l = len(primes)
for i in range(l):
for j in range(i+1, l):
if primes[j] % primes[i] == 1:
return None
return True
@property
def is_cyclic(self):
r"""
Return ``True`` if the group is Cyclic.
Examples
========
>>> from sympy.combinatorics.named_groups import AbelianGroup
>>> G = AbelianGroup(3, 4)
>>> G.is_cyclic
True
>>> G = AbelianGroup(4, 4)
>>> G.is_cyclic
False
Notes
=====
If the order of a group $n$ can be factored into the distinct
primes $p_1, p_2, \dots , p_s$ and if
.. math::
\forall i, j \in \{1, 2, \dots, s \}:
p_i \not \equiv 1 \pmod {p_j}
holds true, there is only one group of the order $n$ which
is a cyclic group. [1]_ This is a generalization of the lemma
that the group of order $15, 35, ...$ are cyclic.
And also, these additional lemmas can be used to test if a
group is cyclic if the order of the group is already found.
- If the group is abelian and the order of the group is
square-free, the group is cyclic.
- If the order of the group is less than $6$ and is not $4$, the
group is cyclic.
- If the order of the group is prime, the group is cyclic.
References
==========
.. [1] 1978: John S. Rose: A Course on Group Theory,
Introduction to Finite Group Theory: 1.4
"""
if self._is_cyclic is not None:
return self._is_cyclic
if len(self.generators) == 1:
self._is_cyclic = True
self._is_abelian = True
return True
if self._is_abelian is False:
self._is_cyclic = False
return False
order = self.order()
if order < 6:
self._is_abelian = True
if order != 4:
self._is_cyclic = True
return True
factors = factorint(order)
if all(v == 1 for v in factors.values()):
if self._is_abelian:
self._is_cyclic = True
return True
primes = list(factors.keys())
if PermutationGroup._distinct_primes_lemma(primes) is True:
self._is_cyclic = True
self._is_abelian = True
return True
for p in factors:
pgens = []
for g in self.generators:
pgens.append(g**p)
if self.index(self.subgroup(pgens)) != p:
self._is_cyclic = False
return False
self._is_cyclic = True
self._is_abelian = True
return True
def pointwise_stabilizer(self, points, incremental=True):
r"""Return the pointwise stabilizer for a set of points.
Explanation
===========
For a permutation group `G` and a set of points
`\{p_1, p_2,\ldots, p_k\}`, the pointwise stabilizer of
`p_1, p_2, \ldots, p_k` is defined as
`G_{p_1,\ldots, p_k} =
\{g\in G | g(p_i) = p_i \forall i\in\{1, 2,\ldots,k\}\}` ([1],p20).
It is a subgroup of `G`.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(7)
>>> Stab = S.pointwise_stabilizer([2, 3, 5])
>>> Stab.is_subgroup(S.stabilizer(2).stabilizer(3).stabilizer(5))
True
See Also
========
stabilizer, schreier_sims_incremental
Notes
=====
When incremental == True,
rather than the obvious implementation using successive calls to
``.stabilizer()``, this uses the incremental Schreier-Sims algorithm
to obtain a base with starting segment - the given points.
"""
if incremental:
base, strong_gens = self.schreier_sims_incremental(base=points)
stab_gens = []
degree = self.degree
for gen in strong_gens:
if [gen(point) for point in points] == points:
stab_gens.append(gen)
if not stab_gens:
stab_gens = _af_new(list(range(degree)))
return PermutationGroup(stab_gens)
else:
gens = self._generators
degree = self.degree
for x in points:
gens = _stabilizer(degree, gens, x)
return PermutationGroup(gens)
def make_perm(self, n, seed=None):
"""
Multiply ``n`` randomly selected permutations from
pgroup together, starting with the identity
permutation. If ``n`` is a list of integers, those
integers will be used to select the permutations and they
will be applied in L to R order: make_perm((A, B, C)) will
give CBA(I) where I is the identity permutation.
``seed`` is used to set the seed for the random selection
of permutations from pgroup. If this is a list of integers,
the corresponding permutations from pgroup will be selected
in the order give. This is mainly used for testing purposes.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a, b = [Permutation([1, 0, 3, 2]), Permutation([1, 3, 0, 2])]
>>> G = PermutationGroup([a, b])
>>> G.make_perm(1, [0])
(0 1)(2 3)
>>> G.make_perm(3, [0, 1, 0])
(0 2 3 1)
>>> G.make_perm([0, 1, 0])
(0 2 3 1)
See Also
========
random
"""
if is_sequence(n):
if seed is not None:
raise ValueError('If n is a sequence, seed should be None')
n, seed = len(n), n
else:
try:
n = int(n)
except TypeError:
raise ValueError('n must be an integer or a sequence.')
randomrange = _randrange(seed)
# start with the identity permutation
result = Permutation(list(range(self.degree)))
m = len(self)
for _ in range(n):
p = self[randomrange(m)]
result = rmul(result, p)
return result
def random(self, af=False):
"""Return a random group element
"""
rank = randrange(self.order())
return self.coset_unrank(rank, af)
def random_pr(self, gen_count=11, iterations=50, _random_prec=None):
"""Return a random group element using product replacement.
Explanation
===========
For the details of the product replacement algorithm, see
``_random_pr_init`` In ``random_pr`` the actual 'product replacement'
is performed. Notice that if the attribute ``_random_gens``
is empty, it needs to be initialized by ``_random_pr_init``.
See Also
========
_random_pr_init
"""
if self._random_gens == []:
self._random_pr_init(gen_count, iterations)
random_gens = self._random_gens
r = len(random_gens) - 1
# handle randomized input for testing purposes
if _random_prec is None:
s = randrange(r)
t = randrange(r - 1)
if t == s:
t = r - 1
x = choice([1, 2])
e = choice([-1, 1])
else:
s = _random_prec['s']
t = _random_prec['t']
if t == s:
t = r - 1
x = _random_prec['x']
e = _random_prec['e']
if x == 1:
random_gens[s] = _af_rmul(random_gens[s], _af_pow(random_gens[t], e))
random_gens[r] = _af_rmul(random_gens[r], random_gens[s])
else:
random_gens[s] = _af_rmul(_af_pow(random_gens[t], e), random_gens[s])
random_gens[r] = _af_rmul(random_gens[s], random_gens[r])
return _af_new(random_gens[r])
def random_stab(self, alpha, schreier_vector=None, _random_prec=None):
"""Random element from the stabilizer of ``alpha``.
The schreier vector for ``alpha`` is an optional argument used
for speeding up repeated calls. The algorithm is described in [1], p.81
See Also
========
random_pr, orbit_rep
"""
if schreier_vector is None:
schreier_vector = self.schreier_vector(alpha)
if _random_prec is None:
rand = self.random_pr()
else:
rand = _random_prec['rand']
beta = rand(alpha)
h = self.orbit_rep(alpha, beta, schreier_vector)
return rmul(~h, rand)
def schreier_sims(self):
"""Schreier-Sims algorithm.
Explanation
===========
It computes the generators of the chain of stabilizers
`G > G_{b_1} > .. > G_{b1,..,b_r} > 1`
in which `G_{b_1,..,b_i}` stabilizes `b_1,..,b_i`,
and the corresponding ``s`` cosets.
An element of the group can be written as the product
`h_1*..*h_s`.
We use the incremental Schreier-Sims algorithm.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_sims()
>>> G.basic_transversals
[{0: (2)(0 1), 1: (2), 2: (1 2)},
{0: (2), 2: (0 2)}]
"""
if self._transversals:
return
self._schreier_sims()
return
def _schreier_sims(self, base=None):
schreier = self.schreier_sims_incremental(base=base, slp_dict=True)
base, strong_gens = schreier[:2]
self._base = base
self._strong_gens = strong_gens
self._strong_gens_slp = schreier[2]
if not base:
self._transversals = []
self._basic_orbits = []
return
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_orbits, transversals, slps = _orbits_transversals_from_bsgs(base,\
strong_gens_distr, slp=True)
# rewrite the indices stored in slps in terms of strong_gens
for i, slp in enumerate(slps):
gens = strong_gens_distr[i]
for k in slp:
slp[k] = [strong_gens.index(gens[s]) for s in slp[k]]
self._transversals = transversals
self._basic_orbits = [sorted(x) for x in basic_orbits]
self._transversal_slp = slps
def schreier_sims_incremental(self, base=None, gens=None, slp_dict=False):
"""Extend a sequence of points and generating set to a base and strong
generating set.
Parameters
==========
base
The sequence of points to be extended to a base. Optional
parameter with default value ``[]``.
gens
The generating set to be extended to a strong generating set
relative to the base obtained. Optional parameter with default
value ``self.generators``.
slp_dict
If `True`, return a dictionary `{g: gens}` for each strong
generator `g` where `gens` is a list of strong generators
coming before `g` in `strong_gens`, such that the product
of the elements of `gens` is equal to `g`.
Returns
=======
(base, strong_gens)
``base`` is the base obtained, and ``strong_gens`` is the strong
generating set relative to it. The original parameters ``base``,
``gens`` remain unchanged.
Examples
========
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> A = AlternatingGroup(7)
>>> base = [2, 3]
>>> seq = [2, 3]
>>> base, strong_gens = A.schreier_sims_incremental(base=seq)
>>> _verify_bsgs(A, base, strong_gens)
True
>>> base[:2]
[2, 3]
Notes
=====
This version of the Schreier-Sims algorithm runs in polynomial time.
There are certain assumptions in the implementation - if the trivial
group is provided, ``base`` and ``gens`` are returned immediately,
as any sequence of points is a base for the trivial group. If the
identity is present in the generators ``gens``, it is removed as
it is a redundant generator.
The implementation is described in [1], pp. 90-93.
See Also
========
schreier_sims, schreier_sims_random
"""
if base is None:
base = []
if gens is None:
gens = self.generators[:]
degree = self.degree
id_af = list(range(degree))
# handle the trivial group
if len(gens) == 1 and gens[0].is_Identity:
if slp_dict:
return base, gens, {gens[0]: [gens[0]]}
return base, gens
# prevent side effects
_base, _gens = base[:], gens[:]
# remove the identity as a generator
_gens = [x for x in _gens if not x.is_Identity]
# make sure no generator fixes all base points
for gen in _gens:
if all(x == gen._array_form[x] for x in _base):
for new in id_af:
if gen._array_form[new] != new:
break
else:
assert None # can this ever happen?
_base.append(new)
# distribute generators according to basic stabilizers
strong_gens_distr = _distribute_gens_by_base(_base, _gens)
strong_gens_slp = []
# initialize the basic stabilizers, basic orbits and basic transversals
orbs = {}
transversals = {}
slps = {}
base_len = len(_base)
for i in range(base_len):
transversals[i], slps[i] = _orbit_transversal(degree, strong_gens_distr[i],
_base[i], pairs=True, af=True, slp=True)
transversals[i] = dict(transversals[i])
orbs[i] = list(transversals[i].keys())
# main loop: amend the stabilizer chain until we have generators
# for all stabilizers
i = base_len - 1
while i >= 0:
# this flag is used to continue with the main loop from inside
# a nested loop
continue_i = False
# test the generators for being a strong generating set
db = {}
for beta, u_beta in list(transversals[i].items()):
for j, gen in enumerate(strong_gens_distr[i]):
gb = gen._array_form[beta]
u1 = transversals[i][gb]
g1 = _af_rmul(gen._array_form, u_beta)
slp = [(i, g) for g in slps[i][beta]]
slp = [(i, j)] + slp
if g1 != u1:
# test if the schreier generator is in the i+1-th
# would-be basic stabilizer
y = True
try:
u1_inv = db[gb]
except KeyError:
u1_inv = db[gb] = _af_invert(u1)
schreier_gen = _af_rmul(u1_inv, g1)
u1_inv_slp = slps[i][gb][:]
u1_inv_slp.reverse()
u1_inv_slp = [(i, (g,)) for g in u1_inv_slp]
slp = u1_inv_slp + slp
h, j, slp = _strip_af(schreier_gen, _base, orbs, transversals, i, slp=slp, slps=slps)
if j <= base_len:
# new strong generator h at level j
y = False
elif h:
# h fixes all base points
y = False
moved = 0
while h[moved] == moved:
moved += 1
_base.append(moved)
base_len += 1
strong_gens_distr.append([])
if y is False:
# if a new strong generator is found, update the
# data structures and start over
h = _af_new(h)
strong_gens_slp.append((h, slp))
for l in range(i + 1, j):
strong_gens_distr[l].append(h)
transversals[l], slps[l] =\
_orbit_transversal(degree, strong_gens_distr[l],
_base[l], pairs=True, af=True, slp=True)
transversals[l] = dict(transversals[l])
orbs[l] = list(transversals[l].keys())
i = j - 1
# continue main loop using the flag
continue_i = True
if continue_i is True:
break
if continue_i is True:
break
if continue_i is True:
continue
i -= 1
strong_gens = _gens[:]
if slp_dict:
# create the list of the strong generators strong_gens and
# rewrite the indices of strong_gens_slp in terms of the
# elements of strong_gens
for k, slp in strong_gens_slp:
strong_gens.append(k)
for i in range(len(slp)):
s = slp[i]
if isinstance(s[1], tuple):
slp[i] = strong_gens_distr[s[0]][s[1][0]]**-1
else:
slp[i] = strong_gens_distr[s[0]][s[1]]
strong_gens_slp = dict(strong_gens_slp)
# add the original generators
for g in _gens:
strong_gens_slp[g] = [g]
return (_base, strong_gens, strong_gens_slp)
strong_gens.extend([k for k, _ in strong_gens_slp])
return _base, strong_gens
def schreier_sims_random(self, base=None, gens=None, consec_succ=10,
_random_prec=None):
r"""Randomized Schreier-Sims algorithm.
Explanation
===========
The randomized Schreier-Sims algorithm takes the sequence ``base``
and the generating set ``gens``, and extends ``base`` to a base, and
``gens`` to a strong generating set relative to that base with
probability of a wrong answer at most `2^{-consec\_succ}`,
provided the random generators are sufficiently random.
Parameters
==========
base
The sequence to be extended to a base.
gens
The generating set to be extended to a strong generating set.
consec_succ
The parameter defining the probability of a wrong answer.
_random_prec
An internal parameter used for testing purposes.
Returns
=======
(base, strong_gens)
``base`` is the base and ``strong_gens`` is the strong generating
set relative to it.
Examples
========
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(5)
>>> base, strong_gens = S.schreier_sims_random(consec_succ=5)
>>> _verify_bsgs(S, base, strong_gens) #doctest: +SKIP
True
Notes
=====
The algorithm is described in detail in [1], pp. 97-98. It extends
the orbits ``orbs`` and the permutation groups ``stabs`` to
basic orbits and basic stabilizers for the base and strong generating
set produced in the end.
The idea of the extension process
is to "sift" random group elements through the stabilizer chain
and amend the stabilizers/orbits along the way when a sift
is not successful.
The helper function ``_strip`` is used to attempt
to decompose a random group element according to the current
state of the stabilizer chain and report whether the element was
fully decomposed (successful sift) or not (unsuccessful sift). In
the latter case, the level at which the sift failed is reported and
used to amend ``stabs``, ``base``, ``gens`` and ``orbs`` accordingly.
The halting condition is for ``consec_succ`` consecutive successful
sifts to pass. This makes sure that the current ``base`` and ``gens``
form a BSGS with probability at least `1 - 1/\text{consec\_succ}`.
See Also
========
schreier_sims
"""
if base is None:
base = []
if gens is None:
gens = self.generators
base_len = len(base)
n = self.degree
# make sure no generator fixes all base points
for gen in gens:
if all(gen(x) == x for x in base):
new = 0
while gen._array_form[new] == new:
new += 1
base.append(new)
base_len += 1
# distribute generators according to basic stabilizers
strong_gens_distr = _distribute_gens_by_base(base, gens)
# initialize the basic stabilizers, basic transversals and basic orbits
transversals = {}
orbs = {}
for i in range(base_len):
transversals[i] = dict(_orbit_transversal(n, strong_gens_distr[i],
base[i], pairs=True))
orbs[i] = list(transversals[i].keys())
# initialize the number of consecutive elements sifted
c = 0
# start sifting random elements while the number of consecutive sifts
# is less than consec_succ
while c < consec_succ:
if _random_prec is None:
g = self.random_pr()
else:
g = _random_prec['g'].pop()
h, j = _strip(g, base, orbs, transversals)
y = True
# determine whether a new base point is needed
if j <= base_len:
y = False
elif not h.is_Identity:
y = False
moved = 0
while h(moved) == moved:
moved += 1
base.append(moved)
base_len += 1
strong_gens_distr.append([])
# if the element doesn't sift, amend the strong generators and
# associated stabilizers and orbits
if y is False:
for l in range(1, j):
strong_gens_distr[l].append(h)
transversals[l] = dict(_orbit_transversal(n,
strong_gens_distr[l], base[l], pairs=True))
orbs[l] = list(transversals[l].keys())
c = 0
else:
c += 1
# build the strong generating set
strong_gens = strong_gens_distr[0][:]
for gen in strong_gens_distr[1]:
if gen not in strong_gens:
strong_gens.append(gen)
return base, strong_gens
def schreier_vector(self, alpha):
"""Computes the schreier vector for ``alpha``.
Explanation
===========
The Schreier vector efficiently stores information
about the orbit of ``alpha``. It can later be used to quickly obtain
elements of the group that send ``alpha`` to a particular element
in the orbit. Notice that the Schreier vector depends on the order
in which the group generators are listed. For a definition, see [3].
Since list indices start from zero, we adopt the convention to use
"None" instead of 0 to signify that an element doesn't belong
to the orbit.
For the algorithm and its correctness, see [2], pp.78-80.
Examples
========
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> a = Permutation([2, 4, 6, 3, 1, 5, 0])
>>> b = Permutation([0, 1, 3, 5, 4, 6, 2])
>>> G = PermutationGroup([a, b])
>>> G.schreier_vector(0)
[-1, None, 0, 1, None, 1, 0]
See Also
========
orbit
"""
n = self.degree
v = [None]*n
v[alpha] = -1
orb = [alpha]
used = [False]*n
used[alpha] = True
gens = self.generators
r = len(gens)
for b in orb:
for i in range(r):
temp = gens[i]._array_form[b]
if used[temp] is False:
orb.append(temp)
used[temp] = True
v[temp] = i
return v
def stabilizer(self, alpha):
r"""Return the stabilizer subgroup of ``alpha``.
Explanation
===========
The stabilizer of `\alpha` is the group `G_\alpha =
\{g \in G | g(\alpha) = \alpha\}`.
For a proof of correctness, see [1], p.79.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> G.stabilizer(5)
PermutationGroup([
(5)(0 4)(1 3)])
See Also
========
orbit
"""
return PermGroup(_stabilizer(self._degree, self._generators, alpha))
@property
def strong_gens(self):
r"""Return a strong generating set from the Schreier-Sims algorithm.
Explanation
===========
A generating set `S = \{g_1, g_2, \dots, g_t\}` for a permutation group
`G` is a strong generating set relative to the sequence of points
(referred to as a "base") `(b_1, b_2, \dots, b_k)` if, for
`1 \leq i \leq k` we have that the intersection of the pointwise
stabilizer `G^{(i+1)} := G_{b_1, b_2, \dots, b_i}` with `S` generates
the pointwise stabilizer `G^{(i+1)}`. The concepts of a base and
strong generating set and their applications are discussed in depth
in [1], pp. 87-89 and [2], pp. 55-57.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> D = DihedralGroup(4)
>>> D.strong_gens
[(0 1 2 3), (0 3)(1 2), (1 3)]
>>> D.base
[0, 1]
See Also
========
base, basic_transversals, basic_orbits, basic_stabilizers
"""
if self._strong_gens == []:
self.schreier_sims()
return self._strong_gens
def subgroup(self, gens):
"""
Return the subgroup generated by `gens` which is a list of
elements of the group
"""
if not all(g in self for g in gens):
raise ValueError("The group doesn't contain the supplied generators")
G = PermutationGroup(gens)
return G
def subgroup_search(self, prop, base=None, strong_gens=None, tests=None,
init_subgroup=None):
"""Find the subgroup of all elements satisfying the property ``prop``.
Explanation
===========
This is done by a depth-first search with respect to base images that
uses several tests to prune the search tree.
Parameters
==========
prop
The property to be used. Has to be callable on group elements
and always return ``True`` or ``False``. It is assumed that
all group elements satisfying ``prop`` indeed form a subgroup.
base
A base for the supergroup.
strong_gens
A strong generating set for the supergroup.
tests
A list of callables of length equal to the length of ``base``.
These are used to rule out group elements by partial base images,
so that ``tests[l](g)`` returns False if the element ``g`` is known
not to satisfy prop base on where g sends the first ``l + 1`` base
points.
init_subgroup
if a subgroup of the sought group is
known in advance, it can be passed to the function as this
parameter.
Returns
=======
res
The subgroup of all elements satisfying ``prop``. The generating
set for this group is guaranteed to be a strong generating set
relative to the base ``base``.
Examples
========
>>> from sympy.combinatorics.named_groups import (SymmetricGroup,
... AlternatingGroup)
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> S = SymmetricGroup(7)
>>> prop_even = lambda x: x.is_even
>>> base, strong_gens = S.schreier_sims_incremental()
>>> G = S.subgroup_search(prop_even, base=base, strong_gens=strong_gens)
>>> G.is_subgroup(AlternatingGroup(7))
True
>>> _verify_bsgs(G, base, G.generators)
True
Notes
=====
This function is extremely lengthy and complicated and will require
some careful attention. The implementation is described in
[1], pp. 114-117, and the comments for the code here follow the lines
of the pseudocode in the book for clarity.
The complexity is exponential in general, since the search process by
itself visits all members of the supergroup. However, there are a lot
of tests which are used to prune the search tree, and users can define
their own tests via the ``tests`` parameter, so in practice, and for
some computations, it's not terrible.
A crucial part in the procedure is the frequent base change performed
(this is line 11 in the pseudocode) in order to obtain a new basic
stabilizer. The book mentiones that this can be done by using
``.baseswap(...)``, however the current implementation uses a more
straightforward way to find the next basic stabilizer - calling the
function ``.stabilizer(...)`` on the previous basic stabilizer.
"""
# initialize BSGS and basic group properties
def get_reps(orbits):
# get the minimal element in the base ordering
return [min(orbit, key = lambda x: base_ordering[x]) \
for orbit in orbits]
def update_nu(l):
temp_index = len(basic_orbits[l]) + 1 -\
len(res_basic_orbits_init_base[l])
# this corresponds to the element larger than all points
if temp_index >= len(sorted_orbits[l]):
nu[l] = base_ordering[degree]
else:
nu[l] = sorted_orbits[l][temp_index]
if base is None:
base, strong_gens = self.schreier_sims_incremental()
base_len = len(base)
degree = self.degree
identity = _af_new(list(range(degree)))
base_ordering = _base_ordering(base, degree)
# add an element larger than all points
base_ordering.append(degree)
# add an element smaller than all points
base_ordering.append(-1)
# compute BSGS-related structures
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
basic_orbits, transversals = _orbits_transversals_from_bsgs(base,
strong_gens_distr)
# handle subgroup initialization and tests
if init_subgroup is None:
init_subgroup = PermutationGroup([identity])
if tests is None:
trivial_test = lambda x: True
tests = []
for i in range(base_len):
tests.append(trivial_test)
# line 1: more initializations.
res = init_subgroup
f = base_len - 1
l = base_len - 1
# line 2: set the base for K to the base for G
res_base = base[:]
# line 3: compute BSGS and related structures for K
res_base, res_strong_gens = res.schreier_sims_incremental(
base=res_base)
res_strong_gens_distr = _distribute_gens_by_base(res_base,
res_strong_gens)
res_generators = res.generators
res_basic_orbits_init_base = \
[_orbit(degree, res_strong_gens_distr[i], res_base[i])\
for i in range(base_len)]
# initialize orbit representatives
orbit_reps = [None]*base_len
# line 4: orbit representatives for f-th basic stabilizer of K
orbits = _orbits(degree, res_strong_gens_distr[f])
orbit_reps[f] = get_reps(orbits)
# line 5: remove the base point from the representatives to avoid
# getting the identity element as a generator for K
orbit_reps[f].remove(base[f])
# line 6: more initializations
c = [0]*base_len
u = [identity]*base_len
sorted_orbits = [None]*base_len
for i in range(base_len):
sorted_orbits[i] = basic_orbits[i][:]
sorted_orbits[i].sort(key=lambda point: base_ordering[point])
# line 7: initializations
mu = [None]*base_len
nu = [None]*base_len
# this corresponds to the element smaller than all points
mu[l] = degree + 1
update_nu(l)
# initialize computed words
computed_words = [identity]*base_len
# line 8: main loop
while True:
# apply all the tests
while l < base_len - 1 and \
computed_words[l](base[l]) in orbit_reps[l] and \
base_ordering[mu[l]] < \
base_ordering[computed_words[l](base[l])] < \
base_ordering[nu[l]] and \
tests[l](computed_words):
# line 11: change the (partial) base of K
new_point = computed_words[l](base[l])
res_base[l] = new_point
new_stab_gens = _stabilizer(degree, res_strong_gens_distr[l],
new_point)
res_strong_gens_distr[l + 1] = new_stab_gens
# line 12: calculate minimal orbit representatives for the
# l+1-th basic stabilizer
orbits = _orbits(degree, new_stab_gens)
orbit_reps[l + 1] = get_reps(orbits)
# line 13: amend sorted orbits
l += 1
temp_orbit = [computed_words[l - 1](point) for point
in basic_orbits[l]]
temp_orbit.sort(key=lambda point: base_ordering[point])
sorted_orbits[l] = temp_orbit
# lines 14 and 15: update variables used minimality tests
new_mu = degree + 1
for i in range(l):
if base[l] in res_basic_orbits_init_base[i]:
candidate = computed_words[i](base[i])
if base_ordering[candidate] > base_ordering[new_mu]:
new_mu = candidate
mu[l] = new_mu
update_nu(l)
# line 16: determine the new transversal element
c[l] = 0
temp_point = sorted_orbits[l][c[l]]
gamma = computed_words[l - 1]._array_form.index(temp_point)
u[l] = transversals[l][gamma]
# update computed words
computed_words[l] = rmul(computed_words[l - 1], u[l])
# lines 17 & 18: apply the tests to the group element found
g = computed_words[l]
temp_point = g(base[l])
if l == base_len - 1 and \
base_ordering[mu[l]] < \
base_ordering[temp_point] < base_ordering[nu[l]] and \
temp_point in orbit_reps[l] and \
tests[l](computed_words) and \
prop(g):
# line 19: reset the base of K
res_generators.append(g)
res_base = base[:]
# line 20: recalculate basic orbits (and transversals)
res_strong_gens.append(g)
res_strong_gens_distr = _distribute_gens_by_base(res_base,
res_strong_gens)
res_basic_orbits_init_base = \
[_orbit(degree, res_strong_gens_distr[i], res_base[i]) \
for i in range(base_len)]
# line 21: recalculate orbit representatives
# line 22: reset the search depth
orbit_reps[f] = get_reps(orbits)
l = f
# line 23: go up the tree until in the first branch not fully
# searched
while l >= 0 and c[l] == len(basic_orbits[l]) - 1:
l = l - 1
# line 24: if the entire tree is traversed, return K
if l == -1:
return PermutationGroup(res_generators)
# lines 25-27: update orbit representatives
if l < f:
# line 26
f = l
c[l] = 0
# line 27
temp_orbits = _orbits(degree, res_strong_gens_distr[f])
orbit_reps[f] = get_reps(temp_orbits)
# line 28: update variables used for minimality testing
mu[l] = degree + 1
temp_index = len(basic_orbits[l]) + 1 - \
len(res_basic_orbits_init_base[l])
if temp_index >= len(sorted_orbits[l]):
nu[l] = base_ordering[degree]
else:
nu[l] = sorted_orbits[l][temp_index]
# line 29: set the next element from the current branch and update
# accordingly
c[l] += 1
if l == 0:
gamma = sorted_orbits[l][c[l]]
else:
gamma = computed_words[l - 1]._array_form.index(sorted_orbits[l][c[l]])
u[l] = transversals[l][gamma]
if l == 0:
computed_words[l] = u[l]
else:
computed_words[l] = rmul(computed_words[l - 1], u[l])
@property
def transitivity_degree(self):
r"""Compute the degree of transitivity of the group.
Explanation
===========
A permutation group `G` acting on `\Omega = \{0, 1, \dots, n-1\}` is
``k``-fold transitive, if, for any `k` points
`(a_1, a_2, \dots, a_k) \in \Omega` and any `k` points
`(b_1, b_2, \dots, b_k) \in \Omega` there exists `g \in G` such that
`g(a_1) = b_1, g(a_2) = b_2, \dots, g(a_k) = b_k`
The degree of transitivity of `G` is the maximum ``k`` such that
`G` is ``k``-fold transitive. ([8])
Examples
========
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> a = Permutation([1, 2, 0])
>>> b = Permutation([1, 0, 2])
>>> G = PermutationGroup([a, b])
>>> G.transitivity_degree
3
See Also
========
is_transitive, orbit
"""
if self._transitivity_degree is None:
n = self.degree
G = self
# if G is k-transitive, a tuple (a_0,..,a_k)
# can be brought to (b_0,...,b_(k-1), b_k)
# where b_0,...,b_(k-1) are fixed points;
# consider the group G_k which stabilizes b_0,...,b_(k-1)
# if G_k is transitive on the subset excluding b_0,...,b_(k-1)
# then G is (k+1)-transitive
for i in range(n):
orb = G.orbit(i)
if len(orb) != n - i:
self._transitivity_degree = i
return i
G = G.stabilizer(i)
self._transitivity_degree = n
return n
else:
return self._transitivity_degree
def _p_elements_group(self, p):
'''
For an abelian p-group, return the subgroup consisting of
all elements of order p (and the identity)
'''
gens = self.generators[:]
gens = sorted(gens, key=lambda x: x.order(), reverse=True)
gens_p = [g**(g.order()/p) for g in gens]
gens_r = []
for i in range(len(gens)):
x = gens[i]
x_order = x.order()
# x_p has order p
x_p = x**(x_order/p)
if i > 0:
P = PermutationGroup(gens_p[:i])
else:
P = PermutationGroup(self.identity)
if x**(x_order/p) not in P:
gens_r.append(x**(x_order/p))
else:
# replace x by an element of order (x.order()/p)
# so that gens still generates G
g = P.generator_product(x_p, original=True)
for s in g:
x = x*s**-1
x_order = x_order/p
# insert x to gens so that the sorting is preserved
del gens[i]
del gens_p[i]
j = i - 1
while j < len(gens) and gens[j].order() >= x_order:
j += 1
gens = gens[:j] + [x] + gens[j:]
gens_p = gens_p[:j] + [x] + gens_p[j:]
return PermutationGroup(gens_r)
def _sylow_alt_sym(self, p):
'''
Return a p-Sylow subgroup of a symmetric or an
alternating group.
Explanation
===========
The algorithm for this is hinted at in [1], Chapter 4,
Exercise 4.
For Sym(n) with n = p^i, the idea is as follows. Partition
the interval [0..n-1] into p equal parts, each of length p^(i-1):
[0..p^(i-1)-1], [p^(i-1)..2*p^(i-1)-1]...[(p-1)*p^(i-1)..p^i-1].
Find a p-Sylow subgroup of Sym(p^(i-1)) (treated as a subgroup
of ``self``) acting on each of the parts. Call the subgroups
P_1, P_2...P_p. The generators for the subgroups P_2...P_p
can be obtained from those of P_1 by applying a "shifting"
permutation to them, that is, a permutation mapping [0..p^(i-1)-1]
to the second part (the other parts are obtained by using the shift
multiple times). The union of this permutation and the generators
of P_1 is a p-Sylow subgroup of ``self``.
For n not equal to a power of p, partition
[0..n-1] in accordance with how n would be written in base p.
E.g. for p=2 and n=11, 11 = 2^3 + 2^2 + 1 so the partition
is [[0..7], [8..9], {10}]. To generate a p-Sylow subgroup,
take the union of the generators for each of the parts.
For the above example, {(0 1), (0 2)(1 3), (0 4), (1 5)(2 7)}
from the first part, {(8 9)} from the second part and
nothing from the third. This gives 4 generators in total, and
the subgroup they generate is p-Sylow.
Alternating groups are treated the same except when p=2. In this
case, (0 1)(s s+1) should be added for an appropriate s (the start
of a part) for each part in the partitions.
See Also
========
sylow_subgroup, is_alt_sym
'''
n = self.degree
gens = []
identity = Permutation(n-1)
# the case of 2-sylow subgroups of alternating groups
# needs special treatment
alt = p == 2 and all(g.is_even for g in self.generators)
# find the presentation of n in base p
coeffs = []
m = n
while m > 0:
coeffs.append(m % p)
m = m // p
power = len(coeffs)-1
# for a symmetric group, gens[:i] is the generating
# set for a p-Sylow subgroup on [0..p**(i-1)-1]. For
# alternating groups, the same is given by gens[:2*(i-1)]
for i in range(1, power+1):
if i == 1 and alt:
# (0 1) shouldn't be added for alternating groups
continue
gen = Permutation([(j + p**(i-1)) % p**i for j in range(p**i)])
gens.append(identity*gen)
if alt:
gen = Permutation(0, 1)*gen*Permutation(0, 1)*gen
gens.append(gen)
# the first point in the current part (see the algorithm
# description in the docstring)
start = 0
while power > 0:
a = coeffs[power]
# make the permutation shifting the start of the first
# part ([0..p^i-1] for some i) to the current one
for _ in range(a):
shift = Permutation()
if start > 0:
for i in range(p**power):
shift = shift(i, start + i)
if alt:
gen = Permutation(0, 1)*shift*Permutation(0, 1)*shift
gens.append(gen)
j = 2*(power - 1)
else:
j = power
for i, gen in enumerate(gens[:j]):
if alt and i % 2 == 1:
continue
# shift the generator to the start of the
# partition part
gen = shift*gen*shift
gens.append(gen)
start += p**power
power = power-1
return gens
def sylow_subgroup(self, p):
'''
Return a p-Sylow subgroup of the group.
The algorithm is described in [1], Chapter 4, Section 7
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.named_groups import AlternatingGroup
>>> D = DihedralGroup(6)
>>> S = D.sylow_subgroup(2)
>>> S.order()
4
>>> G = SymmetricGroup(6)
>>> S = G.sylow_subgroup(5)
>>> S.order()
5
>>> G1 = AlternatingGroup(3)
>>> G2 = AlternatingGroup(5)
>>> G3 = AlternatingGroup(9)
>>> S1 = G1.sylow_subgroup(3)
>>> S2 = G2.sylow_subgroup(3)
>>> S3 = G3.sylow_subgroup(3)
>>> len1 = len(S1.lower_central_series())
>>> len2 = len(S2.lower_central_series())
>>> len3 = len(S3.lower_central_series())
>>> len1 == len2
True
>>> len1 < len3
True
'''
from sympy.combinatorics.homomorphisms import (
orbit_homomorphism, block_homomorphism)
from sympy.ntheory.primetest import isprime
if not isprime(p):
raise ValueError("p must be a prime")
def is_p_group(G):
# check if the order of G is a power of p
# and return the power
m = G.order()
n = 0
while m % p == 0:
m = m/p
n += 1
if m == 1:
return True, n
return False, n
def _sylow_reduce(mu, nu):
# reduction based on two homomorphisms
# mu and nu with trivially intersecting
# kernels
Q = mu.image().sylow_subgroup(p)
Q = mu.invert_subgroup(Q)
nu = nu.restrict_to(Q)
R = nu.image().sylow_subgroup(p)
return nu.invert_subgroup(R)
order = self.order()
if order % p != 0:
return PermutationGroup([self.identity])
p_group, n = is_p_group(self)
if p_group:
return self
if self.is_alt_sym():
return PermutationGroup(self._sylow_alt_sym(p))
# if there is a non-trivial orbit with size not divisible
# by p, the sylow subgroup is contained in its stabilizer
# (by orbit-stabilizer theorem)
orbits = self.orbits()
non_p_orbits = [o for o in orbits if len(o) % p != 0 and len(o) != 1]
if non_p_orbits:
G = self.stabilizer(list(non_p_orbits[0]).pop())
return G.sylow_subgroup(p)
if not self.is_transitive():
# apply _sylow_reduce to orbit actions
orbits = sorted(orbits, key=len)
omega1 = orbits.pop()
omega2 = orbits[0].union(*orbits)
mu = orbit_homomorphism(self, omega1)
nu = orbit_homomorphism(self, omega2)
return _sylow_reduce(mu, nu)
blocks = self.minimal_blocks()
if len(blocks) > 1:
# apply _sylow_reduce to block system actions
mu = block_homomorphism(self, blocks[0])
nu = block_homomorphism(self, blocks[1])
return _sylow_reduce(mu, nu)
elif len(blocks) == 1:
block = list(blocks)[0]
if any(e != 0 for e in block):
# self is imprimitive
mu = block_homomorphism(self, block)
if not is_p_group(mu.image())[0]:
S = mu.image().sylow_subgroup(p)
return mu.invert_subgroup(S).sylow_subgroup(p)
# find an element of order p
g = self.random()
g_order = g.order()
while g_order % p != 0 or g_order == 0:
g = self.random()
g_order = g.order()
g = g**(g_order // p)
if order % p**2 != 0:
return PermutationGroup(g)
C = self.centralizer(g)
while C.order() % p**n != 0:
S = C.sylow_subgroup(p)
s_order = S.order()
Z = S.center()
P = Z._p_elements_group(p)
h = P.random()
C_h = self.centralizer(h)
while C_h.order() % p*s_order != 0:
h = P.random()
C_h = self.centralizer(h)
C = C_h
return C.sylow_subgroup(p)
def _block_verify(self, L, alpha):
delta = sorted(list(self.orbit(alpha)))
# p[i] will be the number of the block
# delta[i] belongs to
p = [-1]*len(delta)
blocks = [-1]*len(delta)
B = [[]] # future list of blocks
u = [0]*len(delta) # u[i] in L s.t. alpha^u[i] = B[0][i]
t = L.orbit_transversal(alpha, pairs=True)
for a, beta in t:
B[0].append(a)
i_a = delta.index(a)
p[i_a] = 0
blocks[i_a] = alpha
u[i_a] = beta
rho = 0
m = 0 # number of blocks - 1
while rho <= m:
beta = B[rho][0]
for g in self.generators:
d = beta^g
i_d = delta.index(d)
sigma = p[i_d]
if sigma < 0:
# define a new block
m += 1
sigma = m
u[i_d] = u[delta.index(beta)]*g
p[i_d] = sigma
rep = d
blocks[i_d] = rep
newb = [rep]
for gamma in B[rho][1:]:
i_gamma = delta.index(gamma)
d = gamma^g
i_d = delta.index(d)
if p[i_d] < 0:
u[i_d] = u[i_gamma]*g
p[i_d] = sigma
blocks[i_d] = rep
newb.append(d)
else:
# B[rho] is not a block
s = u[i_gamma]*g*u[i_d]**(-1)
return False, s
B.append(newb)
else:
for h in B[rho][1:]:
if h^g not in B[sigma]:
# B[rho] is not a block
s = u[delta.index(beta)]*g*u[i_d]**(-1)
return False, s
rho += 1
return True, blocks
def _verify(H, K, phi, z, alpha):
'''
Return a list of relators ``rels`` in generators ``gens`_h` that
are mapped to ``H.generators`` by ``phi`` so that given a finite
presentation <gens_k | rels_k> of ``K`` on a subset of ``gens_h``
<gens_h | rels_k + rels> is a finite presentation of ``H``.
Explanation
===========
``H`` should be generated by the union of ``K.generators`` and ``z``
(a single generator), and ``H.stabilizer(alpha) == K``; ``phi`` is a
canonical injection from a free group into a permutation group
containing ``H``.
The algorithm is described in [1], Chapter 6.
Examples
========
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.homomorphisms import homomorphism
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup
>>> H = PermutationGroup(Permutation(0, 2), Permutation (1, 5))
>>> K = PermutationGroup(Permutation(5)(0, 2))
>>> F = free_group("x_0 x_1")[0]
>>> gens = F.generators
>>> phi = homomorphism(F, H, F.generators, H.generators)
>>> rels_k = [gens[0]**2] # relators for presentation of K
>>> z= Permutation(1, 5)
>>> check, rels_h = H._verify(K, phi, z, 1)
>>> check
True
>>> rels = rels_k + rels_h
>>> G = FpGroup(F, rels) # presentation of H
>>> G.order() == H.order()
True
See also
========
strong_presentation, presentation, stabilizer
'''
orbit = H.orbit(alpha)
beta = alpha^(z**-1)
K_beta = K.stabilizer(beta)
# orbit representatives of K_beta
gammas = [alpha, beta]
orbits = list({tuple(K_beta.orbit(o)) for o in orbit})
orbit_reps = [orb[0] for orb in orbits]
for rep in orbit_reps:
if rep not in gammas:
gammas.append(rep)
# orbit transversal of K
betas = [alpha, beta]
transversal = {alpha: phi.invert(H.identity), beta: phi.invert(z**-1)}
for s, g in K.orbit_transversal(beta, pairs=True):
if s not in transversal:
transversal[s] = transversal[beta]*phi.invert(g)
union = K.orbit(alpha).union(K.orbit(beta))
while (len(union) < len(orbit)):
for gamma in gammas:
if gamma in union:
r = gamma^z
if r not in union:
betas.append(r)
transversal[r] = transversal[gamma]*phi.invert(z)
for s, g in K.orbit_transversal(r, pairs=True):
if s not in transversal:
transversal[s] = transversal[r]*phi.invert(g)
union = union.union(K.orbit(r))
break
# compute relators
rels = []
for b in betas:
k_gens = K.stabilizer(b).generators
for y in k_gens:
new_rel = transversal[b]
gens = K.generator_product(y, original=True)
for g in gens[::-1]:
new_rel = new_rel*phi.invert(g)
new_rel = new_rel*transversal[b]**-1
perm = phi(new_rel)
try:
gens = K.generator_product(perm, original=True)
except ValueError:
return False, perm
for g in gens:
new_rel = new_rel*phi.invert(g)**-1
if new_rel not in rels:
rels.append(new_rel)
for gamma in gammas:
new_rel = transversal[gamma]*phi.invert(z)*transversal[gamma^z]**-1
perm = phi(new_rel)
try:
gens = K.generator_product(perm, original=True)
except ValueError:
return False, perm
for g in gens:
new_rel = new_rel*phi.invert(g)**-1
if new_rel not in rels:
rels.append(new_rel)
return True, rels
def strong_presentation(self):
'''
Return a strong finite presentation of group. The generators
of the returned group are in the same order as the strong
generators of group.
The algorithm is based on Sims' Verify algorithm described
in [1], Chapter 6.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> P = DihedralGroup(4)
>>> G = P.strong_presentation()
>>> P.order() == G.order()
True
See Also
========
presentation, _verify
'''
from sympy.combinatorics.fp_groups import (FpGroup,
simplify_presentation)
from sympy.combinatorics.free_groups import free_group
from sympy.combinatorics.homomorphisms import (block_homomorphism,
homomorphism, GroupHomomorphism)
strong_gens = self.strong_gens[:]
stabs = self.basic_stabilizers[:]
base = self.base[:]
# injection from a free group on len(strong_gens)
# generators into G
gen_syms = [('x_%d'%i) for i in range(len(strong_gens))]
F = free_group(', '.join(gen_syms))[0]
phi = homomorphism(F, self, F.generators, strong_gens)
H = PermutationGroup(self.identity)
while stabs:
alpha = base.pop()
K = H
H = stabs.pop()
new_gens = [g for g in H.generators if g not in K]
if K.order() == 1:
z = new_gens.pop()
rels = [F.generators[-1]**z.order()]
intermediate_gens = [z]
K = PermutationGroup(intermediate_gens)
# add generators one at a time building up from K to H
while new_gens:
z = new_gens.pop()
intermediate_gens = [z] + intermediate_gens
K_s = PermutationGroup(intermediate_gens)
orbit = K_s.orbit(alpha)
orbit_k = K.orbit(alpha)
# split into cases based on the orbit of K_s
if orbit_k == orbit:
if z in K:
rel = phi.invert(z)
perm = z
else:
t = K.orbit_rep(alpha, alpha^z)
rel = phi.invert(z)*phi.invert(t)**-1
perm = z*t**-1
for g in K.generator_product(perm, original=True):
rel = rel*phi.invert(g)**-1
new_rels = [rel]
elif len(orbit_k) == 1:
# `success` is always true because `strong_gens`
# and `base` are already a verified BSGS. Later
# this could be changed to start with a randomly
# generated (potential) BSGS, and then new elements
# would have to be appended to it when `success`
# is false.
success, new_rels = K_s._verify(K, phi, z, alpha)
else:
# K.orbit(alpha) should be a block
# under the action of K_s on K_s.orbit(alpha)
check, block = K_s._block_verify(K, alpha)
if check:
# apply _verify to the action of K_s
# on the block system; for convenience,
# add the blocks as additional points
# that K_s should act on
t = block_homomorphism(K_s, block)
m = t.codomain.degree # number of blocks
d = K_s.degree
# conjugating with p will shift
# permutations in t.image() to
# higher numbers, e.g.
# p*(0 1)*p = (m m+1)
p = Permutation()
for i in range(m):
p *= Permutation(i, i+d)
t_img = t.images
# combine generators of K_s with their
# action on the block system
images = {g: g*p*t_img[g]*p for g in t_img}
for g in self.strong_gens[:-len(K_s.generators)]:
images[g] = g
K_s_act = PermutationGroup(list(images.values()))
f = GroupHomomorphism(self, K_s_act, images)
K_act = PermutationGroup([f(g) for g in K.generators])
success, new_rels = K_s_act._verify(K_act, f.compose(phi), f(z), d)
for n in new_rels:
if n not in rels:
rels.append(n)
K = K_s
group = FpGroup(F, rels)
return simplify_presentation(group)
def presentation(self, eliminate_gens=True):
'''
Return an `FpGroup` presentation of the group.
The algorithm is described in [1], Chapter 6.1.
'''
from sympy.combinatorics.fp_groups import (FpGroup,
simplify_presentation)
from sympy.combinatorics.coset_table import CosetTable
from sympy.combinatorics.free_groups import free_group
from sympy.combinatorics.homomorphisms import homomorphism
from itertools import product
if self._fp_presentation:
return self._fp_presentation
def _factor_group_by_rels(G, rels):
if isinstance(G, FpGroup):
rels.extend(G.relators)
return FpGroup(G.free_group, list(set(rels)))
return FpGroup(G, rels)
gens = self.generators
len_g = len(gens)
if len_g == 1:
order = gens[0].order()
# handle the trivial group
if order == 1:
return free_group([])[0]
F, x = free_group('x')
return FpGroup(F, [x**order])
if self.order() > 20:
half_gens = self.generators[0:(len_g+1)//2]
else:
half_gens = []
H = PermutationGroup(half_gens)
H_p = H.presentation()
len_h = len(H_p.generators)
C = self.coset_table(H)
n = len(C) # subgroup index
gen_syms = [('x_%d'%i) for i in range(len(gens))]
F = free_group(', '.join(gen_syms))[0]
# mapping generators of H_p to those of F
images = [F.generators[i] for i in range(len_h)]
R = homomorphism(H_p, F, H_p.generators, images, check=False)
# rewrite relators
rels = R(H_p.relators)
G_p = FpGroup(F, rels)
# injective homomorphism from G_p into self
T = homomorphism(G_p, self, G_p.generators, gens)
C_p = CosetTable(G_p, [])
C_p.table = [[None]*(2*len_g) for i in range(n)]
# initiate the coset transversal
transversal = [None]*n
transversal[0] = G_p.identity
# fill in the coset table as much as possible
for i in range(2*len_h):
C_p.table[0][i] = 0
gamma = 1
for alpha, x in product(range(0, n), range(2*len_g)):
beta = C[alpha][x]
if beta == gamma:
gen = G_p.generators[x//2]**((-1)**(x % 2))
transversal[beta] = transversal[alpha]*gen
C_p.table[alpha][x] = beta
C_p.table[beta][x + (-1)**(x % 2)] = alpha
gamma += 1
if gamma == n:
break
C_p.p = list(range(n))
beta = x = 0
while not C_p.is_complete():
# find the first undefined entry
while C_p.table[beta][x] == C[beta][x]:
x = (x + 1) % (2*len_g)
if x == 0:
beta = (beta + 1) % n
# define a new relator
gen = G_p.generators[x//2]**((-1)**(x % 2))
new_rel = transversal[beta]*gen*transversal[C[beta][x]]**-1
perm = T(new_rel)
nxt = G_p.identity
for s in H.generator_product(perm, original=True):
nxt = nxt*T.invert(s)**-1
new_rel = new_rel*nxt
# continue coset enumeration
G_p = _factor_group_by_rels(G_p, [new_rel])
C_p.scan_and_fill(0, new_rel)
C_p = G_p.coset_enumeration([], strategy="coset_table",
draft=C_p, max_cosets=n, incomplete=True)
self._fp_presentation = simplify_presentation(G_p)
return self._fp_presentation
def polycyclic_group(self):
"""
Return the PolycyclicGroup instance with below parameters:
Explanation
===========
* ``pc_sequence`` : Polycyclic sequence is formed by collecting all
the missing generators between the adjacent groups in the
derived series of given permutation group.
* ``pc_series`` : Polycyclic series is formed by adding all the missing
generators of ``der[i+1]`` in ``der[i]``, where ``der`` represents
the derived series.
* ``relative_order`` : A list, computed by the ratio of adjacent groups in
pc_series.
"""
from sympy.combinatorics.pc_groups import PolycyclicGroup
if not self.is_polycyclic:
raise ValueError("The group must be solvable")
der = self.derived_series()
pc_series = []
pc_sequence = []
relative_order = []
pc_series.append(der[-1])
der.reverse()
for i in range(len(der)-1):
H = der[i]
for g in der[i+1].generators:
if g not in H:
H = PermutationGroup([g] + H.generators)
pc_series.insert(0, H)
pc_sequence.insert(0, g)
G1 = pc_series[0].order()
G2 = pc_series[1].order()
relative_order.insert(0, G1 // G2)
return PolycyclicGroup(pc_sequence, pc_series, relative_order, collector=None)
def _orbit(degree, generators, alpha, action='tuples'):
r"""Compute the orbit of alpha `\{g(\alpha) | g \in G\}` as a set.
Explanation
===========
The time complexity of the algorithm used here is `O(|Orb|*r)` where
`|Orb|` is the size of the orbit and ``r`` is the number of generators of
the group. For a more detailed analysis, see [1], p.78, [2], pp. 19-21.
Here alpha can be a single point, or a list of points.
If alpha is a single point, the ordinary orbit is computed.
if alpha is a list of points, there are three available options:
'union' - computes the union of the orbits of the points in the list
'tuples' - computes the orbit of the list interpreted as an ordered
tuple under the group action ( i.e., g((1, 2, 3)) = (g(1), g(2), g(3)) )
'sets' - computes the orbit of the list interpreted as a sets
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup, _orbit
>>> a = Permutation([1, 2, 0, 4, 5, 6, 3])
>>> G = PermutationGroup([a])
>>> _orbit(G.degree, G.generators, 0)
{0, 1, 2}
>>> _orbit(G.degree, G.generators, [0, 4], 'union')
{0, 1, 2, 3, 4, 5, 6}
See Also
========
orbit, orbit_transversal
"""
if not hasattr(alpha, '__getitem__'):
alpha = [alpha]
gens = [x._array_form for x in generators]
if len(alpha) == 1 or action == 'union':
orb = alpha
used = [False]*degree
for el in alpha:
used[el] = True
for b in orb:
for gen in gens:
temp = gen[b]
if used[temp] == False:
orb.append(temp)
used[temp] = True
return set(orb)
elif action == 'tuples':
alpha = tuple(alpha)
orb = [alpha]
used = {alpha}
for b in orb:
for gen in gens:
temp = tuple([gen[x] for x in b])
if temp not in used:
orb.append(temp)
used.add(temp)
return set(orb)
elif action == 'sets':
alpha = frozenset(alpha)
orb = [alpha]
used = {alpha}
for b in orb:
for gen in gens:
temp = frozenset([gen[x] for x in b])
if temp not in used:
orb.append(temp)
used.add(temp)
return {tuple(x) for x in orb}
def _orbits(degree, generators):
"""Compute the orbits of G.
If ``rep=False`` it returns a list of sets else it returns a list of
representatives of the orbits
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.perm_groups import _orbits
>>> a = Permutation([0, 2, 1])
>>> b = Permutation([1, 0, 2])
>>> _orbits(a.size, [a, b])
[{0, 1, 2}]
"""
orbs = []
sorted_I = list(range(degree))
I = set(sorted_I)
while I:
i = sorted_I[0]
orb = _orbit(degree, generators, i)
orbs.append(orb)
# remove all indices that are in this orbit
I -= orb
sorted_I = [i for i in sorted_I if i not in orb]
return orbs
def _orbit_transversal(degree, generators, alpha, pairs, af=False, slp=False):
r"""Computes a transversal for the orbit of ``alpha`` as a set.
Explanation
===========
generators generators of the group ``G``
For a permutation group ``G``, a transversal for the orbit
`Orb = \{g(\alpha) | g \in G\}` is a set
`\{g_\beta | g_\beta(\alpha) = \beta\}` for `\beta \in Orb`.
Note that there may be more than one possible transversal.
If ``pairs`` is set to ``True``, it returns the list of pairs
`(\beta, g_\beta)`. For a proof of correctness, see [1], p.79
if ``af`` is ``True``, the transversal elements are given in
array form.
If `slp` is `True`, a dictionary `{beta: slp_beta}` is returned
for `\beta \in Orb` where `slp_beta` is a list of indices of the
generators in `generators` s.t. if `slp_beta = [i_1 \dots i_n]`
`g_\beta = generators[i_n] \times \dots \times generators[i_1]`.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.perm_groups import _orbit_transversal
>>> G = DihedralGroup(6)
>>> _orbit_transversal(G.degree, G.generators, 0, False)
[(5), (0 1 2 3 4 5), (0 5)(1 4)(2 3), (0 2 4)(1 3 5), (5)(0 4)(1 3), (0 3)(1 4)(2 5)]
"""
tr = [(alpha, list(range(degree)))]
slp_dict = {alpha: []}
used = [False]*degree
used[alpha] = True
gens = [x._array_form for x in generators]
for x, px in tr:
px_slp = slp_dict[x]
for gen in gens:
temp = gen[x]
if used[temp] == False:
slp_dict[temp] = [gens.index(gen)] + px_slp
tr.append((temp, _af_rmul(gen, px)))
used[temp] = True
if pairs:
if not af:
tr = [(x, _af_new(y)) for x, y in tr]
if not slp:
return tr
return tr, slp_dict
if af:
tr = [y for _, y in tr]
if not slp:
return tr
return tr, slp_dict
tr = [_af_new(y) for _, y in tr]
if not slp:
return tr
return tr, slp_dict
def _stabilizer(degree, generators, alpha):
r"""Return the stabilizer subgroup of ``alpha``.
Explanation
===========
The stabilizer of `\alpha` is the group `G_\alpha =
\{g \in G | g(\alpha) = \alpha\}`.
For a proof of correctness, see [1], p.79.
degree : degree of G
generators : generators of G
Examples
========
>>> from sympy.combinatorics.perm_groups import _stabilizer
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> G = DihedralGroup(6)
>>> _stabilizer(G.degree, G.generators, 5)
[(5)(0 4)(1 3), (5)]
See Also
========
orbit
"""
orb = [alpha]
table = {alpha: list(range(degree))}
table_inv = {alpha: list(range(degree))}
used = [False]*degree
used[alpha] = True
gens = [x._array_form for x in generators]
stab_gens = []
for b in orb:
for gen in gens:
temp = gen[b]
if used[temp] is False:
gen_temp = _af_rmul(gen, table[b])
orb.append(temp)
table[temp] = gen_temp
table_inv[temp] = _af_invert(gen_temp)
used[temp] = True
else:
schreier_gen = _af_rmuln(table_inv[temp], gen, table[b])
if schreier_gen not in stab_gens:
stab_gens.append(schreier_gen)
return [_af_new(x) for x in stab_gens]
PermGroup = PermutationGroup
class SymmetricPermutationGroup(Basic):
"""
The class defining the lazy form of SymmetricGroup.
deg : int
"""
def __new__(cls, deg):
deg = _sympify(deg)
obj = Basic.__new__(cls, deg)
return obj
def __init__(self, *args, **kwargs):
self._deg = self.args[0]
self._order = None
def __contains__(self, i):
"""Return ``True`` if *i* is contained in SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import Permutation, SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> Permutation(1, 2, 3) in G
True
"""
if not isinstance(i, Permutation):
raise TypeError("A SymmetricPermutationGroup contains only Permutations as "
"elements, not elements of type %s" % type(i))
return i.size == self.degree
def order(self):
"""
Return the order of the SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> G.order()
24
"""
if self._order is not None:
return self._order
n = self._deg
self._order = factorial(n)
return self._order
@property
def degree(self):
"""
Return the degree of the SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> G.degree
4
"""
return self._deg
@property
def identity(self):
'''
Return the identity element of the SymmetricPermutationGroup.
Examples
========
>>> from sympy.combinatorics import SymmetricPermutationGroup
>>> G = SymmetricPermutationGroup(4)
>>> G.identity()
(3)
'''
return _af_new(list(range(self._deg)))
class Coset(Basic):
"""A left coset of a permutation group with respect to an element.
Parameters
==========
g : Permutation
H : PermutationGroup
dir : "+" or "-", If not specified by default it will be "+"
here ``dir`` specified the type of coset "+" represent the
right coset and "-" represent the left coset.
G : PermutationGroup, optional
The group which contains *H* as its subgroup and *g* as its
element.
If not specified, it would automatically become a symmetric
group ``SymmetricPermutationGroup(g.size)`` and
``SymmetricPermutationGroup(H.degree)`` if ``g.size`` and ``H.degree``
are matching.``SymmetricPermutationGroup`` is a lazy form of SymmetricGroup
used for representation purpose.
"""
def __new__(cls, g, H, G=None, dir="+"):
g = _sympify(g)
if not isinstance(g, Permutation):
raise NotImplementedError
H = _sympify(H)
if not isinstance(H, PermutationGroup):
raise NotImplementedError
if G is not None:
G = _sympify(G)
if not isinstance(G, PermutationGroup) and not isinstance(G, SymmetricPermutationGroup):
raise NotImplementedError
if not H.is_subgroup(G):
raise ValueError("{} must be a subgroup of {}.".format(H, G))
if g not in G:
raise ValueError("{} must be an element of {}.".format(g, G))
else:
g_size = g.size
h_degree = H.degree
if g_size != h_degree:
raise ValueError(
"The size of the permutation {} and the degree of "
"the permutation group {} should be matching "
.format(g, H))
G = SymmetricPermutationGroup(g.size)
if isinstance(dir, str):
dir = Symbol(dir)
elif not isinstance(dir, Symbol):
raise TypeError("dir must be of type basestring or "
"Symbol, not %s" % type(dir))
if str(dir) not in ('+', '-'):
raise ValueError("dir must be one of '+' or '-' not %s" % dir)
obj = Basic.__new__(cls, g, H, G, dir)
return obj
def __init__(self, *args, **kwargs):
self._dir = self.args[3]
@property
def is_left_coset(self):
"""
Check if the coset is left coset that is ``gH``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup, Coset
>>> a = Permutation(1, 2)
>>> b = Permutation(0, 1)
>>> G = PermutationGroup([a, b])
>>> cst = Coset(a, G, dir="-")
>>> cst.is_left_coset
True
"""
return str(self._dir) == '-'
@property
def is_right_coset(self):
"""
Check if the coset is right coset that is ``Hg``.
Examples
========
>>> from sympy.combinatorics import Permutation, PermutationGroup, Coset
>>> a = Permutation(1, 2)
>>> b = Permutation(0, 1)
>>> G = PermutationGroup([a, b])
>>> cst = Coset(a, G, dir="+")
>>> cst.is_right_coset
True
"""
return str(self._dir) == '+'
def as_list(self):
"""
Return all the elements of coset in the form of list.
"""
g = self.args[0]
H = self.args[1]
cst = []
if str(self._dir) == '+':
for h in H.elements:
cst.append(h*g)
else:
for h in H.elements:
cst.append(g*h)
return cst
|
c3c8ee4ee1f0b473c8af790896e2824a5605185904ec54a23acbc7a34a2bfd95 | from sympy.core import Basic, Integer
import random
class GrayCode(Basic):
"""
A Gray code is essentially a Hamiltonian walk on
a n-dimensional cube with edge length of one.
The vertices of the cube are represented by vectors
whose values are binary. The Hamilton walk visits
each vertex exactly once. The Gray code for a 3d
cube is ['000','100','110','010','011','111','101',
'001'].
A Gray code solves the problem of sequentially
generating all possible subsets of n objects in such
a way that each subset is obtained from the previous
one by either deleting or adding a single object.
In the above example, 1 indicates that the object is
present, and 0 indicates that its absent.
Gray codes have applications in statistics as well when
we want to compute various statistics related to subsets
in an efficient manner.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> list(a.generate_gray())
['000', '001', '011', '010', '110', '111', '101', '100']
>>> a = GrayCode(4)
>>> list(a.generate_gray())
['0000', '0001', '0011', '0010', '0110', '0111', '0101', '0100', \
'1100', '1101', '1111', '1110', '1010', '1011', '1001', '1000']
References
==========
.. [1] Nijenhuis,A. and Wilf,H.S.(1978).
Combinatorial Algorithms. Academic Press.
.. [2] Knuth, D. (2011). The Art of Computer Programming, Vol 4
Addison Wesley
"""
_skip = False
_current = 0
_rank = None
def __new__(cls, n, *args, **kw_args):
"""
Default constructor.
It takes a single argument ``n`` which gives the dimension of the Gray
code. The starting Gray code string (``start``) or the starting ``rank``
may also be given; the default is to start at rank = 0 ('0...0').
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> a
GrayCode(3)
>>> a.n
3
>>> a = GrayCode(3, start='100')
>>> a.current
'100'
>>> a = GrayCode(4, rank=4)
>>> a.current
'0110'
>>> a.rank
4
"""
if n < 1 or int(n) != n:
raise ValueError(
'Gray code dimension must be a positive integer, not %i' % n)
n = Integer(n)
args = (n,) + args
obj = Basic.__new__(cls, *args)
if 'start' in kw_args:
obj._current = kw_args["start"]
if len(obj._current) > n:
raise ValueError('Gray code start has length %i but '
'should not be greater than %i' % (len(obj._current), n))
elif 'rank' in kw_args:
if int(kw_args["rank"]) != kw_args["rank"]:
raise ValueError('Gray code rank must be a positive integer, '
'not %i' % kw_args["rank"])
obj._rank = int(kw_args["rank"]) % obj.selections
obj._current = obj.unrank(n, obj._rank)
return obj
def next(self, delta=1):
"""
Returns the Gray code a distance ``delta`` (default = 1) from the
current value in canonical order.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3, start='110')
>>> a.next().current
'111'
>>> a.next(-1).current
'010'
"""
return GrayCode(self.n, rank=(self.rank + delta) % self.selections)
@property
def selections(self):
"""
Returns the number of bit vectors in the Gray code.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> a.selections
8
"""
return 2**self.n
@property
def n(self):
"""
Returns the dimension of the Gray code.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(5)
>>> a.n
5
"""
return self.args[0]
def generate_gray(self, **hints):
"""
Generates the sequence of bit vectors of a Gray Code.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> list(a.generate_gray())
['000', '001', '011', '010', '110', '111', '101', '100']
>>> list(a.generate_gray(start='011'))
['011', '010', '110', '111', '101', '100']
>>> list(a.generate_gray(rank=4))
['110', '111', '101', '100']
See Also
========
skip
References
==========
.. [1] Knuth, D. (2011). The Art of Computer Programming,
Vol 4, Addison Wesley
"""
bits = self.n
start = None
if "start" in hints:
start = hints["start"]
elif "rank" in hints:
start = GrayCode.unrank(self.n, hints["rank"])
if start is not None:
self._current = start
current = self.current
graycode_bin = gray_to_bin(current)
if len(graycode_bin) > self.n:
raise ValueError('Gray code start has length %i but should '
'not be greater than %i' % (len(graycode_bin), bits))
self._current = int(current, 2)
graycode_int = int(''.join(graycode_bin), 2)
for i in range(graycode_int, 1 << bits):
if self._skip:
self._skip = False
else:
yield self.current
bbtc = (i ^ (i + 1))
gbtc = (bbtc ^ (bbtc >> 1))
self._current = (self._current ^ gbtc)
self._current = 0
def skip(self):
"""
Skips the bit generation.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> for i in a.generate_gray():
... if i == '010':
... a.skip()
... print(i)
...
000
001
011
010
111
101
100
See Also
========
generate_gray
"""
self._skip = True
@property
def rank(self):
"""
Ranks the Gray code.
A ranking algorithm determines the position (or rank)
of a combinatorial object among all the objects w.r.t.
a given order. For example, the 4 bit binary reflected
Gray code (BRGC) '0101' has a rank of 6 as it appears in
the 6th position in the canonical ordering of the family
of 4 bit Gray codes.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> a = GrayCode(3)
>>> list(a.generate_gray())
['000', '001', '011', '010', '110', '111', '101', '100']
>>> GrayCode(3, start='100').rank
7
>>> GrayCode(3, rank=7).current
'100'
See Also
========
unrank
References
==========
.. [1] http://statweb.stanford.edu/~susan/courses/s208/node12.html
"""
if self._rank is None:
self._rank = int(gray_to_bin(self.current), 2)
return self._rank
@property
def current(self):
"""
Returns the currently referenced Gray code as a bit string.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> GrayCode(3, start='100').current
'100'
"""
rv = self._current or '0'
if not isinstance(rv, str):
rv = bin(rv)[2:]
return rv.rjust(self.n, '0')
@classmethod
def unrank(self, n, rank):
"""
Unranks an n-bit sized Gray code of rank k. This method exists
so that a derivative GrayCode class can define its own code of
a given rank.
The string here is generated in reverse order to allow for tail-call
optimization.
Examples
========
>>> from sympy.combinatorics.graycode import GrayCode
>>> GrayCode(5, rank=3).current
'00010'
>>> GrayCode.unrank(5, 3)
'00010'
See Also
========
rank
"""
def _unrank(k, n):
if n == 1:
return str(k % 2)
m = 2**(n - 1)
if k < m:
return '0' + _unrank(k, n - 1)
return '1' + _unrank(m - (k % m) - 1, n - 1)
return _unrank(rank, n)
def random_bitstring(n):
"""
Generates a random bitlist of length n.
Examples
========
>>> from sympy.combinatorics.graycode import random_bitstring
>>> random_bitstring(3) # doctest: +SKIP
100
"""
return ''.join([random.choice('01') for i in range(n)])
def gray_to_bin(bin_list):
"""
Convert from Gray coding to binary coding.
We assume big endian encoding.
Examples
========
>>> from sympy.combinatorics.graycode import gray_to_bin
>>> gray_to_bin('100')
'111'
See Also
========
bin_to_gray
"""
b = [bin_list[0]]
for i in range(1, len(bin_list)):
b += str(int(b[i - 1] != bin_list[i]))
return ''.join(b)
def bin_to_gray(bin_list):
"""
Convert from binary coding to gray coding.
We assume big endian encoding.
Examples
========
>>> from sympy.combinatorics.graycode import bin_to_gray
>>> bin_to_gray('111')
'100'
See Also
========
gray_to_bin
"""
b = [bin_list[0]]
for i in range(1, len(bin_list)):
b += str(int(bin_list[i]) ^ int(bin_list[i - 1]))
return ''.join(b)
def get_subset_from_bitstring(super_set, bitstring):
"""
Gets the subset defined by the bitstring.
Examples
========
>>> from sympy.combinatorics.graycode import get_subset_from_bitstring
>>> get_subset_from_bitstring(['a', 'b', 'c', 'd'], '0011')
['c', 'd']
>>> get_subset_from_bitstring(['c', 'a', 'c', 'c'], '1100')
['c', 'a']
See Also
========
graycode_subsets
"""
if len(super_set) != len(bitstring):
raise ValueError("The sizes of the lists are not equal")
return [super_set[i] for i, j in enumerate(bitstring)
if bitstring[i] == '1']
def graycode_subsets(gray_code_set):
"""
Generates the subsets as enumerated by a Gray code.
Examples
========
>>> from sympy.combinatorics.graycode import graycode_subsets
>>> list(graycode_subsets(['a', 'b', 'c']))
[[], ['c'], ['b', 'c'], ['b'], ['a', 'b'], ['a', 'b', 'c'], \
['a', 'c'], ['a']]
>>> list(graycode_subsets(['a', 'b', 'c', 'c']))
[[], ['c'], ['c', 'c'], ['c'], ['b', 'c'], ['b', 'c', 'c'], \
['b', 'c'], ['b'], ['a', 'b'], ['a', 'b', 'c'], ['a', 'b', 'c', 'c'], \
['a', 'b', 'c'], ['a', 'c'], ['a', 'c', 'c'], ['a', 'c'], ['a']]
See Also
========
get_subset_from_bitstring
"""
for bitstring in list(GrayCode(len(gray_code_set)).generate_gray()):
yield get_subset_from_bitstring(gray_code_set, bitstring)
|
33350f8ae91d4f58418d958eaa68852a713e2d5144f9af90d755c30113daab16 | from sympy.combinatorics.rewritingsystem_fsm import StateMachine
class RewritingSystem:
'''
A class implementing rewriting systems for `FpGroup`s.
References
==========
.. [1] Epstein, D., Holt, D. and Rees, S. (1991).
The use of Knuth-Bendix methods to solve the word problem in automatic groups.
Journal of Symbolic Computation, 12(4-5), pp.397-414.
.. [2] GAP's Manual on its KBMAG package
https://www.gap-system.org/Manuals/pkg/kbmag-1.5.3/doc/manual.pdf
'''
def __init__(self, group):
from collections import deque
self.group = group
self.alphabet = group.generators
self._is_confluent = None
# these values are taken from [2]
self.maxeqns = 32767 # max rules
self.tidyint = 100 # rules before tidying
# _max_exceeded is True if maxeqns is exceeded
# at any point
self._max_exceeded = False
# Reduction automaton
self.reduction_automaton = None
self._new_rules = {}
# dictionary of reductions
self.rules = {}
self.rules_cache = deque([], 50)
self._init_rules()
# All the transition symbols in the automaton
generators = list(self.alphabet)
generators += [gen**-1 for gen in generators]
# Create a finite state machine as an instance of the StateMachine object
self.reduction_automaton = StateMachine('Reduction automaton for '+ repr(self.group), generators)
self.construct_automaton()
def set_max(self, n):
'''
Set the maximum number of rules that can be defined
'''
if n > self.maxeqns:
self._max_exceeded = False
self.maxeqns = n
return
@property
def is_confluent(self):
'''
Return `True` if the system is confluent
'''
if self._is_confluent is None:
self._is_confluent = self._check_confluence()
return self._is_confluent
def _init_rules(self):
identity = self.group.free_group.identity
for r in self.group.relators:
self.add_rule(r, identity)
self._remove_redundancies()
return
def _add_rule(self, r1, r2):
'''
Add the rule r1 -> r2 with no checking or further
deductions
'''
if len(self.rules) + 1 > self.maxeqns:
self._is_confluent = self._check_confluence()
self._max_exceeded = True
raise RuntimeError("Too many rules were defined.")
self.rules[r1] = r2
# Add the newly added rule to the `new_rules` dictionary.
if self.reduction_automaton:
self._new_rules[r1] = r2
def add_rule(self, w1, w2, check=False):
new_keys = set()
if w1 == w2:
return new_keys
if w1 < w2:
w1, w2 = w2, w1
if (w1, w2) in self.rules_cache:
return new_keys
self.rules_cache.append((w1, w2))
s1, s2 = w1, w2
# The following is the equivalent of checking
# s1 for overlaps with the implicit reductions
# {g*g**-1 -> <identity>} and {g**-1*g -> <identity>}
# for any generator g without installing the
# redundant rules that would result from processing
# the overlaps. See [1], Section 3 for details.
if len(s1) - len(s2) < 3:
if s1 not in self.rules:
new_keys.add(s1)
if not check:
self._add_rule(s1, s2)
if s2**-1 > s1**-1 and s2**-1 not in self.rules:
new_keys.add(s2**-1)
if not check:
self._add_rule(s2**-1, s1**-1)
# overlaps on the right
while len(s1) - len(s2) > -1:
g = s1[len(s1)-1]
s1 = s1.subword(0, len(s1)-1)
s2 = s2*g**-1
if len(s1) - len(s2) < 0:
if s2 not in self.rules:
if not check:
self._add_rule(s2, s1)
new_keys.add(s2)
elif len(s1) - len(s2) < 3:
new = self.add_rule(s1, s2, check)
new_keys.update(new)
# overlaps on the left
while len(w1) - len(w2) > -1:
g = w1[0]
w1 = w1.subword(1, len(w1))
w2 = g**-1*w2
if len(w1) - len(w2) < 0:
if w2 not in self.rules:
if not check:
self._add_rule(w2, w1)
new_keys.add(w2)
elif len(w1) - len(w2) < 3:
new = self.add_rule(w1, w2, check)
new_keys.update(new)
return new_keys
def _remove_redundancies(self, changes=False):
'''
Reduce left- and right-hand sides of reduction rules
and remove redundant equations (i.e. those for which
lhs == rhs). If `changes` is `True`, return a set
containing the removed keys and a set containing the
added keys
'''
removed = set()
added = set()
rules = self.rules.copy()
for r in rules:
v = self.reduce(r, exclude=r)
w = self.reduce(rules[r])
if v != r:
del self.rules[r]
removed.add(r)
if v > w:
added.add(v)
self.rules[v] = w
elif v < w:
added.add(w)
self.rules[w] = v
else:
self.rules[v] = w
if changes:
return removed, added
return
def make_confluent(self, check=False):
'''
Try to make the system confluent using the Knuth-Bendix
completion algorithm
'''
if self._max_exceeded:
return self._is_confluent
lhs = list(self.rules.keys())
def _overlaps(r1, r2):
len1 = len(r1)
len2 = len(r2)
result = []
for j in range(1, len1 + len2):
if (r1.subword(len1 - j, len1 + len2 - j, strict=False)
== r2.subword(j - len1, j, strict=False)):
a = r1.subword(0, len1-j, strict=False)
a = a*r2.subword(0, j-len1, strict=False)
b = r2.subword(j-len1, j, strict=False)
c = r2.subword(j, len2, strict=False)
c = c*r1.subword(len1 + len2 - j, len1, strict=False)
result.append(a*b*c)
return result
def _process_overlap(w, r1, r2, check):
s = w.eliminate_word(r1, self.rules[r1])
s = self.reduce(s)
t = w.eliminate_word(r2, self.rules[r2])
t = self.reduce(t)
if s != t:
if check:
# system not confluent
return [0]
try:
new_keys = self.add_rule(t, s, check)
return new_keys
except RuntimeError:
return False
return
added = 0
i = 0
while i < len(lhs):
r1 = lhs[i]
i += 1
# j could be i+1 to not
# check each pair twice but lhs
# is extended in the loop and the new
# elements have to be checked with the
# preceding ones. there is probably a better way
# to handle this
j = 0
while j < len(lhs):
r2 = lhs[j]
j += 1
if r1 == r2:
continue
overlaps = _overlaps(r1, r2)
overlaps.extend(_overlaps(r1**-1, r2))
if not overlaps:
continue
for w in overlaps:
new_keys = _process_overlap(w, r1, r2, check)
if new_keys:
if check:
return False
lhs.extend(new_keys)
added += len(new_keys)
elif new_keys == False:
# too many rules were added so the process
# couldn't complete
return self._is_confluent
if added > self.tidyint and not check:
# tidy up
r, a = self._remove_redundancies(changes=True)
added = 0
if r:
# reset i since some elements were removed
i = min([lhs.index(s) for s in r])
lhs = [l for l in lhs if l not in r]
lhs.extend(a)
if r1 in r:
# r1 was removed as redundant
break
self._is_confluent = True
if not check:
self._remove_redundancies()
return True
def _check_confluence(self):
return self.make_confluent(check=True)
def reduce(self, word, exclude=None):
'''
Apply reduction rules to `word` excluding the reduction rule
for the lhs equal to `exclude`
'''
rules = {r: self.rules[r] for r in self.rules if r != exclude}
# the following is essentially `eliminate_words()` code from the
# `FreeGroupElement` class, the only difference being the first
# "if" statement
again = True
new = word
while again:
again = False
for r in rules:
prev = new
if rules[r]**-1 > r**-1:
new = new.eliminate_word(r, rules[r], _all=True, inverse=False)
else:
new = new.eliminate_word(r, rules[r], _all=True)
if new != prev:
again = True
return new
def _compute_inverse_rules(self, rules):
'''
Compute the inverse rules for a given set of rules.
The inverse rules are used in the automaton for word reduction.
Arguments:
rules (dictionary): Rules for which the inverse rules are to computed.
Returns:
Dictionary of inverse_rules.
'''
inverse_rules = {}
for r in rules:
rule_key_inverse = r**-1
rule_value_inverse = (rules[r])**-1
if (rule_value_inverse < rule_key_inverse):
inverse_rules[rule_key_inverse] = rule_value_inverse
else:
inverse_rules[rule_value_inverse] = rule_key_inverse
return inverse_rules
def construct_automaton(self):
'''
Construct the automaton based on the set of reduction rules of the system.
Automata Design:
The accept states of the automaton are the proper prefixes of the left hand side of the rules.
The complete left hand side of the rules are the dead states of the automaton.
'''
self._add_to_automaton(self.rules)
def _add_to_automaton(self, rules):
'''
Add new states and transitions to the automaton.
Summary:
States corresponding to the new rules added to the system are computed and added to the automaton.
Transitions in the previously added states are also modified if necessary.
Arguments:
rules (dictionary) -- Dictionary of the newly added rules.
'''
# Automaton variables
automaton_alphabet = []
proper_prefixes = {}
# compute the inverses of all the new rules added
all_rules = rules
inverse_rules = self._compute_inverse_rules(all_rules)
all_rules.update(inverse_rules)
# Keep track of the accept_states.
accept_states = []
for rule in all_rules:
# The symbols present in the new rules are the symbols to be verified at each state.
# computes the automaton_alphabet, as the transitions solely depend upon the new states.
automaton_alphabet += rule.letter_form_elm
# Compute the proper prefixes for every rule.
proper_prefixes[rule] = []
letter_word_array = [s for s in rule.letter_form_elm]
len_letter_word_array = len(letter_word_array)
for i in range (1, len_letter_word_array):
letter_word_array[i] = letter_word_array[i-1]*letter_word_array[i]
# Add accept states.
elem = letter_word_array[i-1]
if elem not in self.reduction_automaton.states:
self.reduction_automaton.add_state(elem, state_type='a')
accept_states.append(elem)
proper_prefixes[rule] = letter_word_array
# Check for overlaps between dead and accept states.
if rule in accept_states:
self.reduction_automaton.states[rule].state_type = 'd'
self.reduction_automaton.states[rule].rh_rule = all_rules[rule]
accept_states.remove(rule)
# Add dead states
if rule not in self.reduction_automaton.states:
self.reduction_automaton.add_state(rule, state_type='d', rh_rule=all_rules[rule])
automaton_alphabet = set(automaton_alphabet)
# Add new transitions for every state.
for state in self.reduction_automaton.states:
current_state_name = state
current_state_type = self.reduction_automaton.states[state].state_type
# Transitions will be modified only when suffixes of the current_state
# belongs to the proper_prefixes of the new rules.
# The rest are ignored if they cannot lead to a dead state after a finite number of transisitons.
if current_state_type == 's':
for letter in automaton_alphabet:
if letter in self.reduction_automaton.states:
self.reduction_automaton.states[state].add_transition(letter, letter)
else:
self.reduction_automaton.states[state].add_transition(letter, current_state_name)
elif current_state_type == 'a':
# Check if the transition to any new state in possible.
for letter in automaton_alphabet:
_next = current_state_name*letter
while len(_next) and _next not in self.reduction_automaton.states:
_next = _next.subword(1, len(_next))
if not len(_next):
_next = 'start'
self.reduction_automaton.states[state].add_transition(letter, _next)
# Add transitions for new states. All symbols used in the automaton are considered here.
# Ignore this if `reduction_automaton.automaton_alphabet` = `automaton_alphabet`.
if len(self.reduction_automaton.automaton_alphabet) != len(automaton_alphabet):
for state in accept_states:
current_state_name = state
for letter in self.reduction_automaton.automaton_alphabet:
_next = current_state_name*letter
while len(_next) and _next not in self.reduction_automaton.states:
_next = _next.subword(1, len(_next))
if not len(_next):
_next = 'start'
self.reduction_automaton.states[state].add_transition(letter, _next)
def reduce_using_automaton(self, word):
'''
Reduce a word using an automaton.
Summary:
All the symbols of the word are stored in an array and are given as the input to the automaton.
If the automaton reaches a dead state that subword is replaced and the automaton is run from the beginning.
The complete word has to be replaced when the word is read and the automaton reaches a dead state.
So, this process is repeated until the word is read completely and the automaton reaches the accept state.
Arguments:
word (instance of FreeGroupElement) -- Word that needs to be reduced.
'''
# Modify the automaton if new rules are found.
if self._new_rules:
self._add_to_automaton(self._new_rules)
self._new_rules = {}
flag = 1
while flag:
flag = 0
current_state = self.reduction_automaton.states['start']
word_array = [s for s in word.letter_form_elm]
for i in range (0, len(word_array)):
next_state_name = current_state.transitions[word_array[i]]
next_state = self.reduction_automaton.states[next_state_name]
if next_state.state_type == 'd':
subst = next_state.rh_rule
word = word.substituted_word(i - len(next_state_name) + 1, i+1, subst)
flag = 1
break
current_state = next_state
return word
|
bab6c06c711f35bbfa4e6a09513553745da9a7ed53f4041a286d454cde128f7d | import random
from collections import defaultdict
from collections.abc import Iterable
from functools import reduce
from sympy.core.parameters import global_parameters
from sympy.core.basic import Atom
from sympy.core.expr import Expr
from sympy.core.numbers import Integer
from sympy.core.sympify import _sympify
from sympy.matrices import zeros
from sympy.polys.polytools import lcm
from sympy.utilities.iterables import (flatten, has_variety, minlex,
has_dups, runs, is_sequence)
from sympy.utilities.misc import as_int
from mpmath.libmp.libintmath import ifac
from sympy.multipledispatch import dispatch
def _af_rmul(a, b):
"""
Return the product b*a; input and output are array forms. The ith value
is a[b[i]].
Examples
========
>>> from sympy.combinatorics.permutations import _af_rmul, Permutation
>>> a, b = [1, 0, 2], [0, 2, 1]
>>> _af_rmul(a, b)
[1, 2, 0]
>>> [a[b[i]] for i in range(3)]
[1, 2, 0]
This handles the operands in reverse order compared to the ``*`` operator:
>>> a = Permutation(a)
>>> b = Permutation(b)
>>> list(a*b)
[2, 0, 1]
>>> [b(a(i)) for i in range(3)]
[2, 0, 1]
See Also
========
rmul, _af_rmuln
"""
return [a[i] for i in b]
def _af_rmuln(*abc):
"""
Given [a, b, c, ...] return the product of ...*c*b*a using array forms.
The ith value is a[b[c[i]]].
Examples
========
>>> from sympy.combinatorics.permutations import _af_rmul, Permutation
>>> a, b = [1, 0, 2], [0, 2, 1]
>>> _af_rmul(a, b)
[1, 2, 0]
>>> [a[b[i]] for i in range(3)]
[1, 2, 0]
This handles the operands in reverse order compared to the ``*`` operator:
>>> a = Permutation(a); b = Permutation(b)
>>> list(a*b)
[2, 0, 1]
>>> [b(a(i)) for i in range(3)]
[2, 0, 1]
See Also
========
rmul, _af_rmul
"""
a = abc
m = len(a)
if m == 3:
p0, p1, p2 = a
return [p0[p1[i]] for i in p2]
if m == 4:
p0, p1, p2, p3 = a
return [p0[p1[p2[i]]] for i in p3]
if m == 5:
p0, p1, p2, p3, p4 = a
return [p0[p1[p2[p3[i]]]] for i in p4]
if m == 6:
p0, p1, p2, p3, p4, p5 = a
return [p0[p1[p2[p3[p4[i]]]]] for i in p5]
if m == 7:
p0, p1, p2, p3, p4, p5, p6 = a
return [p0[p1[p2[p3[p4[p5[i]]]]]] for i in p6]
if m == 8:
p0, p1, p2, p3, p4, p5, p6, p7 = a
return [p0[p1[p2[p3[p4[p5[p6[i]]]]]]] for i in p7]
if m == 1:
return a[0][:]
if m == 2:
a, b = a
return [a[i] for i in b]
if m == 0:
raise ValueError("String must not be empty")
p0 = _af_rmuln(*a[:m//2])
p1 = _af_rmuln(*a[m//2:])
return [p0[i] for i in p1]
def _af_parity(pi):
"""
Computes the parity of a permutation in array form.
Explanation
===========
The parity of a permutation reflects the parity of the
number of inversions in the permutation, i.e., the
number of pairs of x and y such that x > y but p[x] < p[y].
Examples
========
>>> from sympy.combinatorics.permutations import _af_parity
>>> _af_parity([0, 1, 2, 3])
0
>>> _af_parity([3, 2, 0, 1])
1
See Also
========
Permutation
"""
n = len(pi)
a = [0] * n
c = 0
for j in range(n):
if a[j] == 0:
c += 1
a[j] = 1
i = j
while pi[i] != j:
i = pi[i]
a[i] = 1
return (n - c) % 2
def _af_invert(a):
"""
Finds the inverse, ~A, of a permutation, A, given in array form.
Examples
========
>>> from sympy.combinatorics.permutations import _af_invert, _af_rmul
>>> A = [1, 2, 0, 3]
>>> _af_invert(A)
[2, 0, 1, 3]
>>> _af_rmul(_, A)
[0, 1, 2, 3]
See Also
========
Permutation, __invert__
"""
inv_form = [0] * len(a)
for i, ai in enumerate(a):
inv_form[ai] = i
return inv_form
def _af_pow(a, n):
"""
Routine for finding powers of a permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation, _af_pow
>>> p = Permutation([2, 0, 3, 1])
>>> p.order()
4
>>> _af_pow(p._array_form, 4)
[0, 1, 2, 3]
"""
if n == 0:
return list(range(len(a)))
if n < 0:
return _af_pow(_af_invert(a), -n)
if n == 1:
return a[:]
elif n == 2:
b = [a[i] for i in a]
elif n == 3:
b = [a[a[i]] for i in a]
elif n == 4:
b = [a[a[a[i]]] for i in a]
else:
# use binary multiplication
b = list(range(len(a)))
while 1:
if n & 1:
b = [b[i] for i in a]
n -= 1
if not n:
break
if n % 4 == 0:
a = [a[a[a[i]]] for i in a]
n = n // 4
elif n % 2 == 0:
a = [a[i] for i in a]
n = n // 2
return b
def _af_commutes_with(a, b):
"""
Checks if the two permutations with array forms
given by ``a`` and ``b`` commute.
Examples
========
>>> from sympy.combinatorics.permutations import _af_commutes_with
>>> _af_commutes_with([1, 2, 0], [0, 2, 1])
False
See Also
========
Permutation, commutes_with
"""
return not any(a[b[i]] != b[a[i]] for i in range(len(a) - 1))
class Cycle(dict):
"""
Wrapper around dict which provides the functionality of a disjoint cycle.
Explanation
===========
A cycle shows the rule to use to move subsets of elements to obtain
a permutation. The Cycle class is more flexible than Permutation in
that 1) all elements need not be present in order to investigate how
multiple cycles act in sequence and 2) it can contain singletons:
>>> from sympy.combinatorics.permutations import Perm, Cycle
A Cycle will automatically parse a cycle given as a tuple on the rhs:
>>> Cycle(1, 2)(2, 3)
(1 3 2)
The identity cycle, Cycle(), can be used to start a product:
>>> Cycle()(1, 2)(2, 3)
(1 3 2)
The array form of a Cycle can be obtained by calling the list
method (or passing it to the list function) and all elements from
0 will be shown:
>>> a = Cycle(1, 2)
>>> a.list()
[0, 2, 1]
>>> list(a)
[0, 2, 1]
If a larger (or smaller) range is desired use the list method and
provide the desired size -- but the Cycle cannot be truncated to
a size smaller than the largest element that is out of place:
>>> b = Cycle(2, 4)(1, 2)(3, 1, 4)(1, 3)
>>> b.list()
[0, 2, 1, 3, 4]
>>> b.list(b.size + 1)
[0, 2, 1, 3, 4, 5]
>>> b.list(-1)
[0, 2, 1]
Singletons are not shown when printing with one exception: the largest
element is always shown -- as a singleton if necessary:
>>> Cycle(1, 4, 10)(4, 5)
(1 5 4 10)
>>> Cycle(1, 2)(4)(5)(10)
(1 2)(10)
The array form can be used to instantiate a Permutation so other
properties of the permutation can be investigated:
>>> Perm(Cycle(1, 2)(3, 4).list()).transpositions()
[(1, 2), (3, 4)]
Notes
=====
The underlying structure of the Cycle is a dictionary and although
the __iter__ method has been redefined to give the array form of the
cycle, the underlying dictionary items are still available with the
such methods as items():
>>> list(Cycle(1, 2).items())
[(1, 2), (2, 1)]
See Also
========
Permutation
"""
def __missing__(self, arg):
"""Enter arg into dictionary and return arg."""
return as_int(arg)
def __iter__(self):
yield from self.list()
def __call__(self, *other):
"""Return product of cycles processed from R to L.
Examples
========
>>> from sympy.combinatorics.permutations import Cycle as C
>>> C(1, 2)(2, 3)
(1 3 2)
An instance of a Cycle will automatically parse list-like
objects and Permutations that are on the right. It is more
flexible than the Permutation in that all elements need not
be present:
>>> a = C(1, 2)
>>> a(2, 3)
(1 3 2)
>>> a(2, 3)(4, 5)
(1 3 2)(4 5)
"""
rv = Cycle(*other)
for k, v in zip(list(self.keys()), [rv[self[k]] for k in self.keys()]):
rv[k] = v
return rv
def list(self, size=None):
"""Return the cycles as an explicit list starting from 0 up
to the greater of the largest value in the cycles and size.
Truncation of trailing unmoved items will occur when size
is less than the maximum element in the cycle; if this is
desired, setting ``size=-1`` will guarantee such trimming.
Examples
========
>>> from sympy.combinatorics.permutations import Cycle
>>> p = Cycle(2, 3)(4, 5)
>>> p.list()
[0, 1, 3, 2, 5, 4]
>>> p.list(10)
[0, 1, 3, 2, 5, 4, 6, 7, 8, 9]
Passing a length too small will trim trailing, unchanged elements
in the permutation:
>>> Cycle(2, 4)(1, 2, 4).list(-1)
[0, 2, 1]
"""
if not self and size is None:
raise ValueError('must give size for empty Cycle')
if size is not None:
big = max([i for i in self.keys() if self[i] != i] + [0])
size = max(size, big + 1)
else:
size = self.size
return [self[i] for i in range(size)]
def __repr__(self):
"""We want it to print as a Cycle, not as a dict.
Examples
========
>>> from sympy.combinatorics import Cycle
>>> Cycle(1, 2)
(1 2)
>>> print(_)
(1 2)
>>> list(Cycle(1, 2).items())
[(1, 2), (2, 1)]
"""
if not self:
return 'Cycle()'
cycles = Permutation(self).cyclic_form
s = ''.join(str(tuple(c)) for c in cycles)
big = self.size - 1
if not any(i == big for c in cycles for i in c):
s += '(%s)' % big
return 'Cycle%s' % s
def __str__(self):
"""We want it to be printed in a Cycle notation with no
comma in-between.
Examples
========
>>> from sympy.combinatorics import Cycle
>>> Cycle(1, 2)
(1 2)
>>> Cycle(1, 2, 4)(5, 6)
(1 2 4)(5 6)
"""
if not self:
return '()'
cycles = Permutation(self).cyclic_form
s = ''.join(str(tuple(c)) for c in cycles)
big = self.size - 1
if not any(i == big for c in cycles for i in c):
s += '(%s)' % big
s = s.replace(',', '')
return s
def __init__(self, *args):
"""Load up a Cycle instance with the values for the cycle.
Examples
========
>>> from sympy.combinatorics.permutations import Cycle
>>> Cycle(1, 2, 6)
(1 2 6)
"""
if not args:
return
if len(args) == 1:
if isinstance(args[0], Permutation):
for c in args[0].cyclic_form:
self.update(self(*c))
return
elif isinstance(args[0], Cycle):
for k, v in args[0].items():
self[k] = v
return
args = [as_int(a) for a in args]
if any(i < 0 for i in args):
raise ValueError('negative integers are not allowed in a cycle.')
if has_dups(args):
raise ValueError('All elements must be unique in a cycle.')
for i in range(-len(args), 0):
self[args[i]] = args[i + 1]
@property
def size(self):
if not self:
return 0
return max(self.keys()) + 1
def copy(self):
return Cycle(self)
class Permutation(Atom):
"""
A permutation, alternatively known as an 'arrangement number' or 'ordering'
is an arrangement of the elements of an ordered list into a one-to-one
mapping with itself. The permutation of a given arrangement is given by
indicating the positions of the elements after re-arrangement [2]_. For
example, if one started with elements [x, y, a, b] (in that order) and
they were reordered as [x, y, b, a] then the permutation would be
[0, 1, 3, 2]. Notice that (in SymPy) the first element is always referred
to as 0 and the permutation uses the indices of the elements in the
original ordering, not the elements (a, b, etc...) themselves.
>>> from sympy.combinatorics import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
Permutations Notation
=====================
Permutations are commonly represented in disjoint cycle or array forms.
Array Notation and 2-line Form
------------------------------------
In the 2-line form, the elements and their final positions are shown
as a matrix with 2 rows:
[0 1 2 ... n-1]
[p(0) p(1) p(2) ... p(n-1)]
Since the first line is always range(n), where n is the size of p,
it is sufficient to represent the permutation by the second line,
referred to as the "array form" of the permutation. This is entered
in brackets as the argument to the Permutation class:
>>> p = Permutation([0, 2, 1]); p
Permutation([0, 2, 1])
Given i in range(p.size), the permutation maps i to i^p
>>> [i^p for i in range(p.size)]
[0, 2, 1]
The composite of two permutations p*q means first apply p, then q, so
i^(p*q) = (i^p)^q which is i^p^q according to Python precedence rules:
>>> q = Permutation([2, 1, 0])
>>> [i^p^q for i in range(3)]
[2, 0, 1]
>>> [i^(p*q) for i in range(3)]
[2, 0, 1]
One can use also the notation p(i) = i^p, but then the composition
rule is (p*q)(i) = q(p(i)), not p(q(i)):
>>> [(p*q)(i) for i in range(p.size)]
[2, 0, 1]
>>> [q(p(i)) for i in range(p.size)]
[2, 0, 1]
>>> [p(q(i)) for i in range(p.size)]
[1, 2, 0]
Disjoint Cycle Notation
-----------------------
In disjoint cycle notation, only the elements that have shifted are
indicated.
For example, [1, 3, 2, 0] can be represented as (0, 1, 3)(2).
This can be understood from the 2 line format of the given permutation.
In the 2-line form,
[0 1 2 3]
[1 3 2 0]
The element in the 0th position is 1, so 0 -> 1. The element in the 1st
position is three, so 1 -> 3. And the element in the third position is again
0, so 3 -> 0. Thus, 0 -> 1 -> 3 -> 0, and 2 -> 2. Thus, this can be represented
as 2 cycles: (0, 1, 3)(2).
In common notation, singular cycles are not explicitly written as they can be
inferred implicitly.
Only the relative ordering of elements in a cycle matter:
>>> Permutation(1,2,3) == Permutation(2,3,1) == Permutation(3,1,2)
True
The disjoint cycle notation is convenient when representing
permutations that have several cycles in them:
>>> Permutation(1, 2)(3, 5) == Permutation([[1, 2], [3, 5]])
True
It also provides some economy in entry when computing products of
permutations that are written in disjoint cycle notation:
>>> Permutation(1, 2)(1, 3)(2, 3)
Permutation([0, 3, 2, 1])
>>> _ == Permutation([[1, 2]])*Permutation([[1, 3]])*Permutation([[2, 3]])
True
Caution: when the cycles have common elements between them then the order
in which the permutations are applied matters. This module applies
the permutations from *left to right*.
>>> Permutation(1, 2)(2, 3) == Permutation([(1, 2), (2, 3)])
True
>>> Permutation(1, 2)(2, 3).list()
[0, 3, 1, 2]
In the above case, (1,2) is computed before (2,3).
As 0 -> 0, 0 -> 0, element in position 0 is 0.
As 1 -> 2, 2 -> 3, element in position 1 is 3.
As 2 -> 1, 1 -> 1, element in position 2 is 1.
As 3 -> 3, 3 -> 2, element in position 3 is 2.
If the first and second elements had been
swapped first, followed by the swapping of the second
and third, the result would have been [0, 2, 3, 1].
If, you want to apply the cycles in the conventional
right to left order, call the function with arguments in reverse order
as demonstrated below:
>>> Permutation([(1, 2), (2, 3)][::-1]).list()
[0, 2, 3, 1]
Entering a singleton in a permutation is a way to indicate the size of the
permutation. The ``size`` keyword can also be used.
Array-form entry:
>>> Permutation([[1, 2], [9]])
Permutation([0, 2, 1], size=10)
>>> Permutation([[1, 2]], size=10)
Permutation([0, 2, 1], size=10)
Cyclic-form entry:
>>> Permutation(1, 2, size=10)
Permutation([0, 2, 1], size=10)
>>> Permutation(9)(1, 2)
Permutation([0, 2, 1], size=10)
Caution: no singleton containing an element larger than the largest
in any previous cycle can be entered. This is an important difference
in how Permutation and Cycle handle the __call__ syntax. A singleton
argument at the start of a Permutation performs instantiation of the
Permutation and is permitted:
>>> Permutation(5)
Permutation([], size=6)
A singleton entered after instantiation is a call to the permutation
-- a function call -- and if the argument is out of range it will
trigger an error. For this reason, it is better to start the cycle
with the singleton:
The following fails because there is no element 3:
>>> Permutation(1, 2)(3)
Traceback (most recent call last):
...
IndexError: list index out of range
This is ok: only the call to an out of range singleton is prohibited;
otherwise the permutation autosizes:
>>> Permutation(3)(1, 2)
Permutation([0, 2, 1, 3])
>>> Permutation(1, 2)(3, 4) == Permutation(3, 4)(1, 2)
True
Equality testing
----------------
The array forms must be the same in order for permutations to be equal:
>>> Permutation([1, 0, 2, 3]) == Permutation([1, 0])
False
Identity Permutation
--------------------
The identity permutation is a permutation in which no element is out of
place. It can be entered in a variety of ways. All the following create
an identity permutation of size 4:
>>> I = Permutation([0, 1, 2, 3])
>>> all(p == I for p in [
... Permutation(3),
... Permutation(range(4)),
... Permutation([], size=4),
... Permutation(size=4)])
True
Watch out for entering the range *inside* a set of brackets (which is
cycle notation):
>>> I == Permutation([range(4)])
False
Permutation Printing
====================
There are a few things to note about how Permutations are printed.
1) If you prefer one form (array or cycle) over another, you can set
``init_printing`` with the ``perm_cyclic`` flag.
>>> from sympy import init_printing
>>> p = Permutation(1, 2)(4, 5)(3, 4)
>>> p
Permutation([0, 2, 1, 4, 5, 3])
>>> init_printing(perm_cyclic=True, pretty_print=False)
>>> p
(1 2)(3 4 5)
2) Regardless of the setting, a list of elements in the array for cyclic
form can be obtained and either of those can be copied and supplied as
the argument to Permutation:
>>> p.array_form
[0, 2, 1, 4, 5, 3]
>>> p.cyclic_form
[[1, 2], [3, 4, 5]]
>>> Permutation(_) == p
True
3) Printing is economical in that as little as possible is printed while
retaining all information about the size of the permutation:
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> Permutation([1, 0, 2, 3])
Permutation([1, 0, 2, 3])
>>> Permutation([1, 0, 2, 3], size=20)
Permutation([1, 0], size=20)
>>> Permutation([1, 0, 2, 4, 3, 5, 6], size=20)
Permutation([1, 0, 2, 4, 3], size=20)
>>> p = Permutation([1, 0, 2, 3])
>>> init_printing(perm_cyclic=True, pretty_print=False)
>>> p
(3)(0 1)
>>> init_printing(perm_cyclic=False, pretty_print=False)
The 2 was not printed but it is still there as can be seen with the
array_form and size methods:
>>> p.array_form
[1, 0, 2, 3]
>>> p.size
4
Short introduction to other methods
===================================
The permutation can act as a bijective function, telling what element is
located at a given position
>>> q = Permutation([5, 2, 3, 4, 1, 0])
>>> q.array_form[1] # the hard way
2
>>> q(1) # the easy way
2
>>> {i: q(i) for i in range(q.size)} # showing the bijection
{0: 5, 1: 2, 2: 3, 3: 4, 4: 1, 5: 0}
The full cyclic form (including singletons) can be obtained:
>>> p.full_cyclic_form
[[0, 1], [2], [3]]
Any permutation can be factored into transpositions of pairs of elements:
>>> Permutation([[1, 2], [3, 4, 5]]).transpositions()
[(1, 2), (3, 5), (3, 4)]
>>> Permutation.rmul(*[Permutation([ti], size=6) for ti in _]).cyclic_form
[[1, 2], [3, 4, 5]]
The number of permutations on a set of n elements is given by n! and is
called the cardinality.
>>> p.size
4
>>> p.cardinality
24
A given permutation has a rank among all the possible permutations of the
same elements, but what that rank is depends on how the permutations are
enumerated. (There are a number of different methods of doing so.) The
lexicographic rank is given by the rank method and this rank is used to
increment a permutation with addition/subtraction:
>>> p.rank()
6
>>> p + 1
Permutation([1, 0, 3, 2])
>>> p.next_lex()
Permutation([1, 0, 3, 2])
>>> _.rank()
7
>>> p.unrank_lex(p.size, rank=7)
Permutation([1, 0, 3, 2])
The product of two permutations p and q is defined as their composition as
functions, (p*q)(i) = q(p(i)) [6]_.
>>> p = Permutation([1, 0, 2, 3])
>>> q = Permutation([2, 3, 1, 0])
>>> list(q*p)
[2, 3, 0, 1]
>>> list(p*q)
[3, 2, 1, 0]
>>> [q(p(i)) for i in range(p.size)]
[3, 2, 1, 0]
The permutation can be 'applied' to any list-like object, not only
Permutations:
>>> p(['zero', 'one', 'four', 'two'])
['one', 'zero', 'four', 'two']
>>> p('zo42')
['o', 'z', '4', '2']
If you have a list of arbitrary elements, the corresponding permutation
can be found with the from_sequence method:
>>> Permutation.from_sequence('SymPy')
Permutation([1, 3, 2, 0, 4])
Checking if a Permutation is contained in a Group
=================================================
Generally if you have a group of permutations G on n symbols, and
you're checking if a permutation on less than n symbols is part
of that group, the check will fail.
Here is an example for n=5 and we check if the cycle
(1,2,3) is in G:
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=True, pretty_print=False)
>>> from sympy.combinatorics import Cycle, Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> G = PermutationGroup(Cycle(2, 3)(4, 5), Cycle(1, 2, 3, 4, 5))
>>> p1 = Permutation(Cycle(2, 5, 3))
>>> p2 = Permutation(Cycle(1, 2, 3))
>>> a1 = Permutation(Cycle(1, 2, 3).list(6))
>>> a2 = Permutation(Cycle(1, 2, 3)(5))
>>> a3 = Permutation(Cycle(1, 2, 3),size=6)
>>> for p in [p1,p2,a1,a2,a3]: p, G.contains(p)
((2 5 3), True)
((1 2 3), False)
((5)(1 2 3), True)
((5)(1 2 3), True)
((5)(1 2 3), True)
The check for p2 above will fail.
Checking if p1 is in G works because SymPy knows
G is a group on 5 symbols, and p1 is also on 5 symbols
(its largest element is 5).
For ``a1``, the ``.list(6)`` call will extend the permutation to 5
symbols, so the test will work as well. In the case of ``a2`` the
permutation is being extended to 5 symbols by using a singleton,
and in the case of ``a3`` it's extended through the constructor
argument ``size=6``.
There is another way to do this, which is to tell the ``contains``
method that the number of symbols the group is on doesn't need to
match perfectly the number of symbols for the permutation:
>>> G.contains(p2,strict=False)
True
This can be via the ``strict`` argument to the ``contains`` method,
and SymPy will try to extend the permutation on its own and then
perform the containment check.
See Also
========
Cycle
References
==========
.. [1] Skiena, S. 'Permutations.' 1.1 in Implementing Discrete Mathematics
Combinatorics and Graph Theory with Mathematica. Reading, MA:
Addison-Wesley, pp. 3-16, 1990.
.. [2] Knuth, D. E. The Art of Computer Programming, Vol. 4: Combinatorial
Algorithms, 1st ed. Reading, MA: Addison-Wesley, 2011.
.. [3] Wendy Myrvold and Frank Ruskey. 2001. Ranking and unranking
permutations in linear time. Inf. Process. Lett. 79, 6 (September 2001),
281-284. DOI=10.1016/S0020-0190(01)00141-7
.. [4] D. L. Kreher, D. R. Stinson 'Combinatorial Algorithms'
CRC Press, 1999
.. [5] Graham, R. L.; Knuth, D. E.; and Patashnik, O.
Concrete Mathematics: A Foundation for Computer Science, 2nd ed.
Reading, MA: Addison-Wesley, 1994.
.. [6] https://en.wikipedia.org/wiki/Permutation#Product_and_inverse
.. [7] https://en.wikipedia.org/wiki/Lehmer_code
"""
is_Permutation = True
_array_form = None
_cyclic_form = None
_cycle_structure = None
_size = None
_rank = None
def __new__(cls, *args, size=None, **kwargs):
"""
Constructor for the Permutation object from a list or a
list of lists in which all elements of the permutation may
appear only once.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
Permutations entered in array-form are left unaltered:
>>> Permutation([0, 2, 1])
Permutation([0, 2, 1])
Permutations entered in cyclic form are converted to array form;
singletons need not be entered, but can be entered to indicate the
largest element:
>>> Permutation([[4, 5, 6], [0, 1]])
Permutation([1, 0, 2, 3, 5, 6, 4])
>>> Permutation([[4, 5, 6], [0, 1], [19]])
Permutation([1, 0, 2, 3, 5, 6, 4], size=20)
All manipulation of permutations assumes that the smallest element
is 0 (in keeping with 0-based indexing in Python) so if the 0 is
missing when entering a permutation in array form, an error will be
raised:
>>> Permutation([2, 1])
Traceback (most recent call last):
...
ValueError: Integers 0 through 2 must be present.
If a permutation is entered in cyclic form, it can be entered without
singletons and the ``size`` specified so those values can be filled
in, otherwise the array form will only extend to the maximum value
in the cycles:
>>> Permutation([[1, 4], [3, 5, 2]], size=10)
Permutation([0, 4, 3, 5, 1, 2], size=10)
>>> _.array_form
[0, 4, 3, 5, 1, 2, 6, 7, 8, 9]
"""
if size is not None:
size = int(size)
#a) ()
#b) (1) = identity
#c) (1, 2) = cycle
#d) ([1, 2, 3]) = array form
#e) ([[1, 2]]) = cyclic form
#f) (Cycle) = conversion to permutation
#g) (Permutation) = adjust size or return copy
ok = True
if not args: # a
return cls._af_new(list(range(size or 0)))
elif len(args) > 1: # c
return cls._af_new(Cycle(*args).list(size))
if len(args) == 1:
a = args[0]
if isinstance(a, cls): # g
if size is None or size == a.size:
return a
return cls(a.array_form, size=size)
if isinstance(a, Cycle): # f
return cls._af_new(a.list(size))
if not is_sequence(a): # b
if size is not None and a + 1 > size:
raise ValueError('size is too small when max is %s' % a)
return cls._af_new(list(range(a + 1)))
if has_variety(is_sequence(ai) for ai in a):
ok = False
else:
ok = False
if not ok:
raise ValueError("Permutation argument must be a list of ints, "
"a list of lists, Permutation or Cycle.")
# safe to assume args are valid; this also makes a copy
# of the args
args = list(args[0])
is_cycle = args and is_sequence(args[0])
if is_cycle: # e
args = [[int(i) for i in c] for c in args]
else: # d
args = [int(i) for i in args]
# if there are n elements present, 0, 1, ..., n-1 should be present
# unless a cycle notation has been provided. A 0 will be added
# for convenience in case one wants to enter permutations where
# counting starts from 1.
temp = flatten(args)
if has_dups(temp) and not is_cycle:
raise ValueError('there were repeated elements.')
temp = set(temp)
if not is_cycle:
if temp != set(range(len(temp))):
raise ValueError('Integers 0 through %s must be present.' %
max(temp))
if size is not None and temp and max(temp) + 1 > size:
raise ValueError('max element should not exceed %s' % (size - 1))
if is_cycle:
# it's not necessarily canonical so we won't store
# it -- use the array form instead
c = Cycle()
for ci in args:
c = c(*ci)
aform = c.list()
else:
aform = list(args)
if size and size > len(aform):
# don't allow for truncation of permutation which
# might split a cycle and lead to an invalid aform
# but do allow the permutation size to be increased
aform.extend(list(range(len(aform), size)))
return cls._af_new(aform)
@classmethod
def _af_new(cls, perm):
"""A method to produce a Permutation object from a list;
the list is bound to the _array_form attribute, so it must
not be modified; this method is meant for internal use only;
the list ``a`` is supposed to be generated as a temporary value
in a method, so p = Perm._af_new(a) is the only object
to hold a reference to ``a``::
Examples
========
>>> from sympy.combinatorics.permutations import Perm
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> a = [2, 1, 3, 0]
>>> p = Perm._af_new(a)
>>> p
Permutation([2, 1, 3, 0])
"""
p = super().__new__(cls)
p._array_form = perm
p._size = len(perm)
return p
def _hashable_content(self):
# the array_form (a list) is the Permutation arg, so we need to
# return a tuple, instead
return tuple(self.array_form)
@property
def array_form(self):
"""
Return a copy of the attribute _array_form
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([[2, 0], [3, 1]])
>>> p.array_form
[2, 3, 0, 1]
>>> Permutation([[2, 0, 3, 1]]).array_form
[3, 2, 0, 1]
>>> Permutation([2, 0, 3, 1]).array_form
[2, 0, 3, 1]
>>> Permutation([[1, 2], [4, 5]]).array_form
[0, 2, 1, 3, 5, 4]
"""
return self._array_form[:]
def list(self, size=None):
"""Return the permutation as an explicit list, possibly
trimming unmoved elements if size is less than the maximum
element in the permutation; if this is desired, setting
``size=-1`` will guarantee such trimming.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation(2, 3)(4, 5)
>>> p.list()
[0, 1, 3, 2, 5, 4]
>>> p.list(10)
[0, 1, 3, 2, 5, 4, 6, 7, 8, 9]
Passing a length too small will trim trailing, unchanged elements
in the permutation:
>>> Permutation(2, 4)(1, 2, 4).list(-1)
[0, 2, 1]
>>> Permutation(3).list(-1)
[]
"""
if not self and size is None:
raise ValueError('must give size for empty Cycle')
rv = self.array_form
if size is not None:
if size > self.size:
rv.extend(list(range(self.size, size)))
else:
# find first value from rhs where rv[i] != i
i = self.size - 1
while rv:
if rv[-1] != i:
break
rv.pop()
i -= 1
return rv
@property
def cyclic_form(self):
"""
This is used to convert to the cyclic notation
from the canonical notation. Singletons are omitted.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 3, 1, 2])
>>> p.cyclic_form
[[1, 3, 2]]
>>> Permutation([1, 0, 2, 4, 3, 5]).cyclic_form
[[0, 1], [3, 4]]
See Also
========
array_form, full_cyclic_form
"""
if self._cyclic_form is not None:
return list(self._cyclic_form)
array_form = self.array_form
unchecked = [True] * len(array_form)
cyclic_form = []
for i in range(len(array_form)):
if unchecked[i]:
cycle = []
cycle.append(i)
unchecked[i] = False
j = i
while unchecked[array_form[j]]:
j = array_form[j]
cycle.append(j)
unchecked[j] = False
if len(cycle) > 1:
cyclic_form.append(cycle)
assert cycle == list(minlex(cycle))
cyclic_form.sort()
self._cyclic_form = cyclic_form[:]
return cyclic_form
@property
def full_cyclic_form(self):
"""Return permutation in cyclic form including singletons.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation([0, 2, 1]).full_cyclic_form
[[0], [1, 2]]
"""
need = set(range(self.size)) - set(flatten(self.cyclic_form))
rv = self.cyclic_form + [[i] for i in need]
rv.sort()
return rv
@property
def size(self):
"""
Returns the number of elements in the permutation.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation([[3, 2], [0, 1]]).size
4
See Also
========
cardinality, length, order, rank
"""
return self._size
def support(self):
"""Return the elements in permutation, P, for which P[i] != i.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> p = Permutation([[3, 2], [0, 1], [4]])
>>> p.array_form
[1, 0, 3, 2, 4]
>>> p.support()
[0, 1, 2, 3]
"""
a = self.array_form
return [i for i, e in enumerate(a) if a[i] != i]
def __add__(self, other):
"""Return permutation that is other higher in rank than self.
The rank is the lexicographical rank, with the identity permutation
having rank of 0.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> I = Permutation([0, 1, 2, 3])
>>> a = Permutation([2, 1, 3, 0])
>>> I + a.rank() == a
True
See Also
========
__sub__, inversion_vector
"""
rank = (self.rank() + other) % self.cardinality
rv = self.unrank_lex(self.size, rank)
rv._rank = rank
return rv
def __sub__(self, other):
"""Return the permutation that is other lower in rank than self.
See Also
========
__add__
"""
return self.__add__(-other)
@staticmethod
def rmul(*args):
"""
Return product of Permutations [a, b, c, ...] as the Permutation whose
ith value is a(b(c(i))).
a, b, c, ... can be Permutation objects or tuples.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> a, b = [1, 0, 2], [0, 2, 1]
>>> a = Permutation(a); b = Permutation(b)
>>> list(Permutation.rmul(a, b))
[1, 2, 0]
>>> [a(b(i)) for i in range(3)]
[1, 2, 0]
This handles the operands in reverse order compared to the ``*`` operator:
>>> a = Permutation(a); b = Permutation(b)
>>> list(a*b)
[2, 0, 1]
>>> [b(a(i)) for i in range(3)]
[2, 0, 1]
Notes
=====
All items in the sequence will be parsed by Permutation as
necessary as long as the first item is a Permutation:
>>> Permutation.rmul(a, [0, 2, 1]) == Permutation.rmul(a, b)
True
The reverse order of arguments will raise a TypeError.
"""
rv = args[0]
for i in range(1, len(args)):
rv = args[i]*rv
return rv
@classmethod
def rmul_with_af(cls, *args):
"""
same as rmul, but the elements of args are Permutation objects
which have _array_form
"""
a = [x._array_form for x in args]
rv = cls._af_new(_af_rmuln(*a))
return rv
def mul_inv(self, other):
"""
other*~self, self and other have _array_form
"""
a = _af_invert(self._array_form)
b = other._array_form
return self._af_new(_af_rmul(a, b))
def __rmul__(self, other):
"""This is needed to coerce other to Permutation in rmul."""
cls = type(self)
return cls(other)*self
def __mul__(self, other):
"""
Return the product a*b as a Permutation; the ith value is b(a(i)).
Examples
========
>>> from sympy.combinatorics.permutations import _af_rmul, Permutation
>>> a, b = [1, 0, 2], [0, 2, 1]
>>> a = Permutation(a); b = Permutation(b)
>>> list(a*b)
[2, 0, 1]
>>> [b(a(i)) for i in range(3)]
[2, 0, 1]
This handles operands in reverse order compared to _af_rmul and rmul:
>>> al = list(a); bl = list(b)
>>> _af_rmul(al, bl)
[1, 2, 0]
>>> [al[bl[i]] for i in range(3)]
[1, 2, 0]
It is acceptable for the arrays to have different lengths; the shorter
one will be padded to match the longer one:
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> b*Permutation([1, 0])
Permutation([1, 2, 0])
>>> Permutation([1, 0])*b
Permutation([2, 0, 1])
It is also acceptable to allow coercion to handle conversion of a
single list to the left of a Permutation:
>>> [0, 1]*a # no change: 2-element identity
Permutation([1, 0, 2])
>>> [[0, 1]]*a # exchange first two elements
Permutation([0, 1, 2])
You cannot use more than 1 cycle notation in a product of cycles
since coercion can only handle one argument to the left. To handle
multiple cycles it is convenient to use Cycle instead of Permutation:
>>> [[1, 2]]*[[2, 3]]*Permutation([]) # doctest: +SKIP
>>> from sympy.combinatorics.permutations import Cycle
>>> Cycle(1, 2)(2, 3)
(1 3 2)
"""
from sympy.combinatorics.perm_groups import PermutationGroup, Coset
if isinstance(other, PermutationGroup):
return Coset(self, other, dir='-')
a = self.array_form
# __rmul__ makes sure the other is a Permutation
b = other.array_form
if not b:
perm = a
else:
b.extend(list(range(len(b), len(a))))
perm = [b[i] for i in a] + b[len(a):]
return self._af_new(perm)
def commutes_with(self, other):
"""
Checks if the elements are commuting.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> a = Permutation([1, 4, 3, 0, 2, 5])
>>> b = Permutation([0, 1, 2, 3, 4, 5])
>>> a.commutes_with(b)
True
>>> b = Permutation([2, 3, 5, 4, 1, 0])
>>> a.commutes_with(b)
False
"""
a = self.array_form
b = other.array_form
return _af_commutes_with(a, b)
def __pow__(self, n):
"""
Routine for finding powers of a permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> p = Permutation([2, 0, 3, 1])
>>> p.order()
4
>>> p**4
Permutation([0, 1, 2, 3])
"""
if isinstance(n, Permutation):
raise NotImplementedError(
'p**p is not defined; do you mean p^p (conjugate)?')
n = int(n)
return self._af_new(_af_pow(self.array_form, n))
def __rxor__(self, i):
"""Return self(i) when ``i`` is an int.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> p = Permutation(1, 2, 9)
>>> 2^p == p(2) == 9
True
"""
if int(i) == i:
return self(i)
else:
raise NotImplementedError(
"i^p = p(i) when i is an integer, not %s." % i)
def __xor__(self, h):
"""Return the conjugate permutation ``~h*self*h` `.
Explanation
===========
If ``a`` and ``b`` are conjugates, ``a = h*b*~h`` and
``b = ~h*a*h`` and both have the same cycle structure.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation(1, 2, 9)
>>> q = Permutation(6, 9, 8)
>>> p*q != q*p
True
Calculate and check properties of the conjugate:
>>> c = p^q
>>> c == ~q*p*q and p == q*c*~q
True
The expression q^p^r is equivalent to q^(p*r):
>>> r = Permutation(9)(4, 6, 8)
>>> q^p^r == q^(p*r)
True
If the term to the left of the conjugate operator, i, is an integer
then this is interpreted as selecting the ith element from the
permutation to the right:
>>> all(i^p == p(i) for i in range(p.size))
True
Note that the * operator as higher precedence than the ^ operator:
>>> q^r*p^r == q^(r*p)^r == Permutation(9)(1, 6, 4)
True
Notes
=====
In Python the precedence rule is p^q^r = (p^q)^r which differs
in general from p^(q^r)
>>> q^p^r
(9)(1 4 8)
>>> q^(p^r)
(9)(1 8 6)
For a given r and p, both of the following are conjugates of p:
~r*p*r and r*p*~r. But these are not necessarily the same:
>>> ~r*p*r == r*p*~r
True
>>> p = Permutation(1, 2, 9)(5, 6)
>>> ~r*p*r == r*p*~r
False
The conjugate ~r*p*r was chosen so that ``p^q^r`` would be equivalent
to ``p^(q*r)`` rather than ``p^(r*q)``. To obtain r*p*~r, pass ~r to
this method:
>>> p^~r == r*p*~r
True
"""
if self.size != h.size:
raise ValueError("The permutations must be of equal size.")
a = [None]*self.size
h = h._array_form
p = self._array_form
for i in range(self.size):
a[h[i]] = h[p[i]]
return self._af_new(a)
def transpositions(self):
"""
Return the permutation decomposed into a list of transpositions.
Explanation
===========
It is always possible to express a permutation as the product of
transpositions, see [1]
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([[1, 2, 3], [0, 4, 5, 6, 7]])
>>> t = p.transpositions()
>>> t
[(0, 7), (0, 6), (0, 5), (0, 4), (1, 3), (1, 2)]
>>> print(''.join(str(c) for c in t))
(0, 7)(0, 6)(0, 5)(0, 4)(1, 3)(1, 2)
>>> Permutation.rmul(*[Permutation([ti], size=p.size) for ti in t]) == p
True
References
==========
.. [1] https://en.wikipedia.org/wiki/Transposition_%28mathematics%29#Properties
"""
a = self.cyclic_form
res = []
for x in a:
nx = len(x)
if nx == 2:
res.append(tuple(x))
elif nx > 2:
first = x[0]
for y in x[nx - 1:0:-1]:
res.append((first, y))
return res
@classmethod
def from_sequence(self, i, key=None):
"""Return the permutation needed to obtain ``i`` from the sorted
elements of ``i``. If custom sorting is desired, a key can be given.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation.from_sequence('SymPy')
(4)(0 1 3)
>>> _(sorted("SymPy"))
['S', 'y', 'm', 'P', 'y']
>>> Permutation.from_sequence('SymPy', key=lambda x: x.lower())
(4)(0 2)(1 3)
"""
ic = list(zip(i, list(range(len(i)))))
if key:
ic.sort(key=lambda x: key(x[0]))
else:
ic.sort()
return ~Permutation([i[1] for i in ic])
def __invert__(self):
"""
Return the inverse of the permutation.
A permutation multiplied by its inverse is the identity permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> p = Permutation([[2, 0], [3, 1]])
>>> ~p
Permutation([2, 3, 0, 1])
>>> _ == p**-1
True
>>> p*~p == ~p*p == Permutation([0, 1, 2, 3])
True
"""
return self._af_new(_af_invert(self._array_form))
def __iter__(self):
"""Yield elements from array form.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> list(Permutation(range(3)))
[0, 1, 2]
"""
yield from self.array_form
def __repr__(self):
from sympy.printing.repr import srepr
return srepr(self)
def __call__(self, *i):
"""
Allows applying a permutation instance as a bijective function.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([[2, 0], [3, 1]])
>>> p.array_form
[2, 3, 0, 1]
>>> [p(i) for i in range(4)]
[2, 3, 0, 1]
If an array is given then the permutation selects the items
from the array (i.e. the permutation is applied to the array):
>>> from sympy.abc import x
>>> p([x, 1, 0, x**2])
[0, x**2, x, 1]
"""
# list indices can be Integer or int; leave this
# as it is (don't test or convert it) because this
# gets called a lot and should be fast
if len(i) == 1:
i = i[0]
if not isinstance(i, Iterable):
i = as_int(i)
if i < 0 or i > self.size:
raise TypeError(
"{} should be an integer between 0 and {}"
.format(i, self.size-1))
return self._array_form[i]
# P([a, b, c])
if len(i) != self.size:
raise TypeError(
"{} should have the length {}.".format(i, self.size))
return [i[j] for j in self._array_form]
# P(1, 2, 3)
return self*Permutation(Cycle(*i), size=self.size)
def atoms(self):
"""
Returns all the elements of a permutation
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation([0, 1, 2, 3, 4, 5]).atoms()
{0, 1, 2, 3, 4, 5}
>>> Permutation([[0, 1], [2, 3], [4, 5]]).atoms()
{0, 1, 2, 3, 4, 5}
"""
return set(self.array_form)
def apply(self, i):
r"""Apply the permutation to an expression.
Parameters
==========
i : Expr
It should be an integer between $0$ and $n-1$ where $n$
is the size of the permutation.
If it is a symbol or a symbolic expression that can
have integer values, an ``AppliedPermutation`` object
will be returned which can represent an unevaluated
function.
Notes
=====
Any permutation can be defined as a bijective function
$\sigma : \{ 0, 1, \dots, n-1 \} \rightarrow \{ 0, 1, \dots, n-1 \}$
where $n$ denotes the size of the permutation.
The definition may even be extended for any set with distinctive
elements, such that the permutation can even be applied for
real numbers or such, however, it is not implemented for now for
computational reasons and the integrity with the group theory
module.
This function is similar to the ``__call__`` magic, however,
``__call__`` magic already has some other applications like
permuting an array or attatching new cycles, which would
not always be mathematically consistent.
This also guarantees that the return type is a SymPy integer,
which guarantees the safety to use assumptions.
"""
i = _sympify(i)
if i.is_integer is False:
raise NotImplementedError("{} should be an integer.".format(i))
n = self.size
if (i < 0) == True or (i >= n) == True:
raise NotImplementedError(
"{} should be an integer between 0 and {}".format(i, n-1))
if i.is_Integer:
return Integer(self._array_form[i])
return AppliedPermutation(self, i)
def next_lex(self):
"""
Returns the next permutation in lexicographical order.
If self is the last permutation in lexicographical order
it returns None.
See [4] section 2.4.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([2, 3, 1, 0])
>>> p = Permutation([2, 3, 1, 0]); p.rank()
17
>>> p = p.next_lex(); p.rank()
18
See Also
========
rank, unrank_lex
"""
perm = self.array_form[:]
n = len(perm)
i = n - 2
while perm[i + 1] < perm[i]:
i -= 1
if i == -1:
return None
else:
j = n - 1
while perm[j] < perm[i]:
j -= 1
perm[j], perm[i] = perm[i], perm[j]
i += 1
j = n - 1
while i < j:
perm[j], perm[i] = perm[i], perm[j]
i += 1
j -= 1
return self._af_new(perm)
@classmethod
def unrank_nonlex(self, n, r):
"""
This is a linear time unranking algorithm that does not
respect lexicographic order [3].
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> Permutation.unrank_nonlex(4, 5)
Permutation([2, 0, 3, 1])
>>> Permutation.unrank_nonlex(4, -1)
Permutation([0, 1, 2, 3])
See Also
========
next_nonlex, rank_nonlex
"""
def _unrank1(n, r, a):
if n > 0:
a[n - 1], a[r % n] = a[r % n], a[n - 1]
_unrank1(n - 1, r//n, a)
id_perm = list(range(n))
n = int(n)
r = r % ifac(n)
_unrank1(n, r, id_perm)
return self._af_new(id_perm)
def rank_nonlex(self, inv_perm=None):
"""
This is a linear time ranking algorithm that does not
enforce lexicographic order [3].
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3])
>>> p.rank_nonlex()
23
See Also
========
next_nonlex, unrank_nonlex
"""
def _rank1(n, perm, inv_perm):
if n == 1:
return 0
s = perm[n - 1]
t = inv_perm[n - 1]
perm[n - 1], perm[t] = perm[t], s
inv_perm[n - 1], inv_perm[s] = inv_perm[s], t
return s + n*_rank1(n - 1, perm, inv_perm)
if inv_perm is None:
inv_perm = (~self).array_form
if not inv_perm:
return 0
perm = self.array_form[:]
r = _rank1(len(perm), perm, inv_perm)
return r
def next_nonlex(self):
"""
Returns the next permutation in nonlex order [3].
If self is the last permutation in this order it returns None.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> p = Permutation([2, 0, 3, 1]); p.rank_nonlex()
5
>>> p = p.next_nonlex(); p
Permutation([3, 0, 1, 2])
>>> p.rank_nonlex()
6
See Also
========
rank_nonlex, unrank_nonlex
"""
r = self.rank_nonlex()
if r == ifac(self.size) - 1:
return None
return self.unrank_nonlex(self.size, r + 1)
def rank(self):
"""
Returns the lexicographic rank of the permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3])
>>> p.rank()
0
>>> p = Permutation([3, 2, 1, 0])
>>> p.rank()
23
See Also
========
next_lex, unrank_lex, cardinality, length, order, size
"""
if self._rank is not None:
return self._rank
rank = 0
rho = self.array_form[:]
n = self.size - 1
size = n + 1
psize = int(ifac(n))
for j in range(size - 1):
rank += rho[j]*psize
for i in range(j + 1, size):
if rho[i] > rho[j]:
rho[i] -= 1
psize //= n
n -= 1
self._rank = rank
return rank
@property
def cardinality(self):
"""
Returns the number of all possible permutations.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3])
>>> p.cardinality
24
See Also
========
length, order, rank, size
"""
return int(ifac(self.size))
def parity(self):
"""
Computes the parity of a permutation.
Explanation
===========
The parity of a permutation reflects the parity of the
number of inversions in the permutation, i.e., the
number of pairs of x and y such that ``x > y`` but ``p[x] < p[y]``.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3])
>>> p.parity()
0
>>> p = Permutation([3, 2, 0, 1])
>>> p.parity()
1
See Also
========
_af_parity
"""
if self._cyclic_form is not None:
return (self.size - self.cycles) % 2
return _af_parity(self.array_form)
@property
def is_even(self):
"""
Checks if a permutation is even.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3])
>>> p.is_even
True
>>> p = Permutation([3, 2, 1, 0])
>>> p.is_even
True
See Also
========
is_odd
"""
return not self.is_odd
@property
def is_odd(self):
"""
Checks if a permutation is odd.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3])
>>> p.is_odd
False
>>> p = Permutation([3, 2, 0, 1])
>>> p.is_odd
True
See Also
========
is_even
"""
return bool(self.parity() % 2)
@property
def is_Singleton(self):
"""
Checks to see if the permutation contains only one number and is
thus the only possible permutation of this set of numbers
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation([0]).is_Singleton
True
>>> Permutation([0, 1]).is_Singleton
False
See Also
========
is_Empty
"""
return self.size == 1
@property
def is_Empty(self):
"""
Checks to see if the permutation is a set with zero elements
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation([]).is_Empty
True
>>> Permutation([0]).is_Empty
False
See Also
========
is_Singleton
"""
return self.size == 0
@property
def is_identity(self):
return self.is_Identity
@property
def is_Identity(self):
"""
Returns True if the Permutation is an identity permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([])
>>> p.is_Identity
True
>>> p = Permutation([[0], [1], [2]])
>>> p.is_Identity
True
>>> p = Permutation([0, 1, 2])
>>> p.is_Identity
True
>>> p = Permutation([0, 2, 1])
>>> p.is_Identity
False
See Also
========
order
"""
af = self.array_form
return not af or all(i == af[i] for i in range(self.size))
def ascents(self):
"""
Returns the positions of ascents in a permutation, ie, the location
where p[i] < p[i+1]
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([4, 0, 1, 3, 2])
>>> p.ascents()
[1, 2]
See Also
========
descents, inversions, min, max
"""
a = self.array_form
pos = [i for i in range(len(a) - 1) if a[i] < a[i + 1]]
return pos
def descents(self):
"""
Returns the positions of descents in a permutation, ie, the location
where p[i] > p[i+1]
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([4, 0, 1, 3, 2])
>>> p.descents()
[0, 3]
See Also
========
ascents, inversions, min, max
"""
a = self.array_form
pos = [i for i in range(len(a) - 1) if a[i] > a[i + 1]]
return pos
def max(self):
"""
The maximum element moved by the permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([1, 0, 2, 3, 4])
>>> p.max()
1
See Also
========
min, descents, ascents, inversions
"""
max = 0
a = self.array_form
for i in range(len(a)):
if a[i] != i and a[i] > max:
max = a[i]
return max
def min(self):
"""
The minimum element moved by the permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 4, 3, 2])
>>> p.min()
2
See Also
========
max, descents, ascents, inversions
"""
a = self.array_form
min = len(a)
for i in range(len(a)):
if a[i] != i and a[i] < min:
min = a[i]
return min
def inversions(self):
"""
Computes the number of inversions of a permutation.
Explanation
===========
An inversion is where i > j but p[i] < p[j].
For small length of p, it iterates over all i and j
values and calculates the number of inversions.
For large length of p, it uses a variation of merge
sort to calculate the number of inversions.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3, 4, 5])
>>> p.inversions()
0
>>> Permutation([3, 2, 1, 0]).inversions()
6
See Also
========
descents, ascents, min, max
References
==========
.. [1] http://www.cp.eng.chula.ac.th/~piak/teaching/algo/algo2008/count-inv.htm
"""
inversions = 0
a = self.array_form
n = len(a)
if n < 130:
for i in range(n - 1):
b = a[i]
for c in a[i + 1:]:
if b > c:
inversions += 1
else:
k = 1
right = 0
arr = a[:]
temp = a[:]
while k < n:
i = 0
while i + k < n:
right = i + k * 2 - 1
if right >= n:
right = n - 1
inversions += _merge(arr, temp, i, i + k, right)
i = i + k * 2
k = k * 2
return inversions
def commutator(self, x):
"""Return the commutator of ``self`` and ``x``: ``~x*~self*x*self``
If f and g are part of a group, G, then the commutator of f and g
is the group identity iff f and g commute, i.e. fg == gf.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> p = Permutation([0, 2, 3, 1])
>>> x = Permutation([2, 0, 3, 1])
>>> c = p.commutator(x); c
Permutation([2, 1, 3, 0])
>>> c == ~x*~p*x*p
True
>>> I = Permutation(3)
>>> p = [I + i for i in range(6)]
>>> for i in range(len(p)):
... for j in range(len(p)):
... c = p[i].commutator(p[j])
... if p[i]*p[j] == p[j]*p[i]:
... assert c == I
... else:
... assert c != I
...
References
==========
.. [1] https://en.wikipedia.org/wiki/Commutator
"""
a = self.array_form
b = x.array_form
n = len(a)
if len(b) != n:
raise ValueError("The permutations must be of equal size.")
inva = [None]*n
for i in range(n):
inva[a[i]] = i
invb = [None]*n
for i in range(n):
invb[b[i]] = i
return self._af_new([a[b[inva[i]]] for i in invb])
def signature(self):
"""
Gives the signature of the permutation needed to place the
elements of the permutation in canonical order.
The signature is calculated as (-1)^<number of inversions>
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2])
>>> p.inversions()
0
>>> p.signature()
1
>>> q = Permutation([0,2,1])
>>> q.inversions()
1
>>> q.signature()
-1
See Also
========
inversions
"""
if self.is_even:
return 1
return -1
def order(self):
"""
Computes the order of a permutation.
When the permutation is raised to the power of its
order it equals the identity permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> p = Permutation([3, 1, 5, 2, 4, 0])
>>> p.order()
4
>>> (p**(p.order()))
Permutation([], size=6)
See Also
========
identity, cardinality, length, rank, size
"""
return reduce(lcm, [len(cycle) for cycle in self.cyclic_form], 1)
def length(self):
"""
Returns the number of integers moved by a permutation.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation([0, 3, 2, 1]).length()
2
>>> Permutation([[0, 1], [2, 3]]).length()
4
See Also
========
min, max, support, cardinality, order, rank, size
"""
return len(self.support())
@property
def cycle_structure(self):
"""Return the cycle structure of the permutation as a dictionary
indicating the multiplicity of each cycle length.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation(3).cycle_structure
{1: 4}
>>> Permutation(0, 4, 3)(1, 2)(5, 6).cycle_structure
{2: 2, 3: 1}
"""
if self._cycle_structure:
rv = self._cycle_structure
else:
rv = defaultdict(int)
singletons = self.size
for c in self.cyclic_form:
rv[len(c)] += 1
singletons -= len(c)
if singletons:
rv[1] = singletons
self._cycle_structure = rv
return dict(rv) # make a copy
@property
def cycles(self):
"""
Returns the number of cycles contained in the permutation
(including singletons).
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation([0, 1, 2]).cycles
3
>>> Permutation([0, 1, 2]).full_cyclic_form
[[0], [1], [2]]
>>> Permutation(0, 1)(2, 3).cycles
2
See Also
========
sympy.functions.combinatorial.numbers.stirling
"""
return len(self.full_cyclic_form)
def index(self):
"""
Returns the index of a permutation.
The index of a permutation is the sum of all subscripts j such
that p[j] is greater than p[j+1].
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([3, 0, 2, 1, 4])
>>> p.index()
2
"""
a = self.array_form
return sum([j for j in range(len(a) - 1) if a[j] > a[j + 1]])
def runs(self):
"""
Returns the runs of a permutation.
An ascending sequence in a permutation is called a run [5].
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([2, 5, 7, 3, 6, 0, 1, 4, 8])
>>> p.runs()
[[2, 5, 7], [3, 6], [0, 1, 4, 8]]
>>> q = Permutation([1,3,2,0])
>>> q.runs()
[[1, 3], [2], [0]]
"""
return runs(self.array_form)
def inversion_vector(self):
"""Return the inversion vector of the permutation.
The inversion vector consists of elements whose value
indicates the number of elements in the permutation
that are lesser than it and lie on its right hand side.
The inversion vector is the same as the Lehmer encoding of a
permutation.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([4, 8, 0, 7, 1, 5, 3, 6, 2])
>>> p.inversion_vector()
[4, 7, 0, 5, 0, 2, 1, 1]
>>> p = Permutation([3, 2, 1, 0])
>>> p.inversion_vector()
[3, 2, 1]
The inversion vector increases lexicographically with the rank
of the permutation, the -ith element cycling through 0..i.
>>> p = Permutation(2)
>>> while p:
... print('%s %s %s' % (p, p.inversion_vector(), p.rank()))
... p = p.next_lex()
(2) [0, 0] 0
(1 2) [0, 1] 1
(2)(0 1) [1, 0] 2
(0 1 2) [1, 1] 3
(0 2 1) [2, 0] 4
(0 2) [2, 1] 5
See Also
========
from_inversion_vector
"""
self_array_form = self.array_form
n = len(self_array_form)
inversion_vector = [0] * (n - 1)
for i in range(n - 1):
val = 0
for j in range(i + 1, n):
if self_array_form[j] < self_array_form[i]:
val += 1
inversion_vector[i] = val
return inversion_vector
def rank_trotterjohnson(self):
"""
Returns the Trotter Johnson rank, which we get from the minimal
change algorithm. See [4] section 2.4.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 1, 2, 3])
>>> p.rank_trotterjohnson()
0
>>> p = Permutation([0, 2, 1, 3])
>>> p.rank_trotterjohnson()
7
See Also
========
unrank_trotterjohnson, next_trotterjohnson
"""
if self.array_form == [] or self.is_Identity:
return 0
if self.array_form == [1, 0]:
return 1
perm = self.array_form
n = self.size
rank = 0
for j in range(1, n):
k = 1
i = 0
while perm[i] != j:
if perm[i] < j:
k += 1
i += 1
j1 = j + 1
if rank % 2 == 0:
rank = j1*rank + j1 - k
else:
rank = j1*rank + k - 1
return rank
@classmethod
def unrank_trotterjohnson(cls, size, rank):
"""
Trotter Johnson permutation unranking. See [4] section 2.4.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> Permutation.unrank_trotterjohnson(5, 10)
Permutation([0, 3, 1, 2, 4])
See Also
========
rank_trotterjohnson, next_trotterjohnson
"""
perm = [0]*size
r2 = 0
n = ifac(size)
pj = 1
for j in range(2, size + 1):
pj *= j
r1 = (rank * pj) // n
k = r1 - j*r2
if r2 % 2 == 0:
for i in range(j - 1, j - k - 1, -1):
perm[i] = perm[i - 1]
perm[j - k - 1] = j - 1
else:
for i in range(j - 1, k, -1):
perm[i] = perm[i - 1]
perm[k] = j - 1
r2 = r1
return cls._af_new(perm)
def next_trotterjohnson(self):
"""
Returns the next permutation in Trotter-Johnson order.
If self is the last permutation it returns None.
See [4] section 2.4. If it is desired to generate all such
permutations, they can be generated in order more quickly
with the ``generate_bell`` function.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> p = Permutation([3, 0, 2, 1])
>>> p.rank_trotterjohnson()
4
>>> p = p.next_trotterjohnson(); p
Permutation([0, 3, 2, 1])
>>> p.rank_trotterjohnson()
5
See Also
========
rank_trotterjohnson, unrank_trotterjohnson, sympy.utilities.iterables.generate_bell
"""
pi = self.array_form[:]
n = len(pi)
st = 0
rho = pi[:]
done = False
m = n-1
while m > 0 and not done:
d = rho.index(m)
for i in range(d, m):
rho[i] = rho[i + 1]
par = _af_parity(rho[:m])
if par == 1:
if d == m:
m -= 1
else:
pi[st + d], pi[st + d + 1] = pi[st + d + 1], pi[st + d]
done = True
else:
if d == 0:
m -= 1
st += 1
else:
pi[st + d], pi[st + d - 1] = pi[st + d - 1], pi[st + d]
done = True
if m == 0:
return None
return self._af_new(pi)
def get_precedence_matrix(self):
"""
Gets the precedence matrix. This is used for computing the
distance between two permutations.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> p = Permutation.josephus(3, 6, 1)
>>> p
Permutation([2, 5, 3, 1, 4, 0])
>>> p.get_precedence_matrix()
Matrix([
[0, 0, 0, 0, 0, 0],
[1, 0, 0, 0, 1, 0],
[1, 1, 0, 1, 1, 1],
[1, 1, 0, 0, 1, 0],
[1, 0, 0, 0, 0, 0],
[1, 1, 0, 1, 1, 0]])
See Also
========
get_precedence_distance, get_adjacency_matrix, get_adjacency_distance
"""
m = zeros(self.size)
perm = self.array_form
for i in range(m.rows):
for j in range(i + 1, m.cols):
m[perm[i], perm[j]] = 1
return m
def get_precedence_distance(self, other):
"""
Computes the precedence distance between two permutations.
Explanation
===========
Suppose p and p' represent n jobs. The precedence metric
counts the number of times a job j is preceded by job i
in both p and p'. This metric is commutative.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([2, 0, 4, 3, 1])
>>> q = Permutation([3, 1, 2, 4, 0])
>>> p.get_precedence_distance(q)
7
>>> q.get_precedence_distance(p)
7
See Also
========
get_precedence_matrix, get_adjacency_matrix, get_adjacency_distance
"""
if self.size != other.size:
raise ValueError("The permutations must be of equal size.")
self_prec_mat = self.get_precedence_matrix()
other_prec_mat = other.get_precedence_matrix()
n_prec = 0
for i in range(self.size):
for j in range(self.size):
if i == j:
continue
if self_prec_mat[i, j] * other_prec_mat[i, j] == 1:
n_prec += 1
d = self.size * (self.size - 1)//2 - n_prec
return d
def get_adjacency_matrix(self):
"""
Computes the adjacency matrix of a permutation.
Explanation
===========
If job i is adjacent to job j in a permutation p
then we set m[i, j] = 1 where m is the adjacency
matrix of p.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation.josephus(3, 6, 1)
>>> p.get_adjacency_matrix()
Matrix([
[0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1],
[0, 1, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0]])
>>> q = Permutation([0, 1, 2, 3])
>>> q.get_adjacency_matrix()
Matrix([
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 1],
[0, 0, 0, 0]])
See Also
========
get_precedence_matrix, get_precedence_distance, get_adjacency_distance
"""
m = zeros(self.size)
perm = self.array_form
for i in range(self.size - 1):
m[perm[i], perm[i + 1]] = 1
return m
def get_adjacency_distance(self, other):
"""
Computes the adjacency distance between two permutations.
Explanation
===========
This metric counts the number of times a pair i,j of jobs is
adjacent in both p and p'. If n_adj is this quantity then
the adjacency distance is n - n_adj - 1 [1]
[1] Reeves, Colin R. Landscapes, Operators and Heuristic search, Annals
of Operational Research, 86, pp 473-490. (1999)
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 3, 1, 2, 4])
>>> q = Permutation.josephus(4, 5, 2)
>>> p.get_adjacency_distance(q)
3
>>> r = Permutation([0, 2, 1, 4, 3])
>>> p.get_adjacency_distance(r)
4
See Also
========
get_precedence_matrix, get_precedence_distance, get_adjacency_matrix
"""
if self.size != other.size:
raise ValueError("The permutations must be of the same size.")
self_adj_mat = self.get_adjacency_matrix()
other_adj_mat = other.get_adjacency_matrix()
n_adj = 0
for i in range(self.size):
for j in range(self.size):
if i == j:
continue
if self_adj_mat[i, j] * other_adj_mat[i, j] == 1:
n_adj += 1
d = self.size - n_adj - 1
return d
def get_positional_distance(self, other):
"""
Computes the positional distance between two permutations.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> p = Permutation([0, 3, 1, 2, 4])
>>> q = Permutation.josephus(4, 5, 2)
>>> r = Permutation([3, 1, 4, 0, 2])
>>> p.get_positional_distance(q)
12
>>> p.get_positional_distance(r)
12
See Also
========
get_precedence_distance, get_adjacency_distance
"""
a = self.array_form
b = other.array_form
if len(a) != len(b):
raise ValueError("The permutations must be of the same size.")
return sum([abs(a[i] - b[i]) for i in range(len(a))])
@classmethod
def josephus(cls, m, n, s=1):
"""Return as a permutation the shuffling of range(n) using the Josephus
scheme in which every m-th item is selected until all have been chosen.
The returned permutation has elements listed by the order in which they
were selected.
The parameter ``s`` stops the selection process when there are ``s``
items remaining and these are selected by continuing the selection,
counting by 1 rather than by ``m``.
Consider selecting every 3rd item from 6 until only 2 remain::
choices chosen
======== ======
012345
01 345 2
01 34 25
01 4 253
0 4 2531
0 25314
253140
Examples
========
>>> from sympy.combinatorics import Permutation
>>> Permutation.josephus(3, 6, 2).array_form
[2, 5, 3, 1, 4, 0]
References
==========
.. [1] https://en.wikipedia.org/wiki/Flavius_Josephus
.. [2] https://en.wikipedia.org/wiki/Josephus_problem
.. [3] http://www.wou.edu/~burtonl/josephus.html
"""
from collections import deque
m -= 1
Q = deque(list(range(n)))
perm = []
while len(Q) > max(s, 1):
for dp in range(m):
Q.append(Q.popleft())
perm.append(Q.popleft())
perm.extend(list(Q))
return cls(perm)
@classmethod
def from_inversion_vector(cls, inversion):
"""
Calculates the permutation from the inversion vector.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> Permutation.from_inversion_vector([3, 2, 1, 0, 0])
Permutation([3, 2, 1, 0, 4, 5])
"""
size = len(inversion)
N = list(range(size + 1))
perm = []
try:
for k in range(size):
val = N[inversion[k]]
perm.append(val)
N.remove(val)
except IndexError:
raise ValueError("The inversion vector is not valid.")
perm.extend(N)
return cls._af_new(perm)
@classmethod
def random(cls, n):
"""
Generates a random permutation of length ``n``.
Uses the underlying Python pseudo-random number generator.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> Permutation.random(2) in (Permutation([1, 0]), Permutation([0, 1]))
True
"""
perm_array = list(range(n))
random.shuffle(perm_array)
return cls._af_new(perm_array)
@classmethod
def unrank_lex(cls, size, rank):
"""
Lexicographic permutation unranking.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy import init_printing
>>> init_printing(perm_cyclic=False, pretty_print=False)
>>> a = Permutation.unrank_lex(5, 10)
>>> a.rank()
10
>>> a
Permutation([0, 2, 4, 1, 3])
See Also
========
rank, next_lex
"""
perm_array = [0] * size
psize = 1
for i in range(size):
new_psize = psize*(i + 1)
d = (rank % new_psize) // psize
rank -= d*psize
perm_array[size - i - 1] = d
for j in range(size - i, size):
if perm_array[j] > d - 1:
perm_array[j] += 1
psize = new_psize
return cls._af_new(perm_array)
def resize(self, n):
"""Resize the permutation to the new size ``n``.
Parameters
==========
n : int
The new size of the permutation.
Raises
======
ValueError
If the permutation cannot be resized to the given size.
This may only happen when resized to a smaller size than
the original.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
Increasing the size of a permutation:
>>> p = Permutation(0, 1, 2)
>>> p = p.resize(5)
>>> p
(4)(0 1 2)
Decreasing the size of the permutation:
>>> p = p.resize(4)
>>> p
(3)(0 1 2)
If resizing to the specific size breaks the cycles:
>>> p.resize(2)
Traceback (most recent call last):
...
ValueError: The permutation cannot be resized to 2 because the
cycle (0, 1, 2) may break.
"""
aform = self.array_form
l = len(aform)
if n > l:
aform += list(range(l, n))
return Permutation._af_new(aform)
elif n < l:
cyclic_form = self.full_cyclic_form
new_cyclic_form = []
for cycle in cyclic_form:
cycle_min = min(cycle)
cycle_max = max(cycle)
if cycle_min <= n-1:
if cycle_max > n-1:
raise ValueError(
"The permutation cannot be resized to {} "
"because the cycle {} may break."
.format(n, tuple(cycle)))
new_cyclic_form.append(cycle)
return Permutation(new_cyclic_form)
return self
# XXX Deprecated flag
print_cyclic = None
def _merge(arr, temp, left, mid, right):
"""
Merges two sorted arrays and calculates the inversion count.
Helper function for calculating inversions. This method is
for internal use only.
"""
i = k = left
j = mid
inv_count = 0
while i < mid and j <= right:
if arr[i] < arr[j]:
temp[k] = arr[i]
k += 1
i += 1
else:
temp[k] = arr[j]
k += 1
j += 1
inv_count += (mid -i)
while i < mid:
temp[k] = arr[i]
k += 1
i += 1
if j <= right:
k += right - j + 1
j += right - j + 1
arr[left:k + 1] = temp[left:k + 1]
else:
arr[left:right + 1] = temp[left:right + 1]
return inv_count
Perm = Permutation
_af_new = Perm._af_new
class AppliedPermutation(Expr):
"""A permutation applied to a symbolic variable.
Parameters
==========
perm : Permutation
x : Expr
Examples
========
>>> from sympy import Symbol
>>> from sympy.combinatorics import Permutation
Creating a symbolic permutation function application:
>>> x = Symbol('x')
>>> p = Permutation(0, 1, 2)
>>> p.apply(x)
AppliedPermutation((0 1 2), x)
>>> _.subs(x, 1)
2
"""
def __new__(cls, perm, x, evaluate=None):
if evaluate is None:
evaluate = global_parameters.evaluate
perm = _sympify(perm)
x = _sympify(x)
if not isinstance(perm, Permutation):
raise ValueError("{} must be a Permutation instance."
.format(perm))
if evaluate:
if x.is_Integer:
return perm.apply(x)
obj = super().__new__(cls, perm, x)
return obj
@dispatch(Permutation, Permutation)
def _eval_is_eq(lhs, rhs):
if lhs._size != rhs._size:
return None
return lhs._array_form == rhs._array_form
|
fd2b6bf4faf83d8847cbfdd7675187fd65f1579dfb0cdfc469b7d751c4a7ae07 | from itertools import combinations
from sympy.combinatorics.graycode import GrayCode
class Subset():
"""
Represents a basic subset object.
Explanation
===========
We generate subsets using essentially two techniques,
binary enumeration and lexicographic enumeration.
The Subset class takes two arguments, the first one
describes the initial subset to consider and the second
describes the superset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
['b']
>>> a.prev_binary().subset
['c']
"""
_rank_binary = None
_rank_lex = None
_rank_graycode = None
_subset = None
_superset = None
def __new__(cls, subset, superset):
"""
Default constructor.
It takes the ``subset`` and its ``superset`` as its parameters.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.subset
['c', 'd']
>>> a.superset
['a', 'b', 'c', 'd']
>>> a.size
2
"""
if len(subset) > len(superset):
raise ValueError('Invalid arguments have been provided. The '
'superset must be larger than the subset.')
for elem in subset:
if elem not in superset:
raise ValueError('The superset provided is invalid as it does '
'not contain the element {}'.format(elem))
obj = object.__new__(cls)
obj._subset = subset
obj._superset = superset
return obj
def __eq__(self, other):
"""Return a boolean indicating whether a == b on the basis of
whether both objects are of the class Subset and if the values
of the subset and superset attributes are the same.
"""
if not isinstance(other, Subset):
return NotImplemented
return self.subset == other.subset and self.superset == other.superset
def iterate_binary(self, k):
"""
This is a helper function. It iterates over the
binary subsets by ``k`` steps. This variable can be
both positive or negative.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.iterate_binary(-2).subset
['d']
>>> a = Subset(['a', 'b', 'c'], ['a', 'b', 'c', 'd'])
>>> a.iterate_binary(2).subset
[]
See Also
========
next_binary, prev_binary
"""
bin_list = Subset.bitlist_from_subset(self.subset, self.superset)
n = (int(''.join(bin_list), 2) + k) % 2**self.superset_size
bits = bin(n)[2:].rjust(self.superset_size, '0')
return Subset.subset_from_bitlist(self.superset, bits)
def next_binary(self):
"""
Generates the next binary ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
['b']
>>> a = Subset(['a', 'b', 'c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
[]
See Also
========
prev_binary, iterate_binary
"""
return self.iterate_binary(1)
def prev_binary(self):
"""
Generates the previous binary ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([], ['a', 'b', 'c', 'd'])
>>> a.prev_binary().subset
['a', 'b', 'c', 'd']
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.prev_binary().subset
['c']
See Also
========
next_binary, iterate_binary
"""
return self.iterate_binary(-1)
def next_lexicographic(self):
"""
Generates the next lexicographically ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_lexicographic().subset
['d']
>>> a = Subset(['d'], ['a', 'b', 'c', 'd'])
>>> a.next_lexicographic().subset
[]
See Also
========
prev_lexicographic
"""
i = self.superset_size - 1
indices = Subset.subset_indices(self.subset, self.superset)
if i in indices:
if i - 1 in indices:
indices.remove(i - 1)
else:
indices.remove(i)
i = i - 1
while i >= 0 and i not in indices:
i = i - 1
if i >= 0:
indices.remove(i)
indices.append(i+1)
else:
while i not in indices and i >= 0:
i = i - 1
indices.append(i + 1)
ret_set = []
super_set = self.superset
for i in indices:
ret_set.append(super_set[i])
return Subset(ret_set, super_set)
def prev_lexicographic(self):
"""
Generates the previous lexicographically ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([], ['a', 'b', 'c', 'd'])
>>> a.prev_lexicographic().subset
['d']
>>> a = Subset(['c','d'], ['a', 'b', 'c', 'd'])
>>> a.prev_lexicographic().subset
['c']
See Also
========
next_lexicographic
"""
i = self.superset_size - 1
indices = Subset.subset_indices(self.subset, self.superset)
while i >= 0 and i not in indices:
i = i - 1
if i == 0 or i - 1 in indices:
indices.remove(i)
else:
if i >= 0:
indices.remove(i)
indices.append(i - 1)
indices.append(self.superset_size - 1)
ret_set = []
super_set = self.superset
for i in indices:
ret_set.append(super_set[i])
return Subset(ret_set, super_set)
def iterate_graycode(self, k):
"""
Helper function used for prev_gray and next_gray.
It performs ``k`` step overs to get the respective Gray codes.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
>>> a.iterate_graycode(3).subset
[1, 4]
>>> a.iterate_graycode(-2).subset
[1, 2, 4]
See Also
========
next_gray, prev_gray
"""
unranked_code = GrayCode.unrank(self.superset_size,
(self.rank_gray + k) % self.cardinality)
return Subset.subset_from_bitlist(self.superset,
unranked_code)
def next_gray(self):
"""
Generates the next Gray code ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
>>> a.next_gray().subset
[1, 3]
See Also
========
iterate_graycode, prev_gray
"""
return self.iterate_graycode(1)
def prev_gray(self):
"""
Generates the previous Gray code ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([2, 3, 4], [1, 2, 3, 4, 5])
>>> a.prev_gray().subset
[2, 3, 4, 5]
See Also
========
iterate_graycode, next_gray
"""
return self.iterate_graycode(-1)
@property
def rank_binary(self):
"""
Computes the binary ordered rank.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([], ['a','b','c','d'])
>>> a.rank_binary
0
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.rank_binary
3
See Also
========
iterate_binary, unrank_binary
"""
if self._rank_binary is None:
self._rank_binary = int("".join(
Subset.bitlist_from_subset(self.subset,
self.superset)), 2)
return self._rank_binary
@property
def rank_lexicographic(self):
"""
Computes the lexicographic ranking of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.rank_lexicographic
14
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
>>> a.rank_lexicographic
43
"""
if self._rank_lex is None:
def _ranklex(self, subset_index, i, n):
if subset_index == [] or i > n:
return 0
if i in subset_index:
subset_index.remove(i)
return 1 + _ranklex(self, subset_index, i + 1, n)
return 2**(n - i - 1) + _ranklex(self, subset_index, i + 1, n)
indices = Subset.subset_indices(self.subset, self.superset)
self._rank_lex = _ranklex(self, indices, 0, self.superset_size)
return self._rank_lex
@property
def rank_gray(self):
"""
Computes the Gray code ranking of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.rank_gray
2
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
>>> a.rank_gray
27
See Also
========
iterate_graycode, unrank_gray
"""
if self._rank_graycode is None:
bits = Subset.bitlist_from_subset(self.subset, self.superset)
self._rank_graycode = GrayCode(len(bits), start=bits).rank
return self._rank_graycode
@property
def subset(self):
"""
Gets the subset represented by the current instance.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.subset
['c', 'd']
See Also
========
superset, size, superset_size, cardinality
"""
return self._subset
@property
def size(self):
"""
Gets the size of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.size
2
See Also
========
subset, superset, superset_size, cardinality
"""
return len(self.subset)
@property
def superset(self):
"""
Gets the superset of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.superset
['a', 'b', 'c', 'd']
See Also
========
subset, size, superset_size, cardinality
"""
return self._superset
@property
def superset_size(self):
"""
Returns the size of the superset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.superset_size
4
See Also
========
subset, superset, size, cardinality
"""
return len(self.superset)
@property
def cardinality(self):
"""
Returns the number of all possible subsets.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.cardinality
16
See Also
========
subset, superset, size, superset_size
"""
return 2**(self.superset_size)
@classmethod
def subset_from_bitlist(self, super_set, bitlist):
"""
Gets the subset defined by the bitlist.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.subset_from_bitlist(['a', 'b', 'c', 'd'], '0011').subset
['c', 'd']
See Also
========
bitlist_from_subset
"""
if len(super_set) != len(bitlist):
raise ValueError("The sizes of the lists are not equal")
ret_set = []
for i in range(len(bitlist)):
if bitlist[i] == '1':
ret_set.append(super_set[i])
return Subset(ret_set, super_set)
@classmethod
def bitlist_from_subset(self, subset, superset):
"""
Gets the bitlist corresponding to a subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.bitlist_from_subset(['c', 'd'], ['a', 'b', 'c', 'd'])
'0011'
See Also
========
subset_from_bitlist
"""
bitlist = ['0'] * len(superset)
if isinstance(subset, Subset):
subset = subset.subset
for i in Subset.subset_indices(subset, superset):
bitlist[i] = '1'
return ''.join(bitlist)
@classmethod
def unrank_binary(self, rank, superset):
"""
Gets the binary ordered subset of the specified rank.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.unrank_binary(4, ['a', 'b', 'c', 'd']).subset
['b']
See Also
========
iterate_binary, rank_binary
"""
bits = bin(rank)[2:].rjust(len(superset), '0')
return Subset.subset_from_bitlist(superset, bits)
@classmethod
def unrank_gray(self, rank, superset):
"""
Gets the Gray code ordered subset of the specified rank.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.unrank_gray(4, ['a', 'b', 'c']).subset
['a', 'b']
>>> Subset.unrank_gray(0, ['a', 'b', 'c']).subset
[]
See Also
========
iterate_graycode, rank_gray
"""
graycode_bitlist = GrayCode.unrank(len(superset), rank)
return Subset.subset_from_bitlist(superset, graycode_bitlist)
@classmethod
def subset_indices(self, subset, superset):
"""Return indices of subset in superset in a list; the list is empty
if all elements of ``subset`` are not in ``superset``.
Examples
========
>>> from sympy.combinatorics import Subset
>>> superset = [1, 3, 2, 5, 4]
>>> Subset.subset_indices([3, 2, 1], superset)
[1, 2, 0]
>>> Subset.subset_indices([1, 6], superset)
[]
>>> Subset.subset_indices([], superset)
[]
"""
a, b = superset, subset
sb = set(b)
d = {}
for i, ai in enumerate(a):
if ai in sb:
d[ai] = i
sb.remove(ai)
if not sb:
break
else:
return list()
return [d[bi] for bi in b]
def ksubsets(superset, k):
"""
Finds the subsets of size ``k`` in lexicographic order.
This uses the itertools generator.
Examples
========
>>> from sympy.combinatorics.subsets import ksubsets
>>> list(ksubsets([1, 2, 3], 2))
[(1, 2), (1, 3), (2, 3)]
>>> list(ksubsets([1, 2, 3, 4, 5], 2))
[(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), \
(2, 5), (3, 4), (3, 5), (4, 5)]
See Also
========
Subset
"""
return combinations(superset, k)
|
60117c31f998697fe9c486a82b53aea78a3934d759b40383aa8f7079ab713650 | from sympy.combinatorics.permutations import Permutation, _af_rmul, \
_af_invert, _af_new
from sympy.combinatorics.perm_groups import PermutationGroup, _orbit, \
_orbit_transversal
from sympy.combinatorics.util import _distribute_gens_by_base, \
_orbits_transversals_from_bsgs
"""
References for tensor canonicalization:
[1] R. Portugal "Algorithmic simplification of tensor expressions",
J. Phys. A 32 (1999) 7779-7789
[2] R. Portugal, B.F. Svaiter "Group-theoretic Approach for Symbolic
Tensor Manipulation: I. Free Indices"
arXiv:math-ph/0107031v1
[3] L.R.U. Manssur, R. Portugal "Group-theoretic Approach for Symbolic
Tensor Manipulation: II. Dummy Indices"
arXiv:math-ph/0107032v1
[4] xperm.c part of XPerm written by J. M. Martin-Garcia
http://www.xact.es/index.html
"""
def dummy_sgs(dummies, sym, n):
"""
Return the strong generators for dummy indices.
Parameters
==========
dummies : List of dummy indices.
`dummies[2k], dummies[2k+1]` are paired indices.
In base form, the dummy indices are always in
consecutive positions.
sym : symmetry under interchange of contracted dummies::
* None no symmetry
* 0 commuting
* 1 anticommuting
n : number of indices
Examples
========
>>> from sympy.combinatorics.tensor_can import dummy_sgs
>>> dummy_sgs(list(range(2, 8)), 0, 8)
[[0, 1, 3, 2, 4, 5, 6, 7, 8, 9], [0, 1, 2, 3, 5, 4, 6, 7, 8, 9],
[0, 1, 2, 3, 4, 5, 7, 6, 8, 9], [0, 1, 4, 5, 2, 3, 6, 7, 8, 9],
[0, 1, 2, 3, 6, 7, 4, 5, 8, 9]]
"""
if len(dummies) > n:
raise ValueError("List too large")
res = []
# exchange of contravariant and covariant indices
if sym is not None:
for j in dummies[::2]:
a = list(range(n + 2))
if sym == 1:
a[n] = n + 1
a[n + 1] = n
a[j], a[j + 1] = a[j + 1], a[j]
res.append(a)
# rename dummy indices
for j in dummies[:-3:2]:
a = list(range(n + 2))
a[j:j + 4] = a[j + 2], a[j + 3], a[j], a[j + 1]
res.append(a)
return res
def _min_dummies(dummies, sym, indices):
"""
Return list of minima of the orbits of indices in group of dummies.
See ``double_coset_can_rep`` for the description of ``dummies`` and ``sym``.
``indices`` is the initial list of dummy indices.
Examples
========
>>> from sympy.combinatorics.tensor_can import _min_dummies
>>> _min_dummies([list(range(2, 8))], [0], list(range(10)))
[0, 1, 2, 2, 2, 2, 2, 2, 8, 9]
"""
num_types = len(sym)
m = []
for dx in dummies:
if dx:
m.append(min(dx))
else:
m.append(None)
res = indices[:]
for i in range(num_types):
for c, i in enumerate(indices):
for j in range(num_types):
if i in dummies[j]:
res[c] = m[j]
break
return res
def _trace_S(s, j, b, S_cosets):
"""
Return the representative h satisfying s[h[b]] == j
If there is not such a representative return None
"""
for h in S_cosets[b]:
if s[h[b]] == j:
return h
return None
def _trace_D(gj, p_i, Dxtrav):
"""
Return the representative h satisfying h[gj] == p_i
If there is not such a representative return None
"""
for h in Dxtrav:
if h[gj] == p_i:
return h
return None
def _dumx_remove(dumx, dumx_flat, p0):
"""
remove p0 from dumx
"""
res = []
for dx in dumx:
if p0 not in dx:
res.append(dx)
continue
k = dx.index(p0)
if k % 2 == 0:
p0_paired = dx[k + 1]
else:
p0_paired = dx[k - 1]
dx.remove(p0)
dx.remove(p0_paired)
dumx_flat.remove(p0)
dumx_flat.remove(p0_paired)
res.append(dx)
def transversal2coset(size, base, transversal):
a = []
j = 0
for i in range(size):
if i in base:
a.append(sorted(transversal[j].values()))
j += 1
else:
a.append([list(range(size))])
j = len(a) - 1
while a[j] == [list(range(size))]:
j -= 1
return a[:j + 1]
def double_coset_can_rep(dummies, sym, b_S, sgens, S_transversals, g):
r"""
Butler-Portugal algorithm for tensor canonicalization with dummy indices.
Parameters
==========
dummies
list of lists of dummy indices,
one list for each type of index;
the dummy indices are put in order contravariant, covariant
[d0, -d0, d1, -d1, ...].
sym
list of the symmetries of the index metric for each type.
possible symmetries of the metrics
* 0 symmetric
* 1 antisymmetric
* None no symmetry
b_S
base of a minimal slot symmetry BSGS.
sgens
generators of the slot symmetry BSGS.
S_transversals
transversals for the slot BSGS.
g
permutation representing the tensor.
Returns
=======
Return 0 if the tensor is zero, else return the array form of
the permutation representing the canonical form of the tensor.
Notes
=====
A tensor with dummy indices can be represented in a number
of equivalent ways which typically grows exponentially with
the number of indices. To be able to establish if two tensors
with many indices are equal becomes computationally very slow
in absence of an efficient algorithm.
The Butler-Portugal algorithm [3] is an efficient algorithm to
put tensors in canonical form, solving the above problem.
Portugal observed that a tensor can be represented by a permutation,
and that the class of tensors equivalent to it under slot and dummy
symmetries is equivalent to the double coset `D*g*S`
(Note: in this documentation we use the conventions for multiplication
of permutations p, q with (p*q)(i) = p[q[i]] which is opposite
to the one used in the Permutation class)
Using the algorithm by Butler to find a representative of the
double coset one can find a canonical form for the tensor.
To see this correspondence,
let `g` be a permutation in array form; a tensor with indices `ind`
(the indices including both the contravariant and the covariant ones)
can be written as
`t = T(ind[g[0]], \dots, ind[g[n-1]])`,
where `n = len(ind)`;
`g` has size `n + 2`, the last two indices for the sign of the tensor
(trick introduced in [4]).
A slot symmetry transformation `s` is a permutation acting on the slots
`t \rightarrow T(ind[(g*s)[0]], \dots, ind[(g*s)[n-1]])`
A dummy symmetry transformation acts on `ind`
`t \rightarrow T(ind[(d*g)[0]], \dots, ind[(d*g)[n-1]])`
Being interested only in the transformations of the tensor under
these symmetries, one can represent the tensor by `g`, which transforms
as
`g -> d*g*s`, so it belongs to the coset `D*g*S`, or in other words
to the set of all permutations allowed by the slot and dummy symmetries.
Let us explain the conventions by an example.
Given a tensor `T^{d3 d2 d1}{}_{d1 d2 d3}` with the slot symmetries
`T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`
`T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`
and symmetric metric, find the tensor equivalent to it which
is the lowest under the ordering of indices:
lexicographic ordering `d1, d2, d3` and then contravariant
before covariant index; that is the canonical form of the tensor.
The canonical form is `-T^{d1 d2 d3}{}_{d1 d2 d3}`
obtained using `T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`.
To convert this problem in the input for this function,
use the following ordering of the index names
(- for covariant for short) `d1, -d1, d2, -d2, d3, -d3`
`T^{d3 d2 d1}{}_{d1 d2 d3}` corresponds to `g = [4, 2, 0, 1, 3, 5, 6, 7]`
where the last two indices are for the sign
`sgens = [Permutation(0, 2)(6, 7), Permutation(0, 4)(6, 7)]`
sgens[0] is the slot symmetry `-(0, 2)`
`T^{a0 a1 a2 a3 a4 a5} = -T^{a2 a1 a0 a3 a4 a5}`
sgens[1] is the slot symmetry `-(0, 4)`
`T^{a0 a1 a2 a3 a4 a5} = -T^{a4 a1 a2 a3 a0 a5}`
The dummy symmetry group D is generated by the strong base generators
`[(0, 1), (2, 3), (4, 5), (0, 2)(1, 3), (0, 4)(1, 5)]`
where the first three interchange covariant and contravariant
positions of the same index (d1 <-> -d1) and the last two interchange
the dummy indices themselves (d1 <-> d2).
The dummy symmetry acts from the left
`d = [1, 0, 2, 3, 4, 5, 6, 7]` exchange `d1 \leftrightarrow -d1`
`T^{d3 d2 d1}{}_{d1 d2 d3} == T^{d3 d2}{}_{d1}{}^{d1}{}_{d2 d3}`
`g=[4, 2, 0, 1, 3, 5, 6, 7] -> [4, 2, 1, 0, 3, 5, 6, 7] = _af_rmul(d, g)`
which differs from `_af_rmul(g, d)`.
The slot symmetry acts from the right
`s = [2, 1, 0, 3, 4, 5, 7, 6]` exchanges slots 0 and 2 and changes sign
`T^{d3 d2 d1}{}_{d1 d2 d3} == -T^{d1 d2 d3}{}_{d1 d2 d3}`
`g=[4,2,0,1,3,5,6,7] -> [0, 2, 4, 1, 3, 5, 7, 6] = _af_rmul(g, s)`
Example in which the tensor is zero, same slot symmetries as above:
`T^{d2}{}_{d1 d3}{}^{d1 d3}{}_{d2}`
`= -T^{d3}{}_{d1 d3}{}^{d1 d2}{}_{d2}` under slot symmetry `-(0,4)`;
`= T_{d3 d1}{}^{d3}{}^{d1 d2}{}_{d2}` under slot symmetry `-(0,2)`;
`= T^{d3}{}_{d1 d3}{}^{d1 d2}{}_{d2}` symmetric metric;
`= 0` since two of these lines have tensors differ only for the sign.
The double coset D*g*S consists of permutations `h = d*g*s` corresponding
to equivalent tensors; if there are two `h` which are the same apart
from the sign, return zero; otherwise
choose as representative the tensor with indices
ordered lexicographically according to `[d1, -d1, d2, -d2, d3, -d3]`
that is ``rep = min(D*g*S) = min([d*g*s for d in D for s in S])``
The indices are fixed one by one; first choose the lowest index
for slot 0, then the lowest remaining index for slot 1, etc.
Doing this one obtains a chain of stabilizers
`S \rightarrow S_{b0} \rightarrow S_{b0,b1} \rightarrow \dots` and
`D \rightarrow D_{p0} \rightarrow D_{p0,p1} \rightarrow \dots`
where ``[b0, b1, ...] = range(b)`` is a base of the symmetric group;
the strong base `b_S` of S is an ordered sublist of it;
therefore it is sufficient to compute once the
strong base generators of S using the Schreier-Sims algorithm;
the stabilizers of the strong base generators are the
strong base generators of the stabilizer subgroup.
``dbase = [p0, p1, ...]`` is not in general in lexicographic order,
so that one must recompute the strong base generators each time;
however this is trivial, there is no need to use the Schreier-Sims
algorithm for D.
The algorithm keeps a TAB of elements `(s_i, d_i, h_i)`
where `h_i = d_i \times g \times s_i` satisfying `h_i[j] = p_j` for `0 \le j < i`
starting from `s_0 = id, d_0 = id, h_0 = g`.
The equations `h_0[0] = p_0, h_1[1] = p_1, \dots` are solved in this order,
choosing each time the lowest possible value of p_i
For `j < i`
`d_i*g*s_i*S_{b_0, \dots, b_{i-1}}*b_j = D_{p_0, \dots, p_{i-1}}*p_j`
so that for dx in `D_{p_0,\dots,p_{i-1}}` and sx in
`S_{base[0], \dots, base[i-1]}` one has `dx*d_i*g*s_i*sx*b_j = p_j`
Search for dx, sx such that this equation holds for `j = i`;
it can be written as `s_i*sx*b_j = J, dx*d_i*g*J = p_j`
`sx*b_j = s_i**-1*J; sx = trace(s_i**-1, S_{b_0,...,b_{i-1}})`
`dx**-1*p_j = d_i*g*J; dx = trace(d_i*g*J, D_{p_0,...,p_{i-1}})`
`s_{i+1} = s_i*trace(s_i**-1*J, S_{b_0,...,b_{i-1}})`
`d_{i+1} = trace(d_i*g*J, D_{p_0,...,p_{i-1}})**-1*d_i`
`h_{i+1}*b_i = d_{i+1}*g*s_{i+1}*b_i = p_i`
`h_n*b_j = p_j` for all j, so that `h_n` is the solution.
Add the found `(s, d, h)` to TAB1.
At the end of the iteration sort TAB1 with respect to the `h`;
if there are two consecutive `h` in TAB1 which differ only for the
sign, the tensor is zero, so return 0;
if there are two consecutive `h` which are equal, keep only one.
Then stabilize the slot generators under `i` and the dummy generators
under `p_i`.
Assign `TAB = TAB1` at the end of the iteration step.
At the end `TAB` contains a unique `(s, d, h)`, since all the slots
of the tensor `h` have been fixed to have the minimum value according
to the symmetries. The algorithm returns `h`.
It is important that the slot BSGS has lexicographic minimal base,
otherwise there is an `i` which does not belong to the slot base
for which `p_i` is fixed by the dummy symmetry only, while `i`
is not invariant from the slot stabilizer, so `p_i` is not in
general the minimal value.
This algorithm differs slightly from the original algorithm [3]:
the canonical form is minimal lexicographically, and
the BSGS has minimal base under lexicographic order.
Equal tensors `h` are eliminated from TAB.
Examples
========
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.tensor_can import double_coset_can_rep, get_transversals
>>> gens = [Permutation(x) for x in [[2, 1, 0, 3, 4, 5, 7, 6], [4, 1, 2, 3, 0, 5, 7, 6]]]
>>> base = [0, 2]
>>> g = Permutation([4, 2, 0, 1, 3, 5, 6, 7])
>>> transversals = get_transversals(base, gens)
>>> double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)
[0, 1, 2, 3, 4, 5, 7, 6]
>>> g = Permutation([4, 1, 3, 0, 5, 2, 6, 7])
>>> double_coset_can_rep([list(range(6))], [0], base, gens, transversals, g)
0
"""
size = g.size
g = g.array_form
num_dummies = size - 2
indices = list(range(num_dummies))
all_metrics_with_sym = not any(_ is None for _ in sym)
num_types = len(sym)
dumx = dummies[:]
dumx_flat = []
for dx in dumx:
dumx_flat.extend(dx)
b_S = b_S[:]
sgensx = [h._array_form for h in sgens]
if b_S:
S_transversals = transversal2coset(size, b_S, S_transversals)
# strong generating set for D
dsgsx = []
for i in range(num_types):
dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
idn = list(range(size))
# TAB = list of entries (s, d, h) where h = _af_rmuln(d,g,s)
# for short, in the following d*g*s means _af_rmuln(d,g,s)
TAB = [(idn, idn, g)]
for i in range(size - 2):
b = i
testb = b in b_S and sgensx
if testb:
sgensx1 = [_af_new(_) for _ in sgensx]
deltab = _orbit(size, sgensx1, b)
else:
deltab = {b}
# p1 = min(IMAGES) = min(Union D_p*h*deltab for h in TAB)
if all_metrics_with_sym:
md = _min_dummies(dumx, sym, indices)
else:
md = [min(_orbit(size, [_af_new(
ddx) for ddx in dsgsx], ii)) for ii in range(size - 2)]
p_i = min([min([md[h[x]] for x in deltab]) for s, d, h in TAB])
dsgsx1 = [_af_new(_) for _ in dsgsx]
Dxtrav = _orbit_transversal(size, dsgsx1, p_i, False, af=True) \
if dsgsx else None
if Dxtrav:
Dxtrav = [_af_invert(x) for x in Dxtrav]
# compute the orbit of p_i
for ii in range(num_types):
if p_i in dumx[ii]:
# the orbit is made by all the indices in dum[ii]
if sym[ii] is not None:
deltap = dumx[ii]
else:
# the orbit is made by all the even indices if p_i
# is even, by all the odd indices if p_i is odd
p_i_index = dumx[ii].index(p_i) % 2
deltap = dumx[ii][p_i_index::2]
break
else:
deltap = [p_i]
TAB1 = []
while TAB:
s, d, h = TAB.pop()
if min([md[h[x]] for x in deltab]) != p_i:
continue
deltab1 = [x for x in deltab if md[h[x]] == p_i]
# NEXT = s*deltab1 intersection (d*g)**-1*deltap
dg = _af_rmul(d, g)
dginv = _af_invert(dg)
sdeltab = [s[x] for x in deltab1]
gdeltap = [dginv[x] for x in deltap]
NEXT = [x for x in sdeltab if x in gdeltap]
# d, s satisfy
# d*g*s*base[i-1] = p_{i-1}; using the stabilizers
# d*g*s*S_{base[0],...,base[i-1]}*base[i-1] =
# D_{p_0,...,p_{i-1}}*p_{i-1}
# so that to find d1, s1 satisfying d1*g*s1*b = p_i
# one can look for dx in D_{p_0,...,p_{i-1}} and
# sx in S_{base[0],...,base[i-1]}
# d1 = dx*d; s1 = s*sx
# d1*g*s1*b = dx*d*g*s*sx*b = p_i
for j in NEXT:
if testb:
# solve s1*b = j with s1 = s*sx for some element sx
# of the stabilizer of ..., base[i-1]
# sx*b = s**-1*j; sx = _trace_S(s, j,...)
# s1 = s*trace_S(s**-1*j,...)
s1 = _trace_S(s, j, b, S_transversals)
if not s1:
continue
else:
s1 = [s[ix] for ix in s1]
else:
s1 = s
# assert s1[b] == j # invariant
# solve d1*g*j = p_i with d1 = dx*d for some element dg
# of the stabilizer of ..., p_{i-1}
# dx**-1*p_i = d*g*j; dx**-1 = trace_D(d*g*j,...)
# d1 = trace_D(d*g*j,...)**-1*d
# to save an inversion in the inner loop; notice we did
# Dxtrav = [perm_af_invert(x) for x in Dxtrav] out of the loop
if Dxtrav:
d1 = _trace_D(dg[j], p_i, Dxtrav)
if not d1:
continue
else:
if p_i != dg[j]:
continue
d1 = idn
assert d1[dg[j]] == p_i # invariant
d1 = [d1[ix] for ix in d]
h1 = [d1[g[ix]] for ix in s1]
# assert h1[b] == p_i # invariant
TAB1.append((s1, d1, h1))
# if TAB contains equal permutations, keep only one of them;
# if TAB contains equal permutations up to the sign, return 0
TAB1.sort(key=lambda x: x[-1])
prev = [0] * size
while TAB1:
s, d, h = TAB1.pop()
if h[:-2] == prev[:-2]:
if h[-1] != prev[-1]:
return 0
else:
TAB.append((s, d, h))
prev = h
# stabilize the SGS
sgensx = [h for h in sgensx if h[b] == b]
if b in b_S:
b_S.remove(b)
_dumx_remove(dumx, dumx_flat, p_i)
dsgsx = []
for i in range(num_types):
dsgsx.extend(dummy_sgs(dumx[i], sym[i], num_dummies))
return TAB[0][-1]
def canonical_free(base, gens, g, num_free):
"""
Canonicalization of a tensor with respect to free indices
choosing the minimum with respect to lexicographical ordering
in the free indices.
Explanation
===========
``base``, ``gens`` BSGS for slot permutation group
``g`` permutation representing the tensor
``num_free`` number of free indices
The indices must be ordered with first the free indices
See explanation in double_coset_can_rep
The algorithm is a variation of the one given in [2].
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import canonical_free
>>> gens = [[1, 0, 2, 3, 5, 4], [2, 3, 0, 1, 4, 5],[0, 1, 3, 2, 5, 4]]
>>> gens = [Permutation(h) for h in gens]
>>> base = [0, 2]
>>> g = Permutation([2, 1, 0, 3, 4, 5])
>>> canonical_free(base, gens, g, 4)
[0, 3, 1, 2, 5, 4]
Consider the product of Riemann tensors
``T = R^{a}_{d0}^{d1,d2}*R_{d2,d1}^{d0,b}``
The order of the indices is ``[a, b, d0, -d0, d1, -d1, d2, -d2]``
The permutation corresponding to the tensor is
``g = [0, 3, 4, 6, 7, 5, 2, 1, 8, 9]``
In particular ``a`` is position ``0``, ``b`` is in position ``9``.
Use the slot symmetries to get `T` is a form which is the minimal
in lexicographic order in the free indices ``a`` and ``b``, e.g.
``-R^{a}_{d0}^{d1,d2}*R^{b,d0}_{d2,d1}`` corresponding to
``[0, 3, 4, 6, 1, 2, 7, 5, 9, 8]``
>>> from sympy.combinatorics.tensor_can import riemann_bsgs, tensor_gens
>>> base, gens = riemann_bsgs
>>> size, sbase, sgens = tensor_gens(base, gens, [[], []], 0)
>>> g = Permutation([0, 3, 4, 6, 7, 5, 2, 1, 8, 9])
>>> canonical_free(sbase, [Permutation(h) for h in sgens], g, 2)
[0, 3, 4, 6, 1, 2, 7, 5, 9, 8]
"""
g = g.array_form
size = len(g)
if not base:
return g[:]
transversals = get_transversals(base, gens)
for x in sorted(g[:-2]):
if x not in base:
base.append(x)
h = g
for i, transv in enumerate(transversals):
h_i = [size]*num_free
# find the element s in transversals[i] such that
# _af_rmul(h, s) has its free elements with the lowest position in h
s = None
for sk in transv.values():
h1 = _af_rmul(h, sk)
hi = [h1.index(ix) for ix in range(num_free)]
if hi < h_i:
h_i = hi
s = sk
if s:
h = _af_rmul(h, s)
return h
def _get_map_slots(size, fixed_slots):
res = list(range(size))
pos = 0
for i in range(size):
if i in fixed_slots:
continue
res[i] = pos
pos += 1
return res
def _lift_sgens(size, fixed_slots, free, s):
a = []
j = k = 0
fd = list(zip(fixed_slots, free))
fd = [y for x, y in sorted(fd)]
num_free = len(free)
for i in range(size):
if i in fixed_slots:
a.append(fd[k])
k += 1
else:
a.append(s[j] + num_free)
j += 1
return a
def canonicalize(g, dummies, msym, *v):
"""
canonicalize tensor formed by tensors
Parameters
==========
g : permutation representing the tensor
dummies : list representing the dummy indices
it can be a list of dummy indices of the same type
or a list of lists of dummy indices, one list for each
type of index;
the dummy indices must come after the free indices,
and put in order contravariant, covariant
[d0, -d0, d1,-d1,...]
msym : symmetry of the metric(s)
it can be an integer or a list;
in the first case it is the symmetry of the dummy index metric;
in the second case it is the list of the symmetries of the
index metric for each type
v : list, (base_i, gens_i, n_i, sym_i) for tensors of type `i`
base_i, gens_i : BSGS for tensors of this type.
The BSGS should have minimal base under lexicographic ordering;
if not, an attempt is made do get the minimal BSGS;
in case of failure,
canonicalize_naive is used, which is much slower.
n_i : number of tensors of type `i`.
sym_i : symmetry under exchange of component tensors of type `i`.
Both for msym and sym_i the cases are
* None no symmetry
* 0 commuting
* 1 anticommuting
Returns
=======
0 if the tensor is zero, else return the array form of
the permutation representing the canonical form of the tensor.
Algorithm
=========
First one uses canonical_free to get the minimum tensor under
lexicographic order, using only the slot symmetries.
If the component tensors have not minimal BSGS, it is attempted
to find it; if the attempt fails canonicalize_naive
is used instead.
Compute the residual slot symmetry keeping fixed the free indices
using tensor_gens(base, gens, list_free_indices, sym).
Reduce the problem eliminating the free indices.
Then use double_coset_can_rep and lift back the result reintroducing
the free indices.
Examples
========
one type of index with commuting metric;
`A_{a b}` and `B_{a b}` antisymmetric and commuting
`T = A_{d0 d1} * B^{d0}{}_{d2} * B^{d2 d1}`
`ord = [d0,-d0,d1,-d1,d2,-d2]` order of the indices
g = [1, 3, 0, 5, 4, 2, 6, 7]
`T_c = 0`
>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, canonicalize, bsgs_direct_product
>>> from sympy.combinatorics import Permutation
>>> base2a, gens2a = get_symmetric_group_sgs(2, 1)
>>> t0 = (base2a, gens2a, 1, 0)
>>> t1 = (base2a, gens2a, 2, 0)
>>> g = Permutation([1, 3, 0, 5, 4, 2, 6, 7])
>>> canonicalize(g, range(6), 0, t0, t1)
0
same as above, but with `B_{a b}` anticommuting
`T_c = -A^{d0 d1} * B_{d0}{}^{d2} * B_{d1 d2}`
can = [0,2,1,4,3,5,7,6]
>>> t1 = (base2a, gens2a, 2, 1)
>>> canonicalize(g, range(6), 0, t0, t1)
[0, 2, 1, 4, 3, 5, 7, 6]
two types of indices `[a,b,c,d,e,f]` and `[m,n]`, in this order,
both with commuting metric
`f^{a b c}` antisymmetric, commuting
`A_{m a}` no symmetry, commuting
`T = f^c{}_{d a} * f^f{}_{e b} * A_m{}^d * A^{m b} * A_n{}^a * A^{n e}`
ord = [c,f,a,-a,b,-b,d,-d,e,-e,m,-m,n,-n]
g = [0,7,3, 1,9,5, 11,6, 10,4, 13,2, 12,8, 14,15]
The canonical tensor is
`T_c = -f^{c a b} * f^{f d e} * A^m{}_a * A_{m d} * A^n{}_b * A_{n e}`
can = [0,2,4, 1,6,8, 10,3, 11,7, 12,5, 13,9, 15,14]
>>> base_f, gens_f = get_symmetric_group_sgs(3, 1)
>>> base1, gens1 = get_symmetric_group_sgs(1)
>>> base_A, gens_A = bsgs_direct_product(base1, gens1, base1, gens1)
>>> t0 = (base_f, gens_f, 2, 0)
>>> t1 = (base_A, gens_A, 4, 0)
>>> dummies = [range(2, 10), range(10, 14)]
>>> g = Permutation([0, 7, 3, 1, 9, 5, 11, 6, 10, 4, 13, 2, 12, 8, 14, 15])
>>> canonicalize(g, dummies, [0, 0], t0, t1)
[0, 2, 4, 1, 6, 8, 10, 3, 11, 7, 12, 5, 13, 9, 15, 14]
"""
from sympy.combinatorics.testutil import canonicalize_naive
if not isinstance(msym, list):
if msym not in (0, 1, None):
raise ValueError('msym must be 0, 1 or None')
num_types = 1
else:
num_types = len(msym)
if not all(msymx in (0, 1, None) for msymx in msym):
raise ValueError('msym entries must be 0, 1 or None')
if len(dummies) != num_types:
raise ValueError(
'dummies and msym must have the same number of elements')
size = g.size
num_tensors = 0
v1 = []
for i in range(len(v)):
base_i, gens_i, n_i, sym_i = v[i]
# check that the BSGS is minimal;
# this property is used in double_coset_can_rep;
# if it is not minimal use canonicalize_naive
if not _is_minimal_bsgs(base_i, gens_i):
mbsgs = get_minimal_bsgs(base_i, gens_i)
if not mbsgs:
can = canonicalize_naive(g, dummies, msym, *v)
return can
base_i, gens_i = mbsgs
v1.append((base_i, gens_i, [[]] * n_i, sym_i))
num_tensors += n_i
if num_types == 1 and not isinstance(msym, list):
dummies = [dummies]
msym = [msym]
flat_dummies = []
for dumx in dummies:
flat_dummies.extend(dumx)
if flat_dummies and flat_dummies != list(range(flat_dummies[0], flat_dummies[-1] + 1)):
raise ValueError('dummies is not valid')
# slot symmetry of the tensor
size1, sbase, sgens = gens_products(*v1)
if size != size1:
raise ValueError(
'g has size %d, generators have size %d' % (size, size1))
free = [i for i in range(size - 2) if i not in flat_dummies]
num_free = len(free)
# g1 minimal tensor under slot symmetry
g1 = canonical_free(sbase, sgens, g, num_free)
if not flat_dummies:
return g1
# save the sign of g1
sign = 0 if g1[-1] == size - 1 else 1
# the free indices are kept fixed.
# Determine free_i, the list of slots of tensors which are fixed
# since they are occupied by free indices, which are fixed.
start = 0
for i in range(len(v)):
free_i = []
base_i, gens_i, n_i, sym_i = v[i]
len_tens = gens_i[0].size - 2
# for each component tensor get a list od fixed islots
for j in range(n_i):
# get the elements corresponding to the component tensor
h = g1[start:(start + len_tens)]
fr = []
# get the positions of the fixed elements in h
for k in free:
if k in h:
fr.append(h.index(k))
free_i.append(fr)
start += len_tens
v1[i] = (base_i, gens_i, free_i, sym_i)
# BSGS of the tensor with fixed free indices
# if tensor_gens fails in gens_product, use canonicalize_naive
size, sbase, sgens = gens_products(*v1)
# reduce the permutations getting rid of the free indices
pos_free = [g1.index(x) for x in range(num_free)]
size_red = size - num_free
g1_red = [x - num_free for x in g1 if x in flat_dummies]
if sign:
g1_red.extend([size_red - 1, size_red - 2])
else:
g1_red.extend([size_red - 2, size_red - 1])
map_slots = _get_map_slots(size, pos_free)
sbase_red = [map_slots[i] for i in sbase if i not in pos_free]
sgens_red = [_af_new([map_slots[i] for i in y._array_form if i not in pos_free]) for y in sgens]
dummies_red = [[x - num_free for x in y] for y in dummies]
transv_red = get_transversals(sbase_red, sgens_red)
g1_red = _af_new(g1_red)
g2 = double_coset_can_rep(
dummies_red, msym, sbase_red, sgens_red, transv_red, g1_red)
if g2 == 0:
return 0
# lift to the case with the free indices
g3 = _lift_sgens(size, pos_free, free, g2)
return g3
def perm_af_direct_product(gens1, gens2, signed=True):
"""
Direct products of the generators gens1 and gens2.
Examples
========
>>> from sympy.combinatorics.tensor_can import perm_af_direct_product
>>> gens1 = [[1, 0, 2, 3], [0, 1, 3, 2]]
>>> gens2 = [[1, 0]]
>>> perm_af_direct_product(gens1, gens2, False)
[[1, 0, 2, 3, 4, 5], [0, 1, 3, 2, 4, 5], [0, 1, 2, 3, 5, 4]]
>>> gens1 = [[1, 0, 2, 3, 5, 4], [0, 1, 3, 2, 4, 5]]
>>> gens2 = [[1, 0, 2, 3]]
>>> perm_af_direct_product(gens1, gens2, True)
[[1, 0, 2, 3, 4, 5, 7, 6], [0, 1, 3, 2, 4, 5, 6, 7], [0, 1, 2, 3, 5, 4, 6, 7]]
"""
gens1 = [list(x) for x in gens1]
gens2 = [list(x) for x in gens2]
s = 2 if signed else 0
n1 = len(gens1[0]) - s
n2 = len(gens2[0]) - s
start = list(range(n1))
end = list(range(n1, n1 + n2))
if signed:
gens1 = [gen[:-2] + end + [gen[-2] + n2, gen[-1] + n2]
for gen in gens1]
gens2 = [start + [x + n1 for x in gen] for gen in gens2]
else:
gens1 = [gen + end for gen in gens1]
gens2 = [start + [x + n1 for x in gen] for gen in gens2]
res = gens1 + gens2
return res
def bsgs_direct_product(base1, gens1, base2, gens2, signed=True):
"""
Direct product of two BSGS.
Parameters
==========
base1 : base of the first BSGS.
gens1 : strong generating sequence of the first BSGS.
base2, gens2 : similarly for the second BSGS.
signed : flag for signed permutations.
Examples
========
>>> from sympy.combinatorics.tensor_can import (get_symmetric_group_sgs, bsgs_direct_product)
>>> base1, gens1 = get_symmetric_group_sgs(1)
>>> base2, gens2 = get_symmetric_group_sgs(2)
>>> bsgs_direct_product(base1, gens1, base2, gens2)
([1], [(4)(1 2)])
"""
s = 2 if signed else 0
n1 = gens1[0].size - s
base = list(base1)
base += [x + n1 for x in base2]
gens1 = [h._array_form for h in gens1]
gens2 = [h._array_form for h in gens2]
gens = perm_af_direct_product(gens1, gens2, signed)
size = len(gens[0])
id_af = list(range(size))
gens = [h for h in gens if h != id_af]
if not gens:
gens = [id_af]
return base, [_af_new(h) for h in gens]
def get_symmetric_group_sgs(n, antisym=False):
"""
Return base, gens of the minimal BSGS for (anti)symmetric tensor
Parameters
==========
``n``: rank of the tensor
``antisym`` : bool
``antisym = False`` symmetric tensor
``antisym = True`` antisymmetric tensor
Examples
========
>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs
>>> get_symmetric_group_sgs(3)
([0, 1], [(4)(0 1), (4)(1 2)])
"""
if n == 1:
return [], [_af_new(list(range(3)))]
gens = [Permutation(n - 1)(i, i + 1)._array_form for i in range(n - 1)]
if antisym == 0:
gens = [x + [n, n + 1] for x in gens]
else:
gens = [x + [n + 1, n] for x in gens]
base = list(range(n - 1))
return base, [_af_new(h) for h in gens]
riemann_bsgs = [0, 2], [Permutation(0, 1)(4, 5), Permutation(2, 3)(4, 5),
Permutation(5)(0, 2)(1, 3)]
def get_transversals(base, gens):
"""
Return transversals for the group with BSGS base, gens
"""
if not base:
return []
stabs = _distribute_gens_by_base(base, gens)
orbits, transversals = _orbits_transversals_from_bsgs(base, stabs)
transversals = [{x: h._array_form for x, h in y.items()} for y in
transversals]
return transversals
def _is_minimal_bsgs(base, gens):
"""
Check if the BSGS has minimal base under lexigographic order.
base, gens BSGS
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import riemann_bsgs, _is_minimal_bsgs
>>> _is_minimal_bsgs(*riemann_bsgs)
True
>>> riemann_bsgs1 = ([2, 0], ([Permutation(5)(0, 1)(4, 5), Permutation(5)(0, 2)(1, 3)]))
>>> _is_minimal_bsgs(*riemann_bsgs1)
False
"""
base1 = []
sgs1 = gens[:]
size = gens[0].size
for i in range(size):
if not all(h._array_form[i] == i for h in sgs1):
base1.append(i)
sgs1 = [h for h in sgs1 if h._array_form[i] == i]
return base1 == base
def get_minimal_bsgs(base, gens):
"""
Compute a minimal GSGS
base, gens BSGS
If base, gens is a minimal BSGS return it; else return a minimal BSGS
if it fails in finding one, it returns None
TODO: use baseswap in the case in which if it fails in finding a
minimal BSGS
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.tensor_can import get_minimal_bsgs
>>> riemann_bsgs1 = ([2, 0], ([Permutation(5)(0, 1)(4, 5), Permutation(5)(0, 2)(1, 3)]))
>>> get_minimal_bsgs(*riemann_bsgs1)
([0, 2], [(0 1)(4 5), (5)(0 2)(1 3), (2 3)(4 5)])
"""
G = PermutationGroup(gens)
base, gens = G.schreier_sims_incremental()
if not _is_minimal_bsgs(base, gens):
return None
return base, gens
def tensor_gens(base, gens, list_free_indices, sym=0):
"""
Returns size, res_base, res_gens BSGS for n tensors of the
same type.
Explanation
===========
base, gens BSGS for tensors of this type
list_free_indices list of the slots occupied by fixed indices
for each of the tensors
sym symmetry under commutation of two tensors
sym None no symmetry
sym 0 commuting
sym 1 anticommuting
Examples
========
>>> from sympy.combinatorics.tensor_can import tensor_gens, get_symmetric_group_sgs
two symmetric tensors with 3 indices without free indices
>>> base, gens = get_symmetric_group_sgs(3)
>>> tensor_gens(base, gens, [[], []])
(8, [0, 1, 3, 4], [(7)(0 1), (7)(1 2), (7)(3 4), (7)(4 5), (7)(0 3)(1 4)(2 5)])
two symmetric tensors with 3 indices with free indices in slot 1 and 0
>>> tensor_gens(base, gens, [[1], [0]])
(8, [0, 4], [(7)(0 2), (7)(4 5)])
four symmetric tensors with 3 indices, two of which with free indices
"""
def _get_bsgs(G, base, gens, free_indices):
"""
return the BSGS for G.pointwise_stabilizer(free_indices)
"""
if not free_indices:
return base[:], gens[:]
else:
H = G.pointwise_stabilizer(free_indices)
base, sgs = H.schreier_sims_incremental()
return base, sgs
# if not base there is no slot symmetry for the component tensors
# if list_free_indices.count([]) < 2 there is no commutation symmetry
# so there is no resulting slot symmetry
if not base and list_free_indices.count([]) < 2:
n = len(list_free_indices)
size = gens[0].size
size = n * (size - 2) + 2
return size, [], [_af_new(list(range(size)))]
# if any(list_free_indices) one needs to compute the pointwise
# stabilizer, so G is needed
if any(list_free_indices):
G = PermutationGroup(gens)
else:
G = None
# no_free list of lists of indices for component tensors without fixed
# indices
no_free = []
size = gens[0].size
id_af = list(range(size))
num_indices = size - 2
if not list_free_indices[0]:
no_free.append(list(range(num_indices)))
res_base, res_gens = _get_bsgs(G, base, gens, list_free_indices[0])
for i in range(1, len(list_free_indices)):
base1, gens1 = _get_bsgs(G, base, gens, list_free_indices[i])
res_base, res_gens = bsgs_direct_product(res_base, res_gens,
base1, gens1, 1)
if not list_free_indices[i]:
no_free.append(list(range(size - 2, size - 2 + num_indices)))
size += num_indices
nr = size - 2
res_gens = [h for h in res_gens if h._array_form != id_af]
# if sym there are no commuting tensors stop here
if sym is None or not no_free:
if not res_gens:
res_gens = [_af_new(id_af)]
return size, res_base, res_gens
# if the component tensors have moinimal BSGS, so is their direct
# product P; the slot symmetry group is S = P*C, where C is the group
# to (anti)commute the component tensors with no free indices
# a stabilizer has the property S_i = P_i*C_i;
# the BSGS of P*C has SGS_P + SGS_C and the base is
# the ordered union of the bases of P and C.
# If P has minimal BSGS, so has S with this base.
base_comm = []
for i in range(len(no_free) - 1):
ind1 = no_free[i]
ind2 = no_free[i + 1]
a = list(range(ind1[0]))
a.extend(ind2)
a.extend(ind1)
base_comm.append(ind1[0])
a.extend(list(range(ind2[-1] + 1, nr)))
if sym == 0:
a.extend([nr, nr + 1])
else:
a.extend([nr + 1, nr])
res_gens.append(_af_new(a))
res_base = list(res_base)
# each base is ordered; order the union of the two bases
for i in base_comm:
if i not in res_base:
res_base.append(i)
res_base.sort()
if not res_gens:
res_gens = [_af_new(id_af)]
return size, res_base, res_gens
def gens_products(*v):
"""
Returns size, res_base, res_gens BSGS for n tensors of different types.
Explanation
===========
v is a sequence of (base_i, gens_i, free_i, sym_i)
where
base_i, gens_i BSGS of tensor of type `i`
free_i list of the fixed slots for each of the tensors
of type `i`; if there are `n_i` tensors of type `i`
and none of them have fixed slots, `free = [[]]*n_i`
sym 0 (1) if the tensors of type `i` (anti)commute among themselves
Examples
========
>>> from sympy.combinatorics.tensor_can import get_symmetric_group_sgs, gens_products
>>> base, gens = get_symmetric_group_sgs(2)
>>> gens_products((base, gens, [[], []], 0))
(6, [0, 2], [(5)(0 1), (5)(2 3), (5)(0 2)(1 3)])
>>> gens_products((base, gens, [[1], []], 0))
(6, [2], [(5)(2 3)])
"""
res_size, res_base, res_gens = tensor_gens(*v[0])
for i in range(1, len(v)):
size, base, gens = tensor_gens(*v[i])
res_base, res_gens = bsgs_direct_product(res_base, res_gens, base,
gens, 1)
res_size = res_gens[0].size
id_af = list(range(res_size))
res_gens = [h for h in res_gens if h != id_af]
if not res_gens:
res_gens = [id_af]
return res_size, res_base, res_gens
|
5dcfb746b81dfb413e0bc1a79af1dd01c73ff7a51099924eb07dbdc223ce02a0 | from sympy.ntheory.primetest import isprime
from sympy.combinatorics.perm_groups import PermutationGroup
from sympy.printing.defaults import DefaultPrinting
from sympy.combinatorics.free_groups import free_group
class PolycyclicGroup(DefaultPrinting):
is_group = True
is_solvable = True
def __init__(self, pc_sequence, pc_series, relative_order, collector=None):
"""
Parameters
==========
pc_sequence : list
A sequence of elements whose classes generate the cyclic factor
groups of pc_series.
pc_series : list
A subnormal sequence of subgroups where each factor group is cyclic.
relative_order : list
The orders of factor groups of pc_series.
collector : Collector
By default, it is None. Collector class provides the
polycyclic presentation with various other functionalities.
"""
self.pcgs = pc_sequence
self.pc_series = pc_series
self.relative_order = relative_order
self.collector = Collector(self.pcgs, pc_series, relative_order) if not collector else collector
def is_prime_order(self):
return all(isprime(order) for order in self.relative_order)
def length(self):
return len(self.pcgs)
class Collector(DefaultPrinting):
"""
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 8.1.3
"""
def __init__(self, pcgs, pc_series, relative_order, free_group_=None, pc_presentation=None):
"""
Most of the parameters for the Collector class are the same as for PolycyclicGroup.
Others are described below.
Parameters
==========
free_group_ : tuple
free_group_ provides the mapping of polycyclic generating
sequence with the free group elements.
pc_presentation : dict
Provides the presentation of polycyclic groups with the
help of power and conjugate relators.
See Also
========
PolycyclicGroup
"""
self.pcgs = pcgs
self.pc_series = pc_series
self.relative_order = relative_order
self.free_group = free_group('x:{}'.format(len(pcgs)))[0] if not free_group_ else free_group_
self.index = {s: i for i, s in enumerate(self.free_group.symbols)}
self.pc_presentation = self.pc_relators()
def minimal_uncollected_subword(self, word):
r"""
Returns the minimal uncollected subwords.
Explanation
===========
A word ``v`` defined on generators in ``X`` is a minimal
uncollected subword of the word ``w`` if ``v`` is a subword
of ``w`` and it has one of the following form
* `v = {x_{i+1}}^{a_j}x_i`
* `v = {x_{i+1}}^{a_j}{x_i}^{-1}`
* `v = {x_i}^{a_j}`
for `a_j` not in `\{1, \ldots, s-1\}`. Where, ``s`` is the power
exponent of the corresponding generator.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x1, x2 = free_group("x1, x2")
>>> word = x2**2*x1**7
>>> collector.minimal_uncollected_subword(word)
((x2, 2),)
"""
# To handle the case word = <identity>
if not word:
return None
array = word.array_form
re = self.relative_order
index = self.index
for i in range(len(array)):
s1, e1 = array[i]
if re[index[s1]] and (e1 < 0 or e1 > re[index[s1]]-1):
return ((s1, e1), )
for i in range(len(array)-1):
s1, e1 = array[i]
s2, e2 = array[i+1]
if index[s1] > index[s2]:
e = 1 if e2 > 0 else -1
return ((s1, e1), (s2, e))
return None
def relations(self):
"""
Separates the given relators of pc presentation in power and
conjugate relations.
Returns
=======
(power_rel, conj_rel)
Separates pc presentation into power and conjugate relations.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> power_rel, conj_rel = collector.relations()
>>> power_rel
{x0**2: (), x1**3: ()}
>>> conj_rel
{x0**-1*x1*x0: x1**2}
See Also
========
pc_relators
"""
power_relators = {}
conjugate_relators = {}
for key, value in self.pc_presentation.items():
if len(key.array_form) == 1:
power_relators[key] = value
else:
conjugate_relators[key] = value
return power_relators, conjugate_relators
def subword_index(self, word, w):
"""
Returns the start and ending index of a given
subword in a word.
Parameters
==========
word : FreeGroupElement
word defined on free group elements for a
polycyclic group.
w : FreeGroupElement
subword of a given word, whose starting and
ending index to be computed.
Returns
=======
(i, j)
A tuple containing starting and ending index of ``w``
in the given word.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x1, x2 = free_group("x1, x2")
>>> word = x2**2*x1**7
>>> w = x2**2*x1
>>> collector.subword_index(word, w)
(0, 3)
>>> w = x1**7
>>> collector.subword_index(word, w)
(2, 9)
"""
low = -1
high = -1
for i in range(len(word)-len(w)+1):
if word.subword(i, i+len(w)) == w:
low = i
high = i+len(w)
break
if low == high == -1:
return -1, -1
return low, high
def map_relation(self, w):
"""
Return a conjugate relation.
Explanation
===========
Given a word formed by two free group elements, the
corresponding conjugate relation with those free
group elements is formed and mapped with the collected
word in the polycyclic presentation.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x0, x1 = free_group("x0, x1")
>>> w = x1*x0
>>> collector.map_relation(w)
x1**2
See Also
========
pc_presentation
"""
array = w.array_form
s1 = array[0][0]
s2 = array[1][0]
key = ((s2, -1), (s1, 1), (s2, 1))
key = self.free_group.dtype(key)
return self.pc_presentation[key]
def collected_word(self, word):
r"""
Return the collected form of a word.
Explanation
===========
A word ``w`` is called collected, if `w = {x_{i_1}}^{a_1} * \ldots *
{x_{i_r}}^{a_r}` with `i_1 < i_2< \ldots < i_r` and `a_j` is in
`\{1, \ldots, {s_j}-1\}`.
Otherwise w is uncollected.
Parameters
==========
word : FreeGroupElement
An uncollected word.
Returns
=======
word
A collected word of form `w = {x_{i_1}}^{a_1}, \ldots,
{x_{i_r}}^{a_r}` with `i_1, i_2, \ldots, i_r` and `a_j \in
\{1, \ldots, {s_j}-1\}`.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> F, x0, x1, x2, x3 = free_group("x0, x1, x2, x3")
>>> word = x3*x2*x1*x0
>>> collected_word = collector.collected_word(word)
>>> free_to_perm = {}
>>> free_group = collector.free_group
>>> for sym, gen in zip(free_group.symbols, collector.pcgs):
... free_to_perm[sym] = gen
>>> G1 = PermutationGroup()
>>> for w in word:
... sym = w[0]
... perm = free_to_perm[sym]
... G1 = PermutationGroup([perm] + G1.generators)
>>> G2 = PermutationGroup()
>>> for w in collected_word:
... sym = w[0]
... perm = free_to_perm[sym]
... G2 = PermutationGroup([perm] + G2.generators)
The two are not identical, but they are equivalent:
>>> G1.equals(G2), G1 == G2
(True, False)
See Also
========
minimal_uncollected_subword
"""
free_group = self.free_group
while True:
w = self.minimal_uncollected_subword(word)
if not w:
break
low, high = self.subword_index(word, free_group.dtype(w))
if low == -1:
continue
s1, e1 = w[0]
if len(w) == 1:
re = self.relative_order[self.index[s1]]
q = e1 // re
r = e1-q*re
key = ((w[0][0], re), )
key = free_group.dtype(key)
if self.pc_presentation[key]:
presentation = self.pc_presentation[key].array_form
sym, exp = presentation[0]
word_ = ((w[0][0], r), (sym, q*exp))
word_ = free_group.dtype(word_)
else:
if r != 0:
word_ = ((w[0][0], r), )
word_ = free_group.dtype(word_)
else:
word_ = None
word = word.eliminate_word(free_group.dtype(w), word_)
if len(w) == 2 and w[1][1] > 0:
s2, e2 = w[1]
s2 = ((s2, 1), )
s2 = free_group.dtype(s2)
word_ = self.map_relation(free_group.dtype(w))
word_ = s2*word_**e1
word_ = free_group.dtype(word_)
word = word.substituted_word(low, high, word_)
elif len(w) == 2 and w[1][1] < 0:
s2, e2 = w[1]
s2 = ((s2, 1), )
s2 = free_group.dtype(s2)
word_ = self.map_relation(free_group.dtype(w))
word_ = s2**-1*word_**e1
word_ = free_group.dtype(word_)
word = word.substituted_word(low, high, word_)
return word
def pc_relators(self):
r"""
Return the polycyclic presentation.
Explanation
===========
There are two types of relations used in polycyclic
presentation.
* ``Power relations`` : Power relators are of the form `x_i^{re_i}`,
where `i \in \{0, \ldots, \mathrm{len(pcgs)}\}`, ``x`` represents polycyclic
generator and ``re`` is the corresponding relative order.
* ``Conjugate relations`` : Conjugate relators are of the form `x_j^-1x_ix_j`,
where `j < i \in \{0, \ldots, \mathrm{len(pcgs)}\}`.
Returns
=======
A dictionary with power and conjugate relations as key and
their collected form as corresponding values.
Notes
=====
Identity Permutation is mapped with empty ``()``.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> S = SymmetricGroup(49).sylow_subgroup(7)
>>> der = S.derived_series()
>>> G = der[len(der)-2]
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> pcgs = PcGroup.pcgs
>>> len(pcgs)
6
>>> free_group = collector.free_group
>>> pc_resentation = collector.pc_presentation
>>> free_to_perm = {}
>>> for s, g in zip(free_group.symbols, pcgs):
... free_to_perm[s] = g
>>> for k, v in pc_resentation.items():
... k_array = k.array_form
... if v != ():
... v_array = v.array_form
... lhs = Permutation()
... for gen in k_array:
... s = gen[0]
... e = gen[1]
... lhs = lhs*free_to_perm[s]**e
... if v == ():
... assert lhs.is_identity
... continue
... rhs = Permutation()
... for gen in v_array:
... s = gen[0]
... e = gen[1]
... rhs = rhs*free_to_perm[s]**e
... assert lhs == rhs
"""
free_group = self.free_group
rel_order = self.relative_order
pc_relators = {}
perm_to_free = {}
pcgs = self.pcgs
for gen, s in zip(pcgs, free_group.generators):
perm_to_free[gen**-1] = s**-1
perm_to_free[gen] = s
pcgs = pcgs[::-1]
series = self.pc_series[::-1]
rel_order = rel_order[::-1]
collected_gens = []
for i, gen in enumerate(pcgs):
re = rel_order[i]
relation = perm_to_free[gen]**re
G = series[i]
l = G.generator_product(gen**re, original = True)
l.reverse()
word = free_group.identity
for g in l:
word = word*perm_to_free[g]
word = self.collected_word(word)
pc_relators[relation] = word if word else ()
self.pc_presentation = pc_relators
collected_gens.append(gen)
if len(collected_gens) > 1:
conj = collected_gens[len(collected_gens)-1]
conjugator = perm_to_free[conj]
for j in range(len(collected_gens)-1):
conjugated = perm_to_free[collected_gens[j]]
relation = conjugator**-1*conjugated*conjugator
gens = conj**-1*collected_gens[j]*conj
l = G.generator_product(gens, original = True)
l.reverse()
word = free_group.identity
for g in l:
word = word*perm_to_free[g]
word = self.collected_word(word)
pc_relators[relation] = word if word else ()
self.pc_presentation = pc_relators
return pc_relators
def exponent_vector(self, element):
r"""
Return the exponent vector of length equal to the
length of polycyclic generating sequence.
Explanation
===========
For a given generator/element ``g`` of the polycyclic group,
it can be represented as `g = {x_1}^{e_1}, \ldots, {x_n}^{e_n}`,
where `x_i` represents polycyclic generators and ``n`` is
the number of generators in the free_group equal to the length
of pcgs.
Parameters
==========
element : Permutation
Generator of a polycyclic group.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> G = SymmetricGroup(4)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> pcgs = PcGroup.pcgs
>>> collector.exponent_vector(G[0])
[1, 0, 0, 0]
>>> exp = collector.exponent_vector(G[1])
>>> g = Permutation()
>>> for i in range(len(exp)):
... g = g*pcgs[i]**exp[i] if exp[i] else g
>>> assert g == G[1]
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 8.1.1, Definition 8.4
"""
free_group = self.free_group
G = PermutationGroup()
for g in self.pcgs:
G = PermutationGroup([g] + G.generators)
gens = G.generator_product(element, original = True)
gens.reverse()
perm_to_free = {}
for sym, g in zip(free_group.generators, self.pcgs):
perm_to_free[g**-1] = sym**-1
perm_to_free[g] = sym
w = free_group.identity
for g in gens:
w = w*perm_to_free[g]
word = self.collected_word(w)
index = self.index
exp_vector = [0]*len(free_group)
word = word.array_form
for t in word:
exp_vector[index[t[0]]] = t[1]
return exp_vector
def depth(self, element):
r"""
Return the depth of a given element.
Explanation
===========
The depth of a given element ``g`` is defined by
`\mathrm{dep}[g] = i` if `e_1 = e_2 = \ldots = e_{i-1} = 0`
and `e_i != 0`, where ``e`` represents the exponent-vector.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> collector.depth(G[0])
2
>>> collector.depth(G[1])
1
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 8.1.1, Definition 8.5
"""
exp_vector = self.exponent_vector(element)
return next((i+1 for i, x in enumerate(exp_vector) if x), len(self.pcgs)+1)
def leading_exponent(self, element):
r"""
Return the leading non-zero exponent.
Explanation
===========
The leading exponent for a given element `g` is defined
by `\mathrm{leading\_exponent}[g]` `= e_i`, if `\mathrm{depth}[g] = i`.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> G = SymmetricGroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> collector.leading_exponent(G[1])
1
"""
exp_vector = self.exponent_vector(element)
depth = self.depth(element)
if depth != len(self.pcgs)+1:
return exp_vector[depth-1]
return None
def _sift(self, z, g):
h = g
d = self.depth(h)
while d < len(self.pcgs) and z[d-1] != 1:
k = z[d-1]
e = self.leading_exponent(h)*(self.leading_exponent(k))**-1
e = e % self.relative_order[d-1]
h = k**-e*h
d = self.depth(h)
return h
def induced_pcgs(self, gens):
"""
Parameters
==========
gens : list
A list of generators on which polycyclic subgroup
is to be defined.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> S = SymmetricGroup(8)
>>> G = S.sylow_subgroup(2)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> gens = [G[0], G[1]]
>>> ipcgs = collector.induced_pcgs(gens)
>>> [gen.order() for gen in ipcgs]
[2, 2, 2]
>>> G = S.sylow_subgroup(3)
>>> PcGroup = G.polycyclic_group()
>>> collector = PcGroup.collector
>>> gens = [G[0], G[1]]
>>> ipcgs = collector.induced_pcgs(gens)
>>> [gen.order() for gen in ipcgs]
[3]
"""
z = [1]*len(self.pcgs)
G = gens
while G:
g = G.pop(0)
h = self._sift(z, g)
d = self.depth(h)
if d < len(self.pcgs):
for gen in z:
if gen != 1:
G.append(h**-1*gen**-1*h*gen)
z[d-1] = h;
z = [gen for gen in z if gen != 1]
return z
def constructive_membership_test(self, ipcgs, g):
"""
Return the exponent vector for induced pcgs.
"""
e = [0]*len(ipcgs)
h = g
d = self.depth(h)
for i, gen in enumerate(ipcgs):
while self.depth(gen) == d:
f = self.leading_exponent(h)*self.leading_exponent(gen)
f = f % self.relative_order[d-1]
h = gen**(-f)*h
e[i] = f
d = self.depth(h)
if h == 1:
return e
return False
|
183e339fa7341d569b0b03c70c422d643d7d9965b43ef49f6029a81b10783ba2 | import itertools
from sympy.combinatorics.fp_groups import FpGroup, FpSubgroup, simplify_presentation
from sympy.combinatorics.free_groups import FreeGroup
from sympy.combinatorics.perm_groups import PermutationGroup
from sympy.core.numbers import igcd
from sympy.ntheory.factor_ import totient
from sympy.core.singleton import S
class GroupHomomorphism:
'''
A class representing group homomorphisms. Instantiate using `homomorphism()`.
References
==========
.. [1] Holt, D., Eick, B. and O'Brien, E. (2005). Handbook of computational group theory.
'''
def __init__(self, domain, codomain, images):
self.domain = domain
self.codomain = codomain
self.images = images
self._inverses = None
self._kernel = None
self._image = None
def _invs(self):
'''
Return a dictionary with `{gen: inverse}` where `gen` is a rewriting
generator of `codomain` (e.g. strong generator for permutation groups)
and `inverse` is an element of its preimage
'''
image = self.image()
inverses = {}
for k in list(self.images.keys()):
v = self.images[k]
if not (v in inverses
or v.is_identity):
inverses[v] = k
if isinstance(self.codomain, PermutationGroup):
gens = image.strong_gens
else:
gens = image.generators
for g in gens:
if g in inverses or g.is_identity:
continue
w = self.domain.identity
if isinstance(self.codomain, PermutationGroup):
parts = image._strong_gens_slp[g][::-1]
else:
parts = g
for s in parts:
if s in inverses:
w = w*inverses[s]
else:
w = w*inverses[s**-1]**-1
inverses[g] = w
return inverses
def invert(self, g):
'''
Return an element of the preimage of ``g`` or of each element
of ``g`` if ``g`` is a list.
Explanation
===========
If the codomain is an FpGroup, the inverse for equal
elements might not always be the same unless the FpGroup's
rewriting system is confluent. However, making a system
confluent can be time-consuming. If it's important, try
`self.codomain.make_confluent()` first.
'''
from sympy.combinatorics import Permutation
from sympy.combinatorics.free_groups import FreeGroupElement
if isinstance(g, (Permutation, FreeGroupElement)):
if isinstance(self.codomain, FpGroup):
g = self.codomain.reduce(g)
if self._inverses is None:
self._inverses = self._invs()
image = self.image()
w = self.domain.identity
if isinstance(self.codomain, PermutationGroup):
gens = image.generator_product(g)[::-1]
else:
gens = g
# the following can't be "for s in gens:"
# because that would be equivalent to
# "for s in gens.array_form:" when g is
# a FreeGroupElement. On the other hand,
# when you call gens by index, the generator
# (or inverse) at position i is returned.
for i in range(len(gens)):
s = gens[i]
if s.is_identity:
continue
if s in self._inverses:
w = w*self._inverses[s]
else:
w = w*self._inverses[s**-1]**-1
return w
elif isinstance(g, list):
return [self.invert(e) for e in g]
def kernel(self):
'''
Compute the kernel of `self`.
'''
if self._kernel is None:
self._kernel = self._compute_kernel()
return self._kernel
def _compute_kernel(self):
G = self.domain
G_order = G.order()
if G_order is S.Infinity:
raise NotImplementedError(
"Kernel computation is not implemented for infinite groups")
gens = []
if isinstance(G, PermutationGroup):
K = PermutationGroup(G.identity)
else:
K = FpSubgroup(G, gens, normal=True)
i = self.image().order()
while K.order()*i != G_order:
r = G.random()
k = r*self.invert(self(r))**-1
if k not in K:
gens.append(k)
if isinstance(G, PermutationGroup):
K = PermutationGroup(gens)
else:
K = FpSubgroup(G, gens, normal=True)
return K
def image(self):
'''
Compute the image of `self`.
'''
if self._image is None:
values = list(set(self.images.values()))
if isinstance(self.codomain, PermutationGroup):
self._image = self.codomain.subgroup(values)
else:
self._image = FpSubgroup(self.codomain, values)
return self._image
def _apply(self, elem):
'''
Apply `self` to `elem`.
'''
if elem not in self.domain:
if isinstance(elem, (list, tuple)):
return [self._apply(e) for e in elem]
raise ValueError("The supplied element doesn't belong to the domain")
if elem.is_identity:
return self.codomain.identity
else:
images = self.images
value = self.codomain.identity
if isinstance(self.domain, PermutationGroup):
gens = self.domain.generator_product(elem, original=True)
for g in gens:
if g in self.images:
value = images[g]*value
else:
value = images[g**-1]**-1*value
else:
i = 0
for _, p in elem.array_form:
if p < 0:
g = elem[i]**-1
else:
g = elem[i]
value = value*images[g]**p
i += abs(p)
return value
def __call__(self, elem):
return self._apply(elem)
def is_injective(self):
'''
Check if the homomorphism is injective
'''
return self.kernel().order() == 1
def is_surjective(self):
'''
Check if the homomorphism is surjective
'''
im = self.image().order()
oth = self.codomain.order()
if im is S.Infinity and oth is S.Infinity:
return None
else:
return im == oth
def is_isomorphism(self):
'''
Check if `self` is an isomorphism.
'''
return self.is_injective() and self.is_surjective()
def is_trivial(self):
'''
Check is `self` is a trivial homomorphism, i.e. all elements
are mapped to the identity.
'''
return self.image().order() == 1
def compose(self, other):
'''
Return the composition of `self` and `other`, i.e.
the homomorphism phi such that for all g in the domain
of `other`, phi(g) = self(other(g))
'''
if not other.image().is_subgroup(self.domain):
raise ValueError("The image of `other` must be a subgroup of "
"the domain of `self`")
images = {g: self(other(g)) for g in other.images}
return GroupHomomorphism(other.domain, self.codomain, images)
def restrict_to(self, H):
'''
Return the restriction of the homomorphism to the subgroup `H`
of the domain.
'''
if not isinstance(H, PermutationGroup) or not H.is_subgroup(self.domain):
raise ValueError("Given H is not a subgroup of the domain")
domain = H
images = {g: self(g) for g in H.generators}
return GroupHomomorphism(domain, self.codomain, images)
def invert_subgroup(self, H):
'''
Return the subgroup of the domain that is the inverse image
of the subgroup ``H`` of the homomorphism image
'''
if not H.is_subgroup(self.image()):
raise ValueError("Given H is not a subgroup of the image")
gens = []
P = PermutationGroup(self.image().identity)
for h in H.generators:
h_i = self.invert(h)
if h_i not in P:
gens.append(h_i)
P = PermutationGroup(gens)
for k in self.kernel().generators:
if k*h_i not in P:
gens.append(k*h_i)
P = PermutationGroup(gens)
return P
def homomorphism(domain, codomain, gens, images=(), check=True):
'''
Create (if possible) a group homomorphism from the group ``domain``
to the group ``codomain`` defined by the images of the domain's
generators ``gens``. ``gens`` and ``images`` can be either lists or tuples
of equal sizes. If ``gens`` is a proper subset of the group's generators,
the unspecified generators will be mapped to the identity. If the
images are not specified, a trivial homomorphism will be created.
If the given images of the generators do not define a homomorphism,
an exception is raised.
If ``check`` is ``False``, do not check whether the given images actually
define a homomorphism.
'''
if not isinstance(domain, (PermutationGroup, FpGroup, FreeGroup)):
raise TypeError("The domain must be a group")
if not isinstance(codomain, (PermutationGroup, FpGroup, FreeGroup)):
raise TypeError("The codomain must be a group")
generators = domain.generators
if not all(g in generators for g in gens):
raise ValueError("The supplied generators must be a subset of the domain's generators")
if not all(g in codomain for g in images):
raise ValueError("The images must be elements of the codomain")
if images and len(images) != len(gens):
raise ValueError("The number of images must be equal to the number of generators")
gens = list(gens)
images = list(images)
images.extend([codomain.identity]*(len(generators)-len(images)))
gens.extend([g for g in generators if g not in gens])
images = dict(zip(gens,images))
if check and not _check_homomorphism(domain, codomain, images):
raise ValueError("The given images do not define a homomorphism")
return GroupHomomorphism(domain, codomain, images)
def _check_homomorphism(domain, codomain, images):
if hasattr(domain, 'relators'):
rels = domain.relators
else:
gens = domain.presentation().generators
rels = domain.presentation().relators
identity = codomain.identity
def _image(r):
if r.is_identity:
return identity
else:
w = identity
r_arr = r.array_form
i = 0
j = 0
# i is the index for r and j is for
# r_arr. r_arr[j] is the tuple (sym, p)
# where sym is the generator symbol
# and p is the power to which it is
# raised while r[i] is a generator
# (not just its symbol) or the inverse of
# a generator - hence the need for
# both indices
while i < len(r):
power = r_arr[j][1]
if isinstance(domain, PermutationGroup) and r[i] in gens:
s = domain.generators[gens.index(r[i])]
else:
s = r[i]
if s in images:
w = w*images[s]**power
elif s**-1 in images:
w = w*images[s**-1]**power
i += abs(power)
j += 1
return w
for r in rels:
if isinstance(codomain, FpGroup):
s = codomain.equals(_image(r), identity)
if s is None:
# only try to make the rewriting system
# confluent when it can't determine the
# truth of equality otherwise
success = codomain.make_confluent()
s = codomain.equals(_image(r), identity)
if s is None and not success:
raise RuntimeError("Can't determine if the images "
"define a homomorphism. Try increasing "
"the maximum number of rewriting rules "
"(group._rewriting_system.set_max(new_value); "
"the current value is stored in group._rewriting"
"_system.maxeqns)")
else:
s = _image(r).is_identity
if not s:
return False
return True
def orbit_homomorphism(group, omega):
'''
Return the homomorphism induced by the action of the permutation
group ``group`` on the set ``omega`` that is closed under the action.
'''
from sympy.combinatorics import Permutation
from sympy.combinatorics.named_groups import SymmetricGroup
codomain = SymmetricGroup(len(omega))
identity = codomain.identity
omega = list(omega)
images = {g: identity*Permutation([omega.index(o^g) for o in omega]) for g in group.generators}
group._schreier_sims(base=omega)
H = GroupHomomorphism(group, codomain, images)
if len(group.basic_stabilizers) > len(omega):
H._kernel = group.basic_stabilizers[len(omega)]
else:
H._kernel = PermutationGroup([group.identity])
return H
def block_homomorphism(group, blocks):
'''
Return the homomorphism induced by the action of the permutation
group ``group`` on the block system ``blocks``. The latter should be
of the same form as returned by the ``minimal_block`` method for
permutation groups, namely a list of length ``group.degree`` where
the i-th entry is a representative of the block i belongs to.
'''
from sympy.combinatorics import Permutation
from sympy.combinatorics.named_groups import SymmetricGroup
n = len(blocks)
# number the blocks; m is the total number,
# b is such that b[i] is the number of the block i belongs to,
# p is the list of length m such that p[i] is the representative
# of the i-th block
m = 0
p = []
b = [None]*n
for i in range(n):
if blocks[i] == i:
p.append(i)
b[i] = m
m += 1
for i in range(n):
b[i] = b[blocks[i]]
codomain = SymmetricGroup(m)
# the list corresponding to the identity permutation in codomain
identity = range(m)
images = {g: Permutation([b[p[i]^g] for i in identity]) for g in group.generators}
H = GroupHomomorphism(group, codomain, images)
return H
def group_isomorphism(G, H, isomorphism=True):
'''
Compute an isomorphism between 2 given groups.
Parameters
==========
G : A finite ``FpGroup`` or a ``PermutationGroup``.
First group.
H : A finite ``FpGroup`` or a ``PermutationGroup``
Second group.
isomorphism : bool
This is used to avoid the computation of homomorphism
when the user only wants to check if there exists
an isomorphism between the groups.
Returns
=======
If isomorphism = False -- Returns a boolean.
If isomorphism = True -- Returns a boolean and an isomorphism between `G` and `H`.
Examples
========
>>> from sympy.combinatorics import Permutation
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup
>>> from sympy.combinatorics.homomorphisms import group_isomorphism
>>> from sympy.combinatorics.named_groups import DihedralGroup, AlternatingGroup
>>> D = DihedralGroup(8)
>>> p = Permutation(0, 1, 2, 3, 4, 5, 6, 7)
>>> P = PermutationGroup(p)
>>> group_isomorphism(D, P)
(False, None)
>>> F, a, b = free_group("a, b")
>>> G = FpGroup(F, [a**3, b**3, (a*b)**2])
>>> H = AlternatingGroup(4)
>>> (check, T) = group_isomorphism(G, H)
>>> check
True
>>> T(b*a*b**-1*a**-1*b**-1)
(0 2 3)
Notes
=====
Uses the approach suggested by Robert Tarjan to compute the isomorphism between two groups.
First, the generators of ``G`` are mapped to the elements of ``H`` and
we check if the mapping induces an isomorphism.
'''
if not isinstance(G, (PermutationGroup, FpGroup)):
raise TypeError("The group must be a PermutationGroup or an FpGroup")
if not isinstance(H, (PermutationGroup, FpGroup)):
raise TypeError("The group must be a PermutationGroup or an FpGroup")
if isinstance(G, FpGroup) and isinstance(H, FpGroup):
G = simplify_presentation(G)
H = simplify_presentation(H)
# Two infinite FpGroups with the same generators are isomorphic
# when the relators are same but are ordered differently.
if G.generators == H.generators and (G.relators).sort() == (H.relators).sort():
if not isomorphism:
return True
return (True, homomorphism(G, H, G.generators, H.generators))
# `_H` is the permutation group isomorphic to `H`.
_H = H
g_order = G.order()
h_order = H.order()
if g_order is S.Infinity:
raise NotImplementedError("Isomorphism methods are not implemented for infinite groups.")
if isinstance(H, FpGroup):
if h_order is S.Infinity:
raise NotImplementedError("Isomorphism methods are not implemented for infinite groups.")
_H, h_isomorphism = H._to_perm_group()
if (g_order != h_order) or (G.is_abelian != H.is_abelian):
if not isomorphism:
return False
return (False, None)
if not isomorphism:
# Two groups of the same cyclic numbered order
# are isomorphic to each other.
n = g_order
if (igcd(n, totient(n))) == 1:
return True
# Match the generators of `G` with subsets of `_H`
gens = list(G.generators)
for subset in itertools.permutations(_H, len(gens)):
images = list(subset)
images.extend([_H.identity]*(len(G.generators)-len(images)))
_images = dict(zip(gens,images))
if _check_homomorphism(G, _H, _images):
if isinstance(H, FpGroup):
images = h_isomorphism.invert(images)
T = homomorphism(G, H, G.generators, images, check=False)
if T.is_isomorphism():
# It is a valid isomorphism
if not isomorphism:
return True
return (True, T)
if not isomorphism:
return False
return (False, None)
def is_isomorphic(G, H):
'''
Check if the groups are isomorphic to each other
Parameters
==========
G : A finite ``FpGroup`` or a ``PermutationGroup``
First group.
H : A finite ``FpGroup`` or a ``PermutationGroup``
Second group.
Returns
=======
boolean
'''
return group_isomorphism(G, H, isomorphism=False)
|
5b2b3f8fd0fe5e8a7b942c718273b75720eb490ecd540b84130564c8253eebdb | from sympy.core import Basic, Dict, sympify, Tuple
from sympy.core.numbers import Integer
from sympy.core.sorting import default_sort_key
from sympy.core.sympify import _sympify
from sympy.functions.combinatorial.numbers import bell
from sympy.matrices import zeros
from sympy.sets.sets import FiniteSet, Union
from sympy.utilities.iterables import flatten, group
from sympy.utilities.misc import as_int
from collections import defaultdict
class Partition(FiniteSet):
"""
This class represents an abstract partition.
A partition is a set of disjoint sets whose union equals a given set.
See Also
========
sympy.utilities.iterables.partitions,
sympy.utilities.iterables.multiset_partitions
"""
_rank = None
_partition = None
def __new__(cls, *partition):
"""
Generates a new partition object.
This method also verifies if the arguments passed are
valid and raises a ValueError if they are not.
Examples
========
Creating Partition from Python lists:
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3])
>>> a
Partition({3}, {1, 2})
>>> a.partition
[[1, 2], [3]]
>>> len(a)
2
>>> a.members
(1, 2, 3)
Creating Partition from Python sets:
>>> Partition({1, 2, 3}, {4, 5})
Partition({4, 5}, {1, 2, 3})
Creating Partition from SymPy finite sets:
>>> from sympy import FiniteSet
>>> a = FiniteSet(1, 2, 3)
>>> b = FiniteSet(4, 5)
>>> Partition(a, b)
Partition({4, 5}, {1, 2, 3})
"""
args = []
dups = False
for arg in partition:
if isinstance(arg, list):
as_set = set(arg)
if len(as_set) < len(arg):
dups = True
break # error below
arg = as_set
args.append(_sympify(arg))
if not all(isinstance(part, FiniteSet) for part in args):
raise ValueError(
"Each argument to Partition should be " \
"a list, set, or a FiniteSet")
# sort so we have a canonical reference for RGS
U = Union(*args)
if dups or len(U) < sum(len(arg) for arg in args):
raise ValueError("Partition contained duplicate elements.")
obj = FiniteSet.__new__(cls, *args)
obj.members = tuple(U)
obj.size = len(U)
return obj
def sort_key(self, order=None):
"""Return a canonical key that can be used for sorting.
Ordering is based on the size and sorted elements of the partition
and ties are broken with the rank.
Examples
========
>>> from sympy import default_sort_key
>>> from sympy.combinatorics.partitions import Partition
>>> from sympy.abc import x
>>> a = Partition([1, 2])
>>> b = Partition([3, 4])
>>> c = Partition([1, x])
>>> d = Partition(list(range(4)))
>>> l = [d, b, a + 1, a, c]
>>> l.sort(key=default_sort_key); l
[Partition({1, 2}), Partition({1}, {2}), Partition({1, x}), Partition({3, 4}), Partition({0, 1, 2, 3})]
"""
if order is None:
members = self.members
else:
members = tuple(sorted(self.members,
key=lambda w: default_sort_key(w, order)))
return tuple(map(default_sort_key, (self.size, members, self.rank)))
@property
def partition(self):
"""Return partition as a sorted list of lists.
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> Partition([1], [2, 3]).partition
[[1], [2, 3]]
"""
if self._partition is None:
self._partition = sorted([sorted(p, key=default_sort_key)
for p in self.args])
return self._partition
def __add__(self, other):
"""
Return permutation whose rank is ``other`` greater than current rank,
(mod the maximum rank for the set).
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3])
>>> a.rank
1
>>> (a + 1).rank
2
>>> (a + 100).rank
1
"""
other = as_int(other)
offset = self.rank + other
result = RGS_unrank((offset) %
RGS_enum(self.size),
self.size)
return Partition.from_rgs(result, self.members)
def __sub__(self, other):
"""
Return permutation whose rank is ``other`` less than current rank,
(mod the maximum rank for the set).
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3])
>>> a.rank
1
>>> (a - 1).rank
0
>>> (a - 100).rank
1
"""
return self.__add__(-other)
def __le__(self, other):
"""
Checks if a partition is less than or equal to
the other based on rank.
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3, 4, 5])
>>> b = Partition([1], [2, 3], [4], [5])
>>> a.rank, b.rank
(9, 34)
>>> a <= a
True
>>> a <= b
True
"""
return self.sort_key() <= sympify(other).sort_key()
def __lt__(self, other):
"""
Checks if a partition is less than the other.
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3, 4, 5])
>>> b = Partition([1], [2, 3], [4], [5])
>>> a.rank, b.rank
(9, 34)
>>> a < b
True
"""
return self.sort_key() < sympify(other).sort_key()
@property
def rank(self):
"""
Gets the rank of a partition.
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3], [4, 5])
>>> a.rank
13
"""
if self._rank is not None:
return self._rank
self._rank = RGS_rank(self.RGS)
return self._rank
@property
def RGS(self):
"""
Returns the "restricted growth string" of the partition.
Explanation
===========
The RGS is returned as a list of indices, L, where L[i] indicates
the block in which element i appears. For example, in a partition
of 3 elements (a, b, c) into 2 blocks ([c], [a, b]) the RGS is
[1, 1, 0]: "a" is in block 1, "b" is in block 1 and "c" is in block 0.
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> a = Partition([1, 2], [3], [4, 5])
>>> a.members
(1, 2, 3, 4, 5)
>>> a.RGS
(0, 0, 1, 2, 2)
>>> a + 1
Partition({3}, {4}, {5}, {1, 2})
>>> _.RGS
(0, 0, 1, 2, 3)
"""
rgs = {}
partition = self.partition
for i, part in enumerate(partition):
for j in part:
rgs[j] = i
return tuple([rgs[i] for i in sorted(
[i for p in partition for i in p], key=default_sort_key)])
@classmethod
def from_rgs(self, rgs, elements):
"""
Creates a set partition from a restricted growth string.
Explanation
===========
The indices given in rgs are assumed to be the index
of the element as given in elements *as provided* (the
elements are not sorted by this routine). Block numbering
starts from 0. If any block was not referenced in ``rgs``
an error will be raised.
Examples
========
>>> from sympy.combinatorics.partitions import Partition
>>> Partition.from_rgs([0, 1, 2, 0, 1], list('abcde'))
Partition({c}, {a, d}, {b, e})
>>> Partition.from_rgs([0, 1, 2, 0, 1], list('cbead'))
Partition({e}, {a, c}, {b, d})
>>> a = Partition([1, 4], [2], [3, 5])
>>> Partition.from_rgs(a.RGS, a.members)
Partition({2}, {1, 4}, {3, 5})
"""
if len(rgs) != len(elements):
raise ValueError('mismatch in rgs and element lengths')
max_elem = max(rgs) + 1
partition = [[] for i in range(max_elem)]
j = 0
for i in rgs:
partition[i].append(elements[j])
j += 1
if not all(p for p in partition):
raise ValueError('some blocks of the partition were empty.')
return Partition(*partition)
class IntegerPartition(Basic):
"""
This class represents an integer partition.
Explanation
===========
In number theory and combinatorics, a partition of a positive integer,
``n``, also called an integer partition, is a way of writing ``n`` as a
list of positive integers that sum to n. Two partitions that differ only
in the order of summands are considered to be the same partition; if order
matters then the partitions are referred to as compositions. For example,
4 has five partitions: [4], [3, 1], [2, 2], [2, 1, 1], and [1, 1, 1, 1];
the compositions [1, 2, 1] and [1, 1, 2] are the same as partition
[2, 1, 1].
See Also
========
sympy.utilities.iterables.partitions,
sympy.utilities.iterables.multiset_partitions
References
==========
.. [1] https://en.wikipedia.org/wiki/Partition_%28number_theory%29
"""
_dict = None
_keys = None
def __new__(cls, partition, integer=None):
"""
Generates a new IntegerPartition object from a list or dictionary.
Explantion
==========
The partition can be given as a list of positive integers or a
dictionary of (integer, multiplicity) items. If the partition is
preceded by an integer an error will be raised if the partition
does not sum to that given integer.
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> a = IntegerPartition([5, 4, 3, 1, 1])
>>> a
IntegerPartition(14, (5, 4, 3, 1, 1))
>>> print(a)
[5, 4, 3, 1, 1]
>>> IntegerPartition({1:3, 2:1})
IntegerPartition(5, (2, 1, 1, 1))
If the value that the partition should sum to is given first, a check
will be made to see n error will be raised if there is a discrepancy:
>>> IntegerPartition(10, [5, 4, 3, 1])
Traceback (most recent call last):
...
ValueError: The partition is not valid
"""
if integer is not None:
integer, partition = partition, integer
if isinstance(partition, (dict, Dict)):
_ = []
for k, v in sorted(list(partition.items()), reverse=True):
if not v:
continue
k, v = as_int(k), as_int(v)
_.extend([k]*v)
partition = tuple(_)
else:
partition = tuple(sorted(map(as_int, partition), reverse=True))
sum_ok = False
if integer is None:
integer = sum(partition)
sum_ok = True
else:
integer = as_int(integer)
if not sum_ok and sum(partition) != integer:
raise ValueError("Partition did not add to %s" % integer)
if any(i < 1 for i in partition):
raise ValueError("All integer summands must be greater than one")
obj = Basic.__new__(cls, Integer(integer), Tuple(*partition))
obj.partition = list(partition)
obj.integer = integer
return obj
def prev_lex(self):
"""Return the previous partition of the integer, n, in lexical order,
wrapping around to [1, ..., 1] if the partition is [n].
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> p = IntegerPartition([4])
>>> print(p.prev_lex())
[3, 1]
>>> p.partition > p.prev_lex().partition
True
"""
d = defaultdict(int)
d.update(self.as_dict())
keys = self._keys
if keys == [1]:
return IntegerPartition({self.integer: 1})
if keys[-1] != 1:
d[keys[-1]] -= 1
if keys[-1] == 2:
d[1] = 2
else:
d[keys[-1] - 1] = d[1] = 1
else:
d[keys[-2]] -= 1
left = d[1] + keys[-2]
new = keys[-2]
d[1] = 0
while left:
new -= 1
if left - new >= 0:
d[new] += left//new
left -= d[new]*new
return IntegerPartition(self.integer, d)
def next_lex(self):
"""Return the next partition of the integer, n, in lexical order,
wrapping around to [n] if the partition is [1, ..., 1].
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> p = IntegerPartition([3, 1])
>>> print(p.next_lex())
[4]
>>> p.partition < p.next_lex().partition
True
"""
d = defaultdict(int)
d.update(self.as_dict())
key = self._keys
a = key[-1]
if a == self.integer:
d.clear()
d[1] = self.integer
elif a == 1:
if d[a] > 1:
d[a + 1] += 1
d[a] -= 2
else:
b = key[-2]
d[b + 1] += 1
d[1] = (d[b] - 1)*b
d[b] = 0
else:
if d[a] > 1:
if len(key) == 1:
d.clear()
d[a + 1] = 1
d[1] = self.integer - a - 1
else:
a1 = a + 1
d[a1] += 1
d[1] = d[a]*a - a1
d[a] = 0
else:
b = key[-2]
b1 = b + 1
d[b1] += 1
need = d[b]*b + d[a]*a - b1
d[a] = d[b] = 0
d[1] = need
return IntegerPartition(self.integer, d)
def as_dict(self):
"""Return the partition as a dictionary whose keys are the
partition integers and the values are the multiplicity of that
integer.
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> IntegerPartition([1]*3 + [2] + [3]*4).as_dict()
{1: 3, 2: 1, 3: 4}
"""
if self._dict is None:
groups = group(self.partition, multiple=False)
self._keys = [g[0] for g in groups]
self._dict = dict(groups)
return self._dict
@property
def conjugate(self):
"""
Computes the conjugate partition of itself.
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> a = IntegerPartition([6, 3, 3, 2, 1])
>>> a.conjugate
[5, 4, 3, 1, 1, 1]
"""
j = 1
temp_arr = list(self.partition) + [0]
k = temp_arr[0]
b = [0]*k
while k > 0:
while k > temp_arr[j]:
b[k - 1] = j
k -= 1
j += 1
return b
def __lt__(self, other):
"""Return True if self is less than other when the partition
is listed from smallest to biggest.
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> a = IntegerPartition([3, 1])
>>> a < a
False
>>> b = a.next_lex()
>>> a < b
True
>>> a == b
False
"""
return list(reversed(self.partition)) < list(reversed(other.partition))
def __le__(self, other):
"""Return True if self is less than other when the partition
is listed from smallest to biggest.
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> a = IntegerPartition([4])
>>> a <= a
True
"""
return list(reversed(self.partition)) <= list(reversed(other.partition))
def as_ferrers(self, char='#'):
"""
Prints the ferrer diagram of a partition.
Examples
========
>>> from sympy.combinatorics.partitions import IntegerPartition
>>> print(IntegerPartition([1, 1, 5]).as_ferrers())
#####
#
#
"""
return "\n".join([char*i for i in self.partition])
def __str__(self):
return str(list(self.partition))
def random_integer_partition(n, seed=None):
"""
Generates a random integer partition summing to ``n`` as a list
of reverse-sorted integers.
Examples
========
>>> from sympy.combinatorics.partitions import random_integer_partition
For the following, a seed is given so a known value can be shown; in
practice, the seed would not be given.
>>> random_integer_partition(100, seed=[1, 1, 12, 1, 2, 1, 85, 1])
[85, 12, 2, 1]
>>> random_integer_partition(10, seed=[1, 2, 3, 1, 5, 1])
[5, 3, 1, 1]
>>> random_integer_partition(1)
[1]
"""
from sympy.core.random import _randint
n = as_int(n)
if n < 1:
raise ValueError('n must be a positive integer')
randint = _randint(seed)
partition = []
while (n > 0):
k = randint(1, n)
mult = randint(1, n//k)
partition.append((k, mult))
n -= k*mult
partition.sort(reverse=True)
partition = flatten([[k]*m for k, m in partition])
return partition
def RGS_generalized(m):
"""
Computes the m + 1 generalized unrestricted growth strings
and returns them as rows in matrix.
Examples
========
>>> from sympy.combinatorics.partitions import RGS_generalized
>>> RGS_generalized(6)
Matrix([
[ 1, 1, 1, 1, 1, 1, 1],
[ 1, 2, 3, 4, 5, 6, 0],
[ 2, 5, 10, 17, 26, 0, 0],
[ 5, 15, 37, 77, 0, 0, 0],
[ 15, 52, 151, 0, 0, 0, 0],
[ 52, 203, 0, 0, 0, 0, 0],
[203, 0, 0, 0, 0, 0, 0]])
"""
d = zeros(m + 1)
for i in range(0, m + 1):
d[0, i] = 1
for i in range(1, m + 1):
for j in range(m):
if j <= m - i:
d[i, j] = j * d[i - 1, j] + d[i - 1, j + 1]
else:
d[i, j] = 0
return d
def RGS_enum(m):
"""
RGS_enum computes the total number of restricted growth strings
possible for a superset of size m.
Examples
========
>>> from sympy.combinatorics.partitions import RGS_enum
>>> from sympy.combinatorics.partitions import Partition
>>> RGS_enum(4)
15
>>> RGS_enum(5)
52
>>> RGS_enum(6)
203
We can check that the enumeration is correct by actually generating
the partitions. Here, the 15 partitions of 4 items are generated:
>>> a = Partition(list(range(4)))
>>> s = set()
>>> for i in range(20):
... s.add(a)
... a += 1
...
>>> assert len(s) == 15
"""
if (m < 1):
return 0
elif (m == 1):
return 1
else:
return bell(m)
def RGS_unrank(rank, m):
"""
Gives the unranked restricted growth string for a given
superset size.
Examples
========
>>> from sympy.combinatorics.partitions import RGS_unrank
>>> RGS_unrank(14, 4)
[0, 1, 2, 3]
>>> RGS_unrank(0, 4)
[0, 0, 0, 0]
"""
if m < 1:
raise ValueError("The superset size must be >= 1")
if rank < 0 or RGS_enum(m) <= rank:
raise ValueError("Invalid arguments")
L = [1] * (m + 1)
j = 1
D = RGS_generalized(m)
for i in range(2, m + 1):
v = D[m - i, j]
cr = j*v
if cr <= rank:
L[i] = j + 1
rank -= cr
j += 1
else:
L[i] = int(rank / v + 1)
rank %= v
return [x - 1 for x in L[1:]]
def RGS_rank(rgs):
"""
Computes the rank of a restricted growth string.
Examples
========
>>> from sympy.combinatorics.partitions import RGS_rank, RGS_unrank
>>> RGS_rank([0, 1, 2, 1, 3])
42
>>> RGS_rank(RGS_unrank(4, 7))
4
"""
rgs_size = len(rgs)
rank = 0
D = RGS_generalized(rgs_size)
for i in range(1, rgs_size):
n = len(rgs[(i + 1):])
m = max(rgs[0:i])
rank += D[n, m + 1] * rgs[i]
return rank
|
51f566a8449cf9a4a556f4f4031c24c33e8531a392e6efae19dcb9945dac3956 | from sympy.combinatorics.permutations import Permutation, _af_invert, _af_rmul
from sympy.ntheory import isprime
rmul = Permutation.rmul
_af_new = Permutation._af_new
############################################
#
# Utilities for computational group theory
#
############################################
def _base_ordering(base, degree):
r"""
Order `\{0, 1, \dots, n-1\}` so that base points come first and in order.
Parameters
==========
``base`` : the base
``degree`` : the degree of the associated permutation group
Returns
=======
A list ``base_ordering`` such that ``base_ordering[point]`` is the
number of ``point`` in the ordering.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.util import _base_ordering
>>> S = SymmetricGroup(4)
>>> S.schreier_sims()
>>> _base_ordering(S.base, S.degree)
[0, 1, 2, 3]
Notes
=====
This is used in backtrack searches, when we define a relation `\ll` on
the underlying set for a permutation group of degree `n`,
`\{0, 1, \dots, n-1\}`, so that if `(b_1, b_2, \dots, b_k)` is a base we
have `b_i \ll b_j` whenever `i<j` and `b_i \ll a` for all
`i\in\{1,2, \dots, k\}` and `a` is not in the base. The idea is developed
and applied to backtracking algorithms in [1], pp.108-132. The points
that are not in the base are taken in increasing order.
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
"""
base_len = len(base)
ordering = [0]*degree
for i in range(base_len):
ordering[base[i]] = i
current = base_len
for i in range(degree):
if i not in base:
ordering[i] = current
current += 1
return ordering
def _check_cycles_alt_sym(perm):
"""
Checks for cycles of prime length p with n/2 < p < n-2.
Explanation
===========
Here `n` is the degree of the permutation. This is a helper function for
the function is_alt_sym from sympy.combinatorics.perm_groups.
Examples
========
>>> from sympy.combinatorics.util import _check_cycles_alt_sym
>>> from sympy.combinatorics.permutations import Permutation
>>> a = Permutation([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [11, 12]])
>>> _check_cycles_alt_sym(a)
False
>>> b = Permutation([[0, 1, 2, 3, 4, 5, 6], [7, 8, 9, 10]])
>>> _check_cycles_alt_sym(b)
True
See Also
========
sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym
"""
n = perm.size
af = perm.array_form
current_len = 0
total_len = 0
used = set()
for i in range(n//2):
if i not in used and i < n//2 - total_len:
current_len = 1
used.add(i)
j = i
while af[j] != i:
current_len += 1
j = af[j]
used.add(j)
total_len += current_len
if current_len > n//2 and current_len < n - 2 and isprime(current_len):
return True
return False
def _distribute_gens_by_base(base, gens):
r"""
Distribute the group elements ``gens`` by membership in basic stabilizers.
Explanation
===========
Notice that for a base `(b_1, b_2, \dots, b_k)`, the basic stabilizers
are defined as `G^{(i)} = G_{b_1, \dots, b_{i-1}}` for
`i \in\{1, 2, \dots, k\}`.
Parameters
==========
``base`` : a sequence of points in `\{0, 1, \dots, n-1\}`
``gens`` : a list of elements of a permutation group of degree `n`.
Returns
=======
List of length `k`, where `k` is
the length of ``base``. The `i`-th entry contains those elements in
``gens`` which fix the first `i` elements of ``base`` (so that the
`0`-th entry is equal to ``gens`` itself). If no element fixes the first
`i` elements of ``base``, the `i`-th element is set to a list containing
the identity element.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.util import _distribute_gens_by_base
>>> D = DihedralGroup(3)
>>> D.schreier_sims()
>>> D.strong_gens
[(0 1 2), (0 2), (1 2)]
>>> D.base
[0, 1]
>>> _distribute_gens_by_base(D.base, D.strong_gens)
[[(0 1 2), (0 2), (1 2)],
[(1 2)]]
See Also
========
_strong_gens_from_distr, _orbits_transversals_from_bsgs,
_handle_precomputed_bsgs
"""
base_len = len(base)
degree = gens[0].size
stabs = [[] for _ in range(base_len)]
max_stab_index = 0
for gen in gens:
j = 0
while j < base_len - 1 and gen._array_form[base[j]] == base[j]:
j += 1
if j > max_stab_index:
max_stab_index = j
for k in range(j + 1):
stabs[k].append(gen)
for i in range(max_stab_index + 1, base_len):
stabs[i].append(_af_new(list(range(degree))))
return stabs
def _handle_precomputed_bsgs(base, strong_gens, transversals=None,
basic_orbits=None, strong_gens_distr=None):
"""
Calculate BSGS-related structures from those present.
Explanation
===========
The base and strong generating set must be provided; if any of the
transversals, basic orbits or distributed strong generators are not
provided, they will be calculated from the base and strong generating set.
Parameters
==========
``base`` - the base
``strong_gens`` - the strong generators
``transversals`` - basic transversals
``basic_orbits`` - basic orbits
``strong_gens_distr`` - strong generators distributed by membership in basic
stabilizers
Returns
=======
``(transversals, basic_orbits, strong_gens_distr)`` where ``transversals``
are the basic transversals, ``basic_orbits`` are the basic orbits, and
``strong_gens_distr`` are the strong generators distributed by membership
in basic stabilizers.
Examples
========
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.util import _handle_precomputed_bsgs
>>> D = DihedralGroup(3)
>>> D.schreier_sims()
>>> _handle_precomputed_bsgs(D.base, D.strong_gens,
... basic_orbits=D.basic_orbits)
([{0: (2), 1: (0 1 2), 2: (0 2)}, {1: (2), 2: (1 2)}], [[0, 1, 2], [1, 2]], [[(0 1 2), (0 2), (1 2)], [(1 2)]])
See Also
========
_orbits_transversals_from_bsgs, _distribute_gens_by_base
"""
if strong_gens_distr is None:
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
if transversals is None:
if basic_orbits is None:
basic_orbits, transversals = \
_orbits_transversals_from_bsgs(base, strong_gens_distr)
else:
transversals = \
_orbits_transversals_from_bsgs(base, strong_gens_distr,
transversals_only=True)
else:
if basic_orbits is None:
base_len = len(base)
basic_orbits = [None]*base_len
for i in range(base_len):
basic_orbits[i] = list(transversals[i].keys())
return transversals, basic_orbits, strong_gens_distr
def _orbits_transversals_from_bsgs(base, strong_gens_distr,
transversals_only=False, slp=False):
"""
Compute basic orbits and transversals from a base and strong generating set.
Explanation
===========
The generators are provided as distributed across the basic stabilizers.
If the optional argument ``transversals_only`` is set to True, only the
transversals are returned.
Parameters
==========
``base`` - The base.
``strong_gens_distr`` - Strong generators distributed by membership in basic
stabilizers.
``transversals_only`` - bool
A flag switching between returning only the
transversals and both orbits and transversals.
``slp`` -
If ``True``, return a list of dictionaries containing the
generator presentations of the elements of the transversals,
i.e. the list of indices of generators from ``strong_gens_distr[i]``
such that their product is the relevant transversal element.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.util import _distribute_gens_by_base
>>> S = SymmetricGroup(3)
>>> S.schreier_sims()
>>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens)
>>> (S.base, strong_gens_distr)
([0, 1], [[(0 1 2), (2)(0 1), (1 2)], [(1 2)]])
See Also
========
_distribute_gens_by_base, _handle_precomputed_bsgs
"""
from sympy.combinatorics.perm_groups import _orbit_transversal
base_len = len(base)
degree = strong_gens_distr[0][0].size
transversals = [None]*base_len
slps = [None]*base_len
if transversals_only is False:
basic_orbits = [None]*base_len
for i in range(base_len):
transversals[i], slps[i] = _orbit_transversal(degree, strong_gens_distr[i],
base[i], pairs=True, slp=True)
transversals[i] = dict(transversals[i])
if transversals_only is False:
basic_orbits[i] = list(transversals[i].keys())
if transversals_only:
return transversals
else:
if not slp:
return basic_orbits, transversals
return basic_orbits, transversals, slps
def _remove_gens(base, strong_gens, basic_orbits=None, strong_gens_distr=None):
"""
Remove redundant generators from a strong generating set.
Parameters
==========
``base`` - a base
``strong_gens`` - a strong generating set relative to ``base``
``basic_orbits`` - basic orbits
``strong_gens_distr`` - strong generators distributed by membership in basic
stabilizers
Returns
=======
A strong generating set with respect to ``base`` which is a subset of
``strong_gens``.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.util import _remove_gens
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> S = SymmetricGroup(15)
>>> base, strong_gens = S.schreier_sims_incremental()
>>> new_gens = _remove_gens(base, strong_gens)
>>> len(new_gens)
14
>>> _verify_bsgs(S, base, new_gens)
True
Notes
=====
This procedure is outlined in [1],p.95.
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
"""
from sympy.combinatorics.perm_groups import _orbit
base_len = len(base)
degree = strong_gens[0].size
if strong_gens_distr is None:
strong_gens_distr = _distribute_gens_by_base(base, strong_gens)
if basic_orbits is None:
basic_orbits = []
for i in range(base_len):
basic_orbit = _orbit(degree, strong_gens_distr[i], base[i])
basic_orbits.append(basic_orbit)
strong_gens_distr.append([])
res = strong_gens[:]
for i in range(base_len - 1, -1, -1):
gens_copy = strong_gens_distr[i][:]
for gen in strong_gens_distr[i]:
if gen not in strong_gens_distr[i + 1]:
temp_gens = gens_copy[:]
temp_gens.remove(gen)
if temp_gens == []:
continue
temp_orbit = _orbit(degree, temp_gens, base[i])
if temp_orbit == basic_orbits[i]:
gens_copy.remove(gen)
res.remove(gen)
return res
def _strip(g, base, orbits, transversals):
"""
Attempt to decompose a permutation using a (possibly partial) BSGS
structure.
Explanation
===========
This is done by treating the sequence ``base`` as an actual base, and
the orbits ``orbits`` and transversals ``transversals`` as basic orbits and
transversals relative to it.
This process is called "sifting". A sift is unsuccessful when a certain
orbit element is not found or when after the sift the decomposition
doesn't end with the identity element.
The argument ``transversals`` is a list of dictionaries that provides
transversal elements for the orbits ``orbits``.
Parameters
==========
``g`` - permutation to be decomposed
``base`` - sequence of points
``orbits`` - a list in which the ``i``-th entry is an orbit of ``base[i]``
under some subgroup of the pointwise stabilizer of `
`base[0], base[1], ..., base[i - 1]``. The groups themselves are implicit
in this function since the only information we need is encoded in the orbits
and transversals
``transversals`` - a list of orbit transversals associated with the orbits
``orbits``.
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.util import _strip
>>> S = SymmetricGroup(5)
>>> S.schreier_sims()
>>> g = Permutation([0, 2, 3, 1, 4])
>>> _strip(g, S.base, S.basic_orbits, S.basic_transversals)
((4), 5)
Notes
=====
The algorithm is described in [1],pp.89-90. The reason for returning
both the current state of the element being decomposed and the level
at which the sifting ends is that they provide important information for
the randomized version of the Schreier-Sims algorithm.
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E."Handbook of computational group theory"
See Also
========
sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims
sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random
"""
h = g._array_form
base_len = len(base)
for i in range(base_len):
beta = h[base[i]]
if beta == base[i]:
continue
if beta not in orbits[i]:
return _af_new(h), i + 1
u = transversals[i][beta]._array_form
h = _af_rmul(_af_invert(u), h)
return _af_new(h), base_len + 1
def _strip_af(h, base, orbits, transversals, j, slp=[], slps={}):
"""
optimized _strip, with h, transversals and result in array form
if the stripped elements is the identity, it returns False, base_len + 1
j h[base[i]] == base[i] for i <= j
"""
base_len = len(base)
for i in range(j+1, base_len):
beta = h[base[i]]
if beta == base[i]:
continue
if beta not in orbits[i]:
if not slp:
return h, i + 1
return h, i + 1, slp
u = transversals[i][beta]
if h == u:
if not slp:
return False, base_len + 1
return False, base_len + 1, slp
h = _af_rmul(_af_invert(u), h)
if slp:
u_slp = slps[i][beta][:]
u_slp.reverse()
u_slp = [(i, (g,)) for g in u_slp]
slp = u_slp + slp
if not slp:
return h, base_len + 1
return h, base_len + 1, slp
def _strong_gens_from_distr(strong_gens_distr):
"""
Retrieve strong generating set from generators of basic stabilizers.
This is just the union of the generators of the first and second basic
stabilizers.
Parameters
==========
``strong_gens_distr`` - strong generators distributed by membership in basic
stabilizers
Examples
========
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.util import (_strong_gens_from_distr,
... _distribute_gens_by_base)
>>> S = SymmetricGroup(3)
>>> S.schreier_sims()
>>> S.strong_gens
[(0 1 2), (2)(0 1), (1 2)]
>>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens)
>>> _strong_gens_from_distr(strong_gens_distr)
[(0 1 2), (2)(0 1), (1 2)]
See Also
========
_distribute_gens_by_base
"""
if len(strong_gens_distr) == 1:
return strong_gens_distr[0][:]
else:
result = strong_gens_distr[0]
for gen in strong_gens_distr[1]:
if gen not in result:
result.append(gen)
return result
|
41c464faa64730c96f5e6257b42dbeb1bf8042a853151f16329614b99337ac14 | from sympy.combinatorics.free_groups import free_group
from sympy.printing.defaults import DefaultPrinting
from itertools import chain, product
from bisect import bisect_left
###############################################################################
# COSET TABLE #
###############################################################################
class CosetTable(DefaultPrinting):
# coset_table: Mathematically a coset table
# represented using a list of lists
# alpha: Mathematically a coset (precisely, a live coset)
# represented by an integer between i with 1 <= i <= n
# alpha in c
# x: Mathematically an element of "A" (set of generators and
# their inverses), represented using "FpGroupElement"
# fp_grp: Finitely Presented Group with < X|R > as presentation.
# H: subgroup of fp_grp.
# NOTE: We start with H as being only a list of words in generators
# of "fp_grp". Since `.subgroup` method has not been implemented.
r"""
Properties
==========
[1] `0 \in \Omega` and `\tau(1) = \epsilon`
[2] `\alpha^x = \beta \Leftrightarrow \beta^{x^{-1}} = \alpha`
[3] If `\alpha^x = \beta`, then `H \tau(\alpha)x = H \tau(\beta)`
[4] `\forall \alpha \in \Omega, 1^{\tau(\alpha)} = \alpha`
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
.. [2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490.
"Implementation and Analysis of the Todd-Coxeter Algorithm"
"""
# default limit for the number of cosets allowed in a
# coset enumeration.
coset_table_max_limit = 4096000
# limit for the current instance
coset_table_limit = None
# maximum size of deduction stack above or equal to
# which it is emptied
max_stack_size = 100
def __init__(self, fp_grp, subgroup, max_cosets=None):
if not max_cosets:
max_cosets = CosetTable.coset_table_max_limit
self.fp_group = fp_grp
self.subgroup = subgroup
self.coset_table_limit = max_cosets
# "p" is setup independent of Omega and n
self.p = [0]
# a list of the form `[gen_1, gen_1^{-1}, ... , gen_k, gen_k^{-1}]`
self.A = list(chain.from_iterable((gen, gen**-1) \
for gen in self.fp_group.generators))
#P[alpha, x] Only defined when alpha^x is defined.
self.P = [[None]*len(self.A)]
# the mathematical coset table which is a list of lists
self.table = [[None]*len(self.A)]
self.A_dict = {x: self.A.index(x) for x in self.A}
self.A_dict_inv = {}
for x, index in self.A_dict.items():
if index % 2 == 0:
self.A_dict_inv[x] = self.A_dict[x] + 1
else:
self.A_dict_inv[x] = self.A_dict[x] - 1
# used in the coset-table based method of coset enumeration. Each of
# the element is called a "deduction" which is the form (alpha, x) whenever
# a value is assigned to alpha^x during a definition or "deduction process"
self.deduction_stack = []
# Attributes for modified methods.
H = self.subgroup
self._grp = free_group(', ' .join(["a_%d" % i for i in range(len(H))]))[0]
self.P = [[None]*len(self.A)]
self.p_p = {}
@property
def omega(self):
"""Set of live cosets. """
return [coset for coset in range(len(self.p)) if self.p[coset] == coset]
def copy(self):
"""
Return a shallow copy of Coset Table instance ``self``.
"""
self_copy = self.__class__(self.fp_group, self.subgroup)
self_copy.table = [list(perm_rep) for perm_rep in self.table]
self_copy.p = list(self.p)
self_copy.deduction_stack = list(self.deduction_stack)
return self_copy
def __str__(self):
return "Coset Table on %s with %s as subgroup generators" \
% (self.fp_group, self.subgroup)
__repr__ = __str__
@property
def n(self):
"""The number `n` represents the length of the sublist containing the
live cosets.
"""
if not self.table:
return 0
return max(self.omega) + 1
# Pg. 152 [1]
def is_complete(self):
r"""
The coset table is called complete if it has no undefined entries
on the live cosets; that is, `\alpha^x` is defined for all
`\alpha \in \Omega` and `x \in A`.
"""
return not any(None in self.table[coset] for coset in self.omega)
# Pg. 153 [1]
def define(self, alpha, x, modified=False):
r"""
This routine is used in the relator-based strategy of Todd-Coxeter
algorithm if some `\alpha^x` is undefined. We check whether there is
space available for defining a new coset. If there is enough space
then we remedy this by adjoining a new coset `\beta` to `\Omega`
(i.e to set of live cosets) and put that equal to `\alpha^x`, then
make an assignment satisfying Property[1]. If there is not enough space
then we halt the Coset Table creation. The maximum amount of space that
can be used by Coset Table can be manipulated using the class variable
``CosetTable.coset_table_max_limit``.
See Also
========
define_c
"""
A = self.A
table = self.table
len_table = len(table)
if len_table >= self.coset_table_limit:
# abort the further generation of cosets
raise ValueError("the coset enumeration has defined more than "
"%s cosets. Try with a greater value max number of cosets "
% self.coset_table_limit)
table.append([None]*len(A))
self.P.append([None]*len(self.A))
# beta is the new coset generated
beta = len_table
self.p.append(beta)
table[alpha][self.A_dict[x]] = beta
table[beta][self.A_dict_inv[x]] = alpha
# P[alpha][x] = epsilon, P[beta][x**-1] = epsilon
if modified:
self.P[alpha][self.A_dict[x]] = self._grp.identity
self.P[beta][self.A_dict_inv[x]] = self._grp.identity
self.p_p[beta] = self._grp.identity
def define_c(self, alpha, x):
r"""
A variation of ``define`` routine, described on Pg. 165 [1], used in
the coset table-based strategy of Todd-Coxeter algorithm. It differs
from ``define`` routine in that for each definition it also adds the
tuple `(\alpha, x)` to the deduction stack.
See Also
========
define
"""
A = self.A
table = self.table
len_table = len(table)
if len_table >= self.coset_table_limit:
# abort the further generation of cosets
raise ValueError("the coset enumeration has defined more than "
"%s cosets. Try with a greater value max number of cosets "
% self.coset_table_limit)
table.append([None]*len(A))
# beta is the new coset generated
beta = len_table
self.p.append(beta)
table[alpha][self.A_dict[x]] = beta
table[beta][self.A_dict_inv[x]] = alpha
# append to deduction stack
self.deduction_stack.append((alpha, x))
def scan_c(self, alpha, word):
"""
A variation of ``scan`` routine, described on pg. 165 of [1], which
puts at tuple, whenever a deduction occurs, to deduction stack.
See Also
========
scan, scan_check, scan_and_fill, scan_and_fill_c
"""
# alpha is an integer representing a "coset"
# since scanning can be in two cases
# 1. for alpha=0 and w in Y (i.e generating set of H)
# 2. alpha in Omega (set of live cosets), w in R (relators)
A_dict = self.A_dict
A_dict_inv = self.A_dict_inv
table = self.table
f = alpha
i = 0
r = len(word)
b = alpha
j = r - 1
# list of union of generators and their inverses
while i <= j and table[f][A_dict[word[i]]] is not None:
f = table[f][A_dict[word[i]]]
i += 1
if i > j:
if f != b:
self.coincidence_c(f, b)
return
while j >= i and table[b][A_dict_inv[word[j]]] is not None:
b = table[b][A_dict_inv[word[j]]]
j -= 1
if j < i:
# we have an incorrect completed scan with coincidence f ~ b
# run the "coincidence" routine
self.coincidence_c(f, b)
elif j == i:
# deduction process
table[f][A_dict[word[i]]] = b
table[b][A_dict_inv[word[i]]] = f
self.deduction_stack.append((f, word[i]))
# otherwise scan is incomplete and yields no information
# alpha, beta coincide, i.e. alpha, beta represent the pair of cosets where
# coincidence occurs
def coincidence_c(self, alpha, beta):
"""
A variation of ``coincidence`` routine used in the coset-table based
method of coset enumeration. The only difference being on addition of
a new coset in coset table(i.e new coset introduction), then it is
appended to ``deduction_stack``.
See Also
========
coincidence
"""
A_dict = self.A_dict
A_dict_inv = self.A_dict_inv
table = self.table
# behaves as a queue
q = []
self.merge(alpha, beta, q)
while len(q) > 0:
gamma = q.pop(0)
for x in A_dict:
delta = table[gamma][A_dict[x]]
if delta is not None:
table[delta][A_dict_inv[x]] = None
# only line of difference from ``coincidence`` routine
self.deduction_stack.append((delta, x**-1))
mu = self.rep(gamma)
nu = self.rep(delta)
if table[mu][A_dict[x]] is not None:
self.merge(nu, table[mu][A_dict[x]], q)
elif table[nu][A_dict_inv[x]] is not None:
self.merge(mu, table[nu][A_dict_inv[x]], q)
else:
table[mu][A_dict[x]] = nu
table[nu][A_dict_inv[x]] = mu
def scan(self, alpha, word, y=None, fill=False, modified=False):
r"""
``scan`` performs a scanning process on the input ``word``.
It first locates the largest prefix ``s`` of ``word`` for which
`\alpha^s` is defined (i.e is not ``None``), ``s`` may be empty. Let
``word=sv``, let ``t`` be the longest suffix of ``v`` for which
`\alpha^{t^{-1}}` is defined, and let ``v=ut``. Then three
possibilities are there:
1. If ``t=v``, then we say that the scan completes, and if, in addition
`\alpha^s = \alpha^{t^{-1}}`, then we say that the scan completes
correctly.
2. It can also happen that scan does not complete, but `|u|=1`; that
is, the word ``u`` consists of a single generator `x \in A`. In that
case, if `\alpha^s = \beta` and `\alpha^{t^{-1}} = \gamma`, then we can
set `\beta^x = \gamma` and `\gamma^{x^{-1}} = \beta`. These assignments
are known as deductions and enable the scan to complete correctly.
3. See ``coicidence`` routine for explanation of third condition.
Notes
=====
The code for the procedure of scanning `\alpha \in \Omega`
under `w \in A*` is defined on pg. 155 [1]
See Also
========
scan_c, scan_check, scan_and_fill, scan_and_fill_c
Scan and Fill
=============
Performed when the default argument fill=True.
Modified Scan
=============
Performed when the default argument modified=True
"""
# alpha is an integer representing a "coset"
# since scanning can be in two cases
# 1. for alpha=0 and w in Y (i.e generating set of H)
# 2. alpha in Omega (set of live cosets), w in R (relators)
A_dict = self.A_dict
A_dict_inv = self.A_dict_inv
table = self.table
f = alpha
i = 0
r = len(word)
b = alpha
j = r - 1
b_p = y
if modified:
f_p = self._grp.identity
flag = 0
while fill or flag == 0:
flag = 1
while i <= j and table[f][A_dict[word[i]]] is not None:
if modified:
f_p = f_p*self.P[f][A_dict[word[i]]]
f = table[f][A_dict[word[i]]]
i += 1
if i > j:
if f != b:
if modified:
self.modified_coincidence(f, b, f_p**-1*y)
else:
self.coincidence(f, b)
return
while j >= i and table[b][A_dict_inv[word[j]]] is not None:
if modified:
b_p = b_p*self.P[b][self.A_dict_inv[word[j]]]
b = table[b][A_dict_inv[word[j]]]
j -= 1
if j < i:
# we have an incorrect completed scan with coincidence f ~ b
# run the "coincidence" routine
if modified:
self.modified_coincidence(f, b, f_p**-1*b_p)
else:
self.coincidence(f, b)
elif j == i:
# deduction process
table[f][A_dict[word[i]]] = b
table[b][A_dict_inv[word[i]]] = f
if modified:
self.P[f][self.A_dict[word[i]]] = f_p**-1*b_p
self.P[b][self.A_dict_inv[word[i]]] = b_p**-1*f_p
return
elif fill:
self.define(f, word[i], modified=modified)
# otherwise scan is incomplete and yields no information
# used in the low-index subgroups algorithm
def scan_check(self, alpha, word):
r"""
Another version of ``scan`` routine, described on, it checks whether
`\alpha` scans correctly under `word`, it is a straightforward
modification of ``scan``. ``scan_check`` returns ``False`` (rather than
calling ``coincidence``) if the scan completes incorrectly; otherwise
it returns ``True``.
See Also
========
scan, scan_c, scan_and_fill, scan_and_fill_c
"""
# alpha is an integer representing a "coset"
# since scanning can be in two cases
# 1. for alpha=0 and w in Y (i.e generating set of H)
# 2. alpha in Omega (set of live cosets), w in R (relators)
A_dict = self.A_dict
A_dict_inv = self.A_dict_inv
table = self.table
f = alpha
i = 0
r = len(word)
b = alpha
j = r - 1
while i <= j and table[f][A_dict[word[i]]] is not None:
f = table[f][A_dict[word[i]]]
i += 1
if i > j:
return f == b
while j >= i and table[b][A_dict_inv[word[j]]] is not None:
b = table[b][A_dict_inv[word[j]]]
j -= 1
if j < i:
# we have an incorrect completed scan with coincidence f ~ b
# return False, instead of calling coincidence routine
return False
elif j == i:
# deduction process
table[f][A_dict[word[i]]] = b
table[b][A_dict_inv[word[i]]] = f
return True
def merge(self, k, lamda, q, w=None, modified=False):
"""
Merge two classes with representatives ``k`` and ``lamda``, described
on Pg. 157 [1] (for pseudocode), start by putting ``p[k] = lamda``.
It is more efficient to choose the new representative from the larger
of the two classes being merged, i.e larger among ``k`` and ``lamda``.
procedure ``merge`` performs the merging operation, adds the deleted
class representative to the queue ``q``.
Parameters
==========
'k', 'lamda' being the two class representatives to be merged.
Notes
=====
Pg. 86-87 [1] contains a description of this method.
See Also
========
coincidence, rep
"""
p = self.p
rep = self.rep
phi = rep(k, modified=modified)
psi = rep(lamda, modified=modified)
if phi != psi:
mu = min(phi, psi)
v = max(phi, psi)
p[v] = mu
if modified:
if v == phi:
self.p_p[phi] = self.p_p[k]**-1*w*self.p_p[lamda]
else:
self.p_p[psi] = self.p_p[lamda]**-1*w**-1*self.p_p[k]
q.append(v)
def rep(self, k, modified=False):
r"""
Parameters
==========
`k \in [0 \ldots n-1]`, as for ``self`` only array ``p`` is used
Returns
=======
Representative of the class containing ``k``.
Returns the representative of `\sim` class containing ``k``, it also
makes some modification to array ``p`` of ``self`` to ease further
computations, described on Pg. 157 [1].
The information on classes under `\sim` is stored in array `p` of
``self`` argument, which will always satisfy the property:
`p[\alpha] \sim \alpha` and `p[\alpha]=\alpha \iff \alpha=rep(\alpha)`
`\forall \in [0 \ldots n-1]`.
So, for `\alpha \in [0 \ldots n-1]`, we find `rep(self, \alpha)` by
continually replacing `\alpha` by `p[\alpha]` until it becomes
constant (i.e satisfies `p[\alpha] = \alpha`):w
To increase the efficiency of later ``rep`` calculations, whenever we
find `rep(self, \alpha)=\beta`, we set
`p[\gamma] = \beta \forall \gamma \in p-chain` from `\alpha` to `\beta`
Notes
=====
``rep`` routine is also described on Pg. 85-87 [1] in Atkinson's
algorithm, this results from the fact that ``coincidence`` routine
introduces functionality similar to that introduced by the
``minimal_block`` routine on Pg. 85-87 [1].
See Also
========
coincidence, merge
"""
p = self.p
lamda = k
rho = p[lamda]
if modified:
s = p[:]
while rho != lamda:
if modified:
s[rho] = lamda
lamda = rho
rho = p[lamda]
if modified:
rho = s[lamda]
while rho != k:
mu = rho
rho = s[mu]
p[rho] = lamda
self.p_p[rho] = self.p_p[rho]*self.p_p[mu]
else:
mu = k
rho = p[mu]
while rho != lamda:
p[mu] = lamda
mu = rho
rho = p[mu]
return lamda
# alpha, beta coincide, i.e. alpha, beta represent the pair of cosets
# where coincidence occurs
def coincidence(self, alpha, beta, w=None, modified=False):
r"""
The third situation described in ``scan`` routine is handled by this
routine, described on Pg. 156-161 [1].
The unfortunate situation when the scan completes but not correctly,
then ``coincidence`` routine is run. i.e when for some `i` with
`1 \le i \le r+1`, we have `w=st` with `s = x_1 x_2 \dots x_{i-1}`,
`t = x_i x_{i+1} \dots x_r`, and `\beta = \alpha^s` and
`\gamma = \alpha^{t-1}` are defined but unequal. This means that
`\beta` and `\gamma` represent the same coset of `H` in `G`. Described
on Pg. 156 [1]. ``rep``
See Also
========
scan
"""
A_dict = self.A_dict
A_dict_inv = self.A_dict_inv
table = self.table
# behaves as a queue
q = []
if modified:
self.modified_merge(alpha, beta, w, q)
else:
self.merge(alpha, beta, q)
while len(q) > 0:
gamma = q.pop(0)
for x in A_dict:
delta = table[gamma][A_dict[x]]
if delta is not None:
table[delta][A_dict_inv[x]] = None
mu = self.rep(gamma, modified=modified)
nu = self.rep(delta, modified=modified)
if table[mu][A_dict[x]] is not None:
if modified:
v = self.p_p[delta]**-1*self.P[gamma][self.A_dict[x]]**-1
v = v*self.p_p[gamma]*self.P[mu][self.A_dict[x]]
self.modified_merge(nu, table[mu][self.A_dict[x]], v, q)
else:
self.merge(nu, table[mu][A_dict[x]], q)
elif table[nu][A_dict_inv[x]] is not None:
if modified:
v = self.p_p[gamma]**-1*self.P[gamma][self.A_dict[x]]
v = v*self.p_p[delta]*self.P[mu][self.A_dict_inv[x]]
self.modified_merge(mu, table[nu][self.A_dict_inv[x]], v, q)
else:
self.merge(mu, table[nu][A_dict_inv[x]], q)
else:
table[mu][A_dict[x]] = nu
table[nu][A_dict_inv[x]] = mu
if modified:
v = self.p_p[gamma]**-1*self.P[gamma][self.A_dict[x]]*self.p_p[delta]
self.P[mu][self.A_dict[x]] = v
self.P[nu][self.A_dict_inv[x]] = v**-1
# method used in the HLT strategy
def scan_and_fill(self, alpha, word):
"""
A modified version of ``scan`` routine used in the relator-based
method of coset enumeration, described on pg. 162-163 [1], which
follows the idea that whenever the procedure is called and the scan
is incomplete then it makes new definitions to enable the scan to
complete; i.e it fills in the gaps in the scan of the relator or
subgroup generator.
"""
self.scan(alpha, word, fill=True)
def scan_and_fill_c(self, alpha, word):
"""
A modified version of ``scan`` routine, described on Pg. 165 second
para. [1], with modification similar to that of ``scan_anf_fill`` the
only difference being it calls the coincidence procedure used in the
coset-table based method i.e. the routine ``coincidence_c`` is used.
See Also
========
scan, scan_and_fill
"""
A_dict = self.A_dict
A_dict_inv = self.A_dict_inv
table = self.table
r = len(word)
f = alpha
i = 0
b = alpha
j = r - 1
# loop until it has filled the alpha row in the table.
while True:
# do the forward scanning
while i <= j and table[f][A_dict[word[i]]] is not None:
f = table[f][A_dict[word[i]]]
i += 1
if i > j:
if f != b:
self.coincidence_c(f, b)
return
# forward scan was incomplete, scan backwards
while j >= i and table[b][A_dict_inv[word[j]]] is not None:
b = table[b][A_dict_inv[word[j]]]
j -= 1
if j < i:
self.coincidence_c(f, b)
elif j == i:
table[f][A_dict[word[i]]] = b
table[b][A_dict_inv[word[i]]] = f
self.deduction_stack.append((f, word[i]))
else:
self.define_c(f, word[i])
# method used in the HLT strategy
def look_ahead(self):
"""
When combined with the HLT method this is known as HLT+Lookahead
method of coset enumeration, described on pg. 164 [1]. Whenever
``define`` aborts due to lack of space available this procedure is
executed. This routine helps in recovering space resulting from
"coincidence" of cosets.
"""
R = self.fp_group.relators
p = self.p
# complete scan all relators under all cosets(obviously live)
# without making new definitions
for beta in self.omega:
for w in R:
self.scan(beta, w)
if p[beta] < beta:
break
# Pg. 166
def process_deductions(self, R_c_x, R_c_x_inv):
"""
Processes the deductions that have been pushed onto ``deduction_stack``,
described on Pg. 166 [1] and is used in coset-table based enumeration.
See Also
========
deduction_stack
"""
p = self.p
table = self.table
while len(self.deduction_stack) > 0:
if len(self.deduction_stack) >= CosetTable.max_stack_size:
self.look_ahead()
del self.deduction_stack[:]
continue
else:
alpha, x = self.deduction_stack.pop()
if p[alpha] == alpha:
for w in R_c_x:
self.scan_c(alpha, w)
if p[alpha] < alpha:
break
beta = table[alpha][self.A_dict[x]]
if beta is not None and p[beta] == beta:
for w in R_c_x_inv:
self.scan_c(beta, w)
if p[beta] < beta:
break
def process_deductions_check(self, R_c_x, R_c_x_inv):
"""
A variation of ``process_deductions``, this calls ``scan_check``
wherever ``process_deductions`` calls ``scan``, described on Pg. [1].
See Also
========
process_deductions
"""
table = self.table
while len(self.deduction_stack) > 0:
alpha, x = self.deduction_stack.pop()
for w in R_c_x:
if not self.scan_check(alpha, w):
return False
beta = table[alpha][self.A_dict[x]]
if beta is not None:
for w in R_c_x_inv:
if not self.scan_check(beta, w):
return False
return True
def switch(self, beta, gamma):
r"""Switch the elements `\beta, \gamma \in \Omega` of ``self``, used
by the ``standardize`` procedure, described on Pg. 167 [1].
See Also
========
standardize
"""
A = self.A
A_dict = self.A_dict
table = self.table
for x in A:
z = table[gamma][A_dict[x]]
table[gamma][A_dict[x]] = table[beta][A_dict[x]]
table[beta][A_dict[x]] = z
for alpha in range(len(self.p)):
if self.p[alpha] == alpha:
if table[alpha][A_dict[x]] == beta:
table[alpha][A_dict[x]] = gamma
elif table[alpha][A_dict[x]] == gamma:
table[alpha][A_dict[x]] = beta
def standardize(self):
r"""
A coset table is standardized if when running through the cosets and
within each coset through the generator images (ignoring generator
inverses), the cosets appear in order of the integers
`0, 1, \dots, n`. "Standardize" reorders the elements of `\Omega`
such that, if we scan the coset table first by elements of `\Omega`
and then by elements of A, then the cosets occur in ascending order.
``standardize()`` is used at the end of an enumeration to permute the
cosets so that they occur in some sort of standard order.
Notes
=====
procedure is described on pg. 167-168 [1], it also makes use of the
``switch`` routine to replace by smaller integer value.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r
>>> F, x, y = free_group("x, y")
# Example 5.3 from [1]
>>> f = FpGroup(F, [x**2*y**2, x**3*y**5])
>>> C = coset_enumeration_r(f, [])
>>> C.compress()
>>> C.table
[[1, 3, 1, 3], [2, 0, 2, 0], [3, 1, 3, 1], [0, 2, 0, 2]]
>>> C.standardize()
>>> C.table
[[1, 2, 1, 2], [3, 0, 3, 0], [0, 3, 0, 3], [2, 1, 2, 1]]
"""
A = self.A
A_dict = self.A_dict
gamma = 1
for alpha, x in product(range(self.n), A):
beta = self.table[alpha][A_dict[x]]
if beta >= gamma:
if beta > gamma:
self.switch(gamma, beta)
gamma += 1
if gamma == self.n:
return
# Compression of a Coset Table
def compress(self):
"""Removes the non-live cosets from the coset table, described on
pg. 167 [1].
"""
gamma = -1
A = self.A
A_dict = self.A_dict
A_dict_inv = self.A_dict_inv
table = self.table
chi = tuple([i for i in range(len(self.p)) if self.p[i] != i])
for alpha in self.omega:
gamma += 1
if gamma != alpha:
# replace alpha by gamma in coset table
for x in A:
beta = table[alpha][A_dict[x]]
table[gamma][A_dict[x]] = beta
table[beta][A_dict_inv[x]] == gamma
# all the cosets in the table are live cosets
self.p = list(range(gamma + 1))
# delete the useless columns
del table[len(self.p):]
# re-define values
for row in table:
for j in range(len(self.A)):
row[j] -= bisect_left(chi, row[j])
def conjugates(self, R):
R_c = list(chain.from_iterable((rel.cyclic_conjugates(), \
(rel**-1).cyclic_conjugates()) for rel in R))
R_set = set()
for conjugate in R_c:
R_set = R_set.union(conjugate)
R_c_list = []
for x in self.A:
r = {word for word in R_set if word[0] == x}
R_c_list.append(r)
R_set.difference_update(r)
return R_c_list
def coset_representative(self, coset):
'''
Compute the coset representative of a given coset.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
>>> C = coset_enumeration_r(f, [x])
>>> C.compress()
>>> C.table
[[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]]
>>> C.coset_representative(0)
<identity>
>>> C.coset_representative(1)
y
>>> C.coset_representative(2)
y**-1
'''
for x in self.A:
gamma = self.table[coset][self.A_dict[x]]
if coset == 0:
return self.fp_group.identity
if gamma < coset:
return self.coset_representative(gamma)*x**-1
##############################
# Modified Methods #
##############################
def modified_define(self, alpha, x):
r"""
Define a function p_p from from [1..n] to A* as
an additional component of the modified coset table.
Parameters
==========
\alpha \in \Omega
x \in A*
See Also
========
define
"""
self.define(alpha, x, modified=True)
def modified_scan(self, alpha, w, y, fill=False):
r"""
Parameters
==========
\alpha \in \Omega
w \in A*
y \in (YUY^-1)
fill -- `modified_scan_and_fill` when set to True.
See Also
========
scan
"""
self.scan(alpha, w, y=y, fill=fill, modified=True)
def modified_scan_and_fill(self, alpha, w, y):
self.modified_scan(alpha, w, y, fill=True)
def modified_merge(self, k, lamda, w, q):
r"""
Parameters
==========
'k', 'lamda' -- the two class representatives to be merged.
q -- queue of length l of elements to be deleted from `\Omega` *.
w -- Word in (YUY^-1)
See Also
========
merge
"""
self.merge(k, lamda, q, w=w, modified=True)
def modified_rep(self, k):
r"""
Parameters
==========
`k \in [0 \ldots n-1]`
See Also
========
rep
"""
self.rep(k, modified=True)
def modified_coincidence(self, alpha, beta, w):
r"""
Parameters
==========
A coincident pair `\alpha, \beta \in \Omega, w \in Y \cup Y^{-1}`
See Also
========
coincidence
"""
self.coincidence(alpha, beta, w=w, modified=True)
###############################################################################
# COSET ENUMERATION #
###############################################################################
# relator-based method
def coset_enumeration_r(fp_grp, Y, max_cosets=None, draft=None,
incomplete=False, modified=False):
"""
This is easier of the two implemented methods of coset enumeration.
and is often called the HLT method, after Hazelgrove, Leech, Trotter
The idea is that we make use of ``scan_and_fill`` makes new definitions
whenever the scan is incomplete to enable the scan to complete; this way
we fill in the gaps in the scan of the relator or subgroup generator,
that's why the name relator-based method.
An instance of `CosetTable` for `fp_grp` can be passed as the keyword
argument `draft` in which case the coset enumeration will start with
that instance and attempt to complete it.
When `incomplete` is `True` and the function is unable to complete for
some reason, the partially complete table will be returned.
# TODO: complete the docstring
See Also
========
scan_and_fill,
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r
>>> F, x, y = free_group("x, y")
# Example 5.1 from [1]
>>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
>>> C = coset_enumeration_r(f, [x])
>>> for i in range(len(C.p)):
... if C.p[i] == i:
... print(C.table[i])
[0, 0, 1, 2]
[1, 1, 2, 0]
[2, 2, 0, 1]
>>> C.p
[0, 1, 2, 1, 1]
# Example from exercises Q2 [1]
>>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
>>> C = coset_enumeration_r(f, [])
>>> C.compress(); C.standardize()
>>> C.table
[[1, 2, 3, 4],
[5, 0, 6, 7],
[0, 5, 7, 6],
[7, 6, 5, 0],
[6, 7, 0, 5],
[2, 1, 4, 3],
[3, 4, 2, 1],
[4, 3, 1, 2]]
# Example 5.2
>>> f = FpGroup(F, [x**2, y**3, (x*y)**3])
>>> Y = [x*y]
>>> C = coset_enumeration_r(f, Y)
>>> for i in range(len(C.p)):
... if C.p[i] == i:
... print(C.table[i])
[1, 1, 2, 1]
[0, 0, 0, 2]
[3, 3, 1, 0]
[2, 2, 3, 3]
# Example 5.3
>>> f = FpGroup(F, [x**2*y**2, x**3*y**5])
>>> Y = []
>>> C = coset_enumeration_r(f, Y)
>>> for i in range(len(C.p)):
... if C.p[i] == i:
... print(C.table[i])
[1, 3, 1, 3]
[2, 0, 2, 0]
[3, 1, 3, 1]
[0, 2, 0, 2]
# Example 5.4
>>> F, a, b, c, d, e = free_group("a, b, c, d, e")
>>> f = FpGroup(F, [a*b*c**-1, b*c*d**-1, c*d*e**-1, d*e*a**-1, e*a*b**-1])
>>> Y = [a]
>>> C = coset_enumeration_r(f, Y)
>>> for i in range(len(C.p)):
... if C.p[i] == i:
... print(C.table[i])
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
# example of "compress" method
>>> C.compress()
>>> C.table
[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]
# Exercises Pg. 161, Q2.
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
>>> Y = []
>>> C = coset_enumeration_r(f, Y)
>>> C.compress()
>>> C.standardize()
>>> C.table
[[1, 2, 3, 4],
[5, 0, 6, 7],
[0, 5, 7, 6],
[7, 6, 5, 0],
[6, 7, 0, 5],
[2, 1, 4, 3],
[3, 4, 2, 1],
[4, 3, 1, 2]]
# John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
# Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490
# from 1973chwd.pdf
# Table 1. Ex. 1
>>> F, r, s, t = free_group("r, s, t")
>>> E1 = FpGroup(F, [t**-1*r*t*r**-2, r**-1*s*r*s**-2, s**-1*t*s*t**-2])
>>> C = coset_enumeration_r(E1, [r])
>>> for i in range(len(C.p)):
... if C.p[i] == i:
... print(C.table[i])
[0, 0, 0, 0, 0, 0]
Ex. 2
>>> F, a, b = free_group("a, b")
>>> Cox = FpGroup(F, [a**6, b**6, (a*b)**2, (a**2*b**2)**2, (a**3*b**3)**5])
>>> C = coset_enumeration_r(Cox, [a])
>>> index = 0
>>> for i in range(len(C.p)):
... if C.p[i] == i:
... index += 1
>>> index
500
# Ex. 3
>>> F, a, b = free_group("a, b")
>>> B_2_4 = FpGroup(F, [a**4, b**4, (a*b)**4, (a**-1*b)**4, (a**2*b)**4, \
(a*b**2)**4, (a**2*b**2)**4, (a**-1*b*a*b)**4, (a*b**-1*a*b)**4])
>>> C = coset_enumeration_r(B_2_4, [a])
>>> index = 0
>>> for i in range(len(C.p)):
... if C.p[i] == i:
... index += 1
>>> index
1024
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of computational group theory"
"""
# 1. Initialize a coset table C for < X|R >
C = CosetTable(fp_grp, Y, max_cosets=max_cosets)
# Define coset table methods.
if modified:
_scan_and_fill = C.modified_scan_and_fill
_define = C.modified_define
else:
_scan_and_fill = C.scan_and_fill
_define = C.define
if draft:
C.table = draft.table[:]
C.p = draft.p[:]
R = fp_grp.relators
A_dict = C.A_dict
p = C.p
for i in range(0, len(Y)):
if modified:
_scan_and_fill(0, Y[i], C._grp.generators[i])
else:
_scan_and_fill(0, Y[i])
alpha = 0
while alpha < C.n:
if p[alpha] == alpha:
try:
for w in R:
if modified:
_scan_and_fill(alpha, w, C._grp.identity)
else:
_scan_and_fill(alpha, w)
# if alpha was eliminated during the scan then break
if p[alpha] < alpha:
break
if p[alpha] == alpha:
for x in A_dict:
if C.table[alpha][A_dict[x]] is None:
_define(alpha, x)
except ValueError as e:
if incomplete:
return C
raise e
alpha += 1
return C
def modified_coset_enumeration_r(fp_grp, Y, max_cosets=None, draft=None,
incomplete=False):
r"""
Introduce a new set of symbols y \in Y that correspond to the
generators of the subgroup. Store the elements of Y as a
word P[\alpha, x] and compute the coset table similar to that of
the regular coset enumeration methods.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup
>>> from sympy.combinatorics.coset_table import modified_coset_enumeration_r
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
>>> C = modified_coset_enumeration_r(f, [x])
>>> C.table
[[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1], [None, 1, None, None], [1, 3, None, None]]
See Also
========
coset_enumertation_r
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.,
"Handbook of Computational Group Theory",
Section 5.3.2
"""
return coset_enumeration_r(fp_grp, Y, max_cosets=max_cosets, draft=draft,
incomplete=incomplete, modified=True)
# Pg. 166
# coset-table based method
def coset_enumeration_c(fp_grp, Y, max_cosets=None, draft=None,
incomplete=False):
"""
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_c
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
>>> C = coset_enumeration_c(f, [x])
>>> C.table
[[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]]
"""
# Initialize a coset table C for < X|R >
X = fp_grp.generators
R = fp_grp.relators
C = CosetTable(fp_grp, Y, max_cosets=max_cosets)
if draft:
C.table = draft.table[:]
C.p = draft.p[:]
C.deduction_stack = draft.deduction_stack
for alpha, x in product(range(len(C.table)), X):
if C.table[alpha][C.A_dict[x]] is not None:
C.deduction_stack.append((alpha, x))
A = C.A
# replace all the elements by cyclic reductions
R_cyc_red = [rel.identity_cyclic_reduction() for rel in R]
R_c = list(chain.from_iterable((rel.cyclic_conjugates(), (rel**-1).cyclic_conjugates()) \
for rel in R_cyc_red))
R_set = set()
for conjugate in R_c:
R_set = R_set.union(conjugate)
# a list of subsets of R_c whose words start with "x".
R_c_list = []
for x in C.A:
r = {word for word in R_set if word[0] == x}
R_c_list.append(r)
R_set.difference_update(r)
for w in Y:
C.scan_and_fill_c(0, w)
for x in A:
C.process_deductions(R_c_list[C.A_dict[x]], R_c_list[C.A_dict_inv[x]])
alpha = 0
while alpha < len(C.table):
if C.p[alpha] == alpha:
try:
for x in C.A:
if C.p[alpha] != alpha:
break
if C.table[alpha][C.A_dict[x]] is None:
C.define_c(alpha, x)
C.process_deductions(R_c_list[C.A_dict[x]], R_c_list[C.A_dict_inv[x]])
except ValueError as e:
if incomplete:
return C
raise e
alpha += 1
return C
|
bf069204a5e613d329e8b4ca4835709193f6d96f585c31b3412d2950c7ae1aa4 | from sympy.core import Basic
from sympy.core.containers import Tuple
from sympy.tensor.array import Array
from sympy.core.sympify import _sympify
from sympy.utilities.iterables import flatten, iterable
from sympy.utilities.misc import as_int
from collections import defaultdict
class Prufer(Basic):
"""
The Prufer correspondence is an algorithm that describes the
bijection between labeled trees and the Prufer code. A Prufer
code of a labeled tree is unique up to isomorphism and has
a length of n - 2.
Prufer sequences were first used by Heinz Prufer to give a
proof of Cayley's formula.
References
==========
.. [1] http://mathworld.wolfram.com/LabeledTree.html
"""
_prufer_repr = None
_tree_repr = None
_nodes = None
_rank = None
@property
def prufer_repr(self):
"""Returns Prufer sequence for the Prufer object.
This sequence is found by removing the highest numbered vertex,
recording the node it was attached to, and continuing until only
two vertices remain. The Prufer sequence is the list of recorded nodes.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).prufer_repr
[3, 3, 3, 4]
>>> Prufer([1, 0, 0]).prufer_repr
[1, 0, 0]
See Also
========
to_prufer
"""
if self._prufer_repr is None:
self._prufer_repr = self.to_prufer(self._tree_repr[:], self.nodes)
return self._prufer_repr
@property
def tree_repr(self):
"""Returns the tree representation of the Prufer object.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).tree_repr
[[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]
>>> Prufer([1, 0, 0]).tree_repr
[[1, 2], [0, 1], [0, 3], [0, 4]]
See Also
========
to_tree
"""
if self._tree_repr is None:
self._tree_repr = self.to_tree(self._prufer_repr[:])
return self._tree_repr
@property
def nodes(self):
"""Returns the number of nodes in the tree.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]]).nodes
6
>>> Prufer([1, 0, 0]).nodes
5
"""
return self._nodes
@property
def rank(self):
"""Returns the rank of the Prufer sequence.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> p = Prufer([[0, 3], [1, 3], [2, 3], [3, 4], [4, 5]])
>>> p.rank
778
>>> p.next(1).rank
779
>>> p.prev().rank
777
See Also
========
prufer_rank, next, prev, size
"""
if self._rank is None:
self._rank = self.prufer_rank()
return self._rank
@property
def size(self):
"""Return the number of possible trees of this Prufer object.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer([0]*4).size == Prufer([6]*4).size == 1296
True
See Also
========
prufer_rank, rank, next, prev
"""
return self.prev(self.rank).prev().rank + 1
@staticmethod
def to_prufer(tree, n):
"""Return the Prufer sequence for a tree given as a list of edges where
``n`` is the number of nodes in the tree.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
>>> a.prufer_repr
[0, 0]
>>> Prufer.to_prufer([[0, 1], [0, 2], [0, 3]], 4)
[0, 0]
See Also
========
prufer_repr: returns Prufer sequence of a Prufer object.
"""
d = defaultdict(int)
L = []
for edge in tree:
# Increment the value of the corresponding
# node in the degree list as we encounter an
# edge involving it.
d[edge[0]] += 1
d[edge[1]] += 1
for i in range(n - 2):
# find the smallest leaf
for x in range(n):
if d[x] == 1:
break
# find the node it was connected to
y = None
for edge in tree:
if x == edge[0]:
y = edge[1]
elif x == edge[1]:
y = edge[0]
if y is not None:
break
# record and update
L.append(y)
for j in (x, y):
d[j] -= 1
if not d[j]:
d.pop(j)
tree.remove(edge)
return L
@staticmethod
def to_tree(prufer):
"""Return the tree (as a list of edges) of the given Prufer sequence.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([0, 2], 4)
>>> a.tree_repr
[[0, 1], [0, 2], [2, 3]]
>>> Prufer.to_tree([0, 2])
[[0, 1], [0, 2], [2, 3]]
References
==========
.. [1] https://hamberg.no/erlend/posts/2010-11-06-prufer-sequence-compact-tree-representation.html
See Also
========
tree_repr: returns tree representation of a Prufer object.
"""
tree = []
last = []
n = len(prufer) + 2
d = defaultdict(lambda: 1)
for p in prufer:
d[p] += 1
for i in prufer:
for j in range(n):
# find the smallest leaf (degree = 1)
if d[j] == 1:
break
# (i, j) is the new edge that we append to the tree
# and remove from the degree dictionary
d[i] -= 1
d[j] -= 1
tree.append(sorted([i, j]))
last = [i for i in range(n) if d[i] == 1] or [0, 1]
tree.append(last)
return tree
@staticmethod
def edges(*runs):
"""Return a list of edges and the number of nodes from the given runs
that connect nodes in an integer-labelled tree.
All node numbers will be shifted so that the minimum node is 0. It is
not a problem if edges are repeated in the runs; only unique edges are
returned. There is no assumption made about what the range of the node
labels should be, but all nodes from the smallest through the largest
must be present.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer.edges([1, 2, 3], [2, 4, 5]) # a T
([[0, 1], [1, 2], [1, 3], [3, 4]], 5)
Duplicate edges are removed:
>>> Prufer.edges([0, 1, 2, 3], [1, 4, 5], [1, 4, 6]) # a K
([[0, 1], [1, 2], [1, 4], [2, 3], [4, 5], [4, 6]], 7)
"""
e = set()
nmin = runs[0][0]
for r in runs:
for i in range(len(r) - 1):
a, b = r[i: i + 2]
if b < a:
a, b = b, a
e.add((a, b))
rv = []
got = set()
nmin = nmax = None
for ei in e:
for i in ei:
got.add(i)
nmin = min(ei[0], nmin) if nmin is not None else ei[0]
nmax = max(ei[1], nmax) if nmax is not None else ei[1]
rv.append(list(ei))
missing = set(range(nmin, nmax + 1)) - got
if missing:
missing = [i + nmin for i in missing]
if len(missing) == 1:
msg = 'Node %s is missing.' % missing.pop()
else:
msg = 'Nodes %s are missing.' % list(sorted(missing))
raise ValueError(msg)
if nmin != 0:
for i, ei in enumerate(rv):
rv[i] = [n - nmin for n in ei]
nmax -= nmin
return sorted(rv), nmax + 1
def prufer_rank(self):
"""Computes the rank of a Prufer sequence.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
>>> a.prufer_rank()
0
See Also
========
rank, next, prev, size
"""
r = 0
p = 1
for i in range(self.nodes - 3, -1, -1):
r += p*self.prufer_repr[i]
p *= self.nodes
return r
@classmethod
def unrank(self, rank, n):
"""Finds the unranked Prufer sequence.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> Prufer.unrank(0, 4)
Prufer([0, 0])
"""
n, rank = as_int(n), as_int(rank)
L = defaultdict(int)
for i in range(n - 3, -1, -1):
L[i] = rank % n
rank = (rank - L[i])//n
return Prufer([L[i] for i in range(len(L))])
def __new__(cls, *args, **kw_args):
"""The constructor for the Prufer object.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
A Prufer object can be constructed from a list of edges:
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
>>> a.prufer_repr
[0, 0]
If the number of nodes is given, no checking of the nodes will
be performed; it will be assumed that nodes 0 through n - 1 are
present:
>>> Prufer([[0, 1], [0, 2], [0, 3]], 4)
Prufer([[0, 1], [0, 2], [0, 3]], 4)
A Prufer object can be constructed from a Prufer sequence:
>>> b = Prufer([1, 3])
>>> b.tree_repr
[[0, 1], [1, 3], [2, 3]]
"""
arg0 = Array(args[0]) if args[0] else Tuple()
args = (arg0,) + tuple(_sympify(arg) for arg in args[1:])
ret_obj = Basic.__new__(cls, *args, **kw_args)
args = [list(args[0])]
if args[0] and iterable(args[0][0]):
if not args[0][0]:
raise ValueError(
'Prufer expects at least one edge in the tree.')
if len(args) > 1:
nnodes = args[1]
else:
nodes = set(flatten(args[0]))
nnodes = max(nodes) + 1
if nnodes != len(nodes):
missing = set(range(nnodes)) - nodes
if len(missing) == 1:
msg = 'Node %s is missing.' % missing.pop()
else:
msg = 'Nodes %s are missing.' % list(sorted(missing))
raise ValueError(msg)
ret_obj._tree_repr = [list(i) for i in args[0]]
ret_obj._nodes = nnodes
else:
ret_obj._prufer_repr = args[0]
ret_obj._nodes = len(ret_obj._prufer_repr) + 2
return ret_obj
def next(self, delta=1):
"""Generates the Prufer sequence that is delta beyond the current one.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [0, 2], [0, 3]])
>>> b = a.next(1) # == a.next()
>>> b.tree_repr
[[0, 2], [0, 1], [1, 3]]
>>> b.rank
1
See Also
========
prufer_rank, rank, prev, size
"""
return Prufer.unrank(self.rank + delta, self.nodes)
def prev(self, delta=1):
"""Generates the Prufer sequence that is -delta before the current one.
Examples
========
>>> from sympy.combinatorics.prufer import Prufer
>>> a = Prufer([[0, 1], [1, 2], [2, 3], [1, 4]])
>>> a.rank
36
>>> b = a.prev()
>>> b
Prufer([1, 2, 0])
>>> b.rank
35
See Also
========
prufer_rank, rank, next, size
"""
return Prufer.unrank(self.rank -delta, self.nodes)
|
aca62d2d1139e252ce8a7e1705794cb0dc99c997cbcffa98fa8197ab3aff043d | """Finitely Presented Groups and its algorithms. """
from sympy.core.singleton import S
from sympy.core.symbol import symbols
from sympy.combinatorics.free_groups import (FreeGroup, FreeGroupElement,
free_group)
from sympy.combinatorics.rewritingsystem import RewritingSystem
from sympy.combinatorics.coset_table import (CosetTable,
coset_enumeration_r,
coset_enumeration_c)
from sympy.combinatorics import PermutationGroup
from sympy.printing.defaults import DefaultPrinting
from sympy.utilities import public
from sympy.utilities.magic import pollute
from itertools import product
@public
def fp_group(fr_grp, relators=()):
_fp_group = FpGroup(fr_grp, relators)
return (_fp_group,) + tuple(_fp_group._generators)
@public
def xfp_group(fr_grp, relators=()):
_fp_group = FpGroup(fr_grp, relators)
return (_fp_group, _fp_group._generators)
# Does not work. Both symbols and pollute are undefined. Never tested.
@public
def vfp_group(fr_grpm, relators):
_fp_group = FpGroup(symbols, relators)
pollute([sym.name for sym in _fp_group.symbols], _fp_group.generators)
return _fp_group
def _parse_relators(rels):
"""Parse the passed relators."""
return rels
###############################################################################
# FINITELY PRESENTED GROUPS #
###############################################################################
class FpGroup(DefaultPrinting):
"""
The FpGroup would take a FreeGroup and a list/tuple of relators, the
relators would be specified in such a way that each of them be equal to the
identity of the provided free group.
"""
is_group = True
is_FpGroup = True
is_PermutationGroup = False
def __init__(self, fr_grp, relators):
relators = _parse_relators(relators)
self.free_group = fr_grp
self.relators = relators
self.generators = self._generators()
self.dtype = type("FpGroupElement", (FpGroupElement,), {"group": self})
# CosetTable instance on identity subgroup
self._coset_table = None
# returns whether coset table on identity subgroup
# has been standardized
self._is_standardized = False
self._order = None
self._center = None
self._rewriting_system = RewritingSystem(self)
self._perm_isomorphism = None
return
def _generators(self):
return self.free_group.generators
def make_confluent(self):
'''
Try to make the group's rewriting system confluent
'''
self._rewriting_system.make_confluent()
return
def reduce(self, word):
'''
Return the reduced form of `word` in `self` according to the group's
rewriting system. If it's confluent, the reduced form is the unique normal
form of the word in the group.
'''
return self._rewriting_system.reduce(word)
def equals(self, word1, word2):
'''
Compare `word1` and `word2` for equality in the group
using the group's rewriting system. If the system is
confluent, the returned answer is necessarily correct.
(If it isn't, `False` could be returned in some cases
where in fact `word1 == word2`)
'''
if self.reduce(word1*word2**-1) == self.identity:
return True
elif self._rewriting_system.is_confluent:
return False
return None
@property
def identity(self):
return self.free_group.identity
def __contains__(self, g):
return g in self.free_group
def subgroup(self, gens, C=None, homomorphism=False):
'''
Return the subgroup generated by `gens` using the
Reidemeister-Schreier algorithm
homomorphism -- When set to True, return a dictionary containing the images
of the presentation generators in the original group.
Examples
========
>>> from sympy.combinatorics.fp_groups import FpGroup
>>> from sympy.combinatorics.free_groups import free_group
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**3, y**5, (x*y)**2])
>>> H = [x*y, x**-1*y**-1*x*y*x]
>>> K, T = f.subgroup(H, homomorphism=True)
>>> T(K.generators)
[x*y, x**-1*y**2*x**-1]
'''
if not all(isinstance(g, FreeGroupElement) for g in gens):
raise ValueError("Generators must be `FreeGroupElement`s")
if not all(g.group == self.free_group for g in gens):
raise ValueError("Given generators are not members of the group")
if homomorphism:
g, rels, _gens = reidemeister_presentation(self, gens, C=C, homomorphism=True)
else:
g, rels = reidemeister_presentation(self, gens, C=C)
if g:
g = FpGroup(g[0].group, rels)
else:
g = FpGroup(free_group('')[0], [])
if homomorphism:
from sympy.combinatorics.homomorphisms import homomorphism
return g, homomorphism(g, self, g.generators, _gens, check=False)
return g
def coset_enumeration(self, H, strategy="relator_based", max_cosets=None,
draft=None, incomplete=False):
"""
Return an instance of ``coset table``, when Todd-Coxeter algorithm is
run over the ``self`` with ``H`` as subgroup, using ``strategy``
argument as strategy. The returned coset table is compressed but not
standardized.
An instance of `CosetTable` for `fp_grp` can be passed as the keyword
argument `draft` in which case the coset enumeration will start with
that instance and attempt to complete it.
When `incomplete` is `True` and the function is unable to complete for
some reason, the partially complete table will be returned.
"""
if not max_cosets:
max_cosets = CosetTable.coset_table_max_limit
if strategy == 'relator_based':
C = coset_enumeration_r(self, H, max_cosets=max_cosets,
draft=draft, incomplete=incomplete)
else:
C = coset_enumeration_c(self, H, max_cosets=max_cosets,
draft=draft, incomplete=incomplete)
if C.is_complete():
C.compress()
return C
def standardize_coset_table(self):
"""
Standardized the coset table ``self`` and makes the internal variable
``_is_standardized`` equal to ``True``.
"""
self._coset_table.standardize()
self._is_standardized = True
def coset_table(self, H, strategy="relator_based", max_cosets=None,
draft=None, incomplete=False):
"""
Return the mathematical coset table of ``self`` in ``H``.
"""
if not H:
if self._coset_table is not None:
if not self._is_standardized:
self.standardize_coset_table()
else:
C = self.coset_enumeration([], strategy, max_cosets=max_cosets,
draft=draft, incomplete=incomplete)
self._coset_table = C
self.standardize_coset_table()
return self._coset_table.table
else:
C = self.coset_enumeration(H, strategy, max_cosets=max_cosets,
draft=draft, incomplete=incomplete)
C.standardize()
return C.table
def order(self, strategy="relator_based"):
"""
Returns the order of the finitely presented group ``self``. It uses
the coset enumeration with identity group as subgroup, i.e ``H=[]``.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x, y**2])
>>> f.order(strategy="coset_table_based")
2
"""
from sympy.polys.polytools import gcd
if self._order is not None:
return self._order
if self._coset_table is not None:
self._order = len(self._coset_table.table)
elif len(self.relators) == 0:
self._order = self.free_group.order()
elif len(self.generators) == 1:
self._order = abs(gcd([r.array_form[0][1] for r in self.relators]))
elif self._is_infinite():
self._order = S.Infinity
else:
gens, C = self._finite_index_subgroup()
if C:
ind = len(C.table)
self._order = ind*self.subgroup(gens, C=C).order()
else:
self._order = self.index([])
return self._order
def _is_infinite(self):
'''
Test if the group is infinite. Return `True` if the test succeeds
and `None` otherwise
'''
used_gens = set()
for r in self.relators:
used_gens.update(r.contains_generators())
if not set(self.generators) <= used_gens:
return True
# Abelianisation test: check is the abelianisation is infinite
abelian_rels = []
from sympy.matrices.normalforms import invariant_factors
from sympy.matrices import Matrix
for rel in self.relators:
abelian_rels.append([rel.exponent_sum(g) for g in self.generators])
m = Matrix(Matrix(abelian_rels))
if 0 in invariant_factors(m):
return True
else:
return None
def _finite_index_subgroup(self, s=None):
'''
Find the elements of `self` that generate a finite index subgroup
and, if found, return the list of elements and the coset table of `self` by
the subgroup, otherwise return `(None, None)`
'''
gen = self.most_frequent_generator()
rels = list(self.generators)
rels.extend(self.relators)
if not s:
if len(self.generators) == 2:
s = [gen] + [g for g in self.generators if g != gen]
else:
rand = self.free_group.identity
i = 0
while ((rand in rels or rand**-1 in rels or rand.is_identity)
and i<10):
rand = self.random()
i += 1
s = [gen, rand] + [g for g in self.generators if g != gen]
mid = (len(s)+1)//2
half1 = s[:mid]
half2 = s[mid:]
draft1 = None
draft2 = None
m = 200
C = None
while not C and (m/2 < CosetTable.coset_table_max_limit):
m = min(m, CosetTable.coset_table_max_limit)
draft1 = self.coset_enumeration(half1, max_cosets=m,
draft=draft1, incomplete=True)
if draft1.is_complete():
C = draft1
half = half1
else:
draft2 = self.coset_enumeration(half2, max_cosets=m,
draft=draft2, incomplete=True)
if draft2.is_complete():
C = draft2
half = half2
if not C:
m *= 2
if not C:
return None, None
C.compress()
return half, C
def most_frequent_generator(self):
gens = self.generators
rels = self.relators
freqs = [sum([r.generator_count(g) for r in rels]) for g in gens]
return gens[freqs.index(max(freqs))]
def random(self):
import random
r = self.free_group.identity
for i in range(random.randint(2,3)):
r = r*random.choice(self.generators)**random.choice([1,-1])
return r
def index(self, H, strategy="relator_based"):
"""
Return the index of subgroup ``H`` in group ``self``.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**5, y**4, y*x*y**3*x**3])
>>> f.index([x])
4
"""
# TODO: use |G:H| = |G|/|H| (currently H can't be made into a group)
# when we know |G| and |H|
if H == []:
return self.order()
else:
C = self.coset_enumeration(H, strategy)
return len(C.table)
def __str__(self):
if self.free_group.rank > 30:
str_form = "<fp group with %s generators>" % self.free_group.rank
else:
str_form = "<fp group on the generators %s>" % str(self.generators)
return str_form
__repr__ = __str__
#==============================================================================
# PERMUTATION GROUP METHODS
#==============================================================================
def _to_perm_group(self):
'''
Return an isomorphic permutation group and the isomorphism.
The implementation is dependent on coset enumeration so
will only terminate for finite groups.
'''
from sympy.combinatorics import Permutation
from sympy.combinatorics.homomorphisms import homomorphism
if self.order() is S.Infinity:
raise NotImplementedError("Permutation presentation of infinite "
"groups is not implemented")
if self._perm_isomorphism:
T = self._perm_isomorphism
P = T.image()
else:
C = self.coset_table([])
gens = self.generators
images = [[C[i][2*gens.index(g)] for i in range(len(C))] for g in gens]
images = [Permutation(i) for i in images]
P = PermutationGroup(images)
T = homomorphism(self, P, gens, images, check=False)
self._perm_isomorphism = T
return P, T
def _perm_group_list(self, method_name, *args):
'''
Given the name of a `PermutationGroup` method (returning a subgroup
or a list of subgroups) and (optionally) additional arguments it takes,
return a list or a list of lists containing the generators of this (or
these) subgroups in terms of the generators of `self`.
'''
P, T = self._to_perm_group()
perm_result = getattr(P, method_name)(*args)
single = False
if isinstance(perm_result, PermutationGroup):
perm_result, single = [perm_result], True
result = []
for group in perm_result:
gens = group.generators
result.append(T.invert(gens))
return result[0] if single else result
def derived_series(self):
'''
Return the list of lists containing the generators
of the subgroups in the derived series of `self`.
'''
return self._perm_group_list('derived_series')
def lower_central_series(self):
'''
Return the list of lists containing the generators
of the subgroups in the lower central series of `self`.
'''
return self._perm_group_list('lower_central_series')
def center(self):
'''
Return the list of generators of the center of `self`.
'''
return self._perm_group_list('center')
def derived_subgroup(self):
'''
Return the list of generators of the derived subgroup of `self`.
'''
return self._perm_group_list('derived_subgroup')
def centralizer(self, other):
'''
Return the list of generators of the centralizer of `other`
(a list of elements of `self`) in `self`.
'''
T = self._to_perm_group()[1]
other = T(other)
return self._perm_group_list('centralizer', other)
def normal_closure(self, other):
'''
Return the list of generators of the normal closure of `other`
(a list of elements of `self`) in `self`.
'''
T = self._to_perm_group()[1]
other = T(other)
return self._perm_group_list('normal_closure', other)
def _perm_property(self, attr):
'''
Given an attribute of a `PermutationGroup`, return
its value for a permutation group isomorphic to `self`.
'''
P = self._to_perm_group()[0]
return getattr(P, attr)
@property
def is_abelian(self):
'''
Check if `self` is abelian.
'''
return self._perm_property("is_abelian")
@property
def is_nilpotent(self):
'''
Check if `self` is nilpotent.
'''
return self._perm_property("is_nilpotent")
@property
def is_solvable(self):
'''
Check if `self` is solvable.
'''
return self._perm_property("is_solvable")
@property
def elements(self):
'''
List the elements of `self`.
'''
P, T = self._to_perm_group()
return T.invert(P._elements)
@property
def is_cyclic(self):
"""
Return ``True`` if group is Cyclic.
"""
if len(self.generators) <= 1:
return True
try:
P, T = self._to_perm_group()
except NotImplementedError:
raise NotImplementedError("Check for infinite Cyclic group "
"is not implemented")
return P.is_cyclic
def abelian_invariants(self):
"""
Return Abelian Invariants of a group.
"""
try:
P, T = self._to_perm_group()
except NotImplementedError:
raise NotImplementedError("abelian invariants is not implemented"
"for infinite group")
return P.abelian_invariants()
def composition_series(self):
"""
Return subnormal series of maximum length for a group.
"""
try:
P, T = self._to_perm_group()
except NotImplementedError:
raise NotImplementedError("composition series is not implemented"
"for infinite group")
return P.composition_series()
class FpSubgroup(DefaultPrinting):
'''
The class implementing a subgroup of an FpGroup or a FreeGroup
(only finite index subgroups are supported at this point). This
is to be used if one wishes to check if an element of the original
group belongs to the subgroup
'''
def __init__(self, G, gens, normal=False):
super().__init__()
self.parent = G
self.generators = list({g for g in gens if g != G.identity})
self._min_words = None #for use in __contains__
self.C = None
self.normal = normal
def __contains__(self, g):
if isinstance(self.parent, FreeGroup):
if self._min_words is None:
# make _min_words - a list of subwords such that
# g is in the subgroup if and only if it can be
# partitioned into these subwords. Infinite families of
# subwords are presented by tuples, e.g. (r, w)
# stands for the family of subwords r*w**n*r**-1
def _process(w):
# this is to be used before adding new words
# into _min_words; if the word w is not cyclically
# reduced, it will generate an infinite family of
# subwords so should be written as a tuple;
# if it is, w**-1 should be added to the list
# as well
p, r = w.cyclic_reduction(removed=True)
if not r.is_identity:
return [(r, p)]
else:
return [w, w**-1]
# make the initial list
gens = []
for w in self.generators:
if self.normal:
w = w.cyclic_reduction()
gens.extend(_process(w))
for w1 in gens:
for w2 in gens:
# if w1 and w2 are equal or are inverses, continue
if w1 == w2 or (not isinstance(w1, tuple)
and w1**-1 == w2):
continue
# if the start of one word is the inverse of the
# end of the other, their multiple should be added
# to _min_words because of cancellation
if isinstance(w1, tuple):
# start, end
s1, s2 = w1[0][0], w1[0][0]**-1
else:
s1, s2 = w1[0], w1[len(w1)-1]
if isinstance(w2, tuple):
# start, end
r1, r2 = w2[0][0], w2[0][0]**-1
else:
r1, r2 = w2[0], w2[len(w1)-1]
# p1 and p2 are w1 and w2 or, in case when
# w1 or w2 is an infinite family, a representative
p1, p2 = w1, w2
if isinstance(w1, tuple):
p1 = w1[0]*w1[1]*w1[0]**-1
if isinstance(w2, tuple):
p2 = w2[0]*w2[1]*w2[0]**-1
# add the product of the words to the list is necessary
if r1**-1 == s2 and not (p1*p2).is_identity:
new = _process(p1*p2)
if new not in gens:
gens.extend(new)
if r2**-1 == s1 and not (p2*p1).is_identity:
new = _process(p2*p1)
if new not in gens:
gens.extend(new)
self._min_words = gens
min_words = self._min_words
def _is_subword(w):
# check if w is a word in _min_words or one of
# the infinite families in it
w, r = w.cyclic_reduction(removed=True)
if r.is_identity or self.normal:
return w in min_words
else:
t = [s[1] for s in min_words if isinstance(s, tuple)
and s[0] == r]
return [s for s in t if w.power_of(s)] != []
# store the solution of words for which the result of
# _word_break (below) is known
known = {}
def _word_break(w):
# check if w can be written as a product of words
# in min_words
if len(w) == 0:
return True
i = 0
while i < len(w):
i += 1
prefix = w.subword(0, i)
if not _is_subword(prefix):
continue
rest = w.subword(i, len(w))
if rest not in known:
known[rest] = _word_break(rest)
if known[rest]:
return True
return False
if self.normal:
g = g.cyclic_reduction()
return _word_break(g)
else:
if self.C is None:
C = self.parent.coset_enumeration(self.generators)
self.C = C
i = 0
C = self.C
for j in range(len(g)):
i = C.table[i][C.A_dict[g[j]]]
return i == 0
def order(self):
if not self.generators:
return S.One
if isinstance(self.parent, FreeGroup):
return S.Infinity
if self.C is None:
C = self.parent.coset_enumeration(self.generators)
self.C = C
# This is valid because `len(self.C.table)` (the index of the subgroup)
# will always be finite - otherwise coset enumeration doesn't terminate
return self.parent.order()/len(self.C.table)
def to_FpGroup(self):
if isinstance(self.parent, FreeGroup):
gen_syms = [('x_%d'%i) for i in range(len(self.generators))]
return free_group(', '.join(gen_syms))[0]
return self.parent.subgroup(C=self.C)
def __str__(self):
if len(self.generators) > 30:
str_form = "<fp subgroup with %s generators>" % len(self.generators)
else:
str_form = "<fp subgroup on the generators %s>" % str(self.generators)
return str_form
__repr__ = __str__
###############################################################################
# LOW INDEX SUBGROUPS #
###############################################################################
def low_index_subgroups(G, N, Y=()):
"""
Implements the Low Index Subgroups algorithm, i.e find all subgroups of
``G`` upto a given index ``N``. This implements the method described in
[Sim94]. This procedure involves a backtrack search over incomplete Coset
Tables, rather than over forced coincidences.
Parameters
==========
G: An FpGroup < X|R >
N: positive integer, representing the maximum index value for subgroups
Y: (an optional argument) specifying a list of subgroup generators, such
that each of the resulting subgroup contains the subgroup generated by Y.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, low_index_subgroups
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**2, y**3, (x*y)**4])
>>> L = low_index_subgroups(f, 4)
>>> for coset_table in L:
... print(coset_table.table)
[[0, 0, 0, 0]]
[[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]]
[[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]]
[[1, 1, 0, 0], [0, 0, 1, 1]]
References
==========
.. [1] Holt, D., Eick, B., O'Brien, E.
"Handbook of Computational Group Theory"
Section 5.4
.. [2] Marston Conder and Peter Dobcsanyi
"Applications and Adaptions of the Low Index Subgroups Procedure"
"""
C = CosetTable(G, [])
R = G.relators
# length chosen for the length of the short relators
len_short_rel = 5
# elements of R2 only checked at the last step for complete
# coset tables
R2 = {rel for rel in R if len(rel) > len_short_rel}
# elements of R1 are used in inner parts of the process to prune
# branches of the search tree,
R1 = {rel.identity_cyclic_reduction() for rel in set(R) - R2}
R1_c_list = C.conjugates(R1)
S = []
descendant_subgroups(S, C, R1_c_list, C.A[0], R2, N, Y)
return S
def descendant_subgroups(S, C, R1_c_list, x, R2, N, Y):
A_dict = C.A_dict
A_dict_inv = C.A_dict_inv
if C.is_complete():
# if C is complete then it only needs to test
# whether the relators in R2 are satisfied
for w, alpha in product(R2, C.omega):
if not C.scan_check(alpha, w):
return
# relators in R2 are satisfied, append the table to list
S.append(C)
else:
# find the first undefined entry in Coset Table
for alpha, x in product(range(len(C.table)), C.A):
if C.table[alpha][A_dict[x]] is None:
# this is "x" in pseudo-code (using "y" makes it clear)
undefined_coset, undefined_gen = alpha, x
break
# for filling up the undefine entry we try all possible values
# of beta in Omega or beta = n where beta^(undefined_gen^-1) is undefined
reach = C.omega + [C.n]
for beta in reach:
if beta < N:
if beta == C.n or C.table[beta][A_dict_inv[undefined_gen]] is None:
try_descendant(S, C, R1_c_list, R2, N, undefined_coset, \
undefined_gen, beta, Y)
def try_descendant(S, C, R1_c_list, R2, N, alpha, x, beta, Y):
r"""
Solves the problem of trying out each individual possibility
for `\alpha^x.
"""
D = C.copy()
if beta == D.n and beta < N:
D.table.append([None]*len(D.A))
D.p.append(beta)
D.table[alpha][D.A_dict[x]] = beta
D.table[beta][D.A_dict_inv[x]] = alpha
D.deduction_stack.append((alpha, x))
if not D.process_deductions_check(R1_c_list[D.A_dict[x]], \
R1_c_list[D.A_dict_inv[x]]):
return
for w in Y:
if not D.scan_check(0, w):
return
if first_in_class(D, Y):
descendant_subgroups(S, D, R1_c_list, x, R2, N, Y)
def first_in_class(C, Y=()):
"""
Checks whether the subgroup ``H=G1`` corresponding to the Coset Table
could possibly be the canonical representative of its conjugacy class.
Parameters
==========
C: CosetTable
Returns
=======
bool: True/False
If this returns False, then no descendant of C can have that property, and
so we can abandon C. If it returns True, then we need to process further
the node of the search tree corresponding to C, and so we call
``descendant_subgroups`` recursively on C.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, first_in_class
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**2, y**3, (x*y)**4])
>>> C = CosetTable(f, [])
>>> C.table = [[0, 0, None, None]]
>>> first_in_class(C)
True
>>> C.table = [[1, 1, 1, None], [0, 0, None, 1]]; C.p = [0, 1]
>>> first_in_class(C)
True
>>> C.table = [[1, 1, 2, 1], [0, 0, 0, None], [None, None, None, 0]]
>>> C.p = [0, 1, 2]
>>> first_in_class(C)
False
>>> C.table = [[1, 1, 1, 2], [0, 0, 2, 0], [2, None, 0, 1]]
>>> first_in_class(C)
False
# TODO:: Sims points out in [Sim94] that performance can be improved by
# remembering some of the information computed by ``first_in_class``. If
# the ``continue alpha`` statement is executed at line 14, then the same thing
# will happen for that value of alpha in any descendant of the table C, and so
# the values the values of alpha for which this occurs could profitably be
# stored and passed through to the descendants of C. Of course this would
# make the code more complicated.
# The code below is taken directly from the function on page 208 of [Sim94]
# nu[alpha]
"""
n = C.n
# lamda is the largest numbered point in Omega_c_alpha which is currently defined
lamda = -1
# for alpha in Omega_c, nu[alpha] is the point in Omega_c_alpha corresponding to alpha
nu = [None]*n
# for alpha in Omega_c_alpha, mu[alpha] is the point in Omega_c corresponding to alpha
mu = [None]*n
# mutually nu and mu are the mutually-inverse equivalence maps between
# Omega_c_alpha and Omega_c
next_alpha = False
# For each 0!=alpha in [0 .. nc-1], we start by constructing the equivalent
# standardized coset table C_alpha corresponding to H_alpha
for alpha in range(1, n):
# reset nu to "None" after previous value of alpha
for beta in range(lamda+1):
nu[mu[beta]] = None
# we only want to reject our current table in favour of a preceding
# table in the ordering in which 1 is replaced by alpha, if the subgroup
# G_alpha corresponding to this preceding table definitely contains the
# given subgroup
for w in Y:
# TODO: this should support input of a list of general words
# not just the words which are in "A" (i.e gen and gen^-1)
if C.table[alpha][C.A_dict[w]] != alpha:
# continue with alpha
next_alpha = True
break
if next_alpha:
next_alpha = False
continue
# try alpha as the new point 0 in Omega_C_alpha
mu[0] = alpha
nu[alpha] = 0
# compare corresponding entries in C and C_alpha
lamda = 0
for beta in range(n):
for x in C.A:
gamma = C.table[beta][C.A_dict[x]]
delta = C.table[mu[beta]][C.A_dict[x]]
# if either of the entries is undefined,
# we move with next alpha
if gamma is None or delta is None:
# continue with alpha
next_alpha = True
break
if nu[delta] is None:
# delta becomes the next point in Omega_C_alpha
lamda += 1
nu[delta] = lamda
mu[lamda] = delta
if nu[delta] < gamma:
return False
if nu[delta] > gamma:
# continue with alpha
next_alpha = True
break
if next_alpha:
next_alpha = False
break
return True
#========================================================================
# Simplifying Presentation
#========================================================================
def simplify_presentation(*args, change_gens=False):
'''
For an instance of `FpGroup`, return a simplified isomorphic copy of
the group (e.g. remove redundant generators or relators). Alternatively,
a list of generators and relators can be passed in which case the
simplified lists will be returned.
By default, the generators of the group are unchanged. If you would
like to remove redundant generators, set the keyword argument
`change_gens = True`.
'''
if len(args) == 1:
if not isinstance(args[0], FpGroup):
raise TypeError("The argument must be an instance of FpGroup")
G = args[0]
gens, rels = simplify_presentation(G.generators, G.relators,
change_gens=change_gens)
if gens:
return FpGroup(gens[0].group, rels)
return FpGroup(FreeGroup([]), [])
elif len(args) == 2:
gens, rels = args[0][:], args[1][:]
if not gens:
return gens, rels
identity = gens[0].group.identity
else:
if len(args) == 0:
m = "Not enough arguments"
else:
m = "Too many arguments"
raise RuntimeError(m)
prev_gens = []
prev_rels = []
while not set(prev_rels) == set(rels):
prev_rels = rels
while change_gens and not set(prev_gens) == set(gens):
prev_gens = gens
gens, rels = elimination_technique_1(gens, rels, identity)
rels = _simplify_relators(rels, identity)
if change_gens:
syms = [g.array_form[0][0] for g in gens]
F = free_group(syms)[0]
identity = F.identity
gens = F.generators
subs = dict(zip(syms, gens))
for j, r in enumerate(rels):
a = r.array_form
rel = identity
for sym, p in a:
rel = rel*subs[sym]**p
rels[j] = rel
return gens, rels
def _simplify_relators(rels, identity):
"""Relies upon ``_simplification_technique_1`` for its functioning. """
rels = rels[:]
rels = list(set(_simplification_technique_1(rels)))
rels.sort()
rels = [r.identity_cyclic_reduction() for r in rels]
try:
rels.remove(identity)
except ValueError:
pass
return rels
# Pg 350, section 2.5.1 from [2]
def elimination_technique_1(gens, rels, identity):
rels = rels[:]
# the shorter relators are examined first so that generators selected for
# elimination will have shorter strings as equivalent
rels.sort()
gens = gens[:]
redundant_gens = {}
redundant_rels = []
used_gens = set()
# examine each relator in relator list for any generator occurring exactly
# once
for rel in rels:
# don't look for a redundant generator in a relator which
# depends on previously found ones
contained_gens = rel.contains_generators()
if any(g in contained_gens for g in redundant_gens):
continue
contained_gens = list(contained_gens)
contained_gens.sort(reverse = True)
for gen in contained_gens:
if rel.generator_count(gen) == 1 and gen not in used_gens:
k = rel.exponent_sum(gen)
gen_index = rel.index(gen**k)
bk = rel.subword(gen_index + 1, len(rel))
fw = rel.subword(0, gen_index)
chi = bk*fw
redundant_gens[gen] = chi**(-1*k)
used_gens.update(chi.contains_generators())
redundant_rels.append(rel)
break
rels = [r for r in rels if r not in redundant_rels]
# eliminate the redundant generators from remaining relators
rels = [r.eliminate_words(redundant_gens, _all = True).identity_cyclic_reduction() for r in rels]
rels = list(set(rels))
try:
rels.remove(identity)
except ValueError:
pass
gens = [g for g in gens if g not in redundant_gens]
return gens, rels
def _simplification_technique_1(rels):
"""
All relators are checked to see if they are of the form `gen^n`. If any
such relators are found then all other relators are processed for strings
in the `gen` known order.
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import _simplification_technique_1
>>> F, x, y = free_group("x, y")
>>> w1 = [x**2*y**4, x**3]
>>> _simplification_technique_1(w1)
[x**-1*y**4, x**3]
>>> w2 = [x**2*y**-4*x**5, x**3, x**2*y**8, y**5]
>>> _simplification_technique_1(w2)
[x**-1*y*x**-1, x**3, x**-1*y**-2, y**5]
>>> w3 = [x**6*y**4, x**4]
>>> _simplification_technique_1(w3)
[x**2*y**4, x**4]
"""
from sympy.polys.polytools import gcd
rels = rels[:]
# dictionary with "gen: n" where gen^n is one of the relators
exps = {}
for i in range(len(rels)):
rel = rels[i]
if rel.number_syllables() == 1:
g = rel[0]
exp = abs(rel.array_form[0][1])
if rel.array_form[0][1] < 0:
rels[i] = rels[i]**-1
g = g**-1
if g in exps:
exp = gcd(exp, exps[g].array_form[0][1])
exps[g] = g**exp
one_syllables_words = exps.values()
# decrease some of the exponents in relators, making use of the single
# syllable relators
for i in range(len(rels)):
rel = rels[i]
if rel in one_syllables_words:
continue
rel = rel.eliminate_words(one_syllables_words, _all = True)
# if rels[i] contains g**n where abs(n) is greater than half of the power p
# of g in exps, g**n can be replaced by g**(n-p) (or g**(p-n) if n<0)
for g in rel.contains_generators():
if g in exps:
exp = exps[g].array_form[0][1]
max_exp = (exp + 1)//2
rel = rel.eliminate_word(g**(max_exp), g**(max_exp-exp), _all = True)
rel = rel.eliminate_word(g**(-max_exp), g**(-(max_exp-exp)), _all = True)
rels[i] = rel
rels = [r.identity_cyclic_reduction() for r in rels]
return rels
###############################################################################
# SUBGROUP PRESENTATIONS #
###############################################################################
# Pg 175 [1]
def define_schreier_generators(C, homomorphism=False):
'''
Parameters
==========
C -- Coset table.
homomorphism -- When set to True, return a dictionary containing the images
of the presentation generators in the original group.
'''
y = []
gamma = 1
f = C.fp_group
X = f.generators
if homomorphism:
# `_gens` stores the elements of the parent group to
# to which the schreier generators correspond to.
_gens = {}
# compute the schreier Traversal
tau = {}
tau[0] = f.identity
C.P = [[None]*len(C.A) for i in range(C.n)]
for alpha, x in product(C.omega, C.A):
beta = C.table[alpha][C.A_dict[x]]
if beta == gamma:
C.P[alpha][C.A_dict[x]] = "<identity>"
C.P[beta][C.A_dict_inv[x]] = "<identity>"
gamma += 1
if homomorphism:
tau[beta] = tau[alpha]*x
elif x in X and C.P[alpha][C.A_dict[x]] is None:
y_alpha_x = '%s_%s' % (x, alpha)
y.append(y_alpha_x)
C.P[alpha][C.A_dict[x]] = y_alpha_x
if homomorphism:
_gens[y_alpha_x] = tau[alpha]*x*tau[beta]**-1
grp_gens = list(free_group(', '.join(y)))
C._schreier_free_group = grp_gens.pop(0)
C._schreier_generators = grp_gens
if homomorphism:
C._schreier_gen_elem = _gens
# replace all elements of P by, free group elements
for i, j in product(range(len(C.P)), range(len(C.A))):
# if equals "<identity>", replace by identity element
if C.P[i][j] == "<identity>":
C.P[i][j] = C._schreier_free_group.identity
elif isinstance(C.P[i][j], str):
r = C._schreier_generators[y.index(C.P[i][j])]
C.P[i][j] = r
beta = C.table[i][j]
C.P[beta][j + 1] = r**-1
def reidemeister_relators(C):
R = C.fp_group.relators
rels = [rewrite(C, coset, word) for word in R for coset in range(C.n)]
order_1_gens = {i for i in rels if len(i) == 1}
# remove all the order 1 generators from relators
rels = list(filter(lambda rel: rel not in order_1_gens, rels))
# replace order 1 generators by identity element in reidemeister relators
for i in range(len(rels)):
w = rels[i]
w = w.eliminate_words(order_1_gens, _all=True)
rels[i] = w
C._schreier_generators = [i for i in C._schreier_generators
if not (i in order_1_gens or i**-1 in order_1_gens)]
# Tietze transformation 1 i.e TT_1
# remove cyclic conjugate elements from relators
i = 0
while i < len(rels):
w = rels[i]
j = i + 1
while j < len(rels):
if w.is_cyclic_conjugate(rels[j]):
del rels[j]
else:
j += 1
i += 1
C._reidemeister_relators = rels
def rewrite(C, alpha, w):
"""
Parameters
==========
C: CosetTable
alpha: A live coset
w: A word in `A*`
Returns
=======
rho(tau(alpha), w)
Examples
========
>>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, define_schreier_generators, rewrite
>>> from sympy.combinatorics.free_groups import free_group
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**2, y**3, (x*y)**6])
>>> C = CosetTable(f, [])
>>> C.table = [[1, 1, 2, 3], [0, 0, 4, 5], [4, 4, 3, 0], [5, 5, 0, 2], [2, 2, 5, 1], [3, 3, 1, 4]]
>>> C.p = [0, 1, 2, 3, 4, 5]
>>> define_schreier_generators(C)
>>> rewrite(C, 0, (x*y)**6)
x_4*y_2*x_3*x_1*x_2*y_4*x_5
"""
v = C._schreier_free_group.identity
for i in range(len(w)):
x_i = w[i]
v = v*C.P[alpha][C.A_dict[x_i]]
alpha = C.table[alpha][C.A_dict[x_i]]
return v
# Pg 350, section 2.5.2 from [2]
def elimination_technique_2(C):
"""
This technique eliminates one generator at a time. Heuristically this
seems superior in that we may select for elimination the generator with
shortest equivalent string at each stage.
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r, \
reidemeister_relators, define_schreier_generators, elimination_technique_2
>>> F, x, y = free_group("x, y")
>>> f = FpGroup(F, [x**3, y**5, (x*y)**2]); H = [x*y, x**-1*y**-1*x*y*x]
>>> C = coset_enumeration_r(f, H)
>>> C.compress(); C.standardize()
>>> define_schreier_generators(C)
>>> reidemeister_relators(C)
>>> elimination_technique_2(C)
([y_1, y_2], [y_2**-3, y_2*y_1*y_2*y_1*y_2*y_1, y_1**2])
"""
rels = C._reidemeister_relators
rels.sort(reverse=True)
gens = C._schreier_generators
for i in range(len(gens) - 1, -1, -1):
rel = rels[i]
for j in range(len(gens) - 1, -1, -1):
gen = gens[j]
if rel.generator_count(gen) == 1:
k = rel.exponent_sum(gen)
gen_index = rel.index(gen**k)
bk = rel.subword(gen_index + 1, len(rel))
fw = rel.subword(0, gen_index)
rep_by = (bk*fw)**(-1*k)
del rels[i]; del gens[j]
for l in range(len(rels)):
rels[l] = rels[l].eliminate_word(gen, rep_by)
break
C._reidemeister_relators = rels
C._schreier_generators = gens
return C._schreier_generators, C._reidemeister_relators
def reidemeister_presentation(fp_grp, H, C=None, homomorphism=False):
"""
Parameters
==========
fp_group: A finitely presented group, an instance of FpGroup
H: A subgroup whose presentation is to be found, given as a list
of words in generators of `fp_grp`
homomorphism: When set to True, return a homomorphism from the subgroup
to the parent group
Examples
========
>>> from sympy.combinatorics.free_groups import free_group
>>> from sympy.combinatorics.fp_groups import FpGroup, reidemeister_presentation
>>> F, x, y = free_group("x, y")
Example 5.6 Pg. 177 from [1]
>>> f = FpGroup(F, [x**3, y**5, (x*y)**2])
>>> H = [x*y, x**-1*y**-1*x*y*x]
>>> reidemeister_presentation(f, H)
((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))
Example 5.8 Pg. 183 from [1]
>>> f = FpGroup(F, [x**3, y**3, (x*y)**3])
>>> H = [x*y, x*y**-1]
>>> reidemeister_presentation(f, H)
((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))
Exercises Q2. Pg 187 from [1]
>>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
>>> H = [x]
>>> reidemeister_presentation(f, H)
((x_0,), (x_0**4,))
Example 5.9 Pg. 183 from [1]
>>> f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
>>> H = [x]
>>> reidemeister_presentation(f, H)
((x_0,), (x_0**6,))
"""
if not C:
C = coset_enumeration_r(fp_grp, H)
C.compress(); C.standardize()
define_schreier_generators(C, homomorphism=homomorphism)
reidemeister_relators(C)
gens, rels = C._schreier_generators, C._reidemeister_relators
gens, rels = simplify_presentation(gens, rels, change_gens=True)
C.schreier_generators = tuple(gens)
C.reidemeister_relators = tuple(rels)
if homomorphism:
_gens = []
for gen in gens:
_gens.append(C._schreier_gen_elem[str(gen)])
return C.schreier_generators, C.reidemeister_relators, _gens
return C.schreier_generators, C.reidemeister_relators
FpGroupElement = FreeGroupElement
|
7daa54aac95910d61e560561dfc68412694b4c215d7598a09f42720b15b8bb9b | from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.expr import Expr
from sympy.core.function import AppliedUndef, UndefinedFunction
from sympy.core.mul import Mul
from sympy.core.relational import Equality, Relational
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, Dummy
from sympy.core.sympify import sympify
from sympy.functions.elementary.piecewise import (piecewise_fold,
Piecewise)
from sympy.logic.boolalg import BooleanFunction
from sympy.matrices.matrices import MatrixBase
from sympy.sets.sets import Interval, Set
from sympy.sets.fancysets import Range
from sympy.tensor.indexed import Idx
from sympy.utilities import flatten
from sympy.utilities.iterables import sift, is_sequence
from sympy.utilities.exceptions import SymPyDeprecationWarning
def _common_new(cls, function, *symbols, discrete, **assumptions):
"""Return either a special return value or the tuple,
(function, limits, orientation). This code is common to
both ExprWithLimits and AddWithLimits."""
function = sympify(function)
if isinstance(function, Equality):
# This transforms e.g. Integral(Eq(x, y)) to Eq(Integral(x), Integral(y))
# but that is only valid for definite integrals.
limits, orientation = _process_limits(*symbols, discrete=discrete)
if not (limits and all(len(limit) == 3 for limit in limits)):
SymPyDeprecationWarning(
feature='Integral(Eq(x, y))',
useinstead='Eq(Integral(x, z), Integral(y, z))',
issue=18053,
deprecated_since_version=1.6,
).warn()
lhs = function.lhs
rhs = function.rhs
return Equality(cls(lhs, *symbols, **assumptions), \
cls(rhs, *symbols, **assumptions))
if function is S.NaN:
return S.NaN
if symbols:
limits, orientation = _process_limits(*symbols, discrete=discrete)
for i, li in enumerate(limits):
if len(li) == 4:
function = function.subs(li[0], li[-1])
limits[i] = Tuple(*li[:-1])
else:
# symbol not provided -- we can still try to compute a general form
free = function.free_symbols
if len(free) != 1:
raise ValueError(
"specify dummy variables for %s" % function)
limits, orientation = [Tuple(s) for s in free], 1
# denest any nested calls
while cls == type(function):
limits = list(function.limits) + limits
function = function.function
# Any embedded piecewise functions need to be brought out to the
# top level. We only fold Piecewise that contain the integration
# variable.
reps = {}
symbols_of_integration = {i[0] for i in limits}
for p in function.atoms(Piecewise):
if not p.has(*symbols_of_integration):
reps[p] = Dummy()
# mask off those that don't
function = function.xreplace(reps)
# do the fold
function = piecewise_fold(function)
# remove the masking
function = function.xreplace({v: k for k, v in reps.items()})
return function, limits, orientation
def _process_limits(*symbols, discrete=None):
"""Process the list of symbols and convert them to canonical limits,
storing them as Tuple(symbol, lower, upper). The orientation of
the function is also returned when the upper limit is missing
so (x, 1, None) becomes (x, None, 1) and the orientation is changed.
In the case that a limit is specified as (symbol, Range), a list of
length 4 may be returned if a change of variables is needed; the
expression that should replace the symbol in the expression is
the fourth element in the list.
"""
limits = []
orientation = 1
if discrete is None:
err_msg = 'discrete must be True or False'
elif discrete:
err_msg = 'use Range, not Interval or Relational'
else:
err_msg = 'use Interval or Relational, not Range'
for V in symbols:
if isinstance(V, (Relational, BooleanFunction)):
if discrete:
raise TypeError(err_msg)
variable = V.atoms(Symbol).pop()
V = (variable, V.as_set())
elif isinstance(V, Symbol) or getattr(V, '_diff_wrt', False):
if isinstance(V, Idx):
if V.lower is None or V.upper is None:
limits.append(Tuple(V))
else:
limits.append(Tuple(V, V.lower, V.upper))
else:
limits.append(Tuple(V))
continue
if is_sequence(V) and not isinstance(V, Set):
if len(V) == 2 and isinstance(V[1], Set):
V = list(V)
if isinstance(V[1], Interval): # includes Reals
if discrete:
raise TypeError(err_msg)
V[1:] = V[1].inf, V[1].sup
elif isinstance(V[1], Range):
if not discrete:
raise TypeError(err_msg)
lo = V[1].inf
hi = V[1].sup
dx = abs(V[1].step) # direction doesn't matter
if dx == 1:
V[1:] = [lo, hi]
else:
if lo is not S.NegativeInfinity:
V = [V[0]] + [0, (hi - lo)//dx, dx*V[0] + lo]
else:
V = [V[0]] + [0, S.Infinity, -dx*V[0] + hi]
else:
# more complicated sets would require splitting, e.g.
# Union(Interval(1, 3), interval(6,10))
raise NotImplementedError(
'expecting Range' if discrete else
'Relational or single Interval' )
V = sympify(flatten(V)) # list of sympified elements/None
if isinstance(V[0], (Symbol, Idx)) or getattr(V[0], '_diff_wrt', False):
newsymbol = V[0]
if len(V) == 3:
# general case
if V[2] is None and V[1] is not None:
orientation *= -1
V = [newsymbol] + [i for i in V[1:] if i is not None]
lenV = len(V)
if not isinstance(newsymbol, Idx) or lenV == 3:
if lenV == 4:
limits.append(Tuple(*V))
continue
if lenV == 3:
if isinstance(newsymbol, Idx):
# Idx represents an integer which may have
# specified values it can take on; if it is
# given such a value, an error is raised here
# if the summation would try to give it a larger
# or smaller value than permitted. None and Symbolic
# values will not raise an error.
lo, hi = newsymbol.lower, newsymbol.upper
try:
if lo is not None and not bool(V[1] >= lo):
raise ValueError("Summation will set Idx value too low.")
except TypeError:
pass
try:
if hi is not None and not bool(V[2] <= hi):
raise ValueError("Summation will set Idx value too high.")
except TypeError:
pass
limits.append(Tuple(*V))
continue
if lenV == 1 or (lenV == 2 and V[1] is None):
limits.append(Tuple(newsymbol))
continue
elif lenV == 2:
limits.append(Tuple(newsymbol, V[1]))
continue
raise ValueError('Invalid limits given: %s' % str(symbols))
return limits, orientation
class ExprWithLimits(Expr):
__slots__ = ('is_commutative',)
def __new__(cls, function, *symbols, **assumptions):
from sympy.concrete.products import Product
pre = _common_new(cls, function, *symbols,
discrete=issubclass(cls, Product), **assumptions)
if isinstance(pre, tuple):
function, limits, _ = pre
else:
return pre
# limits must have upper and lower bounds; the indefinite form
# is not supported. This restriction does not apply to AddWithLimits
if any(len(l) != 3 or None in l for l in limits):
raise ValueError('ExprWithLimits requires values for lower and upper bounds.')
obj = Expr.__new__(cls, **assumptions)
arglist = [function]
arglist.extend(limits)
obj._args = tuple(arglist)
obj.is_commutative = function.is_commutative # limits already checked
return obj
@property
def function(self):
"""Return the function applied across limits.
Examples
========
>>> from sympy import Integral
>>> from sympy.abc import x
>>> Integral(x**2, (x,)).function
x**2
See Also
========
limits, variables, free_symbols
"""
return self._args[0]
@property
def kind(self):
return self.function.kind
@property
def limits(self):
"""Return the limits of expression.
Examples
========
>>> from sympy import Integral
>>> from sympy.abc import x, i
>>> Integral(x**i, (i, 1, 3)).limits
((i, 1, 3),)
See Also
========
function, variables, free_symbols
"""
return self._args[1:]
@property
def variables(self):
"""Return a list of the limit variables.
>>> from sympy import Sum
>>> from sympy.abc import x, i
>>> Sum(x**i, (i, 1, 3)).variables
[i]
See Also
========
function, limits, free_symbols
as_dummy : Rename dummy variables
sympy.integrals.integrals.Integral.transform : Perform mapping on the dummy variable
"""
return [l[0] for l in self.limits]
@property
def bound_symbols(self):
"""Return only variables that are dummy variables.
Examples
========
>>> from sympy import Integral
>>> from sympy.abc import x, i, j, k
>>> Integral(x**i, (i, 1, 3), (j, 2), k).bound_symbols
[i, j]
See Also
========
function, limits, free_symbols
as_dummy : Rename dummy variables
sympy.integrals.integrals.Integral.transform : Perform mapping on the dummy variable
"""
return [l[0] for l in self.limits if len(l) != 1]
@property
def free_symbols(self):
"""
This method returns the symbols in the object, excluding those
that take on a specific value (i.e. the dummy symbols).
Examples
========
>>> from sympy import Sum
>>> from sympy.abc import x, y
>>> Sum(x, (x, y, 1)).free_symbols
{y}
"""
# don't test for any special values -- nominal free symbols
# should be returned, e.g. don't return set() if the
# function is zero -- treat it like an unevaluated expression.
function, limits = self.function, self.limits
# mask off non-symbol integration variables that have
# more than themself as a free symbol
reps = {i[0]: i[0] if i[0].free_symbols == {i[0]} else Dummy()
for i in self.limits}
function = function.xreplace(reps)
isyms = function.free_symbols
for xab in limits:
v = reps[xab[0]]
if len(xab) == 1:
isyms.add(v)
continue
# take out the target symbol
if v in isyms:
isyms.remove(v)
# add in the new symbols
for i in xab[1:]:
isyms.update(i.free_symbols)
reps = {v: k for k, v in reps.items()}
return set([reps.get(_, _) for _ in isyms])
@property
def is_number(self):
"""Return True if the Sum has no free symbols, else False."""
return not self.free_symbols
def _eval_interval(self, x, a, b):
limits = [(i if i[0] != x else (x, a, b)) for i in self.limits]
integrand = self.function
return self.func(integrand, *limits)
def _eval_subs(self, old, new):
"""
Perform substitutions over non-dummy variables
of an expression with limits. Also, can be used
to specify point-evaluation of an abstract antiderivative.
Examples
========
>>> from sympy import Sum, oo
>>> from sympy.abc import s, n
>>> Sum(1/n**s, (n, 1, oo)).subs(s, 2)
Sum(n**(-2), (n, 1, oo))
>>> from sympy import Integral
>>> from sympy.abc import x, a
>>> Integral(a*x**2, x).subs(x, 4)
Integral(a*x**2, (x, 4))
See Also
========
variables : Lists the integration variables
transform : Perform mapping on the dummy variable for integrals
change_index : Perform mapping on the sum and product dummy variables
"""
func, limits = self.function, list(self.limits)
# If one of the expressions we are replacing is used as a func index
# one of two things happens.
# - the old variable first appears as a free variable
# so we perform all free substitutions before it becomes
# a func index.
# - the old variable first appears as a func index, in
# which case we ignore. See change_index.
# Reorder limits to match standard mathematical practice for scoping
limits.reverse()
if not isinstance(old, Symbol) or \
old.free_symbols.intersection(self.free_symbols):
sub_into_func = True
for i, xab in enumerate(limits):
if 1 == len(xab) and old == xab[0]:
if new._diff_wrt:
xab = (new,)
else:
xab = (old, old)
limits[i] = Tuple(xab[0], *[l._subs(old, new) for l in xab[1:]])
if len(xab[0].free_symbols.intersection(old.free_symbols)) != 0:
sub_into_func = False
break
if isinstance(old, (AppliedUndef, UndefinedFunction)):
sy2 = set(self.variables).intersection(set(new.atoms(Symbol)))
sy1 = set(self.variables).intersection(set(old.args))
if not sy2.issubset(sy1):
raise ValueError(
"substitution cannot create dummy dependencies")
sub_into_func = True
if sub_into_func:
func = func.subs(old, new)
else:
# old is a Symbol and a dummy variable of some limit
for i, xab in enumerate(limits):
if len(xab) == 3:
limits[i] = Tuple(xab[0], *[l._subs(old, new) for l in xab[1:]])
if old == xab[0]:
break
# simplify redundant limits (x, x) to (x, )
for i, xab in enumerate(limits):
if len(xab) == 2 and (xab[0] - xab[1]).is_zero:
limits[i] = Tuple(xab[0], )
# Reorder limits back to representation-form
limits.reverse()
return self.func(func, *limits)
@property
def has_finite_limits(self):
"""
Returns True if the limits are known to be finite, either by the
explicit bounds, assumptions on the bounds, or assumptions on the
variables. False if known to be infinite, based on the bounds.
None if not enough information is available to determine.
Examples
========
>>> from sympy import Sum, Integral, Product, oo, Symbol
>>> x = Symbol('x')
>>> Sum(x, (x, 1, 8)).has_finite_limits
True
>>> Integral(x, (x, 1, oo)).has_finite_limits
False
>>> M = Symbol('M')
>>> Sum(x, (x, 1, M)).has_finite_limits
>>> N = Symbol('N', integer=True)
>>> Product(x, (x, 1, N)).has_finite_limits
True
See Also
========
has_reversed_limits
"""
ret_None = False
for lim in self.limits:
if len(lim) == 3:
if any(l.is_infinite for l in lim[1:]):
# Any of the bounds are +/-oo
return False
elif any(l.is_infinite is None for l in lim[1:]):
# Maybe there are assumptions on the variable?
if lim[0].is_infinite is None:
ret_None = True
else:
if lim[0].is_infinite is None:
ret_None = True
if ret_None:
return None
return True
@property
def has_reversed_limits(self):
"""
Returns True if the limits are known to be in reversed order, either
by the explicit bounds, assumptions on the bounds, or assumptions on the
variables. False if known to be in normal order, based on the bounds.
None if not enough information is available to determine.
Examples
========
>>> from sympy import Sum, Integral, Product, oo, Symbol
>>> x = Symbol('x')
>>> Sum(x, (x, 8, 1)).has_reversed_limits
True
>>> Sum(x, (x, 1, oo)).has_reversed_limits
False
>>> M = Symbol('M')
>>> Integral(x, (x, 1, M)).has_reversed_limits
>>> N = Symbol('N', integer=True, positive=True)
>>> Sum(x, (x, 1, N)).has_reversed_limits
False
>>> Product(x, (x, 2, N)).has_reversed_limits
>>> Product(x, (x, 2, N)).subs(N, N + 2).has_reversed_limits
False
See Also
========
sympy.concrete.expr_with_intlimits.ExprWithIntLimits.has_empty_sequence
"""
ret_None = False
for lim in self.limits:
if len(lim) == 3:
var, a, b = lim
dif = b - a
if dif.is_extended_negative:
return True
elif dif.is_extended_nonnegative:
continue
else:
ret_None = True
else:
return None
if ret_None:
return None
return False
class AddWithLimits(ExprWithLimits):
r"""Represents unevaluated oriented additions.
Parent class for Integral and Sum.
"""
def __new__(cls, function, *symbols, **assumptions):
from sympy.concrete.summations import Sum
pre = _common_new(cls, function, *symbols,
discrete=issubclass(cls, Sum), **assumptions)
if isinstance(pre, tuple):
function, limits, orientation = pre
else:
return pre
obj = Expr.__new__(cls, **assumptions)
arglist = [orientation*function] # orientation not used in ExprWithLimits
arglist.extend(limits)
obj._args = tuple(arglist)
obj.is_commutative = function.is_commutative # limits already checked
return obj
def _eval_adjoint(self):
if all(x.is_real for x in flatten(self.limits)):
return self.func(self.function.adjoint(), *self.limits)
return None
def _eval_conjugate(self):
if all(x.is_real for x in flatten(self.limits)):
return self.func(self.function.conjugate(), *self.limits)
return None
def _eval_transpose(self):
if all(x.is_real for x in flatten(self.limits)):
return self.func(self.function.transpose(), *self.limits)
return None
def _eval_factor(self, **hints):
if 1 == len(self.limits):
summand = self.function.factor(**hints)
if summand.is_Mul:
out = sift(summand.args, lambda w: w.is_commutative \
and not set(self.variables) & w.free_symbols)
return Mul(*out[True])*self.func(Mul(*out[False]), \
*self.limits)
else:
summand = self.func(self.function, *self.limits[0:-1]).factor()
if not summand.has(self.variables[-1]):
return self.func(1, [self.limits[-1]]).doit()*summand
elif isinstance(summand, Mul):
return self.func(summand, self.limits[-1]).factor()
return self
def _eval_expand_basic(self, **hints):
summand = self.function.expand(**hints)
if summand.is_Add and summand.is_commutative:
return Add(*[self.func(i, *self.limits) for i in summand.args])
elif isinstance(summand, MatrixBase):
return summand.applyfunc(lambda x: self.func(x, *self.limits))
elif summand != self.function:
return self.func(summand, *self.limits)
return self
|
ebe4825aa7da4152cfb9d65f3a998854bc13e6a3b001302bd4ea8f75179c7c84 | from typing import Tuple as tTuple
from sympy.calculus.singularities import is_decreasing
from sympy.calculus.accumulationbounds import AccumulationBounds
from .expr_with_intlimits import ExprWithIntLimits
from .expr_with_limits import AddWithLimits
from .gosper import gosper_sum
from sympy.core.expr import Expr
from sympy.core.add import Add
from sympy.core.containers import Tuple
from sympy.core.function import Derivative, expand
from sympy.core.mul import Mul
from sympy.core.numbers import Float
from sympy.core.relational import Eq
from sympy.core.singleton import S
from sympy.core.sorting import ordered
from sympy.core.symbol import Dummy, Wild, Symbol, symbols
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.combinatorial.numbers import bernoulli, harmonic
from sympy.functions.elementary.exponential import log
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary.trigonometric import cot, csc
from sympy.functions.special.hyper import hyper
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.functions.special.zeta_functions import zeta
from sympy.integrals.integrals import Integral
from sympy.logic.boolalg import And
from sympy.polys.partfrac import apart
from sympy.polys.polyerrors import PolynomialError, PolificationFailed
from sympy.polys.polytools import parallel_poly_from_expr, Poly, factor
from sympy.polys.rationaltools import together
from sympy.series.limitseq import limit_seq
from sympy.series.order import O
from sympy.series.residues import residue
from sympy.sets.sets import FiniteSet, Interval
from sympy.simplify.combsimp import combsimp
from sympy.simplify.hyperexpand import hyperexpand
from sympy.simplify.powsimp import powsimp
from sympy.simplify.radsimp import denom, fraction
from sympy.simplify.simplify import (factor_sum, sum_combine, simplify,
nsimplify, hypersimp)
from sympy.solvers.solvers import solve
from sympy.solvers.solveset import solveset
from sympy.utilities.iterables import sift
import itertools
class Sum(AddWithLimits, ExprWithIntLimits):
r"""
Represents unevaluated summation.
Explanation
===========
``Sum`` represents a finite or infinite series, with the first argument
being the general form of terms in the series, and the second argument
being ``(dummy_variable, start, end)``, with ``dummy_variable`` taking
all integer values from ``start`` through ``end``. In accordance with
long-standing mathematical convention, the end term is included in the
summation.
Finite sums
===========
For finite sums (and sums with symbolic limits assumed to be finite) we
follow the summation convention described by Karr [1], especially
definition 3 of section 1.4. The sum:
.. math::
\sum_{m \leq i < n} f(i)
has *the obvious meaning* for `m < n`, namely:
.. math::
\sum_{m \leq i < n} f(i) = f(m) + f(m+1) + \ldots + f(n-2) + f(n-1)
with the upper limit value `f(n)` excluded. The sum over an empty set is
zero if and only if `m = n`:
.. math::
\sum_{m \leq i < n} f(i) = 0 \quad \mathrm{for} \quad m = n
Finally, for all other sums over empty sets we assume the following
definition:
.. math::
\sum_{m \leq i < n} f(i) = - \sum_{n \leq i < m} f(i) \quad \mathrm{for} \quad m > n
It is important to note that Karr defines all sums with the upper
limit being exclusive. This is in contrast to the usual mathematical notation,
but does not affect the summation convention. Indeed we have:
.. math::
\sum_{m \leq i < n} f(i) = \sum_{i = m}^{n - 1} f(i)
where the difference in notation is intentional to emphasize the meaning,
with limits typeset on the top being inclusive.
Examples
========
>>> from sympy.abc import i, k, m, n, x
>>> from sympy import Sum, factorial, oo, IndexedBase, Function
>>> Sum(k, (k, 1, m))
Sum(k, (k, 1, m))
>>> Sum(k, (k, 1, m)).doit()
m**2/2 + m/2
>>> Sum(k**2, (k, 1, m))
Sum(k**2, (k, 1, m))
>>> Sum(k**2, (k, 1, m)).doit()
m**3/3 + m**2/2 + m/6
>>> Sum(x**k, (k, 0, oo))
Sum(x**k, (k, 0, oo))
>>> Sum(x**k, (k, 0, oo)).doit()
Piecewise((1/(1 - x), Abs(x) < 1), (Sum(x**k, (k, 0, oo)), True))
>>> Sum(x**k/factorial(k), (k, 0, oo)).doit()
exp(x)
Here are examples to do summation with symbolic indices. You
can use either Function of IndexedBase classes:
>>> f = Function('f')
>>> Sum(f(n), (n, 0, 3)).doit()
f(0) + f(1) + f(2) + f(3)
>>> Sum(f(n), (n, 0, oo)).doit()
Sum(f(n), (n, 0, oo))
>>> f = IndexedBase('f')
>>> Sum(f[n]**2, (n, 0, 3)).doit()
f[0]**2 + f[1]**2 + f[2]**2 + f[3]**2
An example showing that the symbolic result of a summation is still
valid for seemingly nonsensical values of the limits. Then the Karr
convention allows us to give a perfectly valid interpretation to
those sums by interchanging the limits according to the above rules:
>>> S = Sum(i, (i, 1, n)).doit()
>>> S
n**2/2 + n/2
>>> S.subs(n, -4)
6
>>> Sum(i, (i, 1, -4)).doit()
6
>>> Sum(-i, (i, -3, 0)).doit()
6
An explicit example of the Karr summation convention:
>>> S1 = Sum(i**2, (i, m, m+n-1)).doit()
>>> S1
m**2*n + m*n**2 - m*n + n**3/3 - n**2/2 + n/6
>>> S2 = Sum(i**2, (i, m+n, m-1)).doit()
>>> S2
-m**2*n - m*n**2 + m*n - n**3/3 + n**2/2 - n/6
>>> S1 + S2
0
>>> S3 = Sum(i, (i, m, m-1)).doit()
>>> S3
0
See Also
========
summation
Product, sympy.concrete.products.product
References
==========
.. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM,
Volume 28 Issue 2, April 1981, Pages 305-350
http://dl.acm.org/citation.cfm?doid=322248.322255
.. [2] https://en.wikipedia.org/wiki/Summation#Capital-sigma_notation
.. [3] https://en.wikipedia.org/wiki/Empty_sum
"""
__slots__ = ('is_commutative',)
limits: tTuple[tTuple[Symbol, Expr, Expr]]
def __new__(cls, function, *symbols, **assumptions):
obj = AddWithLimits.__new__(cls, function, *symbols, **assumptions)
if not hasattr(obj, 'limits'):
return obj
if any(len(l) != 3 or None in l for l in obj.limits):
raise ValueError('Sum requires values for lower and upper bounds.')
return obj
def _eval_is_zero(self):
# a Sum is only zero if its function is zero or if all terms
# cancel out. This only answers whether the summand is zero; if
# not then None is returned since we don't analyze whether all
# terms cancel out.
if self.function.is_zero or self.has_empty_sequence:
return True
def _eval_is_extended_real(self):
if self.has_empty_sequence:
return True
return self.function.is_extended_real
def _eval_is_positive(self):
if self.has_finite_limits and self.has_reversed_limits is False:
return self.function.is_positive
def _eval_is_negative(self):
if self.has_finite_limits and self.has_reversed_limits is False:
return self.function.is_negative
def _eval_is_finite(self):
if self.has_finite_limits and self.function.is_finite:
return True
def doit(self, **hints):
if hints.get('deep', True):
f = self.function.doit(**hints)
else:
f = self.function
# first make sure any definite limits have summation
# variables with matching assumptions
reps = {}
for xab in self.limits:
d = _dummy_with_inherited_properties_concrete(xab)
if d:
reps[xab[0]] = d
if reps:
undo = {v: k for k, v in reps.items()}
did = self.xreplace(reps).doit(**hints)
if isinstance(did, tuple): # when separate=True
did = tuple([i.xreplace(undo) for i in did])
elif did is not None:
did = did.xreplace(undo)
else:
did = self
return did
if self.function.is_Matrix:
expanded = self.expand()
if self != expanded:
return expanded.doit()
return _eval_matrix_sum(self)
for n, limit in enumerate(self.limits):
i, a, b = limit
dif = b - a
if dif == -1:
# Any summation over an empty set is zero
return S.Zero
if dif.is_integer and dif.is_negative:
a, b = b + 1, a - 1
f = -f
newf = eval_sum(f, (i, a, b))
if newf is None:
if f == self.function:
zeta_function = self.eval_zeta_function(f, (i, a, b))
if zeta_function is not None:
return zeta_function
return self
else:
return self.func(f, *self.limits[n:])
f = newf
if hints.get('deep', True):
# eval_sum could return partially unevaluated
# result with Piecewise. In this case we won't
# doit() recursively.
if not isinstance(f, Piecewise):
return f.doit(**hints)
return f
def eval_zeta_function(self, f, limits):
"""
Check whether the function matches with the zeta function.
If it matches, then return a `Piecewise` expression because
zeta function does not converge unless `s > 1` and `q > 0`
"""
i, a, b = limits
w, y, z = Wild('w', exclude=[i]), Wild('y', exclude=[i]), Wild('z', exclude=[i])
result = f.match((w * i + y) ** (-z))
if result is not None and b is S.Infinity:
coeff = 1 / result[w] ** result[z]
s = result[z]
q = result[y] / result[w] + a
return Piecewise((coeff * zeta(s, q), And(q > 0, s > 1)), (self, True))
def _eval_derivative(self, x):
"""
Differentiate wrt x as long as x is not in the free symbols of any of
the upper or lower limits.
Explanation
===========
Sum(a*b*x, (x, 1, a)) can be differentiated wrt x or b but not `a`
since the value of the sum is discontinuous in `a`. In a case
involving a limit variable, the unevaluated derivative is returned.
"""
# diff already confirmed that x is in the free symbols of self, but we
# don't want to differentiate wrt any free symbol in the upper or lower
# limits
# XXX remove this test for free_symbols when the default _eval_derivative is in
if isinstance(x, Symbol) and x not in self.free_symbols:
return S.Zero
# get limits and the function
f, limits = self.function, list(self.limits)
limit = limits.pop(-1)
if limits: # f is the argument to a Sum
f = self.func(f, *limits)
_, a, b = limit
if x in a.free_symbols or x in b.free_symbols:
return None
df = Derivative(f, x, evaluate=True)
rv = self.func(df, limit)
return rv
def _eval_difference_delta(self, n, step):
k, _, upper = self.args[-1]
new_upper = upper.subs(n, n + step)
if len(self.args) == 2:
f = self.args[0]
else:
f = self.func(*self.args[:-1])
return Sum(f, (k, upper + 1, new_upper)).doit()
def _eval_simplify(self, **kwargs):
# split the function into adds
terms = Add.make_args(expand(self.function))
s_t = [] # Sum Terms
o_t = [] # Other Terms
for term in terms:
if term.has(Sum):
# if there is an embedded sum here
# it is of the form x * (Sum(whatever))
# hence we make a Mul out of it, and simplify all interior sum terms
subterms = Mul.make_args(expand(term))
out_terms = []
for subterm in subterms:
# go through each term
if isinstance(subterm, Sum):
# if it's a sum, simplify it
out_terms.append(subterm._eval_simplify())
else:
# otherwise, add it as is
out_terms.append(subterm)
# turn it back into a Mul
s_t.append(Mul(*out_terms))
else:
o_t.append(term)
# next try to combine any interior sums for further simplification
result = Add(sum_combine(s_t), *o_t)
return factor_sum(result, limits=self.limits)
def is_convergent(self):
r"""
Checks for the convergence of a Sum.
Explanation
===========
We divide the study of convergence of infinite sums and products in
two parts.
First Part:
One part is the question whether all the terms are well defined, i.e.,
they are finite in a sum and also non-zero in a product. Zero
is the analogy of (minus) infinity in products as
:math:`e^{-\infty} = 0`.
Second Part:
The second part is the question of convergence after infinities,
and zeros in products, have been omitted assuming that their number
is finite. This means that we only consider the tail of the sum or
product, starting from some point after which all terms are well
defined.
For example, in a sum of the form:
.. math::
\sum_{1 \leq i < \infty} \frac{1}{n^2 + an + b}
where a and b are numbers. The routine will return true, even if there
are infinities in the term sequence (at most two). An analogous
product would be:
.. math::
\prod_{1 \leq i < \infty} e^{\frac{1}{n^2 + an + b}}
This is how convergence is interpreted. It is concerned with what
happens at the limit. Finding the bad terms is another independent
matter.
Note: It is responsibility of user to see that the sum or product
is well defined.
There are various tests employed to check the convergence like
divergence test, root test, integral test, alternating series test,
comparison tests, Dirichlet tests. It returns true if Sum is convergent
and false if divergent and NotImplementedError if it cannot be checked.
References
==========
.. [1] https://en.wikipedia.org/wiki/Convergence_tests
Examples
========
>>> from sympy import factorial, S, Sum, Symbol, oo
>>> n = Symbol('n', integer=True)
>>> Sum(n/(n - 1), (n, 4, 7)).is_convergent()
True
>>> Sum(n/(2*n + 1), (n, 1, oo)).is_convergent()
False
>>> Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent()
False
>>> Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent()
True
See Also
========
Sum.is_absolutely_convergent()
sympy.concrete.products.Product.is_convergent()
"""
p, q, r = symbols('p q r', cls=Wild)
sym = self.limits[0][0]
lower_limit = self.limits[0][1]
upper_limit = self.limits[0][2]
sequence_term = self.function.simplify()
if len(sequence_term.free_symbols) > 1:
raise NotImplementedError("convergence checking for more than one symbol "
"containing series is not handled")
if lower_limit.is_finite and upper_limit.is_finite:
return S.true
# transform sym -> -sym and swap the upper_limit = S.Infinity
# and lower_limit = - upper_limit
if lower_limit is S.NegativeInfinity:
if upper_limit is S.Infinity:
return Sum(sequence_term, (sym, 0, S.Infinity)).is_convergent() and \
Sum(sequence_term, (sym, S.NegativeInfinity, 0)).is_convergent()
sequence_term = simplify(sequence_term.xreplace({sym: -sym}))
lower_limit = -upper_limit
upper_limit = S.Infinity
sym_ = Dummy(sym.name, integer=True, positive=True)
sequence_term = sequence_term.xreplace({sym: sym_})
sym = sym_
interval = Interval(lower_limit, upper_limit)
# Piecewise function handle
if sequence_term.is_Piecewise:
for func, cond in sequence_term.args:
# see if it represents something going to oo
if cond == True or cond.as_set().sup is S.Infinity:
s = Sum(func, (sym, lower_limit, upper_limit))
return s.is_convergent()
return S.true
### -------- Divergence test ----------- ###
try:
lim_val = limit_seq(sequence_term, sym)
if lim_val is not None and lim_val.is_zero is False:
return S.false
except NotImplementedError:
pass
try:
lim_val_abs = limit_seq(abs(sequence_term), sym)
if lim_val_abs is not None and lim_val_abs.is_zero is False:
return S.false
except NotImplementedError:
pass
order = O(sequence_term, (sym, S.Infinity))
### --------- p-series test (1/n**p) ---------- ###
p_series_test = order.expr.match(sym**p)
if p_series_test is not None:
if p_series_test[p] < -1:
return S.true
if p_series_test[p] >= -1:
return S.false
### ------------- comparison test ------------- ###
# 1/(n**p*log(n)**q*log(log(n))**r) comparison
n_log_test = order.expr.match(1/(sym**p*log(sym)**q*log(log(sym))**r))
if n_log_test is not None:
if (n_log_test[p] > 1 or
(n_log_test[p] == 1 and n_log_test[q] > 1) or
(n_log_test[p] == n_log_test[q] == 1 and n_log_test[r] > 1)):
return S.true
return S.false
### ------------- Limit comparison test -----------###
# (1/n) comparison
try:
lim_comp = limit_seq(sym*sequence_term, sym)
if lim_comp is not None and lim_comp.is_number and lim_comp > 0:
return S.false
except NotImplementedError:
pass
### ----------- ratio test ---------------- ###
next_sequence_term = sequence_term.xreplace({sym: sym + 1})
ratio = combsimp(powsimp(next_sequence_term/sequence_term))
try:
lim_ratio = limit_seq(ratio, sym)
if lim_ratio is not None and lim_ratio.is_number:
if abs(lim_ratio) > 1:
return S.false
if abs(lim_ratio) < 1:
return S.true
except NotImplementedError:
lim_ratio = None
### ---------- Raabe's test -------------- ###
if lim_ratio == 1: # ratio test inconclusive
test_val = sym*(sequence_term/
sequence_term.subs(sym, sym + 1) - 1)
test_val = test_val.gammasimp()
try:
lim_val = limit_seq(test_val, sym)
if lim_val is not None and lim_val.is_number:
if lim_val > 1:
return S.true
if lim_val < 1:
return S.false
except NotImplementedError:
pass
### ----------- root test ---------------- ###
# lim = Limit(abs(sequence_term)**(1/sym), sym, S.Infinity)
try:
lim_evaluated = limit_seq(abs(sequence_term)**(1/sym), sym)
if lim_evaluated is not None and lim_evaluated.is_number:
if lim_evaluated < 1:
return S.true
if lim_evaluated > 1:
return S.false
except NotImplementedError:
pass
### ------------- alternating series test ----------- ###
dict_val = sequence_term.match(S.NegativeOne**(sym + p)*q)
if not dict_val[p].has(sym) and is_decreasing(dict_val[q], interval):
return S.true
### ------------- integral test -------------- ###
check_interval = None
maxima = solveset(sequence_term.diff(sym), sym, interval)
if not maxima:
check_interval = interval
elif isinstance(maxima, FiniteSet) and maxima.sup.is_number:
check_interval = Interval(maxima.sup, interval.sup)
if (check_interval is not None and
(is_decreasing(sequence_term, check_interval) or
is_decreasing(-sequence_term, check_interval))):
integral_val = Integral(
sequence_term, (sym, lower_limit, upper_limit))
try:
integral_val_evaluated = integral_val.doit()
if integral_val_evaluated.is_number:
return S(integral_val_evaluated.is_finite)
except NotImplementedError:
pass
### ----- Dirichlet and bounded times convergent tests ----- ###
# TODO
#
# Dirichlet_test
# https://en.wikipedia.org/wiki/Dirichlet%27s_test
#
# Bounded times convergent test
# It is based on comparison theorems for series.
# In particular, if the general term of a series can
# be written as a product of two terms a_n and b_n
# and if a_n is bounded and if Sum(b_n) is absolutely
# convergent, then the original series Sum(a_n * b_n)
# is absolutely convergent and so convergent.
#
# The following code can grows like 2**n where n is the
# number of args in order.expr
# Possibly combined with the potentially slow checks
# inside the loop, could make this test extremely slow
# for larger summation expressions.
if order.expr.is_Mul:
args = order.expr.args
argset = set(args)
### -------------- Dirichlet tests -------------- ###
m = Dummy('m', integer=True)
def _dirichlet_test(g_n):
try:
ing_val = limit_seq(Sum(g_n, (sym, interval.inf, m)).doit(), m)
if ing_val is not None and ing_val.is_finite:
return S.true
except NotImplementedError:
pass
### -------- bounded times convergent test ---------###
def _bounded_convergent_test(g1_n, g2_n):
try:
lim_val = limit_seq(g1_n, sym)
if lim_val is not None and (lim_val.is_finite or (
isinstance(lim_val, AccumulationBounds)
and (lim_val.max - lim_val.min).is_finite)):
if Sum(g2_n, (sym, lower_limit, upper_limit)).is_absolutely_convergent():
return S.true
except NotImplementedError:
pass
for n in range(1, len(argset)):
for a_tuple in itertools.combinations(args, n):
b_set = argset - set(a_tuple)
a_n = Mul(*a_tuple)
b_n = Mul(*b_set)
if is_decreasing(a_n, interval):
dirich = _dirichlet_test(b_n)
if dirich is not None:
return dirich
bc_test = _bounded_convergent_test(a_n, b_n)
if bc_test is not None:
return bc_test
_sym = self.limits[0][0]
sequence_term = sequence_term.xreplace({sym: _sym})
raise NotImplementedError("The algorithm to find the Sum convergence of %s "
"is not yet implemented" % (sequence_term))
def is_absolutely_convergent(self):
"""
Checks for the absolute convergence of an infinite series.
Same as checking convergence of absolute value of sequence_term of
an infinite series.
References
==========
.. [1] https://en.wikipedia.org/wiki/Absolute_convergence
Examples
========
>>> from sympy import Sum, Symbol, oo
>>> n = Symbol('n', integer=True)
>>> Sum((-1)**n, (n, 1, oo)).is_absolutely_convergent()
False
>>> Sum((-1)**n/n**2, (n, 1, oo)).is_absolutely_convergent()
True
See Also
========
Sum.is_convergent()
"""
return Sum(abs(self.function), self.limits).is_convergent()
def euler_maclaurin(self, m=0, n=0, eps=0, eval_integral=True):
"""
Return an Euler-Maclaurin approximation of self, where m is the
number of leading terms to sum directly and n is the number of
terms in the tail.
With m = n = 0, this is simply the corresponding integral
plus a first-order endpoint correction.
Returns (s, e) where s is the Euler-Maclaurin approximation
and e is the estimated error (taken to be the magnitude of
the first omitted term in the tail):
>>> from sympy.abc import k, a, b
>>> from sympy import Sum
>>> Sum(1/k, (k, 2, 5)).doit().evalf()
1.28333333333333
>>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin()
>>> s
-log(2) + 7/20 + log(5)
>>> from sympy import sstr
>>> print(sstr((s.evalf(), e.evalf()), full_prec=True))
(1.26629073187415, 0.0175000000000000)
The endpoints may be symbolic:
>>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin()
>>> s
-log(a) + log(b) + 1/(2*b) + 1/(2*a)
>>> e
Abs(1/(12*b**2) - 1/(12*a**2))
If the function is a polynomial of degree at most 2n+1, the
Euler-Maclaurin formula becomes exact (and e = 0 is returned):
>>> Sum(k, (k, 2, b)).euler_maclaurin()
(b**2/2 + b/2 - 1, 0)
>>> Sum(k, (k, 2, b)).doit()
b**2/2 + b/2 - 1
With a nonzero eps specified, the summation is ended
as soon as the remainder term is less than the epsilon.
"""
m = int(m)
n = int(n)
f = self.function
if len(self.limits) != 1:
raise ValueError("More than 1 limit")
i, a, b = self.limits[0]
if (a > b) == True:
if a - b == 1:
return S.Zero, S.Zero
a, b = b + 1, a - 1
f = -f
s = S.Zero
if m:
if b.is_Integer and a.is_Integer:
m = min(m, b - a + 1)
if not eps or f.is_polynomial(i):
for k in range(m):
s += f.subs(i, a + k)
else:
term = f.subs(i, a)
if term:
test = abs(term.evalf(3)) < eps
if test == True:
return s, abs(term)
elif not (test == False):
# a symbolic Relational class, can't go further
return term, S.Zero
s += term
for k in range(1, m):
term = f.subs(i, a + k)
if abs(term.evalf(3)) < eps and term != 0:
return s, abs(term)
s += term
if b - a + 1 == m:
return s, S.Zero
a += m
x = Dummy('x')
I = Integral(f.subs(i, x), (x, a, b))
if eval_integral:
I = I.doit()
s += I
def fpoint(expr):
if b is S.Infinity:
return expr.subs(i, a), 0
return expr.subs(i, a), expr.subs(i, b)
fa, fb = fpoint(f)
iterm = (fa + fb)/2
g = f.diff(i)
for k in range(1, n + 2):
ga, gb = fpoint(g)
term = bernoulli(2*k)/factorial(2*k)*(gb - ga)
if k > n:
break
if eps and term:
term_evalf = term.evalf(3)
if term_evalf is S.NaN:
return S.NaN, S.NaN
if abs(term_evalf) < eps:
break
s += term
g = g.diff(i, 2, simplify=False)
return s + iterm, abs(term)
def reverse_order(self, *indices):
"""
Reverse the order of a limit in a Sum.
Explanation
===========
``reverse_order(self, *indices)`` reverses some limits in the expression
``self`` which can be either a ``Sum`` or a ``Product``. The selectors in
the argument ``indices`` specify some indices whose limits get reversed.
These selectors are either variable names or numerical indices counted
starting from the inner-most limit tuple.
Examples
========
>>> from sympy import Sum
>>> from sympy.abc import x, y, a, b, c, d
>>> Sum(x, (x, 0, 3)).reverse_order(x)
Sum(-x, (x, 4, -1))
>>> Sum(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(x, y)
Sum(x*y, (x, 6, 0), (y, 7, -1))
>>> Sum(x, (x, a, b)).reverse_order(x)
Sum(-x, (x, b + 1, a - 1))
>>> Sum(x, (x, a, b)).reverse_order(0)
Sum(-x, (x, b + 1, a - 1))
While one should prefer variable names when specifying which limits
to reverse, the index counting notation comes in handy in case there
are several symbols with the same name.
>>> S = Sum(x**2, (x, a, b), (x, c, d))
>>> S
Sum(x**2, (x, a, b), (x, c, d))
>>> S0 = S.reverse_order(0)
>>> S0
Sum(-x**2, (x, b + 1, a - 1), (x, c, d))
>>> S1 = S0.reverse_order(1)
>>> S1
Sum(x**2, (x, b + 1, a - 1), (x, d + 1, c - 1))
Of course we can mix both notations:
>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
>>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x)
Sum(x*y, (x, b + 1, a - 1), (y, 6, 1))
See Also
========
sympy.concrete.expr_with_intlimits.ExprWithIntLimits.index, reorder_limit,
sympy.concrete.expr_with_intlimits.ExprWithIntLimits.reorder
References
==========
.. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM,
Volume 28 Issue 2, April 1981, Pages 305-350
http://dl.acm.org/citation.cfm?doid=322248.322255
"""
l_indices = list(indices)
for i, indx in enumerate(l_indices):
if not isinstance(indx, int):
l_indices[i] = self.index(indx)
e = 1
limits = []
for i, limit in enumerate(self.limits):
l = limit
if i in l_indices:
e = -e
l = (limit[0], limit[2] + 1, limit[1] - 1)
limits.append(l)
return Sum(e * self.function, *limits)
def summation(f, *symbols, **kwargs):
r"""
Compute the summation of f with respect to symbols.
Explanation
===========
The notation for symbols is similar to the notation used in Integral.
summation(f, (i, a, b)) computes the sum of f with respect to i from a to b,
i.e.,
::
b
____
\ `
summation(f, (i, a, b)) = ) f
/___,
i = a
If it cannot compute the sum, it returns an unevaluated Sum object.
Repeated sums can be computed by introducing additional symbols tuples::
Examples
========
>>> from sympy import summation, oo, symbols, log
>>> i, n, m = symbols('i n m', integer=True)
>>> summation(2*i - 1, (i, 1, n))
n**2
>>> summation(1/2**i, (i, 0, oo))
2
>>> summation(1/log(n)**n, (n, 2, oo))
Sum(log(n)**(-n), (n, 2, oo))
>>> summation(i, (i, 0, n), (n, 0, m))
m**3/6 + m**2/2 + m/3
>>> from sympy.abc import x
>>> from sympy import factorial
>>> summation(x**n/factorial(n), (n, 0, oo))
exp(x)
See Also
========
Sum
Product, sympy.concrete.products.product
"""
return Sum(f, *symbols, **kwargs).doit(deep=False)
def telescopic_direct(L, R, n, limits):
"""
Returns the direct summation of the terms of a telescopic sum
Explanation
===========
L is the term with lower index
R is the term with higher index
n difference between the indexes of L and R
Examples
========
>>> from sympy.concrete.summations import telescopic_direct
>>> from sympy.abc import k, a, b
>>> telescopic_direct(1/k, -1/(k+2), 2, (k, a, b))
-1/(b + 2) - 1/(b + 1) + 1/(a + 1) + 1/a
"""
(i, a, b) = limits
s = 0
for m in range(n):
s += L.subs(i, a + m) + R.subs(i, b - m)
return s
def telescopic(L, R, limits):
'''
Tries to perform the summation using the telescopic property.
Return None if not possible.
'''
(i, a, b) = limits
if L.is_Add or R.is_Add:
return None
# We want to solve(L.subs(i, i + m) + R, m)
# First we try a simple match since this does things that
# solve doesn't do, e.g. solve(f(k+m)-f(k), m) fails
k = Wild("k")
sol = (-R).match(L.subs(i, i + k))
s = None
if sol and k in sol:
s = sol[k]
if not (s.is_Integer and L.subs(i, i + s) == -R):
# sometimes match fail(f(x+2).match(-f(x+k))->{k: -2 - 2x}))
s = None
# But there are things that match doesn't do that solve
# can do, e.g. determine that 1/(x + m) = 1/(1 - x) when m = 1
if s is None:
m = Dummy('m')
try:
sol = solve(L.subs(i, i + m) + R, m) or []
except NotImplementedError:
return None
sol = [si for si in sol if si.is_Integer and
(L.subs(i, i + si) + R).expand().is_zero]
if len(sol) != 1:
return None
s = sol[0]
if s < 0:
return telescopic_direct(R, L, abs(s), (i, a, b))
elif s > 0:
return telescopic_direct(L, R, s, (i, a, b))
def eval_sum(f, limits):
(i, a, b) = limits
if f.is_zero:
return S.Zero
if i not in f.free_symbols:
return f*(b - a + 1)
if a == b:
return f.subs(i, a)
if isinstance(f, Piecewise):
if not any(i in arg.args[1].free_symbols for arg in f.args):
# Piecewise conditions do not depend on the dummy summation variable,
# therefore we can fold: Sum(Piecewise((e, c), ...), limits)
# --> Piecewise((Sum(e, limits), c), ...)
newargs = []
for arg in f.args:
newexpr = eval_sum(arg.expr, limits)
if newexpr is None:
return None
newargs.append((newexpr, arg.cond))
return f.func(*newargs)
if f.has(KroneckerDelta):
from .delta import deltasummation, _has_simple_delta
f = f.replace(
lambda x: isinstance(x, Sum),
lambda x: x.factor()
)
if _has_simple_delta(f, limits[0]):
return deltasummation(f, limits)
dif = b - a
definite = dif.is_Integer
# Doing it directly may be faster if there are very few terms.
if definite and (dif < 100):
return eval_sum_direct(f, (i, a, b))
if isinstance(f, Piecewise):
return None
# Try to do it symbolically. Even when the number of terms is known,
# this can save time when b-a is big.
# We should try to transform to partial fractions
value = eval_sum_symbolic(f.expand(), (i, a, b))
if value is not None:
return value
# Do it directly
if definite:
return eval_sum_direct(f, (i, a, b))
def eval_sum_direct(expr, limits):
"""
Evaluate expression directly, but perform some simple checks first
to possibly result in a smaller expression and faster execution.
"""
(i, a, b) = limits
dif = b - a
# Linearity
if expr.is_Mul:
# Try factor out everything not including i
without_i, with_i = expr.as_independent(i)
if without_i != 1:
s = eval_sum_direct(with_i, (i, a, b))
if s:
r = without_i*s
if r is not S.NaN:
return r
else:
# Try term by term
L, R = expr.as_two_terms()
if not L.has(i):
sR = eval_sum_direct(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_direct(L, (i, a, b))
if sL:
return sL*R
try:
expr = apart(expr, i) # see if it becomes an Add
except PolynomialError:
pass
if expr.is_Add:
# Try factor out everything not including i
without_i, with_i = expr.as_independent(i)
if without_i != 0:
s = eval_sum_direct(with_i, (i, a, b))
if s:
r = without_i*(dif + 1) + s
if r is not S.NaN:
return r
else:
# Try term by term
L, R = expr.as_two_terms()
lsum = eval_sum_direct(L, (i, a, b))
rsum = eval_sum_direct(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if r is not S.NaN:
return r
return Add(*[expr.subs(i, a + j) for j in range(dif + 1)])
def eval_sum_symbolic(f, limits):
f_orig = f
(i, a, b) = limits
if not f.has(i):
return f*(b - a + 1)
# Linearity
if f.is_Mul:
# Try factor out everything not including i
without_i, with_i = f.as_independent(i)
if without_i != 1:
s = eval_sum_symbolic(with_i, (i, a, b))
if s:
r = without_i*s
if r is not S.NaN:
return r
else:
# Try term by term
L, R = f.as_two_terms()
if not L.has(i):
sR = eval_sum_symbolic(R, (i, a, b))
if sR:
return L*sR
if not R.has(i):
sL = eval_sum_symbolic(L, (i, a, b))
if sL:
return sL*R
try:
f = apart(f, i) # see if it becomes an Add
except PolynomialError:
pass
if f.is_Add:
L, R = f.as_two_terms()
lrsum = telescopic(L, R, (i, a, b))
if lrsum:
return lrsum
# Try factor out everything not including i
without_i, with_i = f.as_independent(i)
if without_i != 0:
s = eval_sum_symbolic(with_i, (i, a, b))
if s:
r = without_i*(b - a + 1) + s
if r is not S.NaN:
return r
else:
# Try term by term
lsum = eval_sum_symbolic(L, (i, a, b))
rsum = eval_sum_symbolic(R, (i, a, b))
if None not in (lsum, rsum):
r = lsum + rsum
if r is not S.NaN:
return r
# Polynomial terms with Faulhaber's formula
n = Wild('n')
result = f.match(i**n)
if result is not None:
n = result[n]
if n.is_Integer:
if n >= 0:
if (b is S.Infinity and a is not S.NegativeInfinity) or \
(a is S.NegativeInfinity and b is not S.Infinity):
return S.Infinity
return ((bernoulli(n + 1, b + 1) - bernoulli(n + 1, a))/(n + 1)).expand()
elif a.is_Integer and a >= 1:
if n == -1:
return harmonic(b) - harmonic(a - 1)
else:
return harmonic(b, abs(n)) - harmonic(a - 1, abs(n))
if not (a.has(S.Infinity, S.NegativeInfinity) or
b.has(S.Infinity, S.NegativeInfinity)):
# Geometric terms
c1 = Wild('c1', exclude=[i])
c2 = Wild('c2', exclude=[i])
c3 = Wild('c3', exclude=[i])
wexp = Wild('wexp')
# Here we first attempt powsimp on f for easier matching with the
# exponential pattern, and attempt expansion on the exponent for easier
# matching with the linear pattern.
e = f.powsimp().match(c1 ** wexp)
if e is not None:
e_exp = e.pop(wexp).expand().match(c2*i + c3)
if e_exp is not None:
e.update(e_exp)
p = (c1**c3).subs(e)
q = (c1**c2).subs(e)
r = p*(q**a - q**(b + 1))/(1 - q)
l = p*(b - a + 1)
return Piecewise((l, Eq(q, S.One)), (r, True))
r = gosper_sum(f, (i, a, b))
if isinstance(r, (Mul,Add)):
non_limit = r.free_symbols - Tuple(*limits[1:]).free_symbols
den = denom(together(r))
den_sym = non_limit & den.free_symbols
args = []
for v in ordered(den_sym):
try:
s = solve(den, v)
m = Eq(v, s[0]) if s else S.false
if m != False:
args.append((Sum(f_orig.subs(*m.args), limits).doit(), m))
break
except NotImplementedError:
continue
args.append((r, True))
return Piecewise(*args)
if r not in (None, S.NaN):
return r
h = eval_sum_hyper(f_orig, (i, a, b))
if h is not None:
return h
r = eval_sum_residue(f_orig, (i, a, b))
if r is not None:
return r
factored = f_orig.factor()
if factored != f_orig:
return eval_sum_symbolic(factored, (i, a, b))
def _eval_sum_hyper(f, i, a):
""" Returns (res, cond). Sums from a to oo. """
if a != 0:
return _eval_sum_hyper(f.subs(i, i + a), i, 0)
if f.subs(i, 0) == 0:
if simplify(f.subs(i, Dummy('i', integer=True, positive=True))) == 0:
return S.Zero, True
return _eval_sum_hyper(f.subs(i, i + 1), i, 0)
hs = hypersimp(f, i)
if hs is None:
return None
if isinstance(hs, Float):
hs = nsimplify(hs)
numer, denom = fraction(factor(hs))
top, topl = numer.as_coeff_mul(i)
bot, botl = denom.as_coeff_mul(i)
ab = [top, bot]
factors = [topl, botl]
params = [[], []]
for k in range(2):
for fac in factors[k]:
mul = 1
if fac.is_Pow:
mul = fac.exp
fac = fac.base
if not mul.is_Integer:
return None
p = Poly(fac, i)
if p.degree() != 1:
return None
m, n = p.all_coeffs()
ab[k] *= m**mul
params[k] += [n/m]*mul
# Add "1" to numerator parameters, to account for implicit n! in
# hypergeometric series.
ap = params[0] + [1]
bq = params[1]
x = ab[0]/ab[1]
h = hyper(ap, bq, x)
f = combsimp(f)
return f.subs(i, 0)*hyperexpand(h), h.convergence_statement
def eval_sum_hyper(f, i_a_b):
i, a, b = i_a_b
if (b - a).is_Integer:
# We are never going to do better than doing the sum in the obvious way
return None
old_sum = Sum(f, (i, a, b))
if b != S.Infinity:
if a is S.NegativeInfinity:
res = _eval_sum_hyper(f.subs(i, -i), i, -b)
if res is not None:
return Piecewise(res, (old_sum, True))
else:
res1 = _eval_sum_hyper(f, i, a)
res2 = _eval_sum_hyper(f, i, b + 1)
if res1 is None or res2 is None:
return None
(res1, cond1), (res2, cond2) = res1, res2
cond = And(cond1, cond2)
if cond == False:
return None
return Piecewise((res1 - res2, cond), (old_sum, True))
if a is S.NegativeInfinity:
res1 = _eval_sum_hyper(f.subs(i, -i), i, 1)
res2 = _eval_sum_hyper(f, i, 0)
if res1 is None or res2 is None:
return None
res1, cond1 = res1
res2, cond2 = res2
cond = And(cond1, cond2)
if cond == False or cond.as_set() == S.EmptySet:
return None
return Piecewise((res1 + res2, cond), (old_sum, True))
# Now b == oo, a != -oo
res = _eval_sum_hyper(f, i, a)
if res is not None:
r, c = res
if c == False:
if r.is_number:
f = f.subs(i, Dummy('i', integer=True, positive=True) + a)
if f.is_positive or f.is_zero:
return S.Infinity
elif f.is_negative:
return S.NegativeInfinity
return None
return Piecewise(res, (old_sum, True))
def eval_sum_residue(f, i_a_b):
r"""Compute the infinite summation with residues
Notes
=====
If $f(n), g(n)$ are polynomials with $\deg(g(n)) - \deg(f(n)) \ge 2$,
some infinite summations can be computed by the following residue
evaluations.
.. math::
\sum_{n=-\infty, g(n) \ne 0}^{\infty} \frac{f(n)}{g(n)} =
-\pi \sum_{\alpha|g(\alpha)=0}
\text{Res}(\cot(\pi x) \frac{f(x)}{g(x)}, \alpha)
.. math::
\sum_{n=-\infty, g(n) \ne 0}^{\infty} (-1)^n \frac{f(n)}{g(n)} =
-\pi \sum_{\alpha|g(\alpha)=0}
\text{Res}(\csc(\pi x) \frac{f(x)}{g(x)}, \alpha)
Examples
========
>>> from sympy import Sum, oo, Symbol
>>> x = Symbol('x')
Doubly infinite series of rational functions.
>>> Sum(1 / (x**2 + 1), (x, -oo, oo)).doit()
pi/tanh(pi)
Doubly infinite alternating series of rational functions.
>>> Sum((-1)**x / (x**2 + 1), (x, -oo, oo)).doit()
pi/sinh(pi)
Infinite series of even rational functions.
>>> Sum(1 / (x**2 + 1), (x, 0, oo)).doit()
1/2 + pi/(2*tanh(pi))
Infinite series of alternating even rational functions.
>>> Sum((-1)**x / (x**2 + 1), (x, 0, oo)).doit()
pi/(2*sinh(pi)) + 1/2
This also have heuristics to transform arbitrarily shifted summand or
arbitrarily shifted summation range to the canonical problem the
formula can handle.
>>> Sum(1 / (x**2 + 2*x + 2), (x, -1, oo)).doit()
1/2 + pi/(2*tanh(pi))
>>> Sum(1 / (x**2 + 4*x + 5), (x, -2, oo)).doit()
1/2 + pi/(2*tanh(pi))
>>> Sum(1 / (x**2 + 1), (x, 1, oo)).doit()
-1/2 + pi/(2*tanh(pi))
>>> Sum(1 / (x**2 + 1), (x, 2, oo)).doit()
-1 + pi/(2*tanh(pi))
References
==========
.. [#] http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf
.. [#] Asmar N.H., Grafakos L. (2018) Residue Theory.
In: Complex Analysis with Applications.
Undergraduate Texts in Mathematics. Springer, Cham.
https://doi.org/10.1007/978-3-319-94063-2_5
"""
i, a, b = i_a_b
def is_even_function(numer, denom):
"""Test if the rational function is an even function"""
numer_even = all(i % 2 == 0 for (i,) in numer.monoms())
denom_even = all(i % 2 == 0 for (i,) in denom.monoms())
numer_odd = all(i % 2 == 1 for (i,) in numer.monoms())
denom_odd = all(i % 2 == 1 for (i,) in denom.monoms())
return (numer_even and denom_even) or (numer_odd and denom_odd)
def match_rational(f, i):
numer, denom = f.as_numer_denom()
try:
(numer, denom), opt = parallel_poly_from_expr((numer, denom), i)
except (PolificationFailed, PolynomialError):
return None
return numer, denom
def get_poles(denom):
roots = denom.sqf_part().all_roots()
roots = sift(roots, lambda x: x.is_integer)
if None in roots:
return None
int_roots, nonint_roots = roots[True], roots[False]
return int_roots, nonint_roots
def get_shift(denom):
n = denom.degree(i)
a = denom.coeff_monomial(i**n)
b = denom.coeff_monomial(i**(n-1))
shift = - b / a / n
return shift
def get_residue_factor(numer, denom, alternating):
if not alternating:
residue_factor = (numer.as_expr() / denom.as_expr()) * cot(S.Pi * i)
else:
residue_factor = (numer.as_expr() / denom.as_expr()) * csc(S.Pi * i)
return residue_factor
# We don't know how to deal with symbolic constants in summand
if f.free_symbols - set([i]):
return None
if not (a.is_Integer or a in (S.Infinity, S.NegativeInfinity)):
return None
if not (b.is_Integer or b in (S.Infinity, S.NegativeInfinity)):
return None
# Quick exit heuristic for the sums which doesn't have infinite range
if a != S.NegativeInfinity and b != S.Infinity:
return None
match = match_rational(f, i)
if match:
alternating = False
numer, denom = match
else:
match = match_rational(f / S.NegativeOne**i, i)
if match:
alternating = True
numer, denom = match
else:
return None
if denom.degree(i) - numer.degree(i) < 2:
return None
if (a, b) == (S.NegativeInfinity, S.Infinity):
poles = get_poles(denom)
if poles is None:
return None
int_roots, nonint_roots = poles
if int_roots:
return None
residue_factor = get_residue_factor(numer, denom, alternating)
residues = [residue(residue_factor, i, root) for root in nonint_roots]
return -S.Pi * sum(residues)
if not (a.is_finite and b is S.Infinity):
return None
if not is_even_function(numer, denom):
# Try shifting summation and check if the summand can be made
# and even function from the origin.
# Sum(f(n), (n, a, b)) => Sum(f(n + s), (n, a - s, b - s))
shift = get_shift(denom)
if not shift.is_Integer:
return None
if shift == 0:
return None
numer = numer.shift(shift)
denom = denom.shift(shift)
if not is_even_function(numer, denom):
return None
if alternating:
f = S.NegativeOne**i * (S.NegativeOne**shift * numer.as_expr() / denom.as_expr())
else:
f = numer.as_expr() / denom.as_expr()
return eval_sum_residue(f, (i, a-shift, b-shift))
poles = get_poles(denom)
if poles is None:
return None
int_roots, nonint_roots = poles
if int_roots:
int_roots = [int(root) for root in int_roots]
int_roots_max = max(int_roots)
int_roots_min = min(int_roots)
# Integer valued poles must be next to each other
# and also symmetric from origin (Because the function is even)
if not len(int_roots) == int_roots_max - int_roots_min + 1:
return None
# Check whether the summation indices contain poles
if a <= max(int_roots):
return None
residue_factor = get_residue_factor(numer, denom, alternating)
residues = [residue(residue_factor, i, root) for root in int_roots + nonint_roots]
full_sum = -S.Pi * sum(residues)
if not int_roots:
# Compute Sum(f, (i, 0, oo)) by adding a extraneous evaluation
# at the origin.
half_sum = (full_sum + f.xreplace({i: 0})) / 2
# Add and subtract extraneous evaluations
extraneous_neg = [f.xreplace({i: i0}) for i0 in range(int(a), 0)]
extraneous_pos = [f.xreplace({i: i0}) for i0 in range(0, int(a))]
result = half_sum + sum(extraneous_neg) - sum(extraneous_pos)
return result
# Compute Sum(f, (i, min(poles) + 1, oo))
half_sum = full_sum / 2
# Subtract extraneous evaluations
extraneous = [f.xreplace({i: i0}) for i0 in range(max(int_roots) + 1, int(a))]
result = half_sum - sum(extraneous)
return result
def _eval_matrix_sum(expression):
f = expression.function
for n, limit in enumerate(expression.limits):
i, a, b = limit
dif = b - a
if dif.is_Integer:
if (dif < 0) == True:
a, b = b + 1, a - 1
f = -f
newf = eval_sum_direct(f, (i, a, b))
if newf is not None:
return newf.doit()
def _dummy_with_inherited_properties_concrete(limits):
"""
Return a Dummy symbol that inherits as many assumptions as possible
from the provided symbol and limits.
If the symbol already has all True assumption shared by the limits
then return None.
"""
x, a, b = limits
l = [a, b]
assumptions_to_consider = ['extended_nonnegative', 'nonnegative',
'extended_nonpositive', 'nonpositive',
'extended_positive', 'positive',
'extended_negative', 'negative',
'integer', 'rational', 'finite',
'zero', 'real', 'extended_real']
assumptions_to_keep = {}
assumptions_to_add = {}
for assum in assumptions_to_consider:
assum_true = x._assumptions.get(assum, None)
if assum_true:
assumptions_to_keep[assum] = True
elif all(getattr(i, 'is_' + assum) for i in l):
assumptions_to_add[assum] = True
if assumptions_to_add:
assumptions_to_keep.update(assumptions_to_add)
return Dummy('d', **assumptions_to_keep)
|
53cd5f150eccb306c98185b501688f656e0e81a3ad9947bdbe43a078c85b50c8 | """
Limits
======
Implemented according to the PhD thesis
http://www.cybertester.com/data/gruntz.pdf, which contains very thorough
descriptions of the algorithm including many examples. We summarize here
the gist of it.
All functions are sorted according to how rapidly varying they are at
infinity using the following rules. Any two functions f and g can be
compared using the properties of L:
L=lim log|f(x)| / log|g(x)| (for x -> oo)
We define >, < ~ according to::
1. f > g .... L=+-oo
we say that:
- f is greater than any power of g
- f is more rapidly varying than g
- f goes to infinity/zero faster than g
2. f < g .... L=0
we say that:
- f is lower than any power of g
3. f ~ g .... L!=0, +-oo
we say that:
- both f and g are bounded from above and below by suitable integral
powers of the other
Examples
========
::
2 < x < exp(x) < exp(x**2) < exp(exp(x))
2 ~ 3 ~ -5
x ~ x**2 ~ x**3 ~ 1/x ~ x**m ~ -x
exp(x) ~ exp(-x) ~ exp(2x) ~ exp(x)**2 ~ exp(x+exp(-x))
f ~ 1/f
So we can divide all the functions into comparability classes (x and x^2
belong to one class, exp(x) and exp(-x) belong to some other class). In
principle, we could compare any two functions, but in our algorithm, we
do not compare anything below the class 2~3~-5 (for example log(x) is
below this), so we set 2~3~-5 as the lowest comparability class.
Given the function f, we find the list of most rapidly varying (mrv set)
subexpressions of it. This list belongs to the same comparability class.
Let's say it is {exp(x), exp(2x)}. Using the rule f ~ 1/f we find an
element "w" (either from the list or a new one) from the same
comparability class which goes to zero at infinity. In our example we
set w=exp(-x) (but we could also set w=exp(-2x) or w=exp(-3x) ...). We
rewrite the mrv set using w, in our case {1/w, 1/w^2}, and substitute it
into f. Then we expand f into a series in w::
f = c0*w^e0 + c1*w^e1 + ... + O(w^en), where e0<e1<...<en, c0!=0
but for x->oo, lim f = lim c0*w^e0, because all the other terms go to zero,
because w goes to zero faster than the ci and ei. So::
for e0>0, lim f = 0
for e0<0, lim f = +-oo (the sign depends on the sign of c0)
for e0=0, lim f = lim c0
We need to recursively compute limits at several places of the algorithm, but
as is shown in the PhD thesis, it always finishes.
Important functions from the implementation:
compare(a, b, x) compares "a" and "b" by computing the limit L.
mrv(e, x) returns list of most rapidly varying (mrv) subexpressions of "e"
rewrite(e, Omega, x, wsym) rewrites "e" in terms of w
leadterm(f, x) returns the lowest power term in the series of f
mrv_leadterm(e, x) returns the lead term (c0, e0) for e
limitinf(e, x) computes lim e (for x->oo)
limit(e, z, z0) computes any limit by converting it to the case x->oo
All the functions are really simple and straightforward except
rewrite(), which is the most difficult/complex part of the algorithm.
When the algorithm fails, the bugs are usually in the series expansion
(i.e. in SymPy) or in rewrite.
This code is almost exact rewrite of the Maple code inside the Gruntz
thesis.
Debugging
---------
Because the gruntz algorithm is highly recursive, it's difficult to
figure out what went wrong inside a debugger. Instead, turn on nice
debug prints by defining the environment variable SYMPY_DEBUG. For
example:
[user@localhost]: SYMPY_DEBUG=True ./bin/isympy
In [1]: limit(sin(x)/x, x, 0)
limitinf(_x*sin(1/_x), _x) = 1
+-mrv_leadterm(_x*sin(1/_x), _x) = (1, 0)
| +-mrv(_x*sin(1/_x), _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| | +-mrv(sin(1/_x), _x) = set([_x])
| | +-mrv(1/_x, _x) = set([_x])
| | +-mrv(_x, _x) = set([_x])
| +-mrv_leadterm(exp(_x)*sin(exp(-_x)), _x, set([exp(_x)])) = (1, 0)
| +-rewrite(exp(_x)*sin(exp(-_x)), set([exp(_x)]), _x, _w) = (1/_w*sin(_w), -_x)
| +-sign(_x, _x) = 1
| +-mrv_leadterm(1, _x) = (1, 0)
+-sign(0, _x) = 0
+-limitinf(1, _x) = 1
And check manually which line is wrong. Then go to the source code and
debug this function to figure out the exact problem.
"""
from functools import reduce
from sympy.core import Basic, S, Mul, PoleError
from sympy.core.cache import cacheit
from sympy.core.numbers import ilcm, I, oo
from sympy.core.symbol import Dummy, Wild
from sympy.core.traversal import bottom_up
from sympy.functions import log, exp, sign as _sign
from sympy.series.order import Order
from sympy.simplify import logcombine
from sympy.simplify.powsimp import powsimp, powdenest
from sympy.utilities.misc import debug_decorator as debug
from sympy.utilities.timeutils import timethis
timeit = timethis('gruntz')
def compare(a, b, x):
"""Returns "<" if a<b, "=" for a == b, ">" for a>b"""
# log(exp(...)) must always be simplified here for termination
la, lb = log(a), log(b)
if isinstance(a, Basic) and (isinstance(a, exp) or (a.is_Pow and a.base == S.Exp1)):
la = a.exp
if isinstance(b, Basic) and (isinstance(b, exp) or (b.is_Pow and b.base == S.Exp1)):
lb = b.exp
c = limitinf(la/lb, x)
if c == 0:
return "<"
elif c.is_infinite:
return ">"
else:
return "="
class SubsSet(dict):
"""
Stores (expr, dummy) pairs, and how to rewrite expr-s.
Explanation
===========
The gruntz algorithm needs to rewrite certain expressions in term of a new
variable w. We cannot use subs, because it is just too smart for us. For
example::
> Omega=[exp(exp(_p - exp(-_p))/(1 - 1/_p)), exp(exp(_p))]
> O2=[exp(-exp(_p) + exp(-exp(-_p))*exp(_p)/(1 - 1/_p))/_w, 1/_w]
> e = exp(exp(_p - exp(-_p))/(1 - 1/_p)) - exp(exp(_p))
> e.subs(Omega[0],O2[0]).subs(Omega[1],O2[1])
-1/w + exp(exp(p)*exp(-exp(-p))/(1 - 1/p))
is really not what we want!
So we do it the hard way and keep track of all the things we potentially
want to substitute by dummy variables. Consider the expression::
exp(x - exp(-x)) + exp(x) + x.
The mrv set is {exp(x), exp(-x), exp(x - exp(-x))}.
We introduce corresponding dummy variables d1, d2, d3 and rewrite::
d3 + d1 + x.
This class first of all keeps track of the mapping expr->variable, i.e.
will at this stage be a dictionary::
{exp(x): d1, exp(-x): d2, exp(x - exp(-x)): d3}.
[It turns out to be more convenient this way round.]
But sometimes expressions in the mrv set have other expressions from the
mrv set as subexpressions, and we need to keep track of that as well. In
this case, d3 is really exp(x - d2), so rewrites at this stage is::
{d3: exp(x-d2)}.
The function rewrite uses all this information to correctly rewrite our
expression in terms of w. In this case w can be chosen to be exp(-x),
i.e. d2. The correct rewriting then is::
exp(-w)/w + 1/w + x.
"""
def __init__(self):
self.rewrites = {}
def __repr__(self):
return super().__repr__() + ', ' + self.rewrites.__repr__()
def __getitem__(self, key):
if key not in self:
self[key] = Dummy()
return dict.__getitem__(self, key)
def do_subs(self, e):
"""Substitute the variables with expressions"""
for expr, var in self.items():
e = e.xreplace({var: expr})
return e
def meets(self, s2):
"""Tell whether or not self and s2 have non-empty intersection"""
return set(self.keys()).intersection(list(s2.keys())) != set()
def union(self, s2, exps=None):
"""Compute the union of self and s2, adjusting exps"""
res = self.copy()
tr = {}
for expr, var in s2.items():
if expr in self:
if exps:
exps = exps.xreplace({var: res[expr]})
tr[var] = res[expr]
else:
res[expr] = var
for var, rewr in s2.rewrites.items():
res.rewrites[var] = rewr.xreplace(tr)
return res, exps
def copy(self):
"""Create a shallow copy of SubsSet"""
r = SubsSet()
r.rewrites = self.rewrites.copy()
for expr, var in self.items():
r[expr] = var
return r
@debug
def mrv(e, x):
"""Returns a SubsSet of most rapidly varying (mrv) subexpressions of 'e',
and e rewritten in terms of these"""
e = powsimp(e, deep=True, combine='exp')
if not isinstance(e, Basic):
raise TypeError("e should be an instance of Basic")
if not e.has(x):
return SubsSet(), e
elif e == x:
s = SubsSet()
return s, s[x]
elif e.is_Mul or e.is_Add:
i, d = e.as_independent(x) # throw away x-independent terms
if d.func != e.func:
s, expr = mrv(d, x)
return s, e.func(i, expr)
a, b = d.as_two_terms()
s1, e1 = mrv(a, x)
s2, e2 = mrv(b, x)
return mrv_max1(s1, s2, e.func(i, e1, e2), x)
elif e.is_Pow and e.base != S.Exp1:
e1 = S.One
while e.is_Pow:
b1 = e.base
e1 *= e.exp
e = b1
if b1 == 1:
return SubsSet(), b1
if e1.has(x):
base_lim = limitinf(b1, x)
if base_lim is S.One:
return mrv(exp(e1 * (b1 - 1)), x)
return mrv(exp(e1 * log(b1)), x)
else:
s, expr = mrv(b1, x)
return s, expr**e1
elif isinstance(e, log):
s, expr = mrv(e.args[0], x)
return s, log(expr)
elif isinstance(e, exp) or (e.is_Pow and e.base == S.Exp1):
# We know from the theory of this algorithm that exp(log(...)) may always
# be simplified here, and doing so is vital for termination.
if isinstance(e.exp, log):
return mrv(e.exp.args[0], x)
# if a product has an infinite factor the result will be
# infinite if there is no zero, otherwise NaN; here, we
# consider the result infinite if any factor is infinite
li = limitinf(e.exp, x)
if any(_.is_infinite for _ in Mul.make_args(li)):
s1 = SubsSet()
e1 = s1[e]
s2, e2 = mrv(e.exp, x)
su = s1.union(s2)[0]
su.rewrites[e1] = exp(e2)
return mrv_max3(s1, e1, s2, exp(e2), su, e1, x)
else:
s, expr = mrv(e.exp, x)
return s, exp(expr)
elif e.is_Function:
l = [mrv(a, x) for a in e.args]
l2 = [s for (s, _) in l if s != SubsSet()]
if len(l2) != 1:
# e.g. something like BesselJ(x, x)
raise NotImplementedError("MRV set computation for functions in"
" several variables not implemented.")
s, ss = l2[0], SubsSet()
args = [ss.do_subs(x[1]) for x in l]
return s, e.func(*args)
elif e.is_Derivative:
raise NotImplementedError("MRV set computation for derviatives"
" not implemented yet.")
raise NotImplementedError(
"Don't know how to calculate the mrv of '%s'" % e)
def mrv_max3(f, expsf, g, expsg, union, expsboth, x):
"""
Computes the maximum of two sets of expressions f and g, which
are in the same comparability class, i.e. max() compares (two elements of)
f and g and returns either (f, expsf) [if f is larger], (g, expsg)
[if g is larger] or (union, expsboth) [if f, g are of the same class].
"""
if not isinstance(f, SubsSet):
raise TypeError("f should be an instance of SubsSet")
if not isinstance(g, SubsSet):
raise TypeError("g should be an instance of SubsSet")
if f == SubsSet():
return g, expsg
elif g == SubsSet():
return f, expsf
elif f.meets(g):
return union, expsboth
c = compare(list(f.keys())[0], list(g.keys())[0], x)
if c == ">":
return f, expsf
elif c == "<":
return g, expsg
else:
if c != "=":
raise ValueError("c should be =")
return union, expsboth
def mrv_max1(f, g, exps, x):
"""Computes the maximum of two sets of expressions f and g, which
are in the same comparability class, i.e. mrv_max1() compares (two elements of)
f and g and returns the set, which is in the higher comparability class
of the union of both, if they have the same order of variation.
Also returns exps, with the appropriate substitutions made.
"""
u, b = f.union(g, exps)
return mrv_max3(f, g.do_subs(exps), g, f.do_subs(exps),
u, b, x)
@debug
@cacheit
@timeit
def sign(e, x):
"""
Returns a sign of an expression e(x) for x->oo.
::
e > 0 for x sufficiently large ... 1
e == 0 for x sufficiently large ... 0
e < 0 for x sufficiently large ... -1
The result of this function is currently undefined if e changes sign
arbitrarily often for arbitrarily large x (e.g. sin(x)).
Note that this returns zero only if e is *constantly* zero
for x sufficiently large. [If e is constant, of course, this is just
the same thing as the sign of e.]
"""
if not isinstance(e, Basic):
raise TypeError("e should be an instance of Basic")
if e.is_positive:
return 1
elif e.is_negative:
return -1
elif e.is_zero:
return 0
elif not e.has(x):
e = logcombine(e)
return _sign(e)
elif e == x:
return 1
elif e.is_Mul:
a, b = e.as_two_terms()
sa = sign(a, x)
if not sa:
return 0
return sa * sign(b, x)
elif isinstance(e, exp):
return 1
elif e.is_Pow:
if e.base == S.Exp1:
return 1
s = sign(e.base, x)
if s == 1:
return 1
if e.exp.is_Integer:
return s**e.exp
elif isinstance(e, log):
return sign(e.args[0] - 1, x)
# if all else fails, do it the hard way
c0, e0 = mrv_leadterm(e, x)
return sign(c0, x)
@debug
@timeit
@cacheit
def limitinf(e, x, leadsimp=False):
"""Limit e(x) for x-> oo.
Explanation
===========
If ``leadsimp`` is True, an attempt is made to simplify the leading
term of the series expansion of ``e``. That may succeed even if
``e`` cannot be simplified.
"""
# rewrite e in terms of tractable functions only
if not e.has(x):
return e # e is a constant
if e.has(Order):
e = e.expand().removeO()
if not x.is_positive or x.is_integer:
# We make sure that x.is_positive is True and x.is_integer is None
# so we get all the correct mathematical behavior from the expression.
# We need a fresh variable.
p = Dummy('p', positive=True)
e = e.subs(x, p)
x = p
e = e.rewrite('tractable', deep=True, limitvar=x)
e = powdenest(e)
c0, e0 = mrv_leadterm(e, x)
sig = sign(e0, x)
if sig == 1:
return S.Zero # e0>0: lim f = 0
elif sig == -1: # e0<0: lim f = +-oo (the sign depends on the sign of c0)
if c0.match(I*Wild("a", exclude=[I])):
return c0*oo
s = sign(c0, x)
# the leading term shouldn't be 0:
if s == 0:
raise ValueError("Leading term should not be 0")
return s*oo
elif sig == 0:
if leadsimp:
c0 = c0.simplify()
return limitinf(c0, x, leadsimp) # e0=0: lim f = lim c0
else:
raise ValueError("{} could not be evaluated".format(sig))
def moveup2(s, x):
r = SubsSet()
for expr, var in s.items():
r[expr.xreplace({x: exp(x)})] = var
for var, expr in s.rewrites.items():
r.rewrites[var] = s.rewrites[var].xreplace({x: exp(x)})
return r
def moveup(l, x):
return [e.xreplace({x: exp(x)}) for e in l]
@debug
@timeit
def calculate_series(e, x, logx=None):
""" Calculates at least one term of the series of ``e`` in ``x``.
This is a place that fails most often, so it is in its own function.
"""
for t in e.lseries(x, logx=logx):
# bottom_up function is required for a specific case - when e is
# -exp(p/(p + 1)) + exp(-p**2/(p + 1) + p)
t = bottom_up(t, lambda w:
getattr(w, 'normal', lambda: w)())
# And the expression
# `(-sin(1/x) + sin((x + exp(x))*exp(-x)/x))*exp(x)`
# from the first test of test_gruntz_eval_special needs to
# be expanded. But other forms need to be have at least
# factor_terms applied. `factor` accomplishes both and is
# faster than using `factor_terms` for the gruntz suite. It
# does not appear that use of `cancel` is necessary.
# t = cancel(t, expand=False)
t = t.factor()
if t.has(exp) and t.has(log):
t = powdenest(t)
if not t.is_zero:
break
return t
@debug
@timeit
@cacheit
def mrv_leadterm(e, x):
"""Returns (c0, e0) for e."""
Omega = SubsSet()
if not e.has(x):
return (e, S.Zero)
if Omega == SubsSet():
Omega, exps = mrv(e, x)
if not Omega:
# e really does not depend on x after simplification
return exps, S.Zero
if x in Omega:
# move the whole omega up (exponentiate each term):
Omega_up = moveup2(Omega, x)
exps_up = moveup([exps], x)[0]
# NOTE: there is no need to move this down!
Omega = Omega_up
exps = exps_up
#
# The positive dummy, w, is used here so log(w*2) etc. will expand;
# a unique dummy is needed in this algorithm
#
# For limits of complex functions, the algorithm would have to be
# improved, or just find limits of Re and Im components separately.
#
w = Dummy("w", positive=True)
f, logw = rewrite(exps, Omega, x, w)
series = calculate_series(f, w, logx=logw)
try:
lt = series.leadterm(w, logx=logw)
except (ValueError, PoleError):
lt = f.as_coeff_exponent(w)
# as_coeff_exponent won't always split in required form. It may simply
# return (f, 0) when a better form may be obtained. Example (-x)**(-pi)
# can be written as (-1**(-pi), -pi) which as_coeff_exponent does not return
if lt[0].has(w):
base = f.as_base_exp()[0].as_coeff_exponent(w)
ex = f.as_base_exp()[1]
lt = (base[0]**ex, base[1]*ex)
return (lt[0].subs(log(w), logw), lt[1])
def build_expression_tree(Omega, rewrites):
r""" Helper function for rewrite.
We need to sort Omega (mrv set) so that we replace an expression before
we replace any expression in terms of which it has to be rewritten::
e1 ---> e2 ---> e3
\
-> e4
Here we can do e1, e2, e3, e4 or e1, e2, e4, e3.
To do this we assemble the nodes into a tree, and sort them by height.
This function builds the tree, rewrites then sorts the nodes.
"""
class Node:
def __init__(self):
self.before = []
self.expr = None
self.var = None
def ht(self):
return reduce(lambda x, y: x + y,
[x.ht() for x in self.before], 1)
nodes = {}
for expr, v in Omega:
n = Node()
n.var = v
n.expr = expr
nodes[v] = n
for _, v in Omega:
if v in rewrites:
n = nodes[v]
r = rewrites[v]
for _, v2 in Omega:
if r.has(v2):
n.before.append(nodes[v2])
return nodes
@debug
@timeit
def rewrite(e, Omega, x, wsym):
"""e(x) ... the function
Omega ... the mrv set
wsym ... the symbol which is going to be used for w
Returns the rewritten e in terms of w and log(w). See test_rewrite1()
for examples and correct results.
"""
if not isinstance(Omega, SubsSet):
raise TypeError("Omega should be an instance of SubsSet")
if len(Omega) == 0:
raise ValueError("Length cannot be 0")
# all items in Omega must be exponentials
for t in Omega.keys():
if not isinstance(t, exp):
raise ValueError("Value should be exp")
rewrites = Omega.rewrites
Omega = list(Omega.items())
nodes = build_expression_tree(Omega, rewrites)
Omega.sort(key=lambda x: nodes[x[1]].ht(), reverse=True)
# make sure we know the sign of each exp() term; after the loop,
# g is going to be the "w" - the simplest one in the mrv set
for g, _ in Omega:
sig = sign(g.exp, x)
if sig != 1 and sig != -1:
raise NotImplementedError('Result depends on the sign of %s' % sig)
if sig == 1:
wsym = 1/wsym # if g goes to oo, substitute 1/w
# O2 is a list, which results by rewriting each item in Omega using "w"
O2 = []
denominators = []
for f, var in Omega:
c = limitinf(f.exp/g.exp, x)
if c.is_Rational:
denominators.append(c.q)
arg = f.exp
if var in rewrites:
if not isinstance(rewrites[var], exp):
raise ValueError("Value should be exp")
arg = rewrites[var].args[0]
O2.append((var, exp((arg - c*g.exp).expand())*wsym**c))
# Remember that Omega contains subexpressions of "e". So now we find
# them in "e" and substitute them for our rewriting, stored in O2
# the following powsimp is necessary to automatically combine exponentials,
# so that the .xreplace() below succeeds:
# TODO this should not be necessary
f = powsimp(e, deep=True, combine='exp')
for a, b in O2:
f = f.xreplace({a: b})
for _, var in Omega:
assert not f.has(var)
# finally compute the logarithm of w (logw).
logw = g.exp
if sig == 1:
logw = -logw # log(w)->log(1/w)=-log(w)
# Some parts of SymPy have difficulty computing series expansions with
# non-integral exponents. The following heuristic improves the situation:
exponent = reduce(ilcm, denominators, 1)
f = f.subs({wsym: wsym**exponent})
logw /= exponent
return f, logw
def gruntz(e, z, z0, dir="+"):
"""
Compute the limit of e(z) at the point z0 using the Gruntz algorithm.
Explanation
===========
``z0`` can be any expression, including oo and -oo.
For ``dir="+"`` (default) it calculates the limit from the right
(z->z0+) and for ``dir="-"`` the limit from the left (z->z0-). For infinite z0
(oo or -oo), the dir argument doesn't matter.
This algorithm is fully described in the module docstring in the gruntz.py
file. It relies heavily on the series expansion. Most frequently, gruntz()
is only used if the faster limit() function (which uses heuristics) fails.
"""
if not z.is_symbol:
raise NotImplementedError("Second argument must be a Symbol")
# convert all limits to the limit z->oo; sign of z is handled in limitinf
r = None
if z0 == oo:
e0 = e
elif z0 == -oo:
e0 = e.subs(z, -z)
else:
if str(dir) == "-":
e0 = e.subs(z, z0 - 1/z)
elif str(dir) == "+":
e0 = e.subs(z, z0 + 1/z)
else:
raise NotImplementedError("dir must be '+' or '-'")
try:
r = limitinf(e0, z)
except ValueError:
r = limitinf(e0, z, leadsimp=True)
# This is a bit of a heuristic for nice results... we always rewrite
# tractable functions in terms of familiar intractable ones.
# It might be nicer to rewrite the exactly to what they were initially,
# but that would take some work to implement.
return r.rewrite('intractable', deep=True)
|
192244e760b1bf4b574a00229a8bff52d5c32fdff3083839c052f9691af52259 | from sympy.calculus.accumulationbounds import AccumBounds
from sympy.core import S, Symbol, Add, sympify, Expr, PoleError, Mul
from sympy.core.exprtools import factor_terms
from sympy.core.numbers import Float
from sympy.functions.combinatorial.factorials import factorial
from sympy.functions.elementary.complexes import (Abs, sign)
from sympy.functions.elementary.exponential import (exp, log)
from sympy.functions.special.gamma_functions import gamma
from sympy.polys import PolynomialError, factor
from sympy.series.order import Order
from sympy.simplify.powsimp import powsimp
from sympy.simplify.ratsimp import ratsimp
from sympy.simplify.simplify import nsimplify, together
from .gruntz import gruntz
def limit(e, z, z0, dir="+"):
"""Computes the limit of ``e(z)`` at the point ``z0``.
Parameters
==========
e : expression, the limit of which is to be taken
z : symbol representing the variable in the limit.
Other symbols are treated as constants. Multivariate limits
are not supported.
z0 : the value toward which ``z`` tends. Can be any expression,
including ``oo`` and ``-oo``.
dir : string, optional (default: "+")
The limit is bi-directional if ``dir="+-"``, from the right
(z->z0+) if ``dir="+"``, and from the left (z->z0-) if
``dir="-"``. For infinite ``z0`` (``oo`` or ``-oo``), the ``dir``
argument is determined from the direction of the infinity
(i.e., ``dir="-"`` for ``oo``).
Examples
========
>>> from sympy import limit, sin, oo
>>> from sympy.abc import x
>>> limit(sin(x)/x, x, 0)
1
>>> limit(1/x, x, 0) # default dir='+'
oo
>>> limit(1/x, x, 0, dir="-")
-oo
>>> limit(1/x, x, 0, dir='+-')
zoo
>>> limit(1/x, x, oo)
0
Notes
=====
First we try some heuristics for easy and frequent cases like "x", "1/x",
"x**2" and similar, so that it's fast. For all other cases, we use the
Gruntz algorithm (see the gruntz() function).
See Also
========
limit_seq : returns the limit of a sequence.
"""
return Limit(e, z, z0, dir).doit(deep=False)
def heuristics(e, z, z0, dir):
"""Computes the limit of an expression term-wise.
Parameters are the same as for the ``limit`` function.
Works with the arguments of expression ``e`` one by one, computing
the limit of each and then combining the results. This approach
works only for simple limits, but it is fast.
"""
rv = None
if abs(z0) is S.Infinity:
rv = limit(e.subs(z, 1/z), z, S.Zero, "+" if z0 is S.Infinity else "-")
if isinstance(rv, Limit):
return
elif e.is_Mul or e.is_Add or e.is_Pow or e.is_Function:
r = []
for a in e.args:
l = limit(a, z, z0, dir)
if l.has(S.Infinity) and l.is_finite is None:
if isinstance(e, Add):
m = factor_terms(e)
if not isinstance(m, Mul): # try together
m = together(m)
if not isinstance(m, Mul): # try factor if the previous methods failed
m = factor(e)
if isinstance(m, Mul):
return heuristics(m, z, z0, dir)
return
return
elif isinstance(l, Limit):
return
elif l is S.NaN:
return
else:
r.append(l)
if r:
rv = e.func(*r)
if rv is S.NaN and e.is_Mul and any(isinstance(rr, AccumBounds) for rr in r):
r2 = []
e2 = []
for ii in range(len(r)):
if isinstance(r[ii], AccumBounds):
r2.append(r[ii])
else:
e2.append(e.args[ii])
if len(e2) > 0:
e3 = Mul(*e2).simplify()
l = limit(e3, z, z0, dir)
rv = l * Mul(*r2)
if rv is S.NaN:
try:
rat_e = ratsimp(e)
except PolynomialError:
return
if rat_e is S.NaN or rat_e == e:
return
return limit(rat_e, z, z0, dir)
return rv
class Limit(Expr):
"""Represents an unevaluated limit.
Examples
========
>>> from sympy import Limit, sin
>>> from sympy.abc import x
>>> Limit(sin(x)/x, x, 0)
Limit(sin(x)/x, x, 0)
>>> Limit(1/x, x, 0, dir="-")
Limit(1/x, x, 0, dir='-')
"""
def __new__(cls, e, z, z0, dir="+"):
e = sympify(e)
z = sympify(z)
z0 = sympify(z0)
if z0 is S.Infinity:
dir = "-"
elif z0 is S.NegativeInfinity:
dir = "+"
if(z0.has(z)):
raise NotImplementedError("Limits approaching a variable point are"
" not supported (%s -> %s)" % (z, z0))
if isinstance(dir, str):
dir = Symbol(dir)
elif not isinstance(dir, Symbol):
raise TypeError("direction must be of type basestring or "
"Symbol, not %s" % type(dir))
if str(dir) not in ('+', '-', '+-'):
raise ValueError("direction must be one of '+', '-' "
"or '+-', not %s" % dir)
obj = Expr.__new__(cls)
obj._args = (e, z, z0, dir)
return obj
@property
def free_symbols(self):
e = self.args[0]
isyms = e.free_symbols
isyms.difference_update(self.args[1].free_symbols)
isyms.update(self.args[2].free_symbols)
return isyms
def pow_heuristics(self, e):
_, z, z0, _ = self.args
b1, e1 = e.base, e.exp
if not b1.has(z):
res = limit(e1*log(b1), z, z0)
return exp(res)
ex_lim = limit(e1, z, z0)
base_lim = limit(b1, z, z0)
if base_lim is S.One:
if ex_lim in (S.Infinity, S.NegativeInfinity):
res = limit(e1*(b1 - 1), z, z0)
return exp(res)
if base_lim is S.NegativeInfinity and ex_lim is S.Infinity:
return S.ComplexInfinity
def doit(self, **hints):
"""Evaluates the limit.
Parameters
==========
deep : bool, optional (default: True)
Invoke the ``doit`` method of the expressions involved before
taking the limit.
hints : optional keyword arguments
To be passed to ``doit`` methods; only used if deep is True.
"""
e, z, z0, dir = self.args
if z0 is S.ComplexInfinity:
raise NotImplementedError("Limits at complex "
"infinity are not implemented")
if hints.get('deep', True):
e = e.doit(**hints)
z = z.doit(**hints)
z0 = z0.doit(**hints)
if e == z:
return z0
if not e.has(z):
return e
if z0 is S.NaN:
return S.NaN
if e.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity, S.NaN):
return self
if e.is_Order:
return Order(limit(e.expr, z, z0), *e.args[1:])
cdir = 0
if str(dir) == "+":
cdir = 1
elif str(dir) == "-":
cdir = -1
def set_signs(expr):
if not expr.args:
return expr
newargs = tuple(set_signs(arg) for arg in expr.args)
if newargs != expr.args:
expr = expr.func(*newargs)
abs_flag = isinstance(expr, Abs)
sign_flag = isinstance(expr, sign)
if abs_flag or sign_flag:
sig = limit(expr.args[0], z, z0, dir)
if sig.is_zero:
sig = limit(1/expr.args[0], z, z0, dir)
if sig.is_extended_real:
if (sig < 0) == True:
return -expr.args[0] if abs_flag else S.NegativeOne
elif (sig > 0) == True:
return expr.args[0] if abs_flag else S.One
return expr
if e.has(Float):
# Convert floats like 0.5 to exact SymPy numbers like S.Half, to
# prevent rounding errors which can lead to unexpected execution
# of conditional blocks that work on comparisons
# Also see comments in https://github.com/sympy/sympy/issues/19453
e = nsimplify(e)
e = set_signs(e)
if e.is_meromorphic(z, z0):
if abs(z0) is S.Infinity:
newe = e.subs(z, 1/z)
# cdir changes sign as oo- should become 0+
cdir = -cdir
else:
newe = e.subs(z, z + z0)
try:
coeff, ex = newe.leadterm(z, cdir=cdir)
except ValueError:
pass
else:
if ex > 0:
return S.Zero
elif ex == 0:
return coeff
if cdir == 1 or not(int(ex) & 1):
return S.Infinity*sign(coeff)
elif cdir == -1:
return S.NegativeInfinity*sign(coeff)
else:
return S.ComplexInfinity
if abs(z0) is S.Infinity:
if e.is_Mul:
e = factor_terms(e)
newe = e.subs(z, 1/z)
# cdir changes sign as oo- should become 0+
cdir = -cdir
else:
newe = e.subs(z, z + z0)
try:
coeff, ex = newe.leadterm(z, cdir=cdir)
except (ValueError, NotImplementedError, PoleError):
# The NotImplementedError catching is for custom functions
e = powsimp(e)
if e.is_Pow:
r = self.pow_heuristics(e)
if r is not None:
return r
else:
if coeff.has(S.Infinity, S.NegativeInfinity, S.ComplexInfinity):
return self
if not coeff.has(z):
if ex.is_positive:
return S.Zero
elif ex == 0:
return coeff
elif ex.is_negative:
if ex.is_integer:
if cdir == 1 or ex.is_even:
return S.Infinity*sign(coeff)
elif cdir == -1:
return S.NegativeInfinity*sign(coeff)
else:
return S.ComplexInfinity
else:
if cdir == 1:
return S.Infinity*sign(coeff)
elif cdir == -1:
return S.Infinity*sign(coeff)*S.NegativeOne**ex
else:
return S.ComplexInfinity
# gruntz fails on factorials but works with the gamma function
# If no factorial term is present, e should remain unchanged.
# factorial is defined to be zero for negative inputs (which
# differs from gamma) so only rewrite for positive z0.
if z0.is_extended_positive:
e = e.rewrite(factorial, gamma)
l = None
try:
if str(dir) == '+-':
r = gruntz(e, z, z0, '+')
l = gruntz(e, z, z0, '-')
if l != r:
raise ValueError("The limit does not exist since "
"left hand limit = %s and right hand limit = %s"
% (l, r))
else:
r = gruntz(e, z, z0, dir)
if r is S.NaN or l is S.NaN:
raise PoleError()
except (PoleError, ValueError):
if l is not None:
raise
r = heuristics(e, z, z0, dir)
if r is None:
return self
return r
|
58cd4b87745785264cd6041a26b8569d4b5812026e9cbba3204c35826b55858c | """Limits of sequences"""
from sympy.calculus.accumulationbounds import AccumulationBounds
from sympy.core.add import Add
from sympy.core.function import PoleError
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.numbers import fibonacci
from sympy.functions.combinatorial.factorials import factorial, subfactorial
from sympy.functions.special.gamma_functions import gamma
from sympy.functions.elementary.complexes import Abs
from sympy.functions.elementary.miscellaneous import Max, Min
from sympy.functions.elementary.trigonometric import cos, sin
from sympy.series.limits import Limit
def difference_delta(expr, n=None, step=1):
"""Difference Operator.
Explanation
===========
Discrete analog of differential operator. Given a sequence x[n],
returns the sequence x[n + step] - x[n].
Examples
========
>>> from sympy import difference_delta as dd
>>> from sympy.abc import n
>>> dd(n*(n + 1), n)
2*n + 2
>>> dd(n*(n + 1), n, 2)
4*n + 6
References
==========
.. [1] https://reference.wolfram.com/language/ref/DifferenceDelta.html
"""
expr = sympify(expr)
if n is None:
f = expr.free_symbols
if len(f) == 1:
n = f.pop()
elif len(f) == 0:
return S.Zero
else:
raise ValueError("Since there is more than one variable in the"
" expression, a variable must be supplied to"
" take the difference of %s" % expr)
step = sympify(step)
if step.is_number is False or step.is_finite is False:
raise ValueError("Step should be a finite number.")
if hasattr(expr, '_eval_difference_delta'):
result = expr._eval_difference_delta(n, step)
if result:
return result
return expr.subs(n, n + step) - expr
def dominant(expr, n):
"""Finds the dominant term in a sum, that is a term that dominates
every other term.
Explanation
===========
If limit(a/b, n, oo) is oo then a dominates b.
If limit(a/b, n, oo) is 0 then b dominates a.
Otherwise, a and b are comparable.
If there is no unique dominant term, then returns ``None``.
Examples
========
>>> from sympy import Sum
>>> from sympy.series.limitseq import dominant
>>> from sympy.abc import n, k
>>> dominant(5*n**3 + 4*n**2 + n + 1, n)
5*n**3
>>> dominant(2**n + Sum(k, (k, 0, n)), n)
2**n
See Also
========
sympy.series.limitseq.dominant
"""
terms = Add.make_args(expr.expand(func=True))
term0 = terms[-1]
comp = [term0] # comparable terms
for t in terms[:-1]:
r = term0/t
e = r.gammasimp()
if e == r:
e = r.factor()
l = limit_seq(e, n)
if l is None:
return None
elif l.is_zero:
term0 = t
comp = [term0]
elif l not in [S.Infinity, S.NegativeInfinity]:
comp.append(t)
if len(comp) > 1:
return None
return term0
def _limit_inf(expr, n):
try:
return Limit(expr, n, S.Infinity).doit(deep=False)
except (NotImplementedError, PoleError):
return None
def _limit_seq(expr, n, trials):
from sympy.concrete.summations import Sum
for i in range(trials):
if not expr.has(Sum):
result = _limit_inf(expr, n)
if result is not None:
return result
num, den = expr.as_numer_denom()
if not den.has(n) or not num.has(n):
result = _limit_inf(expr.doit(), n)
if result is not None:
return result
return None
num, den = (difference_delta(t.expand(), n) for t in [num, den])
expr = (num / den).gammasimp()
if not expr.has(Sum):
result = _limit_inf(expr, n)
if result is not None:
return result
num, den = expr.as_numer_denom()
num = dominant(num, n)
if num is None:
return None
den = dominant(den, n)
if den is None:
return None
expr = (num / den).gammasimp()
def limit_seq(expr, n=None, trials=5):
"""Finds the limit of a sequence as index ``n`` tends to infinity.
Parameters
==========
expr : Expr
SymPy expression for the ``n-th`` term of the sequence
n : Symbol, optional
The index of the sequence, an integer that tends to positive
infinity. If None, inferred from the expression unless it has
multiple symbols.
trials: int, optional
The algorithm is highly recursive. ``trials`` is a safeguard from
infinite recursion in case the limit is not easily computed by the
algorithm. Try increasing ``trials`` if the algorithm returns ``None``.
Admissible Terms
================
The algorithm is designed for sequences built from rational functions,
indefinite sums, and indefinite products over an indeterminate n. Terms of
alternating sign are also allowed, but more complex oscillatory behavior is
not supported.
Examples
========
>>> from sympy import limit_seq, Sum, binomial
>>> from sympy.abc import n, k, m
>>> limit_seq((5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5), n)
5/3
>>> limit_seq(binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)), n)
3/4
>>> limit_seq(Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n), n)
4
See Also
========
sympy.series.limitseq.dominant
References
==========
.. [1] Computing Limits of Sequences - Manuel Kauers
"""
from sympy.concrete.summations import Sum
if n is None:
free = expr.free_symbols
if len(free) == 1:
n = free.pop()
elif not free:
return expr
else:
raise ValueError("Expression has more than one variable. "
"Please specify a variable.")
elif n not in expr.free_symbols:
return expr
expr = expr.rewrite(fibonacci, S.GoldenRatio)
expr = expr.rewrite(factorial, subfactorial, gamma)
n_ = Dummy("n", integer=True, positive=True)
n1 = Dummy("n", odd=True, positive=True)
n2 = Dummy("n", even=True, positive=True)
# If there is a negative term raised to a power involving n, or a
# trigonometric function, then consider even and odd n separately.
powers = (p.as_base_exp() for p in expr.atoms(Pow))
if (any(b.is_negative and e.has(n) for b, e in powers) or
expr.has(cos, sin)):
L1 = _limit_seq(expr.xreplace({n: n1}), n1, trials)
if L1 is not None:
L2 = _limit_seq(expr.xreplace({n: n2}), n2, trials)
if L1 != L2:
if L1.is_comparable and L2.is_comparable:
return AccumulationBounds(Min(L1, L2), Max(L1, L2))
else:
return None
else:
L1 = _limit_seq(expr.xreplace({n: n_}), n_, trials)
if L1 is not None:
return L1
else:
if expr.is_Add:
limits = [limit_seq(term, n, trials) for term in expr.args]
if any(result is None for result in limits):
return None
else:
return Add(*limits)
# Maybe the absolute value is easier to deal with (though not if
# it has a Sum). If it tends to 0, the limit is 0.
elif not expr.has(Sum):
lim = _limit_seq(Abs(expr.xreplace({n: n_})), n_, trials)
if lim is not None and lim.is_zero:
return S.Zero
|
4dd03d5a9e66f258cb655611ebf10486d1c7adf9a2cfbae054a8efbd84f3d854 | """
Expand Hypergeometric (and Meijer G) functions into named
special functions.
The algorithm for doing this uses a collection of lookup tables of
hypergeometric functions, and various of their properties, to expand
many hypergeometric functions in terms of special functions.
It is based on the following paper:
Kelly B. Roach. Meijer G Function Representations.
In: Proceedings of the 1997 International Symposium on Symbolic and
Algebraic Computation, pages 205-211, New York, 1997. ACM.
It is described in great(er) detail in the Sphinx documentation.
"""
# SUMMARY OF EXTENSIONS FOR MEIJER G FUNCTIONS
#
# o z**rho G(ap, bq; z) = G(ap + rho, bq + rho; z)
#
# o denote z*d/dz by D
#
# o It is helpful to keep in mind that ap and bq play essentially symmetric
# roles: G(1/z) has slightly altered parameters, with ap and bq interchanged.
#
# o There are four shift operators:
# A_J = b_J - D, J = 1, ..., n
# B_J = 1 - a_j + D, J = 1, ..., m
# C_J = -b_J + D, J = m+1, ..., q
# D_J = a_J - 1 - D, J = n+1, ..., p
#
# A_J, C_J increment b_J
# B_J, D_J decrement a_J
#
# o The corresponding four inverse-shift operators are defined if there
# is no cancellation. Thus e.g. an index a_J (upper or lower) can be
# incremented if a_J != b_i for i = 1, ..., q.
#
# o Order reduction: if b_j - a_i is a non-negative integer, where
# j <= m and i > n, the corresponding quotient of gamma functions reduces
# to a polynomial. Hence the G function can be expressed using a G-function
# of lower order.
# Similarly if j > m and i <= n.
#
# Secondly, there are paired index theorems [Adamchik, The evaluation of
# integrals of Bessel functions via G-function identities]. Suppose there
# are three parameters a, b, c, where a is an a_i, i <= n, b is a b_j,
# j <= m and c is a denominator parameter (i.e. a_i, i > n or b_j, j > m).
# Suppose further all three differ by integers.
# Then the order can be reduced.
# TODO work this out in detail.
#
# o An index quadruple is called suitable if its order cannot be reduced.
# If there exists a sequence of shift operators transforming one index
# quadruple into another, we say one is reachable from the other.
#
# o Deciding if one index quadruple is reachable from another is tricky. For
# this reason, we use hand-built routines to match and instantiate formulas.
#
from collections import defaultdict
from itertools import product
from functools import reduce
from sympy import SYMPY_DEBUG
from sympy.core import (S, Dummy, symbols, sympify, Tuple, expand, I, pi, Mul,
EulerGamma, oo, zoo, expand_func, Add, nan, Expr, Rational)
from sympy.core.mod import Mod
from sympy.core.sorting import default_sort_key
from sympy.functions import (exp, sqrt, root, log, lowergamma, cos,
besseli, gamma, uppergamma, expint, erf, sin, besselj, Ei, Ci, Si, Shi,
sinh, cosh, Chi, fresnels, fresnelc, polar_lift, exp_polar, floor, ceiling,
rf, factorial, lerchphi, Piecewise, re, elliptic_k, elliptic_e)
from sympy.functions.elementary.complexes import polarify, unpolarify
from sympy.functions.special.hyper import (hyper, HyperRep_atanh,
HyperRep_power1, HyperRep_power2, HyperRep_log1, HyperRep_asin1,
HyperRep_asin2, HyperRep_sqrts1, HyperRep_sqrts2, HyperRep_log2,
HyperRep_cosasin, HyperRep_sinasin, meijerg)
from sympy.polys import poly, Poly
from sympy.simplify.powsimp import powdenest
from sympy.utilities.iterables import sift
# function to define "buckets"
def _mod1(x):
# TODO see if this can work as Mod(x, 1); this will require
# different handling of the "buckets" since these need to
# be sorted and that fails when there is a mixture of
# integers and expressions with parameters. With the current
# Mod behavior, Mod(k, 1) == Mod(1, 1) == 0 if k is an integer.
# Although the sorting can be done with Basic.compare, this may
# still require different handling of the sorted buckets.
if x.is_Number:
return Mod(x, 1)
c, x = x.as_coeff_Add()
return Mod(c, 1) + x
# leave add formulae at the top for easy reference
def add_formulae(formulae):
""" Create our knowledge base. """
from sympy.matrices import Matrix
a, b, c, z = symbols('a b c, z', cls=Dummy)
def add(ap, bq, res):
func = Hyper_Function(ap, bq)
formulae.append(Formula(func, z, res, (a, b, c)))
def addb(ap, bq, B, C, M):
func = Hyper_Function(ap, bq)
formulae.append(Formula(func, z, None, (a, b, c), B, C, M))
# Luke, Y. L. (1969), The Special Functions and Their Approximations,
# Volume 1, section 6.2
# 0F0
add((), (), exp(z))
# 1F0
add((a, ), (), HyperRep_power1(-a, z))
# 2F1
addb((a, a - S.Half), (2*a, ),
Matrix([HyperRep_power2(a, z),
HyperRep_power2(a + S.Half, z)/2]),
Matrix([[1, 0]]),
Matrix([[(a - S.Half)*z/(1 - z), (S.Half - a)*z/(1 - z)],
[a/(1 - z), a*(z - 2)/(1 - z)]]))
addb((1, 1), (2, ),
Matrix([HyperRep_log1(z), 1]), Matrix([[-1/z, 0]]),
Matrix([[0, z/(z - 1)], [0, 0]]))
addb((S.Half, 1), (S('3/2'), ),
Matrix([HyperRep_atanh(z), 1]),
Matrix([[1, 0]]),
Matrix([[Rational(-1, 2), 1/(1 - z)/2], [0, 0]]))
addb((S.Half, S.Half), (S('3/2'), ),
Matrix([HyperRep_asin1(z), HyperRep_power1(Rational(-1, 2), z)]),
Matrix([[1, 0]]),
Matrix([[Rational(-1, 2), S.Half], [0, z/(1 - z)/2]]))
addb((a, S.Half + a), (S.Half, ),
Matrix([HyperRep_sqrts1(-a, z), -HyperRep_sqrts2(-a - S.Half, z)]),
Matrix([[1, 0]]),
Matrix([[0, -a],
[z*(-2*a - 1)/2/(1 - z), S.Half - z*(-2*a - 1)/(1 - z)]]))
# A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev (1990).
# Integrals and Series: More Special Functions, Vol. 3,.
# Gordon and Breach Science Publisher
addb([a, -a], [S.Half],
Matrix([HyperRep_cosasin(a, z), HyperRep_sinasin(a, z)]),
Matrix([[1, 0]]),
Matrix([[0, -a], [a*z/(1 - z), 1/(1 - z)/2]]))
addb([1, 1], [3*S.Half],
Matrix([HyperRep_asin2(z), 1]), Matrix([[1, 0]]),
Matrix([[(z - S.Half)/(1 - z), 1/(1 - z)/2], [0, 0]]))
# Complete elliptic integrals K(z) and E(z), both a 2F1 function
addb([S.Half, S.Half], [S.One],
Matrix([elliptic_k(z), elliptic_e(z)]),
Matrix([[2/pi, 0]]),
Matrix([[Rational(-1, 2), -1/(2*z-2)],
[Rational(-1, 2), S.Half]]))
addb([Rational(-1, 2), S.Half], [S.One],
Matrix([elliptic_k(z), elliptic_e(z)]),
Matrix([[0, 2/pi]]),
Matrix([[Rational(-1, 2), -1/(2*z-2)],
[Rational(-1, 2), S.Half]]))
# 3F2
addb([Rational(-1, 2), 1, 1], [S.Half, 2],
Matrix([z*HyperRep_atanh(z), HyperRep_log1(z), 1]),
Matrix([[Rational(-2, 3), -S.One/(3*z), Rational(2, 3)]]),
Matrix([[S.Half, 0, z/(1 - z)/2],
[0, 0, z/(z - 1)],
[0, 0, 0]]))
# actually the formula for 3/2 is much nicer ...
addb([Rational(-1, 2), 1, 1], [2, 2],
Matrix([HyperRep_power1(S.Half, z), HyperRep_log2(z), 1]),
Matrix([[Rational(4, 9) - 16/(9*z), 4/(3*z), 16/(9*z)]]),
Matrix([[z/2/(z - 1), 0, 0], [1/(2*(z - 1)), 0, S.Half], [0, 0, 0]]))
# 1F1
addb([1], [b], Matrix([z**(1 - b) * exp(z) * lowergamma(b - 1, z), 1]),
Matrix([[b - 1, 0]]), Matrix([[1 - b + z, 1], [0, 0]]))
addb([a], [2*a],
Matrix([z**(S.Half - a)*exp(z/2)*besseli(a - S.Half, z/2)
* gamma(a + S.Half)/4**(S.Half - a),
z**(S.Half - a)*exp(z/2)*besseli(a + S.Half, z/2)
* gamma(a + S.Half)/4**(S.Half - a)]),
Matrix([[1, 0]]),
Matrix([[z/2, z/2], [z/2, (z/2 - 2*a)]]))
mz = polar_lift(-1)*z
addb([a], [a + 1],
Matrix([mz**(-a)*a*lowergamma(a, mz), a*exp(z)]),
Matrix([[1, 0]]),
Matrix([[-a, 1], [0, z]]))
# This one is redundant.
add([Rational(-1, 2)], [S.Half], exp(z) - sqrt(pi*z)*(-I)*erf(I*sqrt(z)))
# Added to get nice results for Laplace transform of Fresnel functions
# http://functions.wolfram.com/07.22.03.6437.01
# Basic rule
#add([1], [Rational(3, 4), Rational(5, 4)],
# sqrt(pi) * (cos(2*sqrt(polar_lift(-1)*z))*fresnelc(2*root(polar_lift(-1)*z,4)/sqrt(pi)) +
# sin(2*sqrt(polar_lift(-1)*z))*fresnels(2*root(polar_lift(-1)*z,4)/sqrt(pi)))
# / (2*root(polar_lift(-1)*z,4)))
# Manually tuned rule
addb([1], [Rational(3, 4), Rational(5, 4)],
Matrix([ sqrt(pi)*(I*sinh(2*sqrt(z))*fresnels(2*root(z, 4)*exp(I*pi/4)/sqrt(pi))
+ cosh(2*sqrt(z))*fresnelc(2*root(z, 4)*exp(I*pi/4)/sqrt(pi)))
* exp(-I*pi/4)/(2*root(z, 4)),
sqrt(pi)*root(z, 4)*(sinh(2*sqrt(z))*fresnelc(2*root(z, 4)*exp(I*pi/4)/sqrt(pi))
+ I*cosh(2*sqrt(z))*fresnels(2*root(z, 4)*exp(I*pi/4)/sqrt(pi)))
*exp(-I*pi/4)/2,
1 ]),
Matrix([[1, 0, 0]]),
Matrix([[Rational(-1, 4), 1, Rational(1, 4)],
[ z, Rational(1, 4), 0],
[ 0, 0, 0]]))
# 2F2
addb([S.Half, a], [Rational(3, 2), a + 1],
Matrix([a/(2*a - 1)*(-I)*sqrt(pi/z)*erf(I*sqrt(z)),
a/(2*a - 1)*(polar_lift(-1)*z)**(-a)*
lowergamma(a, polar_lift(-1)*z),
a/(2*a - 1)*exp(z)]),
Matrix([[1, -1, 0]]),
Matrix([[Rational(-1, 2), 0, 1], [0, -a, 1], [0, 0, z]]))
# We make a "basis" of four functions instead of three, and give EulerGamma
# an extra slot (it could just be a coefficient to 1). The advantage is
# that this way Polys will not see multivariate polynomials (it treats
# EulerGamma as an indeterminate), which is *way* faster.
addb([1, 1], [2, 2],
Matrix([Ei(z) - log(z), exp(z), 1, EulerGamma]),
Matrix([[1/z, 0, 0, -1/z]]),
Matrix([[0, 1, -1, 0], [0, z, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]]))
# 0F1
add((), (S.Half, ), cosh(2*sqrt(z)))
addb([], [b],
Matrix([gamma(b)*z**((1 - b)/2)*besseli(b - 1, 2*sqrt(z)),
gamma(b)*z**(1 - b/2)*besseli(b, 2*sqrt(z))]),
Matrix([[1, 0]]), Matrix([[0, 1], [z, (1 - b)]]))
# 0F3
x = 4*z**Rational(1, 4)
def fp(a, z):
return besseli(a, x) + besselj(a, x)
def fm(a, z):
return besseli(a, x) - besselj(a, x)
# TODO branching
addb([], [S.Half, a, a + S.Half],
Matrix([fp(2*a - 1, z), fm(2*a, z)*z**Rational(1, 4),
fm(2*a - 1, z)*sqrt(z), fp(2*a, z)*z**Rational(3, 4)])
* 2**(-2*a)*gamma(2*a)*z**((1 - 2*a)/4),
Matrix([[1, 0, 0, 0]]),
Matrix([[0, 1, 0, 0],
[0, S.Half - a, 1, 0],
[0, 0, S.Half, 1],
[z, 0, 0, 1 - a]]))
x = 2*(4*z)**Rational(1, 4)*exp_polar(I*pi/4)
addb([], [a, a + S.Half, 2*a],
(2*sqrt(polar_lift(-1)*z))**(1 - 2*a)*gamma(2*a)**2 *
Matrix([besselj(2*a - 1, x)*besseli(2*a - 1, x),
x*(besseli(2*a, x)*besselj(2*a - 1, x)
- besseli(2*a - 1, x)*besselj(2*a, x)),
x**2*besseli(2*a, x)*besselj(2*a, x),
x**3*(besseli(2*a, x)*besselj(2*a - 1, x)
+ besseli(2*a - 1, x)*besselj(2*a, x))]),
Matrix([[1, 0, 0, 0]]),
Matrix([[0, Rational(1, 4), 0, 0],
[0, (1 - 2*a)/2, Rational(-1, 2), 0],
[0, 0, 1 - 2*a, Rational(1, 4)],
[-32*z, 0, 0, 1 - a]]))
# 1F2
addb([a], [a - S.Half, 2*a],
Matrix([z**(S.Half - a)*besseli(a - S.Half, sqrt(z))**2,
z**(1 - a)*besseli(a - S.Half, sqrt(z))
*besseli(a - Rational(3, 2), sqrt(z)),
z**(Rational(3, 2) - a)*besseli(a - Rational(3, 2), sqrt(z))**2]),
Matrix([[-gamma(a + S.Half)**2/4**(S.Half - a),
2*gamma(a - S.Half)*gamma(a + S.Half)/4**(1 - a),
0]]),
Matrix([[1 - 2*a, 1, 0], [z/2, S.Half - a, S.Half], [0, z, 0]]))
addb([S.Half], [b, 2 - b],
pi*(1 - b)/sin(pi*b)*
Matrix([besseli(1 - b, sqrt(z))*besseli(b - 1, sqrt(z)),
sqrt(z)*(besseli(-b, sqrt(z))*besseli(b - 1, sqrt(z))
+ besseli(1 - b, sqrt(z))*besseli(b, sqrt(z))),
besseli(-b, sqrt(z))*besseli(b, sqrt(z))]),
Matrix([[1, 0, 0]]),
Matrix([[b - 1, S.Half, 0],
[z, 0, z],
[0, S.Half, -b]]))
addb([S.Half], [Rational(3, 2), Rational(3, 2)],
Matrix([Shi(2*sqrt(z))/2/sqrt(z), sinh(2*sqrt(z))/2/sqrt(z),
cosh(2*sqrt(z))]),
Matrix([[1, 0, 0]]),
Matrix([[Rational(-1, 2), S.Half, 0], [0, Rational(-1, 2), S.Half], [0, 2*z, 0]]))
# FresnelS
# Basic rule
#add([Rational(3, 4)], [Rational(3, 2),Rational(7, 4)], 6*fresnels( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) / ( pi * (exp(pi*I/4)*root(z,4)*2/sqrt(pi))**3 ) )
# Manually tuned rule
addb([Rational(3, 4)], [Rational(3, 2), Rational(7, 4)],
Matrix(
[ fresnels(
exp(
pi*I/4)*root(
z, 4)*2/sqrt(
pi) ) / (
pi * (exp(pi*I/4)*root(z, 4)*2/sqrt(pi))**3 ),
sinh(2*sqrt(z))/sqrt(z),
cosh(2*sqrt(z)) ]),
Matrix([[6, 0, 0]]),
Matrix([[Rational(-3, 4), Rational(1, 16), 0],
[ 0, Rational(-1, 2), 1],
[ 0, z, 0]]))
# FresnelC
# Basic rule
#add([Rational(1, 4)], [S.Half,Rational(5, 4)], fresnelc( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) / ( exp(pi*I/4)*root(z,4)*2/sqrt(pi) ) )
# Manually tuned rule
addb([Rational(1, 4)], [S.Half, Rational(5, 4)],
Matrix(
[ sqrt(
pi)*exp(
-I*pi/4)*fresnelc(
2*root(z, 4)*exp(I*pi/4)/sqrt(pi))/(2*root(z, 4)),
cosh(2*sqrt(z)),
sinh(2*sqrt(z))*sqrt(z) ]),
Matrix([[1, 0, 0]]),
Matrix([[Rational(-1, 4), Rational(1, 4), 0 ],
[ 0, 0, 1 ],
[ 0, z, S.Half]]))
# 2F3
# XXX with this five-parameter formula is pretty slow with the current
# Formula.find_instantiations (creates 2!*3!*3**(2+3) ~ 3000
# instantiations ... But it's not too bad.
addb([a, a + S.Half], [2*a, b, 2*a - b + 1],
gamma(b)*gamma(2*a - b + 1) * (sqrt(z)/2)**(1 - 2*a) *
Matrix([besseli(b - 1, sqrt(z))*besseli(2*a - b, sqrt(z)),
sqrt(z)*besseli(b, sqrt(z))*besseli(2*a - b, sqrt(z)),
sqrt(z)*besseli(b - 1, sqrt(z))*besseli(2*a - b + 1, sqrt(z)),
besseli(b, sqrt(z))*besseli(2*a - b + 1, sqrt(z))]),
Matrix([[1, 0, 0, 0]]),
Matrix([[0, S.Half, S.Half, 0],
[z/2, 1 - b, 0, z/2],
[z/2, 0, b - 2*a, z/2],
[0, S.Half, S.Half, -2*a]]))
# (C/f above comment about eulergamma in the basis).
addb([1, 1], [2, 2, Rational(3, 2)],
Matrix([Chi(2*sqrt(z)) - log(2*sqrt(z)),
cosh(2*sqrt(z)), sqrt(z)*sinh(2*sqrt(z)), 1, EulerGamma]),
Matrix([[1/z, 0, 0, 0, -1/z]]),
Matrix([[0, S.Half, 0, Rational(-1, 2), 0],
[0, 0, 1, 0, 0],
[0, z, S.Half, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]]))
# 3F3
# This is rule: http://functions.wolfram.com/07.31.03.0134.01
# Initial reason to add it was a nice solution for
# integrate(erf(a*z)/z**2, z) and same for erfc and erfi.
# Basic rule
# add([1, 1, a], [2, 2, a+1], (a/(z*(a-1)**2)) *
# (1 - (-z)**(1-a) * (gamma(a) - uppergamma(a,-z))
# - (a-1) * (EulerGamma + uppergamma(0,-z) + log(-z))
# - exp(z)))
# Manually tuned rule
addb([1, 1, a], [2, 2, a+1],
Matrix([a*(log(-z) + expint(1, -z) + EulerGamma)/(z*(a**2 - 2*a + 1)),
a*(-z)**(-a)*(gamma(a) - uppergamma(a, -z))/(a - 1)**2,
a*exp(z)/(a**2 - 2*a + 1),
a/(z*(a**2 - 2*a + 1))]),
Matrix([[1-a, 1, -1/z, 1]]),
Matrix([[-1,0,-1/z,1],
[0,-a,1,0],
[0,0,z,0],
[0,0,0,-1]]))
def add_meijerg_formulae(formulae):
from sympy.matrices import Matrix
a, b, c, z = list(map(Dummy, 'abcz'))
rho = Dummy('rho')
def add(an, ap, bm, bq, B, C, M, matcher):
formulae.append(MeijerFormula(an, ap, bm, bq, z, [a, b, c, rho],
B, C, M, matcher))
def detect_uppergamma(func):
x = func.an[0]
y, z = func.bm
swapped = False
if not _mod1((x - y).simplify()):
swapped = True
(y, z) = (z, y)
if _mod1((x - z).simplify()) or x - z > 0:
return None
l = [y, x]
if swapped:
l = [x, y]
return {rho: y, a: x - y}, G_Function([x], [], l, [])
add([a + rho], [], [rho, a + rho], [],
Matrix([gamma(1 - a)*z**rho*exp(z)*uppergamma(a, z),
gamma(1 - a)*z**(a + rho)]),
Matrix([[1, 0]]),
Matrix([[rho + z, -1], [0, a + rho]]),
detect_uppergamma)
def detect_3113(func):
"""http://functions.wolfram.com/07.34.03.0984.01"""
x = func.an[0]
u, v, w = func.bm
if _mod1((u - v).simplify()) == 0:
if _mod1((v - w).simplify()) == 0:
return
sig = (S.Half, S.Half, S.Zero)
x1, x2, y = u, v, w
else:
if _mod1((x - u).simplify()) == 0:
sig = (S.Half, S.Zero, S.Half)
x1, y, x2 = u, v, w
else:
sig = (S.Zero, S.Half, S.Half)
y, x1, x2 = u, v, w
if (_mod1((x - x1).simplify()) != 0 or
_mod1((x - x2).simplify()) != 0 or
_mod1((x - y).simplify()) != S.Half or
x - x1 > 0 or x - x2 > 0):
return
return {a: x}, G_Function([x], [], [x - S.Half + t for t in sig], [])
s = sin(2*sqrt(z))
c_ = cos(2*sqrt(z))
S_ = Si(2*sqrt(z)) - pi/2
C = Ci(2*sqrt(z))
add([a], [], [a, a, a - S.Half], [],
Matrix([sqrt(pi)*z**(a - S.Half)*(c_*S_ - s*C),
sqrt(pi)*z**a*(s*S_ + c_*C),
sqrt(pi)*z**a]),
Matrix([[-2, 0, 0]]),
Matrix([[a - S.Half, -1, 0], [z, a, S.Half], [0, 0, a]]),
detect_3113)
def make_simp(z):
""" Create a function that simplifies rational functions in ``z``. """
def simp(expr):
""" Efficiently simplify the rational function ``expr``. """
numer, denom = expr.as_numer_denom()
numer = numer.expand()
# denom = denom.expand() # is this needed?
c, numer, denom = poly(numer, z).cancel(poly(denom, z))
return c * numer.as_expr() / denom.as_expr()
return simp
def debug(*args):
if SYMPY_DEBUG:
for a in args:
print(a, end="")
print()
class Hyper_Function(Expr):
""" A generalized hypergeometric function. """
def __new__(cls, ap, bq):
obj = super().__new__(cls)
obj.ap = Tuple(*list(map(expand, ap)))
obj.bq = Tuple(*list(map(expand, bq)))
return obj
@property
def args(self):
return (self.ap, self.bq)
@property
def sizes(self):
return (len(self.ap), len(self.bq))
@property
def gamma(self):
"""
Number of upper parameters that are negative integers
This is a transformation invariant.
"""
return sum(bool(x.is_integer and x.is_negative) for x in self.ap)
def _hashable_content(self):
return super()._hashable_content() + (self.ap,
self.bq)
def __call__(self, arg):
return hyper(self.ap, self.bq, arg)
def build_invariants(self):
"""
Compute the invariant vector.
Explanation
===========
The invariant vector is:
(gamma, ((s1, n1), ..., (sk, nk)), ((t1, m1), ..., (tr, mr)))
where gamma is the number of integer a < 0,
s1 < ... < sk
nl is the number of parameters a_i congruent to sl mod 1
t1 < ... < tr
ml is the number of parameters b_i congruent to tl mod 1
If the index pair contains parameters, then this is not truly an
invariant, since the parameters cannot be sorted uniquely mod1.
Examples
========
>>> from sympy.simplify.hyperexpand import Hyper_Function
>>> from sympy import S
>>> ap = (S.Half, S.One/3, S(-1)/2, -2)
>>> bq = (1, 2)
Here gamma = 1,
k = 3, s1 = 0, s2 = 1/3, s3 = 1/2
n1 = 1, n2 = 1, n2 = 2
r = 1, t1 = 0
m1 = 2:
>>> Hyper_Function(ap, bq).build_invariants()
(1, ((0, 1), (1/3, 1), (1/2, 2)), ((0, 2),))
"""
abuckets, bbuckets = sift(self.ap, _mod1), sift(self.bq, _mod1)
def tr(bucket):
bucket = list(bucket.items())
if not any(isinstance(x[0], Mod) for x in bucket):
bucket.sort(key=lambda x: default_sort_key(x[0]))
bucket = tuple([(mod, len(values)) for mod, values in bucket if
values])
return bucket
return (self.gamma, tr(abuckets), tr(bbuckets))
def difficulty(self, func):
""" Estimate how many steps it takes to reach ``func`` from self.
Return -1 if impossible. """
if self.gamma != func.gamma:
return -1
oabuckets, obbuckets, abuckets, bbuckets = [sift(params, _mod1) for
params in (self.ap, self.bq, func.ap, func.bq)]
diff = 0
for bucket, obucket in [(abuckets, oabuckets), (bbuckets, obbuckets)]:
for mod in set(list(bucket.keys()) + list(obucket.keys())):
if (mod not in bucket) or (mod not in obucket) \
or len(bucket[mod]) != len(obucket[mod]):
return -1
l1 = list(bucket[mod])
l2 = list(obucket[mod])
l1.sort()
l2.sort()
for i, j in zip(l1, l2):
diff += abs(i - j)
return diff
def _is_suitable_origin(self):
"""
Decide if ``self`` is a suitable origin.
Explanation
===========
A function is a suitable origin iff:
* none of the ai equals bj + n, with n a non-negative integer
* none of the ai is zero
* none of the bj is a non-positive integer
Note that this gives meaningful results only when none of the indices
are symbolic.
"""
for a in self.ap:
for b in self.bq:
if (a - b).is_integer and (a - b).is_negative is False:
return False
for a in self.ap:
if a == 0:
return False
for b in self.bq:
if b.is_integer and b.is_nonpositive:
return False
return True
class G_Function(Expr):
""" A Meijer G-function. """
def __new__(cls, an, ap, bm, bq):
obj = super().__new__(cls)
obj.an = Tuple(*list(map(expand, an)))
obj.ap = Tuple(*list(map(expand, ap)))
obj.bm = Tuple(*list(map(expand, bm)))
obj.bq = Tuple(*list(map(expand, bq)))
return obj
@property
def args(self):
return (self.an, self.ap, self.bm, self.bq)
def _hashable_content(self):
return super()._hashable_content() + self.args
def __call__(self, z):
return meijerg(self.an, self.ap, self.bm, self.bq, z)
def compute_buckets(self):
"""
Compute buckets for the fours sets of parameters.
Explanation
===========
We guarantee that any two equal Mod objects returned are actually the
same, and that the buckets are sorted by real part (an and bq
descendending, bm and ap ascending).
Examples
========
>>> from sympy.simplify.hyperexpand import G_Function
>>> from sympy.abc import y
>>> from sympy import S
>>> a, b = [1, 3, 2, S(3)/2], [1 + y, y, 2, y + 3]
>>> G_Function(a, b, [2], [y]).compute_buckets()
({0: [3, 2, 1], 1/2: [3/2]},
{0: [2], y: [y, y + 1, y + 3]}, {0: [2]}, {y: [y]})
"""
dicts = pan, pap, pbm, pbq = [defaultdict(list) for i in range(4)]
for dic, lis in zip(dicts, (self.an, self.ap, self.bm, self.bq)):
for x in lis:
dic[_mod1(x)].append(x)
for dic, flip in zip(dicts, (True, False, False, True)):
for m, items in dic.items():
x0 = items[0]
items.sort(key=lambda x: x - x0, reverse=flip)
dic[m] = items
return tuple([dict(w) for w in dicts])
@property
def signature(self):
return (len(self.an), len(self.ap), len(self.bm), len(self.bq))
# Dummy variable.
_x = Dummy('x')
class Formula:
"""
This class represents hypergeometric formulae.
Explanation
===========
Its data members are:
- z, the argument
- closed_form, the closed form expression
- symbols, the free symbols (parameters) in the formula
- func, the function
- B, C, M (see _compute_basis)
Examples
========
>>> from sympy.abc import a, b, z
>>> from sympy.simplify.hyperexpand import Formula, Hyper_Function
>>> func = Hyper_Function((a/2, a/3 + b, (1+a)/2), (a, b, (a+b)/7))
>>> f = Formula(func, z, None, [a, b])
"""
def _compute_basis(self, closed_form):
"""
Compute a set of functions B=(f1, ..., fn), a nxn matrix M
and a 1xn matrix C such that:
closed_form = C B
z d/dz B = M B.
"""
from sympy.matrices import Matrix, eye, zeros
afactors = [_x + a for a in self.func.ap]
bfactors = [_x + b - 1 for b in self.func.bq]
expr = _x*Mul(*bfactors) - self.z*Mul(*afactors)
poly = Poly(expr, _x)
n = poly.degree() - 1
b = [closed_form]
for _ in range(n):
b.append(self.z*b[-1].diff(self.z))
self.B = Matrix(b)
self.C = Matrix([[1] + [0]*n])
m = eye(n)
m = m.col_insert(0, zeros(n, 1))
l = poly.all_coeffs()[1:]
l.reverse()
self.M = m.row_insert(n, -Matrix([l])/poly.all_coeffs()[0])
def __init__(self, func, z, res, symbols, B=None, C=None, M=None):
z = sympify(z)
res = sympify(res)
symbols = [x for x in sympify(symbols) if func.has(x)]
self.z = z
self.symbols = symbols
self.B = B
self.C = C
self.M = M
self.func = func
# TODO with symbolic parameters, it could be advantageous
# (for prettier answers) to compute a basis only *after*
# instantiation
if res is not None:
self._compute_basis(res)
@property
def closed_form(self):
return reduce(lambda s,m: s+m[0]*m[1], zip(self.C, self.B), S.Zero)
def find_instantiations(self, func):
"""
Find substitutions of the free symbols that match ``func``.
Return the substitution dictionaries as a list. Note that the returned
instantiations need not actually match, or be valid!
"""
from sympy.solvers import solve
ap = func.ap
bq = func.bq
if len(ap) != len(self.func.ap) or len(bq) != len(self.func.bq):
raise TypeError('Cannot instantiate other number of parameters')
symbol_values = []
for a in self.symbols:
if a in self.func.ap.args:
symbol_values.append(ap)
elif a in self.func.bq.args:
symbol_values.append(bq)
else:
raise ValueError("At least one of the parameters of the "
"formula must be equal to %s" % (a,))
base_repl = [dict(list(zip(self.symbols, values)))
for values in product(*symbol_values)]
abuckets, bbuckets = [sift(params, _mod1) for params in [ap, bq]]
a_inv, b_inv = [{a: len(vals) for a, vals in bucket.items()}
for bucket in [abuckets, bbuckets]]
critical_values = [[0] for _ in self.symbols]
result = []
_n = Dummy()
for repl in base_repl:
symb_a, symb_b = [sift(params, lambda x: _mod1(x.xreplace(repl)))
for params in [self.func.ap, self.func.bq]]
for bucket, obucket in [(abuckets, symb_a), (bbuckets, symb_b)]:
for mod in set(list(bucket.keys()) + list(obucket.keys())):
if (mod not in bucket) or (mod not in obucket) \
or len(bucket[mod]) != len(obucket[mod]):
break
for a, vals in zip(self.symbols, critical_values):
if repl[a].free_symbols:
continue
exprs = [expr for expr in obucket[mod] if expr.has(a)]
repl0 = repl.copy()
repl0[a] += _n
for expr in exprs:
for target in bucket[mod]:
n0, = solve(expr.xreplace(repl0) - target, _n)
if n0.free_symbols:
raise ValueError("Value should not be true")
vals.append(n0)
else:
values = []
for a, vals in zip(self.symbols, critical_values):
a0 = repl[a]
min_ = floor(min(vals))
max_ = ceiling(max(vals))
values.append([a0 + n for n in range(min_, max_ + 1)])
result.extend(dict(list(zip(self.symbols, l))) for l in product(*values))
return result
class FormulaCollection:
""" A collection of formulae to use as origins. """
def __init__(self):
""" Doing this globally at module init time is a pain ... """
self.symbolic_formulae = {}
self.concrete_formulae = {}
self.formulae = []
add_formulae(self.formulae)
# Now process the formulae into a helpful form.
# These dicts are indexed by (p, q).
for f in self.formulae:
sizes = f.func.sizes
if len(f.symbols) > 0:
self.symbolic_formulae.setdefault(sizes, []).append(f)
else:
inv = f.func.build_invariants()
self.concrete_formulae.setdefault(sizes, {})[inv] = f
def lookup_origin(self, func):
"""
Given the suitable target ``func``, try to find an origin in our
knowledge base.
Examples
========
>>> from sympy.simplify.hyperexpand import (FormulaCollection,
... Hyper_Function)
>>> f = FormulaCollection()
>>> f.lookup_origin(Hyper_Function((), ())).closed_form
exp(_z)
>>> f.lookup_origin(Hyper_Function([1], ())).closed_form
HyperRep_power1(-1, _z)
>>> from sympy import S
>>> i = Hyper_Function([S('1/4'), S('3/4 + 4')], [S.Half])
>>> f.lookup_origin(i).closed_form
HyperRep_sqrts1(-1/4, _z)
"""
inv = func.build_invariants()
sizes = func.sizes
if sizes in self.concrete_formulae and \
inv in self.concrete_formulae[sizes]:
return self.concrete_formulae[sizes][inv]
# We don't have a concrete formula. Try to instantiate.
if sizes not in self.symbolic_formulae:
return None # Too bad...
possible = []
for f in self.symbolic_formulae[sizes]:
repls = f.find_instantiations(func)
for repl in repls:
func2 = f.func.xreplace(repl)
if not func2._is_suitable_origin():
continue
diff = func2.difficulty(func)
if diff == -1:
continue
possible.append((diff, repl, f, func2))
# find the nearest origin
possible.sort(key=lambda x: x[0])
for _, repl, f, func2 in possible:
f2 = Formula(func2, f.z, None, [], f.B.subs(repl),
f.C.subs(repl), f.M.subs(repl))
if not any(e.has(S.NaN, oo, -oo, zoo) for e in [f2.B, f2.M, f2.C]):
return f2
return None
class MeijerFormula:
"""
This class represents a Meijer G-function formula.
Its data members are:
- z, the argument
- symbols, the free symbols (parameters) in the formula
- func, the function
- B, C, M (c/f ordinary Formula)
"""
def __init__(self, an, ap, bm, bq, z, symbols, B, C, M, matcher):
an, ap, bm, bq = [Tuple(*list(map(expand, w))) for w in [an, ap, bm, bq]]
self.func = G_Function(an, ap, bm, bq)
self.z = z
self.symbols = symbols
self._matcher = matcher
self.B = B
self.C = C
self.M = M
@property
def closed_form(self):
return reduce(lambda s,m: s+m[0]*m[1], zip(self.C, self.B), S.Zero)
def try_instantiate(self, func):
"""
Try to instantiate the current formula to (almost) match func.
This uses the _matcher passed on init.
"""
if func.signature != self.func.signature:
return None
res = self._matcher(func)
if res is not None:
subs, newfunc = res
return MeijerFormula(newfunc.an, newfunc.ap, newfunc.bm, newfunc.bq,
self.z, [],
self.B.subs(subs), self.C.subs(subs),
self.M.subs(subs), None)
class MeijerFormulaCollection:
"""
This class holds a collection of meijer g formulae.
"""
def __init__(self):
formulae = []
add_meijerg_formulae(formulae)
self.formulae = defaultdict(list)
for formula in formulae:
self.formulae[formula.func.signature].append(formula)
self.formulae = dict(self.formulae)
def lookup_origin(self, func):
""" Try to find a formula that matches func. """
if func.signature not in self.formulae:
return None
for formula in self.formulae[func.signature]:
res = formula.try_instantiate(func)
if res is not None:
return res
class Operator:
"""
Base class for operators to be applied to our functions.
Explanation
===========
These operators are differential operators. They are by convention
expressed in the variable D = z*d/dz (although this base class does
not actually care).
Note that when the operator is applied to an object, we typically do
*not* blindly differentiate but instead use a different representation
of the z*d/dz operator (see make_derivative_operator).
To subclass from this, define a __init__ method that initializes a
self._poly variable. This variable stores a polynomial. By convention
the generator is z*d/dz, and acts to the right of all coefficients.
Thus this poly
x**2 + 2*z*x + 1
represents the differential operator
(z*d/dz)**2 + 2*z**2*d/dz.
This class is used only in the implementation of the hypergeometric
function expansion algorithm.
"""
def apply(self, obj, op):
"""
Apply ``self`` to the object ``obj``, where the generator is ``op``.
Examples
========
>>> from sympy.simplify.hyperexpand import Operator
>>> from sympy.polys.polytools import Poly
>>> from sympy.abc import x, y, z
>>> op = Operator()
>>> op._poly = Poly(x**2 + z*x + y, x)
>>> op.apply(z**7, lambda f: f.diff(z))
y*z**7 + 7*z**7 + 42*z**5
"""
coeffs = self._poly.all_coeffs()
coeffs.reverse()
diffs = [obj]
for c in coeffs[1:]:
diffs.append(op(diffs[-1]))
r = coeffs[0]*diffs[0]
for c, d in zip(coeffs[1:], diffs[1:]):
r += c*d
return r
class MultOperator(Operator):
""" Simply multiply by a "constant" """
def __init__(self, p):
self._poly = Poly(p, _x)
class ShiftA(Operator):
""" Increment an upper index. """
def __init__(self, ai):
ai = sympify(ai)
if ai == 0:
raise ValueError('Cannot increment zero upper index.')
self._poly = Poly(_x/ai + 1, _x)
def __str__(self):
return '<Increment upper %s.>' % (1/self._poly.all_coeffs()[0])
class ShiftB(Operator):
""" Decrement a lower index. """
def __init__(self, bi):
bi = sympify(bi)
if bi == 1:
raise ValueError('Cannot decrement unit lower index.')
self._poly = Poly(_x/(bi - 1) + 1, _x)
def __str__(self):
return '<Decrement lower %s.>' % (1/self._poly.all_coeffs()[0] + 1)
class UnShiftA(Operator):
""" Decrement an upper index. """
def __init__(self, ap, bq, i, z):
""" Note: i counts from zero! """
ap, bq, i = list(map(sympify, [ap, bq, i]))
self._ap = ap
self._bq = bq
self._i = i
ap = list(ap)
bq = list(bq)
ai = ap.pop(i) - 1
if ai == 0:
raise ValueError('Cannot decrement unit upper index.')
m = Poly(z*ai, _x)
for a in ap:
m *= Poly(_x + a, _x)
A = Dummy('A')
n = D = Poly(ai*A - ai, A)
for b in bq:
n *= D + (b - 1).as_poly(A)
b0 = -n.nth(0)
if b0 == 0:
raise ValueError('Cannot decrement upper index: '
'cancels with lower')
n = Poly(Poly(n.all_coeffs()[:-1], A).as_expr().subs(A, _x/ai + 1), _x)
self._poly = Poly((n - m)/b0, _x)
def __str__(self):
return '<Decrement upper index #%s of %s, %s.>' % (self._i,
self._ap, self._bq)
class UnShiftB(Operator):
""" Increment a lower index. """
def __init__(self, ap, bq, i, z):
""" Note: i counts from zero! """
ap, bq, i = list(map(sympify, [ap, bq, i]))
self._ap = ap
self._bq = bq
self._i = i
ap = list(ap)
bq = list(bq)
bi = bq.pop(i) + 1
if bi == 0:
raise ValueError('Cannot increment -1 lower index.')
m = Poly(_x*(bi - 1), _x)
for b in bq:
m *= Poly(_x + b - 1, _x)
B = Dummy('B')
D = Poly((bi - 1)*B - bi + 1, B)
n = Poly(z, B)
for a in ap:
n *= (D + a.as_poly(B))
b0 = n.nth(0)
if b0 == 0:
raise ValueError('Cannot increment index: cancels with upper')
n = Poly(Poly(n.all_coeffs()[:-1], B).as_expr().subs(
B, _x/(bi - 1) + 1), _x)
self._poly = Poly((m - n)/b0, _x)
def __str__(self):
return '<Increment lower index #%s of %s, %s.>' % (self._i,
self._ap, self._bq)
class MeijerShiftA(Operator):
""" Increment an upper b index. """
def __init__(self, bi):
bi = sympify(bi)
self._poly = Poly(bi - _x, _x)
def __str__(self):
return '<Increment upper b=%s.>' % (self._poly.all_coeffs()[1])
class MeijerShiftB(Operator):
""" Decrement an upper a index. """
def __init__(self, bi):
bi = sympify(bi)
self._poly = Poly(1 - bi + _x, _x)
def __str__(self):
return '<Decrement upper a=%s.>' % (1 - self._poly.all_coeffs()[1])
class MeijerShiftC(Operator):
""" Increment a lower b index. """
def __init__(self, bi):
bi = sympify(bi)
self._poly = Poly(-bi + _x, _x)
def __str__(self):
return '<Increment lower b=%s.>' % (-self._poly.all_coeffs()[1])
class MeijerShiftD(Operator):
""" Decrement a lower a index. """
def __init__(self, bi):
bi = sympify(bi)
self._poly = Poly(bi - 1 - _x, _x)
def __str__(self):
return '<Decrement lower a=%s.>' % (self._poly.all_coeffs()[1] + 1)
class MeijerUnShiftA(Operator):
""" Decrement an upper b index. """
def __init__(self, an, ap, bm, bq, i, z):
""" Note: i counts from zero! """
an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i]))
self._an = an
self._ap = ap
self._bm = bm
self._bq = bq
self._i = i
an = list(an)
ap = list(ap)
bm = list(bm)
bq = list(bq)
bi = bm.pop(i) - 1
m = Poly(1, _x)
for b in bm:
m *= Poly(b - _x, _x)
for b in bq:
m *= Poly(_x - b, _x)
A = Dummy('A')
D = Poly(bi - A, A)
n = Poly(z, A)
for a in an:
n *= (D + 1 - a)
for a in ap:
n *= (-D + a - 1)
b0 = n.nth(0)
if b0 == 0:
raise ValueError('Cannot decrement upper b index (cancels)')
n = Poly(Poly(n.all_coeffs()[:-1], A).as_expr().subs(A, bi - _x), _x)
self._poly = Poly((m - n)/b0, _x)
def __str__(self):
return '<Decrement upper b index #%s of %s, %s, %s, %s.>' % (self._i,
self._an, self._ap, self._bm, self._bq)
class MeijerUnShiftB(Operator):
""" Increment an upper a index. """
def __init__(self, an, ap, bm, bq, i, z):
""" Note: i counts from zero! """
an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i]))
self._an = an
self._ap = ap
self._bm = bm
self._bq = bq
self._i = i
an = list(an)
ap = list(ap)
bm = list(bm)
bq = list(bq)
ai = an.pop(i) + 1
m = Poly(z, _x)
for a in an:
m *= Poly(1 - a + _x, _x)
for a in ap:
m *= Poly(a - 1 - _x, _x)
B = Dummy('B')
D = Poly(B + ai - 1, B)
n = Poly(1, B)
for b in bm:
n *= (-D + b)
for b in bq:
n *= (D - b)
b0 = n.nth(0)
if b0 == 0:
raise ValueError('Cannot increment upper a index (cancels)')
n = Poly(Poly(n.all_coeffs()[:-1], B).as_expr().subs(
B, 1 - ai + _x), _x)
self._poly = Poly((m - n)/b0, _x)
def __str__(self):
return '<Increment upper a index #%s of %s, %s, %s, %s.>' % (self._i,
self._an, self._ap, self._bm, self._bq)
class MeijerUnShiftC(Operator):
""" Decrement a lower b index. """
# XXX this is "essentially" the same as MeijerUnShiftA. This "essentially"
# can be made rigorous using the functional equation G(1/z) = G'(z),
# where G' denotes a G function of slightly altered parameters.
# However, sorting out the details seems harder than just coding it
# again.
def __init__(self, an, ap, bm, bq, i, z):
""" Note: i counts from zero! """
an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i]))
self._an = an
self._ap = ap
self._bm = bm
self._bq = bq
self._i = i
an = list(an)
ap = list(ap)
bm = list(bm)
bq = list(bq)
bi = bq.pop(i) - 1
m = Poly(1, _x)
for b in bm:
m *= Poly(b - _x, _x)
for b in bq:
m *= Poly(_x - b, _x)
C = Dummy('C')
D = Poly(bi + C, C)
n = Poly(z, C)
for a in an:
n *= (D + 1 - a)
for a in ap:
n *= (-D + a - 1)
b0 = n.nth(0)
if b0 == 0:
raise ValueError('Cannot decrement lower b index (cancels)')
n = Poly(Poly(n.all_coeffs()[:-1], C).as_expr().subs(C, _x - bi), _x)
self._poly = Poly((m - n)/b0, _x)
def __str__(self):
return '<Decrement lower b index #%s of %s, %s, %s, %s.>' % (self._i,
self._an, self._ap, self._bm, self._bq)
class MeijerUnShiftD(Operator):
""" Increment a lower a index. """
# XXX This is essentially the same as MeijerUnShiftA.
# See comment at MeijerUnShiftC.
def __init__(self, an, ap, bm, bq, i, z):
""" Note: i counts from zero! """
an, ap, bm, bq, i = list(map(sympify, [an, ap, bm, bq, i]))
self._an = an
self._ap = ap
self._bm = bm
self._bq = bq
self._i = i
an = list(an)
ap = list(ap)
bm = list(bm)
bq = list(bq)
ai = ap.pop(i) + 1
m = Poly(z, _x)
for a in an:
m *= Poly(1 - a + _x, _x)
for a in ap:
m *= Poly(a - 1 - _x, _x)
B = Dummy('B') # - this is the shift operator `D_I`
D = Poly(ai - 1 - B, B)
n = Poly(1, B)
for b in bm:
n *= (-D + b)
for b in bq:
n *= (D - b)
b0 = n.nth(0)
if b0 == 0:
raise ValueError('Cannot increment lower a index (cancels)')
n = Poly(Poly(n.all_coeffs()[:-1], B).as_expr().subs(
B, ai - 1 - _x), _x)
self._poly = Poly((m - n)/b0, _x)
def __str__(self):
return '<Increment lower a index #%s of %s, %s, %s, %s.>' % (self._i,
self._an, self._ap, self._bm, self._bq)
class ReduceOrder(Operator):
""" Reduce Order by cancelling an upper and a lower index. """
def __new__(cls, ai, bj):
""" For convenience if reduction is not possible, return None. """
ai = sympify(ai)
bj = sympify(bj)
n = ai - bj
if not n.is_Integer or n < 0:
return None
if bj.is_integer and bj.is_nonpositive:
return None
expr = Operator.__new__(cls)
p = S.One
for k in range(n):
p *= (_x + bj + k)/(bj + k)
expr._poly = Poly(p, _x)
expr._a = ai
expr._b = bj
return expr
@classmethod
def _meijer(cls, b, a, sign):
""" Cancel b + sign*s and a + sign*s
This is for meijer G functions. """
b = sympify(b)
a = sympify(a)
n = b - a
if n.is_negative or not n.is_Integer:
return None
expr = Operator.__new__(cls)
p = S.One
for k in range(n):
p *= (sign*_x + a + k)
expr._poly = Poly(p, _x)
if sign == -1:
expr._a = b
expr._b = a
else:
expr._b = Add(1, a - 1, evaluate=False)
expr._a = Add(1, b - 1, evaluate=False)
return expr
@classmethod
def meijer_minus(cls, b, a):
return cls._meijer(b, a, -1)
@classmethod
def meijer_plus(cls, a, b):
return cls._meijer(1 - a, 1 - b, 1)
def __str__(self):
return '<Reduce order by cancelling upper %s with lower %s.>' % \
(self._a, self._b)
def _reduce_order(ap, bq, gen, key):
""" Order reduction algorithm used in Hypergeometric and Meijer G """
ap = list(ap)
bq = list(bq)
ap.sort(key=key)
bq.sort(key=key)
nap = []
# we will edit bq in place
operators = []
for a in ap:
op = None
for i in range(len(bq)):
op = gen(a, bq[i])
if op is not None:
bq.pop(i)
break
if op is None:
nap.append(a)
else:
operators.append(op)
return nap, bq, operators
def reduce_order(func):
"""
Given the hypergeometric function ``func``, find a sequence of operators to
reduces order as much as possible.
Explanation
===========
Return (newfunc, [operators]), where applying the operators to the
hypergeometric function newfunc yields func.
Examples
========
>>> from sympy.simplify.hyperexpand import reduce_order, Hyper_Function
>>> reduce_order(Hyper_Function((1, 2), (3, 4)))
(Hyper_Function((1, 2), (3, 4)), [])
>>> reduce_order(Hyper_Function((1,), (1,)))
(Hyper_Function((), ()), [<Reduce order by cancelling upper 1 with lower 1.>])
>>> reduce_order(Hyper_Function((2, 4), (3, 3)))
(Hyper_Function((2,), (3,)), [<Reduce order by cancelling
upper 4 with lower 3.>])
"""
nap, nbq, operators = _reduce_order(func.ap, func.bq, ReduceOrder, default_sort_key)
return Hyper_Function(Tuple(*nap), Tuple(*nbq)), operators
def reduce_order_meijer(func):
"""
Given the Meijer G function parameters, ``func``, find a sequence of
operators that reduces order as much as possible.
Return newfunc, [operators].
Examples
========
>>> from sympy.simplify.hyperexpand import (reduce_order_meijer,
... G_Function)
>>> reduce_order_meijer(G_Function([3, 4], [5, 6], [3, 4], [1, 2]))[0]
G_Function((4, 3), (5, 6), (3, 4), (2, 1))
>>> reduce_order_meijer(G_Function([3, 4], [5, 6], [3, 4], [1, 8]))[0]
G_Function((3,), (5, 6), (3, 4), (1,))
>>> reduce_order_meijer(G_Function([3, 4], [5, 6], [7, 5], [1, 5]))[0]
G_Function((3,), (), (), (1,))
>>> reduce_order_meijer(G_Function([3, 4], [5, 6], [7, 5], [5, 3]))[0]
G_Function((), (), (), ())
"""
nan, nbq, ops1 = _reduce_order(func.an, func.bq, ReduceOrder.meijer_plus,
lambda x: default_sort_key(-x))
nbm, nap, ops2 = _reduce_order(func.bm, func.ap, ReduceOrder.meijer_minus,
default_sort_key)
return G_Function(nan, nap, nbm, nbq), ops1 + ops2
def make_derivative_operator(M, z):
""" Create a derivative operator, to be passed to Operator.apply. """
def doit(C):
r = z*C.diff(z) + C*M
r = r.applyfunc(make_simp(z))
return r
return doit
def apply_operators(obj, ops, op):
"""
Apply the list of operators ``ops`` to object ``obj``, substituting
``op`` for the generator.
"""
res = obj
for o in reversed(ops):
res = o.apply(res, op)
return res
def devise_plan(target, origin, z):
"""
Devise a plan (consisting of shift and un-shift operators) to be applied
to the hypergeometric function ``target`` to yield ``origin``.
Returns a list of operators.
Examples
========
>>> from sympy.simplify.hyperexpand import devise_plan, Hyper_Function
>>> from sympy.abc import z
Nothing to do:
>>> devise_plan(Hyper_Function((1, 2), ()), Hyper_Function((1, 2), ()), z)
[]
>>> devise_plan(Hyper_Function((), (1, 2)), Hyper_Function((), (1, 2)), z)
[]
Very simple plans:
>>> devise_plan(Hyper_Function((2,), ()), Hyper_Function((1,), ()), z)
[<Increment upper 1.>]
>>> devise_plan(Hyper_Function((), (2,)), Hyper_Function((), (1,)), z)
[<Increment lower index #0 of [], [1].>]
Several buckets:
>>> from sympy import S
>>> devise_plan(Hyper_Function((1, S.Half), ()),
... Hyper_Function((2, S('3/2')), ()), z) #doctest: +NORMALIZE_WHITESPACE
[<Decrement upper index #0 of [3/2, 1], [].>,
<Decrement upper index #0 of [2, 3/2], [].>]
A slightly more complicated plan:
>>> devise_plan(Hyper_Function((1, 3), ()), Hyper_Function((2, 2), ()), z)
[<Increment upper 2.>, <Decrement upper index #0 of [2, 2], [].>]
Another more complicated plan: (note that the ap have to be shifted first!)
>>> devise_plan(Hyper_Function((1, -1), (2,)), Hyper_Function((3, -2), (4,)), z)
[<Decrement lower 3.>, <Decrement lower 4.>,
<Decrement upper index #1 of [-1, 2], [4].>,
<Decrement upper index #1 of [-1, 3], [4].>, <Increment upper -2.>]
"""
abuckets, bbuckets, nabuckets, nbbuckets = [sift(params, _mod1) for
params in (target.ap, target.bq, origin.ap, origin.bq)]
if len(list(abuckets.keys())) != len(list(nabuckets.keys())) or \
len(list(bbuckets.keys())) != len(list(nbbuckets.keys())):
raise ValueError('%s not reachable from %s' % (target, origin))
ops = []
def do_shifts(fro, to, inc, dec):
ops = []
for i in range(len(fro)):
if to[i] - fro[i] > 0:
sh = inc
ch = 1
else:
sh = dec
ch = -1
while to[i] != fro[i]:
ops += [sh(fro, i)]
fro[i] += ch
return ops
def do_shifts_a(nal, nbk, al, aother, bother):
""" Shift us from (nal, nbk) to (al, nbk). """
return do_shifts(nal, al, lambda p, i: ShiftA(p[i]),
lambda p, i: UnShiftA(p + aother, nbk + bother, i, z))
def do_shifts_b(nal, nbk, bk, aother, bother):
""" Shift us from (nal, nbk) to (nal, bk). """
return do_shifts(nbk, bk,
lambda p, i: UnShiftB(nal + aother, p + bother, i, z),
lambda p, i: ShiftB(p[i]))
for r in sorted(list(abuckets.keys()) + list(bbuckets.keys()), key=default_sort_key):
al = ()
nal = ()
bk = ()
nbk = ()
if r in abuckets:
al = abuckets[r]
nal = nabuckets[r]
if r in bbuckets:
bk = bbuckets[r]
nbk = nbbuckets[r]
if len(al) != len(nal) or len(bk) != len(nbk):
raise ValueError('%s not reachable from %s' % (target, origin))
al, nal, bk, nbk = [sorted(list(w), key=default_sort_key)
for w in [al, nal, bk, nbk]]
def others(dic, key):
l = []
for k, value in dic.items():
if k != key:
l += list(dic[k])
return l
aother = others(nabuckets, r)
bother = others(nbbuckets, r)
if len(al) == 0:
# there can be no complications, just shift the bs as we please
ops += do_shifts_b([], nbk, bk, aother, bother)
elif len(bk) == 0:
# there can be no complications, just shift the as as we please
ops += do_shifts_a(nal, [], al, aother, bother)
else:
namax = nal[-1]
amax = al[-1]
if nbk[0] - namax <= 0 or bk[0] - amax <= 0:
raise ValueError('Non-suitable parameters.')
if namax - amax > 0:
# we are going to shift down - first do the as, then the bs
ops += do_shifts_a(nal, nbk, al, aother, bother)
ops += do_shifts_b(al, nbk, bk, aother, bother)
else:
# we are going to shift up - first do the bs, then the as
ops += do_shifts_b(nal, nbk, bk, aother, bother)
ops += do_shifts_a(nal, bk, al, aother, bother)
nabuckets[r] = al
nbbuckets[r] = bk
ops.reverse()
return ops
def try_shifted_sum(func, z):
""" Try to recognise a hypergeometric sum that starts from k > 0. """
abuckets, bbuckets = sift(func.ap, _mod1), sift(func.bq, _mod1)
if len(abuckets[S.Zero]) != 1:
return None
r = abuckets[S.Zero][0]
if r <= 0:
return None
if S.Zero not in bbuckets:
return None
l = list(bbuckets[S.Zero])
l.sort()
k = l[0]
if k <= 0:
return None
nap = list(func.ap)
nap.remove(r)
nbq = list(func.bq)
nbq.remove(k)
k -= 1
nap = [x - k for x in nap]
nbq = [x - k for x in nbq]
ops = []
for n in range(r - 1):
ops.append(ShiftA(n + 1))
ops.reverse()
fac = factorial(k)/z**k
for a in nap:
fac /= rf(a, k)
for b in nbq:
fac *= rf(b, k)
ops += [MultOperator(fac)]
p = 0
for n in range(k):
m = z**n/factorial(n)
for a in nap:
m *= rf(a, n)
for b in nbq:
m /= rf(b, n)
p += m
return Hyper_Function(nap, nbq), ops, -p
def try_polynomial(func, z):
""" Recognise polynomial cases. Returns None if not such a case.
Requires order to be fully reduced. """
abuckets, bbuckets = sift(func.ap, _mod1), sift(func.bq, _mod1)
a0 = abuckets[S.Zero]
b0 = bbuckets[S.Zero]
a0.sort()
b0.sort()
al0 = [x for x in a0 if x <= 0]
bl0 = [x for x in b0 if x <= 0]
if bl0 and all(a < bl0[-1] for a in al0):
return oo
if not al0:
return None
a = al0[-1]
fac = 1
res = S.One
for n in Tuple(*list(range(-a))):
fac *= z
fac /= n + 1
for a in func.ap:
fac *= a + n
for b in func.bq:
fac /= b + n
res += fac
return res
def try_lerchphi(func):
"""
Try to find an expression for Hyper_Function ``func`` in terms of Lerch
Transcendents.
Return None if no such expression can be found.
"""
# This is actually quite simple, and is described in Roach's paper,
# section 18.
# We don't need to implement the reduction to polylog here, this
# is handled by expand_func.
from sympy.matrices import Matrix, zeros
from sympy.polys import apart
# First we need to figure out if the summation coefficient is a rational
# function of the summation index, and construct that rational function.
abuckets, bbuckets = sift(func.ap, _mod1), sift(func.bq, _mod1)
paired = {}
for key, value in abuckets.items():
if key != 0 and key not in bbuckets:
return None
bvalue = bbuckets[key]
paired[key] = (list(value), list(bvalue))
bbuckets.pop(key, None)
if bbuckets != {}:
return None
if S.Zero not in abuckets:
return None
aints, bints = paired[S.Zero]
# Account for the additional n! in denominator
paired[S.Zero] = (aints, bints + [1])
t = Dummy('t')
numer = S.One
denom = S.One
for key, (avalue, bvalue) in paired.items():
if len(avalue) != len(bvalue):
return None
# Note that since order has been reduced fully, all the b are
# bigger than all the a they differ from by an integer. In particular
# if there are any negative b left, this function is not well-defined.
for a, b in zip(avalue, bvalue):
if (a - b).is_positive:
k = a - b
numer *= rf(b + t, k)
denom *= rf(b, k)
else:
k = b - a
numer *= rf(a, k)
denom *= rf(a + t, k)
# Now do a partial fraction decomposition.
# We assemble two structures: a list monomials of pairs (a, b) representing
# a*t**b (b a non-negative integer), and a dict terms, where
# terms[a] = [(b, c)] means that there is a term b/(t-a)**c.
part = apart(numer/denom, t)
args = Add.make_args(part)
monomials = []
terms = {}
for arg in args:
numer, denom = arg.as_numer_denom()
if not denom.has(t):
p = Poly(numer, t)
if not p.is_monomial:
raise TypeError("p should be monomial")
((b, ), a) = p.LT()
monomials += [(a/denom, b)]
continue
if numer.has(t):
raise NotImplementedError('Need partial fraction decomposition'
' with linear denominators')
indep, [dep] = denom.as_coeff_mul(t)
n = 1
if dep.is_Pow:
n = dep.exp
dep = dep.base
if dep == t:
a == 0
elif dep.is_Add:
a, tmp = dep.as_independent(t)
b = 1
if tmp != t:
b, _ = tmp.as_independent(t)
if dep != b*t + a:
raise NotImplementedError('unrecognised form %s' % dep)
a /= b
indep *= b**n
else:
raise NotImplementedError('unrecognised form of partial fraction')
terms.setdefault(a, []).append((numer/indep, n))
# Now that we have this information, assemble our formula. All the
# monomials yield rational functions and go into one basis element.
# The terms[a] are related by differentiation. If the largest exponent is
# n, we need lerchphi(z, k, a) for k = 1, 2, ..., n.
# deriv maps a basis to its derivative, expressed as a C(z)-linear
# combination of other basis elements.
deriv = {}
coeffs = {}
z = Dummy('z')
monomials.sort(key=lambda x: x[1])
mon = {0: 1/(1 - z)}
if monomials:
for k in range(monomials[-1][1]):
mon[k + 1] = z*mon[k].diff(z)
for a, n in monomials:
coeffs.setdefault(S.One, []).append(a*mon[n])
for a, l in terms.items():
for c, k in l:
coeffs.setdefault(lerchphi(z, k, a), []).append(c)
l.sort(key=lambda x: x[1])
for k in range(2, l[-1][1] + 1):
deriv[lerchphi(z, k, a)] = [(-a, lerchphi(z, k, a)),
(1, lerchphi(z, k - 1, a))]
deriv[lerchphi(z, 1, a)] = [(-a, lerchphi(z, 1, a)),
(1/(1 - z), S.One)]
trans = {}
for n, b in enumerate([S.One] + list(deriv.keys())):
trans[b] = n
basis = [expand_func(b) for (b, _) in sorted(list(trans.items()),
key=lambda x:x[1])]
B = Matrix(basis)
C = Matrix([[0]*len(B)])
for b, c in coeffs.items():
C[trans[b]] = Add(*c)
M = zeros(len(B))
for b, l in deriv.items():
for c, b2 in l:
M[trans[b], trans[b2]] = c
return Formula(func, z, None, [], B, C, M)
def build_hypergeometric_formula(func):
"""
Create a formula object representing the hypergeometric function ``func``.
"""
# We know that no `ap` are negative integers, otherwise "detect poly"
# would have kicked in. However, `ap` could be empty. In this case we can
# use a different basis.
# I'm not aware of a basis that works in all cases.
from sympy.matrices.dense import (Matrix, eye, zeros)
z = Dummy('z')
if func.ap:
afactors = [_x + a for a in func.ap]
bfactors = [_x + b - 1 for b in func.bq]
expr = _x*Mul(*bfactors) - z*Mul(*afactors)
poly = Poly(expr, _x)
n = poly.degree()
basis = []
M = zeros(n)
for k in range(n):
a = func.ap[0] + k
basis += [hyper([a] + list(func.ap[1:]), func.bq, z)]
if k < n - 1:
M[k, k] = -a
M[k, k + 1] = a
B = Matrix(basis)
C = Matrix([[1] + [0]*(n - 1)])
derivs = [eye(n)]
for k in range(n):
derivs.append(M*derivs[k])
l = poly.all_coeffs()
l.reverse()
res = [0]*n
for k, c in enumerate(l):
for r, d in enumerate(C*derivs[k]):
res[r] += c*d
for k, c in enumerate(res):
M[n - 1, k] = -c/derivs[n - 1][0, n - 1]/poly.all_coeffs()[0]
return Formula(func, z, None, [], B, C, M)
else:
# Since there are no `ap`, none of the `bq` can be non-positive
# integers.
basis = []
bq = list(func.bq[:])
for i in range(len(bq)):
basis += [hyper([], bq, z)]
bq[i] += 1
basis += [hyper([], bq, z)]
B = Matrix(basis)
n = len(B)
C = Matrix([[1] + [0]*(n - 1)])
M = zeros(n)
M[0, n - 1] = z/Mul(*func.bq)
for k in range(1, n):
M[k, k - 1] = func.bq[k - 1]
M[k, k] = -func.bq[k - 1]
return Formula(func, z, None, [], B, C, M)
def hyperexpand_special(ap, bq, z):
"""
Try to find a closed-form expression for hyper(ap, bq, z), where ``z``
is supposed to be a "special" value, e.g. 1.
This function tries various of the classical summation formulae
(Gauss, Saalschuetz, etc).
"""
# This code is very ad-hoc. There are many clever algorithms
# (notably Zeilberger's) related to this problem.
# For now we just want a few simple cases to work.
p, q = len(ap), len(bq)
z_ = z
z = unpolarify(z)
if z == 0:
return S.One
from sympy.simplify.simplify import simplify
if p == 2 and q == 1:
# 2F1
a, b, c = ap + bq
if z == 1:
# Gauss
return gamma(c - a - b)*gamma(c)/gamma(c - a)/gamma(c - b)
if z == -1 and simplify(b - a + c) == 1:
b, a = a, b
if z == -1 and simplify(a - b + c) == 1:
# Kummer
if b.is_integer and b.is_negative:
return 2*cos(pi*b/2)*gamma(-b)*gamma(b - a + 1) \
/gamma(-b/2)/gamma(b/2 - a + 1)
else:
return gamma(b/2 + 1)*gamma(b - a + 1) \
/gamma(b + 1)/gamma(b/2 - a + 1)
# TODO tons of more formulae
# investigate what algorithms exist
return hyper(ap, bq, z_)
_collection = None
def _hyperexpand(func, z, ops0=[], z0=Dummy('z0'), premult=1, prem=0,
rewrite='default'):
"""
Try to find an expression for the hypergeometric function ``func``.
Explanation
===========
The result is expressed in terms of a dummy variable ``z0``. Then it
is multiplied by ``premult``. Then ``ops0`` is applied.
``premult`` must be a*z**prem for some a independent of ``z``.
"""
if z.is_zero:
return S.One
from sympy.simplify.simplify import simplify
z = polarify(z, subs=False)
if rewrite == 'default':
rewrite = 'nonrepsmall'
def carryout_plan(f, ops):
C = apply_operators(f.C.subs(f.z, z0), ops,
make_derivative_operator(f.M.subs(f.z, z0), z0))
from sympy.matrices.dense import eye
C = apply_operators(C, ops0,
make_derivative_operator(f.M.subs(f.z, z0)
+ prem*eye(f.M.shape[0]), z0))
if premult == 1:
C = C.applyfunc(make_simp(z0))
r = reduce(lambda s,m: s+m[0]*m[1], zip(C, f.B.subs(f.z, z0)), S.Zero)*premult
res = r.subs(z0, z)
if rewrite:
res = res.rewrite(rewrite)
return res
# TODO
# The following would be possible:
# *) PFD Duplication (see Kelly Roach's paper)
# *) In a similar spirit, try_lerchphi() can be generalised considerably.
global _collection
if _collection is None:
_collection = FormulaCollection()
debug('Trying to expand hypergeometric function ', func)
# First reduce order as much as possible.
func, ops = reduce_order(func)
if ops:
debug(' Reduced order to ', func)
else:
debug(' Could not reduce order.')
# Now try polynomial cases
res = try_polynomial(func, z0)
if res is not None:
debug(' Recognised polynomial.')
p = apply_operators(res, ops, lambda f: z0*f.diff(z0))
p = apply_operators(p*premult, ops0, lambda f: z0*f.diff(z0))
return unpolarify(simplify(p).subs(z0, z))
# Try to recognise a shifted sum.
p = S.Zero
res = try_shifted_sum(func, z0)
if res is not None:
func, nops, p = res
debug(' Recognised shifted sum, reduced order to ', func)
ops += nops
# apply the plan for poly
p = apply_operators(p, ops, lambda f: z0*f.diff(z0))
p = apply_operators(p*premult, ops0, lambda f: z0*f.diff(z0))
p = simplify(p).subs(z0, z)
# Try special expansions early.
if unpolarify(z) in [1, -1] and (len(func.ap), len(func.bq)) == (2, 1):
f = build_hypergeometric_formula(func)
r = carryout_plan(f, ops).replace(hyper, hyperexpand_special)
if not r.has(hyper):
return r + p
# Try to find a formula in our collection
formula = _collection.lookup_origin(func)
# Now try a lerch phi formula
if formula is None:
formula = try_lerchphi(func)
if formula is None:
debug(' Could not find an origin. ',
'Will return answer in terms of '
'simpler hypergeometric functions.')
formula = build_hypergeometric_formula(func)
debug(' Found an origin: ', formula.closed_form, ' ', formula.func)
# We need to find the operators that convert formula into func.
ops += devise_plan(func, formula.func, z0)
# Now carry out the plan.
r = carryout_plan(formula, ops) + p
return powdenest(r, polar=True).replace(hyper, hyperexpand_special)
def devise_plan_meijer(fro, to, z):
"""
Find operators to convert G-function ``fro`` into G-function ``to``.
Explanation
===========
It is assumed that ``fro`` and ``to`` have the same signatures, and that in fact
any corresponding pair of parameters differs by integers, and a direct path
is possible. I.e. if there are parameters a1 b1 c1 and a2 b2 c2 it is
assumed that a1 can be shifted to a2, etc. The only thing this routine
determines is the order of shifts to apply, nothing clever will be tried.
It is also assumed that ``fro`` is suitable.
Examples
========
>>> from sympy.simplify.hyperexpand import (devise_plan_meijer,
... G_Function)
>>> from sympy.abc import z
Empty plan:
>>> devise_plan_meijer(G_Function([1], [2], [3], [4]),
... G_Function([1], [2], [3], [4]), z)
[]
Very simple plans:
>>> devise_plan_meijer(G_Function([0], [], [], []),
... G_Function([1], [], [], []), z)
[<Increment upper a index #0 of [0], [], [], [].>]
>>> devise_plan_meijer(G_Function([0], [], [], []),
... G_Function([-1], [], [], []), z)
[<Decrement upper a=0.>]
>>> devise_plan_meijer(G_Function([], [1], [], []),
... G_Function([], [2], [], []), z)
[<Increment lower a index #0 of [], [1], [], [].>]
Slightly more complicated plans:
>>> devise_plan_meijer(G_Function([0], [], [], []),
... G_Function([2], [], [], []), z)
[<Increment upper a index #0 of [1], [], [], [].>,
<Increment upper a index #0 of [0], [], [], [].>]
>>> devise_plan_meijer(G_Function([0], [], [0], []),
... G_Function([-1], [], [1], []), z)
[<Increment upper b=0.>, <Decrement upper a=0.>]
Order matters:
>>> devise_plan_meijer(G_Function([0], [], [0], []),
... G_Function([1], [], [1], []), z)
[<Increment upper a index #0 of [0], [], [1], [].>, <Increment upper b=0.>]
"""
# TODO for now, we use the following simple heuristic: inverse-shift
# when possible, shift otherwise. Give up if we cannot make progress.
def try_shift(f, t, shifter, diff, counter):
""" Try to apply ``shifter`` in order to bring some element in ``f``
nearer to its counterpart in ``to``. ``diff`` is +/- 1 and
determines the effect of ``shifter``. Counter is a list of elements
blocking the shift.
Return an operator if change was possible, else None.
"""
for idx, (a, b) in enumerate(zip(f, t)):
if (
(a - b).is_integer and (b - a)/diff > 0 and
all(a != x for x in counter)):
sh = shifter(idx)
f[idx] += diff
return sh
fan = list(fro.an)
fap = list(fro.ap)
fbm = list(fro.bm)
fbq = list(fro.bq)
ops = []
change = True
while change:
change = False
op = try_shift(fan, to.an,
lambda i: MeijerUnShiftB(fan, fap, fbm, fbq, i, z),
1, fbm + fbq)
if op is not None:
ops += [op]
change = True
continue
op = try_shift(fap, to.ap,
lambda i: MeijerUnShiftD(fan, fap, fbm, fbq, i, z),
1, fbm + fbq)
if op is not None:
ops += [op]
change = True
continue
op = try_shift(fbm, to.bm,
lambda i: MeijerUnShiftA(fan, fap, fbm, fbq, i, z),
-1, fan + fap)
if op is not None:
ops += [op]
change = True
continue
op = try_shift(fbq, to.bq,
lambda i: MeijerUnShiftC(fan, fap, fbm, fbq, i, z),
-1, fan + fap)
if op is not None:
ops += [op]
change = True
continue
op = try_shift(fan, to.an, lambda i: MeijerShiftB(fan[i]), -1, [])
if op is not None:
ops += [op]
change = True
continue
op = try_shift(fap, to.ap, lambda i: MeijerShiftD(fap[i]), -1, [])
if op is not None:
ops += [op]
change = True
continue
op = try_shift(fbm, to.bm, lambda i: MeijerShiftA(fbm[i]), 1, [])
if op is not None:
ops += [op]
change = True
continue
op = try_shift(fbq, to.bq, lambda i: MeijerShiftC(fbq[i]), 1, [])
if op is not None:
ops += [op]
change = True
continue
if fan != list(to.an) or fap != list(to.ap) or fbm != list(to.bm) or \
fbq != list(to.bq):
raise NotImplementedError('Could not devise plan.')
ops.reverse()
return ops
_meijercollection = None
def _meijergexpand(func, z0, allow_hyper=False, rewrite='default',
place=None):
"""
Try to find an expression for the Meijer G function specified
by the G_Function ``func``. If ``allow_hyper`` is True, then returning
an expression in terms of hypergeometric functions is allowed.
Currently this just does Slater's theorem.
If expansions exist both at zero and at infinity, ``place``
can be set to ``0`` or ``zoo`` for the preferred choice.
"""
global _meijercollection
if _meijercollection is None:
_meijercollection = MeijerFormulaCollection()
if rewrite == 'default':
rewrite = None
func0 = func
debug('Try to expand Meijer G function corresponding to ', func)
# We will play games with analytic continuation - rather use a fresh symbol
z = Dummy('z')
func, ops = reduce_order_meijer(func)
if ops:
debug(' Reduced order to ', func)
else:
debug(' Could not reduce order.')
# Try to find a direct formula
f = _meijercollection.lookup_origin(func)
if f is not None:
debug(' Found a Meijer G formula: ', f.func)
ops += devise_plan_meijer(f.func, func, z)
# Now carry out the plan.
C = apply_operators(f.C.subs(f.z, z), ops,
make_derivative_operator(f.M.subs(f.z, z), z))
C = C.applyfunc(make_simp(z))
r = C*f.B.subs(f.z, z)
r = r[0].subs(z, z0)
return powdenest(r, polar=True)
debug(" Could not find a direct formula. Trying Slater's theorem.")
# TODO the following would be possible:
# *) Paired Index Theorems
# *) PFD Duplication
# (See Kelly Roach's paper for details on either.)
#
# TODO Also, we tend to create combinations of gamma functions that can be
# simplified.
def can_do(pbm, pap):
""" Test if slater applies. """
for i in pbm:
if len(pbm[i]) > 1:
l = 0
if i in pap:
l = len(pap[i])
if l + 1 < len(pbm[i]):
return False
return True
def do_slater(an, bm, ap, bq, z, zfinal):
# zfinal is the value that will eventually be substituted for z.
# We pass it to _hyperexpand to improve performance.
from sympy.series import residue
func = G_Function(an, bm, ap, bq)
_, pbm, pap, _ = func.compute_buckets()
if not can_do(pbm, pap):
return S.Zero, False
cond = len(an) + len(ap) < len(bm) + len(bq)
if len(an) + len(ap) == len(bm) + len(bq):
cond = abs(z) < 1
if cond is False:
return S.Zero, False
res = S.Zero
for m in pbm:
if len(pbm[m]) == 1:
bh = pbm[m][0]
fac = 1
bo = list(bm)
bo.remove(bh)
for bj in bo:
fac *= gamma(bj - bh)
for aj in an:
fac *= gamma(1 + bh - aj)
for bj in bq:
fac /= gamma(1 + bh - bj)
for aj in ap:
fac /= gamma(aj - bh)
nap = [1 + bh - a for a in list(an) + list(ap)]
nbq = [1 + bh - b for b in list(bo) + list(bq)]
k = polar_lift(S.NegativeOne**(len(ap) - len(bm)))
harg = k*zfinal
# NOTE even though k "is" +-1, this has to be t/k instead of
# t*k ... we are using polar numbers for consistency!
premult = (t/k)**bh
hyp = _hyperexpand(Hyper_Function(nap, nbq), harg, ops,
t, premult, bh, rewrite=None)
res += fac * hyp
else:
b_ = pbm[m][0]
ki = [bi - b_ for bi in pbm[m][1:]]
u = len(ki)
li = [ai - b_ for ai in pap[m][:u + 1]]
bo = list(bm)
for b in pbm[m]:
bo.remove(b)
ao = list(ap)
for a in pap[m][:u]:
ao.remove(a)
lu = li[-1]
di = [l - k for (l, k) in zip(li, ki)]
# We first work out the integrand:
s = Dummy('s')
integrand = z**s
for b in bm:
if not Mod(b, 1) and b.is_Number:
b = int(round(b))
integrand *= gamma(b - s)
for a in an:
integrand *= gamma(1 - a + s)
for b in bq:
integrand /= gamma(1 - b + s)
for a in ap:
integrand /= gamma(a - s)
# Now sum the finitely many residues:
# XXX This speeds up some cases - is it a good idea?
integrand = expand_func(integrand)
for r in range(int(round(lu))):
resid = residue(integrand, s, b_ + r)
resid = apply_operators(resid, ops, lambda f: z*f.diff(z))
res -= resid
# Now the hypergeometric term.
au = b_ + lu
k = polar_lift(S.NegativeOne**(len(ao) + len(bo) + 1))
harg = k*zfinal
premult = (t/k)**au
nap = [1 + au - a for a in list(an) + list(ap)] + [1]
nbq = [1 + au - b for b in list(bm) + list(bq)]
hyp = _hyperexpand(Hyper_Function(nap, nbq), harg, ops,
t, premult, au, rewrite=None)
C = S.NegativeOne**(lu)/factorial(lu)
for i in range(u):
C *= S.NegativeOne**di[i]/rf(lu - li[i] + 1, di[i])
for a in an:
C *= gamma(1 - a + au)
for b in bo:
C *= gamma(b - au)
for a in ao:
C /= gamma(a - au)
for b in bq:
C /= gamma(1 - b + au)
res += C*hyp
return res, cond
t = Dummy('t')
slater1, cond1 = do_slater(func.an, func.bm, func.ap, func.bq, z, z0)
def tr(l):
return [1 - x for x in l]
for op in ops:
op._poly = Poly(op._poly.subs({z: 1/t, _x: -_x}), _x)
slater2, cond2 = do_slater(tr(func.bm), tr(func.an), tr(func.bq), tr(func.ap),
t, 1/z0)
slater1 = powdenest(slater1.subs(z, z0), polar=True)
slater2 = powdenest(slater2.subs(t, 1/z0), polar=True)
if not isinstance(cond2, bool):
cond2 = cond2.subs(t, 1/z)
m = func(z)
if m.delta > 0 or \
(m.delta == 0 and len(m.ap) == len(m.bq) and
(re(m.nu) < -1) is not False and polar_lift(z0) == polar_lift(1)):
# The condition delta > 0 means that the convergence region is
# connected. Any expression we find can be continued analytically
# to the entire convergence region.
# The conditions delta==0, p==q, re(nu) < -1 imply that G is continuous
# on the positive reals, so the values at z=1 agree.
if cond1 is not False:
cond1 = True
if cond2 is not False:
cond2 = True
if cond1 is True:
slater1 = slater1.rewrite(rewrite or 'nonrep')
else:
slater1 = slater1.rewrite(rewrite or 'nonrepsmall')
if cond2 is True:
slater2 = slater2.rewrite(rewrite or 'nonrep')
else:
slater2 = slater2.rewrite(rewrite or 'nonrepsmall')
if cond1 is not False and cond2 is not False:
# If one condition is False, there is no choice.
if place == 0:
cond2 = False
if place == zoo:
cond1 = False
if not isinstance(cond1, bool):
cond1 = cond1.subs(z, z0)
if not isinstance(cond2, bool):
cond2 = cond2.subs(z, z0)
def weight(expr, cond):
if cond is True:
c0 = 0
elif cond is False:
c0 = 1
else:
c0 = 2
if expr.has(oo, zoo, -oo, nan):
# XXX this actually should not happen, but consider
# S('meijerg(((0, -1/2, 0, -1/2, 1/2), ()), ((0,),
# (-1/2, -1/2, -1/2, -1)), exp_polar(I*pi))/4')
c0 = 3
return (c0, expr.count(hyper), expr.count_ops())
w1 = weight(slater1, cond1)
w2 = weight(slater2, cond2)
if min(w1, w2) <= (0, 1, oo):
if w1 < w2:
return slater1
else:
return slater2
if max(w1[0], w2[0]) <= 1 and max(w1[1], w2[1]) <= 1:
return Piecewise((slater1, cond1), (slater2, cond2), (func0(z0), True))
# We couldn't find an expression without hypergeometric functions.
# TODO it would be helpful to give conditions under which the integral
# is known to diverge.
r = Piecewise((slater1, cond1), (slater2, cond2), (func0(z0), True))
if r.has(hyper) and not allow_hyper:
debug(' Could express using hypergeometric functions, '
'but not allowed.')
if not r.has(hyper) or allow_hyper:
return r
return func0(z0)
def hyperexpand(f, allow_hyper=False, rewrite='default', place=None):
"""
Expand hypergeometric functions. If allow_hyper is True, allow partial
simplification (that is a result different from input,
but still containing hypergeometric functions).
If a G-function has expansions both at zero and at infinity,
``place`` can be set to ``0`` or ``zoo`` to indicate the
preferred choice.
Examples
========
>>> from sympy.simplify.hyperexpand import hyperexpand
>>> from sympy.functions import hyper
>>> from sympy.abc import z
>>> hyperexpand(hyper([], [], z))
exp(z)
Non-hyperegeometric parts of the expression and hypergeometric expressions
that are not recognised are left unchanged:
>>> hyperexpand(1 + hyper([1, 1, 1], [], z))
hyper((1, 1, 1), (), z) + 1
"""
f = sympify(f)
def do_replace(ap, bq, z):
r = _hyperexpand(Hyper_Function(ap, bq), z, rewrite=rewrite)
if r is None:
return hyper(ap, bq, z)
else:
return r
def do_meijer(ap, bq, z):
r = _meijergexpand(G_Function(ap[0], ap[1], bq[0], bq[1]), z,
allow_hyper, rewrite=rewrite, place=place)
if not r.has(nan, zoo, oo, -oo):
return r
return f.replace(hyper, do_replace).replace(meijerg, do_meijer)
|
3340f518ee7a08546c8b1cc0e71f8e16e1cb62185a8df91c5bcfe83b6c9b178c | from collections import defaultdict
from sympy.core import (Basic, S, Add, Mul, Pow, Symbol, sympify,
expand_func, Function, Dummy, Expr, factor_terms,
expand_power_exp, Eq)
from sympy.core.exprtools import factor_nc
from sympy.core.parameters import global_parameters
from sympy.core.function import (expand_log, count_ops, _mexpand,
nfloat, expand_mul, expand)
from sympy.core.numbers import Float, I, pi, Rational
from sympy.core.relational import Relational
from sympy.core.rules import Transform
from sympy.core.sorting import ordered
from sympy.core.sympify import _sympify
from sympy.core.traversal import bottom_up as _bottom_up, walk as _walk
from sympy.functions import gamma, exp, sqrt, log, exp_polar, re
from sympy.functions.combinatorial.factorials import CombinatorialFunction
from sympy.functions.elementary.complexes import unpolarify, Abs, sign
from sympy.functions.elementary.exponential import ExpBase
from sympy.functions.elementary.hyperbolic import HyperbolicFunction
from sympy.functions.elementary.integers import ceiling
from sympy.functions.elementary.piecewise import Piecewise, piecewise_fold
from sympy.functions.elementary.trigonometric import TrigonometricFunction
from sympy.functions.special.bessel import (BesselBase, besselj, besseli,
besselk, bessely, jn)
from sympy.functions.special.tensor_functions import KroneckerDelta
from sympy.polys import together, cancel, factor
from sympy.simplify.combsimp import combsimp
from sympy.simplify.cse_opts import sub_pre, sub_post
from sympy.simplify.hyperexpand import hyperexpand
from sympy.simplify.powsimp import powsimp
from sympy.simplify.radsimp import radsimp, fraction, collect_abs
from sympy.simplify.sqrtdenest import sqrtdenest
from sympy.simplify.trigsimp import trigsimp, exptrigsimp
from sympy.utilities.decorator import deprecated
from sympy.utilities.iterables import has_variety, sift, subsets, iterable
from sympy.utilities.misc import as_int
import mpmath
def separatevars(expr, symbols=[], dict=False, force=False):
"""
Separates variables in an expression, if possible. By
default, it separates with respect to all symbols in an
expression and collects constant coefficients that are
independent of symbols.
Explanation
===========
If ``dict=True`` then the separated terms will be returned
in a dictionary keyed to their corresponding symbols.
By default, all symbols in the expression will appear as
keys; if symbols are provided, then all those symbols will
be used as keys, and any terms in the expression containing
other symbols or non-symbols will be returned keyed to the
string 'coeff'. (Passing None for symbols will return the
expression in a dictionary keyed to 'coeff'.)
If ``force=True``, then bases of powers will be separated regardless
of assumptions on the symbols involved.
Notes
=====
The order of the factors is determined by Mul, so that the
separated expressions may not necessarily be grouped together.
Although factoring is necessary to separate variables in some
expressions, it is not necessary in all cases, so one should not
count on the returned factors being factored.
Examples
========
>>> from sympy.abc import x, y, z, alpha
>>> from sympy import separatevars, sin
>>> separatevars((x*y)**y)
(x*y)**y
>>> separatevars((x*y)**y, force=True)
x**y*y**y
>>> e = 2*x**2*z*sin(y)+2*z*x**2
>>> separatevars(e)
2*x**2*z*(sin(y) + 1)
>>> separatevars(e, symbols=(x, y), dict=True)
{'coeff': 2*z, x: x**2, y: sin(y) + 1}
>>> separatevars(e, [x, y, alpha], dict=True)
{'coeff': 2*z, alpha: 1, x: x**2, y: sin(y) + 1}
If the expression is not really separable, or is only partially
separable, separatevars will do the best it can to separate it
by using factoring.
>>> separatevars(x + x*y - 3*x**2)
-x*(3*x - y - 1)
If the expression is not separable then expr is returned unchanged
or (if dict=True) then None is returned.
>>> eq = 2*x + y*sin(x)
>>> separatevars(eq) == eq
True
>>> separatevars(2*x + y*sin(x), symbols=(x, y), dict=True) is None
True
"""
expr = sympify(expr)
if dict:
return _separatevars_dict(_separatevars(expr, force), symbols)
else:
return _separatevars(expr, force)
def _separatevars(expr, force):
if isinstance(expr, Abs):
arg = expr.args[0]
if arg.is_Mul and not arg.is_number:
s = separatevars(arg, dict=True, force=force)
if s is not None:
return Mul(*map(expr.func, s.values()))
else:
return expr
if len(expr.free_symbols) < 2:
return expr
# don't destroy a Mul since much of the work may already be done
if expr.is_Mul:
args = list(expr.args)
changed = False
for i, a in enumerate(args):
args[i] = separatevars(a, force)
changed = changed or args[i] != a
if changed:
expr = expr.func(*args)
return expr
# get a Pow ready for expansion
if expr.is_Pow and expr.base != S.Exp1:
expr = Pow(separatevars(expr.base, force=force), expr.exp)
# First try other expansion methods
expr = expr.expand(mul=False, multinomial=False, force=force)
_expr, reps = posify(expr) if force else (expr, {})
expr = factor(_expr).subs(reps)
if not expr.is_Add:
return expr
# Find any common coefficients to pull out
args = list(expr.args)
commonc = args[0].args_cnc(cset=True, warn=False)[0]
for i in args[1:]:
commonc &= i.args_cnc(cset=True, warn=False)[0]
commonc = Mul(*commonc)
commonc = commonc.as_coeff_Mul()[1] # ignore constants
commonc_set = commonc.args_cnc(cset=True, warn=False)[0]
# remove them
for i, a in enumerate(args):
c, nc = a.args_cnc(cset=True, warn=False)
c = c - commonc_set
args[i] = Mul(*c)*Mul(*nc)
nonsepar = Add(*args)
if len(nonsepar.free_symbols) > 1:
_expr = nonsepar
_expr, reps = posify(_expr) if force else (_expr, {})
_expr = (factor(_expr)).subs(reps)
if not _expr.is_Add:
nonsepar = _expr
return commonc*nonsepar
def _separatevars_dict(expr, symbols):
if symbols:
if not all(t.is_Atom for t in symbols):
raise ValueError("symbols must be Atoms.")
symbols = list(symbols)
elif symbols is None:
return {'coeff': expr}
else:
symbols = list(expr.free_symbols)
if not symbols:
return None
ret = {i: [] for i in symbols + ['coeff']}
for i in Mul.make_args(expr):
expsym = i.free_symbols
intersection = set(symbols).intersection(expsym)
if len(intersection) > 1:
return None
if len(intersection) == 0:
# There are no symbols, so it is part of the coefficient
ret['coeff'].append(i)
else:
ret[intersection.pop()].append(i)
# rebuild
for k, v in ret.items():
ret[k] = Mul(*v)
return ret
def _is_sum_surds(p):
args = p.args if p.is_Add else [p]
for y in args:
if not ((y**2).is_Rational and y.is_extended_real):
return False
return True
def posify(eq):
"""Return ``eq`` (with generic symbols made positive) and a
dictionary containing the mapping between the old and new
symbols.
Explanation
===========
Any symbol that has positive=None will be replaced with a positive dummy
symbol having the same name. This replacement will allow more symbolic
processing of expressions, especially those involving powers and
logarithms.
A dictionary that can be sent to subs to restore ``eq`` to its original
symbols is also returned.
>>> from sympy import posify, Symbol, log, solve
>>> from sympy.abc import x
>>> posify(x + Symbol('p', positive=True) + Symbol('n', negative=True))
(_x + n + p, {_x: x})
>>> eq = 1/x
>>> log(eq).expand()
log(1/x)
>>> log(posify(eq)[0]).expand()
-log(_x)
>>> p, rep = posify(eq)
>>> log(p).expand().subs(rep)
-log(x)
It is possible to apply the same transformations to an iterable
of expressions:
>>> eq = x**2 - 4
>>> solve(eq, x)
[-2, 2]
>>> eq_x, reps = posify([eq, x]); eq_x
[_x**2 - 4, _x]
>>> solve(*eq_x)
[2]
"""
eq = sympify(eq)
if iterable(eq):
f = type(eq)
eq = list(eq)
syms = set()
for e in eq:
syms = syms.union(e.atoms(Symbol))
reps = {}
for s in syms:
reps.update({v: k for k, v in posify(s)[1].items()})
for i, e in enumerate(eq):
eq[i] = e.subs(reps)
return f(eq), {r: s for s, r in reps.items()}
reps = {s: Dummy(s.name, positive=True, **s.assumptions0)
for s in eq.free_symbols if s.is_positive is None}
eq = eq.subs(reps)
return eq, {r: s for s, r in reps.items()}
def hypersimp(f, k):
"""Given combinatorial term f(k) simplify its consecutive term ratio
i.e. f(k+1)/f(k). The input term can be composed of functions and
integer sequences which have equivalent representation in terms
of gamma special function.
Explanation
===========
The algorithm performs three basic steps:
1. Rewrite all functions in terms of gamma, if possible.
2. Rewrite all occurrences of gamma in terms of products
of gamma and rising factorial with integer, absolute
constant exponent.
3. Perform simplification of nested fractions, powers
and if the resulting expression is a quotient of
polynomials, reduce their total degree.
If f(k) is hypergeometric then as result we arrive with a
quotient of polynomials of minimal degree. Otherwise None
is returned.
For more information on the implemented algorithm refer to:
1. W. Koepf, Algorithms for m-fold Hypergeometric Summation,
Journal of Symbolic Computation (1995) 20, 399-417
"""
f = sympify(f)
g = f.subs(k, k + 1) / f
g = g.rewrite(gamma)
if g.has(Piecewise):
g = piecewise_fold(g)
g = g.args[-1][0]
g = expand_func(g)
g = powsimp(g, deep=True, combine='exp')
if g.is_rational_function(k):
return simplify(g, ratio=S.Infinity)
else:
return None
def hypersimilar(f, g, k):
"""
Returns True if ``f`` and ``g`` are hyper-similar.
Explanation
===========
Similarity in hypergeometric sense means that a quotient of
f(k) and g(k) is a rational function in ``k``. This procedure
is useful in solving recurrence relations.
For more information see hypersimp().
"""
f, g = list(map(sympify, (f, g)))
h = (f/g).rewrite(gamma)
h = h.expand(func=True, basic=False)
return h.is_rational_function(k)
def signsimp(expr, evaluate=None):
"""Make all Add sub-expressions canonical wrt sign.
Explanation
===========
If an Add subexpression, ``a``, can have a sign extracted,
as determined by could_extract_minus_sign, it is replaced
with Mul(-1, a, evaluate=False). This allows signs to be
extracted from powers and products.
Examples
========
>>> from sympy import signsimp, exp, symbols
>>> from sympy.abc import x, y
>>> i = symbols('i', odd=True)
>>> n = -1 + 1/x
>>> n/x/(-n)**2 - 1/n/x
(-1 + 1/x)/(x*(1 - 1/x)**2) - 1/(x*(-1 + 1/x))
>>> signsimp(_)
0
>>> x*n + x*-n
x*(-1 + 1/x) + x*(1 - 1/x)
>>> signsimp(_)
0
Since powers automatically handle leading signs
>>> (-2)**i
-2**i
signsimp can be used to put the base of a power with an integer
exponent into canonical form:
>>> n**i
(-1 + 1/x)**i
By default, signsimp doesn't leave behind any hollow simplification:
if making an Add canonical wrt sign didn't change the expression, the
original Add is restored. If this is not desired then the keyword
``evaluate`` can be set to False:
>>> e = exp(y - x)
>>> signsimp(e) == e
True
>>> signsimp(e, evaluate=False)
exp(-(x - y))
"""
if evaluate is None:
evaluate = global_parameters.evaluate
expr = sympify(expr)
if not isinstance(expr, (Expr, Relational)) or expr.is_Atom:
return expr
# get rid of an pre-existing unevaluation regarding sign
e = expr.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x))
e = sub_post(sub_pre(e))
if not isinstance(e, (Expr, Relational)) or e.is_Atom:
return e
if e.is_Add:
rv = e.func(*[signsimp(a) for a in e.args])
if not evaluate and isinstance(rv, Add
) and rv.could_extract_minus_sign():
return Mul(S.NegativeOne, -rv, evaluate=False)
return rv
if evaluate:
e = e.replace(lambda x: x.is_Mul and -(-x) != x, lambda x: -(-x))
return e
def simplify(expr, ratio=1.7, measure=count_ops, rational=False, inverse=False, doit=True, **kwargs):
"""Simplifies the given expression.
Explanation
===========
Simplification is not a well defined term and the exact strategies
this function tries can change in the future versions of SymPy. If
your algorithm relies on "simplification" (whatever it is), try to
determine what you need exactly - is it powsimp()?, radsimp()?,
together()?, logcombine()?, or something else? And use this particular
function directly, because those are well defined and thus your algorithm
will be robust.
Nonetheless, especially for interactive use, or when you do not know
anything about the structure of the expression, simplify() tries to apply
intelligent heuristics to make the input expression "simpler". For
example:
>>> from sympy import simplify, cos, sin
>>> from sympy.abc import x, y
>>> a = (x + x**2)/(x*sin(y)**2 + x*cos(y)**2)
>>> a
(x**2 + x)/(x*sin(y)**2 + x*cos(y)**2)
>>> simplify(a)
x + 1
Note that we could have obtained the same result by using specific
simplification functions:
>>> from sympy import trigsimp, cancel
>>> trigsimp(a)
(x**2 + x)/x
>>> cancel(_)
x + 1
In some cases, applying :func:`simplify` may actually result in some more
complicated expression. The default ``ratio=1.7`` prevents more extreme
cases: if (result length)/(input length) > ratio, then input is returned
unmodified. The ``measure`` parameter lets you specify the function used
to determine how complex an expression is. The function should take a
single argument as an expression and return a number such that if
expression ``a`` is more complex than expression ``b``, then
``measure(a) > measure(b)``. The default measure function is
:func:`~.count_ops`, which returns the total number of operations in the
expression.
For example, if ``ratio=1``, ``simplify`` output cannot be longer
than input.
::
>>> from sympy import sqrt, simplify, count_ops, oo
>>> root = 1/(sqrt(2)+3)
Since ``simplify(root)`` would result in a slightly longer expression,
root is returned unchanged instead::
>>> simplify(root, ratio=1) == root
True
If ``ratio=oo``, simplify will be applied anyway::
>>> count_ops(simplify(root, ratio=oo)) > count_ops(root)
True
Note that the shortest expression is not necessary the simplest, so
setting ``ratio`` to 1 may not be a good idea.
Heuristically, the default value ``ratio=1.7`` seems like a reasonable
choice.
You can easily define your own measure function based on what you feel
should represent the "size" or "complexity" of the input expression. Note
that some choices, such as ``lambda expr: len(str(expr))`` may appear to be
good metrics, but have other problems (in this case, the measure function
may slow down simplify too much for very large expressions). If you do not
know what a good metric would be, the default, ``count_ops``, is a good
one.
For example:
>>> from sympy import symbols, log
>>> a, b = symbols('a b', positive=True)
>>> g = log(a) + log(b) + log(a)*log(1/b)
>>> h = simplify(g)
>>> h
log(a*b**(1 - log(a)))
>>> count_ops(g)
8
>>> count_ops(h)
5
So you can see that ``h`` is simpler than ``g`` using the count_ops metric.
However, we may not like how ``simplify`` (in this case, using
``logcombine``) has created the ``b**(log(1/a) + 1)`` term. A simple way
to reduce this would be to give more weight to powers as operations in
``count_ops``. We can do this by using the ``visual=True`` option:
>>> print(count_ops(g, visual=True))
2*ADD + DIV + 4*LOG + MUL
>>> print(count_ops(h, visual=True))
2*LOG + MUL + POW + SUB
>>> from sympy import Symbol, S
>>> def my_measure(expr):
... POW = Symbol('POW')
... # Discourage powers by giving POW a weight of 10
... count = count_ops(expr, visual=True).subs(POW, 10)
... # Every other operation gets a weight of 1 (the default)
... count = count.replace(Symbol, type(S.One))
... return count
>>> my_measure(g)
8
>>> my_measure(h)
14
>>> 15./8 > 1.7 # 1.7 is the default ratio
True
>>> simplify(g, measure=my_measure)
-log(a)*log(b) + log(a) + log(b)
Note that because ``simplify()`` internally tries many different
simplification strategies and then compares them using the measure
function, we get a completely different result that is still different
from the input expression by doing this.
If ``rational=True``, Floats will be recast as Rationals before simplification.
If ``rational=None``, Floats will be recast as Rationals but the result will
be recast as Floats. If rational=False(default) then nothing will be done
to the Floats.
If ``inverse=True``, it will be assumed that a composition of inverse
functions, such as sin and asin, can be cancelled in any order.
For example, ``asin(sin(x))`` will yield ``x`` without checking whether
x belongs to the set where this relation is true. The default is
False.
Note that ``simplify()`` automatically calls ``doit()`` on the final
expression. You can avoid this behavior by passing ``doit=False`` as
an argument.
Also, it should be noted that simplifying the boolian expression is not
well defined. If the expression prefers automatic evaluation (such as
:obj:`~.Eq()` or :obj:`~.Or()`), simplification will return ``True`` or
``False`` if truth value can be determined. If the expression is not
evaluated by default (such as :obj:`~.Predicate()`), simplification will
not reduce it and you should use :func:`~.refine()` or :func:`~.ask()`
function. This inconsistency will be resolved in future version.
See Also
========
sympy.assumptions.refine.refine : Simplification using assumptions.
sympy.assumptions.ask.ask : Query for boolean expressions using assumptions.
"""
def shorter(*choices):
"""
Return the choice that has the fewest ops. In case of a tie,
the expression listed first is selected.
"""
if not has_variety(choices):
return choices[0]
return min(choices, key=measure)
def done(e):
rv = e.doit() if doit else e
return shorter(rv, collect_abs(rv))
expr = sympify(expr, rational=rational)
kwargs = dict(
ratio=kwargs.get('ratio', ratio),
measure=kwargs.get('measure', measure),
rational=kwargs.get('rational', rational),
inverse=kwargs.get('inverse', inverse),
doit=kwargs.get('doit', doit))
# no routine for Expr needs to check for is_zero
if isinstance(expr, Expr) and expr.is_zero:
return S.Zero if not expr.is_Number else expr
_eval_simplify = getattr(expr, '_eval_simplify', None)
if _eval_simplify is not None:
return _eval_simplify(**kwargs)
original_expr = expr = collect_abs(signsimp(expr))
if not isinstance(expr, Basic) or not expr.args: # XXX: temporary hack
return expr
if inverse and expr.has(Function):
expr = inversecombine(expr)
if not expr.args: # simplified to atomic
return expr
# do deep simplification
handled = Add, Mul, Pow, ExpBase
expr = expr.replace(
# here, checking for x.args is not enough because Basic has
# args but Basic does not always play well with replace, e.g.
# when simultaneous is True found expressions will be masked
# off with a Dummy but not all Basic objects in an expression
# can be replaced with a Dummy
lambda x: isinstance(x, Expr) and x.args and not isinstance(
x, handled),
lambda x: x.func(*[simplify(i, **kwargs) for i in x.args]),
simultaneous=False)
if not isinstance(expr, handled):
return done(expr)
if not expr.is_commutative:
expr = nc_simplify(expr)
# TODO: Apply different strategies, considering expression pattern:
# is it a purely rational function? Is there any trigonometric function?...
# See also https://github.com/sympy/sympy/pull/185.
# rationalize Floats
floats = False
if rational is not False and expr.has(Float):
floats = True
expr = nsimplify(expr, rational=True)
expr = _bottom_up(expr, lambda w: getattr(w, 'normal', lambda: w)())
expr = Mul(*powsimp(expr).as_content_primitive())
_e = cancel(expr)
expr1 = shorter(_e, _mexpand(_e).cancel()) # issue 6829
expr2 = shorter(together(expr, deep=True), together(expr1, deep=True))
if ratio is S.Infinity:
expr = expr2
else:
expr = shorter(expr2, expr1, expr)
if not isinstance(expr, Basic): # XXX: temporary hack
return expr
expr = factor_terms(expr, sign=False)
# must come before `Piecewise` since this introduces more `Piecewise` terms
if expr.has(sign):
expr = expr.rewrite(Abs)
# Deal with Piecewise separately to avoid recursive growth of expressions
if expr.has(Piecewise):
# Fold into a single Piecewise
expr = piecewise_fold(expr)
# Apply doit, if doit=True
expr = done(expr)
# Still a Piecewise?
if expr.has(Piecewise):
# Fold into a single Piecewise, in case doit lead to some
# expressions being Piecewise
expr = piecewise_fold(expr)
# kroneckersimp also affects Piecewise
if expr.has(KroneckerDelta):
expr = kroneckersimp(expr)
# Still a Piecewise?
if expr.has(Piecewise):
from sympy.functions.elementary.piecewise import piecewise_simplify
# Do not apply doit on the segments as it has already
# been done above, but simplify
expr = piecewise_simplify(expr, deep=True, doit=False)
# Still a Piecewise?
if expr.has(Piecewise):
# Try factor common terms
expr = shorter(expr, factor_terms(expr))
# As all expressions have been simplified above with the
# complete simplify, nothing more needs to be done here
return expr
# hyperexpand automatically only works on hypergeometric terms
# Do this after the Piecewise part to avoid recursive expansion
expr = hyperexpand(expr)
if expr.has(KroneckerDelta):
expr = kroneckersimp(expr)
if expr.has(BesselBase):
expr = besselsimp(expr)
if expr.has(TrigonometricFunction, HyperbolicFunction):
expr = trigsimp(expr, deep=True)
if expr.has(log):
expr = shorter(expand_log(expr, deep=True), logcombine(expr))
if expr.has(CombinatorialFunction, gamma):
# expression with gamma functions or non-integer arguments is
# automatically passed to gammasimp
expr = combsimp(expr)
from sympy.concrete.products import Product
from sympy.concrete.summations import Sum
from sympy.integrals.integrals import Integral
if expr.has(Sum):
expr = sum_simplify(expr, **kwargs)
if expr.has(Integral):
expr = expr.xreplace({
i: factor_terms(i) for i in expr.atoms(Integral)})
if expr.has(Product):
expr = product_simplify(expr)
from sympy.physics.units import Quantity
if expr.has(Quantity):
from sympy.physics.units.util import quantity_simplify
expr = quantity_simplify(expr)
short = shorter(powsimp(expr, combine='exp', deep=True), powsimp(expr), expr)
short = shorter(short, cancel(short))
short = shorter(short, factor_terms(short), expand_power_exp(expand_mul(short)))
if short.has(TrigonometricFunction, HyperbolicFunction, ExpBase, exp):
short = exptrigsimp(short)
# get rid of hollow 2-arg Mul factorization
hollow_mul = Transform(
lambda x: Mul(*x.args),
lambda x:
x.is_Mul and
len(x.args) == 2 and
x.args[0].is_Number and
x.args[1].is_Add and
x.is_commutative)
expr = short.xreplace(hollow_mul)
numer, denom = expr.as_numer_denom()
if denom.is_Add:
n, d = fraction(radsimp(1/denom, symbolic=False, max_terms=1))
if n is not S.One:
expr = (numer*n).expand()/d
if expr.could_extract_minus_sign():
n, d = fraction(expr)
if d != 0:
expr = signsimp(-n/(-d))
if measure(expr) > ratio*measure(original_expr):
expr = original_expr
# restore floats
if floats and rational is None:
expr = nfloat(expr, exponent=False)
return done(expr)
def sum_simplify(s, **kwargs):
"""Main function for Sum simplification"""
from sympy.concrete.summations import Sum
if not isinstance(s, Add):
s = s.xreplace({a: sum_simplify(a, **kwargs)
for a in s.atoms(Add) if a.has(Sum)})
s = expand(s)
if not isinstance(s, Add):
return s
terms = s.args
s_t = [] # Sum Terms
o_t = [] # Other Terms
for term in terms:
sum_terms, other = sift(Mul.make_args(term),
lambda i: isinstance(i, Sum), binary=True)
if not sum_terms:
o_t.append(term)
continue
other = [Mul(*other)]
s_t.append(Mul(*(other + [s._eval_simplify(**kwargs) for s in sum_terms])))
result = Add(sum_combine(s_t), *o_t)
return result
def sum_combine(s_t):
"""Helper function for Sum simplification
Attempts to simplify a list of sums, by combining limits / sum function's
returns the simplified sum
"""
from sympy.concrete.summations import Sum
used = [False] * len(s_t)
for method in range(2):
for i, s_term1 in enumerate(s_t):
if not used[i]:
for j, s_term2 in enumerate(s_t):
if not used[j] and i != j:
temp = sum_add(s_term1, s_term2, method)
if isinstance(temp, (Sum, Mul)):
s_t[i] = temp
s_term1 = s_t[i]
used[j] = True
result = S.Zero
for i, s_term in enumerate(s_t):
if not used[i]:
result = Add(result, s_term)
return result
def factor_sum(self, limits=None, radical=False, clear=False, fraction=False, sign=True):
"""Return Sum with constant factors extracted.
If ``limits`` is specified then ``self`` is the summand; the other
keywords are passed to ``factor_terms``.
Examples
========
>>> from sympy import Sum
>>> from sympy.abc import x, y
>>> from sympy.simplify.simplify import factor_sum
>>> s = Sum(x*y, (x, 1, 3))
>>> factor_sum(s)
y*Sum(x, (x, 1, 3))
>>> factor_sum(s.function, s.limits)
y*Sum(x, (x, 1, 3))
"""
# XXX deprecate in favor of direct call to factor_terms
from sympy.concrete.summations import Sum
kwargs = dict(radical=radical, clear=clear,
fraction=fraction, sign=sign)
expr = Sum(self, *limits) if limits else self
return factor_terms(expr, **kwargs)
def sum_add(self, other, method=0):
"""Helper function for Sum simplification"""
from sympy.concrete.summations import Sum
#we know this is something in terms of a constant * a sum
#so we temporarily put the constants inside for simplification
#then simplify the result
def __refactor(val):
args = Mul.make_args(val)
sumv = next(x for x in args if isinstance(x, Sum))
constant = Mul(*[x for x in args if x != sumv])
return Sum(constant * sumv.function, *sumv.limits)
if isinstance(self, Mul):
rself = __refactor(self)
else:
rself = self
if isinstance(other, Mul):
rother = __refactor(other)
else:
rother = other
if type(rself) == type(rother):
if method == 0:
if rself.limits == rother.limits:
return factor_sum(Sum(rself.function + rother.function, *rself.limits))
elif method == 1:
if simplify(rself.function - rother.function) == 0:
if len(rself.limits) == len(rother.limits) == 1:
i = rself.limits[0][0]
x1 = rself.limits[0][1]
y1 = rself.limits[0][2]
j = rother.limits[0][0]
x2 = rother.limits[0][1]
y2 = rother.limits[0][2]
if i == j:
if x2 == y1 + 1:
return factor_sum(Sum(rself.function, (i, x1, y2)))
elif x1 == y2 + 1:
return factor_sum(Sum(rself.function, (i, x2, y1)))
return Add(self, other)
def product_simplify(s):
"""Main function for Product simplification"""
from sympy.concrete.products import Product
terms = Mul.make_args(s)
p_t = [] # Product Terms
o_t = [] # Other Terms
for term in terms:
if isinstance(term, Product):
p_t.append(term)
else:
o_t.append(term)
used = [False] * len(p_t)
for method in range(2):
for i, p_term1 in enumerate(p_t):
if not used[i]:
for j, p_term2 in enumerate(p_t):
if not used[j] and i != j:
if isinstance(product_mul(p_term1, p_term2, method), Product):
p_t[i] = product_mul(p_term1, p_term2, method)
used[j] = True
result = Mul(*o_t)
for i, p_term in enumerate(p_t):
if not used[i]:
result = Mul(result, p_term)
return result
def product_mul(self, other, method=0):
"""Helper function for Product simplification"""
from sympy.concrete.products import Product
if type(self) == type(other):
if method == 0:
if self.limits == other.limits:
return Product(self.function * other.function, *self.limits)
elif method == 1:
if simplify(self.function - other.function) == 0:
if len(self.limits) == len(other.limits) == 1:
i = self.limits[0][0]
x1 = self.limits[0][1]
y1 = self.limits[0][2]
j = other.limits[0][0]
x2 = other.limits[0][1]
y2 = other.limits[0][2]
if i == j:
if x2 == y1 + 1:
return Product(self.function, (i, x1, y2))
elif x1 == y2 + 1:
return Product(self.function, (i, x2, y1))
return Mul(self, other)
def _nthroot_solve(p, n, prec):
"""
helper function for ``nthroot``
It denests ``p**Rational(1, n)`` using its minimal polynomial
"""
from sympy.polys.numberfields.minpoly import _minimal_polynomial_sq
from sympy.solvers import solve
while n % 2 == 0:
p = sqrtdenest(sqrt(p))
n = n // 2
if n == 1:
return p
pn = p**Rational(1, n)
x = Symbol('x')
f = _minimal_polynomial_sq(p, n, x)
if f is None:
return None
sols = solve(f, x)
for sol in sols:
if abs(sol - pn).n() < 1./10**prec:
sol = sqrtdenest(sol)
if _mexpand(sol**n) == p:
return sol
def logcombine(expr, force=False):
"""
Takes logarithms and combines them using the following rules:
- log(x) + log(y) == log(x*y) if both are positive
- a*log(x) == log(x**a) if x is positive and a is real
If ``force`` is ``True`` then the assumptions above will be assumed to hold if
there is no assumption already in place on a quantity. For example, if
``a`` is imaginary or the argument negative, force will not perform a
combination but if ``a`` is a symbol with no assumptions the change will
take place.
Examples
========
>>> from sympy import Symbol, symbols, log, logcombine, I
>>> from sympy.abc import a, x, y, z
>>> logcombine(a*log(x) + log(y) - log(z))
a*log(x) + log(y) - log(z)
>>> logcombine(a*log(x) + log(y) - log(z), force=True)
log(x**a*y/z)
>>> x,y,z = symbols('x,y,z', positive=True)
>>> a = Symbol('a', real=True)
>>> logcombine(a*log(x) + log(y) - log(z))
log(x**a*y/z)
The transformation is limited to factors and/or terms that
contain logs, so the result depends on the initial state of
expansion:
>>> eq = (2 + 3*I)*log(x)
>>> logcombine(eq, force=True) == eq
True
>>> logcombine(eq.expand(), force=True)
log(x**2) + I*log(x**3)
See Also
========
posify: replace all symbols with symbols having positive assumptions
sympy.core.function.expand_log: expand the logarithms of products
and powers; the opposite of logcombine
"""
def f(rv):
if not (rv.is_Add or rv.is_Mul):
return rv
def gooda(a):
# bool to tell whether the leading ``a`` in ``a*log(x)``
# could appear as log(x**a)
return (a is not S.NegativeOne and # -1 *could* go, but we disallow
(a.is_extended_real or force and a.is_extended_real is not False))
def goodlog(l):
# bool to tell whether log ``l``'s argument can combine with others
a = l.args[0]
return a.is_positive or force and a.is_nonpositive is not False
other = []
logs = []
log1 = defaultdict(list)
for a in Add.make_args(rv):
if isinstance(a, log) and goodlog(a):
log1[()].append(([], a))
elif not a.is_Mul:
other.append(a)
else:
ot = []
co = []
lo = []
for ai in a.args:
if ai.is_Rational and ai < 0:
ot.append(S.NegativeOne)
co.append(-ai)
elif isinstance(ai, log) and goodlog(ai):
lo.append(ai)
elif gooda(ai):
co.append(ai)
else:
ot.append(ai)
if len(lo) > 1:
logs.append((ot, co, lo))
elif lo:
log1[tuple(ot)].append((co, lo[0]))
else:
other.append(a)
# if there is only one log in other, put it with the
# good logs
if len(other) == 1 and isinstance(other[0], log):
log1[()].append(([], other.pop()))
# if there is only one log at each coefficient and none have
# an exponent to place inside the log then there is nothing to do
if not logs and all(len(log1[k]) == 1 and log1[k][0] == [] for k in log1):
return rv
# collapse multi-logs as far as possible in a canonical way
# TODO: see if x*log(a)+x*log(a)*log(b) -> x*log(a)*(1+log(b))?
# -- in this case, it's unambiguous, but if it were were a log(c) in
# each term then it's arbitrary whether they are grouped by log(a) or
# by log(c). So for now, just leave this alone; it's probably better to
# let the user decide
for o, e, l in logs:
l = list(ordered(l))
e = log(l.pop(0).args[0]**Mul(*e))
while l:
li = l.pop(0)
e = log(li.args[0]**e)
c, l = Mul(*o), e
if isinstance(l, log): # it should be, but check to be sure
log1[(c,)].append(([], l))
else:
other.append(c*l)
# logs that have the same coefficient can multiply
for k in list(log1.keys()):
log1[Mul(*k)] = log(logcombine(Mul(*[
l.args[0]**Mul(*c) for c, l in log1.pop(k)]),
force=force), evaluate=False)
# logs that have oppositely signed coefficients can divide
for k in ordered(list(log1.keys())):
if k not in log1: # already popped as -k
continue
if -k in log1:
# figure out which has the minus sign; the one with
# more op counts should be the one
num, den = k, -k
if num.count_ops() > den.count_ops():
num, den = den, num
other.append(
num*log(log1.pop(num).args[0]/log1.pop(den).args[0],
evaluate=False))
else:
other.append(k*log1.pop(k))
return Add(*other)
return _bottom_up(expr, f)
def inversecombine(expr):
"""Simplify the composition of a function and its inverse.
Explanation
===========
No attention is paid to whether the inverse is a left inverse or a
right inverse; thus, the result will in general not be equivalent
to the original expression.
Examples
========
>>> from sympy.simplify.simplify import inversecombine
>>> from sympy import asin, sin, log, exp
>>> from sympy.abc import x
>>> inversecombine(asin(sin(x)))
x
>>> inversecombine(2*log(exp(3*x)))
6*x
"""
def f(rv):
if isinstance(rv, log):
if isinstance(rv.args[0], exp) or (rv.args[0].is_Pow and rv.args[0].base == S.Exp1):
rv = rv.args[0].exp
elif rv.is_Function and hasattr(rv, "inverse"):
if (len(rv.args) == 1 and len(rv.args[0].args) == 1 and
isinstance(rv.args[0], rv.inverse(argindex=1))):
rv = rv.args[0].args[0]
if rv.is_Pow and rv.base == S.Exp1:
if isinstance(rv.exp, log):
rv = rv.exp.args[0]
return rv
return _bottom_up(expr, f)
def kroneckersimp(expr):
"""
Simplify expressions with KroneckerDelta.
The only simplification currently attempted is to identify multiplicative cancellation:
Examples
========
>>> from sympy import KroneckerDelta, kroneckersimp
>>> from sympy.abc import i
>>> kroneckersimp(1 + KroneckerDelta(0, i) * KroneckerDelta(1, i))
1
"""
def args_cancel(args1, args2):
for i1 in range(2):
for i2 in range(2):
a1 = args1[i1]
a2 = args2[i2]
a3 = args1[(i1 + 1) % 2]
a4 = args2[(i2 + 1) % 2]
if Eq(a1, a2) is S.true and Eq(a3, a4) is S.false:
return True
return False
def cancel_kronecker_mul(m):
args = m.args
deltas = [a for a in args if isinstance(a, KroneckerDelta)]
for delta1, delta2 in subsets(deltas, 2):
args1 = delta1.args
args2 = delta2.args
if args_cancel(args1, args2):
return S.Zero * m # In case of oo etc
return m
if not expr.has(KroneckerDelta):
return expr
if expr.has(Piecewise):
expr = expr.rewrite(KroneckerDelta)
newexpr = expr
expr = None
while newexpr != expr:
expr = newexpr
newexpr = expr.replace(lambda e: isinstance(e, Mul), cancel_kronecker_mul)
return expr
def besselsimp(expr):
"""
Simplify bessel-type functions.
Explanation
===========
This routine tries to simplify bessel-type functions. Currently it only
works on the Bessel J and I functions, however. It works by looking at all
such functions in turn, and eliminating factors of "I" and "-1" (actually
their polar equivalents) in front of the argument. Then, functions of
half-integer order are rewritten using strigonometric functions and
functions of integer order (> 1) are rewritten using functions
of low order. Finally, if the expression was changed, compute
factorization of the result with factor().
>>> from sympy import besselj, besseli, besselsimp, polar_lift, I, S
>>> from sympy.abc import z, nu
>>> besselsimp(besselj(nu, z*polar_lift(-1)))
exp(I*pi*nu)*besselj(nu, z)
>>> besselsimp(besseli(nu, z*polar_lift(-I)))
exp(-I*pi*nu/2)*besselj(nu, z)
>>> besselsimp(besseli(S(-1)/2, z))
sqrt(2)*cosh(z)/(sqrt(pi)*sqrt(z))
>>> besselsimp(z*besseli(0, z) + z*(besseli(2, z))/2 + besseli(1, z))
3*z*besseli(0, z)/2
"""
# TODO
# - better algorithm?
# - simplify (cos(pi*b)*besselj(b,z) - besselj(-b,z))/sin(pi*b) ...
# - use contiguity relations?
def replacer(fro, to, factors):
factors = set(factors)
def repl(nu, z):
if factors.intersection(Mul.make_args(z)):
return to(nu, z)
return fro(nu, z)
return repl
def torewrite(fro, to):
def tofunc(nu, z):
return fro(nu, z).rewrite(to)
return tofunc
def tominus(fro):
def tofunc(nu, z):
return exp(I*pi*nu)*fro(nu, exp_polar(-I*pi)*z)
return tofunc
orig_expr = expr
ifactors = [I, exp_polar(I*pi/2), exp_polar(-I*pi/2)]
expr = expr.replace(
besselj, replacer(besselj,
torewrite(besselj, besseli), ifactors))
expr = expr.replace(
besseli, replacer(besseli,
torewrite(besseli, besselj), ifactors))
minusfactors = [-1, exp_polar(I*pi)]
expr = expr.replace(
besselj, replacer(besselj, tominus(besselj), minusfactors))
expr = expr.replace(
besseli, replacer(besseli, tominus(besseli), minusfactors))
z0 = Dummy('z')
def expander(fro):
def repl(nu, z):
if (nu % 1) == S.Half:
return simplify(trigsimp(unpolarify(
fro(nu, z0).rewrite(besselj).rewrite(jn).expand(
func=True)).subs(z0, z)))
elif nu.is_Integer and nu > 1:
return fro(nu, z).expand(func=True)
return fro(nu, z)
return repl
expr = expr.replace(besselj, expander(besselj))
expr = expr.replace(bessely, expander(bessely))
expr = expr.replace(besseli, expander(besseli))
expr = expr.replace(besselk, expander(besselk))
def _bessel_simp_recursion(expr):
def _use_recursion(bessel, expr):
while True:
bessels = expr.find(lambda x: isinstance(x, bessel))
try:
for ba in sorted(bessels, key=lambda x: re(x.args[0])):
a, x = ba.args
bap1 = bessel(a+1, x)
bap2 = bessel(a+2, x)
if expr.has(bap1) and expr.has(bap2):
expr = expr.subs(ba, 2*(a+1)/x*bap1 - bap2)
break
else:
return expr
except (ValueError, TypeError):
return expr
if expr.has(besselj):
expr = _use_recursion(besselj, expr)
if expr.has(bessely):
expr = _use_recursion(bessely, expr)
return expr
expr = _bessel_simp_recursion(expr)
if expr != orig_expr:
expr = expr.factor()
return expr
def nthroot(expr, n, max_len=4, prec=15):
"""
Compute a real nth-root of a sum of surds.
Parameters
==========
expr : sum of surds
n : integer
max_len : maximum number of surds passed as constants to ``nsimplify``
Algorithm
=========
First ``nsimplify`` is used to get a candidate root; if it is not a
root the minimal polynomial is computed; the answer is one of its
roots.
Examples
========
>>> from sympy.simplify.simplify import nthroot
>>> from sympy import sqrt
>>> nthroot(90 + 34*sqrt(7), 3)
sqrt(7) + 3
"""
expr = sympify(expr)
n = sympify(n)
p = expr**Rational(1, n)
if not n.is_integer:
return p
if not _is_sum_surds(expr):
return p
surds = []
coeff_muls = [x.as_coeff_Mul() for x in expr.args]
for x, y in coeff_muls:
if not x.is_rational:
return p
if y is S.One:
continue
if not (y.is_Pow and y.exp == S.Half and y.base.is_integer):
return p
surds.append(y)
surds.sort()
surds = surds[:max_len]
if expr < 0 and n % 2 == 1:
p = (-expr)**Rational(1, n)
a = nsimplify(p, constants=surds)
res = a if _mexpand(a**n) == _mexpand(-expr) else p
return -res
a = nsimplify(p, constants=surds)
if _mexpand(a) is not _mexpand(p) and _mexpand(a**n) == _mexpand(expr):
return _mexpand(a)
expr = _nthroot_solve(expr, n, prec)
if expr is None:
return p
return expr
def nsimplify(expr, constants=(), tolerance=None, full=False, rational=None,
rational_conversion='base10'):
"""
Find a simple representation for a number or, if there are free symbols or
if ``rational=True``, then replace Floats with their Rational equivalents. If
no change is made and rational is not False then Floats will at least be
converted to Rationals.
Explanation
===========
For numerical expressions, a simple formula that numerically matches the
given numerical expression is sought (and the input should be possible
to evalf to a precision of at least 30 digits).
Optionally, a list of (rationally independent) constants to
include in the formula may be given.
A lower tolerance may be set to find less exact matches. If no tolerance
is given then the least precise value will set the tolerance (e.g. Floats
default to 15 digits of precision, so would be tolerance=10**-15).
With ``full=True``, a more extensive search is performed
(this is useful to find simpler numbers when the tolerance
is set low).
When converting to rational, if rational_conversion='base10' (the default), then
convert floats to rationals using their base-10 (string) representation.
When rational_conversion='exact' it uses the exact, base-2 representation.
Examples
========
>>> from sympy import nsimplify, sqrt, GoldenRatio, exp, I, pi
>>> nsimplify(4/(1+sqrt(5)), [GoldenRatio])
-2 + 2*GoldenRatio
>>> nsimplify((1/(exp(3*pi*I/5)+1)))
1/2 - I*sqrt(sqrt(5)/10 + 1/4)
>>> nsimplify(I**I, [pi])
exp(-pi/2)
>>> nsimplify(pi, tolerance=0.01)
22/7
>>> nsimplify(0.333333333333333, rational=True, rational_conversion='exact')
6004799503160655/18014398509481984
>>> nsimplify(0.333333333333333, rational=True)
1/3
See Also
========
sympy.core.function.nfloat
"""
try:
return sympify(as_int(expr))
except (TypeError, ValueError):
pass
expr = sympify(expr).xreplace({
Float('inf'): S.Infinity,
Float('-inf'): S.NegativeInfinity,
})
if expr is S.Infinity or expr is S.NegativeInfinity:
return expr
if rational or expr.free_symbols:
return _real_to_rational(expr, tolerance, rational_conversion)
# SymPy's default tolerance for Rationals is 15; other numbers may have
# lower tolerances set, so use them to pick the largest tolerance if None
# was given
if tolerance is None:
tolerance = 10**-min([15] +
[mpmath.libmp.libmpf.prec_to_dps(n._prec)
for n in expr.atoms(Float)])
# XXX should prec be set independent of tolerance or should it be computed
# from tolerance?
prec = 30
bprec = int(prec*3.33)
constants_dict = {}
for constant in constants:
constant = sympify(constant)
v = constant.evalf(prec)
if not v.is_Float:
raise ValueError("constants must be real-valued")
constants_dict[str(constant)] = v._to_mpmath(bprec)
exprval = expr.evalf(prec, chop=True)
re, im = exprval.as_real_imag()
# safety check to make sure that this evaluated to a number
if not (re.is_Number and im.is_Number):
return expr
def nsimplify_real(x):
orig = mpmath.mp.dps
xv = x._to_mpmath(bprec)
try:
# We'll be happy with low precision if a simple fraction
if not (tolerance or full):
mpmath.mp.dps = 15
rat = mpmath.pslq([xv, 1])
if rat is not None:
return Rational(-int(rat[1]), int(rat[0]))
mpmath.mp.dps = prec
newexpr = mpmath.identify(xv, constants=constants_dict,
tol=tolerance, full=full)
if not newexpr:
raise ValueError
if full:
newexpr = newexpr[0]
expr = sympify(newexpr)
if x and not expr: # don't let x become 0
raise ValueError
if expr.is_finite is False and xv not in [mpmath.inf, mpmath.ninf]:
raise ValueError
return expr
finally:
# even though there are returns above, this is executed
# before leaving
mpmath.mp.dps = orig
try:
if re:
re = nsimplify_real(re)
if im:
im = nsimplify_real(im)
except ValueError:
if rational is None:
return _real_to_rational(expr, rational_conversion=rational_conversion)
return expr
rv = re + im*S.ImaginaryUnit
# if there was a change or rational is explicitly not wanted
# return the value, else return the Rational representation
if rv != expr or rational is False:
return rv
return _real_to_rational(expr, rational_conversion=rational_conversion)
def _real_to_rational(expr, tolerance=None, rational_conversion='base10'):
"""
Replace all reals in expr with rationals.
Examples
========
>>> from sympy.simplify.simplify import _real_to_rational
>>> from sympy.abc import x
>>> _real_to_rational(.76 + .1*x**.5)
sqrt(x)/10 + 19/25
If rational_conversion='base10', this uses the base-10 string. If
rational_conversion='exact', the exact, base-2 representation is used.
>>> _real_to_rational(0.333333333333333, rational_conversion='exact')
6004799503160655/18014398509481984
>>> _real_to_rational(0.333333333333333)
1/3
"""
expr = _sympify(expr)
inf = Float('inf')
p = expr
reps = {}
reduce_num = None
if tolerance is not None and tolerance < 1:
reduce_num = ceiling(1/tolerance)
for fl in p.atoms(Float):
key = fl
if reduce_num is not None:
r = Rational(fl).limit_denominator(reduce_num)
elif (tolerance is not None and tolerance >= 1 and
fl.is_Integer is False):
r = Rational(tolerance*round(fl/tolerance)
).limit_denominator(int(tolerance))
else:
if rational_conversion == 'exact':
r = Rational(fl)
reps[key] = r
continue
elif rational_conversion != 'base10':
raise ValueError("rational_conversion must be 'base10' or 'exact'")
r = nsimplify(fl, rational=False)
# e.g. log(3).n() -> log(3) instead of a Rational
if fl and not r:
r = Rational(fl)
elif not r.is_Rational:
if fl in (inf, -inf):
r = S.ComplexInfinity
elif fl < 0:
fl = -fl
d = Pow(10, int(mpmath.log(fl)/mpmath.log(10)))
r = -Rational(str(fl/d))*d
elif fl > 0:
d = Pow(10, int(mpmath.log(fl)/mpmath.log(10)))
r = Rational(str(fl/d))*d
else:
r = S.Zero
reps[key] = r
return p.subs(reps, simultaneous=True)
def clear_coefficients(expr, rhs=S.Zero):
"""Return `p, r` where `p` is the expression obtained when Rational
additive and multiplicative coefficients of `expr` have been stripped
away in a naive fashion (i.e. without simplification). The operations
needed to remove the coefficients will be applied to `rhs` and returned
as `r`.
Examples
========
>>> from sympy.simplify.simplify import clear_coefficients
>>> from sympy.abc import x, y
>>> from sympy import Dummy
>>> expr = 4*y*(6*x + 3)
>>> clear_coefficients(expr - 2)
(y*(2*x + 1), 1/6)
When solving 2 or more expressions like `expr = a`,
`expr = b`, etc..., it is advantageous to provide a Dummy symbol
for `rhs` and simply replace it with `a`, `b`, etc... in `r`.
>>> rhs = Dummy('rhs')
>>> clear_coefficients(expr, rhs)
(y*(2*x + 1), _rhs/12)
>>> _[1].subs(rhs, 2)
1/6
"""
was = None
free = expr.free_symbols
if expr.is_Rational:
return (S.Zero, rhs - expr)
while expr and was != expr:
was = expr
m, expr = (
expr.as_content_primitive()
if free else
factor_terms(expr).as_coeff_Mul(rational=True))
rhs /= m
c, expr = expr.as_coeff_Add(rational=True)
rhs -= c
expr = signsimp(expr, evaluate = False)
if expr.could_extract_minus_sign():
expr = -expr
rhs = -rhs
return expr, rhs
def nc_simplify(expr, deep=True):
'''
Simplify a non-commutative expression composed of multiplication
and raising to a power by grouping repeated subterms into one power.
Priority is given to simplifications that give the fewest number
of arguments in the end (for example, in a*b*a*b*c*a*b*c simplifying
to (a*b)**2*c*a*b*c gives 5 arguments while a*b*(a*b*c)**2 has 3).
If ``expr`` is a sum of such terms, the sum of the simplified terms
is returned.
Keyword argument ``deep`` controls whether or not subexpressions
nested deeper inside the main expression are simplified. See examples
below. Setting `deep` to `False` can save time on nested expressions
that do not need simplifying on all levels.
Examples
========
>>> from sympy import symbols
>>> from sympy.simplify.simplify import nc_simplify
>>> a, b, c = symbols("a b c", commutative=False)
>>> nc_simplify(a*b*a*b*c*a*b*c)
a*b*(a*b*c)**2
>>> expr = a**2*b*a**4*b*a**4
>>> nc_simplify(expr)
a**2*(b*a**4)**2
>>> nc_simplify(a*b*a*b*c**2*(a*b)**2*c**2)
((a*b)**2*c**2)**2
>>> nc_simplify(a*b*a*b + 2*a*c*a**2*c*a**2*c*a)
(a*b)**2 + 2*(a*c*a)**3
>>> nc_simplify(b**-1*a**-1*(a*b)**2)
a*b
>>> nc_simplify(a**-1*b**-1*c*a)
(b*a)**(-1)*c*a
>>> expr = (a*b*a*b)**2*a*c*a*c
>>> nc_simplify(expr)
(a*b)**4*(a*c)**2
>>> nc_simplify(expr, deep=False)
(a*b*a*b)**2*(a*c)**2
'''
from sympy.matrices.expressions import (MatrixExpr, MatAdd, MatMul,
MatPow, MatrixSymbol)
if isinstance(expr, MatrixExpr):
expr = expr.doit(inv_expand=False)
_Add, _Mul, _Pow, _Symbol = MatAdd, MatMul, MatPow, MatrixSymbol
else:
_Add, _Mul, _Pow, _Symbol = Add, Mul, Pow, Symbol
# =========== Auxiliary functions ========================
def _overlaps(args):
# Calculate a list of lists m such that m[i][j] contains the lengths
# of all possible overlaps between args[:i+1] and args[i+1+j:].
# An overlap is a suffix of the prefix that matches a prefix
# of the suffix.
# For example, let expr=c*a*b*a*b*a*b*a*b. Then m[3][0] contains
# the lengths of overlaps of c*a*b*a*b with a*b*a*b. The overlaps
# are a*b*a*b, a*b and the empty word so that m[3][0]=[4,2,0].
# All overlaps rather than only the longest one are recorded
# because this information helps calculate other overlap lengths.
m = [[([1, 0] if a == args[0] else [0]) for a in args[1:]]]
for i in range(1, len(args)):
overlaps = []
j = 0
for j in range(len(args) - i - 1):
overlap = []
for v in m[i-1][j+1]:
if j + i + 1 + v < len(args) and args[i] == args[j+i+1+v]:
overlap.append(v + 1)
overlap += [0]
overlaps.append(overlap)
m.append(overlaps)
return m
def _reduce_inverses(_args):
# replace consecutive negative powers by an inverse
# of a product of positive powers, e.g. a**-1*b**-1*c
# will simplify to (a*b)**-1*c;
# return that new args list and the number of negative
# powers in it (inv_tot)
inv_tot = 0 # total number of inverses
inverses = []
args = []
for arg in _args:
if isinstance(arg, _Pow) and arg.args[1] < 0:
inverses = [arg**-1] + inverses
inv_tot += 1
else:
if len(inverses) == 1:
args.append(inverses[0]**-1)
elif len(inverses) > 1:
args.append(_Pow(_Mul(*inverses), -1))
inv_tot -= len(inverses) - 1
inverses = []
args.append(arg)
if inverses:
args.append(_Pow(_Mul(*inverses), -1))
inv_tot -= len(inverses) - 1
return inv_tot, tuple(args)
def get_score(s):
# compute the number of arguments of s
# (including in nested expressions) overall
# but ignore exponents
if isinstance(s, _Pow):
return get_score(s.args[0])
elif isinstance(s, (_Add, _Mul)):
return sum([get_score(a) for a in s.args])
return 1
def compare(s, alt_s):
# compare two possible simplifications and return a
# "better" one
if s != alt_s and get_score(alt_s) < get_score(s):
return alt_s
return s
# ========================================================
if not isinstance(expr, (_Add, _Mul, _Pow)) or expr.is_commutative:
return expr
args = expr.args[:]
if isinstance(expr, _Pow):
if deep:
return _Pow(nc_simplify(args[0]), args[1]).doit()
else:
return expr
elif isinstance(expr, _Add):
return _Add(*[nc_simplify(a, deep=deep) for a in args]).doit()
else:
# get the non-commutative part
c_args, args = expr.args_cnc()
com_coeff = Mul(*c_args)
if com_coeff != 1:
return com_coeff*nc_simplify(expr/com_coeff, deep=deep)
inv_tot, args = _reduce_inverses(args)
# if most arguments are negative, work with the inverse
# of the expression, e.g. a**-1*b*a**-1*c**-1 will become
# (c*a*b**-1*a)**-1 at the end so can work with c*a*b**-1*a
invert = False
if inv_tot > len(args)/2:
invert = True
args = [a**-1 for a in args[::-1]]
if deep:
args = tuple(nc_simplify(a) for a in args)
m = _overlaps(args)
# simps will be {subterm: end} where `end` is the ending
# index of a sequence of repetitions of subterm;
# this is for not wasting time with subterms that are part
# of longer, already considered sequences
simps = {}
post = 1
pre = 1
# the simplification coefficient is the number of
# arguments by which contracting a given sequence
# would reduce the word; e.g. in a*b*a*b*c*a*b*c,
# contracting a*b*a*b to (a*b)**2 removes 3 arguments
# while a*b*c*a*b*c to (a*b*c)**2 removes 6. It's
# better to contract the latter so simplification
# with a maximum simplification coefficient will be chosen
max_simp_coeff = 0
simp = None # information about future simplification
for i in range(1, len(args)):
simp_coeff = 0
l = 0 # length of a subterm
p = 0 # the power of a subterm
if i < len(args) - 1:
rep = m[i][0]
start = i # starting index of the repeated sequence
end = i+1 # ending index of the repeated sequence
if i == len(args)-1 or rep == [0]:
# no subterm is repeated at this stage, at least as
# far as the arguments are concerned - there may be
# a repetition if powers are taken into account
if (isinstance(args[i], _Pow) and
not isinstance(args[i].args[0], _Symbol)):
subterm = args[i].args[0].args
l = len(subterm)
if args[i-l:i] == subterm:
# e.g. a*b in a*b*(a*b)**2 is not repeated
# in args (= [a, b, (a*b)**2]) but it
# can be matched here
p += 1
start -= l
if args[i+1:i+1+l] == subterm:
# e.g. a*b in (a*b)**2*a*b
p += 1
end += l
if p:
p += args[i].args[1]
else:
continue
else:
l = rep[0] # length of the longest repeated subterm at this point
start -= l - 1
subterm = args[start:end]
p = 2
end += l
if subterm in simps and simps[subterm] >= start:
# the subterm is part of a sequence that
# has already been considered
continue
# count how many times it's repeated
while end < len(args):
if l in m[end-1][0]:
p += 1
end += l
elif isinstance(args[end], _Pow) and args[end].args[0].args == subterm:
# for cases like a*b*a*b*(a*b)**2*a*b
p += args[end].args[1]
end += 1
else:
break
# see if another match can be made, e.g.
# for b*a**2 in b*a**2*b*a**3 or a*b in
# a**2*b*a*b
pre_exp = 0
pre_arg = 1
if start - l >= 0 and args[start-l+1:start] == subterm[1:]:
if isinstance(subterm[0], _Pow):
pre_arg = subterm[0].args[0]
exp = subterm[0].args[1]
else:
pre_arg = subterm[0]
exp = 1
if isinstance(args[start-l], _Pow) and args[start-l].args[0] == pre_arg:
pre_exp = args[start-l].args[1] - exp
start -= l
p += 1
elif args[start-l] == pre_arg:
pre_exp = 1 - exp
start -= l
p += 1
post_exp = 0
post_arg = 1
if end + l - 1 < len(args) and args[end:end+l-1] == subterm[:-1]:
if isinstance(subterm[-1], _Pow):
post_arg = subterm[-1].args[0]
exp = subterm[-1].args[1]
else:
post_arg = subterm[-1]
exp = 1
if isinstance(args[end+l-1], _Pow) and args[end+l-1].args[0] == post_arg:
post_exp = args[end+l-1].args[1] - exp
end += l
p += 1
elif args[end+l-1] == post_arg:
post_exp = 1 - exp
end += l
p += 1
# Consider a*b*a**2*b*a**2*b*a:
# b*a**2 is explicitly repeated, but note
# that in this case a*b*a is also repeated
# so there are two possible simplifications:
# a*(b*a**2)**3*a**-1 or (a*b*a)**3
# The latter is obviously simpler.
# But in a*b*a**2*b**2*a**2 the simplifications are
# a*(b*a**2)**2 and (a*b*a)**3*a in which case
# it's better to stick with the shorter subterm
if post_exp and exp % 2 == 0 and start > 0:
exp = exp/2
_pre_exp = 1
_post_exp = 1
if isinstance(args[start-1], _Pow) and args[start-1].args[0] == post_arg:
_post_exp = post_exp + exp
_pre_exp = args[start-1].args[1] - exp
elif args[start-1] == post_arg:
_post_exp = post_exp + exp
_pre_exp = 1 - exp
if _pre_exp == 0 or _post_exp == 0:
if not pre_exp:
start -= 1
post_exp = _post_exp
pre_exp = _pre_exp
pre_arg = post_arg
subterm = (post_arg**exp,) + subterm[:-1] + (post_arg**exp,)
simp_coeff += end-start
if post_exp:
simp_coeff -= 1
if pre_exp:
simp_coeff -= 1
simps[subterm] = end
if simp_coeff > max_simp_coeff:
max_simp_coeff = simp_coeff
simp = (start, _Mul(*subterm), p, end, l)
pre = pre_arg**pre_exp
post = post_arg**post_exp
if simp:
subterm = _Pow(nc_simplify(simp[1], deep=deep), simp[2])
pre = nc_simplify(_Mul(*args[:simp[0]])*pre, deep=deep)
post = post*nc_simplify(_Mul(*args[simp[3]:]), deep=deep)
simp = pre*subterm*post
if pre != 1 or post != 1:
# new simplifications may be possible but no need
# to recurse over arguments
simp = nc_simplify(simp, deep=False)
else:
simp = _Mul(*args)
if invert:
simp = _Pow(simp, -1)
# see if factor_nc(expr) is simplified better
if not isinstance(expr, MatrixExpr):
f_expr = factor_nc(expr)
if f_expr != expr:
alt_simp = nc_simplify(f_expr, deep=deep)
simp = compare(simp, alt_simp)
else:
simp = simp.doit(inv_expand=False)
return simp
def dotprodsimp(expr, withsimp=False):
"""Simplification for a sum of products targeted at the kind of blowup that
occurs during summation of products. Intended to reduce expression blowup
during matrix multiplication or other similar operations. Only works with
algebraic expressions and does not recurse into non.
Parameters
==========
withsimp : bool, optional
Specifies whether a flag should be returned along with the expression
to indicate roughly whether simplification was successful. It is used
in ``MatrixArithmetic._eval_pow_by_recursion`` to avoid attempting to
simplify an expression repetitively which does not simplify.
"""
def count_ops_alg(expr):
"""Optimized count algebraic operations with no recursion into
non-algebraic args that ``core.function.count_ops`` does. Also returns
whether rational functions may be present according to negative
exponents of powers or non-number fractions.
Returns
=======
ops, ratfunc : int, bool
``ops`` is the number of algebraic operations starting at the top
level expression (not recursing into non-alg children). ``ratfunc``
specifies whether the expression MAY contain rational functions
which ``cancel`` MIGHT optimize.
"""
ops = 0
args = [expr]
ratfunc = False
while args:
a = args.pop()
if not isinstance(a, Basic):
continue
if a.is_Rational:
if a is not S.One: # -1/3 = NEG + DIV
ops += bool (a.p < 0) + bool (a.q != 1)
elif a.is_Mul:
if a.could_extract_minus_sign():
ops += 1
if a.args[0] is S.NegativeOne:
a = a.as_two_terms()[1]
else:
a = -a
n, d = fraction(a)
if n.is_Integer:
ops += 1 + bool (n < 0)
args.append(d) # won't be -Mul but could be Add
elif d is not S.One:
if not d.is_Integer:
args.append(d)
ratfunc=True
ops += 1
args.append(n) # could be -Mul
else:
ops += len(a.args) - 1
args.extend(a.args)
elif a.is_Add:
laargs = len(a.args)
negs = 0
for ai in a.args:
if ai.could_extract_minus_sign():
negs += 1
ai = -ai
args.append(ai)
ops += laargs - (negs != laargs) # -x - y = NEG + SUB
elif a.is_Pow:
ops += 1
args.append(a.base)
if not ratfunc:
ratfunc = a.exp.is_negative is not False
return ops, ratfunc
def nonalg_subs_dummies(expr, dummies):
"""Substitute dummy variables for non-algebraic expressions to avoid
evaluation of non-algebraic terms that ``polys.polytools.cancel`` does.
"""
if not expr.args:
return expr
if expr.is_Add or expr.is_Mul or expr.is_Pow:
args = None
for i, a in enumerate(expr.args):
c = nonalg_subs_dummies(a, dummies)
if c is a:
continue
if args is None:
args = list(expr.args)
args[i] = c
if args is None:
return expr
return expr.func(*args)
return dummies.setdefault(expr, Dummy())
simplified = False # doesn't really mean simplified, rather "can simplify again"
if isinstance(expr, Basic) and (expr.is_Add or expr.is_Mul or expr.is_Pow):
expr2 = expr.expand(deep=True, modulus=None, power_base=False,
power_exp=False, mul=True, log=False, multinomial=True, basic=False)
if expr2 != expr:
expr = expr2
simplified = True
exprops, ratfunc = count_ops_alg(expr)
if exprops >= 6: # empirically tested cutoff for expensive simplification
if ratfunc:
dummies = {}
expr2 = nonalg_subs_dummies(expr, dummies)
if expr2 is expr or count_ops_alg(expr2)[0] >= 6: # check again after substitution
expr3 = cancel(expr2)
if expr3 != expr2:
expr = expr3.subs([(d, e) for e, d in dummies.items()])
simplified = True
# very special case: x/(x-1) - 1/(x-1) -> 1
elif (exprops == 5 and expr.is_Add and expr.args [0].is_Mul and
expr.args [1].is_Mul and expr.args [0].args [-1].is_Pow and
expr.args [1].args [-1].is_Pow and
expr.args [0].args [-1].exp is S.NegativeOne and
expr.args [1].args [-1].exp is S.NegativeOne):
expr2 = together (expr)
expr2ops = count_ops_alg(expr2)[0]
if expr2ops < exprops:
expr = expr2
simplified = True
else:
simplified = True
return (expr, simplified) if withsimp else expr
bottom_up = deprecated(
useinstead="sympy.core.traversal.bottom_up",
deprecated_since_version="1.10", issue=22288)(_bottom_up)
walk = deprecated(
useinstead="sympy.core.traversal.walk",
deprecated_since_version="1.10", issue=22288)(_walk)
|
54bcc2af4960b25afdf622464e8832e4da3486d31876afbab55e219e96df1620 | from collections import defaultdict
from functools import reduce
from sympy.core.function import expand_log, count_ops, _coeff_isneg
from sympy.core import sympify, Basic, Dummy, S, Add, Mul, Pow, expand_mul, factor_terms
from sympy.core.sorting import ordered, default_sort_key
from sympy.core.numbers import Integer, Rational
from sympy.core.mul import prod, _keep_coeff
from sympy.core.rules import Transform
from sympy.functions import exp_polar, exp, log, root, polarify, unpolarify
from sympy.polys import lcm, gcd
from sympy.ntheory.factor_ import multiplicity
def powsimp(expr, deep=False, combine='all', force=False, measure=count_ops):
"""
reduces expression by combining powers with similar bases and exponents.
Explanation
===========
If ``deep`` is ``True`` then powsimp() will also simplify arguments of
functions. By default ``deep`` is set to ``False``.
If ``force`` is ``True`` then bases will be combined without checking for
assumptions, e.g. sqrt(x)*sqrt(y) -> sqrt(x*y) which is not true
if x and y are both negative.
You can make powsimp() only combine bases or only combine exponents by
changing combine='base' or combine='exp'. By default, combine='all',
which does both. combine='base' will only combine::
a a a 2x x
x * y => (x*y) as well as things like 2 => 4
and combine='exp' will only combine
::
a b (a + b)
x * x => x
combine='exp' will strictly only combine exponents in the way that used
to be automatic. Also use deep=True if you need the old behavior.
When combine='all', 'exp' is evaluated first. Consider the first
example below for when there could be an ambiguity relating to this.
This is done so things like the second example can be completely
combined. If you want 'base' combined first, do something like
powsimp(powsimp(expr, combine='base'), combine='exp').
Examples
========
>>> from sympy import powsimp, exp, log, symbols
>>> from sympy.abc import x, y, z, n
>>> powsimp(x**y*x**z*y**z, combine='all')
x**(y + z)*y**z
>>> powsimp(x**y*x**z*y**z, combine='exp')
x**(y + z)*y**z
>>> powsimp(x**y*x**z*y**z, combine='base', force=True)
x**y*(x*y)**z
>>> powsimp(x**z*x**y*n**z*n**y, combine='all', force=True)
(n*x)**(y + z)
>>> powsimp(x**z*x**y*n**z*n**y, combine='exp')
n**(y + z)*x**(y + z)
>>> powsimp(x**z*x**y*n**z*n**y, combine='base', force=True)
(n*x)**y*(n*x)**z
>>> x, y = symbols('x y', positive=True)
>>> powsimp(log(exp(x)*exp(y)))
log(exp(x)*exp(y))
>>> powsimp(log(exp(x)*exp(y)), deep=True)
x + y
Radicals with Mul bases will be combined if combine='exp'
>>> from sympy import sqrt
>>> x, y = symbols('x y')
Two radicals are automatically joined through Mul:
>>> a=sqrt(x*sqrt(y))
>>> a*a**3 == a**4
True
But if an integer power of that radical has been
autoexpanded then Mul does not join the resulting factors:
>>> a**4 # auto expands to a Mul, no longer a Pow
x**2*y
>>> _*a # so Mul doesn't combine them
x**2*y*sqrt(x*sqrt(y))
>>> powsimp(_) # but powsimp will
(x*sqrt(y))**(5/2)
>>> powsimp(x*y*a) # but won't when doing so would violate assumptions
x*y*sqrt(x*sqrt(y))
"""
from sympy.matrices.expressions.matexpr import MatrixSymbol
def recurse(arg, **kwargs):
_deep = kwargs.get('deep', deep)
_combine = kwargs.get('combine', combine)
_force = kwargs.get('force', force)
_measure = kwargs.get('measure', measure)
return powsimp(arg, _deep, _combine, _force, _measure)
expr = sympify(expr)
if (not isinstance(expr, Basic) or isinstance(expr, MatrixSymbol) or (
expr.is_Atom or expr in (exp_polar(0), exp_polar(1)))):
return expr
if deep or expr.is_Add or expr.is_Mul and _y not in expr.args:
expr = expr.func(*[recurse(w) for w in expr.args])
if expr.is_Pow:
return recurse(expr*_y, deep=False)/_y
if not expr.is_Mul:
return expr
# handle the Mul
if combine in ('exp', 'all'):
# Collect base/exp data, while maintaining order in the
# non-commutative parts of the product
c_powers = defaultdict(list)
nc_part = []
newexpr = []
coeff = S.One
for term in expr.args:
if term.is_Rational:
coeff *= term
continue
if term.is_Pow:
term = _denest_pow(term)
if term.is_commutative:
b, e = term.as_base_exp()
if deep:
b, e = [recurse(i) for i in [b, e]]
if b.is_Pow or isinstance(b, exp):
# don't let smthg like sqrt(x**a) split into x**a, 1/2
# or else it will be joined as x**(a/2) later
b, e = b**e, S.One
c_powers[b].append(e)
else:
# This is the logic that combines exponents for equal,
# but non-commutative bases: A**x*A**y == A**(x+y).
if nc_part:
b1, e1 = nc_part[-1].as_base_exp()
b2, e2 = term.as_base_exp()
if (b1 == b2 and
e1.is_commutative and e2.is_commutative):
nc_part[-1] = Pow(b1, Add(e1, e2))
continue
nc_part.append(term)
# add up exponents of common bases
for b, e in ordered(iter(c_powers.items())):
# allow 2**x/4 -> 2**(x - 2); don't do this when b and e are
# Numbers since autoevaluation will undo it, e.g.
# 2**(1/3)/4 -> 2**(1/3 - 2) -> 2**(1/3)/4
if (b and b.is_Rational and not all(ei.is_Number for ei in e) and \
coeff is not S.One and
b not in (S.One, S.NegativeOne)):
m = multiplicity(abs(b), abs(coeff))
if m:
e.append(m)
coeff /= b**m
c_powers[b] = Add(*e)
if coeff is not S.One:
if coeff in c_powers:
c_powers[coeff] += S.One
else:
c_powers[coeff] = S.One
# convert to plain dictionary
c_powers = dict(c_powers)
# check for base and inverted base pairs
be = list(c_powers.items())
skip = set() # skip if we already saw them
for b, e in be:
if b in skip:
continue
bpos = b.is_positive or b.is_polar
if bpos:
binv = 1/b
if b != binv and binv in c_powers:
if b.as_numer_denom()[0] is S.One:
c_powers.pop(b)
c_powers[binv] -= e
else:
skip.add(binv)
e = c_powers.pop(binv)
c_powers[b] -= e
# check for base and negated base pairs
be = list(c_powers.items())
_n = S.NegativeOne
for b, e in be:
if (b.is_Symbol or b.is_Add) and -b in c_powers and b in c_powers:
if (b.is_positive is not None or e.is_integer):
if e.is_integer or b.is_negative:
c_powers[-b] += c_powers.pop(b)
else: # (-b).is_positive so use its e
e = c_powers.pop(-b)
c_powers[b] += e
if _n in c_powers:
c_powers[_n] += e
else:
c_powers[_n] = e
# filter c_powers and convert to a list
c_powers = [(b, e) for b, e in c_powers.items() if e]
# ==============================================================
# check for Mul bases of Rational powers that can be combined with
# separated bases, e.g. x*sqrt(x*y)*sqrt(x*sqrt(x*y)) ->
# (x*sqrt(x*y))**(3/2)
# ---------------- helper functions
def ratq(x):
'''Return Rational part of x's exponent as it appears in the bkey.
'''
return bkey(x)[0][1]
def bkey(b, e=None):
'''Return (b**s, c.q), c.p where e -> c*s. If e is not given then
it will be taken by using as_base_exp() on the input b.
e.g.
x**3/2 -> (x, 2), 3
x**y -> (x**y, 1), 1
x**(2*y/3) -> (x**y, 3), 2
exp(x/2) -> (exp(a), 2), 1
'''
if e is not None: # coming from c_powers or from below
if e.is_Integer:
return (b, S.One), e
elif e.is_Rational:
return (b, Integer(e.q)), Integer(e.p)
else:
c, m = e.as_coeff_Mul(rational=True)
if c is not S.One:
if m.is_integer:
return (b, Integer(c.q)), m*Integer(c.p)
return (b**m, Integer(c.q)), Integer(c.p)
else:
return (b**e, S.One), S.One
else:
return bkey(*b.as_base_exp())
def update(b):
'''Decide what to do with base, b. If its exponent is now an
integer multiple of the Rational denominator, then remove it
and put the factors of its base in the common_b dictionary or
update the existing bases if necessary. If it has been zeroed
out, simply remove the base.
'''
newe, r = divmod(common_b[b], b[1])
if not r:
common_b.pop(b)
if newe:
for m in Mul.make_args(b[0]**newe):
b, e = bkey(m)
if b not in common_b:
common_b[b] = 0
common_b[b] += e
if b[1] != 1:
bases.append(b)
# ---------------- end of helper functions
# assemble a dictionary of the factors having a Rational power
common_b = {}
done = []
bases = []
for b, e in c_powers:
b, e = bkey(b, e)
if b in common_b:
common_b[b] = common_b[b] + e
else:
common_b[b] = e
if b[1] != 1 and b[0].is_Mul:
bases.append(b)
bases.sort(key=default_sort_key) # this makes tie-breaking canonical
bases.sort(key=measure, reverse=True) # handle longest first
for base in bases:
if base not in common_b: # it may have been removed already
continue
b, exponent = base
last = False # True when no factor of base is a radical
qlcm = 1 # the lcm of the radical denominators
while True:
bstart = b
qstart = qlcm
bb = [] # list of factors
ee = [] # (factor's expo. and it's current value in common_b)
for bi in Mul.make_args(b):
bib, bie = bkey(bi)
if bib not in common_b or common_b[bib] < bie:
ee = bb = [] # failed
break
ee.append([bie, common_b[bib]])
bb.append(bib)
if ee:
# find the number of integral extractions possible
# e.g. [(1, 2), (2, 2)] -> min(2/1, 2/2) -> 1
min1 = ee[0][1]//ee[0][0]
for i in range(1, len(ee)):
rat = ee[i][1]//ee[i][0]
if rat < 1:
break
min1 = min(min1, rat)
else:
# update base factor counts
# e.g. if ee = [(2, 5), (3, 6)] then min1 = 2
# and the new base counts will be 5-2*2 and 6-2*3
for i in range(len(bb)):
common_b[bb[i]] -= min1*ee[i][0]
update(bb[i])
# update the count of the base
# e.g. x**2*y*sqrt(x*sqrt(y)) the count of x*sqrt(y)
# will increase by 4 to give bkey (x*sqrt(y), 2, 5)
common_b[base] += min1*qstart*exponent
if (last # no more radicals in base
or len(common_b) == 1 # nothing left to join with
or all(k[1] == 1 for k in common_b) # no rad's in common_b
):
break
# see what we can exponentiate base by to remove any radicals
# so we know what to search for
# e.g. if base were x**(1/2)*y**(1/3) then we should
# exponentiate by 6 and look for powers of x and y in the ratio
# of 2 to 3
qlcm = lcm([ratq(bi) for bi in Mul.make_args(bstart)])
if qlcm == 1:
break # we are done
b = bstart**qlcm
qlcm *= qstart
if all(ratq(bi) == 1 for bi in Mul.make_args(b)):
last = True # we are going to be done after this next pass
# this base no longer can find anything to join with and
# since it was longer than any other we are done with it
b, q = base
done.append((b, common_b.pop(base)*Rational(1, q)))
# update c_powers and get ready to continue with powsimp
c_powers = done
# there may be terms still in common_b that were bases that were
# identified as needing processing, so remove those, too
for (b, q), e in common_b.items():
if (b.is_Pow or isinstance(b, exp)) and \
q is not S.One and not b.exp.is_Rational:
b, be = b.as_base_exp()
b = b**(be/q)
else:
b = root(b, q)
c_powers.append((b, e))
check = len(c_powers)
c_powers = dict(c_powers)
assert len(c_powers) == check # there should have been no duplicates
# ==============================================================
# rebuild the expression
newexpr = expr.func(*(newexpr + [Pow(b, e) for b, e in c_powers.items()]))
if combine == 'exp':
return expr.func(newexpr, expr.func(*nc_part))
else:
return recurse(expr.func(*nc_part), combine='base') * \
recurse(newexpr, combine='base')
elif combine == 'base':
# Build c_powers and nc_part. These must both be lists not
# dicts because exp's are not combined.
c_powers = []
nc_part = []
for term in expr.args:
if term.is_commutative:
c_powers.append(list(term.as_base_exp()))
else:
nc_part.append(term)
# Pull out numerical coefficients from exponent if assumptions allow
# e.g., 2**(2*x) => 4**x
for i in range(len(c_powers)):
b, e = c_powers[i]
if not (all(x.is_nonnegative for x in b.as_numer_denom()) or e.is_integer or force or b.is_polar):
continue
exp_c, exp_t = e.as_coeff_Mul(rational=True)
if exp_c is not S.One and exp_t is not S.One:
c_powers[i] = [Pow(b, exp_c), exp_t]
# Combine bases whenever they have the same exponent and
# assumptions allow
# first gather the potential bases under the common exponent
c_exp = defaultdict(list)
for b, e in c_powers:
if deep:
e = recurse(e)
if e.is_Add and (b.is_positive or e.is_integer):
e = factor_terms(e)
if _coeff_isneg(e):
e = -e
b = 1/b
c_exp[e].append(b)
del c_powers
# Merge back in the results of the above to form a new product
c_powers = defaultdict(list)
for e in c_exp:
bases = c_exp[e]
# calculate the new base for e
if len(bases) == 1:
new_base = bases[0]
elif e.is_integer or force:
new_base = expr.func(*bases)
else:
# see which ones can be joined
unk = []
nonneg = []
neg = []
for bi in bases:
if bi.is_negative:
neg.append(bi)
elif bi.is_nonnegative:
nonneg.append(bi)
elif bi.is_polar:
nonneg.append(
bi) # polar can be treated like non-negative
else:
unk.append(bi)
if len(unk) == 1 and not neg or len(neg) == 1 and not unk:
# a single neg or a single unk can join the rest
nonneg.extend(unk + neg)
unk = neg = []
elif neg:
# their negative signs cancel in groups of 2*q if we know
# that e = p/q else we have to treat them as unknown
israt = False
if e.is_Rational:
israt = True
else:
p, d = e.as_numer_denom()
if p.is_integer and d.is_integer:
israt = True
if israt:
neg = [-w for w in neg]
unk.extend([S.NegativeOne]*len(neg))
else:
unk.extend(neg)
neg = []
del israt
# these shouldn't be joined
for b in unk:
c_powers[b].append(e)
# here is a new joined base
new_base = expr.func(*(nonneg + neg))
# if there are positive parts they will just get separated
# again unless some change is made
def _terms(e):
# return the number of terms of this expression
# when multiplied out -- assuming no joining of terms
if e.is_Add:
return sum([_terms(ai) for ai in e.args])
if e.is_Mul:
return prod([_terms(mi) for mi in e.args])
return 1
xnew_base = expand_mul(new_base, deep=False)
if len(Add.make_args(xnew_base)) < _terms(new_base):
new_base = factor_terms(xnew_base)
c_powers[new_base].append(e)
# break out the powers from c_powers now
c_part = [Pow(b, ei) for b, e in c_powers.items() for ei in e]
# we're done
return expr.func(*(c_part + nc_part))
else:
raise ValueError("combine must be one of ('all', 'exp', 'base').")
def powdenest(eq, force=False, polar=False):
r"""
Collect exponents on powers as assumptions allow.
Explanation
===========
Given ``(bb**be)**e``, this can be simplified as follows:
* if ``bb`` is positive, or
* ``e`` is an integer, or
* ``|be| < 1`` then this simplifies to ``bb**(be*e)``
Given a product of powers raised to a power, ``(bb1**be1 *
bb2**be2...)**e``, simplification can be done as follows:
- if e is positive, the gcd of all bei can be joined with e;
- all non-negative bb can be separated from those that are negative
and their gcd can be joined with e; autosimplification already
handles this separation.
- integer factors from powers that have integers in the denominator
of the exponent can be removed from any term and the gcd of such
integers can be joined with e
Setting ``force`` to ``True`` will make symbols that are not explicitly
negative behave as though they are positive, resulting in more
denesting.
Setting ``polar`` to ``True`` will do simplifications on the Riemann surface of
the logarithm, also resulting in more denestings.
When there are sums of logs in exp() then a product of powers may be
obtained e.g. ``exp(3*(log(a) + 2*log(b)))`` - > ``a**3*b**6``.
Examples
========
>>> from sympy.abc import a, b, x, y, z
>>> from sympy import Symbol, exp, log, sqrt, symbols, powdenest
>>> powdenest((x**(2*a/3))**(3*x))
(x**(2*a/3))**(3*x)
>>> powdenest(exp(3*x*log(2)))
2**(3*x)
Assumptions may prevent expansion:
>>> powdenest(sqrt(x**2))
sqrt(x**2)
>>> p = symbols('p', positive=True)
>>> powdenest(sqrt(p**2))
p
No other expansion is done.
>>> i, j = symbols('i,j', integer=True)
>>> powdenest((x**x)**(i + j)) # -X-> (x**x)**i*(x**x)**j
x**(x*(i + j))
But exp() will be denested by moving all non-log terms outside of
the function; this may result in the collapsing of the exp to a power
with a different base:
>>> powdenest(exp(3*y*log(x)))
x**(3*y)
>>> powdenest(exp(y*(log(a) + log(b))))
(a*b)**y
>>> powdenest(exp(3*(log(a) + log(b))))
a**3*b**3
If assumptions allow, symbols can also be moved to the outermost exponent:
>>> i = Symbol('i', integer=True)
>>> powdenest(((x**(2*i))**(3*y))**x)
((x**(2*i))**(3*y))**x
>>> powdenest(((x**(2*i))**(3*y))**x, force=True)
x**(6*i*x*y)
>>> powdenest(((x**(2*a/3))**(3*y/i))**x)
((x**(2*a/3))**(3*y/i))**x
>>> powdenest((x**(2*i)*y**(4*i))**z, force=True)
(x*y**2)**(2*i*z)
>>> n = Symbol('n', negative=True)
>>> powdenest((x**i)**y, force=True)
x**(i*y)
>>> powdenest((n**i)**x, force=True)
(n**i)**x
"""
from sympy.simplify.simplify import posify
if force:
def _denest(b, e):
if not isinstance(b, (Pow, exp)):
return b.is_positive, Pow(b, e, evaluate=False)
return _denest(b.base, b.exp*e)
reps = []
for p in eq.atoms(Pow, exp):
if isinstance(p.base, (Pow, exp)):
ok, dp = _denest(*p.args)
if ok is not False:
reps.append((p, dp))
if reps:
eq = eq.subs(reps)
eq, reps = posify(eq)
return powdenest(eq, force=False, polar=polar).xreplace(reps)
if polar:
eq, rep = polarify(eq)
return unpolarify(powdenest(unpolarify(eq, exponents_only=True)), rep)
new = powsimp(sympify(eq))
return new.xreplace(Transform(
_denest_pow, filter=lambda m: m.is_Pow or isinstance(m, exp)))
_y = Dummy('y')
def _denest_pow(eq):
"""
Denest powers.
This is a helper function for powdenest that performs the actual
transformation.
"""
from sympy.simplify.simplify import logcombine
b, e = eq.as_base_exp()
if b.is_Pow or isinstance(b, exp) and e != 1:
new = b._eval_power(e)
if new is not None:
eq = new
b, e = new.as_base_exp()
# denest exp with log terms in exponent
if b is S.Exp1 and e.is_Mul:
logs = []
other = []
for ei in e.args:
if any(isinstance(ai, log) for ai in Add.make_args(ei)):
logs.append(ei)
else:
other.append(ei)
logs = logcombine(Mul(*logs))
return Pow(exp(logs), Mul(*other))
_, be = b.as_base_exp()
if be is S.One and not (b.is_Mul or
b.is_Rational and b.q != 1 or
b.is_positive):
return eq
# denest eq which is either pos**e or Pow**e or Mul**e or
# Mul(b1**e1, b2**e2)
# handle polar numbers specially
polars, nonpolars = [], []
for bb in Mul.make_args(b):
if bb.is_polar:
polars.append(bb.as_base_exp())
else:
nonpolars.append(bb)
if len(polars) == 1 and not polars[0][0].is_Mul:
return Pow(polars[0][0], polars[0][1]*e)*powdenest(Mul(*nonpolars)**e)
elif polars:
return Mul(*[powdenest(bb**(ee*e)) for (bb, ee) in polars]) \
*powdenest(Mul(*nonpolars)**e)
if b.is_Integer:
# use log to see if there is a power here
logb = expand_log(log(b))
if logb.is_Mul:
c, logb = logb.args
e *= c
base = logb.args[0]
return Pow(base, e)
# if b is not a Mul or any factor is an atom then there is nothing to do
if not b.is_Mul or any(s.is_Atom for s in Mul.make_args(b)):
return eq
# let log handle the case of the base of the argument being a Mul, e.g.
# sqrt(x**(2*i)*y**(6*i)) -> x**i*y**(3**i) if x and y are positive; we
# will take the log, expand it, and then factor out the common powers that
# now appear as coefficient. We do this manually since terms_gcd pulls out
# fractions, terms_gcd(x+x*y/2) -> x*(y + 2)/2 and we don't want the 1/2;
# gcd won't pull out numerators from a fraction: gcd(3*x, 9*x/2) -> x but
# we want 3*x. Neither work with noncommutatives.
def nc_gcd(aa, bb):
a, b = [i.as_coeff_Mul() for i in [aa, bb]]
c = gcd(a[0], b[0]).as_numer_denom()[0]
g = Mul(*(a[1].args_cnc(cset=True)[0] & b[1].args_cnc(cset=True)[0]))
return _keep_coeff(c, g)
glogb = expand_log(log(b))
if glogb.is_Add:
args = glogb.args
g = reduce(nc_gcd, args)
if g != 1:
cg, rg = g.as_coeff_Mul()
glogb = _keep_coeff(cg, rg*Add(*[a/g for a in args]))
# now put the log back together again
if isinstance(glogb, log) or not glogb.is_Mul:
if glogb.args[0].is_Pow or isinstance(glogb.args[0], exp):
glogb = _denest_pow(glogb.args[0])
if (abs(glogb.exp) < 1) == True:
return Pow(glogb.base, glogb.exp*e)
return eq
# the log(b) was a Mul so join any adds with logcombine
add = []
other = []
for a in glogb.args:
if a.is_Add:
add.append(a)
else:
other.append(a)
return Pow(exp(logcombine(Mul(*add))), e*Mul(*other))
|
f23ef0a04c7549bb96580f6bcc2f08669c44d8ea1d42630969e3b4f133f776ab | from sympy.core import Function, S, Mul, Pow, Add
from sympy.core.sorting import ordered, default_sort_key
from sympy.core.function import expand_func
from sympy.core.symbol import Dummy
from sympy.functions import gamma, sqrt, sin
from sympy.polys import factor, cancel
from sympy.utilities.iterables import sift, uniq
def gammasimp(expr):
r"""
Simplify expressions with gamma functions.
Explanation
===========
This function takes as input an expression containing gamma
functions or functions that can be rewritten in terms of gamma
functions and tries to minimize the number of those functions and
reduce the size of their arguments.
The algorithm works by rewriting all gamma functions as expressions
involving rising factorials (Pochhammer symbols) and applies
recurrence relations and other transformations applicable to rising
factorials, to reduce their arguments, possibly letting the resulting
rising factorial to cancel. Rising factorials with the second argument
being an integer are expanded into polynomial forms and finally all
other rising factorial are rewritten in terms of gamma functions.
Then the following two steps are performed.
1. Reduce the number of gammas by applying the reflection theorem
gamma(x)*gamma(1-x) == pi/sin(pi*x).
2. Reduce the number of gammas by applying the multiplication theorem
gamma(x)*gamma(x+1/n)*...*gamma(x+(n-1)/n) == C*gamma(n*x).
It then reduces the number of prefactors by absorbing them into gammas
where possible and expands gammas with rational argument.
All transformation rules can be found (or were derived from) here:
.. [1] http://functions.wolfram.com/GammaBetaErf/Pochhammer/17/01/02/
.. [2] http://functions.wolfram.com/GammaBetaErf/Pochhammer/27/01/0005/
Examples
========
>>> from sympy.simplify import gammasimp
>>> from sympy import gamma, Symbol
>>> from sympy.abc import x
>>> n = Symbol('n', integer = True)
>>> gammasimp(gamma(x)/gamma(x - 3))
(x - 3)*(x - 2)*(x - 1)
>>> gammasimp(gamma(n + 3))
gamma(n + 3)
"""
expr = expr.rewrite(gamma)
# compute_ST will be looking for Functions and we don't want
# it looking for non-gamma functions: issue 22606
# so we mask free, non-gamma functions
f = expr.atoms(Function)
# take out gammas
gammas = {i for i in f if isinstance(i, gamma)}
if not gammas:
return expr # avoid side effects like factoring
f -= gammas
# keep only those without bound symbols
f = f & expr.as_dummy().atoms(Function)
if f:
dum, fun, simp = zip(*[
(Dummy(), fi, fi.func(*[
_gammasimp(a, as_comb=False) for a in fi.args]))
for fi in ordered(f)])
d = expr.xreplace(dict(zip(fun, dum)))
return _gammasimp(d, as_comb=False).xreplace(dict(zip(dum, simp)))
return _gammasimp(expr, as_comb=False)
def _gammasimp(expr, as_comb):
"""
Helper function for gammasimp and combsimp.
Explanation
===========
Simplifies expressions written in terms of gamma function. If
as_comb is True, it tries to preserve integer arguments. See
docstring of gammasimp for more information. This was part of
combsimp() in combsimp.py.
"""
expr = expr.replace(gamma,
lambda n: _rf(1, (n - 1).expand()))
if as_comb:
expr = expr.replace(_rf,
lambda a, b: gamma(b + 1))
else:
expr = expr.replace(_rf,
lambda a, b: gamma(a + b)/gamma(a))
def rule_gamma(expr, level=0):
""" Simplify products of gamma functions further. """
if expr.is_Atom:
return expr
def gamma_rat(x):
# helper to simplify ratios of gammas
was = x.count(gamma)
xx = x.replace(gamma, lambda n: _rf(1, (n - 1).expand()
).replace(_rf, lambda a, b: gamma(a + b)/gamma(a)))
if xx.count(gamma) < was:
x = xx
return x
def gamma_factor(x):
# return True if there is a gamma factor in shallow args
if isinstance(x, gamma):
return True
if x.is_Add or x.is_Mul:
return any(gamma_factor(xi) for xi in x.args)
if x.is_Pow and (x.exp.is_integer or x.base.is_positive):
return gamma_factor(x.base)
return False
# recursion step
if level == 0:
expr = expr.func(*[rule_gamma(x, level + 1) for x in expr.args])
level += 1
if not expr.is_Mul:
return expr
# non-commutative step
if level == 1:
args, nc = expr.args_cnc()
if not args:
return expr
if nc:
return rule_gamma(Mul._from_args(args), level + 1)*Mul._from_args(nc)
level += 1
# pure gamma handling, not factor absorption
if level == 2:
T, F = sift(expr.args, gamma_factor, binary=True)
gamma_ind = Mul(*F)
d = Mul(*T)
nd, dd = d.as_numer_denom()
for ipass in range(2):
args = list(ordered(Mul.make_args(nd)))
for i, ni in enumerate(args):
if ni.is_Add:
ni, dd = Add(*[
rule_gamma(gamma_rat(a/dd), level + 1) for a in ni.args]
).as_numer_denom()
args[i] = ni
if not dd.has(gamma):
break
nd = Mul(*args)
if ipass == 0 and not gamma_factor(nd):
break
nd, dd = dd, nd # now process in reversed order
expr = gamma_ind*nd/dd
if not (expr.is_Mul and (gamma_factor(dd) or gamma_factor(nd))):
return expr
level += 1
# iteration until constant
if level == 3:
while True:
was = expr
expr = rule_gamma(expr, 4)
if expr == was:
return expr
numer_gammas = []
denom_gammas = []
numer_others = []
denom_others = []
def explicate(p):
if p is S.One:
return None, []
b, e = p.as_base_exp()
if e.is_Integer:
if isinstance(b, gamma):
return True, [b.args[0]]*e
else:
return False, [b]*e
else:
return False, [p]
newargs = list(ordered(expr.args))
while newargs:
n, d = newargs.pop().as_numer_denom()
isg, l = explicate(n)
if isg:
numer_gammas.extend(l)
elif isg is False:
numer_others.extend(l)
isg, l = explicate(d)
if isg:
denom_gammas.extend(l)
elif isg is False:
denom_others.extend(l)
# =========== level 2 work: pure gamma manipulation =========
if not as_comb:
# Try to reduce the number of gamma factors by applying the
# reflection formula gamma(x)*gamma(1-x) = pi/sin(pi*x)
for gammas, numer, denom in [(
numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)]:
new = []
while gammas:
g1 = gammas.pop()
if g1.is_integer:
new.append(g1)
continue
for i, g2 in enumerate(gammas):
n = g1 + g2 - 1
if not n.is_Integer:
continue
numer.append(S.Pi)
denom.append(sin(S.Pi*g1))
gammas.pop(i)
if n > 0:
for k in range(n):
numer.append(1 - g1 + k)
elif n < 0:
for k in range(-n):
denom.append(-g1 - k)
break
else:
new.append(g1)
# /!\ updating IN PLACE
gammas[:] = new
# Try to reduce the number of gammas by using the duplication
# theorem to cancel an upper and lower: gamma(2*s)/gamma(s) =
# 2**(2*s + 1)/(4*sqrt(pi))*gamma(s + 1/2). Although this could
# be done with higher argument ratios like gamma(3*x)/gamma(x),
# this would not reduce the number of gammas as in this case.
for ng, dg, no, do in [(numer_gammas, denom_gammas, numer_others,
denom_others),
(denom_gammas, numer_gammas, denom_others,
numer_others)]:
while True:
for x in ng:
for y in dg:
n = x - 2*y
if n.is_Integer:
break
else:
continue
break
else:
break
ng.remove(x)
dg.remove(y)
if n > 0:
for k in range(n):
no.append(2*y + k)
elif n < 0:
for k in range(-n):
do.append(2*y - 1 - k)
ng.append(y + S.Half)
no.append(2**(2*y - 1))
do.append(sqrt(S.Pi))
# Try to reduce the number of gamma factors by applying the
# multiplication theorem (used when n gammas with args differing
# by 1/n mod 1 are encountered).
#
# run of 2 with args differing by 1/2
#
# >>> gammasimp(gamma(x)*gamma(x+S.Half))
# 2*sqrt(2)*2**(-2*x - 1/2)*sqrt(pi)*gamma(2*x)
#
# run of 3 args differing by 1/3 (mod 1)
#
# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(2)/3))
# 6*3**(-3*x - 1/2)*pi*gamma(3*x)
# >>> gammasimp(gamma(x)*gamma(x+S(1)/3)*gamma(x+S(5)/3))
# 2*3**(-3*x - 1/2)*pi*(3*x + 2)*gamma(3*x)
#
def _run(coeffs):
# find runs in coeffs such that the difference in terms (mod 1)
# of t1, t2, ..., tn is 1/n
u = list(uniq(coeffs))
for i in range(len(u)):
dj = ([((u[j] - u[i]) % 1, j) for j in range(i + 1, len(u))])
for one, j in dj:
if one.p == 1 and one.q != 1:
n = one.q
got = [i]
get = list(range(1, n))
for d, j in dj:
m = n*d
if m.is_Integer and m in get:
get.remove(m)
got.append(j)
if not get:
break
else:
continue
for i, j in enumerate(got):
c = u[j]
coeffs.remove(c)
got[i] = c
return one.q, got[0], got[1:]
def _mult_thm(gammas, numer, denom):
# pull off and analyze the leading coefficient from each gamma arg
# looking for runs in those Rationals
# expr -> coeff + resid -> rats[resid] = coeff
rats = {}
for g in gammas:
c, resid = g.as_coeff_Add()
rats.setdefault(resid, []).append(c)
# look for runs in Rationals for each resid
keys = sorted(rats, key=default_sort_key)
for resid in keys:
coeffs = list(sorted(rats[resid]))
new = []
while True:
run = _run(coeffs)
if run is None:
break
# process the sequence that was found:
# 1) convert all the gamma functions to have the right
# argument (could be off by an integer)
# 2) append the factors corresponding to the theorem
# 3) append the new gamma function
n, ui, other = run
# (1)
for u in other:
con = resid + u - 1
for k in range(int(u - ui)):
numer.append(con - k)
con = n*(resid + ui) # for (2) and (3)
# (2)
numer.append((2*S.Pi)**(S(n - 1)/2)*
n**(S.Half - con))
# (3)
new.append(con)
# restore resid to coeffs
rats[resid] = [resid + c for c in coeffs] + new
# rebuild the gamma arguments
g = []
for resid in keys:
g += rats[resid]
# /!\ updating IN PLACE
gammas[:] = g
for l, numer, denom in [(numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)]:
_mult_thm(l, numer, denom)
# =========== level >= 2 work: factor absorption =========
if level >= 2:
# Try to absorb factors into the gammas: x*gamma(x) -> gamma(x + 1)
# and gamma(x)/(x - 1) -> gamma(x - 1)
# This code (in particular repeated calls to find_fuzzy) can be very
# slow.
def find_fuzzy(l, x):
if not l:
return
S1, T1 = compute_ST(x)
for y in l:
S2, T2 = inv[y]
if T1 != T2 or (not S1.intersection(S2) and
(S1 != set() or S2 != set())):
continue
# XXX we want some simplification (e.g. cancel or
# simplify) but no matter what it's slow.
a = len(cancel(x/y).free_symbols)
b = len(x.free_symbols)
c = len(y.free_symbols)
# TODO is there a better heuristic?
if a == 0 and (b > 0 or c > 0):
return y
# We thus try to avoid expensive calls by building the following
# "invariants": For every factor or gamma function argument
# - the set of free symbols S
# - the set of functional components T
# We will only try to absorb if T1==T2 and (S1 intersect S2 != emptyset
# or S1 == S2 == emptyset)
inv = {}
def compute_ST(expr):
if expr in inv:
return inv[expr]
return (expr.free_symbols, expr.atoms(Function).union(
{e.exp for e in expr.atoms(Pow)}))
def update_ST(expr):
inv[expr] = compute_ST(expr)
for expr in numer_gammas + denom_gammas + numer_others + denom_others:
update_ST(expr)
for gammas, numer, denom in [(
numer_gammas, numer_others, denom_others),
(denom_gammas, denom_others, numer_others)]:
new = []
while gammas:
g = gammas.pop()
cont = True
while cont:
cont = False
y = find_fuzzy(numer, g)
if y is not None:
numer.remove(y)
if y != g:
numer.append(y/g)
update_ST(y/g)
g += 1
cont = True
y = find_fuzzy(denom, g - 1)
if y is not None:
denom.remove(y)
if y != g - 1:
numer.append((g - 1)/y)
update_ST((g - 1)/y)
g -= 1
cont = True
new.append(g)
# /!\ updating IN PLACE
gammas[:] = new
# =========== rebuild expr ==================================
return Mul(*[gamma(g) for g in numer_gammas]) \
/ Mul(*[gamma(g) for g in denom_gammas]) \
* Mul(*numer_others) / Mul(*denom_others)
was = factor(expr)
# (for some reason we cannot use Basic.replace in this case)
expr = rule_gamma(was)
if expr != was:
expr = factor(expr)
expr = expr.replace(gamma,
lambda n: expand_func(gamma(n)) if n.is_Rational else gamma(n))
return expr
class _rf(Function):
@classmethod
def eval(cls, a, b):
if b.is_Integer:
if not b:
return S.One
n, result = int(b), S.One
if n > 0:
for i in range(n):
result *= a + i
return result
elif n < 0:
for i in range(1, -n + 1):
result *= a - i
return 1/result
else:
if b.is_Add:
c, _b = b.as_coeff_Add()
if c.is_Integer:
if c > 0:
return _rf(a, _b)*_rf(a + _b, c)
elif c < 0:
return _rf(a, _b)/_rf(a + _b + c, -c)
if a.is_Add:
c, _a = a.as_coeff_Add()
if c.is_Integer:
if c > 0:
return _rf(_a, b)*_rf(_a + b, c)/_rf(_a, c)
elif c < 0:
return _rf(_a, b)*_rf(_a + c, -c)/_rf(_a + b + c, -c)
|
5e9d713ce1670cb7a88f35c38d4bbceb68d11f401f737578ee89dc9726dffbf6 | from collections import defaultdict
from sympy.core.add import Add
from sympy.core.expr import Expr
from sympy.core.exprtools import Factors, gcd_terms, factor_terms
from sympy.core.function import expand_mul
from sympy.core.mul import Mul
from sympy.core.numbers import pi, I
from sympy.core.power import Pow
from sympy.core.singleton import S
from sympy.core.sorting import ordered
from sympy.core.symbol import Dummy
from sympy.core.sympify import sympify
from sympy.core.traversal import bottom_up
from sympy.functions.combinatorial.factorials import binomial
from sympy.functions.elementary.hyperbolic import (
cosh, sinh, tanh, coth, sech, csch, HyperbolicFunction)
from sympy.functions.elementary.trigonometric import (
cos, sin, tan, cot, sec, csc, sqrt, TrigonometricFunction)
from sympy.ntheory.factor_ import perfect_power
from sympy.polys.polytools import factor
from sympy.strategies.tree import greedy
from sympy.strategies.core import identity, debug
from sympy import SYMPY_DEBUG
# ================== Fu-like tools ===========================
def TR0(rv):
"""Simplification of rational polynomials, trying to simplify
the expression, e.g. combine things like 3*x + 2*x, etc....
"""
# although it would be nice to use cancel, it doesn't work
# with noncommutatives
return rv.normal().factor().expand()
def TR1(rv):
"""Replace sec, csc with 1/cos, 1/sin
Examples
========
>>> from sympy.simplify.fu import TR1, sec, csc
>>> from sympy.abc import x
>>> TR1(2*csc(x) + sec(x))
1/cos(x) + 2/sin(x)
"""
def f(rv):
if isinstance(rv, sec):
a = rv.args[0]
return S.One/cos(a)
elif isinstance(rv, csc):
a = rv.args[0]
return S.One/sin(a)
return rv
return bottom_up(rv, f)
def TR2(rv):
"""Replace tan and cot with sin/cos and cos/sin
Examples
========
>>> from sympy.simplify.fu import TR2
>>> from sympy.abc import x
>>> from sympy import tan, cot, sin, cos
>>> TR2(tan(x))
sin(x)/cos(x)
>>> TR2(cot(x))
cos(x)/sin(x)
>>> TR2(tan(tan(x) - sin(x)/cos(x)))
0
"""
def f(rv):
if isinstance(rv, tan):
a = rv.args[0]
return sin(a)/cos(a)
elif isinstance(rv, cot):
a = rv.args[0]
return cos(a)/sin(a)
return rv
return bottom_up(rv, f)
def TR2i(rv, half=False):
"""Converts ratios involving sin and cos as follows::
sin(x)/cos(x) -> tan(x)
sin(x)/(cos(x) + 1) -> tan(x/2) if half=True
Examples
========
>>> from sympy.simplify.fu import TR2i
>>> from sympy.abc import x, a
>>> from sympy import sin, cos
>>> TR2i(sin(x)/cos(x))
tan(x)
Powers of the numerator and denominator are also recognized
>>> TR2i(sin(x)**2/(cos(x) + 1)**2, half=True)
tan(x/2)**2
The transformation does not take place unless assumptions allow
(i.e. the base must be positive or the exponent must be an integer
for both numerator and denominator)
>>> TR2i(sin(x)**a/(cos(x) + 1)**a)
sin(x)**a/(cos(x) + 1)**a
"""
def f(rv):
if not rv.is_Mul:
return rv
n, d = rv.as_numer_denom()
if n.is_Atom or d.is_Atom:
return rv
def ok(k, e):
# initial filtering of factors
return (
(e.is_integer or k.is_positive) and (
k.func in (sin, cos) or (half and
k.is_Add and
len(k.args) >= 2 and
any(any(isinstance(ai, cos) or ai.is_Pow and ai.base is cos
for ai in Mul.make_args(a)) for a in k.args))))
n = n.as_powers_dict()
ndone = [(k, n.pop(k)) for k in list(n.keys()) if not ok(k, n[k])]
if not n:
return rv
d = d.as_powers_dict()
ddone = [(k, d.pop(k)) for k in list(d.keys()) if not ok(k, d[k])]
if not d:
return rv
# factoring if necessary
def factorize(d, ddone):
newk = []
for k in d:
if k.is_Add and len(k.args) > 1:
knew = factor(k) if half else factor_terms(k)
if knew != k:
newk.append((k, knew))
if newk:
for i, (k, knew) in enumerate(newk):
del d[k]
newk[i] = knew
newk = Mul(*newk).as_powers_dict()
for k in newk:
v = d[k] + newk[k]
if ok(k, v):
d[k] = v
else:
ddone.append((k, v))
del newk
factorize(n, ndone)
factorize(d, ddone)
# joining
t = []
for k in n:
if isinstance(k, sin):
a = cos(k.args[0], evaluate=False)
if a in d and d[a] == n[k]:
t.append(tan(k.args[0])**n[k])
n[k] = d[a] = None
elif half:
a1 = 1 + a
if a1 in d and d[a1] == n[k]:
t.append((tan(k.args[0]/2))**n[k])
n[k] = d[a1] = None
elif isinstance(k, cos):
a = sin(k.args[0], evaluate=False)
if a in d and d[a] == n[k]:
t.append(tan(k.args[0])**-n[k])
n[k] = d[a] = None
elif half and k.is_Add and k.args[0] is S.One and \
isinstance(k.args[1], cos):
a = sin(k.args[1].args[0], evaluate=False)
if a in d and d[a] == n[k] and (d[a].is_integer or \
a.is_positive):
t.append(tan(a.args[0]/2)**-n[k])
n[k] = d[a] = None
if t:
rv = Mul(*(t + [b**e for b, e in n.items() if e]))/\
Mul(*[b**e for b, e in d.items() if e])
rv *= Mul(*[b**e for b, e in ndone])/Mul(*[b**e for b, e in ddone])
return rv
return bottom_up(rv, f)
def TR3(rv):
"""Induced formula: example sin(-a) = -sin(a)
Examples
========
>>> from sympy.simplify.fu import TR3
>>> from sympy.abc import x, y
>>> from sympy import pi
>>> from sympy import cos
>>> TR3(cos(y - x*(y - x)))
cos(x*(x - y) + y)
>>> cos(pi/2 + x)
-sin(x)
>>> cos(30*pi/2 + x)
-cos(x)
"""
from sympy.simplify.simplify import signsimp
# Negative argument (already automatic for funcs like sin(-x) -> -sin(x)
# but more complicated expressions can use it, too). Also, trig angles
# between pi/4 and pi/2 are not reduced to an angle between 0 and pi/4.
# The following are automatically handled:
# Argument of type: pi/2 +/- angle
# Argument of type: pi +/- angle
# Argument of type : 2k*pi +/- angle
def f(rv):
if not isinstance(rv, TrigonometricFunction):
return rv
rv = rv.func(signsimp(rv.args[0]))
if not isinstance(rv, TrigonometricFunction):
return rv
if (rv.args[0] - S.Pi/4).is_positive is (S.Pi/2 - rv.args[0]).is_positive is True:
fmap = {cos: sin, sin: cos, tan: cot, cot: tan, sec: csc, csc: sec}
rv = fmap[type(rv)](S.Pi/2 - rv.args[0])
return rv
return bottom_up(rv, f)
def TR4(rv):
"""Identify values of special angles.
a= 0 pi/6 pi/4 pi/3 pi/2
----------------------------------------------------
sin(a) 0 1/2 sqrt(2)/2 sqrt(3)/2 1
cos(a) 1 sqrt(3)/2 sqrt(2)/2 1/2 0
tan(a) 0 sqt(3)/3 1 sqrt(3) --
Examples
========
>>> from sympy import pi
>>> from sympy import cos, sin, tan, cot
>>> for s in (0, pi/6, pi/4, pi/3, pi/2):
... print('%s %s %s %s' % (cos(s), sin(s), tan(s), cot(s)))
...
1 0 0 zoo
sqrt(3)/2 1/2 sqrt(3)/3 sqrt(3)
sqrt(2)/2 sqrt(2)/2 1 1
1/2 sqrt(3)/2 sqrt(3) sqrt(3)/3
0 1 zoo 0
"""
# special values at 0, pi/6, pi/4, pi/3, pi/2 already handled
return rv
def _TR56(rv, f, g, h, max, pow):
"""Helper for TR5 and TR6 to replace f**2 with h(g**2)
Options
=======
max : controls size of exponent that can appear on f
e.g. if max=4 then f**4 will be changed to h(g**2)**2.
pow : controls whether the exponent must be a perfect power of 2
e.g. if pow=True (and max >= 6) then f**6 will not be changed
but f**8 will be changed to h(g**2)**4
>>> from sympy.simplify.fu import _TR56 as T
>>> from sympy.abc import x
>>> from sympy import sin, cos
>>> h = lambda x: 1 - x
>>> T(sin(x)**3, sin, cos, h, 4, False)
(1 - cos(x)**2)*sin(x)
>>> T(sin(x)**6, sin, cos, h, 6, False)
(1 - cos(x)**2)**3
>>> T(sin(x)**6, sin, cos, h, 6, True)
sin(x)**6
>>> T(sin(x)**8, sin, cos, h, 10, True)
(1 - cos(x)**2)**4
"""
def _f(rv):
# I'm not sure if this transformation should target all even powers
# or only those expressible as powers of 2. Also, should it only
# make the changes in powers that appear in sums -- making an isolated
# change is not going to allow a simplification as far as I can tell.
if not (rv.is_Pow and rv.base.func == f):
return rv
if not rv.exp.is_real:
return rv
if (rv.exp < 0) == True:
return rv
if (rv.exp > max) == True:
return rv
if rv.exp == 1:
return rv
if rv.exp == 2:
return h(g(rv.base.args[0])**2)
else:
if rv.exp % 2 == 1:
e = rv.exp//2
return f(rv.base.args[0])*h(g(rv.base.args[0])**2)**e
elif rv.exp == 4:
e = 2
elif not pow:
if rv.exp % 2:
return rv
e = rv.exp//2
else:
p = perfect_power(rv.exp)
if not p:
return rv
e = rv.exp//2
return h(g(rv.base.args[0])**2)**e
return bottom_up(rv, _f)
def TR5(rv, max=4, pow=False):
"""Replacement of sin**2 with 1 - cos(x)**2.
See _TR56 docstring for advanced use of ``max`` and ``pow``.
Examples
========
>>> from sympy.simplify.fu import TR5
>>> from sympy.abc import x
>>> from sympy import sin
>>> TR5(sin(x)**2)
1 - cos(x)**2
>>> TR5(sin(x)**-2) # unchanged
sin(x)**(-2)
>>> TR5(sin(x)**4)
(1 - cos(x)**2)**2
"""
return _TR56(rv, sin, cos, lambda x: 1 - x, max=max, pow=pow)
def TR6(rv, max=4, pow=False):
"""Replacement of cos**2 with 1 - sin(x)**2.
See _TR56 docstring for advanced use of ``max`` and ``pow``.
Examples
========
>>> from sympy.simplify.fu import TR6
>>> from sympy.abc import x
>>> from sympy import cos
>>> TR6(cos(x)**2)
1 - sin(x)**2
>>> TR6(cos(x)**-2) #unchanged
cos(x)**(-2)
>>> TR6(cos(x)**4)
(1 - sin(x)**2)**2
"""
return _TR56(rv, cos, sin, lambda x: 1 - x, max=max, pow=pow)
def TR7(rv):
"""Lowering the degree of cos(x)**2.
Examples
========
>>> from sympy.simplify.fu import TR7
>>> from sympy.abc import x
>>> from sympy import cos
>>> TR7(cos(x)**2)
cos(2*x)/2 + 1/2
>>> TR7(cos(x)**2 + 1)
cos(2*x)/2 + 3/2
"""
def f(rv):
if not (rv.is_Pow and rv.base.func == cos and rv.exp == 2):
return rv
return (1 + cos(2*rv.base.args[0]))/2
return bottom_up(rv, f)
def TR8(rv, first=True):
"""Converting products of ``cos`` and/or ``sin`` to a sum or
difference of ``cos`` and or ``sin`` terms.
Examples
========
>>> from sympy.simplify.fu import TR8
>>> from sympy import cos, sin
>>> TR8(cos(2)*cos(3))
cos(5)/2 + cos(1)/2
>>> TR8(cos(2)*sin(3))
sin(5)/2 + sin(1)/2
>>> TR8(sin(2)*sin(3))
-cos(5)/2 + cos(1)/2
"""
def f(rv):
if not (
rv.is_Mul or
rv.is_Pow and
rv.base.func in (cos, sin) and
(rv.exp.is_integer or rv.base.is_positive)):
return rv
if first:
n, d = [expand_mul(i) for i in rv.as_numer_denom()]
newn = TR8(n, first=False)
newd = TR8(d, first=False)
if newn != n or newd != d:
rv = gcd_terms(newn/newd)
if rv.is_Mul and rv.args[0].is_Rational and \
len(rv.args) == 2 and rv.args[1].is_Add:
rv = Mul(*rv.as_coeff_Mul())
return rv
args = {cos: [], sin: [], None: []}
for a in ordered(Mul.make_args(rv)):
if a.func in (cos, sin):
args[type(a)].append(a.args[0])
elif (a.is_Pow and a.exp.is_Integer and a.exp > 0 and \
a.base.func in (cos, sin)):
# XXX this is ok but pathological expression could be handled
# more efficiently as in TRmorrie
args[type(a.base)].extend([a.base.args[0]]*a.exp)
else:
args[None].append(a)
c = args[cos]
s = args[sin]
if not (c and s or len(c) > 1 or len(s) > 1):
return rv
args = args[None]
n = min(len(c), len(s))
for i in range(n):
a1 = s.pop()
a2 = c.pop()
args.append((sin(a1 + a2) + sin(a1 - a2))/2)
while len(c) > 1:
a1 = c.pop()
a2 = c.pop()
args.append((cos(a1 + a2) + cos(a1 - a2))/2)
if c:
args.append(cos(c.pop()))
while len(s) > 1:
a1 = s.pop()
a2 = s.pop()
args.append((-cos(a1 + a2) + cos(a1 - a2))/2)
if s:
args.append(sin(s.pop()))
return TR8(expand_mul(Mul(*args)))
return bottom_up(rv, f)
def TR9(rv):
"""Sum of ``cos`` or ``sin`` terms as a product of ``cos`` or ``sin``.
Examples
========
>>> from sympy.simplify.fu import TR9
>>> from sympy import cos, sin
>>> TR9(cos(1) + cos(2))
2*cos(1/2)*cos(3/2)
>>> TR9(cos(1) + 2*sin(1) + 2*sin(2))
cos(1) + 4*sin(3/2)*cos(1/2)
If no change is made by TR9, no re-arrangement of the
expression will be made. For example, though factoring
of common term is attempted, if the factored expression
wasn't changed, the original expression will be returned:
>>> TR9(cos(3) + cos(3)*cos(2))
cos(3) + cos(2)*cos(3)
"""
def f(rv):
if not rv.is_Add:
return rv
def do(rv, first=True):
# cos(a)+/-cos(b) can be combined into a product of cosines and
# sin(a)+/-sin(b) can be combined into a product of cosine and
# sine.
#
# If there are more than two args, the pairs which "work" will
# have a gcd extractable and the remaining two terms will have
# the above structure -- all pairs must be checked to find the
# ones that work. args that don't have a common set of symbols
# are skipped since this doesn't lead to a simpler formula and
# also has the arbitrariness of combining, for example, the x
# and y term instead of the y and z term in something like
# cos(x) + cos(y) + cos(z).
if not rv.is_Add:
return rv
args = list(ordered(rv.args))
if len(args) != 2:
hit = False
for i in range(len(args)):
ai = args[i]
if ai is None:
continue
for j in range(i + 1, len(args)):
aj = args[j]
if aj is None:
continue
was = ai + aj
new = do(was)
if new != was:
args[i] = new # update in place
args[j] = None
hit = True
break # go to next i
if hit:
rv = Add(*[_f for _f in args if _f])
if rv.is_Add:
rv = do(rv)
return rv
# two-arg Add
split = trig_split(*args)
if not split:
return rv
gcd, n1, n2, a, b, iscos = split
# application of rule if possible
if iscos:
if n1 == n2:
return gcd*n1*2*cos((a + b)/2)*cos((a - b)/2)
if n1 < 0:
a, b = b, a
return -2*gcd*sin((a + b)/2)*sin((a - b)/2)
else:
if n1 == n2:
return gcd*n1*2*sin((a + b)/2)*cos((a - b)/2)
if n1 < 0:
a, b = b, a
return 2*gcd*cos((a + b)/2)*sin((a - b)/2)
return process_common_addends(rv, do) # DON'T sift by free symbols
return bottom_up(rv, f)
def TR10(rv, first=True):
"""Separate sums in ``cos`` and ``sin``.
Examples
========
>>> from sympy.simplify.fu import TR10
>>> from sympy.abc import a, b, c
>>> from sympy import cos, sin
>>> TR10(cos(a + b))
-sin(a)*sin(b) + cos(a)*cos(b)
>>> TR10(sin(a + b))
sin(a)*cos(b) + sin(b)*cos(a)
>>> TR10(sin(a + b + c))
(-sin(a)*sin(b) + cos(a)*cos(b))*sin(c) + \
(sin(a)*cos(b) + sin(b)*cos(a))*cos(c)
"""
def f(rv):
if rv.func not in (cos, sin):
return rv
f = rv.func
arg = rv.args[0]
if arg.is_Add:
if first:
args = list(ordered(arg.args))
else:
args = list(arg.args)
a = args.pop()
b = Add._from_args(args)
if b.is_Add:
if f == sin:
return sin(a)*TR10(cos(b), first=False) + \
cos(a)*TR10(sin(b), first=False)
else:
return cos(a)*TR10(cos(b), first=False) - \
sin(a)*TR10(sin(b), first=False)
else:
if f == sin:
return sin(a)*cos(b) + cos(a)*sin(b)
else:
return cos(a)*cos(b) - sin(a)*sin(b)
return rv
return bottom_up(rv, f)
def TR10i(rv):
"""Sum of products to function of sum.
Examples
========
>>> from sympy.simplify.fu import TR10i
>>> from sympy import cos, sin, sqrt
>>> from sympy.abc import x
>>> TR10i(cos(1)*cos(3) + sin(1)*sin(3))
cos(2)
>>> TR10i(cos(1)*sin(3) + sin(1)*cos(3) + cos(3))
cos(3) + sin(4)
>>> TR10i(sqrt(2)*cos(x)*x + sqrt(6)*sin(x)*x)
2*sqrt(2)*x*sin(x + pi/6)
"""
global _ROOT2, _ROOT3, _invROOT3
if _ROOT2 is None:
_roots()
def f(rv):
if not rv.is_Add:
return rv
def do(rv, first=True):
# args which can be expressed as A*(cos(a)*cos(b)+/-sin(a)*sin(b))
# or B*(cos(a)*sin(b)+/-cos(b)*sin(a)) can be combined into
# A*f(a+/-b) where f is either sin or cos.
#
# If there are more than two args, the pairs which "work" will have
# a gcd extractable and the remaining two terms will have the above
# structure -- all pairs must be checked to find the ones that
# work.
if not rv.is_Add:
return rv
args = list(ordered(rv.args))
if len(args) != 2:
hit = False
for i in range(len(args)):
ai = args[i]
if ai is None:
continue
for j in range(i + 1, len(args)):
aj = args[j]
if aj is None:
continue
was = ai + aj
new = do(was)
if new != was:
args[i] = new # update in place
args[j] = None
hit = True
break # go to next i
if hit:
rv = Add(*[_f for _f in args if _f])
if rv.is_Add:
rv = do(rv)
return rv
# two-arg Add
split = trig_split(*args, two=True)
if not split:
return rv
gcd, n1, n2, a, b, same = split
# identify and get c1 to be cos then apply rule if possible
if same: # coscos, sinsin
gcd = n1*gcd
if n1 == n2:
return gcd*cos(a - b)
return gcd*cos(a + b)
else: #cossin, cossin
gcd = n1*gcd
if n1 == n2:
return gcd*sin(a + b)
return gcd*sin(b - a)
rv = process_common_addends(
rv, do, lambda x: tuple(ordered(x.free_symbols)))
# need to check for inducible pairs in ratio of sqrt(3):1 that
# appeared in different lists when sorting by coefficient
while rv.is_Add:
byrad = defaultdict(list)
for a in rv.args:
hit = 0
if a.is_Mul:
for ai in a.args:
if ai.is_Pow and ai.exp is S.Half and \
ai.base.is_Integer:
byrad[ai].append(a)
hit = 1
break
if not hit:
byrad[S.One].append(a)
# no need to check all pairs -- just check for the onees
# that have the right ratio
args = []
for a in byrad:
for b in [_ROOT3*a, _invROOT3]:
if b in byrad:
for i in range(len(byrad[a])):
if byrad[a][i] is None:
continue
for j in range(len(byrad[b])):
if byrad[b][j] is None:
continue
was = Add(byrad[a][i] + byrad[b][j])
new = do(was)
if new != was:
args.append(new)
byrad[a][i] = None
byrad[b][j] = None
break
if args:
rv = Add(*(args + [Add(*[_f for _f in v if _f])
for v in byrad.values()]))
else:
rv = do(rv) # final pass to resolve any new inducible pairs
break
return rv
return bottom_up(rv, f)
def TR11(rv, base=None):
"""Function of double angle to product. The ``base`` argument can be used
to indicate what is the un-doubled argument, e.g. if 3*pi/7 is the base
then cosine and sine functions with argument 6*pi/7 will be replaced.
Examples
========
>>> from sympy.simplify.fu import TR11
>>> from sympy import cos, sin, pi
>>> from sympy.abc import x
>>> TR11(sin(2*x))
2*sin(x)*cos(x)
>>> TR11(cos(2*x))
-sin(x)**2 + cos(x)**2
>>> TR11(sin(4*x))
4*(-sin(x)**2 + cos(x)**2)*sin(x)*cos(x)
>>> TR11(sin(4*x/3))
4*(-sin(x/3)**2 + cos(x/3)**2)*sin(x/3)*cos(x/3)
If the arguments are simply integers, no change is made
unless a base is provided:
>>> TR11(cos(2))
cos(2)
>>> TR11(cos(4), 2)
-sin(2)**2 + cos(2)**2
There is a subtle issue here in that autosimplification will convert
some higher angles to lower angles
>>> cos(6*pi/7) + cos(3*pi/7)
-cos(pi/7) + cos(3*pi/7)
The 6*pi/7 angle is now pi/7 but can be targeted with TR11 by supplying
the 3*pi/7 base:
>>> TR11(_, 3*pi/7)
-sin(3*pi/7)**2 + cos(3*pi/7)**2 + cos(3*pi/7)
"""
def f(rv):
if rv.func not in (cos, sin):
return rv
if base:
f = rv.func
t = f(base*2)
co = S.One
if t.is_Mul:
co, t = t.as_coeff_Mul()
if t.func not in (cos, sin):
return rv
if rv.args[0] == t.args[0]:
c = cos(base)
s = sin(base)
if f is cos:
return (c**2 - s**2)/co
else:
return 2*c*s/co
return rv
elif not rv.args[0].is_Number:
# make a change if the leading coefficient's numerator is
# divisible by 2
c, m = rv.args[0].as_coeff_Mul(rational=True)
if c.p % 2 == 0:
arg = c.p//2*m/c.q
c = TR11(cos(arg))
s = TR11(sin(arg))
if rv.func == sin:
rv = 2*s*c
else:
rv = c**2 - s**2
return rv
return bottom_up(rv, f)
def _TR11(rv):
"""
Helper for TR11 to find half-arguments for sin in factors of
num/den that appear in cos or sin factors in the den/num.
Examples
========
>>> from sympy.simplify.fu import TR11, _TR11
>>> from sympy import cos, sin
>>> from sympy.abc import x
>>> TR11(sin(x/3)/(cos(x/6)))
sin(x/3)/cos(x/6)
>>> _TR11(sin(x/3)/(cos(x/6)))
2*sin(x/6)
>>> TR11(sin(x/6)/(sin(x/3)))
sin(x/6)/sin(x/3)
>>> _TR11(sin(x/6)/(sin(x/3)))
1/(2*cos(x/6))
"""
def f(rv):
if not isinstance(rv, Expr):
return rv
def sincos_args(flat):
# find arguments of sin and cos that
# appears as bases in args of flat
# and have Integer exponents
args = defaultdict(set)
for fi in Mul.make_args(flat):
b, e = fi.as_base_exp()
if e.is_Integer and e > 0:
if b.func in (cos, sin):
args[type(b)].add(b.args[0])
return args
num_args, den_args = map(sincos_args, rv.as_numer_denom())
def handle_match(rv, num_args, den_args):
# for arg in sin args of num_args, look for arg/2
# in den_args and pass this half-angle to TR11
# for handling in rv
for narg in num_args[sin]:
half = narg/2
if half in den_args[cos]:
func = cos
elif half in den_args[sin]:
func = sin
else:
continue
rv = TR11(rv, half)
den_args[func].remove(half)
return rv
# sin in num, sin or cos in den
rv = handle_match(rv, num_args, den_args)
# sin in den, sin or cos in num
rv = handle_match(rv, den_args, num_args)
return rv
return bottom_up(rv, f)
def TR12(rv, first=True):
"""Separate sums in ``tan``.
Examples
========
>>> from sympy.abc import x, y
>>> from sympy import tan
>>> from sympy.simplify.fu import TR12
>>> TR12(tan(x + y))
(tan(x) + tan(y))/(-tan(x)*tan(y) + 1)
"""
def f(rv):
if not rv.func == tan:
return rv
arg = rv.args[0]
if arg.is_Add:
if first:
args = list(ordered(arg.args))
else:
args = list(arg.args)
a = args.pop()
b = Add._from_args(args)
if b.is_Add:
tb = TR12(tan(b), first=False)
else:
tb = tan(b)
return (tan(a) + tb)/(1 - tan(a)*tb)
return rv
return bottom_up(rv, f)
def TR12i(rv):
"""Combine tan arguments as
(tan(y) + tan(x))/(tan(x)*tan(y) - 1) -> -tan(x + y).
Examples
========
>>> from sympy.simplify.fu import TR12i
>>> from sympy import tan
>>> from sympy.abc import a, b, c
>>> ta, tb, tc = [tan(i) for i in (a, b, c)]
>>> TR12i((ta + tb)/(-ta*tb + 1))
tan(a + b)
>>> TR12i((ta + tb)/(ta*tb - 1))
-tan(a + b)
>>> TR12i((-ta - tb)/(ta*tb - 1))
tan(a + b)
>>> eq = (ta + tb)/(-ta*tb + 1)**2*(-3*ta - 3*tc)/(2*(ta*tc - 1))
>>> TR12i(eq.expand())
-3*tan(a + b)*tan(a + c)/(2*(tan(a) + tan(b) - 1))
"""
def f(rv):
if not (rv.is_Add or rv.is_Mul or rv.is_Pow):
return rv
n, d = rv.as_numer_denom()
if not d.args or not n.args:
return rv
dok = {}
def ok(di):
m = as_f_sign_1(di)
if m:
g, f, s = m
if s is S.NegativeOne and f.is_Mul and len(f.args) == 2 and \
all(isinstance(fi, tan) for fi in f.args):
return g, f
d_args = list(Mul.make_args(d))
for i, di in enumerate(d_args):
m = ok(di)
if m:
g, t = m
s = Add(*[_.args[0] for _ in t.args])
dok[s] = S.One
d_args[i] = g
continue
if di.is_Add:
di = factor(di)
if di.is_Mul:
d_args.extend(di.args)
d_args[i] = S.One
elif di.is_Pow and (di.exp.is_integer or di.base.is_positive):
m = ok(di.base)
if m:
g, t = m
s = Add(*[_.args[0] for _ in t.args])
dok[s] = di.exp
d_args[i] = g**di.exp
else:
di = factor(di)
if di.is_Mul:
d_args.extend(di.args)
d_args[i] = S.One
if not dok:
return rv
def ok(ni):
if ni.is_Add and len(ni.args) == 2:
a, b = ni.args
if isinstance(a, tan) and isinstance(b, tan):
return a, b
n_args = list(Mul.make_args(factor_terms(n)))
hit = False
for i, ni in enumerate(n_args):
m = ok(ni)
if not m:
m = ok(-ni)
if m:
n_args[i] = S.NegativeOne
else:
if ni.is_Add:
ni = factor(ni)
if ni.is_Mul:
n_args.extend(ni.args)
n_args[i] = S.One
continue
elif ni.is_Pow and (
ni.exp.is_integer or ni.base.is_positive):
m = ok(ni.base)
if m:
n_args[i] = S.One
else:
ni = factor(ni)
if ni.is_Mul:
n_args.extend(ni.args)
n_args[i] = S.One
continue
else:
continue
else:
n_args[i] = S.One
hit = True
s = Add(*[_.args[0] for _ in m])
ed = dok[s]
newed = ed.extract_additively(S.One)
if newed is not None:
if newed:
dok[s] = newed
else:
dok.pop(s)
n_args[i] *= -tan(s)
if hit:
rv = Mul(*n_args)/Mul(*d_args)/Mul(*[(Add(*[
tan(a) for a in i.args]) - 1)**e for i, e in dok.items()])
return rv
return bottom_up(rv, f)
def TR13(rv):
"""Change products of ``tan`` or ``cot``.
Examples
========
>>> from sympy.simplify.fu import TR13
>>> from sympy import tan, cot
>>> TR13(tan(3)*tan(2))
-tan(2)/tan(5) - tan(3)/tan(5) + 1
>>> TR13(cot(3)*cot(2))
cot(2)*cot(5) + 1 + cot(3)*cot(5)
"""
def f(rv):
if not rv.is_Mul:
return rv
# XXX handle products of powers? or let power-reducing handle it?
args = {tan: [], cot: [], None: []}
for a in ordered(Mul.make_args(rv)):
if a.func in (tan, cot):
args[type(a)].append(a.args[0])
else:
args[None].append(a)
t = args[tan]
c = args[cot]
if len(t) < 2 and len(c) < 2:
return rv
args = args[None]
while len(t) > 1:
t1 = t.pop()
t2 = t.pop()
args.append(1 - (tan(t1)/tan(t1 + t2) + tan(t2)/tan(t1 + t2)))
if t:
args.append(tan(t.pop()))
while len(c) > 1:
t1 = c.pop()
t2 = c.pop()
args.append(1 + cot(t1)*cot(t1 + t2) + cot(t2)*cot(t1 + t2))
if c:
args.append(cot(c.pop()))
return Mul(*args)
return bottom_up(rv, f)
def TRmorrie(rv):
"""Returns cos(x)*cos(2*x)*...*cos(2**(k-1)*x) -> sin(2**k*x)/(2**k*sin(x))
Examples
========
>>> from sympy.simplify.fu import TRmorrie, TR8, TR3
>>> from sympy.abc import x
>>> from sympy import Mul, cos, pi
>>> TRmorrie(cos(x)*cos(2*x))
sin(4*x)/(4*sin(x))
>>> TRmorrie(7*Mul(*[cos(x) for x in range(10)]))
7*sin(12)*sin(16)*cos(5)*cos(7)*cos(9)/(64*sin(1)*sin(3))
Sometimes autosimplification will cause a power to be
not recognized. e.g. in the following, cos(4*pi/7) automatically
simplifies to -cos(3*pi/7) so only 2 of the 3 terms are
recognized:
>>> TRmorrie(cos(pi/7)*cos(2*pi/7)*cos(4*pi/7))
-sin(3*pi/7)*cos(3*pi/7)/(4*sin(pi/7))
A touch by TR8 resolves the expression to a Rational
>>> TR8(_)
-1/8
In this case, if eq is unsimplified, the answer is obtained
directly:
>>> eq = cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9)
>>> TRmorrie(eq)
1/16
But if angles are made canonical with TR3 then the answer
is not simplified without further work:
>>> TR3(eq)
sin(pi/18)*cos(pi/9)*cos(2*pi/9)/2
>>> TRmorrie(_)
sin(pi/18)*sin(4*pi/9)/(8*sin(pi/9))
>>> TR8(_)
cos(7*pi/18)/(16*sin(pi/9))
>>> TR3(_)
1/16
The original expression would have resolve to 1/16 directly with TR8,
however:
>>> TR8(eq)
1/16
References
==========
.. [1] https://en.wikipedia.org/wiki/Morrie%27s_law
"""
def f(rv, first=True):
if not rv.is_Mul:
return rv
if first:
n, d = rv.as_numer_denom()
return f(n, 0)/f(d, 0)
args = defaultdict(list)
coss = {}
other = []
for c in rv.args:
b, e = c.as_base_exp()
if e.is_Integer and isinstance(b, cos):
co, a = b.args[0].as_coeff_Mul()
args[a].append(co)
coss[b] = e
else:
other.append(c)
new = []
for a in args:
c = args[a]
c.sort()
while c:
k = 0
cc = ci = c[0]
while cc in c:
k += 1
cc *= 2
if k > 1:
newarg = sin(2**k*ci*a)/2**k/sin(ci*a)
# see how many times this can be taken
take = None
ccs = []
for i in range(k):
cc /= 2
key = cos(a*cc, evaluate=False)
ccs.append(cc)
take = min(coss[key], take or coss[key])
# update exponent counts
for i in range(k):
cc = ccs.pop()
key = cos(a*cc, evaluate=False)
coss[key] -= take
if not coss[key]:
c.remove(cc)
new.append(newarg**take)
else:
b = cos(c.pop(0)*a)
other.append(b**coss[b])
if new:
rv = Mul(*(new + other + [
cos(k*a, evaluate=False) for a in args for k in args[a]]))
return rv
return bottom_up(rv, f)
def TR14(rv, first=True):
"""Convert factored powers of sin and cos identities into simpler
expressions.
Examples
========
>>> from sympy.simplify.fu import TR14
>>> from sympy.abc import x, y
>>> from sympy import cos, sin
>>> TR14((cos(x) - 1)*(cos(x) + 1))
-sin(x)**2
>>> TR14((sin(x) - 1)*(sin(x) + 1))
-cos(x)**2
>>> p1 = (cos(x) + 1)*(cos(x) - 1)
>>> p2 = (cos(y) - 1)*2*(cos(y) + 1)
>>> p3 = (3*(cos(y) - 1))*(3*(cos(y) + 1))
>>> TR14(p1*p2*p3*(x - 1))
-18*(x - 1)*sin(x)**2*sin(y)**4
"""
def f(rv):
if not rv.is_Mul:
return rv
if first:
# sort them by location in numerator and denominator
# so the code below can just deal with positive exponents
n, d = rv.as_numer_denom()
if d is not S.One:
newn = TR14(n, first=False)
newd = TR14(d, first=False)
if newn != n or newd != d:
rv = newn/newd
return rv
other = []
process = []
for a in rv.args:
if a.is_Pow:
b, e = a.as_base_exp()
if not (e.is_integer or b.is_positive):
other.append(a)
continue
a = b
else:
e = S.One
m = as_f_sign_1(a)
if not m or m[1].func not in (cos, sin):
if e is S.One:
other.append(a)
else:
other.append(a**e)
continue
g, f, si = m
process.append((g, e.is_Number, e, f, si, a))
# sort them to get like terms next to each other
process = list(ordered(process))
# keep track of whether there was any change
nother = len(other)
# access keys
keys = (g, t, e, f, si, a) = list(range(6))
while process:
A = process.pop(0)
if process:
B = process[0]
if A[e].is_Number and B[e].is_Number:
# both exponents are numbers
if A[f] == B[f]:
if A[si] != B[si]:
B = process.pop(0)
take = min(A[e], B[e])
# reinsert any remainder
# the B will likely sort after A so check it first
if B[e] != take:
rem = [B[i] for i in keys]
rem[e] -= take
process.insert(0, rem)
elif A[e] != take:
rem = [A[i] for i in keys]
rem[e] -= take
process.insert(0, rem)
if isinstance(A[f], cos):
t = sin
else:
t = cos
other.append((-A[g]*B[g]*t(A[f].args[0])**2)**take)
continue
elif A[e] == B[e]:
# both exponents are equal symbols
if A[f] == B[f]:
if A[si] != B[si]:
B = process.pop(0)
take = A[e]
if isinstance(A[f], cos):
t = sin
else:
t = cos
other.append((-A[g]*B[g]*t(A[f].args[0])**2)**take)
continue
# either we are done or neither condition above applied
other.append(A[a]**A[e])
if len(other) != nother:
rv = Mul(*other)
return rv
return bottom_up(rv, f)
def TR15(rv, max=4, pow=False):
"""Convert sin(x)**-2 to 1 + cot(x)**2.
See _TR56 docstring for advanced use of ``max`` and ``pow``.
Examples
========
>>> from sympy.simplify.fu import TR15
>>> from sympy.abc import x
>>> from sympy import sin
>>> TR15(1 - 1/sin(x)**2)
-cot(x)**2
"""
def f(rv):
if not (isinstance(rv, Pow) and isinstance(rv.base, sin)):
return rv
e = rv.exp
if e % 2 == 1:
return TR15(rv.base**(e + 1))/rv.base
ia = 1/rv
a = _TR56(ia, sin, cot, lambda x: 1 + x, max=max, pow=pow)
if a != ia:
rv = a
return rv
return bottom_up(rv, f)
def TR16(rv, max=4, pow=False):
"""Convert cos(x)**-2 to 1 + tan(x)**2.
See _TR56 docstring for advanced use of ``max`` and ``pow``.
Examples
========
>>> from sympy.simplify.fu import TR16
>>> from sympy.abc import x
>>> from sympy import cos
>>> TR16(1 - 1/cos(x)**2)
-tan(x)**2
"""
def f(rv):
if not (isinstance(rv, Pow) and isinstance(rv.base, cos)):
return rv
e = rv.exp
if e % 2 == 1:
return TR15(rv.base**(e + 1))/rv.base
ia = 1/rv
a = _TR56(ia, cos, tan, lambda x: 1 + x, max=max, pow=pow)
if a != ia:
rv = a
return rv
return bottom_up(rv, f)
def TR111(rv):
"""Convert f(x)**-i to g(x)**i where either ``i`` is an integer
or the base is positive and f, g are: tan, cot; sin, csc; or cos, sec.
Examples
========
>>> from sympy.simplify.fu import TR111
>>> from sympy.abc import x
>>> from sympy import tan
>>> TR111(1 - 1/tan(x)**2)
1 - cot(x)**2
"""
def f(rv):
if not (
isinstance(rv, Pow) and
(rv.base.is_positive or rv.exp.is_integer and rv.exp.is_negative)):
return rv
if isinstance(rv.base, tan):
return cot(rv.base.args[0])**-rv.exp
elif isinstance(rv.base, sin):
return csc(rv.base.args[0])**-rv.exp
elif isinstance(rv.base, cos):
return sec(rv.base.args[0])**-rv.exp
return rv
return bottom_up(rv, f)
def TR22(rv, max=4, pow=False):
"""Convert tan(x)**2 to sec(x)**2 - 1 and cot(x)**2 to csc(x)**2 - 1.
See _TR56 docstring for advanced use of ``max`` and ``pow``.
Examples
========
>>> from sympy.simplify.fu import TR22
>>> from sympy.abc import x
>>> from sympy import tan, cot
>>> TR22(1 + tan(x)**2)
sec(x)**2
>>> TR22(1 + cot(x)**2)
csc(x)**2
"""
def f(rv):
if not (isinstance(rv, Pow) and rv.base.func in (cot, tan)):
return rv
rv = _TR56(rv, tan, sec, lambda x: x - 1, max=max, pow=pow)
rv = _TR56(rv, cot, csc, lambda x: x - 1, max=max, pow=pow)
return rv
return bottom_up(rv, f)
def TRpower(rv):
"""Convert sin(x)**n and cos(x)**n with positive n to sums.
Examples
========
>>> from sympy.simplify.fu import TRpower
>>> from sympy.abc import x
>>> from sympy import cos, sin
>>> TRpower(sin(x)**6)
-15*cos(2*x)/32 + 3*cos(4*x)/16 - cos(6*x)/32 + 5/16
>>> TRpower(sin(x)**3*cos(2*x)**4)
(3*sin(x)/4 - sin(3*x)/4)*(cos(4*x)/2 + cos(8*x)/8 + 3/8)
References
==========
.. [1] https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulae
"""
def f(rv):
if not (isinstance(rv, Pow) and isinstance(rv.base, (sin, cos))):
return rv
b, n = rv.as_base_exp()
x = b.args[0]
if n.is_Integer and n.is_positive:
if n.is_odd and isinstance(b, cos):
rv = 2**(1-n)*Add(*[binomial(n, k)*cos((n - 2*k)*x)
for k in range((n + 1)/2)])
elif n.is_odd and isinstance(b, sin):
rv = 2**(1-n)*S.NegativeOne**((n-1)/2)*Add(*[binomial(n, k)*
S.NegativeOne**k*sin((n - 2*k)*x) for k in range((n + 1)/2)])
elif n.is_even and isinstance(b, cos):
rv = 2**(1-n)*Add(*[binomial(n, k)*cos((n - 2*k)*x)
for k in range(n/2)])
elif n.is_even and isinstance(b, sin):
rv = 2**(1-n)*S.NegativeOne**(n/2)*Add(*[binomial(n, k)*
S.NegativeOne**k*cos((n - 2*k)*x) for k in range(n/2)])
if n.is_even:
rv += 2**(-n)*binomial(n, n/2)
return rv
return bottom_up(rv, f)
def L(rv):
"""Return count of trigonometric functions in expression.
Examples
========
>>> from sympy.simplify.fu import L
>>> from sympy.abc import x
>>> from sympy import cos, sin
>>> L(cos(x)+sin(x))
2
"""
return S(rv.count(TrigonometricFunction))
# ============== end of basic Fu-like tools =====================
if SYMPY_DEBUG:
(TR0, TR1, TR2, TR3, TR4, TR5, TR6, TR7, TR8, TR9, TR10, TR11, TR12, TR13,
TR2i, TRmorrie, TR14, TR15, TR16, TR12i, TR111, TR22
)= list(map(debug,
(TR0, TR1, TR2, TR3, TR4, TR5, TR6, TR7, TR8, TR9, TR10, TR11, TR12, TR13,
TR2i, TRmorrie, TR14, TR15, TR16, TR12i, TR111, TR22)))
# tuples are chains -- (f, g) -> lambda x: g(f(x))
# lists are choices -- [f, g] -> lambda x: min(f(x), g(x), key=objective)
CTR1 = [(TR5, TR0), (TR6, TR0), identity]
CTR2 = (TR11, [(TR5, TR0), (TR6, TR0), TR0])
CTR3 = [(TRmorrie, TR8, TR0), (TRmorrie, TR8, TR10i, TR0), identity]
CTR4 = [(TR4, TR10i), identity]
RL1 = (TR4, TR3, TR4, TR12, TR4, TR13, TR4, TR0)
# XXX it's a little unclear how this one is to be implemented
# see Fu paper of reference, page 7. What is the Union symbol referring to?
# The diagram shows all these as one chain of transformations, but the
# text refers to them being applied independently. Also, a break
# if L starts to increase has not been implemented.
RL2 = [
(TR4, TR3, TR10, TR4, TR3, TR11),
(TR5, TR7, TR11, TR4),
(CTR3, CTR1, TR9, CTR2, TR4, TR9, TR9, CTR4),
identity,
]
def fu(rv, measure=lambda x: (L(x), x.count_ops())):
"""Attempt to simplify expression by using transformation rules given
in the algorithm by Fu et al.
:func:`fu` will try to minimize the objective function ``measure``.
By default this first minimizes the number of trig terms and then minimizes
the number of total operations.
Examples
========
>>> from sympy.simplify.fu import fu
>>> from sympy import cos, sin, tan, pi, S, sqrt
>>> from sympy.abc import x, y, a, b
>>> fu(sin(50)**2 + cos(50)**2 + sin(pi/6))
3/2
>>> fu(sqrt(6)*cos(x) + sqrt(2)*sin(x))
2*sqrt(2)*sin(x + pi/3)
CTR1 example
>>> eq = sin(x)**4 - cos(y)**2 + sin(y)**2 + 2*cos(x)**2
>>> fu(eq)
cos(x)**4 - 2*cos(y)**2 + 2
CTR2 example
>>> fu(S.Half - cos(2*x)/2)
sin(x)**2
CTR3 example
>>> fu(sin(a)*(cos(b) - sin(b)) + cos(a)*(sin(b) + cos(b)))
sqrt(2)*sin(a + b + pi/4)
CTR4 example
>>> fu(sqrt(3)*cos(x)/2 + sin(x)/2)
sin(x + pi/3)
Example 1
>>> fu(1-sin(2*x)**2/4-sin(y)**2-cos(x)**4)
-cos(x)**2 + cos(y)**2
Example 2
>>> fu(cos(4*pi/9))
sin(pi/18)
>>> fu(cos(pi/9)*cos(2*pi/9)*cos(3*pi/9)*cos(4*pi/9))
1/16
Example 3
>>> fu(tan(7*pi/18)+tan(5*pi/18)-sqrt(3)*tan(5*pi/18)*tan(7*pi/18))
-sqrt(3)
Objective function example
>>> fu(sin(x)/cos(x)) # default objective function
tan(x)
>>> fu(sin(x)/cos(x), measure=lambda x: -x.count_ops()) # maximize op count
sin(x)/cos(x)
References
==========
.. [1] https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.657.2478&rep=rep1&type=pdf
"""
fRL1 = greedy(RL1, measure)
fRL2 = greedy(RL2, measure)
was = rv
rv = sympify(rv)
if not isinstance(rv, Expr):
return rv.func(*[fu(a, measure=measure) for a in rv.args])
rv = TR1(rv)
if rv.has(tan, cot):
rv1 = fRL1(rv)
if (measure(rv1) < measure(rv)):
rv = rv1
if rv.has(tan, cot):
rv = TR2(rv)
if rv.has(sin, cos):
rv1 = fRL2(rv)
rv2 = TR8(TRmorrie(rv1))
rv = min([was, rv, rv1, rv2], key=measure)
return min(TR2i(rv), rv, key=measure)
def process_common_addends(rv, do, key2=None, key1=True):
"""Apply ``do`` to addends of ``rv`` that (if ``key1=True``) share at least
a common absolute value of their coefficient and the value of ``key2`` when
applied to the argument. If ``key1`` is False ``key2`` must be supplied and
will be the only key applied.
"""
# collect by absolute value of coefficient and key2
absc = defaultdict(list)
if key1:
for a in rv.args:
c, a = a.as_coeff_Mul()
if c < 0:
c = -c
a = -a # put the sign on `a`
absc[(c, key2(a) if key2 else 1)].append(a)
elif key2:
for a in rv.args:
absc[(S.One, key2(a))].append(a)
else:
raise ValueError('must have at least one key')
args = []
hit = False
for k in absc:
v = absc[k]
c, _ = k
if len(v) > 1:
e = Add(*v, evaluate=False)
new = do(e)
if new != e:
e = new
hit = True
args.append(c*e)
else:
args.append(c*v[0])
if hit:
rv = Add(*args)
return rv
fufuncs = '''
TR0 TR1 TR2 TR3 TR4 TR5 TR6 TR7 TR8 TR9 TR10 TR10i TR11
TR12 TR13 L TR2i TRmorrie TR12i
TR14 TR15 TR16 TR111 TR22'''.split()
FU = dict(list(zip(fufuncs, list(map(locals().get, fufuncs)))))
def _roots():
global _ROOT2, _ROOT3, _invROOT3
_ROOT2, _ROOT3 = sqrt(2), sqrt(3)
_invROOT3 = 1/_ROOT3
_ROOT2 = None
def trig_split(a, b, two=False):
"""Return the gcd, s1, s2, a1, a2, bool where
If two is False (default) then::
a + b = gcd*(s1*f(a1) + s2*f(a2)) where f = cos if bool else sin
else:
if bool, a + b was +/- cos(a1)*cos(a2) +/- sin(a1)*sin(a2) and equals
n1*gcd*cos(a - b) if n1 == n2 else
n1*gcd*cos(a + b)
else a + b was +/- cos(a1)*sin(a2) +/- sin(a1)*cos(a2) and equals
n1*gcd*sin(a + b) if n1 = n2 else
n1*gcd*sin(b - a)
Examples
========
>>> from sympy.simplify.fu import trig_split
>>> from sympy.abc import x, y, z
>>> from sympy import cos, sin, sqrt
>>> trig_split(cos(x), cos(y))
(1, 1, 1, x, y, True)
>>> trig_split(2*cos(x), -2*cos(y))
(2, 1, -1, x, y, True)
>>> trig_split(cos(x)*sin(y), cos(y)*sin(y))
(sin(y), 1, 1, x, y, True)
>>> trig_split(cos(x), -sqrt(3)*sin(x), two=True)
(2, 1, -1, x, pi/6, False)
>>> trig_split(cos(x), sin(x), two=True)
(sqrt(2), 1, 1, x, pi/4, False)
>>> trig_split(cos(x), -sin(x), two=True)
(sqrt(2), 1, -1, x, pi/4, False)
>>> trig_split(sqrt(2)*cos(x), -sqrt(6)*sin(x), two=True)
(2*sqrt(2), 1, -1, x, pi/6, False)
>>> trig_split(-sqrt(6)*cos(x), -sqrt(2)*sin(x), two=True)
(-2*sqrt(2), 1, 1, x, pi/3, False)
>>> trig_split(cos(x)/sqrt(6), sin(x)/sqrt(2), two=True)
(sqrt(6)/3, 1, 1, x, pi/6, False)
>>> trig_split(-sqrt(6)*cos(x)*sin(y), -sqrt(2)*sin(x)*sin(y), two=True)
(-2*sqrt(2)*sin(y), 1, 1, x, pi/3, False)
>>> trig_split(cos(x), sin(x))
>>> trig_split(cos(x), sin(z))
>>> trig_split(2*cos(x), -sin(x))
>>> trig_split(cos(x), -sqrt(3)*sin(x))
>>> trig_split(cos(x)*cos(y), sin(x)*sin(z))
>>> trig_split(cos(x)*cos(y), sin(x)*sin(y))
>>> trig_split(-sqrt(6)*cos(x), sqrt(2)*sin(x)*sin(y), two=True)
"""
global _ROOT2, _ROOT3, _invROOT3
if _ROOT2 is None:
_roots()
a, b = [Factors(i) for i in (a, b)]
ua, ub = a.normal(b)
gcd = a.gcd(b).as_expr()
n1 = n2 = 1
if S.NegativeOne in ua.factors:
ua = ua.quo(S.NegativeOne)
n1 = -n1
elif S.NegativeOne in ub.factors:
ub = ub.quo(S.NegativeOne)
n2 = -n2
a, b = [i.as_expr() for i in (ua, ub)]
def pow_cos_sin(a, two):
"""Return ``a`` as a tuple (r, c, s) such that
``a = (r or 1)*(c or 1)*(s or 1)``.
Three arguments are returned (radical, c-factor, s-factor) as
long as the conditions set by ``two`` are met; otherwise None is
returned. If ``two`` is True there will be one or two non-None
values in the tuple: c and s or c and r or s and r or s or c with c
being a cosine function (if possible) else a sine, and s being a sine
function (if possible) else oosine. If ``two`` is False then there
will only be a c or s term in the tuple.
``two`` also require that either two cos and/or sin be present (with
the condition that if the functions are the same the arguments are
different or vice versa) or that a single cosine or a single sine
be present with an optional radical.
If the above conditions dictated by ``two`` are not met then None
is returned.
"""
c = s = None
co = S.One
if a.is_Mul:
co, a = a.as_coeff_Mul()
if len(a.args) > 2 or not two:
return None
if a.is_Mul:
args = list(a.args)
else:
args = [a]
a = args.pop(0)
if isinstance(a, cos):
c = a
elif isinstance(a, sin):
s = a
elif a.is_Pow and a.exp is S.Half: # autoeval doesn't allow -1/2
co *= a
else:
return None
if args:
b = args[0]
if isinstance(b, cos):
if c:
s = b
else:
c = b
elif isinstance(b, sin):
if s:
c = b
else:
s = b
elif b.is_Pow and b.exp is S.Half:
co *= b
else:
return None
return co if co is not S.One else None, c, s
elif isinstance(a, cos):
c = a
elif isinstance(a, sin):
s = a
if c is None and s is None:
return
co = co if co is not S.One else None
return co, c, s
# get the parts
m = pow_cos_sin(a, two)
if m is None:
return
coa, ca, sa = m
m = pow_cos_sin(b, two)
if m is None:
return
cob, cb, sb = m
# check them
if (not ca) and cb or ca and isinstance(ca, sin):
coa, ca, sa, cob, cb, sb = cob, cb, sb, coa, ca, sa
n1, n2 = n2, n1
if not two: # need cos(x) and cos(y) or sin(x) and sin(y)
c = ca or sa
s = cb or sb
if not isinstance(c, s.func):
return None
return gcd, n1, n2, c.args[0], s.args[0], isinstance(c, cos)
else:
if not coa and not cob:
if (ca and cb and sa and sb):
if isinstance(ca, sa.func) is not isinstance(cb, sb.func):
return
args = {j.args for j in (ca, sa)}
if not all(i.args in args for i in (cb, sb)):
return
return gcd, n1, n2, ca.args[0], sa.args[0], isinstance(ca, sa.func)
if ca and sa or cb and sb or \
two and (ca is None and sa is None or cb is None and sb is None):
return
c = ca or sa
s = cb or sb
if c.args != s.args:
return
if not coa:
coa = S.One
if not cob:
cob = S.One
if coa is cob:
gcd *= _ROOT2
return gcd, n1, n2, c.args[0], pi/4, False
elif coa/cob == _ROOT3:
gcd *= 2*cob
return gcd, n1, n2, c.args[0], pi/3, False
elif coa/cob == _invROOT3:
gcd *= 2*coa
return gcd, n1, n2, c.args[0], pi/6, False
def as_f_sign_1(e):
"""If ``e`` is a sum that can be written as ``g*(a + s)`` where
``s`` is ``+/-1``, return ``g``, ``a``, and ``s`` where ``a`` does
not have a leading negative coefficient.
Examples
========
>>> from sympy.simplify.fu import as_f_sign_1
>>> from sympy.abc import x
>>> as_f_sign_1(x + 1)
(1, x, 1)
>>> as_f_sign_1(x - 1)
(1, x, -1)
>>> as_f_sign_1(-x + 1)
(-1, x, -1)
>>> as_f_sign_1(-x - 1)
(-1, x, 1)
>>> as_f_sign_1(2*x + 2)
(2, x, 1)
"""
if not e.is_Add or len(e.args) != 2:
return
# exact match
a, b = e.args
if a in (S.NegativeOne, S.One):
g = S.One
if b.is_Mul and b.args[0].is_Number and b.args[0] < 0:
a, b = -a, -b
g = -g
return g, b, a
# gcd match
a, b = [Factors(i) for i in e.args]
ua, ub = a.normal(b)
gcd = a.gcd(b).as_expr()
if S.NegativeOne in ua.factors:
ua = ua.quo(S.NegativeOne)
n1 = -1
n2 = 1
elif S.NegativeOne in ub.factors:
ub = ub.quo(S.NegativeOne)
n1 = 1
n2 = -1
else:
n1 = n2 = 1
a, b = [i.as_expr() for i in (ua, ub)]
if a is S.One:
a, b = b, a
n1, n2 = n2, n1
if n1 == -1:
gcd = -gcd
n2 = -n2
if b is S.One:
return gcd, a, n2
def _osborne(e, d):
"""Replace all hyperbolic functions with trig functions using
the Osborne rule.
Notes
=====
``d`` is a dummy variable to prevent automatic evaluation
of trigonometric/hyperbolic functions.
References
==========
.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
"""
def f(rv):
if not isinstance(rv, HyperbolicFunction):
return rv
a = rv.args[0]
a = a*d if not a.is_Add else Add._from_args([i*d for i in a.args])
if isinstance(rv, sinh):
return I*sin(a)
elif isinstance(rv, cosh):
return cos(a)
elif isinstance(rv, tanh):
return I*tan(a)
elif isinstance(rv, coth):
return cot(a)/I
elif isinstance(rv, sech):
return sec(a)
elif isinstance(rv, csch):
return csc(a)/I
else:
raise NotImplementedError('unhandled %s' % rv.func)
return bottom_up(e, f)
def _osbornei(e, d):
"""Replace all trig functions with hyperbolic functions using
the Osborne rule.
Notes
=====
``d`` is a dummy variable to prevent automatic evaluation
of trigonometric/hyperbolic functions.
References
==========
.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
"""
def f(rv):
if not isinstance(rv, TrigonometricFunction):
return rv
const, x = rv.args[0].as_independent(d, as_Add=True)
a = x.xreplace({d: S.One}) + const*I
if isinstance(rv, sin):
return sinh(a)/I
elif isinstance(rv, cos):
return cosh(a)
elif isinstance(rv, tan):
return tanh(a)/I
elif isinstance(rv, cot):
return coth(a)*I
elif isinstance(rv, sec):
return sech(a)
elif isinstance(rv, csc):
return csch(a)*I
else:
raise NotImplementedError('unhandled %s' % rv.func)
return bottom_up(e, f)
def hyper_as_trig(rv):
"""Return an expression containing hyperbolic functions in terms
of trigonometric functions. Any trigonometric functions initially
present are replaced with Dummy symbols and the function to undo
the masking and the conversion back to hyperbolics is also returned. It
should always be true that::
t, f = hyper_as_trig(expr)
expr == f(t)
Examples
========
>>> from sympy.simplify.fu import hyper_as_trig, fu
>>> from sympy.abc import x
>>> from sympy import cosh, sinh
>>> eq = sinh(x)**2 + cosh(x)**2
>>> t, f = hyper_as_trig(eq)
>>> f(fu(t))
cosh(2*x)
References
==========
.. [1] https://en.wikipedia.org/wiki/Hyperbolic_function
"""
from sympy.simplify.simplify import signsimp
from sympy.simplify.radsimp import collect
# mask off trig functions
trigs = rv.atoms(TrigonometricFunction)
reps = [(t, Dummy()) for t in trigs]
masked = rv.xreplace(dict(reps))
# get inversion substitutions in place
reps = [(v, k) for k, v in reps]
d = Dummy()
return _osborne(masked, d), lambda x: collect(signsimp(
_osbornei(x, d).xreplace(dict(reps))), S.ImaginaryUnit)
def sincos_to_sum(expr):
"""Convert products and powers of sin and cos to sums.
Explanation
===========
Applied power reduction TRpower first, then expands products, and
converts products to sums with TR8.
Examples
========
>>> from sympy.simplify.fu import sincos_to_sum
>>> from sympy.abc import x
>>> from sympy import cos, sin
>>> sincos_to_sum(16*sin(x)**3*cos(2*x)**2)
7*sin(x) - 5*sin(3*x) + 3*sin(5*x) - sin(7*x)
"""
if not expr.has(cos, sin):
return expr
else:
return TR8(expand_mul(TRpower(expr)))
|
834643a32c3f104d15ee0d1d2dd79eaf3a1fd16ceb0fd0c3f608530cb2dbaf8a | from typing import Any, Set as tSet
from functools import reduce
from itertools import permutations
from sympy.combinatorics import Permutation
from sympy.core import (
Basic, Expr, Function, diff,
Pow, Mul, Add, Lambda, S, Tuple, Dict
)
from sympy.core.cache import cacheit
from sympy.core.symbol import Symbol, Dummy
from sympy.core.symbol import Str
from sympy.core.sympify import _sympify
from sympy.functions import factorial
from sympy.matrices import ImmutableDenseMatrix as Matrix
from sympy.solvers import solve
from sympy.utilities.exceptions import SymPyDeprecationWarning
# TODO you are a bit excessive in the use of Dummies
# TODO dummy point, literal field
# TODO too often one needs to call doit or simplify on the output, check the
# tests and find out why
from sympy.tensor.array import ImmutableDenseNDimArray
class Manifold(Basic):
"""
A mathematical manifold.
Explanation
===========
A manifold is a topological space that locally resembles
Euclidean space near each point [1].
This class does not provide any means to study the topological
characteristics of the manifold that it represents, though.
Parameters
==========
name : str
The name of the manifold.
dim : int
The dimension of the manifold.
Examples
========
>>> from sympy.diffgeom import Manifold
>>> m = Manifold('M', 2)
>>> m
M
>>> m.dim
2
References
==========
.. [1] https://en.wikipedia.org/wiki/Manifold
"""
def __new__(cls, name, dim, **kwargs):
if not isinstance(name, Str):
name = Str(name)
dim = _sympify(dim)
obj = super().__new__(cls, name, dim)
obj.patches = _deprecated_list(
"Manifold.patches",
"external container for registry",
19321,
"1.7",
[]
)
return obj
@property
def name(self):
return self.args[0]
@property
def dim(self):
return self.args[1]
class Patch(Basic):
"""
A patch on a manifold.
Explanation
===========
Coordinate patch, or patch in short, is a simply-connected open set around
a point in the manifold [1]. On a manifold one can have many patches that
do not always include the whole manifold. On these patches coordinate
charts can be defined that permit the parameterization of any point on the
patch in terms of a tuple of real numbers (the coordinates).
This class does not provide any means to study the topological
characteristics of the patch that it represents.
Parameters
==========
name : str
The name of the patch.
manifold : Manifold
The manifold on which the patch is defined.
Examples
========
>>> from sympy.diffgeom import Manifold, Patch
>>> m = Manifold('M', 2)
>>> p = Patch('P', m)
>>> p
P
>>> p.dim
2
References
==========
.. [1] G. Sussman, J. Wisdom, W. Farr, Functional Differential Geometry
(2013)
"""
def __new__(cls, name, manifold, **kwargs):
if not isinstance(name, Str):
name = Str(name)
obj = super().__new__(cls, name, manifold)
obj.manifold.patches.append(obj) # deprecated
obj.coord_systems = _deprecated_list(
"Patch.coord_systems",
"external container for registry",
19321,
"1.7",
[]
)
return obj
@property
def name(self):
return self.args[0]
@property
def manifold(self):
return self.args[1]
@property
def dim(self):
return self.manifold.dim
class CoordSystem(Basic):
"""
A coordinate system defined on the patch.
Explanation
===========
Coordinate system is a system that uses one or more coordinates to uniquely
determine the position of the points or other geometric elements on a
manifold [1].
By passing ``Symbols`` to *symbols* parameter, user can define the name and
assumptions of coordinate symbols of the coordinate system. If not passed,
these symbols are generated automatically and are assumed to be real valued.
By passing *relations* parameter, user can define the tranform relations of
coordinate systems. Inverse transformation and indirect transformation can
be found automatically. If this parameter is not passed, coordinate
transformation cannot be done.
Parameters
==========
name : str
The name of the coordinate system.
patch : Patch
The patch where the coordinate system is defined.
symbols : list of Symbols, optional
Defines the names and assumptions of coordinate symbols.
relations : dict, optional
Key is a tuple of two strings, who are the names of the systems where
the coordinates transform from and transform to.
Value is a tuple of the symbols before transformation and a tuple of
the expressions after transformation.
Examples
========
We define two-dimensional Cartesian coordinate system and polar coordinate
system.
>>> from sympy import symbols, pi, sqrt, atan2, cos, sin
>>> from sympy.diffgeom import Manifold, Patch, CoordSystem
>>> m = Manifold('M', 2)
>>> p = Patch('P', m)
>>> x, y = symbols('x y', real=True)
>>> r, theta = symbols('r theta', nonnegative=True)
>>> relation_dict = {
... ('Car2D', 'Pol'): [(x, y), (sqrt(x**2 + y**2), atan2(y, x))],
... ('Pol', 'Car2D'): [(r, theta), (r*cos(theta), r*sin(theta))]
... }
>>> Car2D = CoordSystem('Car2D', p, (x, y), relation_dict)
>>> Pol = CoordSystem('Pol', p, (r, theta), relation_dict)
``symbols`` property returns ``CoordinateSymbol`` instances. These symbols
are not same with the symbols used to construct the coordinate system.
>>> Car2D
Car2D
>>> Car2D.dim
2
>>> Car2D.symbols
(x, y)
>>> _[0].func
<class 'sympy.diffgeom.diffgeom.CoordinateSymbol'>
``transformation()`` method returns the transformation function from
one coordinate system to another. ``transform()`` method returns the
transformed coordinates.
>>> Car2D.transformation(Pol)
Lambda((x, y), Matrix([
[sqrt(x**2 + y**2)],
[ atan2(y, x)]]))
>>> Car2D.transform(Pol)
Matrix([
[sqrt(x**2 + y**2)],
[ atan2(y, x)]])
>>> Car2D.transform(Pol, [1, 2])
Matrix([
[sqrt(5)],
[atan(2)]])
``jacobian()`` method returns the Jacobian matrix of coordinate
transformation between two systems. ``jacobian_determinant()`` method
returns the Jacobian determinant of coordinate transformation between two
systems.
>>> Pol.jacobian(Car2D)
Matrix([
[cos(theta), -r*sin(theta)],
[sin(theta), r*cos(theta)]])
>>> Pol.jacobian(Car2D, [1, pi/2])
Matrix([
[0, -1],
[1, 0]])
>>> Car2D.jacobian_determinant(Pol)
1/sqrt(x**2 + y**2)
>>> Car2D.jacobian_determinant(Pol, [1,0])
1
References
==========
.. [1] https://en.wikipedia.org/wiki/Coordinate_system
"""
def __new__(cls, name, patch, symbols=None, relations={}, **kwargs):
if not isinstance(name, Str):
name = Str(name)
# canonicallize the symbols
if symbols is None:
names = kwargs.get('names', None)
if names is None:
symbols = Tuple(
*[Symbol('%s_%s' % (name.name, i), real=True)
for i in range(patch.dim)]
)
else:
SymPyDeprecationWarning(
feature="Class signature 'names' of CoordSystem",
useinstead="class signature 'symbols'",
issue=19321,
deprecated_since_version="1.7"
).warn()
symbols = Tuple(
*[Symbol(n, real=True) for n in names]
)
else:
syms = []
for s in symbols:
if isinstance(s, Symbol):
syms.append(Symbol(s.name, **s._assumptions.generator))
elif isinstance(s, str):
SymPyDeprecationWarning(
feature="Passing str as coordinate symbol's name",
useinstead="Symbol which contains the name and assumption for coordinate symbol",
issue=19321,
deprecated_since_version="1.7"
).warn()
syms.append(Symbol(s, real=True))
symbols = Tuple(*syms)
# canonicallize the relations
rel_temp = {}
for k,v in relations.items():
s1, s2 = k
if not isinstance(s1, Str):
s1 = Str(s1)
if not isinstance(s2, Str):
s2 = Str(s2)
key = Tuple(s1, s2)
# Old version used Lambda as a value.
if isinstance(v, Lambda):
v = (tuple(v.signature), tuple(v.expr))
else:
v = (tuple(v[0]), tuple(v[1]))
rel_temp[key] = v
relations = Dict(rel_temp)
# construct the object
obj = super().__new__(cls, name, patch, symbols, relations)
# Add deprecated attributes
obj.transforms = _deprecated_dict(
"Mutable CoordSystem.transforms",
"'relations' parameter in class signature",
19321,
"1.7",
{}
)
obj._names = [str(n) for n in symbols]
obj.patch.coord_systems.append(obj) # deprecated
obj._dummies = [Dummy(str(n)) for n in symbols] # deprecated
obj._dummy = Dummy()
return obj
@property
def name(self):
return self.args[0]
@property
def patch(self):
return self.args[1]
@property
def manifold(self):
return self.patch.manifold
@property
def symbols(self):
return tuple(CoordinateSymbol(self, i, **s._assumptions.generator)
for i,s in enumerate(self.args[2]))
@property
def relations(self):
return self.args[3]
@property
def dim(self):
return self.patch.dim
##########################################################################
# Finding transformation relation
##########################################################################
def transformation(self, sys):
"""
Return coordinate transformation function from *self* to *sys*.
Parameters
==========
sys : CoordSystem
Returns
=======
sympy.Lambda
Examples
========
>>> from sympy.diffgeom.rn import R2_r, R2_p
>>> R2_r.transformation(R2_p)
Lambda((x, y), Matrix([
[sqrt(x**2 + y**2)],
[ atan2(y, x)]]))
"""
signature = self.args[2]
key = Tuple(self.name, sys.name)
if self == sys:
expr = Matrix(self.symbols)
elif key in self.relations:
expr = Matrix(self.relations[key][1])
elif key[::-1] in self.relations:
expr = Matrix(self._inverse_transformation(sys, self))
else:
expr = Matrix(self._indirect_transformation(self, sys))
return Lambda(signature, expr)
@staticmethod
def _solve_inverse(sym1, sym2, exprs, sys1_name, sys2_name):
ret = solve(
[t[0] - t[1] for t in zip(sym2, exprs)],
list(sym1), dict=True)
if len(ret) == 0:
temp = "Cannot solve inverse relation from {} to {}."
raise NotImplementedError(temp.format(sys1_name, sys2_name))
elif len(ret) > 1:
temp = "Obtained multiple inverse relation from {} to {}."
raise ValueError(temp.format(sys1_name, sys2_name))
return ret[0]
@classmethod
def _inverse_transformation(cls, sys1, sys2):
# Find the transformation relation from sys2 to sys1
forward = sys1.transform(sys2)
inv_results = cls._solve_inverse(sys1.symbols, sys2.symbols, forward,
sys1.name, sys2.name)
signature = tuple(sys1.symbols)
return [inv_results[s] for s in signature]
@classmethod
@cacheit
def _indirect_transformation(cls, sys1, sys2):
# Find the transformation relation between two indirectly connected
# coordinate systems
rel = sys1.relations
path = cls._dijkstra(sys1, sys2)
transforms = []
for s1, s2 in zip(path, path[1:]):
if (s1, s2) in rel:
transforms.append(rel[(s1, s2)])
else:
sym2, inv_exprs = rel[(s2, s1)]
sym1 = tuple(Dummy() for i in sym2)
ret = cls._solve_inverse(sym2, sym1, inv_exprs, s2, s1)
ret = tuple(ret[s] for s in sym2)
transforms.append((sym1, ret))
syms = sys1.args[2]
exprs = syms
for newsyms, newexprs in transforms:
exprs = tuple(e.subs(zip(newsyms, exprs)) for e in newexprs)
return exprs
@staticmethod
def _dijkstra(sys1, sys2):
# Use Dijkstra algorithm to find the shortest path between two indirectly-connected
# coordinate systems
# return value is the list of the names of the systems.
relations = sys1.relations
graph = {}
for s1, s2 in relations.keys():
if s1 not in graph:
graph[s1] = {s2}
else:
graph[s1].add(s2)
if s2 not in graph:
graph[s2] = {s1}
else:
graph[s2].add(s1)
path_dict = {sys:[0, [], 0] for sys in graph} # minimum distance, path, times of visited
def visit(sys):
path_dict[sys][2] = 1
for newsys in graph[sys]:
distance = path_dict[sys][0] + 1
if path_dict[newsys][0] >= distance or not path_dict[newsys][1]:
path_dict[newsys][0] = distance
path_dict[newsys][1] = [i for i in path_dict[sys][1]]
path_dict[newsys][1].append(sys)
visit(sys1.name)
while True:
min_distance = max(path_dict.values(), key=lambda x:x[0])[0]
newsys = None
for sys, lst in path_dict.items():
if 0 < lst[0] <= min_distance and not lst[2]:
min_distance = lst[0]
newsys = sys
if newsys is None:
break
visit(newsys)
result = path_dict[sys2.name][1]
result.append(sys2.name)
if result == [sys2.name]:
raise KeyError("Two coordinate systems are not connected.")
return result
def connect_to(self, to_sys, from_coords, to_exprs, inverse=True, fill_in_gaps=False):
SymPyDeprecationWarning(
feature="CoordSystem.connect_to",
useinstead="new instance generated with new 'transforms' parameter",
issue=19321,
deprecated_since_version="1.7"
).warn()
from_coords, to_exprs = dummyfy(from_coords, to_exprs)
self.transforms[to_sys] = Matrix(from_coords), Matrix(to_exprs)
if inverse:
to_sys.transforms[self] = self._inv_transf(from_coords, to_exprs)
if fill_in_gaps:
self._fill_gaps_in_transformations()
@staticmethod
def _inv_transf(from_coords, to_exprs):
# Will be removed when connect_to is removed
inv_from = [i.as_dummy() for i in from_coords]
inv_to = solve(
[t[0] - t[1] for t in zip(inv_from, to_exprs)],
list(from_coords), dict=True)[0]
inv_to = [inv_to[fc] for fc in from_coords]
return Matrix(inv_from), Matrix(inv_to)
@staticmethod
def _fill_gaps_in_transformations():
# Will be removed when connect_to is removed
raise NotImplementedError
##########################################################################
# Coordinate transformations
##########################################################################
def transform(self, sys, coordinates=None):
"""
Return the result of coordinate transformation from *self* to *sys*.
If coordinates are not given, coordinate symbols of *self* are used.
Parameters
==========
sys : CoordSystem
coordinates : Any iterable, optional.
Returns
=======
sympy.ImmutableDenseMatrix containing CoordinateSymbol
Examples
========
>>> from sympy.diffgeom.rn import R2_r, R2_p
>>> R2_r.transform(R2_p)
Matrix([
[sqrt(x**2 + y**2)],
[ atan2(y, x)]])
>>> R2_r.transform(R2_p, [0, 1])
Matrix([
[ 1],
[pi/2]])
"""
if coordinates is None:
coordinates = self.symbols
if self != sys:
transf = self.transformation(sys)
coordinates = transf(*coordinates)
else:
coordinates = Matrix(coordinates)
return coordinates
def coord_tuple_transform_to(self, to_sys, coords):
"""Transform ``coords`` to coord system ``to_sys``."""
SymPyDeprecationWarning(
feature="CoordSystem.coord_tuple_transform_to",
useinstead="CoordSystem.transform",
issue=19321,
deprecated_since_version="1.7"
).warn()
coords = Matrix(coords)
if self != to_sys:
transf = self.transforms[to_sys]
coords = transf[1].subs(list(zip(transf[0], coords)))
return coords
def jacobian(self, sys, coordinates=None):
"""
Return the jacobian matrix of a transformation on given coordinates.
If coordinates are not given, coordinate symbols of *self* are used.
Parameters
==========
sys : CoordSystem
coordinates : Any iterable, optional.
Returns
=======
sympy.ImmutableDenseMatrix
Examples
========
>>> from sympy.diffgeom.rn import R2_r, R2_p
>>> R2_p.jacobian(R2_r)
Matrix([
[cos(theta), -rho*sin(theta)],
[sin(theta), rho*cos(theta)]])
>>> R2_p.jacobian(R2_r, [1, 0])
Matrix([
[1, 0],
[0, 1]])
"""
result = self.transform(sys).jacobian(self.symbols)
if coordinates is not None:
result = result.subs(list(zip(self.symbols, coordinates)))
return result
jacobian_matrix = jacobian
def jacobian_determinant(self, sys, coordinates=None):
"""
Return the jacobian determinant of a transformation on given
coordinates. If coordinates are not given, coordinate symbols of *self*
are used.
Parameters
==========
sys : CoordSystem
coordinates : Any iterable, optional.
Returns
=======
sympy.Expr
Examples
========
>>> from sympy.diffgeom.rn import R2_r, R2_p
>>> R2_r.jacobian_determinant(R2_p)
1/sqrt(x**2 + y**2)
>>> R2_r.jacobian_determinant(R2_p, [1, 0])
1
"""
return self.jacobian(sys, coordinates).det()
##########################################################################
# Points
##########################################################################
def point(self, coords):
"""Create a ``Point`` with coordinates given in this coord system."""
return Point(self, coords)
def point_to_coords(self, point):
"""Calculate the coordinates of a point in this coord system."""
return point.coords(self)
##########################################################################
# Base fields.
##########################################################################
def base_scalar(self, coord_index):
"""Return ``BaseScalarField`` that takes a point and returns one of the coordinates."""
return BaseScalarField(self, coord_index)
coord_function = base_scalar
def base_scalars(self):
"""Returns a list of all coordinate functions.
For more details see the ``base_scalar`` method of this class."""
return [self.base_scalar(i) for i in range(self.dim)]
coord_functions = base_scalars
def base_vector(self, coord_index):
"""Return a basis vector field.
The basis vector field for this coordinate system. It is also an
operator on scalar fields."""
return BaseVectorField(self, coord_index)
def base_vectors(self):
"""Returns a list of all base vectors.
For more details see the ``base_vector`` method of this class."""
return [self.base_vector(i) for i in range(self.dim)]
def base_oneform(self, coord_index):
"""Return a basis 1-form field.
The basis one-form field for this coordinate system. It is also an
operator on vector fields."""
return Differential(self.coord_function(coord_index))
def base_oneforms(self):
"""Returns a list of all base oneforms.
For more details see the ``base_oneform`` method of this class."""
return [self.base_oneform(i) for i in range(self.dim)]
class CoordinateSymbol(Symbol):
"""A symbol which denotes an abstract value of i-th coordinate of
the coordinate system with given context.
Explanation
===========
Each coordinates in coordinate system are represented by unique symbol,
such as x, y, z in Cartesian coordinate system.
You may not construct this class directly. Instead, use `symbols` method
of CoordSystem.
Parameters
==========
coord_sys : CoordSystem
index : integer
Examples
========
>>> from sympy import symbols, Lambda, Matrix, sqrt, atan2, cos, sin
>>> from sympy.diffgeom import Manifold, Patch, CoordSystem
>>> m = Manifold('M', 2)
>>> p = Patch('P', m)
>>> x, y = symbols('x y', real=True)
>>> r, theta = symbols('r theta', nonnegative=True)
>>> relation_dict = {
... ('Car2D', 'Pol'): Lambda((x, y), Matrix([sqrt(x**2 + y**2), atan2(y, x)])),
... ('Pol', 'Car2D'): Lambda((r, theta), Matrix([r*cos(theta), r*sin(theta)]))
... }
>>> Car2D = CoordSystem('Car2D', p, [x, y], relation_dict)
>>> Pol = CoordSystem('Pol', p, [r, theta], relation_dict)
>>> x, y = Car2D.symbols
``CoordinateSymbol`` contains its coordinate symbol and index.
>>> x.name
'x'
>>> x.coord_sys == Car2D
True
>>> x.index
0
>>> x.is_real
True
You can transform ``CoordinateSymbol`` into other coordinate system using
``rewrite()`` method.
>>> x.rewrite(Pol)
r*cos(theta)
>>> sqrt(x**2 + y**2).rewrite(Pol).simplify()
r
"""
def __new__(cls, coord_sys, index, **assumptions):
name = coord_sys.args[2][index].name
obj = super().__new__(cls, name, **assumptions)
obj.coord_sys = coord_sys
obj.index = index
return obj
def __getnewargs__(self):
return (self.coord_sys, self.index)
def _hashable_content(self):
return (
self.coord_sys, self.index
) + tuple(sorted(self.assumptions0.items()))
def _eval_rewrite(self, rule, args, **hints):
if isinstance(rule, CoordSystem):
return rule.transform(self.coord_sys)[self.index]
return super()._eval_rewrite(rule, args, **hints)
class Point(Basic):
"""Point defined in a coordinate system.
Explanation
===========
Mathematically, point is defined in the manifold and does not have any coordinates
by itself. Coordinate system is what imbues the coordinates to the point by coordinate
chart. However, due to the difficulty of realizing such logic, you must supply
a coordinate system and coordinates to define a Point here.
The usage of this object after its definition is independent of the
coordinate system that was used in order to define it, however due to
limitations in the simplification routines you can arrive at complicated
expressions if you use inappropriate coordinate systems.
Parameters
==========
coord_sys : CoordSystem
coords : list
The coordinates of the point.
Examples
========
>>> from sympy import pi
>>> from sympy.diffgeom import Point
>>> from sympy.diffgeom.rn import R2, R2_r, R2_p
>>> rho, theta = R2_p.symbols
>>> p = Point(R2_p, [rho, 3*pi/4])
>>> p.manifold == R2
True
>>> p.coords()
Matrix([
[ rho],
[3*pi/4]])
>>> p.coords(R2_r)
Matrix([
[-sqrt(2)*rho/2],
[ sqrt(2)*rho/2]])
"""
def __new__(cls, coord_sys, coords, **kwargs):
coords = Matrix(coords)
obj = super().__new__(cls, coord_sys, coords)
obj._coord_sys = coord_sys
obj._coords = coords
return obj
@property
def patch(self):
return self._coord_sys.patch
@property
def manifold(self):
return self._coord_sys.manifold
@property
def dim(self):
return self.manifold.dim
def coords(self, sys=None):
"""
Coordinates of the point in given coordinate system. If coordinate system
is not passed, it returns the coordinates in the coordinate system in which
the poin was defined.
"""
if sys is None:
return self._coords
else:
return self._coord_sys.transform(sys, self._coords)
@property
def free_symbols(self):
return self._coords.free_symbols
class BaseScalarField(Expr):
"""Base scalar field over a manifold for a given coordinate system.
Explanation
===========
A scalar field takes a point as an argument and returns a scalar.
A base scalar field of a coordinate system takes a point and returns one of
the coordinates of that point in the coordinate system in question.
To define a scalar field you need to choose the coordinate system and the
index of the coordinate.
The use of the scalar field after its definition is independent of the
coordinate system in which it was defined, however due to limitations in
the simplification routines you may arrive at more complicated
expression if you use unappropriate coordinate systems.
You can build complicated scalar fields by just building up SymPy
expressions containing ``BaseScalarField`` instances.
Parameters
==========
coord_sys : CoordSystem
index : integer
Examples
========
>>> from sympy import Function, pi
>>> from sympy.diffgeom import BaseScalarField
>>> from sympy.diffgeom.rn import R2_r, R2_p
>>> rho, _ = R2_p.symbols
>>> point = R2_p.point([rho, 0])
>>> fx, fy = R2_r.base_scalars()
>>> ftheta = BaseScalarField(R2_r, 1)
>>> fx(point)
rho
>>> fy(point)
0
>>> (fx**2+fy**2).rcall(point)
rho**2
>>> g = Function('g')
>>> fg = g(ftheta-pi)
>>> fg.rcall(point)
g(-pi)
"""
is_commutative = True
def __new__(cls, coord_sys, index, **kwargs):
index = _sympify(index)
obj = super().__new__(cls, coord_sys, index)
obj._coord_sys = coord_sys
obj._index = index
return obj
@property
def coord_sys(self):
return self.args[0]
@property
def index(self):
return self.args[1]
@property
def patch(self):
return self.coord_sys.patch
@property
def manifold(self):
return self.coord_sys.manifold
@property
def dim(self):
return self.manifold.dim
def __call__(self, *args):
"""Evaluating the field at a point or doing nothing.
If the argument is a ``Point`` instance, the field is evaluated at that
point. The field is returned itself if the argument is any other
object. It is so in order to have working recursive calling mechanics
for all fields (check the ``__call__`` method of ``Expr``).
"""
point = args[0]
if len(args) != 1 or not isinstance(point, Point):
return self
coords = point.coords(self._coord_sys)
# XXX Calling doit is necessary with all the Subs expressions
# XXX Calling simplify is necessary with all the trig expressions
return simplify(coords[self._index]).doit()
# XXX Workaround for limitations on the content of args
free_symbols = set() # type: tSet[Any]
def doit(self):
return self
class BaseVectorField(Expr):
r"""Base vector field over a manifold for a given coordinate system.
Explanation
===========
A vector field is an operator taking a scalar field and returning a
directional derivative (which is also a scalar field).
A base vector field is the same type of operator, however the derivation is
specifically done with respect to a chosen coordinate.
To define a base vector field you need to choose the coordinate system and
the index of the coordinate.
The use of the vector field after its definition is independent of the
coordinate system in which it was defined, however due to limitations in the
simplification routines you may arrive at more complicated expression if you
use unappropriate coordinate systems.
Parameters
==========
coord_sys : CoordSystem
index : integer
Examples
========
>>> from sympy import Function
>>> from sympy.diffgeom.rn import R2_p, R2_r
>>> from sympy.diffgeom import BaseVectorField
>>> from sympy import pprint
>>> x, y = R2_r.symbols
>>> rho, theta = R2_p.symbols
>>> fx, fy = R2_r.base_scalars()
>>> point_p = R2_p.point([rho, theta])
>>> point_r = R2_r.point([x, y])
>>> g = Function('g')
>>> s_field = g(fx, fy)
>>> v = BaseVectorField(R2_r, 1)
>>> pprint(v(s_field))
/ d \|
|---(g(x, xi))||
\dxi /|xi=y
>>> pprint(v(s_field).rcall(point_r).doit())
d
--(g(x, y))
dy
>>> pprint(v(s_field).rcall(point_p))
/ d \|
|---(g(rho*cos(theta), xi))||
\dxi /|xi=rho*sin(theta)
"""
is_commutative = False
def __new__(cls, coord_sys, index, **kwargs):
index = _sympify(index)
obj = super().__new__(cls, coord_sys, index)
obj._coord_sys = coord_sys
obj._index = index
return obj
@property
def coord_sys(self):
return self.args[0]
@property
def index(self):
return self.args[1]
@property
def patch(self):
return self.coord_sys.patch
@property
def manifold(self):
return self.coord_sys.manifold
@property
def dim(self):
return self.manifold.dim
def __call__(self, scalar_field):
"""Apply on a scalar field.
The action of a vector field on a scalar field is a directional
differentiation.
If the argument is not a scalar field an error is raised.
"""
if covariant_order(scalar_field) or contravariant_order(scalar_field):
raise ValueError('Only scalar fields can be supplied as arguments to vector fields.')
if scalar_field is None:
return self
base_scalars = list(scalar_field.atoms(BaseScalarField))
# First step: e_x(x+r**2) -> e_x(x) + 2*r*e_x(r)
d_var = self._coord_sys._dummy
# TODO: you need a real dummy function for the next line
d_funcs = [Function('_#_%s' % i)(d_var) for i,
b in enumerate(base_scalars)]
d_result = scalar_field.subs(list(zip(base_scalars, d_funcs)))
d_result = d_result.diff(d_var)
# Second step: e_x(x) -> 1 and e_x(r) -> cos(atan2(x, y))
coords = self._coord_sys.symbols
d_funcs_deriv = [f.diff(d_var) for f in d_funcs]
d_funcs_deriv_sub = []
for b in base_scalars:
jac = self._coord_sys.jacobian(b._coord_sys, coords)
d_funcs_deriv_sub.append(jac[b._index, self._index])
d_result = d_result.subs(list(zip(d_funcs_deriv, d_funcs_deriv_sub)))
# Remove the dummies
result = d_result.subs(list(zip(d_funcs, base_scalars)))
result = result.subs(list(zip(coords, self._coord_sys.coord_functions())))
return result.doit()
def _find_coords(expr):
# Finds CoordinateSystems existing in expr
fields = expr.atoms(BaseScalarField, BaseVectorField)
result = set()
for f in fields:
result.add(f._coord_sys)
return result
class Commutator(Expr):
r"""Commutator of two vector fields.
Explanation
===========
The commutator of two vector fields `v_1` and `v_2` is defined as the
vector field `[v_1, v_2]` that evaluated on each scalar field `f` is equal
to `v_1(v_2(f)) - v_2(v_1(f))`.
Examples
========
>>> from sympy.diffgeom.rn import R2_p, R2_r
>>> from sympy.diffgeom import Commutator
>>> from sympy import simplify
>>> fx, fy = R2_r.base_scalars()
>>> e_x, e_y = R2_r.base_vectors()
>>> e_r = R2_p.base_vector(0)
>>> c_xy = Commutator(e_x, e_y)
>>> c_xr = Commutator(e_x, e_r)
>>> c_xy
0
Unfortunately, the current code is not able to compute everything:
>>> c_xr
Commutator(e_x, e_rho)
>>> simplify(c_xr(fy**2))
-2*cos(theta)*y**2/(x**2 + y**2)
"""
def __new__(cls, v1, v2):
if (covariant_order(v1) or contravariant_order(v1) != 1
or covariant_order(v2) or contravariant_order(v2) != 1):
raise ValueError(
'Only commutators of vector fields are supported.')
if v1 == v2:
return S.Zero
coord_sys = set().union(*[_find_coords(v) for v in (v1, v2)])
if len(coord_sys) == 1:
# Only one coordinate systems is used, hence it is easy enough to
# actually evaluate the commutator.
if all(isinstance(v, BaseVectorField) for v in (v1, v2)):
return S.Zero
bases_1, bases_2 = [list(v.atoms(BaseVectorField))
for v in (v1, v2)]
coeffs_1 = [v1.expand().coeff(b) for b in bases_1]
coeffs_2 = [v2.expand().coeff(b) for b in bases_2]
res = 0
for c1, b1 in zip(coeffs_1, bases_1):
for c2, b2 in zip(coeffs_2, bases_2):
res += c1*b1(c2)*b2 - c2*b2(c1)*b1
return res
else:
obj = super().__new__(cls, v1, v2)
obj._v1 = v1 # deprecated assignment
obj._v2 = v2 # deprecated assignment
return obj
@property
def v1(self):
return self.args[0]
@property
def v2(self):
return self.args[1]
def __call__(self, scalar_field):
"""Apply on a scalar field.
If the argument is not a scalar field an error is raised.
"""
return self.v1(self.v2(scalar_field)) - self.v2(self.v1(scalar_field))
class Differential(Expr):
r"""Return the differential (exterior derivative) of a form field.
Explanation
===========
The differential of a form (i.e. the exterior derivative) has a complicated
definition in the general case.
The differential `df` of the 0-form `f` is defined for any vector field `v`
as `df(v) = v(f)`.
Examples
========
>>> from sympy import Function
>>> from sympy.diffgeom.rn import R2_r
>>> from sympy.diffgeom import Differential
>>> from sympy import pprint
>>> fx, fy = R2_r.base_scalars()
>>> e_x, e_y = R2_r.base_vectors()
>>> g = Function('g')
>>> s_field = g(fx, fy)
>>> dg = Differential(s_field)
>>> dg
d(g(x, y))
>>> pprint(dg(e_x))
/ d \|
|---(g(xi, y))||
\dxi /|xi=x
>>> pprint(dg(e_y))
/ d \|
|---(g(x, xi))||
\dxi /|xi=y
Applying the exterior derivative operator twice always results in:
>>> Differential(dg)
0
"""
is_commutative = False
def __new__(cls, form_field):
if contravariant_order(form_field):
raise ValueError(
'A vector field was supplied as an argument to Differential.')
if isinstance(form_field, Differential):
return S.Zero
else:
obj = super().__new__(cls, form_field)
obj._form_field = form_field # deprecated assignment
return obj
@property
def form_field(self):
return self.args[0]
def __call__(self, *vector_fields):
"""Apply on a list of vector_fields.
Explanation
===========
If the number of vector fields supplied is not equal to 1 + the order of
the form field inside the differential the result is undefined.
For 1-forms (i.e. differentials of scalar fields) the evaluation is
done as `df(v)=v(f)`. However if `v` is ``None`` instead of a vector
field, the differential is returned unchanged. This is done in order to
permit partial contractions for higher forms.
In the general case the evaluation is done by applying the form field
inside the differential on a list with one less elements than the number
of elements in the original list. Lowering the number of vector fields
is achieved through replacing each pair of fields by their
commutator.
If the arguments are not vectors or ``None``s an error is raised.
"""
if any((contravariant_order(a) != 1 or covariant_order(a)) and a is not None
for a in vector_fields):
raise ValueError('The arguments supplied to Differential should be vector fields or Nones.')
k = len(vector_fields)
if k == 1:
if vector_fields[0]:
return vector_fields[0].rcall(self._form_field)
return self
else:
# For higher form it is more complicated:
# Invariant formula:
# https://en.wikipedia.org/wiki/Exterior_derivative#Invariant_formula
# df(v1, ... vn) = +/- vi(f(v1..no i..vn))
# +/- f([vi,vj],v1..no i, no j..vn)
f = self._form_field
v = vector_fields
ret = 0
for i in range(k):
t = v[i].rcall(f.rcall(*v[:i] + v[i + 1:]))
ret += (-1)**i*t
for j in range(i + 1, k):
c = Commutator(v[i], v[j])
if c: # TODO this is ugly - the Commutator can be Zero and
# this causes the next line to fail
t = f.rcall(*(c,) + v[:i] + v[i + 1:j] + v[j + 1:])
ret += (-1)**(i + j)*t
return ret
class TensorProduct(Expr):
"""Tensor product of forms.
Explanation
===========
The tensor product permits the creation of multilinear functionals (i.e.
higher order tensors) out of lower order fields (e.g. 1-forms and vector
fields). However, the higher tensors thus created lack the interesting
features provided by the other type of product, the wedge product, namely
they are not antisymmetric and hence are not form fields.
Examples
========
>>> from sympy.diffgeom.rn import R2_r
>>> from sympy.diffgeom import TensorProduct
>>> fx, fy = R2_r.base_scalars()
>>> e_x, e_y = R2_r.base_vectors()
>>> dx, dy = R2_r.base_oneforms()
>>> TensorProduct(dx, dy)(e_x, e_y)
1
>>> TensorProduct(dx, dy)(e_y, e_x)
0
>>> TensorProduct(dx, fx*dy)(fx*e_x, e_y)
x**2
>>> TensorProduct(e_x, e_y)(fx**2, fy**2)
4*x*y
>>> TensorProduct(e_y, dx)(fy)
dx
You can nest tensor products.
>>> tp1 = TensorProduct(dx, dy)
>>> TensorProduct(tp1, dx)(e_x, e_y, e_x)
1
You can make partial contraction for instance when 'raising an index'.
Putting ``None`` in the second argument of ``rcall`` means that the
respective position in the tensor product is left as it is.
>>> TP = TensorProduct
>>> metric = TP(dx, dx) + 3*TP(dy, dy)
>>> metric.rcall(e_y, None)
3*dy
Or automatically pad the args with ``None`` without specifying them.
>>> metric.rcall(e_y)
3*dy
"""
def __new__(cls, *args):
scalar = Mul(*[m for m in args if covariant_order(m) + contravariant_order(m) == 0])
multifields = [m for m in args if covariant_order(m) + contravariant_order(m)]
if multifields:
if len(multifields) == 1:
return scalar*multifields[0]
return scalar*super().__new__(cls, *multifields)
else:
return scalar
def __call__(self, *fields):
"""Apply on a list of fields.
If the number of input fields supplied is not equal to the order of
the tensor product field, the list of arguments is padded with ``None``'s.
The list of arguments is divided in sublists depending on the order of
the forms inside the tensor product. The sublists are provided as
arguments to these forms and the resulting expressions are given to the
constructor of ``TensorProduct``.
"""
tot_order = covariant_order(self) + contravariant_order(self)
tot_args = len(fields)
if tot_args != tot_order:
fields = list(fields) + [None]*(tot_order - tot_args)
orders = [covariant_order(f) + contravariant_order(f) for f in self._args]
indices = [sum(orders[:i + 1]) for i in range(len(orders) - 1)]
fields = [fields[i:j] for i, j in zip([0] + indices, indices + [None])]
multipliers = [t[0].rcall(*t[1]) for t in zip(self._args, fields)]
return TensorProduct(*multipliers)
class WedgeProduct(TensorProduct):
"""Wedge product of forms.
Explanation
===========
In the context of integration only completely antisymmetric forms make
sense. The wedge product permits the creation of such forms.
Examples
========
>>> from sympy.diffgeom.rn import R2_r
>>> from sympy.diffgeom import WedgeProduct
>>> fx, fy = R2_r.base_scalars()
>>> e_x, e_y = R2_r.base_vectors()
>>> dx, dy = R2_r.base_oneforms()
>>> WedgeProduct(dx, dy)(e_x, e_y)
1
>>> WedgeProduct(dx, dy)(e_y, e_x)
-1
>>> WedgeProduct(dx, fx*dy)(fx*e_x, e_y)
x**2
>>> WedgeProduct(e_x, e_y)(fy, None)
-e_x
You can nest wedge products.
>>> wp1 = WedgeProduct(dx, dy)
>>> WedgeProduct(wp1, dx)(e_x, e_y, e_x)
0
"""
# TODO the calculation of signatures is slow
# TODO you do not need all these permutations (neither the prefactor)
def __call__(self, *fields):
"""Apply on a list of vector_fields.
The expression is rewritten internally in terms of tensor products and evaluated."""
orders = (covariant_order(e) + contravariant_order(e) for e in self.args)
mul = 1/Mul(*(factorial(o) for o in orders))
perms = permutations(fields)
perms_par = (Permutation(
p).signature() for p in permutations(list(range(len(fields)))))
tensor_prod = TensorProduct(*self.args)
return mul*Add(*[tensor_prod(*p[0])*p[1] for p in zip(perms, perms_par)])
class LieDerivative(Expr):
"""Lie derivative with respect to a vector field.
Explanation
===========
The transport operator that defines the Lie derivative is the pushforward of
the field to be derived along the integral curve of the field with respect
to which one derives.
Examples
========
>>> from sympy.diffgeom.rn import R2_r, R2_p
>>> from sympy.diffgeom import (LieDerivative, TensorProduct)
>>> fx, fy = R2_r.base_scalars()
>>> e_x, e_y = R2_r.base_vectors()
>>> e_rho, e_theta = R2_p.base_vectors()
>>> dx, dy = R2_r.base_oneforms()
>>> LieDerivative(e_x, fy)
0
>>> LieDerivative(e_x, fx)
1
>>> LieDerivative(e_x, e_x)
0
The Lie derivative of a tensor field by another tensor field is equal to
their commutator:
>>> LieDerivative(e_x, e_rho)
Commutator(e_x, e_rho)
>>> LieDerivative(e_x + e_y, fx)
1
>>> tp = TensorProduct(dx, dy)
>>> LieDerivative(e_x, tp)
LieDerivative(e_x, TensorProduct(dx, dy))
>>> LieDerivative(e_x, tp)
LieDerivative(e_x, TensorProduct(dx, dy))
"""
def __new__(cls, v_field, expr):
expr_form_ord = covariant_order(expr)
if contravariant_order(v_field) != 1 or covariant_order(v_field):
raise ValueError('Lie derivatives are defined only with respect to'
' vector fields. The supplied argument was not a '
'vector field.')
if expr_form_ord > 0:
obj = super().__new__(cls, v_field, expr)
# deprecated assignments
obj._v_field = v_field
obj._expr = expr
return obj
if expr.atoms(BaseVectorField):
return Commutator(v_field, expr)
else:
return v_field.rcall(expr)
@property
def v_field(self):
return self.args[0]
@property
def expr(self):
return self.args[1]
def __call__(self, *args):
v = self.v_field
expr = self.expr
lead_term = v(expr(*args))
rest = Add(*[Mul(*args[:i] + (Commutator(v, args[i]),) + args[i + 1:])
for i in range(len(args))])
return lead_term - rest
class BaseCovarDerivativeOp(Expr):
"""Covariant derivative operator with respect to a base vector.
Examples
========
>>> from sympy.diffgeom.rn import R2_r
>>> from sympy.diffgeom import BaseCovarDerivativeOp
>>> from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct
>>> TP = TensorProduct
>>> fx, fy = R2_r.base_scalars()
>>> e_x, e_y = R2_r.base_vectors()
>>> dx, dy = R2_r.base_oneforms()
>>> ch = metric_to_Christoffel_2nd(TP(dx, dx) + TP(dy, dy))
>>> ch
[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
>>> cvd = BaseCovarDerivativeOp(R2_r, 0, ch)
>>> cvd(fx)
1
>>> cvd(fx*e_x)
e_x
"""
def __new__(cls, coord_sys, index, christoffel):
index = _sympify(index)
christoffel = ImmutableDenseNDimArray(christoffel)
obj = super().__new__(cls, coord_sys, index, christoffel)
# deprecated assignments
obj._coord_sys = coord_sys
obj._index = index
obj._christoffel = christoffel
return obj
@property
def coord_sys(self):
return self.args[0]
@property
def index(self):
return self.args[1]
@property
def christoffel(self):
return self.args[2]
def __call__(self, field):
"""Apply on a scalar field.
The action of a vector field on a scalar field is a directional
differentiation.
If the argument is not a scalar field the behaviour is undefined.
"""
if covariant_order(field) != 0:
raise NotImplementedError()
field = vectors_in_basis(field, self._coord_sys)
wrt_vector = self._coord_sys.base_vector(self._index)
wrt_scalar = self._coord_sys.coord_function(self._index)
vectors = list(field.atoms(BaseVectorField))
# First step: replace all vectors with something susceptible to
# derivation and do the derivation
# TODO: you need a real dummy function for the next line
d_funcs = [Function('_#_%s' % i)(wrt_scalar) for i,
b in enumerate(vectors)]
d_result = field.subs(list(zip(vectors, d_funcs)))
d_result = wrt_vector(d_result)
# Second step: backsubstitute the vectors in
d_result = d_result.subs(list(zip(d_funcs, vectors)))
# Third step: evaluate the derivatives of the vectors
derivs = []
for v in vectors:
d = Add(*[(self._christoffel[k, wrt_vector._index, v._index]
*v._coord_sys.base_vector(k))
for k in range(v._coord_sys.dim)])
derivs.append(d)
to_subs = [wrt_vector(d) for d in d_funcs]
# XXX: This substitution can fail when there are Dummy symbols and the
# cache is disabled: https://github.com/sympy/sympy/issues/17794
result = d_result.subs(list(zip(to_subs, derivs)))
# Remove the dummies
result = result.subs(list(zip(d_funcs, vectors)))
return result.doit()
class CovarDerivativeOp(Expr):
"""Covariant derivative operator.
Examples
========
>>> from sympy.diffgeom.rn import R2_r
>>> from sympy.diffgeom import CovarDerivativeOp
>>> from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct
>>> TP = TensorProduct
>>> fx, fy = R2_r.base_scalars()
>>> e_x, e_y = R2_r.base_vectors()
>>> dx, dy = R2_r.base_oneforms()
>>> ch = metric_to_Christoffel_2nd(TP(dx, dx) + TP(dy, dy))
>>> ch
[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
>>> cvd = CovarDerivativeOp(fx*e_x, ch)
>>> cvd(fx)
x
>>> cvd(fx*e_x)
x*e_x
"""
def __new__(cls, wrt, christoffel):
if len({v._coord_sys for v in wrt.atoms(BaseVectorField)}) > 1:
raise NotImplementedError()
if contravariant_order(wrt) != 1 or covariant_order(wrt):
raise ValueError('Covariant derivatives are defined only with '
'respect to vector fields. The supplied argument '
'was not a vector field.')
christoffel = ImmutableDenseNDimArray(christoffel)
obj = super().__new__(cls, wrt, christoffel)
# deprecated assigments
obj._wrt = wrt
obj._christoffel = christoffel
return obj
@property
def wrt(self):
return self.args[0]
@property
def christoffel(self):
return self.args[1]
def __call__(self, field):
vectors = list(self._wrt.atoms(BaseVectorField))
base_ops = [BaseCovarDerivativeOp(v._coord_sys, v._index, self._christoffel)
for v in vectors]
return self._wrt.subs(list(zip(vectors, base_ops))).rcall(field)
###############################################################################
# Integral curves on vector fields
###############################################################################
def intcurve_series(vector_field, param, start_point, n=6, coord_sys=None, coeffs=False):
r"""Return the series expansion for an integral curve of the field.
Explanation
===========
Integral curve is a function `\gamma` taking a parameter in `R` to a point
in the manifold. It verifies the equation:
`V(f)\big(\gamma(t)\big) = \frac{d}{dt}f\big(\gamma(t)\big)`
where the given ``vector_field`` is denoted as `V`. This holds for any
value `t` for the parameter and any scalar field `f`.
This equation can also be decomposed of a basis of coordinate functions
`V(f_i)\big(\gamma(t)\big) = \frac{d}{dt}f_i\big(\gamma(t)\big) \quad \forall i`
This function returns a series expansion of `\gamma(t)` in terms of the
coordinate system ``coord_sys``. The equations and expansions are necessarily
done in coordinate-system-dependent way as there is no other way to
represent movement between points on the manifold (i.e. there is no such
thing as a difference of points for a general manifold).
Parameters
==========
vector_field
the vector field for which an integral curve will be given
param
the argument of the function `\gamma` from R to the curve
start_point
the point which corresponds to `\gamma(0)`
n
the order to which to expand
coord_sys
the coordinate system in which to expand
coeffs (default False) - if True return a list of elements of the expansion
Examples
========
Use the predefined R2 manifold:
>>> from sympy.abc import t, x, y
>>> from sympy.diffgeom.rn import R2_p, R2_r
>>> from sympy.diffgeom import intcurve_series
Specify a starting point and a vector field:
>>> start_point = R2_r.point([x, y])
>>> vector_field = R2_r.e_x
Calculate the series:
>>> intcurve_series(vector_field, t, start_point, n=3)
Matrix([
[t + x],
[ y]])
Or get the elements of the expansion in a list:
>>> series = intcurve_series(vector_field, t, start_point, n=3, coeffs=True)
>>> series[0]
Matrix([
[x],
[y]])
>>> series[1]
Matrix([
[t],
[0]])
>>> series[2]
Matrix([
[0],
[0]])
The series in the polar coordinate system:
>>> series = intcurve_series(vector_field, t, start_point,
... n=3, coord_sys=R2_p, coeffs=True)
>>> series[0]
Matrix([
[sqrt(x**2 + y**2)],
[ atan2(y, x)]])
>>> series[1]
Matrix([
[t*x/sqrt(x**2 + y**2)],
[ -t*y/(x**2 + y**2)]])
>>> series[2]
Matrix([
[t**2*(-x**2/(x**2 + y**2)**(3/2) + 1/sqrt(x**2 + y**2))/2],
[ t**2*x*y/(x**2 + y**2)**2]])
See Also
========
intcurve_diffequ
"""
if contravariant_order(vector_field) != 1 or covariant_order(vector_field):
raise ValueError('The supplied field was not a vector field.')
def iter_vfield(scalar_field, i):
"""Return ``vector_field`` called `i` times on ``scalar_field``."""
return reduce(lambda s, v: v.rcall(s), [vector_field, ]*i, scalar_field)
def taylor_terms_per_coord(coord_function):
"""Return the series for one of the coordinates."""
return [param**i*iter_vfield(coord_function, i).rcall(start_point)/factorial(i)
for i in range(n)]
coord_sys = coord_sys if coord_sys else start_point._coord_sys
coord_functions = coord_sys.coord_functions()
taylor_terms = [taylor_terms_per_coord(f) for f in coord_functions]
if coeffs:
return [Matrix(t) for t in zip(*taylor_terms)]
else:
return Matrix([sum(c) for c in taylor_terms])
def intcurve_diffequ(vector_field, param, start_point, coord_sys=None):
r"""Return the differential equation for an integral curve of the field.
Explanation
===========
Integral curve is a function `\gamma` taking a parameter in `R` to a point
in the manifold. It verifies the equation:
`V(f)\big(\gamma(t)\big) = \frac{d}{dt}f\big(\gamma(t)\big)`
where the given ``vector_field`` is denoted as `V`. This holds for any
value `t` for the parameter and any scalar field `f`.
This function returns the differential equation of `\gamma(t)` in terms of the
coordinate system ``coord_sys``. The equations and expansions are necessarily
done in coordinate-system-dependent way as there is no other way to
represent movement between points on the manifold (i.e. there is no such
thing as a difference of points for a general manifold).
Parameters
==========
vector_field
the vector field for which an integral curve will be given
param
the argument of the function `\gamma` from R to the curve
start_point
the point which corresponds to `\gamma(0)`
coord_sys
the coordinate system in which to give the equations
Returns
=======
a tuple of (equations, initial conditions)
Examples
========
Use the predefined R2 manifold:
>>> from sympy.abc import t
>>> from sympy.diffgeom.rn import R2, R2_p, R2_r
>>> from sympy.diffgeom import intcurve_diffequ
Specify a starting point and a vector field:
>>> start_point = R2_r.point([0, 1])
>>> vector_field = -R2.y*R2.e_x + R2.x*R2.e_y
Get the equation:
>>> equations, init_cond = intcurve_diffequ(vector_field, t, start_point)
>>> equations
[f_1(t) + Derivative(f_0(t), t), -f_0(t) + Derivative(f_1(t), t)]
>>> init_cond
[f_0(0), f_1(0) - 1]
The series in the polar coordinate system:
>>> equations, init_cond = intcurve_diffequ(vector_field, t, start_point, R2_p)
>>> equations
[Derivative(f_0(t), t), Derivative(f_1(t), t) - 1]
>>> init_cond
[f_0(0) - 1, f_1(0) - pi/2]
See Also
========
intcurve_series
"""
if contravariant_order(vector_field) != 1 or covariant_order(vector_field):
raise ValueError('The supplied field was not a vector field.')
coord_sys = coord_sys if coord_sys else start_point._coord_sys
gammas = [Function('f_%d' % i)(param) for i in range(
start_point._coord_sys.dim)]
arbitrary_p = Point(coord_sys, gammas)
coord_functions = coord_sys.coord_functions()
equations = [simplify(diff(cf.rcall(arbitrary_p), param) - vector_field.rcall(cf).rcall(arbitrary_p))
for cf in coord_functions]
init_cond = [simplify(cf.rcall(arbitrary_p).subs(param, 0) - cf.rcall(start_point))
for cf in coord_functions]
return equations, init_cond
###############################################################################
# Helpers
###############################################################################
def dummyfy(args, exprs):
# TODO Is this a good idea?
d_args = Matrix([s.as_dummy() for s in args])
reps = dict(zip(args, d_args))
d_exprs = Matrix([_sympify(expr).subs(reps) for expr in exprs])
return d_args, d_exprs
###############################################################################
# Helpers
###############################################################################
def contravariant_order(expr, _strict=False):
"""Return the contravariant order of an expression.
Examples
========
>>> from sympy.diffgeom import contravariant_order
>>> from sympy.diffgeom.rn import R2
>>> from sympy.abc import a
>>> contravariant_order(a)
0
>>> contravariant_order(a*R2.x + 2)
0
>>> contravariant_order(a*R2.x*R2.e_y + R2.e_x)
1
"""
# TODO move some of this to class methods.
# TODO rewrite using the .as_blah_blah methods
if isinstance(expr, Add):
orders = [contravariant_order(e) for e in expr.args]
if len(set(orders)) != 1:
raise ValueError('Misformed expression containing contravariant fields of varying order.')
return orders[0]
elif isinstance(expr, Mul):
orders = [contravariant_order(e) for e in expr.args]
not_zero = [o for o in orders if o != 0]
if len(not_zero) > 1:
raise ValueError('Misformed expression containing multiplication between vectors.')
return 0 if not not_zero else not_zero[0]
elif isinstance(expr, Pow):
if covariant_order(expr.base) or covariant_order(expr.exp):
raise ValueError(
'Misformed expression containing a power of a vector.')
return 0
elif isinstance(expr, BaseVectorField):
return 1
elif isinstance(expr, TensorProduct):
return sum(contravariant_order(a) for a in expr.args)
elif not _strict or expr.atoms(BaseScalarField):
return 0
else: # If it does not contain anything related to the diffgeom module and it is _strict
return -1
def covariant_order(expr, _strict=False):
"""Return the covariant order of an expression.
Examples
========
>>> from sympy.diffgeom import covariant_order
>>> from sympy.diffgeom.rn import R2
>>> from sympy.abc import a
>>> covariant_order(a)
0
>>> covariant_order(a*R2.x + 2)
0
>>> covariant_order(a*R2.x*R2.dy + R2.dx)
1
"""
# TODO move some of this to class methods.
# TODO rewrite using the .as_blah_blah methods
if isinstance(expr, Add):
orders = [covariant_order(e) for e in expr.args]
if len(set(orders)) != 1:
raise ValueError('Misformed expression containing form fields of varying order.')
return orders[0]
elif isinstance(expr, Mul):
orders = [covariant_order(e) for e in expr.args]
not_zero = [o for o in orders if o != 0]
if len(not_zero) > 1:
raise ValueError('Misformed expression containing multiplication between forms.')
return 0 if not not_zero else not_zero[0]
elif isinstance(expr, Pow):
if covariant_order(expr.base) or covariant_order(expr.exp):
raise ValueError(
'Misformed expression containing a power of a form.')
return 0
elif isinstance(expr, Differential):
return covariant_order(*expr.args) + 1
elif isinstance(expr, TensorProduct):
return sum(covariant_order(a) for a in expr.args)
elif not _strict or expr.atoms(BaseScalarField):
return 0
else: # If it does not contain anything related to the diffgeom module and it is _strict
return -1
###############################################################################
# Coordinate transformation functions
###############################################################################
def vectors_in_basis(expr, to_sys):
"""Transform all base vectors in base vectors of a specified coord basis.
While the new base vectors are in the new coordinate system basis, any
coefficients are kept in the old system.
Examples
========
>>> from sympy.diffgeom import vectors_in_basis
>>> from sympy.diffgeom.rn import R2_r, R2_p
>>> vectors_in_basis(R2_r.e_x, R2_p)
-y*e_theta/(x**2 + y**2) + x*e_rho/sqrt(x**2 + y**2)
>>> vectors_in_basis(R2_p.e_r, R2_r)
sin(theta)*e_y + cos(theta)*e_x
"""
vectors = list(expr.atoms(BaseVectorField))
new_vectors = []
for v in vectors:
cs = v._coord_sys
jac = cs.jacobian(to_sys, cs.coord_functions())
new = (jac.T*Matrix(to_sys.base_vectors()))[v._index]
new_vectors.append(new)
return expr.subs(list(zip(vectors, new_vectors)))
###############################################################################
# Coordinate-dependent functions
###############################################################################
def twoform_to_matrix(expr):
"""Return the matrix representing the twoform.
For the twoform `w` return the matrix `M` such that `M[i,j]=w(e_i, e_j)`,
where `e_i` is the i-th base vector field for the coordinate system in
which the expression of `w` is given.
Examples
========
>>> from sympy.diffgeom.rn import R2
>>> from sympy.diffgeom import twoform_to_matrix, TensorProduct
>>> TP = TensorProduct
>>> twoform_to_matrix(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
Matrix([
[1, 0],
[0, 1]])
>>> twoform_to_matrix(R2.x*TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
Matrix([
[x, 0],
[0, 1]])
>>> twoform_to_matrix(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy) - TP(R2.dx, R2.dy)/2)
Matrix([
[ 1, 0],
[-1/2, 1]])
"""
if covariant_order(expr) != 2 or contravariant_order(expr):
raise ValueError('The input expression is not a two-form.')
coord_sys = _find_coords(expr)
if len(coord_sys) != 1:
raise ValueError('The input expression concerns more than one '
'coordinate systems, hence there is no unambiguous '
'way to choose a coordinate system for the matrix.')
coord_sys = coord_sys.pop()
vectors = coord_sys.base_vectors()
expr = expr.expand()
matrix_content = [[expr.rcall(v1, v2) for v1 in vectors]
for v2 in vectors]
return Matrix(matrix_content)
def metric_to_Christoffel_1st(expr):
"""Return the nested list of Christoffel symbols for the given metric.
This returns the Christoffel symbol of first kind that represents the
Levi-Civita connection for the given metric.
Examples
========
>>> from sympy.diffgeom.rn import R2
>>> from sympy.diffgeom import metric_to_Christoffel_1st, TensorProduct
>>> TP = TensorProduct
>>> metric_to_Christoffel_1st(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
>>> metric_to_Christoffel_1st(R2.x*TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
[[[1/2, 0], [0, 0]], [[0, 0], [0, 0]]]
"""
matrix = twoform_to_matrix(expr)
if not matrix.is_symmetric():
raise ValueError(
'The two-form representing the metric is not symmetric.')
coord_sys = _find_coords(expr).pop()
deriv_matrices = [matrix.applyfunc(d) for d in coord_sys.base_vectors()]
indices = list(range(coord_sys.dim))
christoffel = [[[(deriv_matrices[k][i, j] + deriv_matrices[j][i, k] - deriv_matrices[i][j, k])/2
for k in indices]
for j in indices]
for i in indices]
return ImmutableDenseNDimArray(christoffel)
def metric_to_Christoffel_2nd(expr):
"""Return the nested list of Christoffel symbols for the given metric.
This returns the Christoffel symbol of second kind that represents the
Levi-Civita connection for the given metric.
Examples
========
>>> from sympy.diffgeom.rn import R2
>>> from sympy.diffgeom import metric_to_Christoffel_2nd, TensorProduct
>>> TP = TensorProduct
>>> metric_to_Christoffel_2nd(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
[[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
>>> metric_to_Christoffel_2nd(R2.x*TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
[[[1/(2*x), 0], [0, 0]], [[0, 0], [0, 0]]]
"""
ch_1st = metric_to_Christoffel_1st(expr)
coord_sys = _find_coords(expr).pop()
indices = list(range(coord_sys.dim))
# XXX workaround, inverting a matrix does not work if it contains non
# symbols
#matrix = twoform_to_matrix(expr).inv()
matrix = twoform_to_matrix(expr)
s_fields = set()
for e in matrix:
s_fields.update(e.atoms(BaseScalarField))
s_fields = list(s_fields)
dums = coord_sys.symbols
matrix = matrix.subs(list(zip(s_fields, dums))).inv().subs(list(zip(dums, s_fields)))
# XXX end of workaround
christoffel = [[[Add(*[matrix[i, l]*ch_1st[l, j, k] for l in indices])
for k in indices]
for j in indices]
for i in indices]
return ImmutableDenseNDimArray(christoffel)
def metric_to_Riemann_components(expr):
"""Return the components of the Riemann tensor expressed in a given basis.
Given a metric it calculates the components of the Riemann tensor in the
canonical basis of the coordinate system in which the metric expression is
given.
Examples
========
>>> from sympy import exp
>>> from sympy.diffgeom.rn import R2
>>> from sympy.diffgeom import metric_to_Riemann_components, TensorProduct
>>> TP = TensorProduct
>>> metric_to_Riemann_components(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
[[[[0, 0], [0, 0]], [[0, 0], [0, 0]]], [[[0, 0], [0, 0]], [[0, 0], [0, 0]]]]
>>> non_trivial_metric = exp(2*R2.r)*TP(R2.dr, R2.dr) + \
R2.r**2*TP(R2.dtheta, R2.dtheta)
>>> non_trivial_metric
exp(2*rho)*TensorProduct(drho, drho) + rho**2*TensorProduct(dtheta, dtheta)
>>> riemann = metric_to_Riemann_components(non_trivial_metric)
>>> riemann[0, :, :, :]
[[[0, 0], [0, 0]], [[0, exp(-2*rho)*rho], [-exp(-2*rho)*rho, 0]]]
>>> riemann[1, :, :, :]
[[[0, -1/rho], [1/rho, 0]], [[0, 0], [0, 0]]]
"""
ch_2nd = metric_to_Christoffel_2nd(expr)
coord_sys = _find_coords(expr).pop()
indices = list(range(coord_sys.dim))
deriv_ch = [[[[d(ch_2nd[i, j, k])
for d in coord_sys.base_vectors()]
for k in indices]
for j in indices]
for i in indices]
riemann_a = [[[[deriv_ch[rho][sig][nu][mu] - deriv_ch[rho][sig][mu][nu]
for nu in indices]
for mu in indices]
for sig in indices]
for rho in indices]
riemann_b = [[[[Add(*[ch_2nd[rho, l, mu]*ch_2nd[l, sig, nu] - ch_2nd[rho, l, nu]*ch_2nd[l, sig, mu] for l in indices])
for nu in indices]
for mu in indices]
for sig in indices]
for rho in indices]
riemann = [[[[riemann_a[rho][sig][mu][nu] + riemann_b[rho][sig][mu][nu]
for nu in indices]
for mu in indices]
for sig in indices]
for rho in indices]
return ImmutableDenseNDimArray(riemann)
def metric_to_Ricci_components(expr):
"""Return the components of the Ricci tensor expressed in a given basis.
Given a metric it calculates the components of the Ricci tensor in the
canonical basis of the coordinate system in which the metric expression is
given.
Examples
========
>>> from sympy import exp
>>> from sympy.diffgeom.rn import R2
>>> from sympy.diffgeom import metric_to_Ricci_components, TensorProduct
>>> TP = TensorProduct
>>> metric_to_Ricci_components(TP(R2.dx, R2.dx) + TP(R2.dy, R2.dy))
[[0, 0], [0, 0]]
>>> non_trivial_metric = exp(2*R2.r)*TP(R2.dr, R2.dr) + \
R2.r**2*TP(R2.dtheta, R2.dtheta)
>>> non_trivial_metric
exp(2*rho)*TensorProduct(drho, drho) + rho**2*TensorProduct(dtheta, dtheta)
>>> metric_to_Ricci_components(non_trivial_metric)
[[1/rho, 0], [0, exp(-2*rho)*rho]]
"""
riemann = metric_to_Riemann_components(expr)
coord_sys = _find_coords(expr).pop()
indices = list(range(coord_sys.dim))
ricci = [[Add(*[riemann[k, i, k, j] for k in indices])
for j in indices]
for i in indices]
return ImmutableDenseNDimArray(ricci)
###############################################################################
# Classes for deprecation
###############################################################################
class _deprecated_container:
# This class gives deprecation warning.
# When deprecated features are completely deleted, this should be removed as well.
# See https://github.com/sympy/sympy/pull/19368
def __init__(self, feature, useinstead, issue, version, data):
super().__init__(data)
self.feature = feature
self.useinstead = useinstead
self.issue = issue
self.version = version
def warn(self):
SymPyDeprecationWarning(
feature=self.feature,
useinstead=self.useinstead,
issue=self.issue,
deprecated_since_version=self.version).warn()
def __iter__(self):
self.warn()
return super().__iter__()
def __getitem__(self, key):
self.warn()
return super().__getitem__(key)
def __contains__(self, key):
self.warn()
return super().__contains__(key)
class _deprecated_list(_deprecated_container, list):
pass
class _deprecated_dict(_deprecated_container, dict):
pass
# Import at end to avoid cyclic imports
from sympy.simplify.simplify import simplify
|
c16e092714204780d390f6358d67df9bf484016766a3d1d3de8ea8ddc1b236a8 | """This module provides containers for python objects that are valid
printing targets but are not a subclass of SymPy's Printable.
"""
from sympy.core.containers import Tuple
class List(Tuple):
"""Represents a (frozen) (Python) list (for code printing purposes)."""
def __eq__(self, other):
if isinstance(other, list):
return self == List(*other)
else:
return self.args == other
|
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